/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 207 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 0 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 < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: mult(0, y) -> 0 mult(x, y) -> Cond_mult(x > 0, x, y) Cond_mult(TRUE, x, y) -> mult(x - 1, y) + y mult(x, y) -> Cond_mult1(0 > x, x, y) Cond_mult1(TRUE, x, y) -> -(mult(-(x), y)) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer The ITRS R consists of the following rules: mult(0, y) -> 0 mult(x, y) -> Cond_mult(x > 0, x, y) Cond_mult(TRUE, x, y) -> mult(x - 1, y) + y mult(x, y) -> Cond_mult1(0 > x, x, y) Cond_mult1(TRUE, x, y) -> -(mult(-(x), y)) The integer pair graph contains the following rules and edges: (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) (3): COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (0 > x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (-(x[3]) ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (-(x[3]) ->^* x[2] & y[3] ->^* y[2]) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (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: (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) (3): COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (0 > x[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (-(x[3]) ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (-(x[3]) ->^* x[2] & y[3] ->^* y[2]) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(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@3f895654 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2 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 MULT(x, y) -> COND_MULT(>(x, 0), x, y) the following chains were created: *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[0] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15] = 0 & [(-1)bni_15 + (-1)Bound*bni_15] >= 0 & [(-1)bso_16] >= 0) For Pair COND_MULT(TRUE, x, y) -> MULT(-(x, 1), y) the following chains were created: *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 & -(x[1]1, 1)=x[0]2 & y[1]1=y[0]2 & >(x[0]2, 0)=TRUE & x[0]2=x[1]2 & y[0]2=y[1]2 ==> COND_MULT(TRUE, x[1]1, y[1]1)_>=_NonInfC & COND_MULT(TRUE, x[1]1, y[1]1)_>=_MULT(-(x[1]1, 1), y[1]1) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(-(x[0], 1), 0)=TRUE & >(-(-(x[0], 1), 1), 0)=TRUE ==> COND_MULT(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1), 1), y[0]) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 & -(x[1]1, 1)=x[2] & y[1]1=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_MULT(TRUE, x[1]1, y[1]1)_>=_NonInfC & COND_MULT(TRUE, x[1]1, y[1]1)_>=_MULT(-(x[1]1, 1), y[1]1) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(-(x[0], 1), 0)=TRUE & >(0, -(-(x[0], 1), 1))=TRUE ==> COND_MULT(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1), 1), y[0]) & (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & [1] + [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0] & y[3]=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(0, x[2])=TRUE & >(-(x[2]), 0)=TRUE & >(-(-(x[2]), 1), 0)=TRUE ==> COND_MULT(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2]), 1), y[2]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & [-2] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) ([-1] + x[2] >= 0 & [-1] + x[2] >= 0 & [-2] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0] & y[3]=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2]1 & y[1]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(0, x[2])=TRUE & >(-(x[2]), 0)=TRUE & >(0, -(-(x[2]), 1))=TRUE ==> COND_MULT(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2]), 1), y[2]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[2] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1] + [-1]x[2] >= 0 & [-1] + [-1]x[2] >= 0 & x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). For Pair MULT(x, y) -> COND_MULT1(>(0, x), x, y) the following chains were created: *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] ==> MULT(x[2], y[2])_>=_NonInfC & MULT(x[2], y[2])_>=_COND_MULT1(>(0, x[2]), x[2], y[2]) & (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(0, x[2])=TRUE ==> MULT(x[2], y[2])_>=_NonInfC & MULT(x[2], y[2])_>=_COND_MULT1(>(0, x[2]), x[2], y[2]) & (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[2] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [-2]x[2] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) ([-1] + x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [2]x[2] >= 0) For Pair COND_MULT1(TRUE, x, y) -> MULT(-(x), y) the following chains were created: *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[0]1 & y[3]=y[0]1 & >(x[0]1, 0)=TRUE & x[0]1=x[1]1 & y[0]1=y[1]1 ==> COND_MULT1(TRUE, x[3], y[3])_>=_NonInfC & COND_MULT1(TRUE, x[3], y[3])_>=_MULT(-(x[3]), y[3]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(0, -(x[0], 1))=TRUE & >(-(-(x[0], 1)), 0)=TRUE ==> COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1)), y[0]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-1]x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-1)bni_21] = 0 & [(-3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] & >(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 ==> COND_MULT1(TRUE, x[3], y[3])_>=_NonInfC & COND_MULT1(TRUE, x[3], y[3])_>=_MULT(-(x[3]), y[3]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(0, -(x[0], 1))=TRUE & >(0, -(-(x[0], 1)))=TRUE ==> COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_NonInfC & COND_MULT1(TRUE, -(x[0], 1), y[0])_>=_MULT(-(-(x[0], 1)), y[0]) & (U^Increasing(MULT(-(x[3]), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-3)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 & [-1]x[0] >= 0 & [-2] + x[0] >= 0 ==> (U^Increasing(MULT(-(x[3]), y[3])), >=) & [(-1)bni_21] = 0 & [(-3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[0] >= 0 & [-2 + (-1)bso_22] + [2]x[0] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 & -(x[3]1)=x[0] & y[3]1=y[0] & >(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_MULT1(TRUE, x[3]1, y[3]1)_>=_NonInfC & COND_MULT1(TRUE, x[3]1, y[3]1)_>=_MULT(-(x[3]1), y[3]1) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(0, x[2])=TRUE & >(0, -(x[2]))=TRUE & >(-(-(x[2])), 0)=TRUE ==> COND_MULT1(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT1(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2])), y[2]) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). *We consider the chain MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]), MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]), COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) which results in the following constraint: (1) (>(0, x[2])=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3])=x[2]1 & y[3]=y[2]1 & >(0, x[2]1)=TRUE & x[2]1=x[3]1 & y[2]1=y[3]1 & -(x[3]1)=x[2]2 & y[3]1=y[2]2 & >(0, x[2]2)=TRUE & x[2]2=x[3]2 & y[2]2=y[3]2 ==> COND_MULT1(TRUE, x[3]1, y[3]1)_>=_NonInfC & COND_MULT1(TRUE, x[3]1, y[3]1)_>=_MULT(-(x[3]1), y[3]1) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(0, x[2])=TRUE & >(0, -(x[2]))=TRUE & >(0, -(-(x[2])))=TRUE ==> COND_MULT1(TRUE, -(x[2]), y[2])_>=_NonInfC & COND_MULT1(TRUE, -(x[2]), y[2])_>=_MULT(-(-(x[2])), y[2]) & (U^Increasing(MULT(-(x[3]1), y[3]1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1] + [-1]x[2] >= 0 & [-1] + x[2] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(MULT(-(x[3]1), y[3]1)), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-2)bni_21]x[2] >= 0 & [(-1)bso_22] + [-2]x[2] >= 0) We solved constraint (6) using rule (IDP_SMT_SPLIT). To summarize, we get the following constraints P__>=_ for the following pairs. *MULT(x, y) -> COND_MULT(>(x, 0), x, y) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_15] = 0 & [(-1)bni_15 + (-1)Bound*bni_15] >= 0 & [(-1)bso_16] >= 0) *COND_MULT(TRUE, x, y) -> MULT(-(x, 1), y) *(x[0] + [-1] >= 0 & x[0] + [-2] >= 0 & x[0] + [-3] >= 0 ==> (U^Increasing(MULT(-(x[1]1, 1), y[1]1)), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) *([-1] + x[2] >= 0 & [-1] + x[2] >= 0 & [-2] + x[2] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_17] = 0 & [(-1)bni_17 + (-1)Bound*bni_17] >= 0 & [(-1)bso_18] >= 0) *MULT(x, y) -> COND_MULT1(>(0, x), x, y) *([-1] + x[2] >= 0 ==> (U^Increasing(COND_MULT1(>(0, x[2]), x[2], y[2])), >=) & [(-1)bni_19] = 0 & [(-1)bni_19 + (-1)Bound*bni_19] >= 0 & [(-1)bso_20] + [2]x[2] >= 0) *COND_MULT1(TRUE, x, y) -> MULT(-(x), y) 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) = [2] POL(FALSE) = 0 POL(MULT(x_1, x_2)) = [-1] + [-1]x_2 POL(COND_MULT(x_1, x_2, x_3)) = [-1] + [-1]x_3 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_MULT1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [2]x_2 POL(-(x_1)) = [-1]x_1 The following pairs are in P_>: MULT(x[2], y[2]) -> COND_MULT1(>(0, x[2]), x[2], y[2]) COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) The following pairs are in P_bound: COND_MULT1(TRUE, x[3], y[3]) -> MULT(-(x[3]), y[3]) The following pairs are in P_>=: MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) There are no usable rules. ---------------------------------------- (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 < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (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: (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) (2): MULT(x[2], y[2]) -> COND_MULT1(0 > x[2], x[2], y[2]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(TRUE, x0, x1) ---------------------------------------- (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 < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (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: (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) (0): MULT(x[0], y[0]) -> COND_MULT(x[0] > 0, x[0], y[0]) (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(TRUE, x0, x1) ---------------------------------------- (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@275c190c 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_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) the following chains were created: *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]), MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_MULT(TRUE, x[1], y[1])_>=_NonInfC & COND_MULT(TRUE, x[1], y[1])_>=_MULT(-(x[1], 1), y[1]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> COND_MULT(TRUE, x[0], y[0])_>=_NonInfC & COND_MULT(TRUE, x[0], y[0])_>=_MULT(-(x[0], 1), y[0]) & (U^Increasing(MULT(-(x[1], 1), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) For Pair MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) the following chains were created: *We consider the chain MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]), COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) which results in the following constraint: (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE ==> MULT(x[0], y[0])_>=_NonInfC & MULT(x[0], y[0])_>=_COND_MULT(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) *(x[0] + [-1] >= 0 ==> (U^Increasing(MULT(-(x[1], 1), y[1])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x[0] >= 0 & [(-1)bso_12] >= 0) *MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_MULT(>(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x[0] >= 0 & [2 + (-1)bso_14] >= 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) = 0 POL(COND_MULT(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(MULT(x_1, x_2)) = [1] + [2]x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(>(x_1, x_2)) = [2] POL(0) = 0 The following pairs are in P_>: MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) The following pairs are in P_bound: COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) MULT(x[0], y[0]) -> COND_MULT(>(x[0], 0), x[0], y[0]) The following pairs are in P_>=: COND_MULT(TRUE, x[1], y[1]) -> MULT(-(x[1], 1), y[1]) There are no usable rules. ---------------------------------------- (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 < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer - ~ UnaryMinus: (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: (1): COND_MULT(TRUE, x[1], y[1]) -> MULT(x[1] - 1, y[1]) The set Q consists of the following terms: mult(x0, x1) Cond_mult(TRUE, x0, x1) Cond_mult1(TRUE, x0, x1) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (12) TRUE