5.94/2.45 YES 5.94/2.47 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.94/2.47 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.94/2.47 5.94/2.47 5.94/2.47 Termination of the given ITRS could be proven: 5.94/2.47 5.94/2.47 (0) ITRS 5.94/2.47 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.94/2.47 (2) IDP 5.94/2.47 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.94/2.47 (4) IDP 5.94/2.47 (5) IDPNonInfProof [SOUND, 294 ms] 5.94/2.47 (6) IDP 5.94/2.47 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.94/2.47 (8) IDP 5.94/2.47 (9) IDPNonInfProof [SOUND, 116 ms] 5.94/2.47 (10) IDP 5.94/2.47 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.94/2.47 (12) TRUE 5.94/2.47 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (0) 5.94/2.47 Obligation: 5.94/2.47 ITRS problem: 5.94/2.47 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 The TRS R consists of the following rules: 5.94/2.47 eval_1(x, y, z) -> Cond_eval_1(x = y && x > z, x, y, z) 5.94/2.47 Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) 5.94/2.47 eval_2(x, y, z) -> Cond_eval_2(y > z, x, y, z) 5.94/2.47 Cond_eval_2(TRUE, x, y, z) -> eval_2(x - 1, y - 1, z) 5.94/2.47 eval_2(x, y, z) -> Cond_eval_21(z >= y, x, y, z) 5.94/2.47 Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z) 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (1) ITRStoIDPProof (EQUIVALENT) 5.94/2.47 Added dependency pairs 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (2) 5.94/2.47 Obligation: 5.94/2.47 IDP problem: 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 5.94/2.47 The following domains are used: 5.94/2.47 Boolean, Integer 5.94/2.47 5.94/2.47 The ITRS R consists of the following rules: 5.94/2.47 eval_1(x, y, z) -> Cond_eval_1(x = y && x > z, x, y, z) 5.94/2.47 Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) 5.94/2.47 eval_2(x, y, z) -> Cond_eval_2(y > z, x, y, z) 5.94/2.47 Cond_eval_2(TRUE, x, y, z) -> eval_2(x - 1, y - 1, z) 5.94/2.47 eval_2(x, y, z) -> Cond_eval_21(z >= y, x, y, z) 5.94/2.47 Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z) 5.94/2.47 5.94/2.47 The integer pair graph contains the following rules and edges: 5.94/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) 5.94/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(y[2] > z[2], x[2], y[2], z[2]) 5.94/2.47 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) 5.94/2.47 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) 5.94/2.47 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 5.94/2.47 (0) -> (1), if (x[0] = y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.94/2.47 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.94/2.47 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.94/2.47 (2) -> (3), if (y[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.94/2.47 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 5.94/2.47 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) 5.94/2.47 (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.94/2.47 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 5.94/2.47 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (3) UsableRulesProof (EQUIVALENT) 5.94/2.47 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. 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (4) 5.94/2.47 Obligation: 5.94/2.47 IDP problem: 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 5.94/2.47 The following domains are used: 5.94/2.47 Boolean, Integer 5.94/2.47 5.94/2.47 R is empty. 5.94/2.47 5.94/2.47 The integer pair graph contains the following rules and edges: 5.94/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) 5.94/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(y[2] > z[2], x[2], y[2], z[2]) 5.94/2.47 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) 5.94/2.47 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) 5.94/2.47 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 5.94/2.47 (0) -> (1), if (x[0] = y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.94/2.47 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.94/2.47 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.94/2.47 (2) -> (3), if (y[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.94/2.47 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) 5.94/2.47 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) 5.94/2.47 (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.94/2.47 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 5.94/2.47 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (5) IDPNonInfProof (SOUND) 5.94/2.47 Used the following options for this NonInfProof: 5.94/2.47 5.94/2.47 IDPGPoloSolver: 5.94/2.47 Range: [(-1,2)] 5.94/2.47 IsNat: false 5.94/2.47 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@345ab01b 5.94/2.47 Constraint Generator: NonInfConstraintGenerator: 5.94/2.47 PathGenerator: MetricPathGenerator: 5.94/2.47 Max Left Steps: 1 5.94/2.47 Max Right Steps: 1 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 The constraints were generated the following way: 5.94/2.47 5.94/2.47 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.94/2.47 5.94/2.47 Note that final constraints are written in bold face. 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair EVAL_1(x, y, z) -> COND_EVAL_1(&&(=(x, y), >(x, z)), x, y, z) the following chains were created: 5.94/2.47 *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (&&(=(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(x[0], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]y[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]y[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]y[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]y[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] + [bni_30]z[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) the following chains were created: 5.94/2.47 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (x[1]=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (x[1]=x[4] & y[1]=y[4] & z[1]=z[4] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair EVAL_2(x, y, z) -> COND_EVAL_2(>(y, z), x, y, z) the following chains were created: 5.94/2.47 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(y[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (y[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(2)bni_34 + (-1)Bound*bni_34] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(2)bni_34 + (-1)Bound*bni_34] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(2)bni_34 + (-1)Bound*bni_34] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(-(x, 1), -(y, 1), z) the following chains were created: 5.94/2.47 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1 ==> COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(y[2], z[2])=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_2(-(x[2], 1), -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (y[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] ==> COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(y[2], z[2])=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_2(-(x[2], 1), -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36] + [(-1)bni_36]z[2] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (y[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair EVAL_2(x, y, z) -> COND_EVAL_21(>=(z, y), x, y, z) the following chains were created: 5.94/2.47 *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (>=(z[4], y[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>=(z[4], y[4])=TRUE ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [bni_38]y[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [bni_38]y[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [bni_38]y[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [bni_38]y[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, z) the following chains were created: 5.94/2.47 *We consider the chain COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (x[5]=x[0] & y[5]=y[0] & z[5]=z[0] ==> COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], z[5]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], z[5]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) ((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) ((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) ((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 To summarize, we get the following constraints P__>=_ for the following pairs. 5.94/2.47 5.94/2.47 *EVAL_1(x, y, z) -> COND_EVAL_1(&&(=(x, y), >(x, z)), x, y, z) 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) 5.94/2.47 5.94/2.47 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *EVAL_2(x, y, z) -> COND_EVAL_2(>(y, z), x, y, z) 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(2)bni_34 + (-1)Bound*bni_34] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(2)bni_34 + (-1)Bound*bni_34] + [bni_34]y[2] >= 0 & [1 + (-1)bso_35] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(-(x, 1), -(y, 1), z) 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_36 + bni_36] + [bni_36]y[2] >= 0 & [(-1)bso_37] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *EVAL_2(x, y, z) -> COND_EVAL_21(>=(z, y), x, y, z) 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, z) 5.94/2.47 5.94/2.47 *((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 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. 5.94/2.47 5.94/2.47 Using the following integer polynomial ordering the resulting constraints can be solved 5.94/2.47 5.94/2.47 Polynomial interpretation over integers[POLO]: 5.94/2.47 5.94/2.47 POL(TRUE) = [1] 5.94/2.47 POL(FALSE) = [3] 5.94/2.47 POL(EVAL_1(x_1, x_2, x_3)) = [1] + [-1]x_3 + x_2 5.94/2.47 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [2] + [-1]x_4 + x_3 + [-1]x_1 5.94/2.47 POL(&&(x_1, x_2)) = [1] 5.94/2.47 POL(=(x_1, x_2)) = [-1] 5.94/2.47 POL(>(x_1, x_2)) = [1] 5.94/2.47 POL(EVAL_2(x_1, x_2, x_3)) = [1] + [-1]x_3 + x_2 5.94/2.47 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1]x_4 + x_3 5.94/2.47 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.94/2.47 POL(1) = [1] 5.94/2.47 POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [1] + [-1]x_4 + x_3 5.94/2.47 POL(>=(x_1, x_2)) = 0 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_>: 5.94/2.47 5.94/2.47 5.94/2.47 EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_bound: 5.94/2.47 5.94/2.47 5.94/2.47 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 5.94/2.47 EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) 5.94/2.47 COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_>=: 5.94/2.47 5.94/2.47 5.94/2.47 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 5.94/2.47 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) 5.94/2.47 EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) 5.94/2.47 COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 5.94/2.47 5.94/2.47 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.94/2.47 5.94/2.47 &&(TRUE, TRUE)^1 <-> TRUE^1 5.94/2.47 FALSE^1 -> &&(TRUE, FALSE)^1 5.94/2.47 FALSE^1 -> &&(FALSE, TRUE)^1 5.94/2.47 FALSE^1 -> &&(FALSE, FALSE)^1 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (6) 5.94/2.47 Obligation: 5.94/2.47 IDP problem: 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 5.94/2.47 The following domains are used: 5.94/2.47 Boolean, Integer 5.94/2.47 5.94/2.47 R is empty. 5.94/2.47 5.94/2.47 The integer pair graph contains the following rules and edges: 5.94/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) 5.94/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) 5.94/2.47 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) 5.94/2.47 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 5.94/2.47 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 5.94/2.47 (0) -> (1), if (x[0] = y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.94/2.47 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.94/2.47 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) 5.94/2.47 (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.94/2.47 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (7) IDependencyGraphProof (EQUIVALENT) 5.94/2.47 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (8) 5.94/2.47 Obligation: 5.94/2.47 IDP problem: 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 5.94/2.47 The following domains are used: 5.94/2.47 Integer, Boolean 5.94/2.47 5.94/2.47 R is empty. 5.94/2.47 5.94/2.47 The integer pair graph contains the following rules and edges: 5.94/2.47 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) 5.94/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) 5.94/2.47 5.94/2.47 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 5.94/2.47 (0) -> (1), if (x[0] = y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.94/2.47 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.94/2.47 (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.94/2.47 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (9) IDPNonInfProof (SOUND) 5.94/2.47 Used the following options for this NonInfProof: 5.94/2.47 5.94/2.47 IDPGPoloSolver: 5.94/2.47 Range: [(-1,2)] 5.94/2.47 IsNat: false 5.94/2.47 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@345ab01b 5.94/2.47 Constraint Generator: NonInfConstraintGenerator: 5.94/2.47 PathGenerator: MetricPathGenerator: 5.94/2.47 Max Left Steps: 1 5.94/2.47 Max Right Steps: 1 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 The constraints were generated the following way: 5.94/2.47 5.94/2.47 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.94/2.47 5.94/2.47 Note that final constraints are written in bold face. 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) the following chains were created: 5.94/2.47 *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (>=(z[4], y[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[0] & y[5]=y[0] & z[5]=z[0] ==> COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], z[5]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>=(z[4], y[4])=TRUE ==> COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_EVAL_1(x[4], y[4], z[4]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] >= 0) 5.94/2.47 5.94/2.47 (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) the following chains were created: 5.94/2.47 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (x[1]=x[4] & y[1]=y[4] & z[1]=z[4] & >=(z[4], y[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>=(z[4], y[4])=TRUE ==> EVAL_2(x[1], y[4], z[4])_>=_NonInfC & EVAL_2(x[1], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[1], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] + z[4] + [-1]y[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] >= 0 & [(-1)bso_28] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] >= 0 & [(-1)bso_28] + z[4] >= 0) 5.94/2.47 5.94/2.47 (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] >= 0 & [(-1)bso_28] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) the following chains were created: 5.94/2.47 *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (&&(=(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[4] & y[1]=y[4] & z[1]=z[4] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(x[0], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> COND_EVAL_1(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL_1(TRUE, x[0], y[0], z[0])_>=_EVAL_2(x[0], y[0], z[0]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] + z[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) 5.94/2.47 5.94/2.47 (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) the following chains were created: 5.94/2.47 *We consider the chain COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint: 5.94/2.47 5.94/2.47 (1) (x[5]=x[0] & y[5]=y[0] & z[5]=z[0] & &&(=(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.94/2.47 5.94/2.47 (2) (>(x[0], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.94/2.47 5.94/2.47 (3) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]y[0] >= 0 & [-1 + (-1)bso_32] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.94/2.47 5.94/2.47 (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]y[0] >= 0 & [-1 + (-1)bso_32] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.94/2.47 5.94/2.47 (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]y[0] >= 0 & [-1 + (-1)bso_32] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]y[0] >= 0 & [-1 + (-1)bso_32] + [-1]z[0] + y[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] + [bni_31]z[0] >= 0 & [(-1)bso_32] + x[0] + z[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.94/2.47 5.94/2.47 (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.94/2.47 5.94/2.47 (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + x[0] >= 0) 5.94/2.47 5.94/2.47 (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 To summarize, we get the following constraints P__>=_ for the following pairs. 5.94/2.47 5.94/2.47 *COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] >= 0 & [(-1)bso_28] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]z[4] >= 0 & [(-1)bso_28] + z[4] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + x[0] >= 0) 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 5.94/2.47 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. 5.94/2.47 5.94/2.47 Using the following integer polynomial ordering the resulting constraints can be solved 5.94/2.47 5.94/2.47 Polynomial interpretation over integers[POLO]: 5.94/2.47 5.94/2.47 POL(TRUE) = 0 5.94/2.47 POL(FALSE) = [1] 5.94/2.47 POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [-1] 5.94/2.47 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 5.94/2.47 POL(EVAL_2(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 5.94/2.47 POL(>=(x_1, x_2)) = [-1] 5.94/2.47 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_1 5.94/2.47 POL(&&(x_1, x_2)) = [-1] 5.94/2.47 POL(=(x_1, x_2)) = [-1] 5.94/2.47 POL(>(x_1, x_2)) = [-1] 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_>: 5.94/2.47 5.94/2.47 5.94/2.47 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_bound: 5.94/2.47 5.94/2.47 5.94/2.47 COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) 5.94/2.47 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.94/2.47 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 5.94/2.47 5.94/2.47 5.94/2.47 The following pairs are in P_>=: 5.94/2.47 5.94/2.47 5.94/2.47 COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) 5.94/2.47 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 5.94/2.47 5.94/2.47 5.94/2.47 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.94/2.47 5.94/2.47 TRUE^1 -> &&(TRUE, TRUE)^1 5.94/2.47 FALSE^1 -> &&(TRUE, FALSE)^1 5.94/2.47 FALSE^1 -> &&(FALSE, TRUE)^1 5.94/2.47 FALSE^1 -> &&(FALSE, FALSE)^1 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (10) 5.94/2.47 Obligation: 5.94/2.47 IDP problem: 5.94/2.47 The following function symbols are pre-defined: 5.94/2.47 <<< 5.94/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.94/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.94/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.94/2.47 / ~ Div: (Integer, Integer) -> Integer 5.94/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.94/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.94/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.94/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.94/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.94/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.94/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.94/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.94/2.47 + ~ Add: (Integer, Integer) -> Integer 5.94/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.94/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.94/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.94/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.94/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.94/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.94/2.47 >>> 5.94/2.47 5.94/2.47 5.94/2.47 The following domains are used: 5.94/2.47 Integer, Boolean 5.94/2.47 5.94/2.47 R is empty. 5.94/2.47 5.94/2.47 The integer pair graph contains the following rules and edges: 5.94/2.47 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) 5.94/2.47 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) 5.94/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) 5.94/2.47 5.94/2.47 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) 5.94/2.47 (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.94/2.47 5.94/2.47 The set Q consists of the following terms: 5.94/2.47 eval_1(x0, x1, x2) 5.94/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.94/2.47 eval_2(x0, x1, x2) 5.94/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.94/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.94/2.47 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (11) IDependencyGraphProof (EQUIVALENT) 5.94/2.47 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.94/2.47 ---------------------------------------- 5.94/2.47 5.94/2.47 (12) 5.94/2.47 TRUE 6.14/2.52 EOF