/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.itrs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 291 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 15 ms] (10) IDP (11) IDependencyGraphProof [EQUIVALENT, 0 ms] (12) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x >= 0, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x + 1, 1) eval_2(x, y) -> Cond_eval_2(x >= 0 && y > 0 && x >= y, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x, y + 1) eval_2(x, y) -> Cond_eval_21(x >= 0 && y > 0 && y > x, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x - 2, y) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(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, Boolean The ITRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x >= 0, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x + 1, 1) eval_2(x, y) -> Cond_eval_2(x >= 0 && y > 0 && x >= y, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x, y + 1) eval_2(x, y) -> Cond_eval_21(x >= 0 && y > 0 && y > x, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x - 2, y) The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] >= 0, x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1] + 1, 1) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] >= 0 && y[2] > 0 && x[2] >= y[2], x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] + 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] >= 0 && y[4] > 0 && y[4] > x[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 2, y[5]) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] + 1 ->^* x[2] & 1 ->^* y[2]) (1) -> (4), if (x[1] + 1 ->^* x[4] & 1 ->^* y[4]) (2) -> (3), if (x[2] >= 0 && y[2] > 0 && x[2] >= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] + 1 ->^* y[4]) (4) -> (5), if (x[4] >= 0 && y[4] > 0 && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] - 2 ->^* x[0] & y[5] ->^* y[0]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(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, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] >= 0, x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1] + 1, 1) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] >= 0 && y[2] > 0 && x[2] >= y[2], x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] + 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] >= 0 && y[4] > 0 && y[4] > x[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 2, y[5]) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] + 1 ->^* x[2] & 1 ->^* y[2]) (1) -> (4), if (x[1] + 1 ->^* x[4] & 1 ->^* y[4]) (2) -> (3), if (x[2] >= 0 && y[2] > 0 && x[2] >= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] + 1 ->^* y[4]) (4) -> (5), if (x[4] >= 0 && y[4] > 0 && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] - 2 ->^* x[0] & y[5] ->^* y[0]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(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@133d64fc 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_1(x, y) -> COND_EVAL_1(>=(x, 0), x, y) the following chains were created: *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(+(x[1], 1), 1) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>=(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>=(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 ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>=(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>=(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] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(x[0], 0), x[0], y[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 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_1(>=(x[0], 0), x[0], y[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 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_1(>=(x[0], 0), x[0], y[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 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_1(>=(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(+(x, 1), 1) the following chains were created: *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(+(x[1], 1), 1), EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & +(x[1], 1)=x[2] & 1=y[2] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(+(x[1], 1), 1) & (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> COND_EVAL_1(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL_1(TRUE, x[0], y[0])_>=_EVAL_2(+(x[0], 1), 1) & (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 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_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & 0 = 0 & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & 0 = 0 & [1 + (-1)bso_27] >= 0) *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(+(x[1], 1), 1), EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & +(x[1], 1)=x[4] & 1=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(+(x[1], 1), 1) & (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> COND_EVAL_1(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL_1(TRUE, x[0], y[0])_>=_EVAL_2(+(x[0], 1), 1) & (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 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_2(+(x[1], 1), 1)), >=) & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [1 + (-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & 0 = 0 & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & 0 = 0 & [1 + (-1)bso_27] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_2(&&(&&(>=(x, 0), >(y, 0)), >=(x, y)), x, y) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], y[2])=TRUE & >=(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(x, +(y, 1)) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & +(y[3], 1)=y[2]1 ==> COND_EVAL_2(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3])_>=_EVAL_2(x[3], +(y[3], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], y[2])=TRUE & >=(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2])_>=_EVAL_2(x[2], +(y[2], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)), EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[4] & +(y[3], 1)=y[4] ==> COND_EVAL_2(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3])_>=_EVAL_2(x[3], +(y[3], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], y[2])=TRUE & >=(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2])_>=_EVAL_2(x[2], +(y[2], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_21(&&(&&(>=(x, 0), >(y, 0)), >(y, x)), x, y) the following chains were created: *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 2), y[5]) which results in the following constraint: (1) (&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], x[4])=TRUE & >=(x[4], 0)=TRUE & >(y[4], 0)=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(-(x, 2), y) the following chains were created: *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 2), y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]) which results in the following constraint: (1) (&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 2)=x[0] & y[5]=y[0] ==> COND_EVAL_21(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5])_>=_EVAL_1(-(x[5], 2), y[5]) & (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], x[4])=TRUE & >=(x[4], 0)=TRUE & >(y[4], 0)=TRUE ==> COND_EVAL_21(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4])_>=_EVAL_1(-(x[4], 2), y[4]) & (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x, y) -> COND_EVAL_1(>=(x, 0), x, y) *(x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(x[0], 0), x[0], y[0])), >=) & 0 = 0 & [bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) *COND_EVAL_1(TRUE, x, y) -> EVAL_2(+(x, 1), 1) *(x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & 0 = 0 & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & 0 = 0 & [1 + (-1)bso_27] >= 0) *(x[0] >= 0 ==> (U^Increasing(EVAL_2(+(x[1], 1), 1)), >=) & 0 = 0 & [bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & 0 = 0 & [1 + (-1)bso_27] >= 0) *EVAL_2(x, y) -> COND_EVAL_2(&&(&&(>=(x, 0), >(y, 0)), >=(x, y)), x, y) *(x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) *COND_EVAL_2(TRUE, x, y) -> EVAL_2(x, +(y, 1)) *(x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) *(x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] >= 0) *EVAL_2(x, y) -> COND_EVAL_21(&&(&&(>=(x, 0), >(y, 0)), >(y, x)), x, y) *(y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) *COND_EVAL_21(TRUE, x, y) -> EVAL_1(-(x, 2), y) *(y[4] + [-1] + [-1]x[4] >= 0 & x[4] >= 0 & y[4] + [-1] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 2), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 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_1(x_1, x_2)) = [1] + x_1 POL(COND_EVAL_1(x_1, x_2, x_3)) = [1] + x_2 POL(>=(x_1, x_2)) = 0 POL(0) = 0 POL(EVAL_2(x_1, x_2)) = [-1] + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_2 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(2) = [2] The following pairs are in P_>: COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(+(x[1], 1), 1) The following pairs are in P_bound: EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]) COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(+(x[1], 1), 1) EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 2), y[5]) The following pairs are in P_>=: EVAL_1(x[0], y[0]) -> COND_EVAL_1(>=(x[0], 0), x[0], y[0]) EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(&&(>=(x[4], 0), >(y[4], 0)), >(y[4], x[4])), x[4], y[4]) COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 2), y[5]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] >= 0, x[0], y[0]) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] >= 0 && y[2] > 0 && x[2] >= y[2], x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] + 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] >= 0 && y[4] > 0 && y[4] > x[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 2, y[5]) (5) -> (0), if (x[5] - 2 ->^* x[0] & y[5] ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (2) -> (3), if (x[2] >= 0 && y[2] > 0 && x[2] >= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (4), if (x[3] ->^* x[4] & y[3] + 1 ->^* y[4]) (4) -> (5), if (x[4] >= 0 && y[4] > 0 && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] + 1) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] >= 0 && y[2] > 0 && x[2] >= y[2], x[2], y[2]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (2) -> (3), if (x[2] >= 0 && y[2] > 0 && x[2] >= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(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@133d64fc 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_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & +(y[3], 1)=y[2]1 ==> COND_EVAL_2(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3])_>=_EVAL_2(x[3], +(y[3], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], y[2])=TRUE & >=(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2])_>=_EVAL_2(x[2], +(y[2], 1)) & (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]y[2] + [bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]y[2] + [bni_13]x[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) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]y[2] + [bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0) For Pair EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], y[2])=TRUE & >=(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) *(x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], +(y[3], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]y[2] + [bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0) *EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) *(x[2] + [-1]y[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 + [-1]x_1 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(&&(x_1, x_2)) = 0 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) The following pairs are in P_bound: COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], +(y[3], 1)) EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) The following pairs are in P_>=: EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(&&(>=(x[2], 0), >(y[2], 0)), >=(x[2], y[2])), x[2], y[2]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (10) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] >= 0 && y[2] > 0 && x[2] >= y[2], x[2], y[2]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (12) TRUE