/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 327 ms] (6) AND (7) IDP (8) IDependencyGraphProof [EQUIVALENT, 0 ms] (9) IDP (10) IDPNonInfProof [SOUND, 51 ms] (11) AND (12) IDP (13) IDependencyGraphProof [EQUIVALENT, 0 ms] (14) TRUE (15) IDP (16) IDependencyGraphProof [EQUIVALENT, 0 ms] (17) TRUE (18) IDP (19) IDPNonInfProof [SOUND, 62 ms] (20) AND (21) IDP (22) IDependencyGraphProof [EQUIVALENT, 0 ms] (23) TRUE (24) IDP (25) IDependencyGraphProof [EQUIVALENT, 0 ms] (26) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval_1(x, y, z) -> Cond_eval_1(x > y, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) eval_2(x, y, z) -> Cond_eval_2(x > z, x, y, z) Cond_eval_2(TRUE, x, y, z) -> eval_1(x, y + 1, z) eval_2(x, y, z) -> Cond_eval_21(x > z, x, y, z) Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z + 1) eval_2(x, y, z) -> Cond_eval_22(z >= x, x, y, z) Cond_eval_22(TRUE, x, y, z) -> eval_1(x - 1, y, z) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(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 The ITRS R consists of the following rules: eval_1(x, y, z) -> Cond_eval_1(x > y, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) eval_2(x, y, z) -> Cond_eval_2(x > z, x, y, z) Cond_eval_2(TRUE, x, y, z) -> eval_1(x, y + 1, z) eval_2(x, y, z) -> Cond_eval_21(x > z, x, y, z) Cond_eval_21(TRUE, x, y, z) -> eval_1(x, y, z + 1) eval_2(x, y, z) -> Cond_eval_22(z >= x, x, y, z) Cond_eval_22(TRUE, x, y, z) -> eval_1(x - 1, y, z) The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5] + 1) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (2) -> (3), if (x[2] > z[2] & 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]) (4) -> (5), if (x[4] > z[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] + 1 ->^* z[0]) (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(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 R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5] + 1) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (2) -> (3), if (x[2] > z[2] & 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]) (4) -> (5), if (x[4] > z[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] + 1 ->^* z[0]) (6) -> (7), if (z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(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@7ddf04ea Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_1(x, y, z) -> COND_EVAL_1(>(x, y), x, y, z) the following chains were created: *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[0] + [bni_33]x[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[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(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[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 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [(-1)bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) For Pair COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) the following chains were created: *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: (1) (x[1]=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-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_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) which results in the following constraint: (1) (x[1]=x[4] & y[1]=y[4] & z[1]=z[4] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-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_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) which results in the following constraint: (1) (x[1]=x[6] & y[1]=y[6] & z[1]=z[6] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-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_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) For Pair EVAL_2(x, y, z) -> COND_EVAL_2(>(x, z), x, y, z) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) which results in the following constraint: (1) (>(x[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_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) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[2] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) For Pair COND_EVAL_2(TRUE, x, y, z) -> EVAL_1(x, +(y, 1), z) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[2], z[2])=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_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_1(x[3], +(y[3], 1), z[3]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[2], z[2])=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_1(x[2], +(y[2], 1), z[2]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[2] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) For Pair EVAL_2(x, y, z) -> COND_EVAL_21(>(x, z), x, y, z) the following chains were created: *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)) which results in the following constraint: (1) (>(x[4], z[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[4], z[4])=TRUE ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>(x[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) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)bni_41 + (-1)Bound*bni_41] + [(-1)bni_41]z[4] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) (9) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) For Pair COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, +(z, 1)) the following chains were created: *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[4], z[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[0] & y[5]=y[0] & +(z[5], 1)=z[0] ==> COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], +(z[5], 1)) & (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[4], z[4])=TRUE ==> COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_EVAL_1(x[4], y[4], +(z[4], 1)) & (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[4] + [-1] + [-1]z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]z[4] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) (9) (x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) For Pair EVAL_2(x, y, z) -> COND_EVAL_22(>=(z, x), x, y, z) the following chains were created: *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) which results in the following constraint: (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(z[6], x[6])=TRUE ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(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) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] + [bni_45]x[6] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) For Pair COND_EVAL_22(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z) the following chains were created: *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & -(x[7], 1)=x[0] & y[7]=y[0] & z[7]=z[0] ==> COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_EVAL_1(-(x[7], 1), y[7], z[7]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(z[6], x[6])=TRUE ==> COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_EVAL_1(-(x[6], 1), y[6], z[6]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] + [bni_47]x[6] >= 0 & [1 + (-1)bso_48] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x, y, z) -> COND_EVAL_1(>(x, y), x, y, z) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [(-1)bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_33] = 0 & [(-1)Bound*bni_33] + [bni_33]y[0] + [bni_33]x[0] >= 0 & [(-1)bso_34] >= 0) *COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) *EVAL_2(x, y, z) -> COND_EVAL_2(>(x, z), x, y, z) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_37] + [bni_37]x[2] >= 0 & [(-1)bso_38] >= 0) *COND_EVAL_2(TRUE, x, y, z) -> EVAL_1(x, +(y, 1), z) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_39] + [bni_39]x[2] >= 0 & [(-1)bso_40] >= 0) *EVAL_2(x, y, z) -> COND_EVAL_21(>(x, z), x, y, z) *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4])), >=) & 0 = 0 & [(-1)Bound*bni_41] + [bni_41]x[4] >= 0 & [(-1)bso_42] >= 0) *COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, +(z, 1)) *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) *(x[4] >= 0 & z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], +(z[5], 1))), >=) & 0 = 0 & [(-1)Bound*bni_43] + [bni_43]x[4] >= 0 & [1 + (-1)bso_44] >= 0) *EVAL_2(x, y, z) -> COND_EVAL_22(>=(z, x), x, y, z) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & 0 = 0 & [(-1)bni_45 + (-1)Bound*bni_45] + [(-1)bni_45]z[6] >= 0 & [(-1)bso_46] >= 0) *COND_EVAL_22(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & 0 = 0 & [(-1)bni_47 + (-1)Bound*bni_47] + [(-1)bni_47]z[6] >= 0 & [1 + (-1)bso_48] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(>(x_1, x_2)) = [-1] POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(COND_EVAL_22(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(>=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 The following pairs are in P_>: COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)) COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) The following pairs are in P_bound: EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], +(z[5], 1)) The following pairs are in P_>=: EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>(x[4], z[4]), x[4], y[4], z[4]) EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) There are no usable rules. ---------------------------------------- (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: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4], x[4], y[4], z[4]) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0] & z[3] ->^* z[0]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (8) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (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: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], 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] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (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_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (10) 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@7ddf04ea 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_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[2], z[2])=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_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_1(x[3], +(y[3], 1), z[3]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[2], z[2])=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_1(x[2], +(y[2], 1), z[2]) & (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 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[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) For Pair EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) which results in the following constraint: (1) (>(x[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_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) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[2] + [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) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[2] + [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) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) (9) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [(-1)bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) For Pair COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) the following chains were created: *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: (1) (x[1]=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 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_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [(-1)bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_1(x[3], +(y[3], 1), z[3])), >=) & [(-1)bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]z[2] + [bni_22]x[2] >= 0 & [1 + (-1)bso_23] >= 0) *EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) *(x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_24] = 0 & [(-1)Bound*bni_24] + [(-1)bni_24]z[2] + [bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [(-1)bso_29] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 POL(>(x_1, x_2)) = [-1] POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 The following pairs are in P_>: COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], +(y[3], 1), z[3]) The following pairs are in P_bound: EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) The following pairs are in P_>=: EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(x[2], z[2]), x[2], y[2], z[2]) COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) There are no usable rules. ---------------------------------------- (11) Complex Obligation (AND) ---------------------------------------- (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: Integer R is empty. The integer pair graph contains the following rules and edges: (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (13) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (14) TRUE ---------------------------------------- (15) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3], y[3] + 1, z[3]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2], x[2], y[2], z[2]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (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_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (16) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (17) TRUE ---------------------------------------- (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 R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) (7) -> (0), if (x[7] - 1 ->^* x[0] & y[7] ->^* y[0] & z[7] ->^* z[0]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (6) -> (7), if (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_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(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@7ddf04ea Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: *We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[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]y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[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_1(>(x[0], y[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 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[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 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[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_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) the following chains were created: *We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) which results in the following constraint: (1) (x[1]=x[6] & y[1]=y[6] & z[1]=z[6] ==> COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) For Pair EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) the following chains were created: *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) which results in the following constraint: (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(z[6], x[6])=TRUE ==> EVAL_2(x[6], y[6], z[6])_>=_NonInfC & EVAL_2(x[6], y[6], z[6])_>=_COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL_22(>=(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) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[6] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) For Pair COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) the following chains were created: *We consider the chain EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]), COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>=(z[6], x[6])=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] & -(x[7], 1)=x[0] & y[7]=y[0] & z[7]=z[0] ==> COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL_22(TRUE, x[7], y[7], z[7])_>=_EVAL_1(-(x[7], 1), y[7], z[7]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(z[6], x[6])=TRUE ==> COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL_22(TRUE, x[6], y[6], z[6])_>=_EVAL_1(-(x[6], 1), y[6], z[6]) & (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[6] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) (9) (z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[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 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [(-1)bso_24] >= 0) *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0) *EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6])), >=) & [(-1)bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[6] >= 0 & [(-1)bso_28] >= 0) *COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) *(z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL_1(-(x[7], 1), y[7], z[7])), >=) & [(-1)bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[6] >= 0 & [1 + (-1)bso_30] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 POL(>(x_1, x_2)) = [2] POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 POL(COND_EVAL_22(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 POL(>=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(-(x[7], 1), y[7], z[7]) The following pairs are in P_bound: EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) The following pairs are in P_>=: EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], y[0]), x[0], y[0], z[0]) COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(>=(z[6], x[6]), x[6], y[6], z[6]) There are no usable rules. ---------------------------------------- (20) Complex Obligation (AND) ---------------------------------------- (21) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > y[0], x[0], y[0], z[0]) (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) The set Q consists of the following terms: eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (22) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (23) TRUE ---------------------------------------- (24) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) (6): EVAL_2(x[6], y[6], z[6]) -> COND_EVAL_22(z[6] >= x[6], x[6], y[6], z[6]) (7): COND_EVAL_22(TRUE, x[7], y[7], z[7]) -> EVAL_1(x[7] - 1, y[7], z[7]) (1) -> (6), if (x[1] ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) (6) -> (7), if (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_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) eval_2(x0, x1, x2) Cond_eval_2(TRUE, x0, x1, x2) Cond_eval_21(TRUE, x0, x1, x2) Cond_eval_22(TRUE, x0, x1, x2) ---------------------------------------- (25) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (26) TRUE