/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDependencyGraphProof [EQUIVALENT, 0 ms] (6) IDP (7) IDPNonInfProof [SOUND, 456 ms] (8) IDP (9) IDependencyGraphProof [EQUIVALENT, 0 ms] (10) 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_0(x, y, z) -> Cond_eval_0(y > 0, x, y, z) Cond_eval_0(TRUE, x, y, z) -> eval_1(x, y, z) eval_1(x, y, z) -> Cond_eval_1(y > x && z > y && y > 0, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_1(x + y, y, z) eval_1(x, y, z) -> Cond_eval_11(y > x && z > y && y > 0, x, y, z) Cond_eval_11(TRUE, x, y, z) -> eval_1(x, y, x - y) The set Q consists of the following terms: eval_0(x0, x1, x2) Cond_eval_0(TRUE, x0, x1, x2) eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) Cond_eval_11(TRUE, x0, x1, x2) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean The ITRS R consists of the following rules: eval_0(x, y, z) -> Cond_eval_0(y > 0, x, y, z) Cond_eval_0(TRUE, x, y, z) -> eval_1(x, y, z) eval_1(x, y, z) -> Cond_eval_1(y > x && z > y && y > 0, x, y, z) Cond_eval_1(TRUE, x, y, z) -> eval_1(x + y, y, z) eval_1(x, y, z) -> Cond_eval_11(y > x && z > y && y > 0, x, y, z) Cond_eval_11(TRUE, x, y, z) -> eval_1(x, y, x - y) The integer pair graph contains the following rules and edges: (0): EVAL_0(x[0], y[0], z[0]) -> COND_EVAL_0(y[0] > 0, x[0], y[0], z[0]) (1): COND_EVAL_0(TRUE, x[1], y[1], z[1]) -> EVAL_1(x[1], y[1], z[1]) (2): EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(y[2] > x[2] && z[2] > y[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3] + y[3], y[3], z[3]) (4): EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(y[4] > x[4] && z[4] > y[4] && y[4] > 0, x[4], y[4], z[4]) (5): COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], x[5] - y[5]) (0) -> (1), if (y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (2) -> (3), if (y[2] > x[2] && z[2] > y[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (2), if (x[3] + y[3] ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] + y[3] ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (y[4] > x[4] && z[4] > y[4] && y[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & x[5] - y[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & x[5] - y[5] ->^* z[4]) The set Q consists of the following terms: eval_0(x0, x1, x2) Cond_eval_0(TRUE, x0, x1, x2) eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) Cond_eval_11(TRUE, x0, x1, x2) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_0(x[0], y[0], z[0]) -> COND_EVAL_0(y[0] > 0, x[0], y[0], z[0]) (1): COND_EVAL_0(TRUE, x[1], y[1], z[1]) -> EVAL_1(x[1], y[1], z[1]) (2): EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(y[2] > x[2] && z[2] > y[2] && y[2] > 0, x[2], y[2], z[2]) (3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3] + y[3], y[3], z[3]) (4): EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(y[4] > x[4] && z[4] > y[4] && y[4] > 0, x[4], y[4], z[4]) (5): COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], x[5] - y[5]) (0) -> (1), if (y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) (2) -> (3), if (y[2] > x[2] && z[2] > y[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (2), if (x[3] + y[3] ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (3) -> (4), if (x[3] + y[3] ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) (4) -> (5), if (y[4] > x[4] && z[4] > y[4] && y[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & x[5] - y[5] ->^* z[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & x[5] - y[5] ->^* z[4]) The set Q consists of the following terms: eval_0(x0, x1, x2) Cond_eval_0(TRUE, x0, x1, x2) eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) Cond_eval_11(TRUE, x0, x1, x2) ---------------------------------------- (5) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (5): COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], x[5] - y[5]) (4): EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(y[4] > x[4] && z[4] > y[4] && y[4] > 0, x[4], y[4], z[4]) (3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3] + y[3], y[3], z[3]) (2): EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(y[2] > x[2] && z[2] > y[2] && y[2] > 0, x[2], y[2], z[2]) (3) -> (2), if (x[3] + y[3] ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2] & x[5] - y[5] ->^* z[2]) (2) -> (3), if (y[2] > x[2] && z[2] > y[2] && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (4), if (x[3] + y[3] ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4] & x[5] - y[5] ->^* z[4]) (4) -> (5), if (y[4] > x[4] && z[4] > y[4] && y[4] > 0 & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) The set Q consists of the following terms: eval_0(x0, x1, x2) Cond_eval_0(TRUE, x0, x1, x2) eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) Cond_eval_11(TRUE, x0, x1, x2) ---------------------------------------- (7) 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@592a876a 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_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])) the following chains were created: *We consider the chain EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]), COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])), EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[2] & y[5]=y[2] & -(x[5], y[5])=z[2] ==> COND_EVAL_11(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_11(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], -(x[5], y[5])) & (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], 0)=TRUE & >(y[4], x[4])=TRUE & >(z[4], y[4])=TRUE ==> COND_EVAL_11(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_11(TRUE, x[4], y[4], z[4])_>=_EVAL_1(x[4], y[4], -(x[4], y[4])) & (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) (7) (y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + x[4] >= 0) *We consider the chain EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]), COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])), EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) which results in the following constraint: (1) (&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & x[5]=x[4]1 & y[5]=y[4]1 & -(x[5], y[5])=z[4]1 ==> COND_EVAL_11(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_11(TRUE, x[5], y[5], z[5])_>=_EVAL_1(x[5], y[5], -(x[5], y[5])) & (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], 0)=TRUE & >(y[4], x[4])=TRUE & >(z[4], y[4])=TRUE ==> COND_EVAL_11(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_11(TRUE, x[4], y[4], z[4])_>=_EVAL_1(x[4], y[4], -(x[4], y[4])) & (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) (7) (y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + x[4] >= 0) For Pair EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) the following chains were created: *We consider the chain EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]), COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])) which results in the following constraint: (1) (&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL_1(x[4], y[4], z[4])_>=_NonInfC & EVAL_1(x[4], y[4], z[4])_>=_COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[4], 0)=TRUE & >(y[4], x[4])=TRUE & >(z[4], y[4])=TRUE ==> EVAL_1(x[4], y[4], z[4])_>=_NonInfC & EVAL_1(x[4], y[4], z[4])_>=_COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) (7) (y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) For Pair COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]) the following chains were created: *We consider the chain EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]), EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & +(x[3], y[3])=x[2]1 & y[3]=y[2]1 & z[3]=z[2]1 ==> COND_EVAL_1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_1(TRUE, x[3], y[3], z[3])_>=_EVAL_1(+(x[3], y[3]), y[3], z[3]) & (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(z[2], y[2])=TRUE ==> COND_EVAL_1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_1(TRUE, x[2], y[2], z[2])_>=_EVAL_1(+(x[2], y[2]), y[2], z[2]) & (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) (7) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *We consider the chain EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]), EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) which results in the following constraint: (1) (&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & +(x[3], y[3])=x[4] & y[3]=y[4] & z[3]=z[4] ==> COND_EVAL_1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_1(TRUE, x[3], y[3], z[3])_>=_EVAL_1(+(x[3], y[3]), y[3], z[3]) & (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(z[2], y[2])=TRUE ==> COND_EVAL_1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_1(TRUE, x[2], y[2], z[2])_>=_EVAL_1(+(x[2], y[2]), y[2], z[2]) & (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) (7) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) For Pair EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) the following chains were created: *We consider the chain EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]), COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]) which results in the following constraint: (1) (&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL_1(x[2], y[2], z[2])_>=_NonInfC & EVAL_1(x[2], y[2], z[2])_>=_COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(z[2], y[2])=TRUE ==> EVAL_1(x[2], y[2], z[2])_>=_NonInfC & EVAL_1(x[2], y[2], z[2])_>=_COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) (7) (y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])) *(y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) *(y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + x[4] >= 0) *(y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [(-1)bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + [-1]x[4] >= 0) *(y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5], -(x[5], y[5]))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]z[4] + [(-1)bni_19]y[4] + [bni_19]x[4] >= 0 & [(-1)bso_20] + z[4] + x[4] >= 0) *EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) *(y[4] + [-1] >= 0 & y[4] + [-1] + [-1]x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) *(y[4] + [-1] >= 0 & y[4] + [-1] + x[4] >= 0 & z[4] + [-1] + [-1]y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]z[4] + [bni_21]x[4] >= 0 & [(-1)bso_22] + y[4] >= 0) *COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]) *(y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *(y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *(y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *(y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL_1(+(x[3], y[3]), y[3], z[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]z[2] + [(-1)bni_23]y[2] + [(-1)bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) *EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) *(y[2] + [-1] >= 0 & y[2] + [-1] + x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) *(y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & z[2] + [-1] + [-1]y[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]z[2] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] + y[2] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [3] POL(COND_EVAL_11(x_1, x_2, x_3, x_4)) = [-1] + x_4 + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(EVAL_1(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + x_4 + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(+(x_1, x_2)) = x_1 + x_2 The following pairs are in P_>: COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], -(x[5], y[5])) EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) The following pairs are in P_bound: EVAL_1(x[4], y[4], z[4]) -> COND_EVAL_11(&&(&&(>(y[4], x[4]), >(z[4], y[4])), >(y[4], 0)), x[4], y[4], z[4]) EVAL_1(x[2], y[2], z[2]) -> COND_EVAL_1(&&(&&(>(y[2], x[2]), >(z[2], y[2])), >(y[2], 0)), x[2], y[2], z[2]) The following pairs are in P_>=: COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(+(x[3], y[3]), y[3], z[3]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 <-> TRUE^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (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: (5): COND_EVAL_11(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5], y[5], x[5] - y[5]) (3): COND_EVAL_1(TRUE, x[3], y[3], z[3]) -> EVAL_1(x[3] + y[3], y[3], z[3]) The set Q consists of the following terms: eval_0(x0, x1, x2) Cond_eval_0(TRUE, x0, x1, x2) eval_1(x0, x1, x2) Cond_eval_1(TRUE, x0, x1, x2) Cond_eval_11(TRUE, x0, x1, x2) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (10) TRUE