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