Termination of the given ITRSProblem could successfully be proven:
↳ ITRS
↳ ITRStoIDPProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
eval_2(x, y) → Cond_eval_21(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
Cond_eval_1(TRUE, x, y) → eval_2(x, 1@z)
eval_2(x, y) → Cond_eval_2(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
Cond_eval_2(TRUE, x, y) → eval_2(x, *@z(2@z, y))
Cond_eval_21(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
eval_1(x, y) → Cond_eval_1(>=@z(x, 0@z), x, y)
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(x0, x1)
Added dependency pairs
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
I DP problem:
The following domains are used:
z
The ITRS R consists of the following rules:
eval_2(x, y) → Cond_eval_21(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
Cond_eval_1(TRUE, x, y) → eval_2(x, 1@z)
eval_2(x, y) → Cond_eval_2(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
Cond_eval_2(TRUE, x, y) → eval_2(x, *@z(2@z, y))
Cond_eval_21(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
eval_1(x, y) → Cond_eval_1(>=@z(x, 0@z), x, y)
The integer pair graph contains the following rules and edges:
(0): EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])
(1): COND_EVAL_21(TRUE, x[1], y[1]) → EVAL_1(-@z(x[1], 1@z), y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z)
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
(5): EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(>=@z(x[0], 0@z) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(-@z(x[1], 1@z) →* x[0]))
(2) -> (3), if ((x[2] →* x[3]))
(2) -> (5), if ((x[2] →* x[5]))
(3) -> (1), if ((x[3] →* x[1])∧(y[3] →* y[1])∧(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
(4) -> (3), if ((*@z(2@z, y[4]) →* y[3])∧(x[4] →* x[3]))
(4) -> (5), if ((*@z(2@z, y[4]) →* y[5])∧(x[4] →* x[5]))
(5) -> (4), if ((x[5] →* x[4])∧(y[5] →* y[4])∧(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(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): EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])
(1): COND_EVAL_21(TRUE, x[1], y[1]) → EVAL_1(-@z(x[1], 1@z), y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z)
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
(5): EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(>=@z(x[0], 0@z) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(-@z(x[1], 1@z) →* x[0]))
(2) -> (3), if ((x[2] →* x[3]))
(2) -> (5), if ((x[2] →* x[5]))
(3) -> (1), if ((x[3] →* x[1])∧(y[3] →* y[1])∧(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
(4) -> (3), if ((*@z(2@z, y[4]) →* y[3])∧(x[4] →* x[3]))
(4) -> (5), if ((*@z(2@z, y[4]) →* y[5])∧(x[4] →* x[5]))
(5) -> (4), if ((x[5] →* x[4])∧(y[5] →* y[4])∧(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(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_1(x, y) → COND_EVAL_1(>=@z(x, 0@z), x, y) the following chains were created:
- We consider the chain EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0]) which results in the following constraint:
(1) (EVAL_1(x[0], y[0])≥NonInfC∧EVAL_1(x[0], y[0])≥COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])∧(UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥))
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 = 0)
For Pair COND_EVAL_21(TRUE, x, y) → EVAL_1(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL_21(TRUE, x[1], y[1]) → EVAL_1(-@z(x[1], 1@z), y[1]), EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0]) which results in the following constraint:
(6) (x[3]=x[1]∧&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE∧y[3]=y[1]∧y[1]=y[0]∧-@z(x[1], 1@z)=x[0] ⇒ COND_EVAL_21(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_21(TRUE, x[1], y[1])≥EVAL_1(-@z(x[1], 1@z), y[1])∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>=@z(y[3], x[3])=TRUE∧>=@z(x[3], 0@z)=TRUE∧>@z(y[3], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_21(TRUE, x[3], y[3])≥EVAL_1(-@z(x[3], 1@z), y[3])∧(UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (-1 + x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
For Pair COND_EVAL_1(TRUE, x, y) → EVAL_2(x, 1@z) the following chains were created:
- We consider the chain EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0]), COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z), EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(12) (x[0]=x[2]∧y[0]=y[2]∧x[2]=x[5]∧>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], 1@z)∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥))
We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:
(13) (>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_1(TRUE, x[0], y[0])≥EVAL_2(x[0], 1@z)∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥))
We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14) (x[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧(-1)Bound + x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15) (x[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧(-1)Bound + x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16) (x[0] ≥ 0 ⇒ (-1)Bound + x[0] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥))
We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(17) (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(-1)Bound + x[0] ≥ 0∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧0 = 0)
- We consider the chain EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0]), COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z), EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(18) (x[0]=x[2]∧y[0]=y[2]∧x[2]=x[3]∧>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], 1@z)∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥))
We simplified constraint (18) using rules (III), (IV) which results in the following new constraint:
(19) (>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_1(TRUE, x[0], y[0])≥EVAL_2(x[0], 1@z)∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥))
We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20) (x[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧(-1)Bound + x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(21) (x[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧(-1)Bound + x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(22) (x[0] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧(-1)Bound + x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(23) (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧0 = 0∧(-1)Bound + x[0] ≥ 0)
For Pair EVAL_2(x, y) → COND_EVAL_21(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y) the following chains were created:
- We consider the chain EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(24) (EVAL_2(x[3], y[3])≥NonInfC∧EVAL_2(x[3], y[3])≥COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])∧(UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥))
We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25) ((UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26) ((UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27) (0 ≥ 0∧(UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧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∧(UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL_2(TRUE, x, y) → EVAL_2(x, *@z(2@z, y)) the following chains were created:
- We consider the chain EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4])), EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(29) (y[5]=y[4]∧*@z(2@z, y[4])=y[5]1∧x[4]=x[5]1∧x[5]=x[4]∧&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], *@z(2@z, y[4]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30) (>@z(x[5], y[5])=TRUE∧>=@z(x[5], 0@z)=TRUE∧>@z(y[5], 0@z)=TRUE ⇒ COND_EVAL_2(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_2(TRUE, x[5], y[5])≥EVAL_2(x[5], *@z(2@z, y[5]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33) (x[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34) (1 + y[5] + x[5] ≥ 0∧-1 + y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (2 + y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
- We consider the chain EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4])), EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(36) (y[5]=y[4]∧x[5]=x[4]∧*@z(2@z, y[4])=y[3]∧x[4]=x[3]∧&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], *@z(2@z, y[4]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(37) (>@z(x[5], y[5])=TRUE∧>=@z(x[5], 0@z)=TRUE∧>@z(y[5], 0@z)=TRUE ⇒ COND_EVAL_2(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_2(TRUE, x[5], y[5])≥EVAL_2(x[5], *@z(2@z, y[5]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41) (x[5] ≥ 0∧-1 + y[5] ≥ 0∧1 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42) (x[5] ≥ 0∧y[5] ≥ 0∧2 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
For Pair EVAL_2(x, y) → COND_EVAL_2(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y) the following chains were created:
- We consider the chain EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(43) (EVAL_2(x[5], y[5])≥NonInfC∧EVAL_2(x[5], y[5])≥COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))
We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(44) ((UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(45) ((UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(46) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))
We simplified constraint (46) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(47) (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_1(x, y) → COND_EVAL_1(>=@z(x, 0@z), x, y)
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 = 0)
- COND_EVAL_21(TRUE, x, y) → EVAL_1(-@z(x, 1@z), y)
- (-1 + x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
- COND_EVAL_1(TRUE, x, y) → EVAL_2(x, 1@z)
- (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(-1)Bound + x[0] ≥ 0∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧0 = 0)
- (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[2], 1@z)), ≥)∧0 = 0∧(-1)Bound + x[0] ≥ 0)
- EVAL_2(x, y) → COND_EVAL_21(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
- (0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
- COND_EVAL_2(TRUE, x, y) → EVAL_2(x, *@z(2@z, y))
- (2 + y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
- (x[5] ≥ 0∧y[5] ≥ 0∧2 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧0 ≥ 0∧0 ≥ 0)
- EVAL_2(x, y) → COND_EVAL_2(&&(&&(>=@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 = 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(*@z(x1, x2)) = x1·x2
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL_1(x1, x2)) = x1
POL(2@z) = 2
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = 1
POL(EVAL_2(x1, x2)) = -1 + x1
POL(COND_EVAL_1(x1, x2, x3)) = x2
POL(COND_EVAL_2(x1, x2, x3)) = -1 + x2
POL(COND_EVAL_21(x1, x2, x3)) = x2 + x1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z)
The following pairs are in Pbound:
COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], 1@z)
The following pairs are in P≥:
EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])
COND_EVAL_21(TRUE, x[1], y[1]) → EVAL_1(-@z(x[1], 1@z), y[1])
EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
At least the following rules have been oriented under context sensitive arithmetic replacement:
*@z1 ↔
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): EVAL_1(x[0], y[0]) → COND_EVAL_1(>=@z(x[0], 0@z), x[0], y[0])
(1): COND_EVAL_21(TRUE, x[1], y[1]) → EVAL_1(-@z(x[1], 1@z), y[1])
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
(5): EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
(3) -> (1), if ((x[3] →* x[1])∧(y[3] →* y[1])∧(&&(&&(>=@z(x[3], 0@z), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
(5) -> (4), if ((x[5] →* x[4])∧(y[5] →* y[4])∧(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(-@z(x[1], 1@z) →* x[0]))
(4) -> (5), if ((*@z(2@z, y[4]) →* y[5])∧(x[4] →* x[5]))
(4) -> (3), if ((*@z(2@z, y[4]) →* y[3])∧(x[4] →* x[3]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
(5): EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
(5) -> (4), if ((x[5] →* x[4])∧(y[5] →* y[4])∧(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])) →* TRUE))
(4) -> (5), if ((*@z(2@z, y[4]) →* y[5])∧(x[4] →* x[5]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(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_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4])) the following chains were created:
- We consider the chain EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4])), EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(1) (y[5]=y[4]∧*@z(2@z, y[4])=y[5]1∧x[4]=x[5]1∧x[5]=x[4]∧&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], *@z(2@z, y[4]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(x[5], y[5])=TRUE∧>=@z(x[5], 0@z)=TRUE∧>@z(y[5], 0@z)=TRUE ⇒ COND_EVAL_2(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_2(TRUE, x[5], y[5])≥EVAL_2(x[5], *@z(2@z, y[5]))∧(UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧1 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧-1 + y[5] ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧1 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧-1 + y[5] ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + y[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧1 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧-1 + y[5] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (-1 + y[5] ≥ 0∧x[5] ≥ 0∧1 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧2 + (-1)Bound + x[5] ≥ 0∧-1 + y[5] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (y[5] ≥ 0∧x[5] ≥ 0∧2 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧2 + (-1)Bound + x[5] ≥ 0∧y[5] ≥ 0)
For Pair EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) the following chains were created:
- We consider the chain EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(8) (EVAL_2(x[5], y[5])≥NonInfC∧EVAL_2(x[5], y[5])≥COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
- (y[5] ≥ 0∧x[5] ≥ 0∧2 + y[5] + x[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], *@z(2@z, y[4]))), ≥)∧2 + (-1)Bound + x[5] ≥ 0∧y[5] ≥ 0)
- EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
- (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧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)) = -1
POL(0@z) = 0
POL(*@z(x1, x2)) = x1·x2
POL(EVAL_2(x1, x2)) = 1 + (-1)x2 + x1
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL_2(x1, x2, x3)) = (-1)x3 + x2 + (-1)x1
POL(2@z) = 2
POL(FALSE) = -1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
The following pairs are in Pbound:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], *@z(2@z, y[4]))
The following pairs are in P≥:
EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
At least the following rules have been oriented under context sensitive arithmetic replacement:
*@z1 ↔
&&(FALSE, FALSE)1 ↔ FALSE1
&&(TRUE, TRUE)1 ↔ TRUE1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(TRUE, FALSE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(5): EVAL_2(x[5], y[5]) → COND_EVAL_2(&&(&&(>=@z(x[5], 0@z), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_1(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.