5.61/2.35 YES 5.83/2.37 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.83/2.37 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.83/2.37 5.83/2.37 5.83/2.37 Termination of the given ITRS could be proven: 5.83/2.37 5.83/2.37 (0) ITRS 5.83/2.37 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.83/2.37 (2) IDP 5.83/2.37 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.83/2.37 (4) IDP 5.83/2.37 (5) IDPNonInfProof [SOUND, 188 ms] 5.83/2.37 (6) IDP 5.83/2.37 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.83/2.37 (8) IDP 5.83/2.37 (9) IDPNonInfProof [SOUND, 24 ms] 5.83/2.37 (10) IDP 5.83/2.37 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.83/2.37 (12) TRUE 5.83/2.37 5.83/2.37 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (0) 5.83/2.37 Obligation: 5.83/2.37 ITRS problem: 5.83/2.37 5.83/2.37 The following function symbols are pre-defined: 5.83/2.37 <<< 5.83/2.37 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.37 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.37 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.37 / ~ Div: (Integer, Integer) -> Integer 5.83/2.37 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.37 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.37 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.37 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.37 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.37 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.37 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.37 + ~ Add: (Integer, Integer) -> Integer 5.83/2.37 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.37 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.37 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.37 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.37 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.37 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.37 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.37 >>> 5.83/2.37 5.83/2.37 The TRS R consists of the following rules: 5.83/2.37 mult(0, y) -> 0 5.83/2.37 mult(x, y) -> Cond_mult(x > 0, x, y) 5.83/2.37 Cond_mult(TRUE, x, y) -> mult(x - 1, y) + y 5.83/2.37 mult(x, y) -> Cond_mult1(0 > x, x, y) 5.83/2.37 Cond_mult1(TRUE, x, y) -> -(mult(-(x), y)) 5.83/2.37 The set Q consists of the following terms: 5.83/2.37 mult(x0, x1) 5.83/2.37 Cond_mult(TRUE, x0, x1) 5.83/2.37 Cond_mult1(TRUE, x0, x1) 5.83/2.37 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (1) ITRStoIDPProof (EQUIVALENT) 5.83/2.37 Added dependency pairs 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (2) 5.83/2.37 Obligation: 5.83/2.37 IDP problem: 5.83/2.37 The following function symbols are pre-defined: 5.83/2.37 <<< 5.83/2.37 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.37 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.37 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.37 / ~ Div: (Integer, Integer) -> Integer 5.83/2.37 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.37 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.37 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.37 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.37 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.37 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.37 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.37 + ~ Add: (Integer, Integer) -> Integer 5.83/2.37 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.37 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.37 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.37 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.37 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.37 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.37 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.37 >>> 5.83/2.37 5.83/2.37 5.83/2.37 The following domains are used: 5.83/2.37 Integer 5.83/2.37 5.83/2.37 The ITRS R consists of the following rules: 5.83/2.37 mult(0, y) -> 0 5.83/2.37 mult(x, y) -> Cond_mult(x > 0, x, y) 5.83/2.37 Cond_mult(TRUE, x, y) -> mult(x - 1, y) + y 5.83/2.37 mult(x, y) -> Cond_mult1(0 > x, x, y) 5.83/2.37 Cond_mult1(TRUE, x, y) -> -(mult(-(x), y)) 5.83/2.37 5.83/2.37 The integer pair graph contains the following rules and edges: 5.83/2.37 (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) 5.83/2.37 (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) 5.83/2.37 (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) 5.83/2.37 (3): COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) 5.83/2.37 5.83/2.37 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.83/2.37 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.83/2.37 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 5.83/2.37 (2) -> (3), if (0 > x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.83/2.37 (3) -> (0), if (-(x[3]) ->^* x[0] & y[3] ->^* y[0]) 5.83/2.37 (3) -> (2), if (-(x[3]) ->^* x[2] & y[3] ->^* y[2]) 5.83/2.37 5.83/2.37 The set Q consists of the following terms: 5.83/2.37 mult(x0, x1) 5.83/2.37 Cond_mult(TRUE, x0, x1) 5.83/2.37 Cond_mult1(TRUE, x0, x1) 5.83/2.37 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (3) UsableRulesProof (EQUIVALENT) 5.83/2.37 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. 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (4) 5.83/2.37 Obligation: 5.83/2.37 IDP problem: 5.83/2.37 The following function symbols are pre-defined: 5.83/2.37 <<< 5.83/2.37 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.37 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.37 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.37 / ~ Div: (Integer, Integer) -> Integer 5.83/2.37 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.37 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.37 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.37 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.37 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.37 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.37 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.37 + ~ Add: (Integer, Integer) -> Integer 5.83/2.37 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.37 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.37 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.37 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.37 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.37 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.37 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.37 >>> 5.83/2.37 5.83/2.37 5.83/2.37 The following domains are used: 5.83/2.37 Integer 5.83/2.37 5.83/2.37 R is empty. 5.83/2.37 5.83/2.37 The integer pair graph contains the following rules and edges: 5.83/2.37 (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) 5.83/2.37 (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) 5.83/2.37 (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) 5.83/2.37 (3): COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) 5.83/2.37 5.83/2.37 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.83/2.37 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.83/2.37 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 5.83/2.37 (2) -> (3), if (0 > x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.83/2.37 (3) -> (0), if (-(x[3]) ->^* x[0] & y[3] ->^* y[0]) 5.83/2.37 (3) -> (2), if (-(x[3]) ->^* x[2] & y[3] ->^* y[2]) 5.83/2.37 5.83/2.37 The set Q consists of the following terms: 5.83/2.37 mult(x0, x1) 5.83/2.37 Cond_mult(TRUE, x0, x1) 5.83/2.37 Cond_mult1(TRUE, x0, x1) 5.83/2.37 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (5) IDPNonInfProof (SOUND) 5.83/2.37 Used the following options for this NonInfProof: 5.83/2.37 5.83/2.37 IDPGPoloSolver: 5.83/2.37 Range: [(-1,2)] 5.83/2.37 IsNat: false 5.83/2.37 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@1f2ca6e 5.83/2.37 Constraint Generator: NonInfConstraintGenerator: 5.83/2.37 PathGenerator: MetricPathGenerator: 5.83/2.37 Max Left Steps: 3 5.83/2.37 Max Right Steps: 2 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 The constraints were generated the following way: 5.83/2.37 5.83/2.37 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.83/2.37 5.83/2.37 Note that final constraints are written in bold face. 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 For Pair MULT(x, y) -> COND_MULT(>(x, 0), x, y) the following chains were created: 5.83/2.37 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(x[0], 0)=TRUE ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15] = 0 & [(-1)bni_15 + (-1)Bound*bni_15] >= 0 & [(-1)bso_16] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 For Pair COND_MULT(TRUE, x, y) -> MULT(-(x, 1), y) the following chains were created: 5.83/2.37 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 & -(x[1]1, 1)=x[0]2 & y[1]1=y[0]2 & >(x[0]2, 0)=TRUE & x[0]2=x[1]2 & y[0]2=y[1]2 ==> COND_MULT(TRUE, x[1]1, y[1]1)_>=_NonInfC & COND_MULT(TRUE, x[1]1, y[1]1)_>=_MULT(-(x[1]1, 1), y[1]1) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(x[0], 0)=TRUE & >(-(x[0], 1), 0)=TRUE & >(-(-(x[0], 1), 1), 0)=TRUE ==> COND_MULT(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1), 1), y[0]) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 & -(x[1]1, 1)=x[2] & y[1]1=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_MULT(TRUE, x[1]1, y[1]1)_>=_NonInfC & COND_MULT(TRUE, x[1]1, y[1]1)_>=_MULT(-(x[1]1, 1), y[1]1) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(x[0], 0)=TRUE & >(-(x[0], 1), 0)=TRUE & >(0, -(-(x[0], 1), 1))=TRUE ==> COND_MULT(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1), 1), y[0]) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0] & y[3]=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(0, x[2])=TRUE & >(-(x[2]), 0)=TRUE & >(-(-(x[2]), 1), 0)=TRUE ==> COND_MULT(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2]), 1), y[2]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.83/2.37 5.83/2.37 (7) ([-1] + x[2] >= 0 & [-1] + x[2] >= 0 & [-2] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0] & y[3]=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2]1 & y[1]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(0, x[2])=TRUE & >(-(x[2]), 0)=TRUE & >(0, -(-(x[2]), 1))=TRUE ==> COND_MULT(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2]), 1), y[2]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 For Pair MULT(x, y) -> COND_MULT1(>(0, x), x, y) the following chains were created: 5.83/2.37 *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] ==> MULT(x[2], y[2])_>=_NonInfC & MULT(x[2], y[2])_>=_COND_MULT1(>(0, x[2]), x[2], y[2]) & (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(0, x[2])=TRUE ==> MULT(x[2], y[2])_>=_NonInfC & MULT(x[2], y[2])_>=_COND_MULT1(>(0, x[2]), x[2], y[2]) & (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.83/2.37 5.83/2.37 (7) ([-1] + x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 For Pair COND_MULT1(TRUE, x, y) -> MULT(-(x), y) the following chains were created: 5.83/2.37 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0]1 & y[3]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 ==> COND_MULT1(TRUE, x[3], y[3])_>=_NonInfC & COND_MULT1(TRUE, x[3], y[3])_>=_MULT(-(x[3]), y[3]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(x[0], 0)=TRUE & >(0, -(x[0], 1))=TRUE & >(-(-(x[0], 1)), 0)=TRUE ==> COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1)), y[0]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-1)bni_21] = 0 & [(-3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 ==> COND_MULT1(TRUE, x[3], y[3])_>=_NonInfC & COND_MULT1(TRUE, x[3], y[3])_>=_MULT(-(x[3]), y[3]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(x[0], 0)=TRUE & >(0, -(x[0], 1))=TRUE & >(0, -(-(x[0], 1)))=TRUE ==> COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1)), y[0]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-1)bni_21] = 0 & [(-3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 & -(x[3]1)=x[0] & y[3]1=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_MULT1(TRUE, x[3]1, y[3]1)_>=_NonInfC & COND_MULT1(TRUE, x[3]1, y[3]1)_>=_MULT(-(x[3]1), y[3]1) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(0, x[2])=TRUE & >(0, -(x[2]))=TRUE & >(-(-(x[2])), 0)=TRUE ==> COND_MULT1(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT1(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2])), y[2]) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: 5.83/2.37 5.83/2.37 (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 & -(x[3]1)=x[2]2 & y[3]1=y[2]2 & >(0, x[2]2)=TRUE & x[2]2=x[3]2 & y[2]2=y[3]2 ==> COND_MULT1(TRUE, x[3]1, y[3]1)_>=_NonInfC & COND_MULT1(TRUE, x[3]1, y[3]1)_>=_MULT(-(x[3]1), y[3]1) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.37 5.83/2.37 (2) (>(0, x[2])=TRUE & >(0, -(x[2]))=TRUE & >(0, -(-(x[2])))=TRUE ==> COND_MULT1(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT1(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2])), y[2]) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (3) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.37 5.83/2.37 (4) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.37 5.83/2.37 (5) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.37 5.83/2.37 (6) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 We solved constraint (6) using rule (IDP_SMT_SPLIT). 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 To summarize, we get the following constraints P__>=_ for the following pairs. 5.83/2.37 5.83/2.37 *MULT(x, y) -> COND_MULT(>(x, 0), x, y) 5.83/2.37 5.83/2.37 *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15] = 0 & [(-1)bni_15 + (-1)Bound*bni_15] >= 0 & [(-1)bso_16] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 *COND_MULT(TRUE, x, y) -> MULT(-(x, 1), y) 5.83/2.37 5.83/2.37 *(x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 *([-1] + x[2] >= 0 & [-1] + x[2] >= 0 & [-2] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 *MULT(x, y) -> COND_MULT1(>(0, x), x, y) 5.83/2.37 5.83/2.37 *([-1] + x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [2]x[2] >= 0) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 *COND_MULT1(TRUE, x, y) -> MULT(-(x), y) 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 5.83/2.37 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. 5.83/2.37 5.83/2.37 Using the following integer polynomial ordering the resulting constraints can be solved 5.83/2.37 5.83/2.37 Polynomial interpretation over integers[POLO]: 5.83/2.37 5.83/2.37 POL(TRUE) = [2] 5.83/2.37 POL(FALSE) = 0 5.83/2.37 POL(MULT(x_1, x_2)) = [-1] + [-1]x_2 5.83/2.37 POL(COND_MULT(x_1, x_2, x_3)) = [-1] + [-1]x_3 5.83/2.37 POL(>(x_1, x_2)) = [-1] 5.83/2.37 POL(0) = 0 5.83/2.37 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.83/2.37 POL(1) = [1] 5.83/2.37 POL(COND_MULT1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [2]x_2 5.83/2.37 POL(-(x_1)) = [-1]x_1 5.83/2.37 5.83/2.37 5.83/2.37 The following pairs are in P_>: 5.83/2.37 5.83/2.37 5.83/2.37 MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]) 5.83/2.37 COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) 5.83/2.37 5.83/2.37 5.83/2.37 The following pairs are in P_bound: 5.83/2.37 5.83/2.37 5.83/2.37 COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) 5.83/2.37 5.83/2.37 5.83/2.37 The following pairs are in P_>=: 5.83/2.37 5.83/2.37 5.83/2.37 MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) 5.83/2.37 COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) 5.83/2.37 5.83/2.37 5.83/2.37 There are no usable rules. 5.83/2.37 ---------------------------------------- 5.83/2.37 5.83/2.37 (6) 5.83/2.37 Obligation: 5.83/2.37 IDP problem: 5.83/2.37 The following function symbols are pre-defined: 5.83/2.37 <<< 5.83/2.37 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.37 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.37 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.37 / ~ Div: (Integer, Integer) -> Integer 5.83/2.37 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.37 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.37 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.37 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.37 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.37 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.37 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.37 + ~ Add: (Integer, Integer) -> Integer 5.83/2.38 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.38 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.38 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.38 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.38 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.38 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.38 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.38 >>> 5.83/2.38 5.83/2.38 5.83/2.38 The following domains are used: 5.83/2.38 Integer 5.83/2.38 5.83/2.38 R is empty. 5.83/2.38 5.83/2.38 The integer pair graph contains the following rules and edges: 5.83/2.38 (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) 5.83/2.38 (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) 5.83/2.38 (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) 5.83/2.38 5.83/2.38 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.83/2.38 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.83/2.38 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 5.83/2.38 5.83/2.38 The set Q consists of the following terms: 5.83/2.38 mult(x0, x1) 5.83/2.38 Cond_mult(TRUE, x0, x1) 5.83/2.38 Cond_mult1(TRUE, x0, x1) 5.83/2.38 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (7) IDependencyGraphProof (EQUIVALENT) 5.83/2.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (8) 5.83/2.38 Obligation: 5.83/2.38 IDP problem: 5.83/2.38 The following function symbols are pre-defined: 5.83/2.38 <<< 5.83/2.38 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.38 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.38 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.38 / ~ Div: (Integer, Integer) -> Integer 5.83/2.38 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.38 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.38 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.38 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.38 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.38 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.38 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.38 + ~ Add: (Integer, Integer) -> Integer 5.83/2.38 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.38 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.38 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.38 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.38 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.38 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.38 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.38 >>> 5.83/2.38 5.83/2.38 5.83/2.38 The following domains are used: 5.83/2.38 Integer 5.83/2.38 5.83/2.38 R is empty. 5.83/2.38 5.83/2.38 The integer pair graph contains the following rules and edges: 5.83/2.38 (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) 5.83/2.38 (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) 5.83/2.38 5.83/2.38 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.83/2.38 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.83/2.38 5.83/2.38 The set Q consists of the following terms: 5.83/2.38 mult(x0, x1) 5.83/2.38 Cond_mult(TRUE, x0, x1) 5.83/2.38 Cond_mult1(TRUE, x0, x1) 5.83/2.38 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (9) IDPNonInfProof (SOUND) 5.83/2.38 Used the following options for this NonInfProof: 5.83/2.38 5.83/2.38 IDPGPoloSolver: 5.83/2.38 Range: [(-1,2)] 5.83/2.38 IsNat: false 5.83/2.38 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@44ec7e65 5.83/2.38 Constraint Generator: NonInfConstraintGenerator: 5.83/2.38 PathGenerator: MetricPathGenerator: 5.83/2.38 Max Left Steps: 1 5.83/2.38 Max Right Steps: 1 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 The constraints were generated the following way: 5.83/2.38 5.83/2.38 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.83/2.38 5.83/2.38 Note that final constraints are written in bold face. 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 For Pair COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) the following chains were created: 5.83/2.38 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) which results in the following constraint: 5.83/2.38 5.83/2.38 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.83/2.38 5.83/2.38 (2) (>(x[0], 0)=TRUE ==> COND_MULT(TRUE, x[0], y[0])_>=_NonInfC & COND_MULT(TRUE, x[0], y[0])_>=_MULT(-(x[0], 1), y[0]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.38 5.83/2.38 (3) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.38 5.83/2.38 (4) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.38 5.83/2.38 (5) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.38 5.83/2.38 (6) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 For Pair MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) the following chains were created: 5.83/2.38 *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: 5.83/2.38 5.83/2.38 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.83/2.38 5.83/2.38 (2) (>(x[0], 0)=TRUE ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.83/2.38 5.83/2.38 (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.83/2.38 5.83/2.38 (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.83/2.38 5.83/2.38 (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.83/2.38 5.83/2.38 (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 To summarize, we get the following constraints P__>=_ for the following pairs. 5.83/2.38 5.83/2.38 *COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) 5.83/2.38 5.83/2.38 *(x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 *MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) 5.83/2.38 5.83/2.38 *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 5.83/2.38 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. 5.83/2.38 5.83/2.38 Using the following integer polynomial ordering the resulting constraints can be solved 5.83/2.38 5.83/2.38 Polynomial interpretation over integers[POLO]: 5.83/2.38 5.83/2.38 POL(TRUE) = 0 5.83/2.38 POL(FALSE) = 0 5.83/2.38 POL(COND_MULT(x_1, x_2, x_3)) = [-1] + [2]x_2 5.83/2.38 POL(MULT(x_1, x_2)) = [1] + [2]x_1 5.83/2.38 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.83/2.38 POL(1) = [1] 5.83/2.38 POL(>(x_1, x_2)) = [2] 5.83/2.38 POL(0) = 0 5.83/2.38 5.83/2.38 5.83/2.38 The following pairs are in P_>: 5.83/2.38 5.83/2.38 5.83/2.38 MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) 5.83/2.38 5.83/2.38 5.83/2.38 The following pairs are in P_bound: 5.83/2.38 5.83/2.38 5.83/2.38 COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) 5.83/2.38 MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) 5.83/2.38 5.83/2.38 5.83/2.38 The following pairs are in P_>=: 5.83/2.38 5.83/2.38 5.83/2.38 COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) 5.83/2.38 5.83/2.38 5.83/2.38 There are no usable rules. 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (10) 5.83/2.38 Obligation: 5.83/2.38 IDP problem: 5.83/2.38 The following function symbols are pre-defined: 5.83/2.38 <<< 5.83/2.38 & ~ Bwand: (Integer, Integer) -> Integer 5.83/2.38 >= ~ Ge: (Integer, Integer) -> Boolean 5.83/2.38 | ~ Bwor: (Integer, Integer) -> Integer 5.83/2.38 / ~ Div: (Integer, Integer) -> Integer 5.83/2.38 != ~ Neq: (Integer, Integer) -> Boolean 5.83/2.38 && ~ Land: (Boolean, Boolean) -> Boolean 5.83/2.38 ! ~ Lnot: (Boolean) -> Boolean 5.83/2.38 = ~ Eq: (Integer, Integer) -> Boolean 5.83/2.38 <= ~ Le: (Integer, Integer) -> Boolean 5.83/2.38 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.83/2.38 % ~ Mod: (Integer, Integer) -> Integer 5.83/2.38 + ~ Add: (Integer, Integer) -> Integer 5.83/2.38 > ~ Gt: (Integer, Integer) -> Boolean 5.83/2.38 < ~ Lt: (Integer, Integer) -> Boolean 5.83/2.38 || ~ Lor: (Boolean, Boolean) -> Boolean 5.83/2.38 - ~ Sub: (Integer, Integer) -> Integer 5.83/2.38 - ~ UnaryMinus: (Integer) -> Integer 5.83/2.38 ~ ~ Bwnot: (Integer) -> Integer 5.83/2.38 * ~ Mul: (Integer, Integer) -> Integer 5.83/2.38 >>> 5.83/2.38 5.83/2.38 5.83/2.38 The following domains are used: 5.83/2.38 Integer 5.83/2.38 5.83/2.38 R is empty. 5.83/2.38 5.83/2.38 The integer pair graph contains the following rules and edges: 5.83/2.38 (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) 5.83/2.38 5.83/2.38 5.83/2.38 The set Q consists of the following terms: 5.83/2.38 mult(x0, x1) 5.83/2.38 Cond_mult(TRUE, x0, x1) 5.83/2.38 Cond_mult1(TRUE, x0, x1) 5.83/2.38 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (11) IDependencyGraphProof (EQUIVALENT) 5.83/2.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.83/2.38 ---------------------------------------- 5.83/2.38 5.83/2.38 (12) 5.83/2.38 TRUE 5.83/2.40 EOF