/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 270 ms] (6) AND (7) IDP (8) IDependencyGraphProof [EQUIVALENT, 0 ms] (9) TRUE (10) IDP (11) IDependencyGraphProof [EQUIVALENT, 0 ms] (12) IDP (13) IDPNonInfProof [SOUND, 26 ms] (14) IDP (15) IDependencyGraphProof [EQUIVALENT, 0 ms] (16) 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 = 0) && x > y, x, y) Cond_eval(TRUE, x, y) -> eval(y, y) eval(x, y) -> Cond_eval1(x > 0 && !(x = 0) && x <= y, x, y) Cond_eval1(TRUE, x, y) -> eval(x - 1, y) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(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: Boolean, Integer The ITRS R consists of the following rules: eval(x, y) -> Cond_eval(x > 0 && !(x = 0) && x > y, x, y) Cond_eval(TRUE, x, y) -> eval(y, y) eval(x, y) -> Cond_eval1(x > 0 && !(x = 0) && x <= y, x, y) Cond_eval1(TRUE, x, y) -> eval(x - 1, y) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & 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]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(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: Boolean, Integer 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] = 0) && x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & 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]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(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@31b76675 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, 0))), >(x, y)), x, y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[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], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) (3) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (10) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) (11) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + y[0] + x[0] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL(TRUE, x, y) -> EVAL(y, y) the following chains were created: *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: (1) (y[1]=x[0] & y[1]=y[0] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [1 + (-1)bso_22] >= 0) *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (y[1]=x[2] & y[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [1 + (-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [1 + (-1)bso_22] >= 0) For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), <=(x, y)), x, y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). 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(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), 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], 0))), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=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 constraints: (2) (<=(x[2], y[2])=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])), >=)) (3) (<=(x[2], y[2])=TRUE & >(x[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: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=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 constraints: (2) (<=(x[2], y[2])=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])), >=)) (3) (<=(x[2], y[2])=TRUE & >(x[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: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(x, y)), x, y) *(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + [-1]y[0] + x[0] >= 0) *(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [-1 + (-1)bso_20] + y[0] + x[0] >= 0) *COND_EVAL(TRUE, x, y) -> EVAL(y, y) *((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [1 + (-1)bso_22] >= 0) *((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [1 + (-1)bso_22] >= 0) *EVAL(x, y) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), <=(x, y)), x, y) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 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] + [-1]x_2 + x_1 POL(COND_EVAL(x_1, x_2, x_3)) = 0 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(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(<=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[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 > ~ 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], y[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] > y[0], x[0], y[0]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], 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) ---------------------------------------- (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 > ~ 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: (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & 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) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (12) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, 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(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & 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) ---------------------------------------- (13) 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@31b76675 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(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=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 constraints: (2) (<=(x[2], y[2])=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])), >=)) (3) (<=(x[2], y[2])=TRUE & >(x[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: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 0) We solved constraint (9) using rule (IDP_SMT_SPLIT). 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]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) *EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [1 + (-1)bso_18] >= 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) = [1] POL(FALSE) = [3] POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 POL(EVAL(x_1, x_2)) = [-1] + 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(!(x_1)) = [-1] POL(=(x_1, x_2)) = [-1] POL(<=(x_1, x_2)) = [-1] The following pairs are in P_>: EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) 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(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) The following pairs are in P_>=: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 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 ---------------------------------------- (14) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) ---------------------------------------- (15) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (16) TRUE