6.61/2.64 YES 6.61/2.66 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 6.61/2.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.61/2.66 6.61/2.66 6.61/2.66 Termination of the given ITRS could be proven: 6.61/2.66 6.61/2.66 (0) ITRS 6.61/2.66 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 6.61/2.66 (2) IDP 6.61/2.66 (3) UsableRulesProof [EQUIVALENT, 0 ms] 6.61/2.66 (4) IDP 6.61/2.66 (5) IDPNonInfProof [SOUND, 349 ms] 6.61/2.66 (6) IDP 6.61/2.66 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.61/2.66 (8) IDP 6.61/2.66 (9) IDPNonInfProof [SOUND, 90 ms] 6.61/2.66 (10) IDP 6.61/2.66 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.61/2.66 (12) AND 6.61/2.66 (13) IDP 6.61/2.66 (14) IDPNonInfProof [SOUND, 58 ms] 6.61/2.66 (15) IDP 6.61/2.66 (16) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.61/2.66 (17) TRUE 6.61/2.66 (18) IDP 6.61/2.66 (19) IDPNonInfProof [SOUND, 0 ms] 6.61/2.66 (20) IDP 6.61/2.66 (21) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.61/2.66 (22) TRUE 6.61/2.66 6.61/2.66 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (0) 6.61/2.66 Obligation: 6.61/2.66 ITRS problem: 6.61/2.66 6.61/2.66 The following function symbols are pre-defined: 6.61/2.66 <<< 6.61/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.66 / ~ Div: (Integer, Integer) -> Integer 6.61/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.66 + ~ Add: (Integer, Integer) -> Integer 6.61/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.66 >>> 6.61/2.66 6.61/2.66 The TRS R consists of the following rules: 6.61/2.66 eval_1(x, y) -> Cond_eval_1(x > 0 && y > 0 && x > y, x, y) 6.61/2.66 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 6.61/2.66 eval_1(x, y) -> Cond_eval_11(x > 0 && y > 0 && y >= x, x, y) 6.61/2.66 Cond_eval_11(TRUE, x, y) -> eval_3(x, y) 6.61/2.66 eval_2(x, y) -> Cond_eval_2(x > 0, x, y) 6.61/2.66 Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y) 6.61/2.66 eval_2(x, y) -> Cond_eval_21(0 >= x, x, y) 6.61/2.66 Cond_eval_21(TRUE, x, y) -> eval_1(x, y) 6.61/2.66 eval_3(x, y) -> Cond_eval_3(y > 0, x, y) 6.61/2.66 Cond_eval_3(TRUE, x, y) -> eval_3(x, y - 1) 6.61/2.66 eval_3(x, y) -> Cond_eval_31(0 >= y, x, y) 6.61/2.66 Cond_eval_31(TRUE, x, y) -> eval_1(x, y) 6.61/2.66 The set Q consists of the following terms: 6.61/2.66 eval_1(x0, x1) 6.61/2.66 Cond_eval_1(TRUE, x0, x1) 6.61/2.66 Cond_eval_11(TRUE, x0, x1) 6.61/2.66 eval_2(x0, x1) 6.61/2.66 Cond_eval_2(TRUE, x0, x1) 6.61/2.66 Cond_eval_21(TRUE, x0, x1) 6.61/2.66 eval_3(x0, x1) 6.61/2.66 Cond_eval_3(TRUE, x0, x1) 6.61/2.66 Cond_eval_31(TRUE, x0, x1) 6.61/2.66 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (1) ITRStoIDPProof (EQUIVALENT) 6.61/2.66 Added dependency pairs 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (2) 6.61/2.66 Obligation: 6.61/2.66 IDP problem: 6.61/2.66 The following function symbols are pre-defined: 6.61/2.66 <<< 6.61/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.66 / ~ Div: (Integer, Integer) -> Integer 6.61/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.66 + ~ Add: (Integer, Integer) -> Integer 6.61/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.66 >>> 6.61/2.66 6.61/2.66 6.61/2.66 The following domains are used: 6.61/2.66 Boolean, Integer 6.61/2.66 6.61/2.66 The ITRS R consists of the following rules: 6.61/2.66 eval_1(x, y) -> Cond_eval_1(x > 0 && y > 0 && x > y, x, y) 6.61/2.66 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 6.61/2.66 eval_1(x, y) -> Cond_eval_11(x > 0 && y > 0 && y >= x, x, y) 6.61/2.66 Cond_eval_11(TRUE, x, y) -> eval_3(x, y) 6.61/2.66 eval_2(x, y) -> Cond_eval_2(x > 0, x, y) 6.61/2.66 Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y) 6.61/2.66 eval_2(x, y) -> Cond_eval_21(0 >= x, x, y) 6.61/2.66 Cond_eval_21(TRUE, x, y) -> eval_1(x, y) 6.61/2.66 eval_3(x, y) -> Cond_eval_3(y > 0, x, y) 6.61/2.66 Cond_eval_3(TRUE, x, y) -> eval_3(x, y - 1) 6.61/2.66 eval_3(x, y) -> Cond_eval_31(0 >= y, x, y) 6.61/2.66 Cond_eval_31(TRUE, x, y) -> eval_1(x, y) 6.61/2.66 6.61/2.66 The integer pair graph contains the following rules and edges: 6.61/2.66 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.61/2.66 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.66 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.66 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.66 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.66 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.66 (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) 6.61/2.66 (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.66 (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) 6.61/2.66 (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) 6.61/2.66 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.66 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.66 6.61/2.66 (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.61/2.66 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 6.61/2.66 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) 6.61/2.66 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.66 (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) 6.61/2.66 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.66 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.66 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.66 (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) 6.61/2.66 (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.61/2.66 (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) 6.61/2.66 (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) 6.61/2.66 (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) 6.61/2.66 (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) 6.61/2.66 (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) 6.61/2.66 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.66 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) 6.61/2.66 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.66 6.61/2.66 The set Q consists of the following terms: 6.61/2.66 eval_1(x0, x1) 6.61/2.66 Cond_eval_1(TRUE, x0, x1) 6.61/2.66 Cond_eval_11(TRUE, x0, x1) 6.61/2.66 eval_2(x0, x1) 6.61/2.66 Cond_eval_2(TRUE, x0, x1) 6.61/2.66 Cond_eval_21(TRUE, x0, x1) 6.61/2.66 eval_3(x0, x1) 6.61/2.66 Cond_eval_3(TRUE, x0, x1) 6.61/2.66 Cond_eval_31(TRUE, x0, x1) 6.61/2.66 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (3) UsableRulesProof (EQUIVALENT) 6.61/2.66 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. 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (4) 6.61/2.66 Obligation: 6.61/2.66 IDP problem: 6.61/2.66 The following function symbols are pre-defined: 6.61/2.66 <<< 6.61/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.66 / ~ Div: (Integer, Integer) -> Integer 6.61/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.66 + ~ Add: (Integer, Integer) -> Integer 6.61/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.66 >>> 6.61/2.66 6.61/2.66 6.61/2.66 The following domains are used: 6.61/2.66 Boolean, Integer 6.61/2.66 6.61/2.66 R is empty. 6.61/2.66 6.61/2.66 The integer pair graph contains the following rules and edges: 6.61/2.66 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.61/2.66 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.66 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.66 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.66 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.66 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.66 (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) 6.61/2.66 (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.66 (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) 6.61/2.66 (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) 6.61/2.66 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.66 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.66 6.61/2.66 (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.61/2.66 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 6.61/2.66 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) 6.61/2.66 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.66 (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) 6.61/2.66 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.66 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.66 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.66 (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) 6.61/2.66 (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.61/2.66 (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) 6.61/2.66 (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) 6.61/2.66 (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) 6.61/2.66 (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) 6.61/2.66 (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) 6.61/2.66 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.66 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) 6.61/2.66 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.66 6.61/2.66 The set Q consists of the following terms: 6.61/2.66 eval_1(x0, x1) 6.61/2.66 Cond_eval_1(TRUE, x0, x1) 6.61/2.66 Cond_eval_11(TRUE, x0, x1) 6.61/2.66 eval_2(x0, x1) 6.61/2.66 Cond_eval_2(TRUE, x0, x1) 6.61/2.66 Cond_eval_21(TRUE, x0, x1) 6.61/2.66 eval_3(x0, x1) 6.61/2.66 Cond_eval_3(TRUE, x0, x1) 6.61/2.66 Cond_eval_31(TRUE, x0, x1) 6.61/2.66 6.61/2.66 ---------------------------------------- 6.61/2.66 6.61/2.66 (5) IDPNonInfProof (SOUND) 6.61/2.66 Used the following options for this NonInfProof: 6.61/2.66 6.61/2.66 IDPGPoloSolver: 6.61/2.66 Range: [(-1,2)] 6.61/2.66 IsNat: false 6.61/2.66 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4c82a1e2 6.61/2.66 Constraint Generator: NonInfConstraintGenerator: 6.61/2.66 PathGenerator: MetricPathGenerator: 6.61/2.66 Max Left Steps: 1 6.61/2.66 Max Right Steps: 1 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 The constraints were generated the following way: 6.61/2.66 6.61/2.66 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.61/2.66 6.61/2.66 Note that final constraints are written in bold face. 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_1(x, y) -> COND_EVAL_1(&&(&&(>(x, 0), >(y, 0)), >(x, y)), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE & >(y[0], 0)=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]y[0] >= 0 & [(-1)bso_40] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]y[0] >= 0 & [(-1)bso_40] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]y[0] >= 0 & [(-1)bso_40] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) the following chains were created: 6.61/2.66 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[1]=x[6] & y[1]=y[6] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_1(x, y) -> COND_EVAL_11(&&(&&(>(x, 0), >(y, 0)), >=(y, x)), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]y[2] >= 0 & [-1 + (-1)bso_44] + y[2] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]y[2] >= 0 & [-1 + (-1)bso_44] + y[2] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]y[2] >= 0 & [-1 + (-1)bso_44] + y[2] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_11(TRUE, x, y) -> EVAL_3(x, y) the following chains were created: 6.61/2.66 *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[3]=x[8] & y[3]=y[8] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_2(x, y) -> COND_EVAL_2(>(x, 0), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(x[4], 0)=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(2)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(2)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(2)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(2)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[4]1 & y[5]=y[4]1 ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(2)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[6] & y[5]=y[6] ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(2)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(2)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_2(x, y) -> COND_EVAL_21(>=(0, x), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]), COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>=(0, x[6])=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>=(0, x[6])=TRUE ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [(2)bni_51]y[6] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [(2)bni_51]y[6] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [(2)bni_51]y[6] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(2)bni_51] = 0 & [(-1)bni_51 + (-1)Bound*bni_51] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.66 6.61/2.66 (7) (x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(2)bni_51] = 0 & [(-1)bni_51 + (-1)Bound*bni_51] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: 6.61/2.66 *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[7]=x[0] & y[7]=y[0] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[7]=x[2] & y[7]=y[2] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_3(x, y) -> COND_EVAL_3(>(y, 0), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(y[8], 0)=TRUE ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [bni_55]y[8] >= 0 & [(-1)bso_56] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [bni_55]y[8] >= 0 & [(-1)bso_56] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [bni_55]y[8] >= 0 & [(-1)bso_56] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & 0 = 0 & [(-1)bni_55 + (-1)Bound*bni_55] + [bni_55]y[8] >= 0 & [(-1)bso_56] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_3(TRUE, x, y) -> EVAL_3(x, -(y, 1)) the following chains were created: 6.61/2.66 *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[8]1 & -(y[9], 1)=y[8]1 ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[10] & -(y[9], 1)=y[10] ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair EVAL_3(x, y) -> COND_EVAL_31(>=(0, y), x, y) the following chains were created: 6.61/2.66 *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [bni_59]y[10] >= 0 & [(-1)bso_60] + [-1]y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [bni_59]y[10] >= 0 & [(-1)bso_60] + [-1]y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [bni_59]y[10] >= 0 & [(-1)bso_60] + [-1]y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & 0 = 0 & [(-1)bni_59 + (-1)Bound*bni_59] + [bni_59]y[10] >= 0 & [(-1)bso_60] + [-1]y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.66 6.61/2.66 (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & 0 = 0 & [(-1)bni_59 + (-1)Bound*bni_59] + [(-1)bni_59]y[10] >= 0 & [(-1)bso_60] + y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 For Pair COND_EVAL_31(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: 6.61/2.66 *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[11]=x[0] & y[11]=y[0] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: 6.61/2.66 6.61/2.66 (1) (x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.66 6.61/2.66 (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.66 6.61/2.66 (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.66 6.61/2.66 (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.66 6.61/2.66 (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 To summarize, we get the following constraints P__>=_ for the following pairs. 6.61/2.66 6.61/2.66 *EVAL_1(x, y) -> COND_EVAL_1(&&(&&(>(x, 0), >(y, 0)), >(x, y)), x, y) 6.61/2.66 6.61/2.66 *(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]y[0] >= 0 & [(-1)bso_40] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *EVAL_1(x, y) -> COND_EVAL_11(&&(&&(>(x, 0), >(y, 0)), >=(y, x)), x, y) 6.61/2.66 6.61/2.66 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]y[2] >= 0 & [-1 + (-1)bso_44] + y[2] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_11(TRUE, x, y) -> EVAL_3(x, y) 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *EVAL_2(x, y) -> COND_EVAL_2(>(x, 0), x, y) 6.61/2.66 6.61/2.66 *(x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(2)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), y) 6.61/2.66 6.61/2.66 *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(2)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(2)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *EVAL_2(x, y) -> COND_EVAL_21(>=(0, x), x, y) 6.61/2.66 6.61/2.66 *(x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(2)bni_51] = 0 & [(-1)bni_51 + (-1)Bound*bni_51] >= 0 & [(-1)bso_52] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *EVAL_3(x, y) -> COND_EVAL_3(>(y, 0), x, y) 6.61/2.66 6.61/2.66 *(y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & 0 = 0 & [(-1)bni_55 + (-1)Bound*bni_55] + [bni_55]y[8] >= 0 & [(-1)bso_56] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_3(TRUE, x, y) -> EVAL_3(x, -(y, 1)) 6.61/2.66 6.61/2.66 *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_57 + (-1)Bound*bni_57] + [bni_57]y[8] >= 0 & [1 + (-1)bso_58] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *EVAL_3(x, y) -> COND_EVAL_31(>=(0, y), x, y) 6.61/2.66 6.61/2.66 *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & 0 = 0 & [(-1)bni_59 + (-1)Bound*bni_59] + [(-1)bni_59]y[10] >= 0 & [(-1)bso_60] + y[10] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 *COND_EVAL_31(TRUE, x, y) -> EVAL_1(x, y) 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 6.61/2.66 The constraints for P_> respective P_bound 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 P_bound. 6.61/2.66 6.61/2.66 Using the following integer polynomial ordering the resulting constraints can be solved 6.61/2.66 6.61/2.66 Polynomial interpretation over integers[POLO]: 6.61/2.66 6.61/2.66 POL(TRUE) = 0 6.61/2.66 POL(FALSE) = 0 6.61/2.66 POL(EVAL_1(x_1, x_2)) = [-1] + [2]x_2 6.61/2.66 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + [2]x_3 6.61/2.66 POL(&&(x_1, x_2)) = [-1] 6.61/2.67 POL(>(x_1, x_2)) = [-1] 6.61/2.67 POL(0) = 0 6.61/2.67 POL(EVAL_2(x_1, x_2)) = [-1] + [2]x_2 6.61/2.67 POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_1 6.61/2.67 POL(>=(x_1, x_2)) = [-1] 6.61/2.67 POL(EVAL_3(x_1, x_2)) = [-1] + x_2 6.61/2.67 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [2]x_3 6.61/2.67 POL(-(x_1, x_2)) = x_1 + [-1]x_2 6.61/2.67 POL(1) = [1] 6.61/2.67 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + [2]x_3 6.61/2.67 POL(COND_EVAL_3(x_1, x_2, x_3)) = [-1] + x_3 6.61/2.67 POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + [2]x_3 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>: 6.61/2.67 6.61/2.67 6.61/2.67 COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_bound: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) 6.61/2.67 EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) 6.61/2.67 COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>=: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) 6.61/2.67 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) 6.61/2.67 COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) 6.61/2.67 EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) 6.61/2.67 COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 6.61/2.67 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.61/2.67 6.61/2.67 TRUE^1 -> &&(TRUE, TRUE)^1 6.61/2.67 FALSE^1 -> &&(TRUE, FALSE)^1 6.61/2.67 FALSE^1 -> &&(FALSE, TRUE)^1 6.61/2.67 FALSE^1 -> &&(FALSE, FALSE)^1 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (6) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Boolean, Integer 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.61/2.67 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.67 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.67 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.67 (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) 6.61/2.67 (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) 6.61/2.67 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.67 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) 6.61/2.67 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) 6.61/2.67 (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.61/2.67 (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) 6.61/2.67 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.67 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.67 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 6.61/2.67 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.67 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.67 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) 6.61/2.67 (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) 6.61/2.67 (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.61/2.67 (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) 6.61/2.67 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.67 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (7) IDependencyGraphProof (EQUIVALENT) 6.61/2.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (8) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer, Boolean 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.67 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.67 (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) 6.61/2.67 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.67 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.67 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.61/2.67 6.61/2.67 (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) 6.61/2.67 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) 6.61/2.67 (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.61/2.67 (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) 6.61/2.67 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.67 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.67 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 6.61/2.67 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.67 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.67 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) 6.61/2.67 (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) 6.61/2.67 (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.61/2.67 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.67 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (9) IDPNonInfProof (SOUND) 6.61/2.67 Used the following options for this NonInfProof: 6.61/2.67 6.61/2.67 IDPGPoloSolver: 6.61/2.67 Range: [(-1,2)] 6.61/2.67 IsNat: false 6.61/2.67 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4c82a1e2 6.61/2.67 Constraint Generator: NonInfConstraintGenerator: 6.61/2.67 PathGenerator: MetricPathGenerator: 6.61/2.67 Max Left Steps: 1 6.61/2.67 Max Right Steps: 1 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints were generated the following way: 6.61/2.67 6.61/2.67 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.61/2.67 6.61/2.67 Note that final constraints are written in bold face. 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[11]=x[0] & y[11]=y[0] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[10] + [bni_37]x[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[10] + [bni_37]x[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[10] + [bni_37]x[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [bni_37] = 0 & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.67 6.61/2.67 (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [bni_37] = 0 & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]y[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_39] = 0 & [(-1)bso_40] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_39] = 0 & [(-1)bso_40] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_39] = 0 & [(-1)bso_40] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]y[2] + [bni_41]x[2] >= 0 & [(-1)bso_42] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]y[2] + [bni_41]x[2] >= 0 & [(-1)bso_42] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]y[2] + [bni_41]x[2] >= 0 & [(-1)bso_42] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[7]=x[0] & y[7]=y[0] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[7]=x[2] & y[7]=y[2] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]), COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>=(0, x[6])=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(0, x[6])=TRUE ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]y[6] >= 0 & [(-1)bso_46] + [-1]x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]y[6] >= 0 & [(-1)bso_46] + [-1]x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]y[6] >= 0 & [(-1)bso_46] + [-1]x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45] = 0 & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] + [-1]x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.67 6.61/2.67 (7) (x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45] = 0 & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] + x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[4]1 & y[5]=y[4]1 ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[6] & y[5]=y[6] ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]y[4] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[4], 0)=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(-1)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(-1)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [(-1)bni_49]y[4] >= 0 & [(-1)bso_50] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[1]=x[6] & y[1]=y[6] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE & >(y[0], 0)=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] + [(-1)bni_53]y[0] + [bni_53]x[0] >= 0 & [(-1)bso_54] + x[0] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] + [(-1)bni_53]y[0] + [bni_53]x[0] >= 0 & [(-1)bso_54] + x[0] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] + [(-1)bni_53]y[0] + [bni_53]x[0] >= 0 & [(-1)bso_54] + x[0] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 To summarize, we get the following constraints P__>=_ for the following pairs. 6.61/2.67 6.61/2.67 *COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) 6.61/2.67 6.61/2.67 *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [bni_37] = 0 & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]y[10] >= 0 & [(-1)bso_38] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_39] = 0 & [(-1)bso_40] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 6.61/2.67 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]y[2] + [bni_41]x[2] >= 0 & [(-1)bso_42] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_43] = 0 & [(-1)bso_44] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) 6.61/2.67 6.61/2.67 *(x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_45] = 0 & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] + x[6] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 6.61/2.67 *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_47] = 0 & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 6.61/2.67 *(x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_49] = 0 & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_51] = 0 & [(-1)bso_52] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) 6.61/2.67 6.61/2.67 *(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] + [(-1)bni_53]y[0] + [bni_53]x[0] >= 0 & [(-1)bso_54] + x[0] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints for P_> respective P_bound 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 P_bound. 6.61/2.67 6.61/2.67 Using the following integer polynomial ordering the resulting constraints can be solved 6.61/2.67 6.61/2.67 Polynomial interpretation over integers[POLO]: 6.61/2.67 6.61/2.67 POL(TRUE) = 0 6.61/2.67 POL(FALSE) = [1] 6.61/2.67 POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 6.61/2.67 POL(EVAL_1(x_1, x_2)) = [-1] + [-1]x_2 + x_1 6.61/2.67 POL(EVAL_3(x_1, x_2)) = [-1] + [-1]x_2 + x_1 6.61/2.67 POL(>=(x_1, x_2)) = [-1] 6.61/2.67 POL(0) = 0 6.61/2.67 POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 6.61/2.67 POL(&&(x_1, x_2)) = [-1] 6.61/2.67 POL(>(x_1, x_2)) = [-1] 6.61/2.67 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 6.61/2.67 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 6.61/2.67 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 6.61/2.67 POL(-(x_1, x_2)) = x_1 + [-1]x_2 6.61/2.67 POL(1) = [1] 6.61/2.67 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_bound: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>=: 6.61/2.67 6.61/2.67 6.61/2.67 COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) 6.61/2.67 COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) 6.61/2.67 COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 6.61/2.67 6.61/2.67 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.61/2.67 6.61/2.67 TRUE^1 -> &&(TRUE, TRUE)^1 6.61/2.67 FALSE^1 -> &&(TRUE, FALSE)^1 6.61/2.67 FALSE^1 -> &&(FALSE, TRUE)^1 6.61/2.67 FALSE^1 -> &&(FALSE, FALSE)^1 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (10) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer, Boolean 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.67 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.67 (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) 6.61/2.67 (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) 6.61/2.67 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.67 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.67 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 6.61/2.67 6.61/2.67 (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) 6.61/2.67 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.67 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.67 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 6.61/2.67 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.67 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.67 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) 6.61/2.67 (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) 6.61/2.67 (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.61/2.67 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.67 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (11) IDependencyGraphProof (EQUIVALENT) 6.61/2.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (12) 6.61/2.67 Complex Obligation (AND) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (13) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer, Boolean 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.67 (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) 6.61/2.67 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) 6.61/2.67 (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.61/2.67 (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) 6.61/2.67 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (14) IDPNonInfProof (SOUND) 6.61/2.67 Used the following options for this NonInfProof: 6.61/2.67 6.61/2.67 IDPGPoloSolver: 6.61/2.67 Range: [(-1,2)] 6.61/2.67 IsNat: false 6.61/2.67 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4c82a1e2 6.61/2.67 Constraint Generator: NonInfConstraintGenerator: 6.61/2.67 PathGenerator: MetricPathGenerator: 6.61/2.67 Max Left Steps: 1 6.61/2.67 Max Right Steps: 1 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints were generated the following way: 6.61/2.67 6.61/2.67 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.61/2.67 6.61/2.67 Note that final constraints are written in bold face. 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[3]=x[10] & y[3]=y[10] & >=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[3], y[10])_>=_NonInfC & EVAL_3(x[3], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[3], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.67 6.61/2.67 (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] + [2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_11(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_11(TRUE, x[2], y[2])_>=_EVAL_3(x[2], y[2]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) the following chains were created: 6.61/2.67 *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (x[11]=x[2] & y[11]=y[2] & &&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] & x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>=(0, y[10])=TRUE ==> COND_EVAL_31(TRUE, x[10], y[10])_>=_NonInfC & COND_EVAL_31(TRUE, x[10], y[10])_>=_EVAL_1(x[10], y[10]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.61/2.67 6.61/2.67 (7) (y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-2)bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 To summarize, we get the following constraints P__>=_ for the following pairs. 6.61/2.67 6.61/2.67 *EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) 6.61/2.67 6.61/2.67 *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] + [2]y[10] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 6.61/2.67 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 6.61/2.67 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] + y[2] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 *(y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-2)bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints for P_> respective P_bound 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 P_bound. 6.61/2.67 6.61/2.67 Using the following integer polynomial ordering the resulting constraints can be solved 6.61/2.67 6.61/2.67 Polynomial interpretation over integers[POLO]: 6.61/2.67 6.61/2.67 POL(TRUE) = 0 6.61/2.67 POL(FALSE) = 0 6.61/2.67 POL(EVAL_3(x_1, x_2)) = [-1] + [-1]x_1 6.61/2.67 POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + [2]x_3 + [-1]x_2 6.61/2.67 POL(>=(x_1, x_2)) = [-1] 6.61/2.67 POL(0) = 0 6.61/2.67 POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 + [-1]x_1 6.61/2.67 POL(EVAL_1(x_1, x_2)) = [-1] + [2]x_2 + [-1]x_1 6.61/2.67 POL(&&(x_1, x_2)) = 0 6.61/2.67 POL(>(x_1, x_2)) = [-1] 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>: 6.61/2.67 6.61/2.67 6.61/2.67 COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_bound: 6.61/2.67 6.61/2.67 6.61/2.67 COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) 6.61/2.67 EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>=: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) 6.61/2.67 COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 6.61/2.67 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.61/2.67 6.61/2.67 TRUE^1 -> &&(TRUE, TRUE)^1 6.61/2.67 &&(TRUE, FALSE)^1 <-> FALSE^1 6.61/2.67 &&(FALSE, TRUE)^1 <-> FALSE^1 6.61/2.67 &&(FALSE, FALSE)^1 <-> FALSE^1 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (15) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) 6.61/2.67 (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) 6.61/2.67 6.61/2.67 (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (16) IDependencyGraphProof (EQUIVALENT) 6.61/2.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (17) 6.61/2.67 TRUE 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (18) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.67 (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) 6.61/2.67 6.61/2.67 (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) 6.61/2.67 (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (19) IDPNonInfProof (SOUND) 6.61/2.67 Used the following options for this NonInfProof: 6.61/2.67 6.61/2.67 IDPGPoloSolver: 6.61/2.67 Range: [(-1,2)] 6.61/2.67 IsNat: false 6.61/2.67 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4c82a1e2 6.61/2.67 Constraint Generator: NonInfConstraintGenerator: 6.61/2.67 PathGenerator: MetricPathGenerator: 6.61/2.67 Max Left Steps: 1 6.61/2.67 Max Right Steps: 1 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints were generated the following way: 6.61/2.67 6.61/2.67 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.61/2.67 6.61/2.67 Note that final constraints are written in bold face. 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rule (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[4], 0)=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x[4] >= 0 & [(-1)bso_12] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x[4] >= 0 & [(-1)bso_12] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x[4] >= 0 & [(-1)bso_12] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x[4] >= 0 & [(-1)bso_12] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 For Pair COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) the following chains were created: 6.61/2.67 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: 6.61/2.67 6.61/2.67 (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[4]1 & y[5]=y[4]1 ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 6.61/2.67 6.61/2.67 (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.61/2.67 6.61/2.67 (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.61/2.67 6.61/2.67 (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 6.61/2.67 6.61/2.67 (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 To summarize, we get the following constraints P__>=_ for the following pairs. 6.61/2.67 6.61/2.67 *EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 6.61/2.67 *(x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x[4] >= 0 & [(-1)bso_12] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 *COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 6.61/2.67 *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 6.61/2.67 The constraints for P_> respective P_bound 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 P_bound. 6.61/2.67 6.61/2.67 Using the following integer polynomial ordering the resulting constraints can be solved 6.61/2.67 6.61/2.67 Polynomial interpretation over integers[POLO]: 6.61/2.67 6.61/2.67 POL(TRUE) = 0 6.61/2.67 POL(FALSE) = 0 6.61/2.67 POL(EVAL_2(x_1, x_2)) = [-1] + x_1 6.61/2.67 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_2 6.61/2.67 POL(>(x_1, x_2)) = [-1] 6.61/2.67 POL(0) = 0 6.61/2.67 POL(-(x_1, x_2)) = x_1 + [-1]x_2 6.61/2.67 POL(1) = [1] 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>: 6.61/2.67 6.61/2.67 6.61/2.67 COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_bound: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) 6.61/2.67 6.61/2.67 6.61/2.67 The following pairs are in P_>=: 6.61/2.67 6.61/2.67 6.61/2.67 EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) 6.61/2.67 6.61/2.67 6.61/2.67 There are no usable rules. 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (20) 6.61/2.67 Obligation: 6.61/2.67 IDP problem: 6.61/2.67 The following function symbols are pre-defined: 6.61/2.67 <<< 6.61/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.61/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.61/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.61/2.67 / ~ Div: (Integer, Integer) -> Integer 6.61/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.61/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.61/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.61/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.61/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.61/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.61/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.61/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.61/2.67 + ~ Add: (Integer, Integer) -> Integer 6.61/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.61/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.61/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.61/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.61/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.61/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.61/2.67 >>> 6.61/2.67 6.61/2.67 6.61/2.67 The following domains are used: 6.61/2.67 Integer 6.61/2.67 6.61/2.67 R is empty. 6.61/2.67 6.61/2.67 The integer pair graph contains the following rules and edges: 6.61/2.67 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) 6.61/2.67 6.61/2.67 6.61/2.67 The set Q consists of the following terms: 6.61/2.67 eval_1(x0, x1) 6.61/2.67 Cond_eval_1(TRUE, x0, x1) 6.61/2.67 Cond_eval_11(TRUE, x0, x1) 6.61/2.67 eval_2(x0, x1) 6.61/2.67 Cond_eval_2(TRUE, x0, x1) 6.61/2.67 Cond_eval_21(TRUE, x0, x1) 6.61/2.67 eval_3(x0, x1) 6.61/2.67 Cond_eval_3(TRUE, x0, x1) 6.61/2.67 Cond_eval_31(TRUE, x0, x1) 6.61/2.67 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (21) IDependencyGraphProof (EQUIVALENT) 6.61/2.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 6.61/2.67 ---------------------------------------- 6.61/2.67 6.61/2.67 (22) 6.61/2.67 TRUE 6.80/2.70 EOF