/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 194 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 6 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 > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x = y && x > 0, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x, y) eval_2(x, y) -> Cond_eval_2(y > 0, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y - 1) eval_2(x, y) -> Cond_eval_21(0 >= y, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x, 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 > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval_1(x, y) -> Cond_eval_1(x = y && x > 0, x, y) Cond_eval_1(TRUE, x, y) -> eval_2(x, y) eval_2(x, y) -> Cond_eval_2(y > 0, x, y) Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y - 1) eval_2(x, y) -> Cond_eval_21(0 >= y, x, y) Cond_eval_21(TRUE, x, y) -> eval_1(x, y) The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* 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 > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* 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@185fc1d 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, y), >(x, 0)), x, y) the following chains were created: *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: (1) (&&(=(x[0], y[0]), >(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], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(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 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] + [2]y[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] + [2]y[0] >= 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] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] + [2]y[0] >= 0) For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) the following chains were created: *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) which results in the following constraint: (1) (x[1]=x[2] & y[1]=y[2] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) which results in the following constraint: (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_2(>(y, 0), x, y) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) which results in the following constraint: (1) (>(y[2], 0)=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(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(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_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(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] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), -(y, 1)) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(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], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) which results in the following constraint: (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [bni_31]x[2] >= 0 & [(-1)bso_32] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) For Pair EVAL_2(x, y) -> COND_EVAL_21(>=(0, y), x, y) the following chains were created: *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) which results in the following constraint: (1) (>=(0, y[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(>=(0, y[4]), x[4], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(0, y[4])=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(>=(0, y[4]), x[4], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-2]y[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-2]y[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-2]y[4] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] >= 0 & [(-1)bso_34] + [-2]y[4] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]y[4] >= 0 & [(-1)bso_34] + [2]y[4] >= 0) For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: *We consider the chain COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) which results in the following constraint: (1) (x[5]=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], y[5]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_EVAL_21(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5])_>=_EVAL_1(x[5], y[5]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(x, y) -> COND_EVAL_1(&&(=(x, y), >(x, 0)), x, y) *(x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] + [2]y[0] >= 0) *COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) *EVAL_2(x, y) -> COND_EVAL_2(>(y, 0), x, y) *(y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] >= 0 & [(-1)bso_30] >= 0) *COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), -(y, 1)) *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] >= 0 & [(-1)bso_32] >= 0) *EVAL_2(x, y) -> COND_EVAL_21(>=(0, y), x, y) *(y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]y[4] >= 0 & [(-1)bso_34] + [2]y[4] >= 0) *COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) *((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 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(EVAL_1(x_1, x_2)) = [-1] + x_2 + x_1 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(&&(x_1, x_2)) = 0 POL(=(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 + x_1 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_3 + x_2 POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) The following pairs are in P_bound: EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) The following pairs are in P_>=: COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 -> TRUE^1 FALSE^1 -> &&(FALSE, TRUE)^1 ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) (4) -> (5), if (0 >= y[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 > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer 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] - 1, y[3] - 1) (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) (3) -> (2), if (x[3] - 1 ->^* 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_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@185fc1d 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], 1), -(y[3], 1)) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(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], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & 0 = 0 & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) For Pair EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) the following chains were created: *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) which results in the following constraint: (1) (>(y[2], 0)=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(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(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_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(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] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 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], 1), -(y[3], 1)) *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & 0 = 0 & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) *EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) *(y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 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_EVAL_2(x_1, x_2, x_3)) = [2]x_3 POL(EVAL_2(x_1, x_2)) = [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_>: EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) The following pairs are in P_bound: COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) The following pairs are in P_>=: COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 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 > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer 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] - 1, y[3] - 1) 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