Termination of the given ITRSProblem could successfully be proven:
↳ ITRS
↳ ITRStoIDPProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
Cond_eval_1(TRUE, x, y) → eval_2(x, y)
Cond_eval_21(TRUE, x, y) → eval_2(x, -@z(y, 1@z))
eval_2(x, y) → Cond_eval_21(&&(>@z(x, 0@z), >@z(y, 0@z)), x, y)
Cond_eval_2(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
eval_1(x, y) → Cond_eval_1(>@z(x, 0@z), x, y)
eval_2(x, y) → Cond_eval_2(&&(>@z(x, 0@z), >=@z(0@z, y)), x, y)
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(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:
Cond_eval_1(TRUE, x, y) → eval_2(x, y)
Cond_eval_21(TRUE, x, y) → eval_2(x, -@z(y, 1@z))
eval_2(x, y) → Cond_eval_21(&&(>@z(x, 0@z), >@z(y, 0@z)), x, y)
Cond_eval_2(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
eval_1(x, y) → Cond_eval_1(>@z(x, 0@z), x, y)
eval_2(x, y) → Cond_eval_2(&&(>@z(x, 0@z), >=@z(0@z, y)), x, y)
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
(1): COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
(5): COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z))
(0) -> (4), if ((x[0] →* x[4])∧(y[0] →* y[4])∧(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(x[1] →* x[0]))
(1) -> (3), if ((y[1] →* y[3])∧(x[1] →* x[3]))
(2) -> (1), if ((x[2] →* x[1])∧(y[2] →* y[1])∧(>@z(x[2], 0@z) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(x[5] →* x[0]))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(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_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
(1): COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
(5): COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z))
(0) -> (4), if ((x[0] →* x[4])∧(y[0] →* y[4])∧(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(x[1] →* x[0]))
(1) -> (3), if ((y[1] →* y[3])∧(x[1] →* x[3]))
(2) -> (1), if ((x[2] →* x[1])∧(y[2] →* y[1])∧(>@z(x[2], 0@z) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(5) -> (0), if ((-@z(y[5], 1@z) →* y[0])∧(x[5] →* x[0]))
(5) -> (3), if ((-@z(y[5], 1@z) →* y[3])∧(x[5] →* x[3]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(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_2(x, y) → COND_EVAL_2(&&(>@z(x, 0@z), >=@z(0@z, y)), x, y) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) which results in the following constraint:
(1) (EVAL_2(x[0], y[0])≥NonInfC∧EVAL_2(x[0], y[0])≥COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), 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_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), 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_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), 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∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
For Pair COND_EVAL_1(TRUE, x, y) → EVAL_2(x, y) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]), COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1]), EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(6) (y[2]=y[1]∧y[1]=y[3]∧x[2]=x[1]∧>@z(x[2], 0@z)=TRUE∧x[1]=x[3] ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_2(x[1], y[1])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7) (>@z(x[2], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], 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 + x[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11) (-1 + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]), COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1]), EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) which results in the following constraint:
(13) (y[2]=y[1]∧x[2]=x[1]∧>@z(x[2], 0@z)=TRUE∧y[1]=y[0]∧x[1]=x[0] ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_2(x[1], y[1])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:
(14) (>@z(x[2], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17) (-1 + x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18) (-1 + x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19) (x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0)
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[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]) which results in the following constraint:
(20) (EVAL_1(x[2], y[2])≥NonInfC∧EVAL_1(x[2], y[2])≥COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
For Pair EVAL_2(x, y) → COND_EVAL_21(&&(>@z(x, 0@z), >@z(y, 0@z)), 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)), x[3], y[3]) which results in the following constraint:
(25) (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)), x[3], y[3])∧(UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥))
We simplified constraint (25) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(26) ((UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27) ((UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(28) (0 ≥ 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (28) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(29) (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL_2(TRUE, x, y) → EVAL_1(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]), COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4]), EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]) which results in the following constraint:
(30) (-@z(x[4], 1@z)=x[2]∧y[0]=y[4]∧x[0]=x[4]∧y[4]=y[2]∧&&(>@z(x[0], 0@z), >=@z(0@z, y[0]))=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_1(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(31) (>@z(x[0], 0@z)=TRUE∧>=@z(0@z, y[0])=TRUE ⇒ COND_EVAL_2(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_2(TRUE, x[0], y[0])≥EVAL_1(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(32) (x[0] + -1 ≥ 0∧(-1)y[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (32) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(33) (x[0] + -1 ≥ 0∧(-1)y[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(34) ((-1)y[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35) (y[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(36) (y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
For Pair COND_EVAL_21(TRUE, x, y) → EVAL_2(x, -@z(y, 1@z)) 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)), x[3], y[3]), COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z)), EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3]) which results in the following constraint:
(37) (&&(>@z(x[3], 0@z), >@z(y[3], 0@z))=TRUE∧-@z(y[5], 1@z)=y[3]1∧x[5]=x[3]1∧y[3]=y[5]∧x[3]=x[5] ⇒ COND_EVAL_21(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5])≥EVAL_2(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL_2(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(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_2(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39) (-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40) (-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41) (-1 + y[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42) (-1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(43) (y[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
- We consider the chain EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3]), COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z)), EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) which results in the following constraint:
(44) (&&(>@z(x[3], 0@z), >@z(y[3], 0@z))=TRUE∧-@z(y[5], 1@z)=y[0]∧y[3]=y[5]∧x[3]=x[5]∧x[5]=x[0] ⇒ COND_EVAL_21(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_21(TRUE, x[5], y[5])≥EVAL_2(x[5], -@z(y[5], 1@z))∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (44) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(45) (>@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_2(x[3], -@z(y[3], 1@z))∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(46) (-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(47) (-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧0 ≥ 0)
We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(48) (-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[3] ≥ 0)
We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49) (x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[3] ≥ 0)
We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(50) (x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + y[3] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL_2(x, y) → COND_EVAL_2(&&(>@z(x, 0@z), >=@z(0@z, y)), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
- COND_EVAL_1(TRUE, x, y) → EVAL_2(x, y)
- (x[2] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- (x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0)
- EVAL_1(x, y) → COND_EVAL_1(>@z(x, 0@z), x, y)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
- EVAL_2(x, y) → COND_EVAL_21(&&(>@z(x, 0@z), >@z(y, 0@z)), x, y)
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])), ≥)∧0 = 0∧0 ≥ 0)
- COND_EVAL_2(TRUE, x, y) → EVAL_1(-@z(x, 1@z), y)
- (y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- COND_EVAL_21(TRUE, x, y) → EVAL_2(x, -@z(y, 1@z))
- (y[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + y[3] ≥ 0∧(UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥))
- (x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[5], -@z(y[5], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + y[3] ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = 0
POL(&&(x1, x2)) = 0
POL(EVAL_1(x1, x2)) = -1 + x2
POL(FALSE) = 0
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(EVAL_2(x1, x2)) = -1 + x2
POL(COND_EVAL_1(x1, x2, x3)) = -1 + x3
POL(COND_EVAL_2(x1, x2, x3)) = -1 + x3 + (-1)x1
POL(COND_EVAL_21(x1, x2, x3)) = -1 + x3 + (-1)x1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z))
The following pairs are in Pbound:
COND_EVAL_21(TRUE, x[5], y[5]) → EVAL_2(x[5], -@z(y[5], 1@z))
The following pairs are in P≥:
EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
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
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
(1): COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
(3): EVAL_2(x[3], y[3]) → COND_EVAL_21(&&(>@z(x[3], 0@z), >@z(y[3], 0@z)), x[3], y[3])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
(0) -> (4), if ((x[0] →* x[4])∧(y[0] →* y[4])∧(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])) →* TRUE))
(2) -> (1), if ((x[2] →* x[1])∧(y[2] →* y[1])∧(>@z(x[2], 0@z) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(1) -> (3), if ((y[1] →* y[3])∧(x[1] →* x[3]))
(1) -> (0), if ((y[1] →* y[0])∧(x[1] →* x[0]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(TRUE, x0, x1)
eval_1(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ 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:
(1): COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
(0): EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
(0) -> (4), if ((x[0] →* x[4])∧(y[0] →* y[4])∧(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])) →* TRUE))
(2) -> (1), if ((x[2] →* x[1])∧(y[2] →* y[1])∧(>@z(x[2], 0@z) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(1) -> (0), if ((y[1] →* y[0])∧(x[1] →* x[0]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(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_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]), COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1]), EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) which results in the following constraint:
(1) (y[2]=y[1]∧x[2]=x[1]∧>@z(x[2], 0@z)=TRUE∧y[1]=y[0]∧x[1]=x[0] ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_2(x[1], y[1])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[2], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[2] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
For Pair EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]) the following chains were created:
- We consider the chain EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]) which results in the following constraint:
(8) (EVAL_1(x[2], y[2])≥NonInfC∧EVAL_1(x[2], y[2])≥COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥)∧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_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
For Pair COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]), COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4]), EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2]) which results in the following constraint:
(13) (-@z(x[4], 1@z)=x[2]∧y[0]=y[4]∧x[0]=x[4]∧y[4]=y[2]∧&&(>@z(x[0], 0@z), >=@z(0@z, y[0]))=TRUE ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_1(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (13) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14) (>@z(x[0], 0@z)=TRUE∧>=@z(0@z, y[0])=TRUE ⇒ COND_EVAL_2(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_2(TRUE, x[0], y[0])≥EVAL_1(-@z(x[0], 1@z), y[0])∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15) (x[0] + -1 ≥ 0∧(-1)y[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧-2 + (-1)Bound + (-1)y[0] + (2)x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16) (x[0] + -1 ≥ 0∧(-1)y[0] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧-2 + (-1)Bound + (-1)y[0] + (2)x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17) (x[0] + -1 ≥ 0∧(-1)y[0] ≥ 0 ⇒ -2 + (-1)Bound + (-1)y[0] + (2)x[0] ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18) (x[0] + -1 ≥ 0∧y[0] ≥ 0 ⇒ -2 + (-1)Bound + y[0] + (2)x[0] ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19) (x[0] ≥ 0∧y[0] ≥ 0 ⇒ (-1)Bound + y[0] + (2)x[0] ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
For Pair EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) the following chains were created:
- We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0]) which results in the following constraint:
(20) (EVAL_2(x[0], y[0])≥NonInfC∧EVAL_2(x[0], y[0])≥COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
- (x[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[1], y[1])), ≥))
- EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])), ≥))
- COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
- (x[0] ≥ 0∧y[0] ≥ 0 ⇒ (-1)Bound + y[0] + (2)x[0] ≥ 0∧(UIncreasing(EVAL_1(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])), ≥))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = 1
POL(EVAL_1(x1, x2)) = -1 + (-1)x2 + (2)x1
POL(&&(x1, x2)) = 0
POL(FALSE) = 1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL_1(x1, x2, x3)) = -1 + (-1)x3 + (2)x2
POL(EVAL_2(x1, x2)) = -1 + (-1)x2 + (2)x1
POL(COND_EVAL_2(x1, x2, x3)) = -1 + (-1)x3 + (2)x2 + (-1)x1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_1(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
TRUE1 → &&(TRUE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
↳ 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:
(1): COND_EVAL_1(TRUE, x[1], y[1]) → EVAL_2(x[1], y[1])
(2): EVAL_1(x[2], y[2]) → COND_EVAL_1(>@z(x[2], 0@z), x[2], y[2])
(0): EVAL_2(x[0], y[0]) → COND_EVAL_2(&&(>@z(x[0], 0@z), >=@z(0@z, y[0])), x[0], y[0])
(2) -> (1), if ((x[2] →* x[1])∧(y[2] →* y[1])∧(>@z(x[2], 0@z) →* TRUE))
(1) -> (0), if ((y[1] →* y[0])∧(x[1] →* x[0]))
The set Q consists of the following terms:
Cond_eval_1(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(x0, x1)
Cond_eval_2(TRUE, x0, x1)
eval_1(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.