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)
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(y, 1@z))
Cond_eval_21(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
Cond_eval_1(TRUE, x, y) → eval_2(x, 0@z)
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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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)
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(y, 1@z))
Cond_eval_21(TRUE, x, y) → eval_1(-@z(x, 1@z), y)
Cond_eval_1(TRUE, x, y) → eval_2(x, 0@z)
eval_1(x, y) → Cond_eval_1(>@z(x, 0@z), x, y)
The integer pair graph contains the following rules and edges:
(0): COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@z)
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])
(2): COND_EVAL_21(TRUE, x[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2])
(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(y[4], 1@z))
(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) -> (3), if ((x[0] →* x[3]))
(0) -> (5), if ((x[0] →* x[5]))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(>@z(x[1], 0@z) →* TRUE))
(2) -> (1), if ((y[2] →* y[1])∧(-@z(x[2], 1@z) →* x[1]))
(3) -> (2), if ((x[3] →* x[2])∧(y[3] →* y[2])∧(&&(&&(>@z(x[3], 0@z), >=@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
(4) -> (3), if ((+@z(y[4], 1@z) →* y[3])∧(x[4] →* x[3]))
(4) -> (5), if ((+@z(y[4], 1@z) →* 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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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): COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@z)
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])
(2): COND_EVAL_21(TRUE, x[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2])
(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(y[4], 1@z))
(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) -> (3), if ((x[0] →* x[3]))
(0) -> (5), if ((x[0] →* x[5]))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(>@z(x[1], 0@z) →* TRUE))
(2) -> (1), if ((y[2] →* y[1])∧(-@z(x[2], 1@z) →* x[1]))
(3) -> (2), if ((x[3] →* x[2])∧(y[3] →* y[2])∧(&&(&&(>@z(x[3], 0@z), >=@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
(4) -> (3), if ((+@z(y[4], 1@z) →* y[3])∧(x[4] →* x[3]))
(4) -> (5), if ((+@z(y[4], 1@z) →* 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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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, y) → EVAL_2(x, 0@z) the following chains were created:
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1]), COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@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:
(1) (>@z(x[1], 0@z)=TRUE∧x[0]=x[5]∧y[1]=y[0]∧x[1]=x[0] ⇒ COND_EVAL_1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_1(TRUE, x[0], y[0])≥EVAL_2(x[0], 0@z)∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_2(x[1], 0@z)∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧-1 + (-1)Bound + 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 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧-1 + (-1)Bound + 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 ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 ≥ 0)
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (x[1] + -1 ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 = 0∧0 ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 = 0∧0 ≥ 0)
- We consider the chain EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1]), COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@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:
(8) (>@z(x[1], 0@z)=TRUE∧x[0]=x[3]∧y[1]=y[0]∧x[1]=x[0] ⇒ COND_EVAL_1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_1(TRUE, x[0], y[0])≥EVAL_2(x[0], 0@z)∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[1], 0@z)=TRUE ⇒ COND_EVAL_1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_1(TRUE, x[1], y[1])≥EVAL_2(x[1], 0@z)∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧-1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (x[1] + -1 ≥ 0 ⇒ (UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧-1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (x[1] + -1 ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (x[1] + -1 ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[1] ≥ 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 ≥ 0∧(-1)Bound + x[1] ≥ 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[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1]) which results in the following constraint:
(15) (EVAL_1(x[1], y[1])≥NonInfC∧EVAL_1(x[1], y[1])≥COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])∧(UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥)∧0 = 0∧0 = 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[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2]), EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1]) which results in the following constraint:
(20) (y[2]=y[1]∧&&(&&(>@z(x[3], 0@z), >=@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE∧y[3]=y[2]∧-@z(x[2], 1@z)=x[1]∧x[3]=x[2] ⇒ COND_EVAL_21(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_21(TRUE, x[2], y[2])≥EVAL_1(-@z(x[2], 1@z), y[2])∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥))
We simplified constraint (20) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(21) (>=@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[2], 1@z), y[2])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) (y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (-1 + x[3] ≥ 0∧y[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥))
We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25) (x[3] ≥ 0∧y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥))
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (x[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥))
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:
(27) (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 (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28) ((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 (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29) ((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 (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30) ((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 (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(31) (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(y, 1@z)) 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(y[4], 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:
(32) (y[5]=y[4]∧x[5]=x[4]∧+@z(y[4], 1@z)=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(y[4], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@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(y[5], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + x[5] ≥ 0∧y[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (x[5] ≥ 0∧y[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
- 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(y[4], 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:
(39) (y[5]=y[4]∧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∧+@z(y[4], 1@z)=y[5]1 ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], +@z(y[4], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (39) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(40) (>@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(y[5], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43) (y[5] ≥ 0∧-1 + x[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (y[5] ≥ 0∧x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (y[5] ≥ 0∧y[5] + x[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
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:
(46) (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 (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(47) ((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 (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(48) ((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 (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(49) (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)
We simplified constraint (49) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(50) (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_1(TRUE, x, y) → EVAL_2(x, 0@z)
- (x[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 = 0∧0 ≥ 0)
- (x[1] ≥ 0 ⇒ 0 = 0∧0 = 0∧(UIncreasing(EVAL_2(x[0], 0@z)), ≥)∧0 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- EVAL_1(x, y) → COND_EVAL_1(>@z(x, 0@z), x, y)
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- COND_EVAL_21(TRUE, x, y) → EVAL_1(-@z(x, 1@z), y)
- (x[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_1(-@z(x[2], 1@z), y[2])), ≥))
- 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(y, 1@z))
- (y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
- (y[5] ≥ 0∧y[5] + x[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
- 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∧(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)) = x1 + (-1)x2
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL_1(x1, x2)) = -1 + x1
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)) = -1 + x2
POL(COND_EVAL_2(x1, x2, x3)) = -1 + x2
POL(COND_EVAL_21(x1, x2, x3)) = -1 + x2 + x1
POL(+@z(x1, x2)) = x1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
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])
The following pairs are in Pbound:
COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@z)
The following pairs are in P≥:
COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@z)
EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])
COND_EVAL_21(TRUE, x[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2])
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 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])
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
+@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): COND_EVAL_1(TRUE, x[0], y[0]) → EVAL_2(x[0], 0@z)
(1): EVAL_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])
(2): COND_EVAL_21(TRUE, x[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2])
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
(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])
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(>@z(x[1], 0@z) →* TRUE))
(4) -> (5), if ((+@z(y[4], 1@z) →* 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))
(2) -> (1), if ((y[2] →* y[1])∧(-@z(x[2], 1@z) →* x[1]))
(0) -> (5), if ((x[0] →* x[5]))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
(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])
(4) -> (5), if ((+@z(y[4], 1@z) →* 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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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(y[4], 1@z)) 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(y[4], 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:
(1) (y[5]=y[4]∧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∧+@z(y[4], 1@z)=y[5]1 ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], +@z(y[4], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
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(y[5], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 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∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[5] ≥ 0∧y[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧3 + (-1)Bound + x[5] ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
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:
(7) (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 (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) ((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 (8) using rule (IDP_POLY_SIMPLIFY) 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 (POLY_REMOVE_MIN_MAX) 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 (IDP_UNRESTRICTED_VARS) 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])), ≥)∧0 = 0∧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(y[4], 1@z))
- (y[5] + x[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 0 ≥ 0∧3 + (-1)Bound + x[5] ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 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])
- (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∧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(EVAL_2(x1, x2)) = 2 + (-1)x2 + x1
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL_2(x1, x2, x3)) = 1 + (-1)x3 + 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_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
The following pairs are in Pbound:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
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:
FALSE1 → &&(FALSE, FALSE)1
TRUE1 → &&(TRUE, TRUE)1
+@z1 ↔
FALSE1 → &&(TRUE, FALSE)1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
eval_1(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_1(x[1], y[1]) → COND_EVAL_1(>@z(x[1], 0@z), x[1], y[1])
(2): COND_EVAL_21(TRUE, x[2], y[2]) → EVAL_1(-@z(x[2], 1@z), y[2])
(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(y[4], 1@z))
(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])
(4) -> (3), if ((+@z(y[4], 1@z) →* y[3])∧(x[4] →* x[3]))
(4) -> (5), if ((+@z(y[4], 1@z) →* 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))
(2) -> (1), if ((y[2] →* y[1])∧(-@z(x[2], 1@z) →* x[1]))
(3) -> (2), if ((x[3] →* x[2])∧(y[3] →* y[2])∧(&&(&&(>@z(x[3], 0@z), >=@z(y[3], 0@z)), >=@z(y[3], x[3])) →* TRUE))
The set Q consists of the following terms:
eval_2(x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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
↳ 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_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
(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])
(4) -> (5), if ((+@z(y[4], 1@z) →* 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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(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(y[4], 1@z)) 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(y[4], 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:
(1) (y[5]=y[4]∧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∧+@z(y[4], 1@z)=y[5]1 ⇒ COND_EVAL_2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_2(TRUE, x[4], y[4])≥EVAL_2(x[4], +@z(y[4], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
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(y[5], 1@z))∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 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∧-1 + x[5] ≥ 0∧y[5] ≥ 0 ⇒ (UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥)∧2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[5] + (-1)y[5] ≥ 0∧y[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ 2 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[5] + (-1)y[5] ≥ 0∧y[5] ≥ 0∧x[5] ≥ 0 ⇒ 3 + (-1)Bound + (-1)y[5] + x[5] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[5] ≥ 0∧y[5] ≥ 0∧y[5] + x[5] ≥ 0 ⇒ 3 + (-1)Bound + x[5] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 1@z))), ≥))
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∧(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∧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(y[4], 1@z))
- (x[5] ≥ 0∧y[5] ≥ 0∧y[5] + x[5] ≥ 0 ⇒ 3 + (-1)Bound + x[5] ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL_2(x[4], +@z(y[4], 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])
- (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∧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(EVAL_2(x1, x2)) = 2 + (-1)x2 + x1
POL(TRUE) = 2
POL(&&(x1, x2)) = 2
POL(COND_EVAL_2(x1, x2, x3)) = (-1)x3 + x2 + 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_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
The following pairs are in Pbound:
COND_EVAL_2(TRUE, x[4], y[4]) → EVAL_2(x[4], +@z(y[4], 1@z))
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:
&&(FALSE, FALSE)1 ↔ FALSE1
&&(TRUE, TRUE)1 ↔ TRUE1
+@z1 ↔
&&(FALSE, TRUE)1 ↔ FALSE1
&&(TRUE, FALSE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(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_2(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.