/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 593 ms] (6) IDP (7) PisEmptyProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (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: f(TRUE, x, y, z) -> f(x > y && x > z, x, y + 1, z) f(TRUE, x, y, z) -> f(x > y && x > z, x, y, z + 1) The set Q consists of the following terms: f(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 > ~ 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: f(TRUE, x, y, z) -> f(x > y && x > z, x, y + 1, z) f(TRUE, x, y, z) -> f(x > y && x > z, x, y, z + 1) The integer pair graph contains the following rules and edges: (0): F(TRUE, x[0], y[0], z[0]) -> F(x[0] > y[0] && x[0] > z[0], x[0], y[0] + 1, z[0]) (1): F(TRUE, x[1], y[1], z[1]) -> F(x[1] > y[1] && x[1] > z[1], x[1], y[1], z[1] + 1) (0) -> (0), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[0]' & y[0] + 1 ->^* y[0]' & z[0] ->^* z[0]') (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] + 1 ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] > y[1] && x[1] > z[1] & x[1] ->^* x[0] & y[1] ->^* y[0] & z[1] + 1 ->^* z[0]) (1) -> (1), if (x[1] > y[1] && x[1] > z[1] & x[1] ->^* x[1]' & y[1] ->^* y[1]' & z[1] + 1 ->^* z[1]') The set Q consists of the following terms: f(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 > ~ 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): F(TRUE, x[0], y[0], z[0]) -> F(x[0] > y[0] && x[0] > z[0], x[0], y[0] + 1, z[0]) (1): F(TRUE, x[1], y[1], z[1]) -> F(x[1] > y[1] && x[1] > z[1], x[1], y[1], z[1] + 1) (0) -> (0), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[0]' & y[0] + 1 ->^* y[0]' & z[0] ->^* z[0]') (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] + 1 ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] > y[1] && x[1] > z[1] & x[1] ->^* x[0] & y[1] ->^* y[0] & z[1] + 1 ->^* z[0]) (1) -> (1), if (x[1] > y[1] && x[1] > z[1] & x[1] ->^* x[1]' & y[1] ->^* y[1]' & z[1] + 1 ->^* z[1]') The set Q consists of the following terms: f(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@97bd269 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(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, +(y, 1), z) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 & &&(>(x[0]1, y[0]1), >(x[0]1, z[0]1))=TRUE & x[0]1=x[0]2 & +(y[0]1, 1)=y[0]2 & z[0]1=z[0]2 ==> F(TRUE, x[0]1, y[0]1, z[0]1)_>=_NonInfC & F(TRUE, x[0]1, y[0]1, z[0]1)_>=_F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]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 & >(x[0], +(y[0], 1))=TRUE ==> F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(+(y[0], 1), 1), z[0]) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint: (1) (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0] & y[1]=y[0] & +(z[1], 1)=z[0] & &&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE ==> F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) (9) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) *We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 & &&(>(x[0]1, y[0]1), >(x[0]1, z[0]1))=TRUE & x[0]1=x[1] & +(y[0]1, 1)=y[1] & z[0]1=z[1] ==> F(TRUE, x[0]1, y[0]1, z[0]1)_>=_NonInfC & F(TRUE, x[0]1, y[0]1, z[0]1)_>=_F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]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 & >(x[0], +(y[0], 1))=TRUE ==> F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(+(y[0], 1), 1), z[0]) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[0] + [(-1)bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 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 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]y[0] + [(2)bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint: (1) (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0] & y[1]=y[0] & +(z[1], 1)=z[0] & &&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1]1 & +(y[0], 1)=y[1]1 & z[0]=z[1]1 ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE ==> F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-2)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [(-1)bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]y[1] + [(2)bni_10]x[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) (9) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) For Pair F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, y, +(z, 1)) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & +(y[0], 1)=y[1] & z[0]=z[1] & &&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0]1 & y[1]=y[0]1 & +(z[1], 1)=z[0]1 ==> F(TRUE, x[1], y[1], z[1])_>=_NonInfC & F(TRUE, x[1], y[1], z[1])_>=_F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 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 & >(x[0], +(y[0], 1))=TRUE ==> F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint: (1) (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1 & &&(>(x[1]1, y[1]1), >(x[1]1, z[1]1))=TRUE & x[1]1=x[0] & y[1]1=y[0] & +(z[1]1, 1)=z[0] ==> F(TRUE, x[1]1, y[1]1, z[1]1)_>=_NonInfC & F(TRUE, x[1]1, y[1]1, z[1]1)_>=_F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE ==> F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], y[1], +(+(z[1], 1), 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) (9) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) *We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint: (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & +(y[0], 1)=y[1] & z[0]=z[1] & &&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1 ==> F(TRUE, x[1], y[1], z[1])_>=_NonInfC & F(TRUE, x[1], y[1], z[1])_>=_F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 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 & >(x[0], +(y[0], 1))=TRUE ==> F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & x[0] + [-2] + [-1]y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[0] + [(-1)bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 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 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]y[0] + [(2)bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) (9) (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint: (1) (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1 & &&(>(x[1]1, y[1]1), >(x[1]1, z[1]1))=TRUE & x[1]1=x[1]2 & y[1]1=y[1]2 & +(z[1]1, 1)=z[1]2 ==> F(TRUE, x[1]1, y[1]1, z[1]1)_>=_NonInfC & F(TRUE, x[1]1, y[1]1, z[1]1)_>=_F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE ==> F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], y[1], +(+(z[1], 1), 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [(-1)bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]y[1] + [(2)bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) (9) (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, +(y, 1), z) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [bni_10]x[0] + [bni_10]z[0] >= 0 & [1 + (-1)bso_11] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]x[1] + [bni_10]z[1] >= 0 & [1 + (-1)bso_11] >= 0) *F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, y, +(z, 1)) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[0] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0) *(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0 ==> (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]x[1] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 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(F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [2]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] The following pairs are in P_>: F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) The following pairs are in P_bound: F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) The following pairs are in P_>=: none 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) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: none R is empty. The integer pair graph is empty. The set Q consists of the following terms: f(TRUE, x0, x1, x2) ---------------------------------------- (7) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (8) YES