/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.itrs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 369 ms] (6) AND (7) IDP (8) IDPNonInfProof [SOUND, 129 ms] (9) IDP (10) IDependencyGraphProof [EQUIVALENT, 0 ms] (11) IDP (12) IDPNonInfProof [SOUND, 43 ms] (13) AND (14) IDP (15) IDependencyGraphProof [EQUIVALENT, 0 ms] (16) IDP (17) IDPNonInfProof [SOUND, 17 ms] (18) IDP (19) IDependencyGraphProof [EQUIVALENT, 0 ms] (20) TRUE (21) IDP (22) IDPNonInfProof [SOUND, 18 ms] (23) IDP (24) IDependencyGraphProof [EQUIVALENT, 0 ms] (25) TRUE (26) IDP (27) IDependencyGraphProof [EQUIVALENT, 0 ms] (28) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval(x, y) -> Cond_eval(x > 0, x, y) Cond_eval(TRUE, x, y) -> eval(x - 1, y) eval(x, y) -> Cond_eval1(y > 0 && x > 0, x, y) Cond_eval1(TRUE, x, y) -> eval(x - 1, y) eval(x, y) -> Cond_eval2(x > 0 && 0 >= x && y > 0, x, y) Cond_eval2(TRUE, x, y) -> eval(x, y - 1) eval(x, y) -> Cond_eval3(y > 0 && 0 >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x, y - 1) eval(x, y) -> Cond_eval4(x > 0 && 0 >= x && 0 >= y, x, y) Cond_eval4(TRUE, x, y) -> eval(x, y) eval(x, y) -> Cond_eval5(y > 0 && 0 >= x && 0 >= y, x, y) Cond_eval5(TRUE, x, y) -> eval(x, y) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean The ITRS R consists of the following rules: eval(x, y) -> Cond_eval(x > 0, x, y) Cond_eval(TRUE, x, y) -> eval(x - 1, y) eval(x, y) -> Cond_eval1(y > 0 && x > 0, x, y) Cond_eval1(TRUE, x, y) -> eval(x - 1, y) eval(x, y) -> Cond_eval2(x > 0 && 0 >= x && y > 0, x, y) Cond_eval2(TRUE, x, y) -> eval(x, y - 1) eval(x, y) -> Cond_eval3(y > 0 && 0 >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x, y - 1) eval(x, y) -> Cond_eval4(x > 0 && 0 >= x && 0 >= y, x, y) Cond_eval4(TRUE, x, y) -> eval(x, y) eval(x, y) -> Cond_eval5(y > 0 && 0 >= x && 0 >= y, x, y) Cond_eval5(TRUE, x, y) -> eval(x, y) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > 0 && 0 >= x[4] && y[4] > 0, x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > 0 && 0 >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - 1) (8): EVAL(x[8], y[8]) -> COND_EVAL4(x[8] > 0 && 0 >= x[8] && 0 >= y[8], x[8], y[8]) (9): COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]) (10): EVAL(x[10], y[10]) -> COND_EVAL5(y[10] > 0 && 0 >= x[10] && 0 >= y[10], x[10], y[10]) (11): COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8]) (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8]) (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10]) (4) -> (5), if (x[4] > 0 && 0 >= x[4] && y[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6]) (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8]) (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10]) (6) -> (7), if (y[6] > 0 && 0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6]) (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8]) (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10]) (8) -> (9), if (x[8] > 0 && 0 >= x[8] && 0 >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9]) (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0]) (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2]) (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4]) (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6]) (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10]) (10) -> (11), if (y[10] > 0 && 0 >= x[10] && 0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4]) (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6]) (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8]) (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > 0 && 0 >= x[4] && y[4] > 0, x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > 0 && 0 >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - 1) (8): EVAL(x[8], y[8]) -> COND_EVAL4(x[8] > 0 && 0 >= x[8] && 0 >= y[8], x[8], y[8]) (9): COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]) (10): EVAL(x[10], y[10]) -> COND_EVAL5(y[10] > 0 && 0 >= x[10] && 0 >= y[10], x[10], y[10]) (11): COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8]) (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8]) (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10]) (4) -> (5), if (x[4] > 0 && 0 >= x[4] && y[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6]) (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8]) (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10]) (6) -> (7), if (y[6] > 0 && 0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6]) (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8]) (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10]) (8) -> (9), if (x[8] > 0 && 0 >= x[8] && 0 >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9]) (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0]) (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2]) (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4]) (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6]) (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8]) (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10]) (10) -> (11), if (y[10] > 0 && 0 >= x[10] && 0 >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11]) (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0]) (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2]) (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4]) (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6]) (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8]) (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7a00849f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x, y) -> COND_EVAL(>(x, 0), x, y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] >= 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] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] >= 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] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] >= 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] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [(-1)bni_33 + (-1)Bound*bni_33] >= 0 & [(-1)bso_34] >= 0) For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rule (III) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_35 + (-1)Bound*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) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] >= 0 & [(-1)bso_36] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_35 + (-1)Bound*bni_35] >= 0 & [(-1)bso_36] >= 0) For Pair EVAL(x, y) -> COND_EVAL1(&&(>(y, 0), >(x, 0)), x, y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) For Pair COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) For Pair EVAL(x, y) -> COND_EVAL2(&&(&&(>(x, 0), >=(0, x)), >(y, 0)), x, y) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) which results in the following constraint: (1) (&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], 0)=TRUE & >(x[4], 0)=TRUE & >=(0, x[4])=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] >= 0 & [-1 + (-1)bso_42] + y[4] + x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] >= 0 & [-1 + (-1)bso_42] + y[4] + x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4])), >=) & [(-1)bni_41 + (-1)Bound*bni_41] >= 0 & [-1 + (-1)bso_42] + y[4] + x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) which results in the following constraint: (1) (&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0))=TRUE & x[4]=x[5] & y[4]=y[5] ==> COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], -(y[5], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], 0)=TRUE & >(x[4], 0)=TRUE & >=(0, x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(x[4], -(y[4], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]y[4] + [(-1)bni_43]x[4] >= 0 & [(-1)bso_44] + [-1]y[4] + [-1]x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]y[4] + [(-1)bni_43]x[4] >= 0 & [(-1)bso_44] + [-1]y[4] + [-1]x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] >= 0 & x[4] + [-1] >= 0 & [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_43 + (-1)Bound*bni_43] + [(-1)bni_43]y[4] + [(-1)bni_43]x[4] >= 0 & [(-1)bso_44] + [-1]y[4] + [-1]x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y) -> COND_EVAL3(&&(>(y, 0), >=(0, x)), x, y) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] >= 0) For Pair COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) For Pair EVAL(x, y) -> COND_EVAL4(&&(&&(>(x, 0), >=(0, x)), >=(0, y)), x, y) the following chains were created: *We consider the chain EVAL(x[8], y[8]) -> COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]), COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]) which results in the following constraint: (1) (&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8]))=TRUE & x[8]=x[9] & y[8]=y[9] ==> EVAL(x[8], y[8])_>=_NonInfC & EVAL(x[8], y[8])_>=_COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(0, y[8])=TRUE & >(x[8], 0)=TRUE & >=(0, x[8])=TRUE ==> EVAL(x[8], y[8])_>=_NonInfC & EVAL(x[8], y[8])_>=_COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[8] >= 0 & x[8] + [-1] >= 0 & [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[8] >= 0 & x[8] + [-1] >= 0 & [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[8] >= 0 & x[8] + [-1] >= 0 & [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8])), >=) & [(-1)bni_49 + (-1)Bound*bni_49] >= 0 & [(-1)bso_50] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL4(TRUE, x, y) -> EVAL(x, y) the following chains were created: *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (x[9]=x[0] & y[9]=y[0] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (x[9]=x[2] & y[9]=y[2] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) which results in the following constraint: (1) (x[9]=x[4] & y[9]=y[4] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) which results in the following constraint: (1) (x[9]=x[6] & y[9]=y[6] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[8], y[8]) -> COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) which results in the following constraint: (1) (x[9]=x[8] & y[9]=y[8] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *We consider the chain COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]), EVAL(x[10], y[10]) -> COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) which results in the following constraint: (1) (x[9]=x[10] & y[9]=y[10] ==> COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL4(TRUE, x[9], y[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9])_>=_EVAL(x[9], y[9]) & (U^Increasing(EVAL(x[9], y[9])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) For Pair EVAL(x, y) -> COND_EVAL5(&&(&&(>(y, 0), >=(0, x)), >=(0, y)), x, y) the following chains were created: *We consider the chain EVAL(x[10], y[10]) -> COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]), COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]) which results in the following constraint: (1) (&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10]))=TRUE & x[10]=x[11] & y[10]=y[11] ==> EVAL(x[10], y[10])_>=_NonInfC & EVAL(x[10], y[10])_>=_COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(0, y[10])=TRUE & >(y[10], 0)=TRUE & >=(0, x[10])=TRUE ==> EVAL(x[10], y[10])_>=_NonInfC & EVAL(x[10], y[10])_>=_COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[10] >= 0 & y[10] + [-1] >= 0 & [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] >= 0 & [1 + (-1)bso_54] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[10] >= 0 & y[10] + [-1] >= 0 & [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] >= 0 & [1 + (-1)bso_54] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[10] >= 0 & y[10] + [-1] >= 0 & [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10])), >=) & [(-1)bni_53 + (-1)Bound*bni_53] >= 0 & [1 + (-1)bso_54] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL5(TRUE, x, y) -> EVAL(x, y) the following chains were created: *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (x[11]=x[0] & y[11]=y[0] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (x[11]=x[2] & y[11]=y[2] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) which results in the following constraint: (1) (x[11]=x[4] & y[11]=y[4] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) which results in the following constraint: (1) (x[11]=x[6] & y[11]=y[6] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[8], y[8]) -> COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) which results in the following constraint: (1) (x[11]=x[8] & y[11]=y[8] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *We consider the chain COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]), EVAL(x[10], y[10]) -> COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) which results in the following constraint: (1) (x[11]=x[10] & y[11]=y[10] ==> COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL5(TRUE, x[11], y[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11])_>=_EVAL(x[11], y[11]) & (U^Increasing(EVAL(x[11], y[11])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(>(x, 0), x, y) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [(-1)bni_33 + (-1)Bound*bni_33] >= 0 & [(-1)bso_34] >= 0) *COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_35 + (-1)Bound*bni_35] >= 0 & [(-1)bso_36] >= 0) *EVAL(x, y) -> COND_EVAL1(&&(>(y, 0), >(x, 0)), x, y) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) *EVAL(x, y) -> COND_EVAL2(&&(&&(>(x, 0), >=(0, x)), >(y, 0)), x, y) *COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, 1)) *EVAL(x, y) -> COND_EVAL3(&&(>(y, 0), >=(0, x)), x, y) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_45 + (-1)Bound*bni_45] >= 0 & [(-1)bso_46] >= 0) *COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, 1)) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_47 + (-1)Bound*bni_47] >= 0 & [(-1)bso_48] >= 0) *EVAL(x, y) -> COND_EVAL4(&&(&&(>(x, 0), >=(0, x)), >=(0, y)), x, y) *COND_EVAL4(TRUE, x, y) -> EVAL(x, y) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *((U^Increasing(EVAL(x[9], y[9])), >=) & [bni_51] = 0 & [2 + (-1)bso_52] >= 0) *EVAL(x, y) -> COND_EVAL5(&&(&&(>(y, 0), >=(0, x)), >=(0, y)), x, y) *COND_EVAL5(TRUE, x, y) -> EVAL(x, y) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) *((U^Increasing(EVAL(x[11], y[11])), >=) & [bni_55] = 0 & [1 + (-1)bso_56] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [3] POL(EVAL(x_1, x_2)) = [-1] POL(COND_EVAL(x_1, x_2, x_3)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] POL(&&(x_1, x_2)) = [-1] POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] POL(COND_EVAL4(x_1, x_2, x_3)) = [1] + [2]x_1 POL(COND_EVAL5(x_1, x_2, x_3)) = [2]x_1 The following pairs are in P_>: EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) EVAL(x[8], y[8]) -> COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]) EVAL(x[10], y[10]) -> COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(x[4], 0), >=(0, x[4])), >(y[4], 0)), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) EVAL(x[8], y[8]) -> COND_EVAL4(&&(&&(>(x[8], 0), >=(0, x[8])), >=(0, y[8])), x[8], y[8]) EVAL(x[10], y[10]) -> COND_EVAL5(&&(&&(>(y[10], 0), >=(0, x[10])), >=(0, y[10])), x[10], y[10]) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > 0 && 0 >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - 1) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6]) (6) -> (7), if (y[6] > 0 && 0 >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (8) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7a00849f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] >= 0 & [(-1)bso_22] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[6] & y[1]=y[6] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[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] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[2] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[2] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[2] >= 0 & [(-1)bso_26] >= 0) For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[0] & y[3]=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[6] & y[3]=y[6] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) For Pair EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]y[6] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]y[6] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]y[6] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]y[6] >= 0 & [(-1)bso_30] >= 0) For Pair COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[0] & -(y[7], 1)=y[0] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[2] & -(y[7], 1)=y[2] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) which results in the following constraint: (1) (&&(>(y[6], 0), >=(0, x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[6]1 & -(y[7], 1)=y[6]1 ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], 0)=TRUE & >=(0, x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] >= 0 & [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] >= 0 & [(-1)bso_22] >= 0) *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & [(-1)bso_24] >= 0) *EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[2] >= 0 & [(-1)bso_26] >= 0) *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[2] >= 0 & [(-1)bso_28] >= 0) *EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]y[6] >= 0 & [(-1)bso_30] >= 0) *COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-1)bso_32] >= 0) *(y[6] + [-1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]y[6] >= 0 & [1 + (-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) = [3] POL(EVAL(x_1, x_2)) = [-1] + x_2 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_3 POL(>(x_1, x_2)) = 0 POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 POL(&&(x_1, x_2)) = [-1] POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + x_3 POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) The following pairs are in P_bound: EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], 0), >=(0, x[6])), x[6], y[6]) 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 ---------------------------------------- (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: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > 0 && 0 >= x[6], x[6], y[6]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (10) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (11) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (12) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7a00849f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[0] & y[3]=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)Bound*bni_18] + [bni_18]y[2] + [bni_18]x[2] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)Bound*bni_18] + [bni_18]y[2] + [bni_18]x[2] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)Bound*bni_18] + [bni_18]y[2] + [bni_18]x[2] >= 0 & [(-1)bso_19] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_20] = 0 & [(-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]y[0] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_20] = 0 & [(-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) For Pair EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [(-1)Bound*bni_22] + [bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]y[2] + [bni_16]x[2] >= 0 & [(-1)bso_17] >= 0) *EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)Bound*bni_18] + [bni_18]y[2] + [bni_18]x[2] >= 0 & [(-1)bso_19] >= 0) *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_20] = 0 & [(-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [bni_20] = 0 & [(-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [1 + (-1)bso_21] >= 0) *EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_22] = 0 & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 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_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 + x_2 + [-1]x_1 POL(EVAL(x_1, x_2)) = x_2 + x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(COND_EVAL(x_1, x_2, x_3)) = x_3 + x_2 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) The following pairs are in P_bound: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) The following pairs are in P_>=: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[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 ---------------------------------------- (13) Complex Obligation (AND) ---------------------------------------- (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (15) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (16) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (y[2] > 0 && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (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@7a00849f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] >= 0) For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>(y[2], 0), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] + [bni_14]x[2] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] + [bni_14]x[2] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] + [bni_14]x[2] >= 0 & [1 + (-1)bso_15] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] >= 0) *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) *(y[2] + [-1] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] + [bni_14]x[2] >= 0 & [1 + (-1)bso_15] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [3] POL(EVAL(x_1, x_2)) = [-1] + x_2 + x_1 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 + x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) The following pairs are in P_bound: EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) The following pairs are in P_>=: EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], 0), >(x[2], 0)), x[2], y[2]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > 0 && x[2] > 0, x[2], y[2]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (19) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (20) TRUE ---------------------------------------- (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: (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0, x[0], y[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (22) 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@7a00849f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) For Pair EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) *(x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) *EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(EVAL(x_1, x_2)) = [1] + [2]x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(>(x_1, x_2)) = [2] POL(0) = 0 The following pairs are in P_>: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) The following pairs are in P_bound: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], 0), x[0], y[0]) The following pairs are in P_>=: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) There are no usable rules. ---------------------------------------- (23) 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(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (24) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (25) TRUE ---------------------------------------- (26) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: none R is empty. The integer pair graph contains the following rules and edges: (9): COND_EVAL4(TRUE, x[9], y[9]) -> EVAL(x[9], y[9]) (11): COND_EVAL5(TRUE, x[11], y[11]) -> EVAL(x[11], y[11]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) Cond_eval4(TRUE, x0, x1) Cond_eval5(TRUE, x0, x1) ---------------------------------------- (27) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (28) TRUE