/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 425 ms] (6) AND (7) IDP (8) IDependencyGraphProof [EQUIVALENT, 0 ms] (9) TRUE (10) IDP (11) IDPNonInfProof [SOUND, 72 ms] (12) IDP (13) IDependencyGraphProof [EQUIVALENT, 0 ms] (14) 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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval(x, y, z) -> Cond_eval(y > x && z > x, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x + 1, y, z) eval(x, y, z) -> Cond_eval1(y > x && x >= z, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x, y, z + 1) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval(x, y, z) -> Cond_eval(y > x && z > x, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x + 1, y, z) eval(x, y, z) -> Cond_eval1(y > x && x >= z, x, y, z) Cond_eval1(TRUE, x, y, z) -> eval(x, y, z + 1) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] + 1, y[1], z[1]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) (0) -> (1), if (y[0] > x[0] && z[0] > x[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] + 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] + 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] + 1, y[1], z[1]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], z[3] + 1) (0) -> (1), if (y[0] > x[0] && z[0] > x[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] + 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] + 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (y[2] > x[2] && x[2] >= z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (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@37ced614 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x, y, z) -> COND_EVAL(&&(>(y, x), >(z, x)), x, y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[0] + [(2)bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[0] + [(2)bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[0] + [(2)bni_20]y[0] + [(-1)bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]z[0] + [bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(+(x, 1), y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & +(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(+(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(+(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & +(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(+(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(+(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [(2)bni_22]y[0] + [(-1)bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]z[0] + [bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) For Pair EVAL(x, y, z) -> COND_EVAL1(&&(>(y, x), >=(x, z)), x, y, z) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) which results in the following constraint: (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[2] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[2] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[2] + [(2)bni_24]y[2] + [(-1)bni_24]x[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[2] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(x, y, +(z, 1)) the following chains were created: *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & y[3]=y[0] & +(z[3], 1)=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], y[3], +(z[3], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], y[2], +(z[2], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(y[2], x[2]), >=(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & y[3]=y[2]1 & +(z[3], 1)=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(x[3], y[3], +(z[3], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >=(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(x[2], y[2], +(z[2], 1)) & (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [(2)bni_26]y[2] + [(-1)bni_26]x[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] >= 0 & x[2] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[2] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) (9) (y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y, z) -> COND_EVAL(&&(>(y, x), >(z, x)), x, y, z) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]x[0] + [(2)bni_20]y[0] >= 0 & [(-1)bso_21] >= 0) *COND_EVAL(TRUE, x, y, z) -> EVAL(+(x, 1), y, z) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_22] + [(-1)bni_22]x[0] + [(2)bni_22]y[0] >= 0 & [1 + (-1)bso_23] >= 0) *EVAL(x, y, z) -> COND_EVAL1(&&(>(y, x), >=(x, z)), x, y, z) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[2] + [(2)bni_24]y[2] >= 0 & [(-1)bso_25] >= 0) *COND_EVAL1(TRUE, x, y, z) -> EVAL(x, y, +(z, 1)) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) *(y[2] >= 0 & x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(x[3], y[3], +(z[3], 1))), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[2] + [(2)bni_26]y[2] >= 0 & [1 + (-1)bso_27] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [2] POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [2]x_2 + [-1]x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [2]x_3 + [-1]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [2]x_3 + [-1]x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) The following pairs are in P_bound: EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3], y[3], +(z[3], 1)) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >=(x[2], z[2])), x[2], y[2], z[2]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 <-> TRUE^1 FALSE^1 -> &&(TRUE, FALSE)^1 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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > x[2] && x[2] >= z[2], x[2], y[2], z[2]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (8) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (9) TRUE ---------------------------------------- (10) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] + 1, y[1], z[1]) (1) -> (0), if (x[1] + 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) (0) -> (1), if (y[0] > x[0] && z[0] > x[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (11) 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@37ced614 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) which results in the following constraint: (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [bni_14]z[0] + [(-1)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [bni_14]z[0] + [(-1)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [bni_14]z[0] + [(-1)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [bni_14]z[0] + [(-1)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(y[0], x[0]), >(z[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & +(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(+(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], x[0])=TRUE & >(z[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(+(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[0] + [(-1)bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[0] + [(-1)bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] + [-1]x[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[0] + [(-1)bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[0] >= 0 & z[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[0] + [(-1)bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16 + bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16 + bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) (9) (y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16 + bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16 + bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) *(y[0] >= 0 & x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(+(x[1], 1), y[1], z[1])), >=) & [(-1)Bound*bni_16 + bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(EVAL(x_1, x_2, x_3)) = x_3 + [-1]x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = x_4 + [-1]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) The following pairs are in P_bound: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(+(x[1], 1), y[1], z[1]) The following pairs are in P_>=: EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(y[0], x[0]), >(z[0], x[0])), x[0], y[0], z[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (12) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(y[0] > x[0] && z[0] > x[0], x[0], y[0], z[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1, x2) ---------------------------------------- (13) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (14) TRUE