/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.itrs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 415 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 391 ms] (10) IDP (11) IDependencyGraphProof [EQUIVALENT, 0 ms] (12) IDP (13) IDPNonInfProof [SOUND, 124 ms] (14) IDP (15) IDependencyGraphProof [EQUIVALENT, 0 ms] (16) 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(x, y, z) -> Cond_eval(x > z, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval1(y > z && x > z, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval2(x > z && z >= x && y > z, x, y, z) Cond_eval2(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval3(y > z && z >= x, x, y, z) Cond_eval3(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval4(x > z && z >= x && z >= y, x, y, z) Cond_eval4(TRUE, x, y, z) -> eval(x, y, z) eval(x, y, z) -> Cond_eval5(y > z && z >= x && z >= y, x, y, z) Cond_eval5(TRUE, x, y, z) -> eval(x, y, z) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (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: Integer, Boolean The ITRS R consists of the following rules: eval(x, y, z) -> Cond_eval(x > z, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval1(y > z && x > z, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval2(x > z && z >= x && y > z, x, y, z) Cond_eval2(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval3(y > z && z >= x, x, y, z) Cond_eval3(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval4(x > z && z >= x && z >= y, x, y, z) Cond_eval4(TRUE, x, y, z) -> eval(x, y, z) eval(x, y, z) -> Cond_eval5(y > z && z >= x && z >= y, x, y, z) Cond_eval5(TRUE, x, y, z) -> eval(x, y, z) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] > z[4] && z[4] >= x[4] && y[4] > z[4], x[4], y[4], z[4]) (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5] - 1, z[5]) (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) (8): EVAL(x[8], y[8], z[8]) -> COND_EVAL4(x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8], x[8], y[8], z[8]) (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) (10): EVAL(x[10], y[10], z[10]) -> COND_EVAL5(y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10], x[10], y[10], z[10]) (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8] & z[1] ->^* z[8]) (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10] & z[1] ->^* z[10]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8] & z[3] ->^* z[8]) (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10] & z[3] ->^* z[10]) (4) -> (5), if (x[4] > z[4] && z[4] >= x[4] && y[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0] & z[5] ->^* z[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2] & z[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4] & z[5] ->^* z[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6] & z[5] ->^* z[6]) (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8] & z[5] ->^* z[8]) (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10] & z[5] ->^* z[10]) (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4] & z[7] ->^* z[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8] & z[7] ->^* z[8]) (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10] & z[7] ->^* z[10]) (8) -> (9), if (x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9] & z[8] ->^* z[9]) (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4] & z[9] ->^* z[4]) (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6] & z[9] ->^* z[6]) (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8] & z[9] ->^* z[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10] & z[9] ->^* z[10]) (10) -> (11), if (y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11] & z[10] ->^* z[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4] & z[11] ->^* z[4]) (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6] & z[11] ->^* z[6]) (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8] & z[11] ->^* z[8]) (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10] & z[11] ->^* z[10]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (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: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] > z[4] && z[4] >= x[4] && y[4] > z[4], x[4], y[4], z[4]) (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5] - 1, z[5]) (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) (8): EVAL(x[8], y[8], z[8]) -> COND_EVAL4(x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8], x[8], y[8], z[8]) (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) (10): EVAL(x[10], y[10], z[10]) -> COND_EVAL5(y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10], x[10], y[10], z[10]) (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8] & z[1] ->^* z[8]) (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10] & z[1] ->^* z[10]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8] & z[3] ->^* z[8]) (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10] & z[3] ->^* z[10]) (4) -> (5), if (x[4] > z[4] && z[4] >= x[4] && y[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0] & z[5] ->^* z[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2] & z[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4] & z[5] ->^* z[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6] & z[5] ->^* z[6]) (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8] & z[5] ->^* z[8]) (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10] & z[5] ->^* z[10]) (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4] & z[7] ->^* z[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8] & z[7] ->^* z[8]) (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10] & z[7] ->^* z[10]) (8) -> (9), if (x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9] & z[8] ->^* z[9]) (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4] & z[9] ->^* z[4]) (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6] & z[9] ->^* z[6]) (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8] & z[9] ->^* z[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10] & z[9] ->^* z[10]) (10) -> (11), if (y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11] & z[10] ->^* z[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4] & z[11] ->^* z[4]) (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6] & z[11] ->^* z[6]) (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8] & z[11] ->^* z[8]) (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10] & z[11] ->^* z[10]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (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@5586eb06 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(x, y, z) -> COND_EVAL(>(x, z), x, y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rule (III) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL1(&&(>(y, z), >(x, z)), x, y, z) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [(-1)bni_44]x[2] + [bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]x[2] + [bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [(-1)bni_44]x[2] + [bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [(-1)bni_46]x[2] + [bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]x[2] + [bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [(-1)bni_46]x[2] + [bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL2(&&(&&(>(x, z), >=(z, x)), >(y, z)), x, y, z) the following chains were created: *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) which results in the following constraint: (1) (&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], z[4])=TRUE & >(x[4], z[4])=TRUE & >=(z[4], x[4])=TRUE ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] >= 0 & [(-1)bso_49] + y[4] + x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] >= 0 & [(-1)bso_49] + y[4] + x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] >= 0 & [(-1)bso_49] + y[4] + x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL2(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) which results in the following constraint: (1) (&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], -(y[5], 1), z[5]) & (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], z[4])=TRUE & >(x[4], z[4])=TRUE & >=(z[4], x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4], z[4])_>=_EVAL(x[4], -(y[4], 1), z[4]) & (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-1)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-1]y[4] + [-1]x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-1)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-1]y[4] + [-1]x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-1)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-1]y[4] + [-1]x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y, z) -> COND_EVAL3(&&(>(y, z), >=(z, x)), x, y, z) the following chains were created: *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]x[6] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [bni_52]x[6] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]x[6] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) For Pair COND_EVAL3(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_EVAL(x[7], -(y[7], 1), z[7]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_EVAL(x[6], -(y[6], 1), z[6]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]x[6] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]x[6] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]x[6] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL4(&&(&&(>(x, z), >=(z, x)), >=(z, y)), x, y, z) the following chains were created: *We consider the chain EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]), COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) which results in the following constraint: (1) (&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8]))=TRUE & x[8]=x[9] & y[8]=y[9] & z[8]=z[9] ==> EVAL(x[8], y[8], z[8])_>=_NonInfC & EVAL(x[8], y[8], z[8])_>=_COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(z[8], y[8])=TRUE & >(x[8], z[8])=TRUE & >=(z[8], x[8])=TRUE ==> EVAL(x[8], y[8], z[8])_>=_NonInfC & EVAL(x[8], y[8], z[8])_>=_COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] >= 0 & [(-1)bso_57] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] >= 0 & [(-1)bso_57] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] >= 0 & [(-1)bso_57] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL4(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (x[9]=x[0] & y[9]=y[0] & z[9]=z[0] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (x[9]=x[2] & y[9]=y[2] & z[9]=z[2] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) which results in the following constraint: (1) (x[9]=x[4] & y[9]=y[4] & z[9]=z[4] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: (1) (x[9]=x[6] & y[9]=y[6] & z[9]=z[6] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) which results in the following constraint: (1) (x[9]=x[8] & y[9]=y[8] & z[9]=z[8] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) which results in the following constraint: (1) (x[9]=x[10] & y[9]=y[10] & z[9]=z[10] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL5(&&(&&(>(y, z), >=(z, x)), >=(z, y)), x, y, z) the following chains were created: *We consider the chain EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]), COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) which results in the following constraint: (1) (&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10]))=TRUE & x[10]=x[11] & y[10]=y[11] & z[10]=z[11] ==> EVAL(x[10], y[10], z[10])_>=_NonInfC & EVAL(x[10], y[10], z[10])_>=_COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(z[10], y[10])=TRUE & >(y[10], z[10])=TRUE & >=(z[10], x[10])=TRUE ==> EVAL(x[10], y[10], z[10])_>=_NonInfC & EVAL(x[10], y[10], z[10])_>=_COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] >= 0 & [-1 + (-1)bso_61] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] >= 0 & [-1 + (-1)bso_61] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] >= 0 & [-1 + (-1)bso_61] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL5(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (x[11]=x[0] & y[11]=y[0] & z[11]=z[0] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (x[11]=x[2] & y[11]=y[2] & z[11]=z[2] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) which results in the following constraint: (1) (x[11]=x[4] & y[11]=y[4] & z[11]=z[4] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: (1) (x[11]=x[6] & y[11]=y[6] & z[11]=z[6] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) which results in the following constraint: (1) (x[11]=x[8] & y[11]=y[8] & z[11]=z[8] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) which results in the following constraint: (1) (x[11]=x[10] & y[11]=y[10] & z[11]=z[10] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y, z) -> COND_EVAL(>(x, z), x, y, z) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) *EVAL(x, y, z) -> COND_EVAL1(&&(>(y, z), >(x, z)), x, y, z) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]x[2] + [bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [(-1)bni_44]x[2] + [bni_44]z[2] >= 0 & [(-1)bso_45] >= 0) *COND_EVAL1(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]x[2] + [bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [(-1)bni_46]x[2] + [bni_46]z[2] >= 0 & [(-1)bso_47] >= 0) *EVAL(x, y, z) -> COND_EVAL2(&&(&&(>(x, z), >=(z, x)), >(y, z)), x, y, z) *COND_EVAL2(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) *EVAL(x, y, z) -> COND_EVAL3(&&(>(y, z), >=(z, x)), x, y, z) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [bni_52]x[6] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]x[6] + [(-1)bni_52]z[6] >= 0 & [(-1)bso_53] >= 0) *COND_EVAL3(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]x[6] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]x[6] + [(-1)bni_54]z[6] >= 0 & [(-1)bso_55] >= 0) *EVAL(x, y, z) -> COND_EVAL4(&&(&&(>(x, z), >=(z, x)), >=(z, y)), x, y, z) *COND_EVAL4(TRUE, x, y, z) -> EVAL(x, y, z) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) *EVAL(x, y, z) -> COND_EVAL5(&&(&&(>(y, z), >=(z, x)), >=(z, y)), x, y, z) *COND_EVAL5(TRUE, x, y, z) -> EVAL(x, y, z) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 0) *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [1 + (-1)bso_63] >= 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(x_1, x_2, x_3)) = [-1] + [-1]x_3 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 POL(>(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(COND_EVAL2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] POL(COND_EVAL3(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_1 POL(COND_EVAL4(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [2]x_1 POL(COND_EVAL5(x_1, x_2, x_3, x_4)) = [-1]x_4 + [2]x_1 The following pairs are in P_>: EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) The following pairs are in P_bound: EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 <-> 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: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6] & z[9] ->^* z[6]) (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6] & z[11] ->^* z[6]) (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 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: (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (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@5586eb06 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_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) the following chains were created: *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & x[7]=x[0] & -(y[7], 1)=y[0] & z[7]=z[0] ==> COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_EVAL(x[7], -(y[7], 1), z[7]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_EVAL(x[6], -(y[6], 1), z[6]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & x[7]=x[2] & -(y[7], 1)=y[2] & z[7]=z[2] ==> COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_EVAL(x[7], -(y[7], 1), z[7]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_EVAL(x[6], -(y[6], 1), z[6]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & x[7]=x[6]1 & -(y[7], 1)=y[6]1 & z[7]=z[6]1 ==> COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_EVAL(x[7], -(y[7], 1), z[7]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_EVAL(x[6], -(y[6], 1), z[6]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[6] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) For Pair EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) the following chains were created: *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) which results in the following constraint: (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[6] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[6] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]z[6] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[0] & y[3]=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 & z[3]=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[6] & y[3]=y[6] & z[3]=z[6] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[2] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[6] & y[1]=y[6] & z[1]=z[6] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]y[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] + [bni_35]y[0] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] + [bni_35]y[0] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] + [bni_35]y[0] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]z[0] >= 0 & [(-1)bso_36] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] >= 0 & [(-1)bso_36] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)Bound*bni_25] + [bni_25]y[6] >= 0 & [1 + (-1)bso_26] >= 0) *EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_27] + [bni_27]y[6] >= 0 & [(-1)bso_28] >= 0) *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_29] + [bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_31] + [bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]z[0] >= 0 & [(-1)bso_34] >= 0) *EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]z[0] >= 0 & [(-1)bso_36] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_35] = 0 & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[0] >= 0 & [(-1)bso_36] >= 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(COND_EVAL3(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [-1] POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 The following pairs are in P_>: COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) The following pairs are in P_bound: COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) The following pairs are in P_>=: EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 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) 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: (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (12) 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: (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (13) 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@5586eb06 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(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [(-1)bso_20] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [(-1)bso_20] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[0] & y[3]=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 & z[3]=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [(-1)bso_20] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [(-1)bso_20] >= 0) *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [1 + (-1)bso_22] >= 0) *EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 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(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) The following pairs are in P_bound: EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) The following pairs are in P_>=: EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 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 ---------------------------------------- (14) 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: (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) Cond_eval2(TRUE, x0, x1, x2) Cond_eval3(TRUE, x0, x1, x2) Cond_eval4(TRUE, x0, x1, x2) Cond_eval5(TRUE, x0, x1, x2) ---------------------------------------- (15) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (16) TRUE