/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 327 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 64 ms] (10) AND (11) IDP (12) IDependencyGraphProof [EQUIVALENT, 0 ms] (13) IDP (14) IDPNonInfProof [SOUND, 65 ms] (15) AND (16) IDP (17) IDependencyGraphProof [EQUIVALENT, 0 ms] (18) TRUE (19) IDP (20) IDependencyGraphProof [EQUIVALENT, 0 ms] (21) TRUE (22) IDP (23) IDependencyGraphProof [EQUIVALENT, 0 ms] (24) IDP (25) IDPNonInfProof [SOUND, 18 ms] (26) IDP (27) IDependencyGraphProof [EQUIVALENT, 0 ms] (28) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x > 0 && y > 0 && x > y, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x, y) eval_1(x, y) -> Cond_eval_11(x > 0 && y > 0 && y >= x, x, y) Cond_eval_11(TRUE, x, y) -> eval_3(x, y) eval_2(x, y) -> Cond_eval_2(x > 0, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y) eval_2(x, y) -> Cond_eval_21(0 >= x, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x, y) eval_3(x, y) -> Cond_eval_3(y > 0, x, y) Cond_eval_3(TRUE, x, y) -> eval_3(x, y - 1) eval_3(x, y) -> Cond_eval_31(0 >= y, x, y) Cond_eval_31(TRUE, x, y) -> eval_1(x, y) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x > 0 && y > 0 && x > y, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x, y) eval_1(x, y) -> Cond_eval_11(x > 0 && y > 0 && y >= x, x, y) Cond_eval_11(TRUE, x, y) -> eval_3(x, y) eval_2(x, y) -> Cond_eval_2(x > 0, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y) eval_2(x, y) -> Cond_eval_21(0 >= x, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x, y) eval_3(x, y) -> Cond_eval_3(y > 0, x, y) Cond_eval_3(TRUE, x, y) -> eval_3(x, y - 1) eval_3(x, y) -> Cond_eval_31(0 >= y, x, y) Cond_eval_31(TRUE, x, y) -> eval_1(x, y) The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0 && y[0] > 0 && x[0] > y[0], x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) (5): COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(x[5] - 1, y[5]) (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (0) -> (1), if (x[0] > 0 && y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (4) -> (5), if (x[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (4), if (x[5] - 1 ->^* x[4] & y[5] ->^* y[4]) (5) -> (6), if (x[5] - 1 ->^* x[6] & y[5] ->^* y[6]) (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@12727a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_1(x, y) -> COND_EVAL_1(&&(&&(>(x, 0), >(y, 0)), >(x, y)), x, y) the following chains were created: *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE & >(y[0], 0)=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[0] >= 0 & [(-1)bso_40] + x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[0] >= 0 & [(-1)bso_40] + x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[0] >= 0 & [(-1)bso_40] + x[0] >= 0) For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) the following chains were created: *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: (1) (x[1]=x[6] & y[1]=y[6] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) For Pair EVAL_1(x, y) -> COND_EVAL_11(&&(&&(>(x, 0), >(y, 0)), >=(y, x)), x, y) the following chains were created: *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]x[2] >= 0 & [(-1)bso_44] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]x[2] >= 0 & [(-1)bso_44] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]x[2] >= 0 & [(-1)bso_44] >= 0) For Pair COND_EVAL_11(TRUE, x, y) -> EVAL_3(x, y) the following chains were created: *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: (1) (x[3]=x[8] & y[3]=y[8] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: (1) (x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_2(>(x, 0), x, y) the following chains were created: *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) which results in the following constraint: (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[4], 0)=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_2(>(x[4], 0), x[4], y[4]) & (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]x[4] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]x[4] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]x[4] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]x[4] >= 0 & [(-1)bso_48] >= 0) For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), y) the following chains were created: *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) which results in the following constraint: (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[4]1 & y[5]=y[4]1 ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]), COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]), EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) which results in the following constraint: (1) (>(x[4], 0)=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[6] & y[5]=y[6] ==> COND_EVAL_2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_2(TRUE, x[5], y[5])_>=_EVAL_2(-(x[5], 1), y[5]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[4], 0)=TRUE ==> COND_EVAL_2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_2(TRUE, x[4], y[4])_>=_EVAL_2(-(x[4], 1), y[4]) & (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_21(>=(0, x), x, y) the following chains were created: *We consider the chain EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]), COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) which results in the following constraint: (1) (>=(0, x[6])=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(0, x[6])=TRUE ==> EVAL_2(x[6], y[6])_>=_NonInfC & EVAL_2(x[6], y[6])_>=_COND_EVAL_21(>=(0, x[6]), x[6], y[6]) & (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [bni_51]x[6] >= 0 & [(-1)bso_52] + [-1]x[6] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [bni_51]x[6] >= 0 & [(-1)bso_52] + [-1]x[6] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & [(-1)bni_51 + (-1)Bound*bni_51] + [bni_51]x[6] >= 0 & [(-1)bso_52] + [-1]x[6] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & 0 = 0 & [(-1)bni_51 + (-1)Bound*bni_51] + [bni_51]x[6] >= 0 & [(-1)bso_52] + [-1]x[6] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & 0 = 0 & [(-1)bni_51 + (-1)Bound*bni_51] + [(-1)bni_51]x[6] >= 0 & [(-1)bso_52] + x[6] >= 0) For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: (1) (x[7]=x[0] & y[7]=y[0] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) *We consider the chain COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: (1) (x[7]=x[2] & y[7]=y[2] ==> COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_21(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL_21(TRUE, x[7], y[7])_>=_EVAL_1(x[7], y[7]) & (U^Increasing(EVAL_1(x[7], y[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) For Pair EVAL_3(x, y) -> COND_EVAL_3(>(y, 0), x, y) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [(2)bni_55]x[8] >= 0 & [(-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [(2)bni_55]x[8] >= 0 & [(-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_55 + (-1)Bound*bni_55] + [(2)bni_55]x[8] >= 0 & [(-1)bso_56] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(2)bni_55] = 0 & [(-1)bni_55 + (-1)Bound*bni_55] >= 0 & [(-1)bso_56] >= 0) For Pair COND_EVAL_3(TRUE, x, y) -> EVAL_3(x, -(y, 1)) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[8]1 & -(y[9], 1)=y[8]1 ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(2)bni_57] = 0 & [(-1)bni_57 + (-1)Bound*bni_57] >= 0 & [(-1)bso_58] >= 0) *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[10] & -(y[9], 1)=y[10] ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_57 + (-1)Bound*bni_57] + [(2)bni_57]x[8] >= 0 & [(-1)bso_58] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(2)bni_57] = 0 & [(-1)bni_57 + (-1)Bound*bni_57] >= 0 & [(-1)bso_58] >= 0) For Pair EVAL_3(x, y) -> COND_EVAL_31(>=(0, y), x, y) the following chains were created: *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [(2)bni_59]x[10] >= 0 & [(-1)bso_60] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [(2)bni_59]x[10] >= 0 & [(-1)bso_60] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_59 + (-1)Bound*bni_59] + [(2)bni_59]x[10] >= 0 & [(-1)bso_60] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(2)bni_59] = 0 & [(-1)bni_59 + (-1)Bound*bni_59] >= 0 & [(-1)bso_60] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(2)bni_59] = 0 & [(-1)bni_59 + (-1)Bound*bni_59] >= 0 & [(-1)bso_60] >= 0) For Pair COND_EVAL_31(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: (1) (x[11]=x[0] & y[11]=y[0] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: (1) (x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x, y) -> COND_EVAL_1(&&(&&(>(x, 0), >(y, 0)), >(x, y)), x, y) *(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[0] >= 0 & [(-1)bso_40] + x[0] >= 0) *COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_41] = 0 & [(-1)bso_42] >= 0) *EVAL_1(x, y) -> COND_EVAL_11(&&(&&(>(x, 0), >(y, 0)), >=(y, x)), x, y) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(2)bni_43]x[2] >= 0 & [(-1)bso_44] >= 0) *COND_EVAL_11(TRUE, x, y) -> EVAL_3(x, y) *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_45] = 0 & [(-1)bso_46] >= 0) *EVAL_2(x, y) -> COND_EVAL_2(>(x, 0), x, y) *(x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[4], 0), x[4], y[4])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]x[4] >= 0 & [(-1)bso_48] >= 0) *COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), y) *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) *(x[4] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[5], 1), y[5])), >=) & 0 = 0 & [(-1)bni_49 + (-1)Bound*bni_49] + [bni_49]x[4] >= 0 & [1 + (-1)bso_50] >= 0) *EVAL_2(x, y) -> COND_EVAL_21(>=(0, x), x, y) *(x[6] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, x[6]), x[6], y[6])), >=) & 0 = 0 & [(-1)bni_51 + (-1)Bound*bni_51] + [(-1)bni_51]x[6] >= 0 & [(-1)bso_52] + x[6] >= 0) *COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) *((U^Increasing(EVAL_1(x[7], y[7])), >=) & [bni_53] = 0 & [(-1)bso_54] >= 0) *EVAL_3(x, y) -> COND_EVAL_3(>(y, 0), x, y) *(y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(2)bni_55] = 0 & [(-1)bni_55 + (-1)Bound*bni_55] >= 0 & [(-1)bso_56] >= 0) *COND_EVAL_3(TRUE, x, y) -> EVAL_3(x, -(y, 1)) *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(2)bni_57] = 0 & [(-1)bni_57 + (-1)Bound*bni_57] >= 0 & [(-1)bso_58] >= 0) *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(2)bni_57] = 0 & [(-1)bni_57 + (-1)Bound*bni_57] >= 0 & [(-1)bso_58] >= 0) *EVAL_3(x, y) -> COND_EVAL_31(>=(0, y), x, y) *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(2)bni_59] = 0 & [(-1)bni_59 + (-1)Bound*bni_59] >= 0 & [(-1)bso_60] >= 0) *COND_EVAL_31(TRUE, x, y) -> EVAL_1(x, y) *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_61] = 0 & [(-1)bso_62] >= 0) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = [3] POL(FALSE) = [3] POL(EVAL_1(x_1, x_2)) = [-1] + [2]x_1 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + x_2 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(EVAL_2(x_1, x_2)) = [-1] + x_1 POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(>=(x_1, x_2)) = [-1] POL(EVAL_3(x_1, x_2)) = [-1] + [2]x_1 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(COND_EVAL_3(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + [2]x_2 The following pairs are in P_>: EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) The following pairs are in P_bound: EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(&&(>(x[0], 0), >(y[0], 0)), >(x[0], y[0])), x[0], y[0]) EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) COND_EVAL_2(TRUE, x[5], y[5]) -> EVAL_2(-(x[5], 1), y[5]) The following pairs are in P_>=: COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) EVAL_2(x[4], y[4]) -> COND_EVAL_2(>(x[4], 0), x[4], y[4]) EVAL_2(x[6], y[6]) -> COND_EVAL_21(>=(0, x[6]), x[6], y[6]) COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_2(x[4] > 0, x[4], y[4]) (6): EVAL_2(x[6], y[6]) -> COND_EVAL_21(0 >= x[6], x[6], y[6]) (7): COND_EVAL_21(TRUE, x[7], y[7]) -> EVAL_1(x[7], y[7]) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (7) -> (2), if (x[7] ->^* x[2] & y[7] ->^* y[2]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6]) (6) -> (7), if (0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (9) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@12727a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) the following chains were created: *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: (1) (x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_24] = 0 & [(-1)bso_25] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_24] = 0 & [(-1)bso_25] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_24] = 0 & [(-1)bso_25] >= 0) For Pair EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) the following chains were created: *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] + [-1]y[10] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] + [-1]y[10] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] + [-1]y[10] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] >= 0 & [(-1)bso_27] + [-1]y[10] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[10] >= 0 & [(-1)bso_27] + y[10] >= 0) For Pair COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[8]1 & -(y[9], 1)=y[8]1 ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28] = 0 & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] >= 0 & [1 + (-1)bso_29] >= 0) *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[10] & -(y[9], 1)=y[10] ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] + [(-1)bni_28]x[8] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28] = 0 & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] >= 0 & [1 + (-1)bso_29] >= 0) For Pair EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[8] + [(-1)bni_30]x[8] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[8] + [(-1)bni_30]x[8] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[8] + [(-1)bni_30]x[8] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_30] = 0 & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[8] >= 0 & [(-1)bso_31] >= 0) For Pair COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) the following chains were created: *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: (1) (x[3]=x[8] & y[3]=y[8] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: (1) (x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) For Pair EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) the following chains were created: *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]y[2] + [(-1)bni_34]x[2] >= 0 & [-1 + (-1)bso_35] + y[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]y[2] + [(-1)bni_34]x[2] >= 0 & [-1 + (-1)bso_35] + y[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]y[2] + [(-1)bni_34]x[2] >= 0 & [-1 + (-1)bso_35] + y[2] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) *((U^Increasing(EVAL_1(x[11], y[11])), >=) & [bni_24] = 0 & [(-1)bso_25] >= 0) *EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[10] >= 0 & [(-1)bso_27] + y[10] >= 0) *COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28] = 0 & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] >= 0 & [1 + (-1)bso_29] >= 0) *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_28] = 0 & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[8] >= 0 & [1 + (-1)bso_29] >= 0) *EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) *(y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_30] = 0 & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[8] >= 0 & [(-1)bso_31] >= 0) *COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *((U^Increasing(EVAL_3(x[3], y[3])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]y[2] + [(-1)bni_34]x[2] >= 0 & [-1 + (-1)bso_35] + y[2] >= 0) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [2] POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + [2]x_3 + [-1]x_2 POL(EVAL_1(x_1, x_2)) = [-1] + [2]x_2 + [-1]x_1 POL(EVAL_3(x_1, x_2)) = [-1] + x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 POL(COND_EVAL_3(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(>(x_1, x_2)) = [-1] POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) The following pairs are in P_bound: EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) The following pairs are in P_>=: COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (12) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (13) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (2) -> (3), if (x[2] > 0 && y[2] > 0 && y[2] >= x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (14) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@12727a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) the following chains were created: *We consider the chain COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) which results in the following constraint: (1) (x[3]=x[10] & y[3]=y[10] & >=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL_3(x[10], y[10])_>=_NonInfC & EVAL_3(x[10], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[10], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(0, y[10])=TRUE ==> EVAL_3(x[3], y[10])_>=_NonInfC & EVAL_3(x[3], y[10])_>=_COND_EVAL_31(>=(0, y[10]), x[3], y[10]) & (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[10] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[10] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[10] + [(-1)bni_20]x[3] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[10] >= 0 & [(-1)bso_21] + [-2]y[10] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[10] >= 0 & [(-1)bso_21] + [2]y[10] >= 0) For Pair COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) the following chains were created: *We consider the chain EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]), EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[10] & y[3]=y[10] ==> COND_EVAL_11(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_11(TRUE, x[3], y[3])_>=_EVAL_3(x[3], y[3]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_11(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_11(TRUE, x[2], y[2])_>=_EVAL_3(x[2], y[2]) & (U^Increasing(EVAL_3(x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) For Pair EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) the following chains were created: *We consider the chain COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]), COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) which results in the following constraint: (1) (x[11]=x[2] & y[11]=y[2] & &&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[2], x[2])=TRUE & >(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_1(x[2], y[2])_>=_NonInfC & EVAL_1(x[2], y[2])_>=_COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [-1 + (-1)bso_25] + y[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [-1 + (-1)bso_25] + y[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [-1 + (-1)bso_25] + y[2] >= 0) For Pair COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) the following chains were created: *We consider the chain EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]), COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]), EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) which results in the following constraint: (1) (>=(0, y[10])=TRUE & x[10]=x[11] & y[10]=y[11] & x[11]=x[2] & y[11]=y[2] ==> COND_EVAL_31(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL_31(TRUE, x[11], y[11])_>=_EVAL_1(x[11], y[11]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(0, y[10])=TRUE ==> COND_EVAL_31(TRUE, x[10], y[10])_>=_NonInfC & COND_EVAL_31(TRUE, x[10], y[10])_>=_EVAL_1(x[10], y[10]) & (U^Increasing(EVAL_1(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] + [(-1)bni_26]x[10] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) *(y[10] >= 0 ==> (U^Increasing(COND_EVAL_31(>=(0, y[10]), x[10], y[10])), >=) & [(-1)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[10] >= 0 & [(-1)bso_21] + [2]y[10] >= 0) *COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[3], y[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x[2] >= 0 & [(-1)bso_23] + y[2] >= 0) *EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [-1 + (-1)bso_25] + y[2] >= 0) *COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) *(y[10] >= 0 ==> (U^Increasing(EVAL_1(x[11], y[11])), >=) & [(-1)bni_26] = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[10] >= 0 & [(-1)bso_27] >= 0) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(EVAL_3(x_1, x_2)) = [-1] + [-1]x_2 + [-1]x_1 POL(COND_EVAL_31(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 POL(COND_EVAL_11(x_1, x_2, x_3)) = [-1] + [-1]x_2 + [-1]x_1 POL(EVAL_1(x_1, x_2)) = [-1] + x_2 + [-1]x_1 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) The following pairs are in P_bound: EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) The following pairs are in P_>=: EVAL_3(x[10], y[10]) -> COND_EVAL_31(>=(0, y[10]), x[10], y[10]) EVAL_1(x[2], y[2]) -> COND_EVAL_11(&&(&&(>(x[2], 0), >(y[2], 0)), >=(y[2], x[2])), x[2], y[2]) COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (15) Complex Obligation (AND) ---------------------------------------- (16) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (2): EVAL_1(x[2], y[2]) -> COND_EVAL_11(x[2] > 0 && y[2] > 0 && y[2] >= x[2], x[2], y[2]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (17) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (18) TRUE ---------------------------------------- (19) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (20) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (21) TRUE ---------------------------------------- (22) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (11): COND_EVAL_31(TRUE, x[11], y[11]) -> EVAL_1(x[11], y[11]) (10): EVAL_3(x[10], y[10]) -> COND_EVAL_31(0 >= y[10], x[10], y[10]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (3): COND_EVAL_11(TRUE, x[3], y[3]) -> EVAL_3(x[3], y[3]) (3) -> (8), if (x[3] ->^* x[8] & y[3] ->^* y[8]) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) (3) -> (10), if (x[3] ->^* x[10] & y[3] ->^* y[10]) (9) -> (10), if (x[9] ->^* x[10] & y[9] - 1 ->^* y[10]) (10) -> (11), if (0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (23) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (24) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) (9): COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], y[9] - 1) (9) -> (8), if (x[9] ->^* x[8] & y[9] - 1 ->^* y[8]) (8) -> (9), if (y[8] > 0 & x[8] ->^* x[9] & y[8] ->^* y[9]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (25) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@12727a Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> EVAL_3(x[8], y[8])_>=_NonInfC & EVAL_3(x[8], y[8])_>=_COND_EVAL_3(>(y[8], 0), x[8], y[8]) & (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[8] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[8] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[8] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[8] >= 0 & [(-1)bso_12] >= 0) For Pair COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) the following chains were created: *We consider the chain EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]), COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)), EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) which results in the following constraint: (1) (>(y[8], 0)=TRUE & x[8]=x[9] & y[8]=y[9] & x[9]=x[8]1 & -(y[9], 1)=y[8]1 ==> COND_EVAL_3(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL_3(TRUE, x[9], y[9])_>=_EVAL_3(x[9], -(y[9], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[8], 0)=TRUE ==> COND_EVAL_3(TRUE, x[8], y[8])_>=_NonInfC & COND_EVAL_3(TRUE, x[8], y[8])_>=_EVAL_3(x[8], -(y[8], 1)) & (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[8] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[8] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[8] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[8] >= 0 & [1 + (-1)bso_14] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) *(y[8] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_3(>(y[8], 0), x[8], y[8])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[8] >= 0 & [(-1)bso_12] >= 0) *COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) *(y[8] + [-1] >= 0 ==> (U^Increasing(EVAL_3(x[9], -(y[9], 1))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[8] >= 0 & [1 + (-1)bso_14] >= 0) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL_3(x_1, x_2)) = [-1] + x_2 POL(COND_EVAL_3(x_1, x_2, x_3)) = [-1] + x_3 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) The following pairs are in P_bound: EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) COND_EVAL_3(TRUE, x[9], y[9]) -> EVAL_3(x[9], -(y[9], 1)) The following pairs are in P_>=: EVAL_3(x[8], y[8]) -> COND_EVAL_3(>(y[8], 0), x[8], y[8]) There are no usable rules. ---------------------------------------- (26) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (8): EVAL_3(x[8], y[8]) -> COND_EVAL_3(y[8] > 0, x[8], y[8]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) Cond_eval_11(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) eval_3(x0, x1) Cond_eval_3(TRUE, x0, x1) Cond_eval_31(TRUE, x0, x1) ---------------------------------------- (27) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (28) TRUE