/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 419 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 70 ms] (10) IDP (11) IDependencyGraphProof [EQUIVALENT, 0 ms] (12) 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: f(x, y, z) -> Cond_f(x > y && x > z, x, y, z) Cond_f(TRUE, x, y, z) -> f(x, y + 1, z) f(x, y, z) -> Cond_f1(x > y && x > z, x, y, z) Cond_f1(TRUE, x, y, z) -> f(x, y, z + 1) The set Q consists of the following terms: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(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: f(x, y, z) -> Cond_f(x > y && x > z, x, y, z) Cond_f(TRUE, x, y, z) -> f(x, y + 1, z) f(x, y, z) -> Cond_f1(x > y && x > z, x, y, z) Cond_f1(TRUE, x, y, z) -> f(x, y, z + 1) The integer pair graph contains the following rules and edges: (0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) (1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1]) (2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2]) (3): COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], z[3] + 1) (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (x[2] > y[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: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(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): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) (1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1]) (2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2]) (3): COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], z[3] + 1) (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2]) (2) -> (3), if (x[2] > y[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: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(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@345ab01b 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 F(x, y, z) -> COND_F(&&(>(x, y), >(x, z)), x, y, z) the following chains were created: *We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0) For Pair COND_F(TRUE, x, y, z) -> F(x, +(y, 1), z) the following chains were created: *We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) *We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[2] & +(y[1], 1)=y[2] & z[1]=z[2] ==> COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) For Pair F(x, y, z) -> COND_F1(&&(>(x, y), >(x, z)), x, y, z) the following chains were created: *We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)) which results in the following constraint: (1) (&&(>(x[2], y[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> F(x[2], y[2], z[2])_>=_NonInfC & F(x[2], y[2], z[2])_>=_COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_F1(&&(>(x[2], y[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) (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE ==> F(x[2], y[2], z[2])_>=_NonInfC & F(x[2], y[2], z[2])_>=_COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_F1(&&(>(x[2], y[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) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0) (9) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0) For Pair COND_F1(TRUE, x, y, z) -> F(x, y, +(z, 1)) the following chains were created: *We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(x[2], y[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_F1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_F1(TRUE, x[3], y[3], z[3])_>=_F(x[3], y[3], +(z[3], 1)) & (U^Increasing(F(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) (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_F1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_F1(TRUE, x[2], y[2], z[2])_>=_F(x[2], y[2], +(z[2], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) (9) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)), F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(>(x[2], y[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_F1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_F1(TRUE, x[3], y[3], z[3])_>=_F(x[3], y[3], +(z[3], 1)) & (U^Increasing(F(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) (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_F1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_F1(TRUE, x[2], y[2], z[2])_>=_F(x[2], y[2], +(z[2], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[2] >= 0 & z[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) (9) (x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *F(x, y, z) -> COND_F(&&(>(x, y), >(x, z)), x, y, z) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0) *COND_F(TRUE, x, y, z) -> F(x, +(y, 1), z) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0) *F(x, y, z) -> COND_F1(&&(>(x, y), >(x, z)), x, y, z) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0) *COND_F1(TRUE, x, y, z) -> F(x, y, +(z, 1)) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0) *(x[2] >= 0 & z[2] >= 0 & y[2] >= 0 ==> (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[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) = [1] POL(F(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 POL(COND_F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_F1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 The following pairs are in P_>: COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)) The following pairs are in P_bound: F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)) The following pairs are in P_>=: F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[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^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) 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): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) (1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1]) (2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2]) The set Q consists of the following terms: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(TRUE, x0, x1, x2) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (8) 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: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1]) (0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) The set Q consists of the following terms: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(TRUE, x0, x1, x2) ---------------------------------------- (9) 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@345ab01b 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_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) the following chains were created: *We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) For Pair F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) the following chains were created: *We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) *F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-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) = [3] POL(COND_F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 + [-1]x_1 POL(F(x_1, x_2, x_3)) = [-1] + [-1]x_2 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) The following pairs are in P_bound: COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) The following pairs are in P_>=: F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (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): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) The set Q consists of the following terms: f(x0, x1, x2) Cond_f(TRUE, x0, x1, x2) Cond_f1(TRUE, x0, x1, x2) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (12) TRUE