/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 289 ms] (6) AND (7) IDP (8) IDependencyGraphProof [EQUIVALENT, 0 ms] (9) IDP (10) IDPNonInfProof [SOUND, 100 ms] (11) IDP (12) IDependencyGraphProof [EQUIVALENT, 0 ms] (13) TRUE (14) IDP (15) IDependencyGraphProof [EQUIVALENT, 0 ms] (16) IDP (17) IDPNonInfProof [SOUND, 39 ms] (18) IDP (19) IDependencyGraphProof [EQUIVALENT, 0 ms] (20) 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 > z, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) eval_2(x, y, z) -> Cond_eval_2(y > z, x, y, z) Cond_eval_2(TRUE, x, y, z) -> eval_2(x - 1, y - 1, z) eval_2(x, y, z) -> Cond_eval_21(z >= y, x, y, z) Cond_eval_21(TRUE, x, y, z) -> eval_1(x, 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) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval_1(x, y, z) -> Cond_eval_1(x = y && x > z, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z) eval_2(x, y, z) -> Cond_eval_2(y > z, x, y, z) Cond_eval_2(TRUE, x, y, z) -> eval_2(x - 1, y - 1, z) eval_2(x, y, z) -> Cond_eval_21(z >= y, x, y, z) Cond_eval_21(TRUE, x, y, z) -> eval_1(x, 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] > z[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(y[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[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]) (0) -> (1), if (x[0] = y[0] && x[0] > z[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]) (2) -> (3), if (y[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (z[4] >= 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]) 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) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[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(y[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[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]) (0) -> (1), if (x[0] = y[0] && x[0] > z[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]) (2) -> (3), if (y[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (z[4] >= 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]) 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) ---------------------------------------- (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@526b3a00 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, z)), 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], z[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]), >(x[0], z[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], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[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], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(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], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [(-1)bni_30]y[0] + [(2)bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [(-1)bni_30]y[0] + [(2)bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [(-1)bni_30]y[0] + [(2)bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_30 + (-1)Bound*bni_30] + [bni_30]z[0] + [(-1)bni_30]y[0] + [(2)bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] + [(-1)bni_30]z[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 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(>(y[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_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 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(>=(z[4], y[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_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) For Pair EVAL_2(x, y, z) -> COND_EVAL_2(>(y, z), x, y, z) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) which results in the following constraint: (1) (>(y[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(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [(-1)bni_34]y[2] + [(2)bni_34]x[2] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [(-1)bni_34]y[2] + [(2)bni_34]x[2] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [(-1)bni_34]y[2] + [(2)bni_34]x[2] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-2)bni_34 + (-1)Bound*bni_34] + [(-2)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-2)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-2)bni_34 + (-1)Bound*bni_34] + [(-2)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) For Pair COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(-(x, 1), -(y, 1), z) the following chains were created: *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1 ==> COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[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_2(-(x[2], 1), -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) *We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) which results in the following constraint: (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] ==> COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[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_2(-(x[2], 1), -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] + [(2)bni_36]x[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) For Pair EVAL_2(x, y, z) -> COND_EVAL_21(>=(z, y), x, y, z) the following chains were created: *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[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]) which results in the following constraint: (1) (>=(z[4], y[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(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(z[4], y[4])=TRUE ==> EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [(-1)bni_38]y[4] + [(2)bni_38]x[4] >= 0 & [(-1)bso_39] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [(-1)bni_38]y[4] + [(2)bni_38]x[4] >= 0 & [(-1)bso_39] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [(-1)bni_38]y[4] + [(2)bni_38]x[4] >= 0 & [(-1)bso_39] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]z[4] + [(-1)bni_38]y[4] >= 0 & [(-1)bso_39] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(-2)bni_38]y[4] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]y[4] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(-2)bni_38]y[4] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) For Pair COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, z) the following chains were created: *We consider the chain COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[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_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]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (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]) & (U^Increasing(EVAL_1(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_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x, y, z) -> COND_EVAL_1(&&(=(x, y), >(x, z)), x, y, z) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0) *COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *EVAL_2(x, y, z) -> COND_EVAL_2(>(y, z), x, y, z) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-2)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(2)bni_34] = 0 & [(-2)bni_34 + (-1)Bound*bni_34] + [(-2)bni_34]z[2] + [(-1)bni_34]y[2] >= 0 & [(-1)bso_35] >= 0) *COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(-(x, 1), -(y, 1), z) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(2)bni_36] = 0 & [(-2)bni_36 + (-1)Bound*bni_36] + [(-2)bni_36]z[2] + [(-1)bni_36]y[2] >= 0 & [1 + (-1)bso_37] >= 0) *EVAL_2(x, y, z) -> COND_EVAL_21(>=(z, y), x, y, z) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]y[4] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(2)bni_38] = 0 & [(-1)bni_38 + (-1)Bound*bni_38] + [(-2)bni_38]y[4] + [(-1)bni_38]z[4] >= 0 & [(-1)bso_39] >= 0) *COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(x, y, z) *((U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_40] = 0 & [(-1)bso_41] >= 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_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [2]x_1 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [2]x_2 POL(&&(x_1, x_2)) = [-1] POL(=(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [2]x_1 POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [2]x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [2]x_2 POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(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], z[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], z[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(>(y[2], z[2]), x[2], y[2], z[2]) EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[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]) At least the following rules have been oriented under context sensitive arithmetic replacement: FALSE^1 -> &&(TRUE, 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_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[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(y[2] > z[2], x[2], y[2], z[2]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[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]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (0) -> (1), if (x[0] = y[0] && x[0] > z[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]) (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 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) ---------------------------------------- (8) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) (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] > z[0], x[0], y[0], z[0]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (0) -> (1), if (x[0] = y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 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) ---------------------------------------- (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@526b3a00 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_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) the following chains were created: *We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[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]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (>=(z[4], y[4])=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[0] & y[5]=y[0] & z[5]=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]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(z[4], y[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]) & (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] + [bni_25]x[4] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] + [bni_25]x[4] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] + [bni_25]x[4] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] + [-1]y[4] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] >= 0) (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] >= 0) For Pair EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) 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[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[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]) which results in the following constraint: (1) (x[1]=x[4] & y[1]=y[4] & z[1]=z[4] & >=(z[4], y[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(>=(z[4], y[4]), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(z[4], y[4])=TRUE ==> EVAL_2(x[1], y[4], z[4])_>=_NonInfC & EVAL_2(x[1], y[4], z[4])_>=_COND_EVAL_21(>=(z[4], y[4]), x[1], y[4], z[4]) & (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] + [bni_27]x[1] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] + [bni_27]x[1] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] + [bni_27]x[1] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (z[4] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (z[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] >= 0) (9) (z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[4] >= 0 & [(-1)bso_28] >= 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 EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[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[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) which results in the following constraint: (1) (&&(=(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[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 rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> COND_EVAL_1(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL_1(TRUE, x[0], y[0], z[0])_>=_EVAL_2(x[0], y[0], z[0]) & (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) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] + [-1]z[0] + [2]y[0] + [-1]x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] + [-1]z[0] + [2]y[0] + [-1]x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] + [-1]z[0] + [2]y[0] + [-1]x[0] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [-1 + (-1)bso_30] + [-2]z[0] + [2]y[0] + [-1]x[0] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] + [bni_29]z[0] >= 0 & [1 + (-1)bso_30] + x[0] + [2]z[0] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) the following chains were created: *We consider the chain COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[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[5]=x[0] & y[5]=y[0] & z[5]=z[0] & &&(=(x[0], y[0]), >(x[0], z[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], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], z[0])=TRUE & >=(x[0], y[0])=TRUE & <=(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], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]x[0] >= 0 & [(-1)bso_32] + [-1]y[0] + x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]x[0] >= 0 & [(-1)bso_32] + [-1]y[0] + x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]z[0] + [bni_31]x[0] >= 0 & [(-1)bso_32] + [-1]y[0] + x[0] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & [1] + z[0] + x[0] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]z[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [1 + (-1)bso_32] + [-1]y[0] + z[0] + x[0] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & [-1]z[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] + [-1]z[0] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (x[0] >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] >= 0) (10) (x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] >= 0) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], z[5])), >=) & [bni_25] = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[4] >= 0 & [(-1)bso_26] + z[4] >= 0) *EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[4] >= 0 & [(-1)bso_28] >= 0) *(z[4] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4])), >=) & [bni_27] = 0 & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[4] >= 0 & [(-1)bso_28] >= 0) *COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [(-1)Bound*bni_29] + [bni_29]x[0] >= 0 & [1 + (-1)bso_30] + x[0] >= 0) *EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] >= 0) *(x[0] >= 0 & 0 >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_31] + [bni_31]x[0] >= 0 & [(-1)bso_32] >= 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_EVAL_21(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_3 + x_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_4 + x_3 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(=(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) The following pairs are in P_bound: 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], z[0])), x[0], y[0], z[0]) The following pairs are in P_>=: COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(>=(z[4], y[4]), x[4], y[4], z[4]) EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (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: (5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], z[5]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[4], x[4], y[4], z[4]) (0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > z[0], x[0], y[0], z[0]) (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0]) (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 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) ---------------------------------------- (12) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (13) TRUE ---------------------------------------- (14) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: 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]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(y[2] > z[2], x[2], y[2], z[2]) (3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3] - 1, y[3] - 1, z[3]) (4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(z[4] >= y[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) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (y[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]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (z[4] >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 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) ---------------------------------------- (15) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer 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_2(x[3] - 1, y[3] - 1, z[3]) (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(y[2] > z[2], x[2], y[2], z[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2]) (2) -> (3), if (y[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) ---------------------------------------- (17) 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@526b3a00 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_2(-(x[3], 1), -(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(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) which results in the following constraint: (1) (>(y[2], z[2])=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1 ==> COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(y[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_2(-(x[2], 1), -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[2] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[2] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[2] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[2] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) For Pair EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[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(>(y[2], z[2]), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) which results in the following constraint: (1) (>(y[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(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(y[2], z[2])=TRUE ==> EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]y[2] >= 0 & [(-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] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]z[2] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_15] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_15] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) (9) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_15] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1), z[3])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) *EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_15] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 0) *(y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2])), >=) & 0 = 0 & [(-1)Bound*bni_15] + [bni_15]y[2] >= 0 & [(-1)bso_16] >= 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_4 + x_3 POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) The following pairs are in P_bound: COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1), z[3]) EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) The following pairs are in P_>=: EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(>(y[2], z[2]), x[2], y[2], z[2]) There are no usable rules. ---------------------------------------- (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: (2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(y[2] > z[2], x[2], y[2], 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) ---------------------------------------- (19) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (20) TRUE