Termination of the given ITRSProblem could successfully be proven:
↳ ITRS
↳ ITRStoIDPProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
eval_2(x, y) → Cond_eval_2(>=@z(0@z, x), x, y)
Cond_eval_31(TRUE, x, y) → eval_3(x, -@z(y, 1@z))
Cond_eval_11(TRUE, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(>@z(x, 0@z), x, y)
eval_3(x, y) → Cond_eval_3(>=@z(0@z, y), x, y)
eval_1(x, y) → Cond_eval_11(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
Cond_eval_3(TRUE, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_31(>@z(y, 0@z), x, y)
eval_1(x, y) → Cond_eval_1(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
Cond_eval_21(TRUE, x, y) → eval_2(-@z(x, 1@z), y)
Cond_eval_1(TRUE, x, y) → eval_3(x, y)
Cond_eval_2(TRUE, x, y) → eval_1(x, y)
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Added dependency pairs
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
I DP problem:
The following domains are used:
z
The ITRS R consists of the following rules:
eval_2(x, y) → Cond_eval_2(>=@z(0@z, x), x, y)
Cond_eval_31(TRUE, x, y) → eval_3(x, -@z(y, 1@z))
Cond_eval_11(TRUE, x, y) → eval_2(x, y)
eval_2(x, y) → Cond_eval_21(>@z(x, 0@z), x, y)
eval_3(x, y) → Cond_eval_3(>=@z(0@z, y), x, y)
eval_1(x, y) → Cond_eval_11(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
Cond_eval_3(TRUE, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_31(>@z(y, 0@z), x, y)
eval_1(x, y) → Cond_eval_1(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
Cond_eval_21(TRUE, x, y) → eval_2(-@z(x, 1@z), y)
Cond_eval_1(TRUE, x, y) → eval_3(x, y)
Cond_eval_2(TRUE, x, y) → eval_1(x, y)
The integer pair graph contains the following rules and edges:
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(5): COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(10): COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(1) -> (10), if ((x[1] →* x[10])∧(y[1] →* y[10])∧(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])) →* TRUE))
(2) -> (5), if ((x[2] →* x[5])∧(y[2] →* y[5])∧(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])) →* TRUE))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(5) -> (4), if ((y[5] →* y[4])∧(x[5] →* x[4]))
(5) -> (11), if ((y[5] →* y[11])∧(x[5] →* x[11]))
(6) -> (1), if ((y[6] →* y[1])∧(x[6] →* x[1]))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
(8) -> (2), if ((y[8] →* y[2])∧(x[8] →* x[2]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(10) -> (3), if ((y[10] →* y[3])∧(x[10] →* x[3]))
(10) -> (7), if ((y[10] →* y[7])∧(x[10] →* x[7]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(5): COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(10): COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(1) -> (10), if ((x[1] →* x[10])∧(y[1] →* y[10])∧(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])) →* TRUE))
(2) -> (5), if ((x[2] →* x[5])∧(y[2] →* y[5])∧(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])) →* TRUE))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(5) -> (4), if ((y[5] →* y[4])∧(x[5] →* x[4]))
(5) -> (11), if ((y[5] →* y[11])∧(x[5] →* x[11]))
(6) -> (1), if ((y[6] →* y[1])∧(x[6] →* x[1]))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
(8) -> (2), if ((y[8] →* y[2])∧(x[8] →* x[2]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(10) -> (3), if ((y[10] →* y[3])∧(x[10] →* x[3]))
(10) -> (7), if ((y[10] →* y[7])∧(x[10] →* x[7]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_21(TRUE, x, y) → EVAL_2(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(1) (y[4]=y[0]∧-@z(x[0], 1@z)=x[4]1∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE∧y[0]=y[4]1 ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[4] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[4] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[4] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 = 0)
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(8) (y[0]=y[11]∧y[4]=y[0]∧-@z(x[0], 1@z)=x[11]∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[4] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧1 ≥ 0∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧1 ≥ 0∧0 = 0)
For Pair EVAL_1(x, y) → COND_EVAL_1(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) which results in the following constraint:
(15) (EVAL_1(x[1], y[1])≥NonInfC∧EVAL_1(x[1], y[1])≥COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧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_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL_1(x, y) → COND_EVAL_11(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(20) (EVAL_1(x[2], y[2])≥NonInfC∧EVAL_1(x[2], y[2])≥COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧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_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
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∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
For Pair EVAL_3(x, y) → COND_EVAL_31(>@z(y, 0@z), x, y) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(25) (EVAL_3(x[3], y[3])≥NonInfC∧EVAL_3(x[3], y[3])≥COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥))
We simplified constraint (25) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(26) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(28) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(29) (0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
For Pair EVAL_2(x, y) → COND_EVAL_21(>@z(x, 0@z), x, y) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(30) (EVAL_2(x[4], y[4])≥NonInfC∧EVAL_2(x[4], y[4])≥COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) (0 ≥ 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0)
We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34) (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL_11(TRUE, x, y) → EVAL_2(x, y) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5]), EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(35) (x[5]=x[11]∧x[2]=x[5]∧&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2]))=TRUE∧y[2]=y[5]∧y[5]=y[11] ⇒ COND_EVAL_11(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_11(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (35) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(36) (>@z(x[2], y[2])=TRUE∧>@z(x[2], 0@z)=TRUE∧>@z(y[2], 0@z)=TRUE ⇒ COND_EVAL_11(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_11(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-1 + x[2] ≥ 0)
We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-1 + x[2] ≥ 0)
We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39) (-1 + x[2] + (-1)y[2] ≥ 0∧y[2] + -1 ≥ 0∧-1 + x[2] ≥ 0 ⇒ 0 ≥ 0∧-1 + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40) (x[2] + (-1)y[2] ≥ 0∧y[2] + -1 ≥ 0∧x[2] ≥ 0 ⇒ 0 ≥ 0∧x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41) (x[2] ≥ 0∧y[2] + -1 ≥ 0∧y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42) (x[2] ≥ 0∧y[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧1 + y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(43) (y[5]=y[4]∧x[2]=x[5]∧x[5]=x[4]∧&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2]))=TRUE∧y[2]=y[5] ⇒ COND_EVAL_11(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_11(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(44) (>@z(x[2], y[2])=TRUE∧>@z(x[2], 0@z)=TRUE∧>@z(y[2], 0@z)=TRUE ⇒ COND_EVAL_11(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_11(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-1 + x[2] ≥ 0)
We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-1 + x[2] ≥ 0)
We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(47) (y[2] + -1 ≥ 0∧-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0 ⇒ -1 + x[2] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48) (y[2] + -1 ≥ 0∧x[2] ≥ 0∧y[2] + x[2] ≥ 0 ⇒ y[2] + x[2] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49) (y[2] ≥ 0∧x[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 1 + y[2] + x[2] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
For Pair COND_EVAL_2(TRUE, x, y) → EVAL_1(x, y) the following chains were created:
- We consider the chain EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]), COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6]), EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) which results in the following constraint:
(50) (y[11]=y[6]∧x[11]=x[6]∧x[6]=x[1]∧y[6]=y[1]∧>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL_2(TRUE, x[6], y[6])≥EVAL_1(x[6], y[6])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (50) using rules (III), (IV) which results in the following new constraint:
(51) (>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL_2(TRUE, x[11], y[11])≥EVAL_1(x[11], y[11])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (51) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(52) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(53) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(54) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (54) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(55) ((-1)x[11] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧0 = 0∧(-1)x[11] ≥ 0)
We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(56) (x[11] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧0 = 0∧x[11] ≥ 0)
- We consider the chain EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]), COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6]), EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(57) (y[11]=y[6]∧x[6]=x[2]∧x[11]=x[6]∧y[6]=y[2]∧>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL_2(TRUE, x[6], y[6])≥EVAL_1(x[6], y[6])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (57) using rules (III), (IV) which results in the following new constraint:
(58) (>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL_2(TRUE, x[11], y[11])≥EVAL_1(x[11], y[11])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (58) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(59) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (59) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(60) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (60) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(61) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (61) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(62) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧(-1)x[11] ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(63) (x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧x[11] ≥ 0∧0 = 0∧0 = 0)
For Pair EVAL_3(x, y) → COND_EVAL_3(>=@z(0@z, y), x, y) the following chains were created:
- We consider the chain EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(64) (EVAL_3(x[7], y[7])≥NonInfC∧EVAL_3(x[7], y[7])≥COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])∧(UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥))
We simplified constraint (64) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(65) ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (65) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(66) ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (66) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(67) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥))
We simplified constraint (67) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(68) ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL_3(TRUE, x, y) → EVAL_1(x, y) the following chains were created:
- We consider the chain EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]), COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8]), EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(69) (y[7]=y[8]∧x[7]=x[8]∧y[8]=y[2]∧>=@z(0@z, y[7])=TRUE∧x[8]=x[2] ⇒ COND_EVAL_3(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL_3(TRUE, x[8], y[8])≥EVAL_1(x[8], y[8])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (69) using rules (III), (IV) which results in the following new constraint:
(70) (>=@z(0@z, y[7])=TRUE ⇒ COND_EVAL_3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL_3(TRUE, x[7], y[7])≥EVAL_1(x[7], y[7])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (70) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(71) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (71) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(72) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (72) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(73) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (73) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(74) ((-1)y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
We simplified constraint (74) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(75) (y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- We consider the chain EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]), COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8]), EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) which results in the following constraint:
(76) (y[7]=y[8]∧x[8]=x[1]∧x[7]=x[8]∧y[8]=y[1]∧>=@z(0@z, y[7])=TRUE ⇒ COND_EVAL_3(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL_3(TRUE, x[8], y[8])≥EVAL_1(x[8], y[8])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (76) using rules (III), (IV) which results in the following new constraint:
(77) (>=@z(0@z, y[7])=TRUE ⇒ COND_EVAL_3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL_3(TRUE, x[7], y[7])≥EVAL_1(x[7], y[7])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (77) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(78) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (78) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(79) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (79) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(80) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (80) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(81) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0)
We simplified constraint (81) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(82) (y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_31(TRUE, x, y) → EVAL_3(x, -@z(y, 1@z)) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(83) (>@z(y[3], 0@z)=TRUE∧x[9]=x[3]1∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[3]1 ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (83) using rules (III), (IV) which results in the following new constraint:
(84) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (84) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(85) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (85) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(86) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (86) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(87) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧1 ≥ 0)
We simplified constraint (87) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(88) (-1 + y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (88) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(89) (y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(90) (>@z(y[3], 0@z)=TRUE∧x[9]=x[7]∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[7] ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (90) using rules (III), (IV) which results in the following new constraint:
(91) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (91) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(92) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (92) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(93) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (93) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(94) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (94) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(95) (-1 + y[3] ≥ 0 ⇒ 1 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (95) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(96) (y[3] ≥ 0 ⇒ 1 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
For Pair COND_EVAL_1(TRUE, x, y) → EVAL_3(x, y) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]), COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10]), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(97) (x[10]=x[3]∧y[10]=y[3]∧&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1]))=TRUE∧y[1]=y[10]∧x[1]=x[10] ⇒ COND_EVAL_1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL_1(TRUE, x[10], y[10])≥EVAL_3(x[10], y[10])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (97) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(98) (>=@z(y[1], x[1])=TRUE∧>@z(x[1], 0@z)=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_3(x[1], y[1])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (98) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(99) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (99) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(100) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (100) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(101) (x[1] + -1 ≥ 0∧y[1] + (-1)x[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (101) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(102) (x[1] ≥ 0∧-1 + y[1] + (-1)x[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧1 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (102) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(103) (x[1] ≥ 0∧y[1] + (-1)x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (103) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(104) (x[1] ≥ 0∧y[1] ≥ 0∧x[1] + y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + (3)x[1] + y[1] ≥ 0)
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]), COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10]), EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(105) (&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1]))=TRUE∧y[10]=y[7]∧y[1]=y[10]∧x[10]=x[7]∧x[1]=x[10] ⇒ COND_EVAL_1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL_1(TRUE, x[10], y[10])≥EVAL_3(x[10], y[10])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (105) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(106) (>=@z(y[1], x[1])=TRUE∧>@z(x[1], 0@z)=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_3(x[1], y[1])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (106) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(107) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (107) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(108) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (108) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(109) (-1 + y[1] ≥ 0∧x[1] + -1 ≥ 0∧y[1] + (-1)x[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (109) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(110) (-1 + y[1] ≥ 0∧x[1] ≥ 0∧-1 + y[1] + (-1)x[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧1 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (110) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(111) (y[1] ≥ 0∧x[1] ≥ 0∧y[1] + (-1)x[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + y[1] + (2)x[1] ≥ 0)
We simplified constraint (111) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(112) (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + (3)x[1] + y[1] ≥ 0)
For Pair EVAL_2(x, y) → COND_EVAL_2(>=@z(0@z, x), x, y) the following chains were created:
- We consider the chain EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(113) (EVAL_2(x[11], y[11])≥NonInfC∧EVAL_2(x[11], y[11])≥COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])∧(UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥))
We simplified constraint (113) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(114) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (114) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(115) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (115) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(116) (0 ≥ 0∧(UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0)
We simplified constraint (116) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(117) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧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_21(TRUE, x, y) → EVAL_2(-@z(x, 1@z), y)
- (x[4] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 = 0)
- (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧1 ≥ 0∧0 = 0)
- EVAL_1(x, y) → COND_EVAL_1(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL_1(x, y) → COND_EVAL_11(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
- EVAL_3(x, y) → COND_EVAL_31(>@z(y, 0@z), x, y)
- (0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
- EVAL_2(x, y) → COND_EVAL_21(>@z(x, 0@z), x, y)
- (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- COND_EVAL_11(TRUE, x, y) → EVAL_2(x, y)
- (x[2] ≥ 0∧y[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧1 + y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
- (y[2] ≥ 0∧x[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 1 + y[2] + x[2] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
- COND_EVAL_2(TRUE, x, y) → EVAL_1(x, y)
- (x[11] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧0 = 0∧x[11] ≥ 0)
- (x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧0 ≥ 0∧x[11] ≥ 0∧0 = 0∧0 = 0)
- EVAL_3(x, y) → COND_EVAL_3(>=@z(0@z, y), x, y)
- ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- COND_EVAL_3(TRUE, x, y) → EVAL_1(x, y)
- (y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- (y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_31(TRUE, x, y) → EVAL_3(x, -@z(y, 1@z))
- (y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
- (y[3] ≥ 0 ⇒ 1 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
- COND_EVAL_1(TRUE, x, y) → EVAL_3(x, y)
- (x[1] ≥ 0∧y[1] ≥ 0∧x[1] + y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + (3)x[1] + y[1] ≥ 0)
- (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥)∧2 + (-1)Bound + (3)x[1] + y[1] ≥ 0)
- EVAL_2(x, y) → COND_EVAL_2(>=@z(0@z, x), x, y)
- ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧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) = 0
POL(&&(x1, x2)) = 0
POL(EVAL_1(x1, x2)) = -1 + x2 + (2)x1
POL(FALSE) = 0
POL(COND_EVAL_11(x1, x2, x3)) = -1 + x3 + (2)x2 + (-1)x1
POL(>@z(x1, x2)) = -1
POL(COND_EVAL_3(x1, x2, x3)) = -1 + x3 + (2)x2
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL_1(x1, x2, x3)) = -1 + x3 + (2)x2 + (-1)x1
POL(EVAL_2(x1, x2)) = -1 + x2 + x1
POL(COND_EVAL_2(x1, x2, x3)) = -1 + x3 + x2
POL(COND_EVAL_21(x1, x2, x3)) = -1 + x3 + x2
POL(COND_EVAL_31(x1, x2, x3)) = -1 + x3 + (2)x2
POL(1@z) = 1
POL(undefined) = -1
POL(EVAL_3(x1, x2)) = -1 + x2 + (2)x1
The following pairs are in P>:
COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
The following pairs are in Pbound:
COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
The following pairs are in P≥:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
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
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_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(10): COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(10) -> (3), if ((y[10] →* y[3])∧(x[10] →* x[3]))
(6) -> (1), if ((y[6] →* y[1])∧(x[6] →* x[1]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(1) -> (10), if ((x[1] →* x[10])∧(y[1] →* y[10])∧(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])) →* TRUE))
(10) -> (7), if ((y[10] →* y[7])∧(x[10] →* x[7]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (2), if ((y[8] →* y[2])∧(x[8] →* x[2]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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:
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(10): COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(1) -> (10), if ((x[1] →* x[10])∧(y[1] →* y[10])∧(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])) →* TRUE))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(10) -> (3), if ((y[10] →* y[3])∧(x[10] →* x[3]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(10) -> (7), if ((y[10] →* y[7])∧(x[10] →* x[7]))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(1) (>@z(y[3], 0@z)=TRUE∧x[9]=x[3]1∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[3]1 ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + y[3] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (y[3] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 = 0)
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(8) (>@z(y[3], 0@z)=TRUE∧x[9]=x[7]∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[7] ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (y[3] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
For Pair EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(15) (EVAL_3(x[3], y[3])≥NonInfC∧EVAL_3(x[3], y[3])≥COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧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_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧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_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8]) the following chains were created:
- We consider the chain EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]), COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8]), EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) which results in the following constraint:
(20) (y[7]=y[8]∧x[8]=x[1]∧x[7]=x[8]∧y[8]=y[1]∧>=@z(0@z, y[7])=TRUE ⇒ COND_EVAL_3(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL_3(TRUE, x[8], y[8])≥EVAL_1(x[8], y[8])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (20) using rules (III), (IV) which results in the following new constraint:
(21) (>=@z(0@z, y[7])=TRUE ⇒ COND_EVAL_3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL_3(TRUE, x[7], y[7])≥EVAL_1(x[7], y[7])∧(UIncreasing(EVAL_1(x[8], y[8])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧(-1)y[7] ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧(-1)y[7] ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) ((-1)y[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[8], y[8])), ≥)∧(-1)y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25) ((-1)y[7] ≥ 0 ⇒ (-1)y[7] ≥ 0∧0 = 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (y[7] ≥ 0 ⇒ y[7] ≥ 0∧0 = 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 = 0)
For Pair EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) the following chains were created:
- We consider the chain EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(27) (EVAL_3(x[7], y[7])≥NonInfC∧EVAL_3(x[7], y[7])≥COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])∧(UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥))
We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28) ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[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_EVAL_3(>=@z(0@z, y[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) ((UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(31) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10]) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]), COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10]), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(32) (x[10]=x[3]∧y[10]=y[3]∧&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1]))=TRUE∧y[1]=y[10]∧x[1]=x[10] ⇒ COND_EVAL_1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL_1(TRUE, x[10], y[10])≥EVAL_3(x[10], y[10])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>=@z(y[1], x[1])=TRUE∧>@z(x[1], 0@z)=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_3(x[1], y[1])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + y[1] ≥ 0∧y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (-1 + y[1] ≥ 0∧-1 + y[1] + (-1)x[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧(-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (x[1] + y[1] ≥ 0∧y[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧1 + (-1)Bound + (2)x[1] + y[1] ≥ 0∧x[1] + y[1] ≥ 0)
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]), COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10]), EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7]) which results in the following constraint:
(39) (&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1]))=TRUE∧y[10]=y[7]∧y[1]=y[10]∧x[10]=x[7]∧x[1]=x[10] ⇒ COND_EVAL_1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL_1(TRUE, x[10], y[10])≥EVAL_3(x[10], y[10])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (39) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(40) (>=@z(y[1], x[1])=TRUE∧>@z(x[1], 0@z)=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_3(x[1], y[1])∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42) (y[1] + (-1)x[1] ≥ 0∧x[1] + -1 ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43) (y[1] + (-1)x[1] ≥ 0∧-1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ -1 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (-1 + y[1] + (-1)x[1] ≥ 0∧-1 + y[1] ≥ 0∧x[1] ≥ 0 ⇒ (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (y[1] ≥ 0∧x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] + y[1] ≥ 0∧x[1] + y[1] ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
For Pair EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1]) which results in the following constraint:
(46) (EVAL_1(x[1], y[1])≥NonInfC∧EVAL_1(x[1], y[1])≥COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥))
We simplified constraint (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(47) ((UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(48) ((UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(49) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥))
We simplified constraint (49) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(50) (0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
- (y[3] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 = 0)
- (y[3] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
- EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
- (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
- (y[7] ≥ 0 ⇒ y[7] ≥ 0∧0 = 0∧(UIncreasing(EVAL_1(x[8], y[8])), ≥)∧0 ≥ 0∧0 = 0)
- EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
- (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
- (x[1] + y[1] ≥ 0∧y[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[10], y[10])), ≥)∧1 + (-1)Bound + (2)x[1] + y[1] ≥ 0∧x[1] + y[1] ≥ 0)
- (y[1] ≥ 0∧x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] + y[1] ≥ 0∧x[1] + y[1] ≥ 0∧(UIncreasing(EVAL_3(x[10], y[10])), ≥))
- EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
- (0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])), ≥)∧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(EVAL_1(x1, x2)) = -1 + x2 + x1
POL(&&(x1, x2)) = -1
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(COND_EVAL_3(x1, x2, x3)) = -1 + x2
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL_1(x1, x2, x3)) = -1 + x3 + x2
POL(COND_EVAL_31(x1, x2, x3)) = x2
POL(1@z) = 1
POL(undefined) = -1
POL(EVAL_3(x1, x2)) = x1
The following pairs are in P>:
EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
The following pairs are in Pbound:
COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
The following pairs are in P≥:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
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
↳ AND
↳ 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:
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(1) (EVAL_3(x[3], y[3])≥NonInfC∧EVAL_3(x[3], y[3])≥COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 = 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(6) (>@z(y[3], 0@z)=TRUE∧x[9]=x[3]1∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[3]1 ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0 ⇒ -1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + y[3] ≥ 0 ⇒ 0 = 0∧0 = 0∧-1 + (-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (y[3] ≥ 0 ⇒ 0 = 0∧0 = 0∧(-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
- (0 = 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
- (y[3] ≥ 0 ⇒ 0 = 0∧0 = 0∧(-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 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(0@z) = 0
POL(TRUE) = -1
POL(COND_EVAL_31(x1, x2, x3)) = -1 + x3
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
POL(EVAL_3(x1, x2)) = -1 + x2
The following pairs are in P>:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in Pbound:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in P≥:
EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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
↳ AND
↳ 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:
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(10): COND_EVAL_1(TRUE, x[10], y[10]) → EVAL_3(x[10], y[10])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(1) -> (10), if ((x[1] →* x[10])∧(y[1] →* y[10])∧(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])) →* TRUE))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(10) -> (3), if ((y[10] →* y[3])∧(x[10] →* x[3]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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:
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(1) (EVAL_3(x[3], y[3])≥NonInfC∧EVAL_3(x[3], y[3])≥COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
For Pair COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(6) (>@z(y[3], 0@z)=TRUE∧x[9]=x[3]1∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[3]1 ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧2 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧2 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0 ⇒ 2 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + y[3] ≥ 0 ⇒ 0 = 0∧2 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (y[3] ≥ 0 ⇒ 0 = 0∧4 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
- COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
- (y[3] ≥ 0 ⇒ 0 = 0∧4 + (-1)Bound + (2)y[3] ≥ 0∧1 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
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(COND_EVAL_31(x1, x2, x3)) = 2 + (2)x3
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
POL(EVAL_3(x1, x2)) = 2 + (2)x2
The following pairs are in P>:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in Pbound:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in P≥:
EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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
↳ AND
↳ IDP
↳ 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): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(1) (EVAL_2(x[4], y[4])≥NonInfC∧EVAL_2(x[4], y[4])≥COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(6) (y[4]=y[0]∧-@z(x[0], 1@z)=x[4]1∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE∧y[0]=y[4]1 ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + x[4] ≥ 0 ⇒ -1 + (-1)Bound + x[4] ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + x[4] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧-1 + (-1)Bound + x[4] ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[4] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧(-1)Bound + x[4] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 = 0)
- COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
- (x[4] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧(-1)Bound + x[4] ≥ 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(EVAL_2(x1, x2)) = -1 + x1
POL(TRUE) = 0
POL(COND_EVAL_21(x1, x2, x3)) = -1 + x2
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in P≥:
EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ 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): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(&&(&&(>@z(x[1], 0@z), >@z(y[1], 0@z)), >=@z(y[1], x[1])), x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(5): COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(7): EVAL_3(x[7], y[7]) → COND_EVAL_3(>=@z(0@z, y[7]), x[7], y[7])
(8): COND_EVAL_3(TRUE, x[8], y[8]) → EVAL_1(x[8], y[8])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(6) -> (1), if ((y[6] →* y[1])∧(x[6] →* x[1]))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
(9) -> (7), if ((-@z(y[9], 1@z) →* y[7])∧(x[9] →* x[7]))
(5) -> (4), if ((y[5] →* y[4])∧(x[5] →* x[4]))
(8) -> (1), if ((y[8] →* y[1])∧(x[8] →* x[1]))
(2) -> (5), if ((x[2] →* x[5])∧(y[2] →* y[5])∧(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])) →* TRUE))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(5) -> (11), if ((y[5] →* y[11])∧(x[5] →* x[11]))
(7) -> (8), if ((x[7] →* x[8])∧(y[7] →* y[8])∧(>=@z(0@z, y[7]) →* TRUE))
(8) -> (2), if ((y[8] →* y[2])∧(x[8] →* x[2]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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:
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(5): COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(2) -> (5), if ((x[2] →* x[5])∧(y[2] →* y[5])∧(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])) →* TRUE))
(5) -> (11), if ((y[5] →* y[11])∧(x[5] →* x[11]))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
(5) -> (4), if ((y[5] →* y[4])∧(x[5] →* x[4]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(1) (y[4]=y[0]∧-@z(x[0], 1@z)=x[4]1∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE∧y[0]=y[4]1 ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[4] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[4] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(8) (y[0]=y[11]∧y[4]=y[0]∧-@z(x[0], 1@z)=x[11]∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[4] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(15) (EVAL_2(x[4], y[4])≥NonInfC∧EVAL_2(x[4], y[4])≥COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) the following chains were created:
- We consider the chain EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(20) (EVAL_2(x[11], y[11])≥NonInfC∧EVAL_2(x[11], y[11])≥COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])∧(UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) ((UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 = 0∧(UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5]), EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]) which results in the following constraint:
(25) (x[5]=x[11]∧x[2]=x[5]∧&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2]))=TRUE∧y[2]=y[5]∧y[5]=y[11] ⇒ COND_EVAL_11(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_11(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26) (>@z(x[2], y[2])=TRUE∧>@z(x[2], 0@z)=TRUE∧>@z(y[2], 0@z)=TRUE ⇒ COND_EVAL_11(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_11(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-2 + x[2] ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-2 + x[2] ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) (-1 + x[2] + (-1)y[2] ≥ 0∧y[2] + -1 ≥ 0∧-1 + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥)∧-2 + x[2] ≥ 0)
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (x[2] ≥ 0∧y[2] + -1 ≥ 0∧y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥)∧-1 + y[2] + x[2] ≥ 0)
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31) (x[2] ≥ 0∧y[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥)∧y[2] + x[2] ≥ 0)
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(32) (y[5]=y[4]∧x[2]=x[5]∧x[5]=x[4]∧&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2]))=TRUE∧y[2]=y[5] ⇒ COND_EVAL_11(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_11(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(x[2], y[2])=TRUE∧>@z(x[2], 0@z)=TRUE∧>@z(y[2], 0@z)=TRUE ⇒ COND_EVAL_11(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_11(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-2 + x[2] ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (-1 + x[2] + (-1)y[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], y[5])), ≥)∧0 ≥ 0∧-2 + x[2] ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + x[2] ≥ 0∧-1 + x[2] + (-1)y[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ 0 ≥ 0∧-2 + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (y[2] + x[2] ≥ 0∧x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ 0 ≥ 0∧-1 + y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (1 + y[2] + x[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
For Pair EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(39) (EVAL_1(x[2], y[2])≥NonInfC∧EVAL_1(x[2], y[2])≥COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(40) ((UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(41) ((UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(42) (0 ≥ 0∧1 ≥ 0∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥))
We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(43) (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧1 ≥ 0∧0 = 0)
For Pair COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6]) the following chains were created:
- We consider the chain EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11]), COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6]), EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(44) (y[11]=y[6]∧x[6]=x[2]∧x[11]=x[6]∧y[6]=y[2]∧>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL_2(TRUE, x[6], y[6])≥EVAL_1(x[6], y[6])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (44) using rules (III), (IV) which results in the following new constraint:
(45) (>=@z(0@z, x[11])=TRUE ⇒ COND_EVAL_2(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL_2(TRUE, x[11], y[11])≥EVAL_1(x[11], y[11])∧(UIncreasing(EVAL_1(x[6], y[6])), ≥))
We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(46) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧2 + (-1)Bound ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(47) ((-1)x[11] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[6], y[6])), ≥)∧2 + (-1)Bound ≥ 0∧(-1)x[11] ≥ 0)
We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(48) ((-1)x[11] ≥ 0 ⇒ (-1)x[11] ≥ 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧2 + (-1)Bound ≥ 0)
We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(49) ((-1)x[11] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧(-1)x[11] ≥ 0∧2 + (-1)Bound ≥ 0)
We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(50) (x[11] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧x[11] ≥ 0∧2 + (-1)Bound ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
- (x[4] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
- (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
- (0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
- (0 = 0∧(UIncreasing(COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
- (x[2] ≥ 0∧y[2] ≥ 0∧1 + y[2] + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥)∧y[2] + x[2] ≥ 0)
- (1 + y[2] + x[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[5], y[5])), ≥))
- EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])), ≥)∧1 ≥ 0∧0 = 0)
- COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
- (x[11] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_1(x[6], y[6])), ≥)∧x[11] ≥ 0∧2 + (-1)Bound ≥ 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(EVAL_1(x1, x2)) = 2 + x1
POL(&&(x1, x2)) = 0
POL(FALSE) = -1
POL(COND_EVAL_11(x1, x2, x3)) = 1 + x2 + (2)x1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL_2(x1, x2)) = 2
POL(COND_EVAL_2(x1, x2, x3)) = 2
POL(COND_EVAL_21(x1, x2, x3)) = 2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
The following pairs are in Pbound:
COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
The following pairs are in P≥:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 → FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 → TRUE1
&&(TRUE, FALSE)1 → FALSE1
&&(FALSE, TRUE)1 → FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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:
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(5): COND_EVAL_11(TRUE, x[5], y[5]) → EVAL_2(x[5], y[5])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(2) -> (5), if ((x[2] →* x[5])∧(y[2] →* y[5])∧(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])) →* TRUE))
(5) -> (11), if ((y[5] →* y[11])∧(x[5] →* x[11]))
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(5) -> (4), if ((y[5] →* y[4])∧(x[5] →* x[4]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(1) (EVAL_2(x[4], y[4])≥NonInfC∧EVAL_2(x[4], y[4])≥COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(6) (y[4]=y[0]∧-@z(x[0], 1@z)=x[4]1∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE∧y[0]=y[4]1 ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + (2)x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + (2)x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + (2)x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + x[4] ≥ 0 ⇒ -1 + (-1)Bound + (2)x[4] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[4] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[4] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
- COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
- (x[4] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[4] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[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(EVAL_2(x1, x2)) = (2)x1
POL(TRUE) = -1
POL(COND_EVAL_21(x1, x2, x3)) = -1 + (2)x2
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in P≥:
none
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:none
R is empty.
The integer pair graph is empty.
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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
↳ AND
↳ 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:
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(11): EVAL_2(x[11], y[11]) → COND_EVAL_2(>=@z(0@z, x[11]), x[11], y[11])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_11(&&(&&(>@z(x[2], 0@z), >@z(y[2], 0@z)), >@z(x[2], y[2])), x[2], y[2])
(6): COND_EVAL_2(TRUE, x[6], y[6]) → EVAL_1(x[6], y[6])
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(6) -> (2), if ((y[6] →* y[2])∧(x[6] →* x[2]))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
(0) -> (11), if ((y[0] →* y[11])∧(-@z(x[0], 1@z) →* x[11]))
(11) -> (6), if ((x[11] →* x[6])∧(y[11] →* y[6])∧(>=@z(0@z, x[11]) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ 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): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
(0): COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
(4) -> (0), if ((x[4] →* x[0])∧(y[4] →* y[0])∧(>@z(x[4], 0@z) →* TRUE))
(0) -> (4), if ((y[0] →* y[4])∧(-@z(x[0], 1@z) →* x[4]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(1) (EVAL_2(x[4], y[4])≥NonInfC∧EVAL_2(x[4], y[4])≥COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧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_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]), COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0]), EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4]) which results in the following constraint:
(6) (y[4]=y[0]∧-@z(x[0], 1@z)=x[4]1∧x[4]=x[0]∧>@z(x[4], 0@z)=TRUE∧y[0]=y[4]1 ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + (2)x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + (2)x[4] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + x[4] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[4] ≥ 0∧(UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 + (-1)Bound + (2)x[4] ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧3 + (-1)Bound + (2)x[4] ≥ 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])), ≥)∧0 = 0∧0 ≥ 0)
- COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
- (x[4] ≥ 0 ⇒ (UIncreasing(EVAL_2(-@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧3 + (-1)Bound + (2)x[4] ≥ 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(EVAL_2(x1, x2)) = 2 + (2)x1
POL(TRUE) = 2
POL(COND_EVAL_21(x1, x2, x3)) = 1 + (2)x2
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = 2
The following pairs are in P>:
EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[0], y[0]) → EVAL_2(-@z(x[0], 1@z), y[0])
The following pairs are in P≥:
none
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): EVAL_2(x[4], y[4]) → COND_EVAL_21(>@z(x[4], 0@z), x[4], y[4])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:none
R is empty.
The integer pair graph is empty.
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
(9): COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
(3) -> (9), if ((x[3] →* x[9])∧(y[3] →* y[9])∧(>@z(y[3], 0@z) →* TRUE))
(9) -> (3), if ((-@z(y[9], 1@z) →* y[3])∧(x[9] →* x[3]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(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_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(1) (EVAL_3(x[3], y[3])≥NonInfC∧EVAL_3(x[3], y[3])≥COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) ((UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)) the following chains were created:
- We consider the chain EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]), COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z)), EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3]) which results in the following constraint:
(6) (>@z(y[3], 0@z)=TRUE∧x[9]=x[3]1∧x[3]=x[9]∧y[3]=y[9]∧-@z(y[9], 1@z)=y[3]1 ⇒ COND_EVAL_31(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL_31(TRUE, x[9], y[9])≥EVAL_3(x[9], -@z(y[9], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_31(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_31(TRUE, x[3], y[3])≥EVAL_3(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0)
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + y[3] ≥ 0 ⇒ -1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 = 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (y[3] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 1@z))), ≥)∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
- (y[3] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_3(x[9], -@z(y[9], 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(0@z) = 0
POL(TRUE) = -1
POL(COND_EVAL_31(x1, x2, x3)) = -1 + x3
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
POL(EVAL_3(x1, x2)) = -1 + x2
The following pairs are in P>:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in Pbound:
COND_EVAL_31(TRUE, x[9], y[9]) → EVAL_3(x[9], -@z(y[9], 1@z))
The following pairs are in P≥:
EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ AND
↳ IDP
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL_3(x[3], y[3]) → COND_EVAL_31(>@z(y[3], 0@z), x[3], y[3])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_11(TRUE, x0, x1)
eval_3(x0, x1)
eval_1(x0, x1)
Cond_eval_3(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.