/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 470 ms] (6) AND (7) IDP (8) IDPNonInfProof [SOUND, 168 ms] (9) IDP (10) IDependencyGraphProof [EQUIVALENT, 0 ms] (11) IDP (12) IDPNonInfProof [SOUND, 17 ms] (13) IDP (14) IDependencyGraphProof [EQUIVALENT, 0 ms] (15) TRUE (16) IDP (17) IDependencyGraphProof [EQUIVALENT, 0 ms] (18) IDP (19) IDPNonInfProof [SOUND, 55 ms] (20) IDP (21) IDependencyGraphProof [EQUIVALENT, 0 ms] (22) 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 + y > z && z >= 0 && x > 0, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval1(x + y > z && z >= 0 && 0 >= x && y > 0, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval2(x + y > z && z >= 0 && 0 >= x && 0 >= y, x, y, z) Cond_eval2(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) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval(x, y, z) -> Cond_eval(x + y > z && z >= 0 && x > 0, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) eval(x, y, z) -> Cond_eval1(x + y > z && z >= 0 && 0 >= x && y > 0, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x, y - 1, z) eval(x, y, z) -> Cond_eval2(x + y > z && z >= 0 && 0 >= x && 0 >= y, x, y, z) Cond_eval2(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] + y[0] > z[0] && z[0] >= 0 && x[0] > 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(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4], z[4]) (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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]) (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & z[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & z[5] ->^* z[4]) 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) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4], z[4]) (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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]) (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (x[4] + y[4] > z[4] && z[4] >= 0 && 0 >= x[4] && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & z[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & z[5] ->^* z[4]) 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) ---------------------------------------- (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@256bb0f0 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, y), z), >=(z, 0)), >(x, 0)), x, y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(-2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[4] & y[1]=y[4] & z[1]=z[4] ==> 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL1(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >(y, 0)), x, y, z) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_29] + [(-1)bni_29]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_29] + [(-1)bni_29]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_29] + [(-1)bni_29]z[2] + [(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] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_29] + [(-1)bni_29]z[2] + [(2)bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & -(y[3], 1)=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=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], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(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] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL2(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >=(0, y)), x, y, z) the following chains were created: *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[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], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(0, y[4])=TRUE & >=(0, x[4])=TRUE & >(+(x[4], y[4]), z[4])=TRUE & >=(z[4], 0)=TRUE ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_33] + [(-1)bni_33]z[4] + [(2)bni_33]y[4] >= 0 & [1 + (-1)bso_34] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_33] + [(-1)bni_33]z[4] + [(2)bni_33]y[4] >= 0 & [1 + (-1)bso_34] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[4] >= 0 & [-1]x[4] >= 0 & x[4] + [-1] + y[4] + [-1]z[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_33] + [(-1)bni_33]z[4] + [(2)bni_33]y[4] >= 0 & [1 + (-1)bso_34] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL2(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (x[5]=x[0] & y[5]=y[0] & z[5]=z[0] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (x[5]=x[2] & y[5]=y[2] & z[5]=z[2] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) *We consider the chain COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) which results in the following constraint: (1) (x[5]=x[4] & y[5]=y[4] & z[5]=z[4] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], y[5], z[5]) & (U^Increasing(EVAL(x[5], y[5], z[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y, z) -> COND_EVAL(&&(&&(>(+(x, y), z), >=(z, 0)), >(x, 0)), x, y, z) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[0] + [(-2)bni_25]y[0] >= 0 & [(-1)bso_26] >= 0) *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(-2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_27] + [(-1)bni_27]z[0] + [(2)bni_27]y[0] >= 0 & [(-1)bso_28] >= 0) *EVAL(x, y, z) -> COND_EVAL1(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >(y, 0)), x, y, z) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_29] + [(-1)bni_29]z[2] + [(2)bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *COND_EVAL1(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-2)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[2] + [(2)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *EVAL(x, y, z) -> COND_EVAL2(&&(&&(&&(>(+(x, y), z), >=(z, 0)), >=(0, x)), >=(0, y)), x, y, z) *COND_EVAL2(TRUE, x, y, z) -> EVAL(x, y, z) *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-1)bso_36] >= 0) *((U^Increasing(EVAL(x[5], y[5], z[5])), >=) & [bni_35] = 0 & [3 + (-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) = [1] POL(FALSE) = [1] POL(EVAL(x_1, x_2, x_3)) = [-1]x_3 + [2]x_2 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1]x_4 + [2]x_3 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 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 + [2]x_3 + [-1]x_1 POL(COND_EVAL2(x_1, x_2, x_3, x_4)) = [1] + [-1]x_4 + [2]x_3 + [2]x_1 The following pairs are in P_>: EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) The following pairs are in P_bound: EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(&&(>(+(x[4], y[4]), z[4]), >=(z[4], 0)), >=(0, x[4])), >=(0, y[4])), x[4], y[4], z[4]) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & 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) ---------------------------------------- (8) 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@256bb0f0 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[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & -(y[3], 1)=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=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], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x[2] >= 0 & [(-1)bso_27] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(EVAL(x_1, x_2, x_3)) = [-1] + x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 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] + x_2 + [-1]x_1 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) The following pairs are in P_bound: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) 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 ---------------------------------------- (9) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0] & z[3] ->^* z[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & 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) ---------------------------------------- (10) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (11) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3] - 1, z[3]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0 & 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) ---------------------------------------- (12) 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@256bb0f0 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_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=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], -(y[3], 1), z[3]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]z[2] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]z[2] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]z[2] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]z[2] + [bni_15]y[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) which results in the following constraint: (1) (&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0))=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(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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], 0)=TRUE & >=(0, x[2])=TRUE & >(+(x[2], y[2]), z[2])=TRUE & >=(z[2], 0)=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), 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] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]z[2] + [bni_17]y[2] + [(-1)bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]z[2] + [bni_17]y[2] + [(-1)bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & [-1]x[2] >= 0 & x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]z[2] + [bni_17]y[2] + [(-1)bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]z[2] + [bni_17]y[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]z[2] + [bni_15]y[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) *(y[2] + [-1] >= 0 & x[2] >= 0 & [-1]x[2] + [-1] + y[2] + [-1]z[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]z[2] + [bni_17]y[2] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 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_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + x_4 + x_3 + [-1]x_2 POL(EVAL(x_1, x_2, x_3)) = [-1] + x_3 + x_2 + [-1]x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) The following pairs are in P_bound: COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], -(y[3], 1), z[3]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) The following pairs are in P_>=: EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(&&(&&(>(+(x[2], y[2]), z[2]), >=(z[2], 0)), >=(0, x[2])), >(y[2], 0)), x[2], y[2], z[2]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 -> TRUE^1 FALSE^1 -> &&(TRUE, FALSE)^1 ---------------------------------------- (13) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: 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(x[2] + y[2] > z[2] && z[2] >= 0 && 0 >= x[2] && y[2] > 0, x[2], y[2], z[2]) 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) ---------------------------------------- (14) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (15) TRUE ---------------------------------------- (16) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer 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] + y[0] > z[0] && z[0] >= 0 && x[0] > 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]) (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5], z[5]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 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) ---------------------------------------- (17) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (18) 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: (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] + y[0] > z[0] && z[0] >= 0 && x[0] > 0, x[0], y[0], z[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (0) -> (1), if (x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 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) ---------------------------------------- (19) 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@256bb0f0 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair COND_EVAL(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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(+(x[0], y[0]), z[0])=TRUE & >=(z[0], 0)=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) *EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 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) = [3] POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_2 + [-1]x_1 POL(EVAL(x_1, x_2, x_3)) = [-1] + x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) The following pairs are in P_bound: 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], y[0]), z[0]), >=(z[0], 0)), >(x[0], 0)), x[0], y[0], z[0]) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(&&(>(+(x[0], y[0]), z[0]), >=(z[0], 0)), >(x[0], 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 ---------------------------------------- (20) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] + y[0] > z[0] && z[0] >= 0 && x[0] > 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) ---------------------------------------- (21) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (22) TRUE