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(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval(&&(>@z(+@z(x, y), 0@z), >@z(x, 0@z)), x, y)
Cond_eval1(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
eval(x, y) → Cond_eval1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
Cond_eval2(TRUE, x, y) → eval(x, y)
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Added dependency pairs
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
I DP problem:
The following domains are used:
z
The ITRS R consists of the following rules:
Cond_eval(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval(&&(>@z(+@z(x, y), 0@z), >@z(x, 0@z)), x, y)
Cond_eval1(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
eval(x, y) → Cond_eval1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
Cond_eval2(TRUE, x, y) → eval(x, y)
The integer pair graph contains the following rules and edges:
(0): COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(0) -> (2), if ((y[0] →* y[2])∧(x[0] →* x[2]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(2) -> (0), if ((x[2] →* x[0])∧(y[2] →* y[0])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(5) -> (2), if ((-@z(y[5], 1@z) →* y[2])∧(x[5] →* x[2]))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(0) -> (2), if ((y[0] →* y[2])∧(x[0] →* x[2]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(2) -> (0), if ((x[2] →* x[0])∧(y[2] →* y[0])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(5) -> (2), if ((-@z(y[5], 1@z) →* y[2])∧(x[5] →* x[2]))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_EVAL2(TRUE, x, y) → EVAL(x, y) the following chains were created:
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(1) (y[0]=y[3]∧x[2]=x[0]∧x[0]=x[3]∧y[2]=y[0]∧&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2]))=TRUE ⇒ COND_EVAL2(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL2(TRUE, x[0], y[0])≥EVAL(x[0], y[0])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>=@z(0@z, y[2])=TRUE∧>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(0@z, x[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], y[2])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) ((-1)x[2] ≥ 0∧(-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0 ⇒ -2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We solved constraint (5) using rule (IDP_SMT_SPLIT).
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(6) (x[2]=x[0]∧y[2]=y[0]∧y[0]=y[1]∧&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2]))=TRUE∧x[0]=x[1] ⇒ COND_EVAL2(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL2(TRUE, x[0], y[0])≥EVAL(x[0], y[0])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>=@z(0@z, y[2])=TRUE∧>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(0@z, x[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], y[2])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) ((-1)x[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)y[2] ≥ 0 ⇒ -2 ≥ 0∧-2 + (-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We solved constraint (10) using rule (IDP_SMT_SPLIT).
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0]), EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]) which results in the following constraint:
(11) (x[2]=x[0]∧y[0]=y[2]1∧y[2]=y[0]∧&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2]))=TRUE∧x[0]=x[2]1 ⇒ COND_EVAL2(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL2(TRUE, x[0], y[0])≥EVAL(x[0], y[0])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12) (>=@z(0@z, y[2])=TRUE∧>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(0@z, x[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], y[2])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-2 ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) ((-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ -2 ≥ 0∧-2 + (-1)Bound + x[2] ≥ 0∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We solved constraint (15) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x, y) → COND_EVAL(&&(>@z(+@z(x, y), 0@z), >@z(x, 0@z)), x, y) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(16) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧1 ≥ 0∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
For Pair EVAL(x, y) → COND_EVAL2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y) the following chains were created:
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]) which results in the following constraint:
(21) (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])∧(UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) ((UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (0 ≥ 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥)∧0 ≥ 0)
We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25) (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL(x, y) → COND_EVAL1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(26) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) (0 ≥ 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30) (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
For Pair COND_EVAL(TRUE, x, y) → EVAL(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(31) (-@z(x[4], 1@z)=x[3]∧&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧x[1]=x[4]∧y[1]=y[4]∧y[4]=y[3] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (31) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(32) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(36) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(37) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(38) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(40) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (40) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(41) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(46) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(47) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]) which results in the following constraint:
(49) (-@z(x[4], 1@z)=x[2]∧&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[2]∧x[1]=x[4]∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (49) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(50) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(51) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(52) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(53) (x[1] + -1 ≥ 0∧x[1] + -1 + y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (53) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(54) (x[1] ≥ 0∧x[1] + y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (54) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(55) (x[1] ≥ 0∧x[1] + y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(56) (x[1] ≥ 0∧x[1] + (-1)y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(57) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair COND_EVAL1(TRUE, x, y) → EVAL(x, -@z(y, 1@z)) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2]) which results in the following constraint:
(58) (x[5]=x[2]∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧-@z(y[5], 1@z)=y[2]∧x[3]=x[5] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (58) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(59) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (59) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(60) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (60) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(61) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (61) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(62) (-1 + x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(63) (-1 + (-1)x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (63) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(64) (y[3] ≥ 0∧x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(65) (-@z(y[5], 1@z)=y[3]1∧x[5]=x[3]1∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (65) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(66) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(67) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (67) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(68) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (68) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(69) ((-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(70) (x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(71) (x[3] ≥ 0∧y[3] ≥ 0∧(-1)x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(72) (x[3] ≥ 0∧x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(73) (-@z(y[5], 1@z)=y[1]∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5]∧x[5]=x[1] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (73) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(74) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(75) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(76) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(77) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(78) (-1 + y[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (78) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(79) (y[3] ≥ 0∧(-1)x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (79) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(80) (x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL2(TRUE, x, y) → EVAL(x, y)
- EVAL(x, y) → COND_EVAL(&&(>@z(+@z(x, y), 0@z), >@z(x, 0@z)), x, y)
- ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
- EVAL(x, y) → COND_EVAL2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL(x, y) → COND_EVAL1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
- COND_EVAL(TRUE, x, y) → EVAL(-@z(x, 1@z), y)
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧x[1] + y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- COND_EVAL1(TRUE, x, y) → EVAL(x, -@z(y, 1@z))
- (y[3] ≥ 0∧x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[3] ≥ 0∧x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0)
- (x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + x2
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL(x1, x2, x3)) = -1 + x2 + x1
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL2(x1, x2, x3)) = -1 + x2 + x1
POL(EVAL(x1, x2)) = -1 + x1
POL(+@z(x1, x2)) = x1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])
The following pairs are in Pbound:
COND_EVAL2(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
The following pairs are in P≥:
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(1) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧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(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 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(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(6) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) (0 ≥ 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(11) (-@z(x[4], 1@z)=x[3]∧&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧x[1]=x[4]∧y[1]=y[4]∧y[4]=y[3] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16) (x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(17) (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(18) (x[1] + (-1)y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(20) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (20) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(21) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25) (x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(26) (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(27) (x[1] + (-1)y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(29) (-@z(y[5], 1@z)=y[3]1∧x[5]=x[3]1∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) ((-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34) (x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (x[3] ≥ 0∧x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧1 + (-1)Bound + x[3] + y[3] ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(36) (-@z(y[5], 1@z)=y[1]∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5]∧x[5]=x[1] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(37) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40) (-1 + x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41) (-1 + (-1)x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42) ((-1)x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(43) (y[3] ≥ 0∧x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧1 + (-1)Bound + x[3] + y[3] ≥ 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
- (0 = 0∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
- (x[3] ≥ 0∧x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧1 + (-1)Bound + x[3] + y[3] ≥ 0∧0 ≥ 0)
- (y[3] ≥ 0∧x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧1 + (-1)Bound + x[3] + y[3] ≥ 0∧0 ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + x3 + (-1)x1
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL(x1, x2, x3)) = 1 + x3
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL(x1, x2)) = 1 + x2
POL(+@z(x1, x2)) = x1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
The following pairs are in P≥:
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(1) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7) (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(8) (x[1] + (-1)y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + (-1)y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(10) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧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(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) (0 = 0∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
- (0 = 0∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = x2 + x1
POL(COND_EVAL(x1, x2, x3)) = -1 + x3 + x2
POL(+@z(x1, x2)) = x1 + x2
POL(FALSE) = 1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(1) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧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(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(6) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (x[1] + -1 ≥ 0∧x[1] + -1 + y[1] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (x[1] ≥ 0∧x[1] + y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12) (x[1] ≥ 0∧x[1] + y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(13) (x[1] ≥ 0∧x[1] + (-1)y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
- ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[1] ≥ 0∧x[1] + y[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[1] + x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = 0
POL(&&(x1, x2)) = -1
POL(COND_EVAL(x1, x2, x3)) = -1 + x2
POL(EVAL(x1, x2)) = -1 + x1
POL(+@z(x1, x2)) = x1 + x2
POL(FALSE) = 1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(0@z, x[2])), >=@z(0@z, y[2])), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
(5) -> (2), if ((-@z(y[5], 1@z) →* y[2])∧(x[5] →* x[2]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[3]∧&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧x[1]=x[4]∧y[1]=y[4]∧y[4]=y[3] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(8) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(10) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (10) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(11) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(12) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(13) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(14) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ -1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(16) (x[1] ≥ 0∧(-1)y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(17) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18) (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(19) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(23) (1 ≥ 0∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(24) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥))
We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧1 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(28) (0 = 0∧0 = 0∧0 = 0∧0 = 0∧1 ≥ 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
For Pair COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(29) (-@z(y[5], 1@z)=y[3]1∧x[5]=x[3]1∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) ((-1)x[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34) (x[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (x[3] ≥ 0∧(-1)x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(36) (x[3] ≥ 0∧y[3] ≥ 0∧x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(37) (-@z(y[5], 1@z)=y[1]∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5]∧x[5]=x[1] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (37) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(38) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41) ((-1)x[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42) (x[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(43) (x[3] ≥ 0∧(-1)x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (x[3] ≥ 0∧y[3] ≥ 0∧x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (y[1] + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (2)y[1] + (2)x[1] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
- (1 ≥ 0∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
- EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
- (0 = 0∧0 = 0∧0 = 0∧0 = 0∧1 ≥ 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
- COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
- (x[3] ≥ 0∧y[3] ≥ 0∧x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[3] ≥ 0∧y[3] ≥ 0∧x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + (2)x3 + (2)x2
POL(TRUE) = -1
POL(&&(x1, x2)) = 0
POL(COND_EVAL(x1, x2, x3)) = (2)x3 + (2)x2 + x1
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL(x1, x2)) = 1 + (2)x2 + (2)x1
POL(+@z(x1, x2)) = x1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 → FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 → TRUE1
&&(TRUE, FALSE)1 → FALSE1
&&(FALSE, TRUE)1 → FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)) →* TRUE))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(1) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(6) (-@z(y[5], 1@z)=y[3]1∧x[5]=x[3]1∧y[3]=y[5]∧&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z))=TRUE∧x[3]=x[5] ⇒ COND_EVAL1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL1(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(y[3], 0@z)=TRUE∧>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0∧1 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0∧1 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0∧(-1)x[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0∧1 ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (-1 + y[3] ≥ 0∧x[3] ≥ 0∧-1 + (-1)x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧(-1)Bound + (2)y[3] + x[3] ≥ 0∧1 ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (y[3] ≥ 0∧x[3] ≥ 0∧(-1)x[3] + y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧2 + (-1)Bound + (2)y[3] + x[3] ≥ 0∧1 ≥ 0)
We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13) (x[3] + y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧2 + (-1)Bound + (3)x[3] + (2)y[3] ≥ 0∧1 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0)
- COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
- (x[3] + y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧2 + (-1)Bound + (3)x[3] + (2)y[3] ≥ 0∧1 ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(>=@z(x1, x2)) = -1
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + (2)x3 + (-1)x2 + (-1)x1
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = (2)x2 + (-1)x1
POL(+@z(x1, x2)) = x1 + x2
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
The following pairs are in P≥:
EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(&&(>@z(+@z(x[3], y[3]), 0@z), >=@z(0@z, x[3])), >@z(y[3], 0@z)), x[3], y[3])
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(5): COND_EVAL1(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z))
(5) -> (1), if ((-@z(y[5], 1@z) →* y[1])∧(x[5] →* x[1]))
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
(1) -> (4), if ((x[1] →* x[4])∧(y[1] →* y[4])∧(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]), COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(1) (&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z))=TRUE∧y[4]=y[1]1∧x[1]=x[4]∧-@z(x[4], 1@z)=x[1]1∧y[1]=y[4] ⇒ COND_EVAL(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(+@z(x[1], y[1]), 0@z)=TRUE∧>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[1] + -1 + y[1] ≥ 0∧x[1] + -1 ≥ 0 ⇒ -1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[1] + y[1] ≥ 0∧x[1] ≥ 0 ⇒ 1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7) (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
(8) (x[1] + (-1)y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + (-1)y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9) (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1]) which results in the following constraint:
(10) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧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(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14) ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[1] + y[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[1] ≥ 0∧y[1] + x[1] ≥ 0∧y[1] ≥ 0 ⇒ 1 + (-1)Bound + y[1] + (2)x[1] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
- ((UIncreasing(COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = 2
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = -1 + x2 + (2)x1
POL(COND_EVAL(x1, x2, x3)) = -1 + x3 + (2)x2
POL(+@z(x1, x2)) = x1 + x2
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
TRUE1 → &&(TRUE, TRUE)1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(>@z(+@z(x[1], y[1]), 0@z), >@z(x[1], 0@z)), x[1], y[1])
The set Q consists of the following terms:
Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.