5.95/2.43 YES 5.95/2.46 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.95/2.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.95/2.46 5.95/2.46 5.95/2.46 Termination of the given ITRS could be proven: 5.95/2.46 5.95/2.46 (0) ITRS 5.95/2.46 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.95/2.46 (2) IDP 5.95/2.46 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.95/2.46 (4) IDP 5.95/2.46 (5) IDPNonInfProof [SOUND, 317 ms] 5.95/2.46 (6) AND 5.95/2.46 (7) IDP 5.95/2.46 (8) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.95/2.46 (9) IDP 5.95/2.46 (10) IDPNonInfProof [SOUND, 76 ms] 5.95/2.46 (11) AND 5.95/2.46 (12) IDP 5.95/2.46 (13) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.95/2.46 (14) TRUE 5.95/2.46 (15) IDP 5.95/2.46 (16) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.95/2.46 (17) TRUE 5.95/2.46 (18) IDP 5.95/2.46 (19) IDPNonInfProof [SOUND, 52 ms] 5.95/2.46 (20) AND 5.95/2.46 (21) IDP 5.95/2.46 (22) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.95/2.46 (23) TRUE 5.95/2.46 (24) IDP 5.95/2.46 (25) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.95/2.46 (26) TRUE 5.95/2.46 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (0) 5.95/2.46 Obligation: 5.95/2.46 ITRS problem: 5.95/2.46 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 The TRS R consists of the following rules: 5.95/2.46 eval_1(x, y, z) -> Cond_eval_1(x > y, x, y, z) 5.95/2.46 Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_2(x > z, x, y, z) 5.95/2.46 Cond_eval_2(TRUE, x, y, z) -> eval_1(x, y + 1, z) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_21(x > z, x, y, z) 5.95/2.46 Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z + 1) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_22(z >= x, x, y, z) 5.95/2.46 Cond_eval_22(TRUE, x, y, z) -> eval_1(x - 1, y, z) 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (1) ITRStoIDPProof (EQUIVALENT) 5.95/2.46 Added dependency pairs 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (2) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 The ITRS R consists of the following rules: 5.95/2.46 eval_1(x, y, z) -> Cond_eval_1(x > y, x, y, z) 5.95/2.46 Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_2(x > z, x, y, z) 5.95/2.46 Cond_eval_2(TRUE, x, y, z) -> eval_1(x, y + 1, z) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_21(x > z, x, y, z) 5.95/2.46 Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z + 1) 5.95/2.46 eval_2(x, y, z) -> Cond_eval_22(z >= x, x, y, z) 5.95/2.46 Cond_eval_22(TRUE, x, y, z) -> eval_1(x - 1, y, z) 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) 5.95/2.46 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) 5.95/2.46 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5] + 1) 5.95/2.46 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.46 (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) 5.95/2.46 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.95/2.46 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.46 (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.95/2.46 (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0] & z[3] ->^* z[0]) 5.95/2.46 (4) -> (5), if (x[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.95/2.46 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] + 1 ->^* z[0]) 5.95/2.46 (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 5.95/2.46 (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (3) UsableRulesProof (EQUIVALENT) 5.95/2.46 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. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (4) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) 5.95/2.46 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) 5.95/2.46 (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5] + 1) 5.95/2.46 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.46 (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) 5.95/2.46 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.95/2.46 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.46 (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.95/2.46 (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0] & z[3] ->^* z[0]) 5.95/2.46 (4) -> (5), if (x[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 5.95/2.46 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] + 1 ->^* z[0]) 5.95/2.46 (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 5.95/2.46 (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (5) IDPNonInfProof (SOUND) 5.95/2.46 Used the following options for this NonInfProof: 5.95/2.46 5.95/2.46 IDPGPoloSolver: 5.95/2.46 Range: [(-1,2)] 5.95/2.46 IsNat: false 5.95/2.46 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6d62fdc7 5.95/2.46 Constraint Generator: NonInfConstraintGenerator: 5.95/2.46 PathGenerator: MetricPathGenerator: 5.95/2.46 Max Left Steps: 1 5.95/2.46 Max Right Steps: 1 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 The constraints were generated the following way: 5.95/2.46 5.95/2.46 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.95/2.46 5.95/2.46 Note that final constraints are written in bold face. 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_1(x, y, z) -> COND_EVAL_1(>(x, y), x, y, z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[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: 5.95/2.46 5.95/2.46 (1) (>(x[0], y[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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [(-1)bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) the following chains were created: 5.95/2.46 *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(>(x[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *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(>(x[4], z[4]), x[4], y[4], z[4]) which results in the following constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *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[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (x[1]=x[6] & y[1]=y[6] & z[1]=z[6] ==> 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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_2(x, y, z) -> COND_EVAL_2(>(x, z), x, y, z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[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(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_2(TRUE, x, y, z) -> EVAL_1(x, +(y, 1), z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[2], z[2])=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_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_1(x[3], +(y[3], 1), z[3]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[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_1(x[2], +(y[2], 1), z[2]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_2(x, y, z) -> COND_EVAL_21(>(x, z), x, y, z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[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], 1)) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[4], z[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(>(x[4], z[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[4], z[4])=TRUE ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 (9) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, +(z, 1)) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[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], 1)), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[4], z[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[0] & y[5]=y[0] & +(z[5], 1)=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], 1)) & (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[4], z[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], 1)) & (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 (9) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_2(x, y, z) -> COND_EVAL_22(>=(z, x), x, y, z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>=(z[6], x[6])=TRUE ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (z[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_22(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & -(x[7], 1)=x[0] & y[7]=y[0] & z[7]=z[0] ==> COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_EVAL_1(-(x[7], 1), y[7], z[7]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>=(z[6], x[6])=TRUE ==> COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_EVAL_1(-(x[6], 1), y[6], z[6]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (z[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 To summarize, we get the following constraints P__>=_ for the following pairs. 5.95/2.46 5.95/2.46 *EVAL_1(x, y, z) -> COND_EVAL_1(>(x, y), x, y, z) 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [(-1)bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) 5.95/2.46 5.95/2.46 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_2(x, y, z) -> COND_EVAL_2(>(x, z), x, y, z) 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_2(TRUE, x, y, z) -> EVAL_1(x, +(y, 1), z) 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_2(x, y, z) -> COND_EVAL_21(>(x, z), x, y, z) 5.95/2.46 5.95/2.46 *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, +(z, 1)) 5.95/2.46 5.95/2.46 *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_2(x, y, z) -> COND_EVAL_22(>=(z, x), x, y, z) 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_22(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z) 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 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. 5.95/2.46 5.95/2.46 Using the following integer polynomial ordering the resulting constraints can be solved 5.95/2.46 5.95/2.46 Polynomial interpretation over integers[POLO]: 5.95/2.46 5.95/2.46 POL(TRUE) = 0 5.95/2.46 POL(FALSE) = 0 5.95/2.46 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 5.95/2.46 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.95/2.46 POL(>(x_1, x_2)) = [-1] 5.95/2.46 POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 5.95/2.46 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.95/2.46 POL(+(x_1, x_2)) = x_1 + x_2 5.95/2.46 POL(1) = [1] 5.95/2.46 POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.95/2.46 POL(COND_EVAL_22(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.95/2.46 POL(>=(x_1, x_2)) = [-1] 5.95/2.46 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>: 5.95/2.46 5.95/2.46 5.95/2.46 COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)) 5.95/2.46 COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_bound: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) 5.95/2.46 COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) 5.95/2.46 EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) 5.95/2.46 COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>=: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) 5.95/2.46 COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) 5.95/2.46 EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) 5.95/2.46 EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) 5.95/2.46 5.95/2.46 5.95/2.46 There are no usable rules. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (6) 5.95/2.46 Complex Obligation (AND) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (7) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) 5.95/2.46 (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) 5.95/2.46 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.46 5.95/2.46 (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0] & z[3] ->^* z[0]) 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.95/2.46 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 5.95/2.46 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (8) IDependencyGraphProof (EQUIVALENT) 5.95/2.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (9) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0] & z[3] ->^* z[0]) 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (10) IDPNonInfProof (SOUND) 5.95/2.46 Used the following options for this NonInfProof: 5.95/2.46 5.95/2.46 IDPGPoloSolver: 5.95/2.46 Range: [(-1,2)] 5.95/2.46 IsNat: false 5.95/2.46 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6d62fdc7 5.95/2.46 Constraint Generator: NonInfConstraintGenerator: 5.95/2.46 PathGenerator: MetricPathGenerator: 5.95/2.46 Max Left Steps: 1 5.95/2.46 Max Right Steps: 1 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 The constraints were generated the following way: 5.95/2.46 5.95/2.46 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.95/2.46 5.95/2.46 Note that final constraints are written in bold face. 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[2], z[2])=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_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_1(x[3], +(y[3], 1), z[3]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[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_1(x[2], +(y[2], 1), z[2]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22 + bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22 + bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22 + bni_22] + [(-1)bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>(x[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(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(x[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [(-1)bni_24]y[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [(-1)bni_24]y[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [(-1)bni_24]y[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24 + bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24 + bni_24] + [(-1)bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24 + bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 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: 5.95/2.46 *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(>(x[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[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: 5.95/2.46 5.95/2.46 (1) (>(x[0], y[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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28 + bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28 + bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28 + bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 To summarize, we get the following constraints P__>=_ for the following pairs. 5.95/2.46 5.95/2.46 *COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22 + bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22 + bni_22] + [(-1)bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24 + bni_24] + [(-1)bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24 + bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 5.95/2.46 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28 + bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28 + bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 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. 5.95/2.46 5.95/2.46 Using the following integer polynomial ordering the resulting constraints can be solved 5.95/2.46 5.95/2.46 Polynomial interpretation over integers[POLO]: 5.95/2.46 5.95/2.46 POL(TRUE) = [1] 5.95/2.46 POL(FALSE) = 0 5.95/2.46 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1]x_3 + x_2 5.95/2.46 POL(EVAL_1(x_1, x_2, x_3)) = [-1]x_2 + x_1 5.95/2.46 POL(+(x_1, x_2)) = x_1 + x_2 5.95/2.46 POL(1) = [1] 5.95/2.46 POL(EVAL_2(x_1, x_2, x_3)) = [-1]x_2 + x_1 5.95/2.46 POL(>(x_1, x_2)) = [-1] 5.95/2.46 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1]x_3 + x_2 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>: 5.95/2.46 5.95/2.46 5.95/2.46 COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_bound: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>=: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) 5.95/2.46 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 5.95/2.46 There are no usable rules. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (11) 5.95/2.46 Complex Obligation (AND) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (12) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (13) IDependencyGraphProof (EQUIVALENT) 5.95/2.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (14) 5.95/2.46 TRUE 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (15) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) 5.95/2.46 (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 5.95/2.46 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 5.95/2.46 (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (16) IDependencyGraphProof (EQUIVALENT) 5.95/2.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (17) 5.95/2.46 TRUE 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (18) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.46 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.46 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.46 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.46 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.46 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.46 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.46 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.46 + ~ Add: (Integer, Integer) -> Integer 5.95/2.46 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.46 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.46 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.46 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.46 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.46 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.46 >>> 5.95/2.46 5.95/2.46 5.95/2.46 The following domains are used: 5.95/2.46 Integer 5.95/2.46 5.95/2.46 R is empty. 5.95/2.46 5.95/2.46 The integer pair graph contains the following rules and edges: 5.95/2.46 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.46 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.46 (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) 5.95/2.46 5.95/2.46 (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) 5.95/2.46 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.46 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.46 (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 5.95/2.46 5.95/2.46 The set Q consists of the following terms: 5.95/2.46 eval_1(x0, x1, x2) 5.95/2.46 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.46 eval_2(x0, x1, x2) 5.95/2.46 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.46 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (19) IDPNonInfProof (SOUND) 5.95/2.46 Used the following options for this NonInfProof: 5.95/2.46 5.95/2.46 IDPGPoloSolver: 5.95/2.46 Range: [(-1,2)] 5.95/2.46 IsNat: false 5.95/2.46 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6d62fdc7 5.95/2.46 Constraint Generator: NonInfConstraintGenerator: 5.95/2.46 PathGenerator: MetricPathGenerator: 5.95/2.46 Max Left Steps: 1 5.95/2.46 Max Right Steps: 1 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 The constraints were generated the following way: 5.95/2.46 5.95/2.46 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.95/2.46 5.95/2.46 Note that final constraints are written in bold face. 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[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: 5.95/2.46 5.95/2.46 (1) (>(x[0], y[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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>(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], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 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: 5.95/2.46 *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[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (x[1]=x[6] & y[1]=y[6] & z[1]=z[6] ==> 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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (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])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>=(z[6], x[6])=TRUE ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (z[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 For Pair COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) the following chains were created: 5.95/2.46 *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.95/2.46 5.95/2.46 (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & -(x[7], 1)=x[0] & y[7]=y[0] & z[7]=z[0] ==> COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_EVAL_1(-(x[7], 1), y[7], z[7]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.95/2.46 5.95/2.46 (2) (>=(z[6], x[6])=TRUE ==> COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_EVAL_1(-(x[6], 1), y[6], z[6]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.95/2.46 5.95/2.46 (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.95/2.46 5.95/2.46 (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.95/2.46 5.95/2.46 (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.95/2.46 5.95/2.46 (7) (z[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.95/2.46 5.95/2.46 (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 To summarize, we get the following constraints P__>=_ for the following pairs. 5.95/2.46 5.95/2.46 *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 5.95/2.46 *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 *COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 5.95/2.46 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. 5.95/2.46 5.95/2.46 Using the following integer polynomial ordering the resulting constraints can be solved 5.95/2.46 5.95/2.46 Polynomial interpretation over integers[POLO]: 5.95/2.46 5.95/2.46 POL(TRUE) = 0 5.95/2.46 POL(FALSE) = 0 5.95/2.46 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 5.95/2.46 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 5.95/2.46 POL(>(x_1, x_2)) = [-1] 5.95/2.46 POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 5.95/2.46 POL(COND_EVAL_22(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 5.95/2.46 POL(>=(x_1, x_2)) = [-1] 5.95/2.46 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.95/2.46 POL(1) = [1] 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>: 5.95/2.46 5.95/2.46 5.95/2.46 COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_bound: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 5.95/2.46 5.95/2.46 The following pairs are in P_>=: 5.95/2.46 5.95/2.46 5.95/2.46 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) 5.95/2.46 COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.46 EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) 5.95/2.46 5.95/2.46 5.95/2.46 There are no usable rules. 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (20) 5.95/2.46 Complex Obligation (AND) 5.95/2.46 5.95/2.46 ---------------------------------------- 5.95/2.46 5.95/2.46 (21) 5.95/2.46 Obligation: 5.95/2.46 IDP problem: 5.95/2.46 The following function symbols are pre-defined: 5.95/2.46 <<< 5.95/2.46 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.46 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.46 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.46 / ~ Div: (Integer, Integer) -> Integer 5.95/2.46 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.47 + ~ Add: (Integer, Integer) -> Integer 5.95/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.47 >>> 5.95/2.47 5.95/2.47 5.95/2.47 The following domains are used: 5.95/2.47 Integer 5.95/2.47 5.95/2.47 R is empty. 5.95/2.47 5.95/2.47 The integer pair graph contains the following rules and edges: 5.95/2.47 (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) 5.95/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.47 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.47 5.95/2.47 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.95/2.47 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.47 5.95/2.47 The set Q consists of the following terms: 5.95/2.47 eval_1(x0, x1, x2) 5.95/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.47 eval_2(x0, x1, x2) 5.95/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.47 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.47 5.95/2.47 ---------------------------------------- 5.95/2.47 5.95/2.47 (22) IDependencyGraphProof (EQUIVALENT) 5.95/2.47 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.95/2.47 ---------------------------------------- 5.95/2.47 5.95/2.47 (23) 5.95/2.47 TRUE 5.95/2.47 5.95/2.47 ---------------------------------------- 5.95/2.47 5.95/2.47 (24) 5.95/2.47 Obligation: 5.95/2.47 IDP problem: 5.95/2.47 The following function symbols are pre-defined: 5.95/2.47 <<< 5.95/2.47 & ~ Bwand: (Integer, Integer) -> Integer 5.95/2.47 >= ~ Ge: (Integer, Integer) -> Boolean 5.95/2.47 | ~ Bwor: (Integer, Integer) -> Integer 5.95/2.47 / ~ Div: (Integer, Integer) -> Integer 5.95/2.47 != ~ Neq: (Integer, Integer) -> Boolean 5.95/2.47 && ~ Land: (Boolean, Boolean) -> Boolean 5.95/2.47 ! ~ Lnot: (Boolean) -> Boolean 5.95/2.47 = ~ Eq: (Integer, Integer) -> Boolean 5.95/2.47 <= ~ Le: (Integer, Integer) -> Boolean 5.95/2.47 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.95/2.47 % ~ Mod: (Integer, Integer) -> Integer 5.95/2.47 > ~ Gt: (Integer, Integer) -> Boolean 5.95/2.47 + ~ Add: (Integer, Integer) -> Integer 5.95/2.47 -1 ~ UnaryMinus: (Integer) -> Integer 5.95/2.47 < ~ Lt: (Integer, Integer) -> Boolean 5.95/2.47 || ~ Lor: (Boolean, Boolean) -> Boolean 5.95/2.47 - ~ Sub: (Integer, Integer) -> Integer 5.95/2.47 ~ ~ Bwnot: (Integer) -> Integer 5.95/2.47 * ~ Mul: (Integer, Integer) -> Integer 5.95/2.47 >>> 5.95/2.47 5.95/2.47 5.95/2.47 The following domains are used: 5.95/2.47 Integer 5.95/2.47 5.95/2.47 R is empty. 5.95/2.47 5.95/2.47 The integer pair graph contains the following rules and edges: 5.95/2.47 (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) 5.95/2.47 (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) 5.95/2.47 (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) 5.95/2.47 5.95/2.47 (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 5.95/2.47 (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 5.95/2.47 5.95/2.47 The set Q consists of the following terms: 5.95/2.47 eval_1(x0, x1, x2) 5.95/2.47 Cond_eval_1(TRUE, x0, x1, x2) 5.95/2.47 eval_2(x0, x1, x2) 5.95/2.47 Cond_eval_2(TRUE, x0, x1, x2) 5.95/2.47 Cond_eval_21(TRUE, x0, x1, x2) 5.95/2.47 Cond_eval_22(TRUE, x0, x1, x2) 5.95/2.47 5.95/2.47 ---------------------------------------- 5.95/2.47 5.95/2.47 (25) IDependencyGraphProof (EQUIVALENT) 5.95/2.47 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.95/2.47 ---------------------------------------- 5.95/2.47 5.95/2.47 (26) 5.95/2.47 TRUE 5.95/2.49 EOF