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)
Cond_eval_21(TRUE, x, y, z) → eval_1(x, +@z(y, 1@z), z)
eval_2(x, y, z) → Cond_eval_22(>@z(x, z), x, y, z)
Cond_eval_2(TRUE, x, y, z) → eval_1(-@z(x, 1@z), y, z)
eval_2(x, y, z) → Cond_eval_21(>@z(x, z), x, y, z)
Cond_eval_22(TRUE, x, y, z) → eval_1(x, y, +@z(z, 1@z))
eval_1(x, y, z) → Cond_eval_1(>@z(x, y), x, y, z)
eval_2(x, y, z) → Cond_eval_2(>=@z(z, x), x, y, z)
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(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)
Cond_eval_21(TRUE, x, y, z) → eval_1(x, +@z(y, 1@z), z)
eval_2(x, y, z) → Cond_eval_22(>@z(x, z), x, y, z)
Cond_eval_2(TRUE, x, y, z) → eval_1(-@z(x, 1@z), y, z)
eval_2(x, y, z) → Cond_eval_21(>@z(x, z), x, y, z)
Cond_eval_22(TRUE, x, y, z) → eval_1(x, y, +@z(z, 1@z))
eval_1(x, y, z) → Cond_eval_1(>@z(x, y), x, y, z)
eval_2(x, y, z) → Cond_eval_2(>=@z(z, x), x, y, z)
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0], z[0]) → COND_EVAL_2(>=@z(z[0], x[0]), x[0], y[0], z[0])
(1): COND_EVAL_2(TRUE, x[1], y[1], z[1]) → EVAL_1(-@z(x[1], 1@z), y[1], z[1])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(4): COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(0) -> (1), if ((z[0] →* z[1])∧(x[0] →* x[1])∧(y[0] →* y[1])∧(>=@z(z[0], x[0]) →* TRUE))
(1) -> (2), if ((y[1] →* y[2])∧(z[1] →* z[2])∧(-@z(x[1], 1@z) →* x[2]))
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(3) -> (7), if ((y[3] →* y[7])∧(z[3] →* z[7])∧(x[3] →* x[7]))
(4) -> (2), if ((y[4] →* y[2])∧(+@z(z[4], 1@z) →* z[2])∧(x[4] →* x[2]))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
(7) -> (4), if ((z[7] →* z[4])∧(x[7] →* x[4])∧(y[7] →* y[4])∧(>@z(x[7], z[7]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(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_2(>=@z(z[0], x[0]), x[0], y[0], z[0])
(1): COND_EVAL_2(TRUE, x[1], y[1], z[1]) → EVAL_1(-@z(x[1], 1@z), y[1], z[1])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(4): COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(0) -> (1), if ((z[0] →* z[1])∧(x[0] →* x[1])∧(y[0] →* y[1])∧(>=@z(z[0], x[0]) →* TRUE))
(1) -> (2), if ((y[1] →* y[2])∧(z[1] →* z[2])∧(-@z(x[1], 1@z) →* x[2]))
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(3) -> (7), if ((y[3] →* y[7])∧(z[3] →* z[7])∧(x[3] →* x[7]))
(4) -> (2), if ((y[4] →* y[2])∧(+@z(z[4], 1@z) →* z[2])∧(x[4] →* x[2]))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
(7) -> (4), if ((z[7] →* z[4])∧(x[7] →* x[4])∧(y[7] →* y[4])∧(>@z(x[7], z[7]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(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_2(>=@z(z, x), x, y, z) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0], z[0]) → COND_EVAL_2(>=@z(z[0], x[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_2(>=@z(z[0], x[0]), x[0], y[0], z[0])∧(UIncreasing(COND_EVAL_2(>=@z(z[0], x[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_2(>=@z(z[0], x[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_2(>=@z(z[0], x[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) (0 ≥ 0∧(UIncreasing(COND_EVAL_2(>=@z(z[0], x[0]), x[0], y[0], z[0])), ≥)∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 = 0∧(UIncreasing(COND_EVAL_2(>=@z(z[0], x[0]), x[0], y[0], z[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧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[0], y[0], z[0]) → COND_EVAL_2(>=@z(z[0], x[0]), x[0], y[0], z[0]), COND_EVAL_2(TRUE, x[1], y[1], z[1]) → EVAL_1(-@z(x[1], 1@z), y[1], z[1]), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(6) (z[1]=z[2]∧>=@z(z[0], x[0])=TRUE∧-@z(x[1], 1@z)=x[2]∧y[1]=y[2]∧y[0]=y[1]∧x[0]=x[1]∧z[0]=z[1] ⇒ COND_EVAL_2(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_EVAL_2(TRUE, x[1], y[1], z[1])≥EVAL_1(-@z(x[1], 1@z), y[1], z[1])∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>=@z(z[0], x[0])=TRUE ⇒ COND_EVAL_2(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL_2(TRUE, x[0], y[0], z[0])≥EVAL_1(-@z(x[0], 1@z), y[0], z[0])∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (z[0] + (-1)x[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (z[0] + (-1)x[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (z[0] + (-1)x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (z[0] + (-1)x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (z[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(13) (z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
(14) (z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair EVAL_1(x, y, z) → COND_EVAL_1(>@z(x, y), x, y, z) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(15) (EVAL_1(x[2], y[2], z[2])≥NonInfC∧EVAL_1(x[2], y[2], z[2])≥COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧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[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) (0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19) (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
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[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), 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_2(>=@z(z[0], x[0]), x[0], y[0], z[0]) which results in the following constraint:
(20) (>@z(x[2], y[2])=TRUE∧x[2]=x[3]∧y[2]=y[3]∧x[3]=x[0]∧y[3]=y[0]∧z[2]=z[3]∧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 (20) using rules (III), (IV) which results in the following new constraint:
(21) (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(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) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ -1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0)
We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 = 0∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (x[2] ≥ 0 ⇒ 0 = 0∧(-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(27) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧(-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
(28) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧(-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(29) (>@z(x[2], y[2])=TRUE∧y[3]=y[6]∧x[2]=x[3]∧x[3]=x[6]∧y[2]=y[3]∧z[2]=z[3]∧z[3]=z[6] ⇒ 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[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(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) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0)
We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(36) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0∧0 = 0∧0 = 0)
(37) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0∧0 = 0∧0 = 0)
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]) which results in the following constraint:
(38) (>@z(x[2], y[2])=TRUE∧y[3]=y[7]∧x[2]=x[3]∧y[2]=y[3]∧x[3]=x[7]∧z[3]=z[7]∧z[2]=z[3] ⇒ 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 (38) using rules (III), (IV) which results in the following new constraint:
(39) (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(40) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(41) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(42) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0)
We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(43) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 = 0)
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧(-1)Bound + x[2] ≥ 0∧0 = 0)
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(45) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧(-1)Bound + x[2] ≥ 0∧0 = 0)
(46) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧(-1)Bound + x[2] ≥ 0∧0 = 0)
For Pair COND_EVAL_22(TRUE, x, y, z) → EVAL_1(x, y, +@z(z, 1@z)) the following chains were created:
- We consider the chain EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]), COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z)), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(47) (z[7]=z[4]∧+@z(z[4], 1@z)=z[2]∧x[7]=x[4]∧y[4]=y[2]∧x[4]=x[2]∧>@z(x[7], z[7])=TRUE∧y[7]=y[4] ⇒ COND_EVAL_22(TRUE, x[4], y[4], z[4])≥NonInfC∧COND_EVAL_22(TRUE, x[4], y[4], z[4])≥EVAL_1(x[4], y[4], +@z(z[4], 1@z))∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥))
We simplified constraint (47) using rules (III), (IV) which results in the following new constraint:
(48) (>@z(x[7], z[7])=TRUE ⇒ COND_EVAL_22(TRUE, x[7], y[7], z[7])≥NonInfC∧COND_EVAL_22(TRUE, x[7], y[7], z[7])≥EVAL_1(x[7], y[7], +@z(z[7], 1@z))∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥))
We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(49) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(50) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(51) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (51) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(52) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(53) (x[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
We simplified constraint (53) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(54) (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
(55) (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x, y, z) → EVAL_1(x, +@z(y, 1@z), z) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5]), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(56) (z[5]=z[2]∧x[5]=x[2]∧x[6]=x[5]∧y[6]=y[5]∧z[6]=z[5]∧+@z(y[5], 1@z)=y[2]∧>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_1(x[5], +@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (56) using rules (III), (IV) which results in the following new constraint:
(57) (>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[6], y[6], z[6])≥NonInfC∧COND_EVAL_21(TRUE, x[6], y[6], z[6])≥EVAL_1(x[6], +@z(y[6], 1@z), z[6])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (57) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(58) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (58) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(59) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (59) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(60) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧1 ≥ 0∧0 ≥ 0)
We simplified constraint (60) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(61) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (61) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(62) (x[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(63) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
(64) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
For Pair EVAL_2(x, y, z) → COND_EVAL_21(>@z(x, z), x, y, z) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(65) (EVAL_2(x[6], y[6], z[6])≥NonInfC∧EVAL_2(x[6], y[6], z[6])≥COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥))
We simplified constraint (65) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(66) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (66) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(67) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (67) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(68) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (68) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(69) (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair EVAL_2(x, y, z) → COND_EVAL_22(>@z(x, z), x, y, z) the following chains were created:
- We consider the chain EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]) which results in the following constraint:
(70) (EVAL_2(x[7], y[7], z[7])≥NonInfC∧EVAL_2(x[7], y[7], z[7])≥COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
We simplified constraint (70) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(71) ((UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (71) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(72) ((UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (72) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(73) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
We simplified constraint (73) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(74) (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x, y, z) → COND_EVAL_2(>=@z(z, x), x, y, z)
- (0 = 0∧(UIncreasing(COND_EVAL_2(>=@z(z[0], x[0]), x[0], y[0], z[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
- COND_EVAL_2(TRUE, x, y, z) → EVAL_1(-@z(x, 1@z), y, z)
- (z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- (z[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- EVAL_1(x, y, z) → COND_EVAL_1(>@z(x, y), x, y, z)
- (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL_1(TRUE, x, y, z) → EVAL_2(x, y, z)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧(-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧(-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0∧0 = 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0∧0 = 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧(-1)Bound + x[2] ≥ 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧(-1)Bound + x[2] ≥ 0∧0 = 0)
- COND_EVAL_22(TRUE, x, y, z) → EVAL_1(x, y, +@z(z, 1@z))
- (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
- (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0)
- COND_EVAL_21(TRUE, x, y, z) → EVAL_1(x, +@z(y, 1@z), z)
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 = 0)
- EVAL_2(x, y, z) → COND_EVAL_21(>@z(x, z), x, y, z)
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- EVAL_2(x, y, z) → COND_EVAL_22(>@z(x, z), x, y, z)
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
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)x3 + x2
POL(COND_EVAL_2(x1, x2, x3, x4)) = -1 + (-1)x3 + x2
POL(TRUE) = 0
POL(EVAL_2(x1, x2, x3)) = -1 + (-1)x2 + x1
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x3 + x2
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL_1(x1, x2, x3)) = -1 + (-1)x2 + x1
POL(COND_EVAL_22(x1, x2, x3, x4)) = -1 + (-1)x3 + x2
POL(+@z(x1, x2)) = x1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL_2(TRUE, x[1], y[1], z[1]) → EVAL_1(-@z(x[1], 1@z), y[1], z[1])
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_2(>=@z(z[0], x[0]), x[0], y[0], z[0])
EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
+@z1 ↔
↳ 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_2(>=@z(z[0], x[0]), x[0], y[0], z[0])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(4): COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(7) -> (4), if ((z[7] →* z[4])∧(x[7] →* x[4])∧(y[7] →* y[4])∧(>@z(x[7], z[7]) →* TRUE))
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(3) -> (0), if ((y[3] →* y[0])∧(z[3] →* z[0])∧(x[3] →* x[0]))
(4) -> (2), if ((y[4] →* y[2])∧(+@z(z[4], 1@z) →* z[2])∧(x[4] →* x[2]))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(3) -> (7), if ((y[3] →* y[7])∧(z[3] →* z[7])∧(x[3] →* x[7]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(7) -> (4), if ((z[7] →* z[4])∧(x[7] →* x[4])∧(y[7] →* y[4])∧(>@z(x[7], z[7]) →* TRUE))
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(+@z(z[4], 1@z) →* z[2])∧(x[4] →* x[2]))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(3) -> (7), if ((y[3] →* y[7])∧(z[3] →* z[7])∧(x[3] →* x[7]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(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_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z)) the following chains were created:
- We consider the chain EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]), COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z)), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(1) (z[7]=z[4]∧+@z(z[4], 1@z)=z[2]∧x[7]=x[4]∧y[4]=y[2]∧x[4]=x[2]∧>@z(x[7], z[7])=TRUE∧y[7]=y[4] ⇒ COND_EVAL_22(TRUE, x[4], y[4], z[4])≥NonInfC∧COND_EVAL_22(TRUE, x[4], y[4], z[4])≥EVAL_1(x[4], y[4], +@z(z[4], 1@z))∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[7], z[7])=TRUE ⇒ COND_EVAL_22(TRUE, x[7], y[7], z[7])≥NonInfC∧COND_EVAL_22(TRUE, x[7], y[7], z[7])≥EVAL_1(x[7], y[7], +@z(z[7], 1@z))∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧-1 + (-1)Bound + (-1)z[7] + x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧-1 + (-1)Bound + (-1)z[7] + x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + (-1)z[7] + x[7] ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[7] + (-1)z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧-1 + (-1)Bound + (-1)z[7] + x[7] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧(-1)Bound + x[7] ≥ 0)
We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8) (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧(-1)Bound + x[7] ≥ 0)
(9) (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧(-1)Bound + x[7] ≥ 0)
For Pair EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]) the following chains were created:
- We consider the chain EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]) which results in the following constraint:
(10) (EVAL_2(x[7], y[7], z[7])≥NonInfC∧EVAL_2(x[7], y[7], z[7])≥COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥))
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 = 0)
For Pair COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(15) (>@z(x[2], y[2])=TRUE∧y[3]=y[6]∧x[2]=x[3]∧x[3]=x[6]∧y[2]=y[3]∧z[2]=z[3]∧z[3]=z[6] ⇒ 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 (15) using rules (III), (IV) which results in the following new constraint:
(16) (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21) (x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (21) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(22) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
(23) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7]) which results in the following constraint:
(24) (>@z(x[2], y[2])=TRUE∧y[3]=y[7]∧x[2]=x[3]∧y[2]=y[3]∧x[3]=x[7]∧z[3]=z[7]∧z[2]=z[3] ⇒ 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 (24) using rules (III), (IV) which results in the following new constraint:
(25) (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (25) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(26) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(28) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(29) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(31) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
(32) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(33) (EVAL_1(x[2], y[2], z[2])≥NonInfC∧EVAL_1(x[2], y[2], z[2])≥COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) ((UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) ((UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0)
We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(37) (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5]) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5]), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(38) (z[5]=z[2]∧x[5]=x[2]∧x[6]=x[5]∧y[6]=y[5]∧z[6]=z[5]∧+@z(y[5], 1@z)=y[2]∧>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_1(x[5], +@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (38) using rules (III), (IV) which results in the following new constraint:
(39) (>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[6], y[6], z[6])≥NonInfC∧COND_EVAL_21(TRUE, x[6], y[6], z[6])≥EVAL_1(x[6], +@z(y[6], 1@z), z[6])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(40) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(41) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(42) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(43) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (x[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(45) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
(46) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(47) (EVAL_2(x[6], y[6], z[6])≥NonInfC∧EVAL_2(x[6], y[6], z[6])≥COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥))
We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50) (0 ≥ 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0)
We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(51) (0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
- (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧(-1)Bound + x[7] ≥ 0)
- (x[7] ≥ 0∧z[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[4], y[4], +@z(z[4], 1@z))), ≥)∧0 = 0∧0 = 0∧(-1)Bound + x[7] ≥ 0)
- EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])), ≥)∧0 = 0)
- COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)
- EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
- (0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧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(EVAL_1(x1, x2, x3)) = -1 + (-1)x3 + x1
POL(COND_EVAL_1(x1, x2, x3, x4)) = -1 + (-1)x4 + x2
POL(TRUE) = -1
POL(EVAL_2(x1, x2, x3)) = -1 + (-1)x3 + x1
POL(COND_EVAL_22(x1, x2, x3, x4)) = -1 + (-1)x4 + x2
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x4 + x2
POL(+@z(x1, x2)) = x1 + x2
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = 0
The following pairs are in P>:
COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
The following pairs are in Pbound:
COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
The following pairs are in P≥:
EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
At least the following rules have been oriented under context sensitive arithmetic replacement:
+@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(3) -> (7), if ((y[3] →* y[7])∧(z[3] →* z[7])∧(x[3] →* x[7]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(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_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3]), EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(1) (>@z(x[2], y[2])=TRUE∧y[3]=y[6]∧x[2]=x[3]∧x[3]=x[6]∧y[2]=y[3]∧z[2]=z[3]∧z[3]=z[6] ⇒ 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 (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2], z[2])≥EVAL_2(x[2], y[2], z[2])∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ -1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧0 ≥ 0)
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 = 0)
We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 = 0)
(9) (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 = 0)
For Pair EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(10) (EVAL_1(x[2], y[2], z[2])≥NonInfC∧EVAL_1(x[2], y[2], z[2])≥COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧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_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5]) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5]), EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2]) which results in the following constraint:
(15) (z[5]=z[2]∧x[5]=x[2]∧x[6]=x[5]∧y[6]=y[5]∧z[6]=z[5]∧+@z(y[5], 1@z)=y[2]∧>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[5], y[5], z[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5], z[5])≥EVAL_1(x[5], +@z(y[5], 1@z), z[5])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16) (>@z(x[6], z[6])=TRUE ⇒ COND_EVAL_21(TRUE, x[6], y[6], z[6])≥NonInfC∧COND_EVAL_21(TRUE, x[6], y[6], z[6])≥EVAL_1(x[6], +@z(y[6], 1@z), z[6])∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ (UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (x[6] + -1 + (-1)z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21) (x[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
We simplified constraint (21) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(22) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
(23) (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
For Pair EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) the following chains were created:
- We consider the chain EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6]) which results in the following constraint:
(24) (EVAL_2(x[6], y[6], z[6])≥NonInfC∧EVAL_2(x[6], y[6], z[6])≥COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥))
We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27) ((UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(28) (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧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_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 = 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[3], y[3], z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 = 0)
- EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0)
- COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
- (x[6] ≥ 0∧z[6] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(x[5], +@z(y[5], 1@z), z[5])), ≥)∧0 = 0∧0 = 0)
- EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])), ≥)∧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(EVAL_1(x1, x2, x3)) = -1 + (-1)x2 + x1
POL(COND_EVAL_1(x1, x2, x3, x4)) = -1 + (-1)x3 + x2
POL(TRUE) = 2
POL(EVAL_2(x1, x2, x3)) = -1 + (-1)x2 + x1
POL(+@z(x1, x2)) = x1 + x2
POL(COND_EVAL_21(x1, x2, x3, x4)) = -1 + (-1)x3 + 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[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
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≥:
COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
At least the following rules have been oriented under context sensitive arithmetic replacement:
+@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) → EVAL_2(x[3], y[3], z[3])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(2) -> (3), if ((z[2] →* z[3])∧(x[2] →* x[3])∧(y[2] →* y[3])∧(>@z(x[2], y[2]) →* TRUE))
(3) -> (6), if ((y[3] →* y[6])∧(z[3] →* z[6])∧(x[3] →* x[6]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
↳ 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_2(>=@z(z[0], x[0]), x[0], y[0], z[0])
(1): COND_EVAL_2(TRUE, x[1], y[1], z[1]) → EVAL_1(-@z(x[1], 1@z), y[1], z[1])
(2): EVAL_1(x[2], y[2], z[2]) → COND_EVAL_1(>@z(x[2], y[2]), x[2], y[2], z[2])
(4): COND_EVAL_22(TRUE, x[4], y[4], z[4]) → EVAL_1(x[4], y[4], +@z(z[4], 1@z))
(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) → EVAL_1(x[5], +@z(y[5], 1@z), z[5])
(6): EVAL_2(x[6], y[6], z[6]) → COND_EVAL_21(>@z(x[6], z[6]), x[6], y[6], z[6])
(7): EVAL_2(x[7], y[7], z[7]) → COND_EVAL_22(>@z(x[7], z[7]), x[7], y[7], z[7])
(7) -> (4), if ((z[7] →* z[4])∧(x[7] →* x[4])∧(y[7] →* y[4])∧(>@z(x[7], z[7]) →* TRUE))
(1) -> (2), if ((y[1] →* y[2])∧(z[1] →* z[2])∧(-@z(x[1], 1@z) →* x[2]))
(4) -> (2), if ((y[4] →* y[2])∧(+@z(z[4], 1@z) →* z[2])∧(x[4] →* x[2]))
(0) -> (1), if ((z[0] →* z[1])∧(x[0] →* x[1])∧(y[0] →* y[1])∧(>=@z(z[0], x[0]) →* TRUE))
(5) -> (2), if ((+@z(y[5], 1@z) →* y[2])∧(z[5] →* z[2])∧(x[5] →* x[2]))
(6) -> (5), if ((z[6] →* z[5])∧(x[6] →* x[5])∧(y[6] →* y[5])∧(>@z(x[6], z[6]) →* TRUE))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1, x2)
Cond_eval_21(TRUE, x0, x1, x2)
eval_2(x0, x1, x2)
Cond_eval_2(TRUE, x0, x1, x2)
Cond_eval_22(TRUE, x0, x1, x2)
eval_1(x0, x1, x2)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes.