/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 227 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 24 ms] (10) IDP (11) PisEmptyProof [EQUIVALENT, 0 ms] (12) 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 + ~ 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: eval(x, y) -> Cond_eval(x >= 0, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) eval(x, y) -> Cond_eval1(y >= 0, x, y) Cond_eval1(TRUE, x, y) -> eval(x, y - 1) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ 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 The ITRS R consists of the following rules: eval(x, y) -> Cond_eval(x >= 0, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) eval(x, y) -> Cond_eval1(y >= 0, x, y) Cond_eval1(TRUE, x, y) -> eval(x, y - 1) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ 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 R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@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 EVAL(x, y) -> COND_EVAL(>=(x, 0), x, y, z) the following chains were created: *We consider the chain COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: (1) (-(x[1], 1)=x[0] & z[1]=y[0] & >=(x[0], 0)=TRUE & x[0]=x[1]1 & y[0]=y[1]1 & z[0]=z[1]1 ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(-(x[1], 1), 0)=TRUE ==> EVAL(-(x[1], 1), z[1])_>=_NonInfC & EVAL(-(x[1], 1), z[1])_>=_COND_EVAL(>=(-(x[1], 1), 0), -(x[1], 1), z[1], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: (1) (x[3]=x[0] & -(y[3], 1)=y[0] & >=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> EVAL(x[0], -(y[3], 1))_>=_NonInfC & EVAL(x[0], -(y[3], 1))_>=_COND_EVAL(>=(x[0], 0), x[0], -(y[3], 1), z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 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] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)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] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)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] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & z[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & z[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) For Pair EVAL(x, y) -> COND_EVAL1(>=(y, 0), x, y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(2)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) For Pair COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) which results in the following constraint: (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[0] & -(y[3], 1)=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(>=(x, 0), x, y, z) *(x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) *(x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) *(x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) *(x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) *EVAL(x, y) -> COND_EVAL1(>=(y, 0), x, y) *(y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(2)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) *COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 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(EVAL(x_1, x_2)) = [-1] + [2]x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [2]x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [2]x_2 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 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 -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(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 -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(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@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_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) For Pair EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) *EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) *(y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-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_EVAL1(x_1, x_2, x_3)) = [-1] + [2]x_3 POL(EVAL(x_1, x_2)) = [2]x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(>=(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) The following pairs are in P_bound: COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) The following pairs are in P_>=: none 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 -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: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2) Cond_eval1(TRUE, x0, x1) ---------------------------------------- (11) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (12) YES