6.91/2.79 YES 7.20/2.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 7.20/2.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.20/2.82 7.20/2.82 7.20/2.82 Termination of the given ITRS could be proven: 7.20/2.82 7.20/2.82 (0) ITRS 7.20/2.82 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 7.20/2.82 (2) IDP 7.20/2.82 (3) UsableRulesProof [EQUIVALENT, 0 ms] 7.20/2.82 (4) IDP 7.20/2.82 (5) IDPNonInfProof [SOUND, 535 ms] 7.20/2.82 (6) IDP 7.20/2.82 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.20/2.82 (8) IDP 7.20/2.82 (9) IDPNonInfProof [SOUND, 164 ms] 7.20/2.82 (10) IDP 7.20/2.82 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.20/2.82 (12) TRUE 7.20/2.82 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (0) 7.20/2.82 Obligation: 7.20/2.82 ITRS problem: 7.20/2.82 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 The TRS R consists of the following rules: 7.20/2.82 eval(x, y, z) -> Cond_eval(x + y > z && z >= 0 && x > 0, x, y, z) 7.20/2.82 Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) 7.20/2.82 eval(x, y, z) -> Cond_eval1(x + y > z && z >= 0 && 0 >= x && y > 0, x, y, z) 7.20/2.82 Cond_eval1(TRUE, x, y, z) -> eval(x, y - 1, z) 7.20/2.82 eval(x, y, z) -> Cond_eval2(x + y > z && z >= 0 && 0 >= x && 0 >= y, x, y, z) 7.20/2.82 Cond_eval2(TRUE, x, y, z) -> eval(x, y, z) 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (1) ITRStoIDPProof (EQUIVALENT) 7.20/2.82 Added dependency pairs 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (2) 7.20/2.82 Obligation: 7.20/2.82 IDP problem: 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 7.20/2.82 The following domains are used: 7.20/2.82 Boolean, Integer 7.20/2.82 7.20/2.82 The ITRS R consists of the following rules: 7.20/2.82 eval(x, y, z) -> Cond_eval(x + y > z && z >= 0 && x > 0, x, y, z) 7.20/2.82 Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) 7.20/2.82 eval(x, y, z) -> Cond_eval1(x + y > z && z >= 0 && 0 >= x && y > 0, x, y, z) 7.20/2.82 Cond_eval1(TRUE, x, y, z) -> eval(x, y - 1, z) 7.20/2.82 eval(x, y, z) -> Cond_eval2(x + y > z && z >= 0 && 0 >= x && 0 >= y, x, y, z) 7.20/2.82 Cond_eval2(TRUE, x, y, z) -> eval(x, y, z) 7.20/2.82 7.20/2.82 The integer pair graph contains the following rules and edges: 7.20/2.82 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) 7.20/2.82 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.20/2.82 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 7.20/2.82 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) 7.20/2.82 (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4], z[4]) 7.20/2.82 (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) 7.20/2.82 7.20/2.82 (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.20/2.82 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.20/2.82 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.20/2.82 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 7.20/2.82 (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.20/2.82 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) 7.20/2.82 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 7.20/2.82 (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) 7.20/2.82 (4) -> (5), if (x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 7.20/2.82 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 7.20/2.82 (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & z[5] ->^* z[2]) 7.20/2.82 (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & z[5] ->^* z[4]) 7.20/2.82 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (3) UsableRulesProof (EQUIVALENT) 7.20/2.82 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. 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (4) 7.20/2.82 Obligation: 7.20/2.82 IDP problem: 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 7.20/2.82 The following domains are used: 7.20/2.82 Boolean, Integer 7.20/2.82 7.20/2.82 R is empty. 7.20/2.82 7.20/2.82 The integer pair graph contains the following rules and edges: 7.20/2.82 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) 7.20/2.82 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.20/2.82 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 7.20/2.82 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) 7.20/2.82 (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4], z[4]) 7.20/2.82 (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) 7.20/2.82 7.20/2.82 (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.20/2.82 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.20/2.82 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.20/2.82 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 7.20/2.82 (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.20/2.82 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) 7.20/2.82 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 7.20/2.82 (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) 7.20/2.82 (4) -> (5), if (x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 7.20/2.82 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 7.20/2.82 (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & z[5] ->^* z[2]) 7.20/2.82 (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & z[5] ->^* z[4]) 7.20/2.82 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (5) IDPNonInfProof (SOUND) 7.20/2.82 Used the following options for this NonInfProof: 7.20/2.82 7.20/2.82 IDPGPoloSolver: 7.20/2.82 Range: [(-1,2)] 7.20/2.82 IsNat: false 7.20/2.82 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@370e7604 7.20/2.82 Constraint Generator: NonInfConstraintGenerator: 7.20/2.82 PathGenerator: MetricPathGenerator: 7.20/2.82 Max Left Steps: 1 7.20/2.82 Max Right Steps: 1 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 The constraints were generated the following way: 7.20/2.82 7.20/2.82 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 7.20/2.82 7.20/2.82 Note that final constraints are written in bold face. 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair EVAL(x, y, z) -> COND_EVAL(&&(&&(>(+(x, y), z), >=(z, 0)), >(x, 0)), x, y, z) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[4] & y[1]=y[4] & z[1]=z[4] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair EVAL(x, y, z) -> COND_EVAL1(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >(y, 0)), x, y, z) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] >= 0 & [(-1)bso_30] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] >= 0 & [(-1)bso_30] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] >= 0 & [(-1)bso_30] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] >= 0 & [(-1)bso_30] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & -(y[3], 1)=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair EVAL(x, y, z) -> COND_EVAL2(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >=(0, y)), x, y, z) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>=(0, y[4])=TRUE & >=(0, x[4])=TRUE & >(+(x[4], y[4]), z[4])=TRUE & >=(z[4], 0)=TRUE ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[4] >= 0 & [-1 + (-1)bso_34] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[4] >= 0 & [-1 + (-1)bso_34] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[4] >= 0 & [-1 + (-1)bso_34] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We solved constraint (5) using rule (IDP_SMT_SPLIT). 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair COND_EVAL2(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: 7.20/2.82 *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (x[5]=x[0] & y[5]=y[0] & z[5]=z[0] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (x[5]=x[2] & y[5]=y[2] & z[5]=z[2] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (x[5]=x[4] & y[5]=y[4] & z[5]=z[4] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 To summarize, we get the following constraints P__>=_ for the following pairs. 7.20/2.82 7.20/2.82 *EVAL(x, y, z) -> COND_EVAL(&&(&&(>(+(x, y), z), >=(z, 0)), >(x, 0)), x, y, z) 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] >= 0 & [(-1)bso_26] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[0] >= 0 & [(-1)bso_28] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *EVAL(x, y, z) -> COND_EVAL1(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >(y, 0)), x, y, z) 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] >= 0 & [(-1)bso_30] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *COND_EVAL1(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] >= 0 & [(-1)bso_32] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *EVAL(x, y, z) -> COND_EVAL2(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >=(0, y)), x, y, z) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *COND_EVAL2(TRUE, x, y, z) -> EVAL(x, y, z) 7.20/2.82 7.20/2.82 *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [1 + (-1)bso_36] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 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. 7.20/2.82 7.20/2.82 Using the following integer polynomial ordering the resulting constraints can be solved 7.20/2.82 7.20/2.82 Polynomial interpretation over integers[POLO]: 7.20/2.82 7.20/2.82 POL(TRUE) = 0 7.20/2.82 POL(FALSE) = [1] 7.20/2.82 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 7.20/2.82 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_1 7.20/2.82 POL(&&(x_1, x_2)) = 0 7.20/2.82 POL(>(x_1, x_2)) = [-1] 7.20/2.82 POL(+(x_1, x_2)) = x_1 + x_2 7.20/2.82 POL(>=(x_1, x_2)) = [-1] 7.20/2.82 POL(0) = 0 7.20/2.82 POL(-(x_1, x_2)) = x_1 + [-1]x_2 7.20/2.82 POL(1) = [1] 7.20/2.82 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_1 7.20/2.82 POL(COND_EVAL2(x_1, x_2, x_3, x_4)) = [-1]x_4 + [2]x_1 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_>: 7.20/2.82 7.20/2.82 7.20/2.82 EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) 7.20/2.82 COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_bound: 7.20/2.82 7.20/2.82 7.20/2.82 EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_>=: 7.20/2.82 7.20/2.82 7.20/2.82 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) 7.20/2.82 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.20/2.82 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) 7.20/2.82 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) 7.20/2.82 7.20/2.82 7.20/2.82 At least the following rules have been oriented under context sensitive arithmetic replacement: 7.20/2.82 7.20/2.82 &&(TRUE, TRUE)^1 <-> TRUE^1 7.20/2.82 FALSE^1 -> &&(TRUE, FALSE)^1 7.20/2.82 FALSE^1 -> &&(FALSE, TRUE)^1 7.20/2.82 FALSE^1 -> &&(FALSE, FALSE)^1 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (6) 7.20/2.82 Obligation: 7.20/2.82 IDP problem: 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 7.20/2.82 The following domains are used: 7.20/2.82 Boolean, Integer 7.20/2.82 7.20/2.82 R is empty. 7.20/2.82 7.20/2.82 The integer pair graph contains the following rules and edges: 7.20/2.82 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) 7.20/2.82 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.20/2.82 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 7.20/2.82 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) 7.20/2.82 (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) 7.20/2.82 7.20/2.82 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.20/2.82 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) 7.20/2.82 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 7.20/2.82 (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.20/2.82 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.20/2.82 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 7.20/2.82 (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & z[5] ->^* z[2]) 7.20/2.82 (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.20/2.82 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (7) IDependencyGraphProof (EQUIVALENT) 7.20/2.82 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (8) 7.20/2.82 Obligation: 7.20/2.82 IDP problem: 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 7.20/2.82 The following domains are used: 7.20/2.82 Integer, Boolean 7.20/2.82 7.20/2.82 R is empty. 7.20/2.82 7.20/2.82 The integer pair graph contains the following rules and edges: 7.20/2.82 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) 7.20/2.82 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 7.20/2.82 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.20/2.82 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) 7.20/2.82 7.20/2.82 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.20/2.82 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) 7.20/2.82 (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.20/2.82 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.20/2.82 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 7.20/2.82 (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.20/2.82 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (9) IDPNonInfProof (SOUND) 7.20/2.82 Used the following options for this NonInfProof: 7.20/2.82 7.20/2.82 IDPGPoloSolver: 7.20/2.82 Range: [(-1,2)] 7.20/2.82 IsNat: false 7.20/2.82 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@370e7604 7.20/2.82 Constraint Generator: NonInfConstraintGenerator: 7.20/2.82 PathGenerator: MetricPathGenerator: 7.20/2.82 Max Left Steps: 1 7.20/2.82 Max Right Steps: 1 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 The constraints were generated the following way: 7.20/2.82 7.20/2.82 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 7.20/2.82 7.20/2.82 Note that final constraints are written in bold face. 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & -(y[3], 1)=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [(-1)bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [(-1)bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]y[2] + [bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]y[2] + [bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]y[2] + [bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.20/2.82 7.20/2.82 (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [(-1)bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [(-1)bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) the following chains were created: 7.20/2.82 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: 7.20/2.82 7.20/2.82 (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.20/2.82 7.20/2.82 (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.20/2.82 7.20/2.82 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.20/2.82 7.20/2.82 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.20/2.82 7.20/2.82 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.20/2.82 7.20/2.82 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [(-1)bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 To summarize, we get the following constraints P__>=_ for the following pairs. 7.20/2.82 7.20/2.82 *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [(-1)bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[2] + [bni_20]y[2] + [(-1)bni_20]x[2] >= 0 & [1 + (-1)bso_21] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) 7.20/2.82 7.20/2.82 *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]y[2] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [(-1)bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[0] + [(-1)bni_24]y[0] + [bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 *EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [(-1)bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[0] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 7.20/2.82 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. 7.20/2.82 7.20/2.82 Using the following integer polynomial ordering the resulting constraints can be solved 7.20/2.82 7.20/2.82 Polynomial interpretation over integers[POLO]: 7.20/2.82 7.20/2.82 POL(TRUE) = 0 7.20/2.82 POL(FALSE) = 0 7.20/2.82 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 + x_2 + [-1]x_1 7.20/2.82 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 + x_1 7.20/2.82 POL(-(x_1, x_2)) = x_1 + [-1]x_2 7.20/2.82 POL(1) = [1] 7.20/2.82 POL(&&(x_1, x_2)) = 0 7.20/2.82 POL(>(x_1, x_2)) = [-1] 7.20/2.82 POL(+(x_1, x_2)) = x_1 + x_2 7.20/2.82 POL(>=(x_1, x_2)) = [-1] 7.20/2.82 POL(0) = 0 7.20/2.82 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 + x_2 + [-1]x_1 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_>: 7.20/2.82 7.20/2.82 7.20/2.82 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) 7.20/2.82 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_bound: 7.20/2.82 7.20/2.82 7.20/2.82 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) 7.20/2.82 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) 7.20/2.82 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.20/2.82 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) 7.20/2.82 7.20/2.82 7.20/2.82 The following pairs are in P_>=: 7.20/2.82 7.20/2.82 7.20/2.82 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) 7.20/2.82 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) 7.20/2.82 7.20/2.82 7.20/2.82 At least the following rules have been oriented under context sensitive arithmetic replacement: 7.20/2.82 7.20/2.82 &&(TRUE, TRUE)^1 <-> TRUE^1 7.20/2.82 &&(TRUE, FALSE)^1 <-> FALSE^1 7.20/2.82 &&(FALSE, TRUE)^1 <-> FALSE^1 7.20/2.82 &&(FALSE, FALSE)^1 <-> FALSE^1 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (10) 7.20/2.82 Obligation: 7.20/2.82 IDP problem: 7.20/2.82 The following function symbols are pre-defined: 7.20/2.82 <<< 7.20/2.82 & ~ Bwand: (Integer, Integer) -> Integer 7.20/2.82 >= ~ Ge: (Integer, Integer) -> Boolean 7.20/2.82 | ~ Bwor: (Integer, Integer) -> Integer 7.20/2.82 / ~ Div: (Integer, Integer) -> Integer 7.20/2.82 != ~ Neq: (Integer, Integer) -> Boolean 7.20/2.82 && ~ Land: (Boolean, Boolean) -> Boolean 7.20/2.82 ! ~ Lnot: (Boolean) -> Boolean 7.20/2.82 = ~ Eq: (Integer, Integer) -> Boolean 7.20/2.82 <= ~ Le: (Integer, Integer) -> Boolean 7.20/2.82 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.20/2.82 % ~ Mod: (Integer, Integer) -> Integer 7.20/2.82 > ~ Gt: (Integer, Integer) -> Boolean 7.20/2.82 + ~ Add: (Integer, Integer) -> Integer 7.20/2.82 -1 ~ UnaryMinus: (Integer) -> Integer 7.20/2.82 < ~ Lt: (Integer, Integer) -> Boolean 7.20/2.82 || ~ Lor: (Boolean, Boolean) -> Boolean 7.20/2.82 - ~ Sub: (Integer, Integer) -> Integer 7.20/2.82 ~ ~ Bwnot: (Integer) -> Integer 7.20/2.82 * ~ Mul: (Integer, Integer) -> Integer 7.20/2.82 >>> 7.20/2.82 7.20/2.82 7.20/2.82 The following domains are used: 7.20/2.82 Boolean, Integer 7.20/2.82 7.20/2.82 R is empty. 7.20/2.82 7.20/2.82 The integer pair graph contains the following rules and edges: 7.20/2.82 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 7.20/2.82 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) 7.20/2.82 7.20/2.82 7.20/2.82 The set Q consists of the following terms: 7.20/2.82 eval(x0, x1, x2) 7.20/2.82 Cond_eval(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval1(TRUE, x0, x1, x2) 7.20/2.82 Cond_eval2(TRUE, x0, x1, x2) 7.20/2.82 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (11) IDependencyGraphProof (EQUIVALENT) 7.20/2.82 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 7.20/2.82 ---------------------------------------- 7.20/2.82 7.20/2.82 (12) 7.20/2.82 TRUE 7.20/2.84 EOF