6.30/2.64 YES 6.44/2.66 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 6.44/2.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.44/2.66 6.44/2.66 6.44/2.66 Termination of the given ITRS could be proven: 6.44/2.66 6.44/2.66 (0) ITRS 6.44/2.66 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 6.44/2.66 (2) IDP 6.44/2.66 (3) UsableRulesProof [EQUIVALENT, 0 ms] 6.44/2.66 (4) IDP 6.44/2.66 (5) IDPNonInfProof [SOUND, 386 ms] 6.44/2.66 (6) IDP 6.44/2.66 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.44/2.66 (8) IDP 6.44/2.66 (9) IDPNonInfProof [SOUND, 95 ms] 6.44/2.66 (10) IDP 6.44/2.66 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.44/2.66 (12) TRUE 6.44/2.66 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (0) 6.44/2.66 Obligation: 6.44/2.66 ITRS problem: 6.44/2.66 6.44/2.66 The following function symbols are pre-defined: 6.44/2.66 <<< 6.44/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.66 / ~ Div: (Integer, Integer) -> Integer 6.44/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.66 + ~ Add: (Integer, Integer) -> Integer 6.44/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.66 >>> 6.44/2.66 6.44/2.66 The TRS R consists of the following rules: 6.44/2.66 eval(x, y, z) -> Cond_eval(y > x && z > x, x, y, z) 6.44/2.66 Cond_eval(TRUE, x, y, z) -> eval(x + 1, y, z) 6.44/2.66 eval(x, y, z) -> Cond_eval1(y > x && x >= z, x, y, z) 6.44/2.66 Cond_eval1(TRUE, x, y, z) -> eval(x, y, z + 1) 6.44/2.66 The set Q consists of the following terms: 6.44/2.66 eval(x0, x1, x2) 6.44/2.66 Cond_eval(TRUE, x0, x1, x2) 6.44/2.66 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (1) ITRStoIDPProof (EQUIVALENT) 6.44/2.66 Added dependency pairs 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (2) 6.44/2.66 Obligation: 6.44/2.66 IDP problem: 6.44/2.66 The following function symbols are pre-defined: 6.44/2.66 <<< 6.44/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.66 / ~ Div: (Integer, Integer) -> Integer 6.44/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.66 + ~ Add: (Integer, Integer) -> Integer 6.44/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.66 >>> 6.44/2.66 6.44/2.66 6.44/2.66 The following domains are used: 6.44/2.66 Boolean, Integer 6.44/2.66 6.44/2.66 The ITRS R consists of the following rules: 6.44/2.66 eval(x, y, z) -> Cond_eval(y > x && z > x, x, y, z) 6.44/2.66 Cond_eval(TRUE, x, y, z) -> eval(x + 1, y, z) 6.44/2.66 eval(x, y, z) -> Cond_eval1(y > x && x >= z, x, y, z) 6.44/2.66 Cond_eval1(TRUE, x, y, z) -> eval(x, y, z + 1) 6.44/2.66 6.44/2.66 The integer pair graph contains the following rules and edges: 6.44/2.66 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) 6.44/2.66 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] + 1, y[1], z[1]) 6.44/2.66 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) 6.44/2.66 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) 6.44/2.66 6.44/2.66 (0) -> (1), if (y[0] > x[0] && z[0] > x[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 6.44/2.66 (1) -> (0), if (x[1] + 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 6.44/2.66 (1) -> (2), if (x[1] + 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 6.44/2.66 (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 6.44/2.66 (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0]) 6.44/2.66 (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) 6.44/2.66 6.44/2.66 The set Q consists of the following terms: 6.44/2.66 eval(x0, x1, x2) 6.44/2.66 Cond_eval(TRUE, x0, x1, x2) 6.44/2.66 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (3) UsableRulesProof (EQUIVALENT) 6.44/2.66 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (4) 6.44/2.66 Obligation: 6.44/2.66 IDP problem: 6.44/2.66 The following function symbols are pre-defined: 6.44/2.66 <<< 6.44/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.66 / ~ Div: (Integer, Integer) -> Integer 6.44/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.66 + ~ Add: (Integer, Integer) -> Integer 6.44/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.66 >>> 6.44/2.66 6.44/2.66 6.44/2.66 The following domains are used: 6.44/2.66 Boolean, Integer 6.44/2.66 6.44/2.66 R is empty. 6.44/2.66 6.44/2.66 The integer pair graph contains the following rules and edges: 6.44/2.66 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) 6.44/2.66 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] + 1, y[1], z[1]) 6.44/2.66 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) 6.44/2.66 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) 6.44/2.66 6.44/2.66 (0) -> (1), if (y[0] > x[0] && z[0] > x[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 6.44/2.66 (1) -> (0), if (x[1] + 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 6.44/2.66 (1) -> (2), if (x[1] + 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 6.44/2.66 (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 6.44/2.66 (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0]) 6.44/2.66 (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) 6.44/2.66 6.44/2.66 The set Q consists of the following terms: 6.44/2.66 eval(x0, x1, x2) 6.44/2.66 Cond_eval(TRUE, x0, x1, x2) 6.44/2.66 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (5) IDPNonInfProof (SOUND) 6.44/2.66 Used the following options for this NonInfProof: 6.44/2.66 6.44/2.66 IDPGPoloSolver: 6.44/2.66 Range: [(-1,2)] 6.44/2.66 IsNat: false 6.44/2.66 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@67cd6dd4 6.44/2.66 Constraint Generator: NonInfConstraintGenerator: 6.44/2.66 PathGenerator: MetricPathGenerator: 6.44/2.66 Max Left Steps: 1 6.44/2.66 Max Right Steps: 1 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 The constraints were generated the following way: 6.44/2.66 6.44/2.66 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.44/2.66 6.44/2.66 Note that final constraints are written in bold face. 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair EVAL(x, y, z) -> COND_EVAL(&&(>(y, x), >(z, x)), x, y, z) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(+(x, 1), y, z) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & +(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(+(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(+(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & +(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(+(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(+(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair EVAL(x, y, z) -> COND_EVAL1(&&(>(y, x), >=(x, z)), x, y, z) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(x, y, +(z, 1)) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & y[3]=y[0] & +(z[3], 1)=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], y[3], +(z[3], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], y[2], +(z[2], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & y[3]=y[2]1 & +(z[3], 1)=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], y[3], +(z[3], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], y[2], +(z[2], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 To summarize, we get the following constraints P__>=_ for the following pairs. 6.44/2.66 6.44/2.66 *EVAL(x, y, z) -> COND_EVAL(&&(>(y, x), >(z, x)), x, y, z) 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *COND_EVAL(TRUE, x, y, z) -> EVAL(+(x, 1), y, z) 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *EVAL(x, y, z) -> COND_EVAL1(&&(>(y, x), >=(x, z)), x, y, z) 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_24] + [bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *COND_EVAL1(TRUE, x, y, z) -> EVAL(x, y, +(z, 1)) 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_26] + [bni_26]y[2] >= 0 & [(-1)bso_27] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 6.44/2.66 6.44/2.66 Using the following integer polynomial ordering the resulting constraints can be solved 6.44/2.66 6.44/2.66 Polynomial interpretation over integers[POLO]: 6.44/2.66 6.44/2.66 POL(TRUE) = 0 6.44/2.66 POL(FALSE) = 0 6.44/2.66 POL(EVAL(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 6.44/2.66 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_3 + [-1]x_2 6.44/2.66 POL(&&(x_1, x_2)) = [-1] 6.44/2.66 POL(>(x_1, x_2)) = [-1] 6.44/2.66 POL(+(x_1, x_2)) = x_1 + x_2 6.44/2.66 POL(1) = [1] 6.44/2.66 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + x_3 + [-1]x_2 6.44/2.66 POL(>=(x_1, x_2)) = [-1] 6.44/2.66 6.44/2.66 6.44/2.66 The following pairs are in P_>: 6.44/2.66 6.44/2.66 6.44/2.66 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) 6.44/2.66 6.44/2.66 6.44/2.66 The following pairs are in P_bound: 6.44/2.66 6.44/2.66 6.44/2.66 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) 6.44/2.66 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) 6.44/2.66 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) 6.44/2.66 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) 6.44/2.66 6.44/2.66 6.44/2.66 The following pairs are in P_>=: 6.44/2.66 6.44/2.66 6.44/2.66 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) 6.44/2.66 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) 6.44/2.66 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) 6.44/2.66 6.44/2.66 6.44/2.66 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.44/2.66 6.44/2.66 TRUE^1 -> &&(TRUE, TRUE)^1 6.44/2.66 FALSE^1 -> &&(TRUE, FALSE)^1 6.44/2.66 FALSE^1 -> &&(FALSE, TRUE)^1 6.44/2.66 FALSE^1 -> &&(FALSE, FALSE)^1 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (6) 6.44/2.66 Obligation: 6.44/2.66 IDP problem: 6.44/2.66 The following function symbols are pre-defined: 6.44/2.66 <<< 6.44/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.66 / ~ Div: (Integer, Integer) -> Integer 6.44/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.66 + ~ Add: (Integer, Integer) -> Integer 6.44/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.66 >>> 6.44/2.66 6.44/2.66 6.44/2.66 The following domains are used: 6.44/2.66 Boolean, Integer 6.44/2.66 6.44/2.66 R is empty. 6.44/2.66 6.44/2.66 The integer pair graph contains the following rules and edges: 6.44/2.66 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) 6.44/2.66 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) 6.44/2.66 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) 6.44/2.66 6.44/2.66 (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0]) 6.44/2.66 (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) 6.44/2.66 (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 6.44/2.66 6.44/2.66 The set Q consists of the following terms: 6.44/2.66 eval(x0, x1, x2) 6.44/2.66 Cond_eval(TRUE, x0, x1, x2) 6.44/2.66 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (7) IDependencyGraphProof (EQUIVALENT) 6.44/2.66 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (8) 6.44/2.66 Obligation: 6.44/2.66 IDP problem: 6.44/2.66 The following function symbols are pre-defined: 6.44/2.66 <<< 6.44/2.66 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.66 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.66 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.66 / ~ Div: (Integer, Integer) -> Integer 6.44/2.66 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.66 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.66 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.66 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.66 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.66 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.66 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.66 + ~ Add: (Integer, Integer) -> Integer 6.44/2.66 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.66 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.66 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.66 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.66 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.66 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.66 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.66 >>> 6.44/2.66 6.44/2.66 6.44/2.66 The following domains are used: 6.44/2.66 Integer, Boolean 6.44/2.66 6.44/2.66 R is empty. 6.44/2.66 6.44/2.66 The integer pair graph contains the following rules and edges: 6.44/2.66 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) 6.44/2.66 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) 6.44/2.66 6.44/2.66 (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) 6.44/2.66 (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 6.44/2.66 6.44/2.66 The set Q consists of the following terms: 6.44/2.66 eval(x0, x1, x2) 6.44/2.66 Cond_eval(TRUE, x0, x1, x2) 6.44/2.66 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.66 6.44/2.66 ---------------------------------------- 6.44/2.66 6.44/2.66 (9) IDPNonInfProof (SOUND) 6.44/2.66 Used the following options for this NonInfProof: 6.44/2.66 6.44/2.66 IDPGPoloSolver: 6.44/2.66 Range: [(-1,2)] 6.44/2.66 IsNat: false 6.44/2.66 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@67cd6dd4 6.44/2.66 Constraint Generator: NonInfConstraintGenerator: 6.44/2.66 PathGenerator: MetricPathGenerator: 6.44/2.66 Max Left Steps: 1 6.44/2.66 Max Right Steps: 1 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 The constraints were generated the following way: 6.44/2.66 6.44/2.66 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.44/2.66 6.44/2.66 Note that final constraints are written in bold face. 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & y[3]=y[2]1 & +(z[3], 1)=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], y[3], +(z[3], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], y[2], +(z[2], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) the following chains were created: 6.44/2.66 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) which results in the following constraint: 6.44/2.66 6.44/2.66 (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.44/2.66 6.44/2.66 (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.44/2.66 6.44/2.66 (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]z[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.44/2.66 6.44/2.66 (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]z[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.44/2.66 6.44/2.66 (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]z[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]z[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.44/2.66 6.44/2.66 (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.44/2.66 6.44/2.66 (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 To summarize, we get the following constraints P__>=_ for the following pairs. 6.44/2.66 6.44/2.66 *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 6.44/2.66 *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) 6.44/2.66 6.44/2.66 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.67 6.44/2.67 6.44/2.67 *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 6.44/2.67 The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 6.44/2.67 6.44/2.67 Using the following integer polynomial ordering the resulting constraints can be solved 6.44/2.67 6.44/2.67 Polynomial interpretation over integers[POLO]: 6.44/2.67 6.44/2.67 POL(TRUE) = 0 6.44/2.67 POL(FALSE) = [1] 6.44/2.67 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 + [-1]x_1 6.44/2.67 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 6.44/2.67 POL(+(x_1, x_2)) = x_1 + x_2 6.44/2.67 POL(1) = [1] 6.44/2.67 POL(&&(x_1, x_2)) = 0 6.44/2.67 POL(>(x_1, x_2)) = [-1] 6.44/2.67 POL(>=(x_1, x_2)) = [-1] 6.44/2.67 6.44/2.67 6.44/2.67 The following pairs are in P_>: 6.44/2.67 6.44/2.67 6.44/2.67 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) 6.44/2.67 6.44/2.67 6.44/2.67 The following pairs are in P_bound: 6.44/2.67 6.44/2.67 6.44/2.67 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) 6.44/2.67 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) 6.44/2.67 6.44/2.67 6.44/2.67 The following pairs are in P_>=: 6.44/2.67 6.44/2.67 6.44/2.67 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) 6.44/2.67 6.44/2.67 6.44/2.67 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.44/2.67 6.44/2.67 TRUE^1 -> &&(TRUE, TRUE)^1 6.44/2.67 FALSE^1 -> &&(TRUE, FALSE)^1 6.44/2.67 FALSE^1 -> &&(FALSE, TRUE)^1 6.44/2.67 FALSE^1 -> &&(FALSE, FALSE)^1 6.44/2.67 6.44/2.67 ---------------------------------------- 6.44/2.67 6.44/2.67 (10) 6.44/2.67 Obligation: 6.44/2.67 IDP problem: 6.44/2.67 The following function symbols are pre-defined: 6.44/2.67 <<< 6.44/2.67 & ~ Bwand: (Integer, Integer) -> Integer 6.44/2.67 >= ~ Ge: (Integer, Integer) -> Boolean 6.44/2.67 | ~ Bwor: (Integer, Integer) -> Integer 6.44/2.67 / ~ Div: (Integer, Integer) -> Integer 6.44/2.67 != ~ Neq: (Integer, Integer) -> Boolean 6.44/2.67 && ~ Land: (Boolean, Boolean) -> Boolean 6.44/2.67 ! ~ Lnot: (Boolean) -> Boolean 6.44/2.67 = ~ Eq: (Integer, Integer) -> Boolean 6.44/2.67 <= ~ Le: (Integer, Integer) -> Boolean 6.44/2.67 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.44/2.67 % ~ Mod: (Integer, Integer) -> Integer 6.44/2.67 + ~ Add: (Integer, Integer) -> Integer 6.44/2.67 > ~ Gt: (Integer, Integer) -> Boolean 6.44/2.67 -1 ~ UnaryMinus: (Integer) -> Integer 6.44/2.67 < ~ Lt: (Integer, Integer) -> Boolean 6.44/2.67 || ~ Lor: (Boolean, Boolean) -> Boolean 6.44/2.67 - ~ Sub: (Integer, Integer) -> Integer 6.44/2.67 ~ ~ Bwnot: (Integer) -> Integer 6.44/2.67 * ~ Mul: (Integer, Integer) -> Integer 6.44/2.67 >>> 6.44/2.67 6.44/2.67 6.44/2.67 The following domains are used: 6.44/2.67 Boolean, Integer 6.44/2.67 6.44/2.67 R is empty. 6.44/2.67 6.44/2.67 The integer pair graph contains the following rules and edges: 6.44/2.67 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) 6.44/2.67 6.44/2.67 6.44/2.67 The set Q consists of the following terms: 6.44/2.67 eval(x0, x1, x2) 6.44/2.67 Cond_eval(TRUE, x0, x1, x2) 6.44/2.67 Cond_eval1(TRUE, x0, x1, x2) 6.44/2.67 6.44/2.67 ---------------------------------------- 6.44/2.67 6.44/2.67 (11) IDependencyGraphProof (EQUIVALENT) 6.44/2.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 6.44/2.67 ---------------------------------------- 6.44/2.67 6.44/2.67 (12) 6.44/2.67 TRUE 6.44/2.73 EOF