/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 254 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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) -> Cond_eval(x > 0 && !(x = 0) && x % 2 > 0, x) Cond_eval(TRUE, x) -> eval(x - 1) eval(x) -> Cond_eval1(x > 0 && !(x = 0) && x % 2 = 0, x) Cond_eval1(TRUE, x) -> eval(x / 2) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) Cond_eval1(TRUE, x0) ---------------------------------------- (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 / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval(x) -> Cond_eval(x > 0 && !(x = 0) && x % 2 > 0, x) Cond_eval(TRUE, x) -> eval(x - 1) eval(x) -> Cond_eval1(x > 0 && !(x = 0) && x % 2 = 0, x) Cond_eval1(TRUE, x) -> eval(x / 2) The integer pair graph contains the following rules and edges: (0): EVAL(x[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0, x[0]) (1): COND_EVAL(TRUE, x[1]) -> EVAL(x[1] - 1) (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) (3): COND_EVAL1(TRUE, x[3]) -> EVAL(x[3] / 2) (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0 & x[0] ->^* x[1]) (1) -> (0), if (x[1] - 1 ->^* x[0]) (1) -> (2), if (x[1] - 1 ->^* x[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0 & x[2] ->^* x[3]) (3) -> (0), if (x[3] / 2 ->^* x[0]) (3) -> (2), if (x[3] / 2 ->^* x[2]) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) Cond_eval1(TRUE, x0) ---------------------------------------- (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 / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0, x[0]) (1): COND_EVAL(TRUE, x[1]) -> EVAL(x[1] - 1) (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) (3): COND_EVAL1(TRUE, x[3]) -> EVAL(x[3] / 2) (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0 & x[0] ->^* x[1]) (1) -> (0), if (x[1] - 1 ->^* x[0]) (1) -> (2), if (x[1] - 1 ->^* x[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0 & x[2] ->^* x[3]) (3) -> (0), if (x[3] / 2 ->^* x[0]) (3) -> (2), if (x[3] / 2 ->^* x[2]) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) Cond_eval1(TRUE, x0) ---------------------------------------- (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@2fa436b2 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) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(%(x, 2), 0)), x) the following chains were created: *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) which results in the following constraint: (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) For Pair COND_EVAL(TRUE, x) -> EVAL(-(x, 1)) the following chains were created: *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)), EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) which results in the following constraint: (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] & -(x[1], 1)=x[0]1 ==> COND_EVAL(TRUE, x[1])_>=_NonInfC & COND_EVAL(TRUE, x[1])_>=_EVAL(-(x[1], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)), EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) which results in the following constraint: (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] & -(x[1], 1)=x[2] ==> COND_EVAL(TRUE, x[1])_>=_NonInfC & COND_EVAL(TRUE, x[1])_>=_EVAL(-(x[1], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) For Pair EVAL(x) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), =(%(x, 2), 0)), x) the following chains were created: *We consider the chain EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]), COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0))=TRUE & x[2]=x[3] ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) (3) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[2] + [-1] >= 0 & [4] >= 0 & [2] >= 0 & [2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) For Pair COND_EVAL1(TRUE, x) -> EVAL(/(x, 2)) the following chains were created: *We consider the chain EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]), COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0))=TRUE & x[2]=x[3] ==> COND_EVAL1(TRUE, x[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3])_>=_EVAL(/(x[3], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE ==> COND_EVAL1(TRUE, x[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2])_>=_EVAL(/(x[2], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) (3) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE & <(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2])_>=_EVAL(/(x[2], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] + x[2] + [-1]max{x[2], [-1]x[2]} >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] + x[2] + [-1]max{x[2], [-1]x[2]} >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] + x[2] + [-1]max{x[2], [-1]x[2]} >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] + x[2] + [-1]max{x[2], [-1]x[2]} >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[2] + [-1] >= 0 & [4] >= 0 & [2] >= 0 & [2] >= 0 & [2]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 0) We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(%(x, 2), 0)), x) *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) *COND_EVAL(TRUE, x) -> EVAL(-(x, 1)) *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [1 + (-1)bso_18] >= 0) *EVAL(x) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), =(%(x, 2), 0)), x) *(x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) *COND_EVAL1(TRUE, x) -> EVAL(/(x, 2)) *(x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 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)) = [-1] + x_1 POL(COND_EVAL(x_1, x_2)) = [-1] + x_2 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(!(x_1)) = [-1] POL(=(x_1, x_2)) = [-1] POL(2) = [2] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2)) = [-1] + x_2 Polynomial Interpretations with Context Sensitive Arithemetic Replacement POL(Term^CSAR-Mode @ Context) POL(%(x_1, 2)^1 @ {}) = max{x_2, [-1]x_2} POL(%(x_1, 2)^-1 @ {}) = min{x_2, [-1]x_2} POL(/(x_1, 2)^1 @ {EVAL_1/0}) = max{x_1, [-1]x_1} + [-1] The following pairs are in P_>: COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) The following pairs are in P_bound: EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) The following pairs are in P_>=: EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[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 /^1 -> ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0, x[0]) (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) Cond_eval1(TRUE, x0) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (8) TRUE