Termination of the given ITRSProblem could successfully be proven:
↳ ITRS
↳ ITRStoIDPProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
Cond_eval_1(TRUE, x, y, z) → eval_2(x, y, z)
eval_2(x, y, z) → Cond_eval_21(&&(>@z(x, z), >@z(y, z)), x, y, z)
Cond_eval_21(TRUE, x, y, z) → eval_2(x, -@z(y, 1@z), z)
Cond_eval_2(TRUE, x, y, z) → eval_1(-@z(x, 1@z), y, z)
eval_1(x, y, z) → Cond_eval_1(>@z(x, z), x, y, z)
eval_2(x, y, z) → Cond_eval_2(&&(>@z(x, z), >=@z(z, y)), x, y, z)
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
Added dependency pairs
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
I DP problem:
The following domains are used:
z
The ITRS R consists of the following rules:
Cond_eval_1(TRUE, x, y, z) → eval_2(x, y, z)
eval_2(x, y, z) → Cond_eval_21(&&(>@z(x, z), >@z(y, z)), x, y, z)
Cond_eval_21(TRUE, x, y, z) → eval_2(x, -@z(y, 1@z), z)
Cond_eval_2(TRUE, x, y, z) → eval_1(-@z(x, 1@z), y, z)
eval_1(x, y, z) → Cond_eval_1(>@z(x, z), x, y, z)
eval_2(x, y, z) → Cond_eval_2(&&(>@z(x, z), >=@z(z, y)), x, y, z)
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(1): EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])
(2): COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(4): EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
(1) -> (3), if ((z[1] →* z[3])∧(x[1] →* x[3])∧(y[1] →* y[3])∧(>@z(x[1], z[1]) →* TRUE))
(2) -> (1), if ((y[2] →* y[1])∧(z[2] →* z[1])∧(-@z(x[2], 1@z) →* x[1]))
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(3) -> (4), if ((y[3] →* y[4])∧(z[3] →* z[4])∧(x[3] →* x[4]))
(4) -> (2), if ((z[4] →* z[2])∧(x[4] →* x[2])∧(y[4] →* y[2])∧(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])) →* TRUE))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(5) -> (4), if ((-@z(y[5], 1@z) →* y[4])∧(z[5] →* z[4])∧(x[5] →* x[4]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(1): EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])
(2): COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(4): EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
(1) -> (3), if ((z[1] →* z[3])∧(x[1] →* x[3])∧(y[1] →* y[3])∧(>@z(x[1], z[1]) →* TRUE))
(2) -> (1), if ((y[2] →* y[1])∧(z[2] →* z[1])∧(-@z(x[2], 1@z) →* x[1]))
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(3) -> (4), if ((y[3] →* y[4])∧(z[3] →* z[4])∧(x[3] →* x[4]))
(4) -> (2), if ((z[4] →* z[2])∧(x[4] →* x[2])∧(y[4] →* y[2])∧(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])) →* TRUE))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(5) -> (4), if ((-@z(y[5], 1@z) →* y[4])∧(z[5] →* z[4])∧(x[5] →* x[4]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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, y, z) → COND_EVAL_21(&&(>@z(x, z), >@z(y, z)), x, y, z) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(1) (EVAL_2(x[0], y[0], z[0])≥NonInfC∧EVAL_2(x[0], y[0], z[0])≥COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧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[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧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[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧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_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair EVAL_1(x, y, z) → COND_EVAL_1(>@z(x, z), x, y, z) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1]) which results in the following constraint:
(6) (EVAL_1(x[1], y[1], z[1])≥NonInfC∧EVAL_1(x[1], y[1], z[1])≥COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])∧(UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) (0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥)∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) (0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥)∧0 = 0)
For Pair COND_EVAL_2(TRUE, x, y, z) → EVAL_1(-@z(x, 1@z), y, z) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4]), COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2]), EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1]) which results in the following constraint:
(11) (y[2]=y[1]∧z[4]=z[2]∧z[2]=z[1]∧y[4]=y[2]∧&&(>@z(x[4], z[4]), >=@z(z[4], y[4]))=TRUE∧-@z(x[2], 1@z)=x[1]∧x[4]=x[2] ⇒ COND_EVAL_2(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_2(TRUE, x[2], y[2], z[2])≥EVAL_1(-@z(x[2], 1@z), y[2], z[2])∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12) (>@z(x[4], z[4])=TRUE∧>=@z(z[4], y[4])=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4], z[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4], z[4])≥EVAL_1(-@z(x[4], 1@z), y[4], z[4])∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) (-1 + x[4] + (-1)z[4] ≥ 0∧z[4] + (-1)y[4] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) (-1 + x[4] + (-1)z[4] ≥ 0∧z[4] + (-1)y[4] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) (-1 + x[4] + (-1)z[4] ≥ 0∧z[4] + (-1)y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16) (x[4] ≥ 0∧z[4] + (-1)y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17) (x[4] ≥ 0∧z[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(18) (x[4] ≥ 0∧z[4] ≥ 0∧y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
(19) (x[4] ≥ 0∧z[4] ≥ 0∧y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
For Pair COND_EVAL_1(TRUE, x, y, z) → EVAL_2(x, y, z) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
(20) (z[3]=z[4]∧y[1]=y[3]∧y[3]=y[4]∧x[3]=x[4]∧z[1]=z[3]∧x[1]=x[3]∧>@z(x[1], z[1])=TRUE ⇒ COND_EVAL_1(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_EVAL_1(TRUE, x[3], y[3], z[3])≥EVAL_2(x[3], y[3], z[3])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (20) using rules (III), (IV) which results in the following new constraint:
(21) (>@z(x[1], z[1])=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1], z[1])≥EVAL_2(x[1], y[1], z[1])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0)
We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (x[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(27) (x[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[1] ≥ 0∧0 ≥ 0)
(28) (x[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[1] ≥ 0∧0 ≥ 0)
- We consider the chain EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(29) (y[1]=y[3]∧x[3]=x[0]∧y[3]=y[0]∧z[1]=z[3]∧x[1]=x[3]∧>@z(x[1], z[1])=TRUE∧z[3]=z[0] ⇒ COND_EVAL_1(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_EVAL_1(TRUE, x[3], y[3], z[3])≥EVAL_2(x[3], y[3], z[3])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (29) using rules (III), (IV) which results in the following new constraint:
(30) (>@z(x[1], z[1])=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1], z[1])≥EVAL_2(x[1], y[1], z[1])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ -1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34) (x[1] + -1 + (-1)z[1] ≥ 0 ⇒ -1 + (-1)Bound + (-1)z[1] + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (x[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(36) (x[1] ≥ 0∧z[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
(37) (x[1] ≥ 0∧z[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
For Pair EVAL_2(x, y, z) → COND_EVAL_2(&&(>@z(x, z), >=@z(z, y)), x, y, z) the following chains were created:
- We consider the chain EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
(38) (EVAL_2(x[4], y[4], z[4])≥NonInfC∧EVAL_2(x[4], y[4], z[4])≥COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])∧(UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥))
We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39) ((UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40) ((UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41) ((UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(42) (0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x, y, z) → EVAL_2(x, -@z(y, 1@z), z) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]), EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(43) (&&(>@z(x[0], z[0]), >@z(y[0], z[0]))=TRUE∧x[0]=x[5]∧y[0]=y[5]∧z[0]=z[5]∧-@z(y[5], 1@z)=y[0]1∧z[5]=z[0]1∧x[5]=x[0]1 ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_2(x[5], -@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(44) (>@z(x[0], z[0])=TRUE∧>@z(y[0], z[0])=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0], z[0])≥EVAL_2(x[0], -@z(y[0], 1@z), z[0])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(47) (-1 + y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48) (y[0] ≥ 0∧x[0] + -1 + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49) (y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(50) (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
(51) (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]), EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
(52) (&&(>@z(x[0], z[0]), >@z(y[0], z[0]))=TRUE∧x[0]=x[5]∧y[0]=y[5]∧x[5]=x[4]∧z[0]=z[5]∧-@z(y[5], 1@z)=y[4]∧z[5]=z[4] ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_2(x[5], -@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (52) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(53) (>@z(x[0], z[0])=TRUE∧>@z(y[0], z[0])=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0], z[0])≥EVAL_2(x[0], -@z(y[0], 1@z), z[0])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56) (-1 + y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(57) (y[0] ≥ 0∧x[0] + -1 + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(58) (y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(59) (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
(60) (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x, y, z) → COND_EVAL_21(&&(>@z(x, z), >@z(y, z)), x, y, z)
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- EVAL_1(x, y, z) → COND_EVAL_1(>@z(x, z), x, y, z)
- (0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])), ≥)∧0 = 0)
- COND_EVAL_2(TRUE, x, y, z) → EVAL_1(-@z(x, 1@z), y, z)
- (x[4] ≥ 0∧z[4] ≥ 0∧y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
- (x[4] ≥ 0∧z[4] ≥ 0∧y[4] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2], z[2])), ≥))
- COND_EVAL_1(TRUE, x, y, z) → EVAL_2(x, y, z)
- (x[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[1] ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[1] ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧z[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
- (x[1] ≥ 0∧z[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
- EVAL_2(x, y, z) → COND_EVAL_2(&&(>@z(x, z), >=@z(z, y)), x, y, z)
- (0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL_21(TRUE, x, y, z) → EVAL_2(x, -@z(y, 1@z), z)
- (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
- (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
- (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
- (y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(COND_EVAL_1(x1, x2, x3, x4)) = -1 + (-1)x4 + x2
POL(COND_EVAL_2(x1, x2, x3, x4)) = -1 + (-1)x4 + x2 + x1
POL(TRUE) = -1
POL(EVAL_2(x1, x2, x3)) = -1 + (-1)x3 + x1
POL(&&(x1, x2)) = -1
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x4 + x2
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL_1(x1, x2, x3)) = -1 + (-1)x3 + x1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])
The following pairs are in Pbound:
COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
The following pairs are in P≥:
EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])
COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2])
COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
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): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(1): EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])
(2): COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2])
(4): EVAL_2(x[4], y[4], z[4]) → COND_EVAL_2(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])), x[4], y[4], z[4])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(4) -> (2), if ((z[4] →* z[2])∧(x[4] →* x[2])∧(y[4] →* y[2])∧(&&(>@z(x[4], z[4]), >=@z(z[4], y[4])) →* TRUE))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(2) -> (1), if ((y[2] →* y[1])∧(z[2] →* z[1])∧(-@z(x[2], 1@z) →* x[1]))
(5) -> (4), if ((-@z(y[5], 1@z) →* y[4])∧(z[5] →* z[4])∧(x[5] →* x[4]))
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]), EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(1) (&&(>@z(x[0], z[0]), >@z(y[0], z[0]))=TRUE∧x[0]=x[5]∧y[0]=y[5]∧z[0]=z[5]∧-@z(y[5], 1@z)=y[0]1∧z[5]=z[0]1∧x[5]=x[0]1 ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_2(x[5], -@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(x[0], z[0])=TRUE∧>@z(y[0], z[0])=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0], z[0])≥EVAL_2(x[0], -@z(y[0], 1@z), z[0])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧-1 + (-1)Bound + (-1)z[0] + y[0] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧-1 + (-1)Bound + (-1)z[0] + y[0] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧-1 + (-1)Bound + (-1)z[0] + y[0] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[0] + -1 + (-1)z[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (z[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8) (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
(9) (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
For Pair EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(10) (EVAL_2(x[0], y[0], z[0])≥NonInfC∧EVAL_2(x[0], y[0], z[0])≥COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
- (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
- (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + y[0] ≥ 0)
- EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[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(TRUE) = -1
POL(&&(x1, x2)) = 0
POL(EVAL_2(x1, x2, x3)) = (-1)x3 + x2
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x4 + x3
POL(FALSE) = 1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
The following pairs are in P≥:
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(1): EVAL_1(x[1], y[1], z[1]) → COND_EVAL_1(>@z(x[1], z[1]), x[1], y[1], z[1])
(2): COND_EVAL_2(TRUE, x[2], y[2], z[2]) → EVAL_1(-@z(x[2], 1@z), y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(2) -> (1), if ((y[2] →* y[1])∧(z[2] →* z[1])∧(-@z(x[2], 1@z) →* x[1]))
(1) -> (3), if ((z[1] →* z[3])∧(x[1] →* x[3])∧(y[1] →* y[3])∧(>@z(x[1], z[1]) →* TRUE))
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(z[5] →* z[0])∧(x[5] →* x[0]))
(0) -> (5), if ((z[0] →* z[5])∧(x[0] →* x[5])∧(y[0] →* y[5])∧(&&(>@z(x[0], z[0]), >@z(y[0], z[0])) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
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[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5]), EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(1) (&&(>@z(x[0], z[0]), >@z(y[0], z[0]))=TRUE∧x[0]=x[5]∧y[0]=y[5]∧z[0]=z[5]∧-@z(y[5], 1@z)=y[0]1∧z[5]=z[0]1∧x[5]=x[0]1 ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_2(x[5], -@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(x[0], z[0])=TRUE∧>@z(y[0], z[0])=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0], z[0])≥EVAL_2(x[0], -@z(y[0], 1@z), z[0])∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + (-1)z[0] + y[0] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + (-1)z[0] + y[0] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[0] + -1 + (-1)z[0] ≥ 0∧-1 + y[0] + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧(-1)Bound + (-1)z[0] + y[0] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[0] + -1 + (-1)z[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (z[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8) (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
(9) (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
For Pair EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
(10) (EVAL_2(x[0], y[0], z[0])≥NonInfC∧EVAL_2(x[0], y[0], z[0])≥COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13) (0 ≥ 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[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[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
- (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
- (z[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z), z[5])), ≥)∧1 + (-1)Bound + y[0] ≥ 0)
- EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
- (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[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(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL_2(x1, x2, x3)) = (-1)x3 + x2
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x4 + x3 + (-1)x1
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[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_2(x[5], -@z(y[5], 1@z), z[5])
The following pairs are in P≥:
EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
TRUE1 → &&(TRUE, TRUE)1
&&(TRUE, FALSE)1 ↔ FALSE1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_21(&&(>@z(x[0], z[0]), >@z(y[0], z[0])), x[0], y[0], z[0])
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.