/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 288 ms] (6) IDP (7) IDPNonInfProof [SOUND, 133 ms] (8) IDP (9) IDependencyGraphProof [EQUIVALENT, 0 ms] (10) IDP (11) IDPNonInfProof [SOUND, 0 ms] (12) IDP (13) IDependencyGraphProof [EQUIVALENT, 0 ms] (14) 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(x, y) -> Cond_eval(x > y && x > 0 && y > 0, x, y) Cond_eval(TRUE, x, y) -> eval(x - y, y) eval(x, y) -> Cond_eval1(y > x && x > 0 && y > 0 && x > y, x, y) Cond_eval1(TRUE, x, y) -> eval(x - y, y) eval(x, y) -> Cond_eval2(x > y && x > 0 && y > 0 && y >= x, x, y) Cond_eval2(TRUE, x, y) -> eval(x, y - x) eval(x, y) -> Cond_eval3(y > x && x > 0 && y > 0 && y >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x, y - x) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(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(x, y) -> Cond_eval(x > y && x > 0 && y > 0, x, y) Cond_eval(TRUE, x, y) -> eval(x - y, y) eval(x, y) -> Cond_eval1(y > x && x > 0 && y > 0 && x > y, x, y) Cond_eval1(TRUE, x, y) -> eval(x - y, y) eval(x, y) -> Cond_eval2(x > y && x > 0 && y > 0 && y >= x, x, y) Cond_eval2(TRUE, x, y) -> eval(x, y - x) eval(x, y) -> Cond_eval3(y > x && x > 0 && y > 0 && y >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x, y - x) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - y[3], y[3]) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4], x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - x[5]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] - y[1] ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) (2) -> (3), if (y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] - y[3] ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (x[3] - y[3] ->^* x[2] & y[3] ->^* y[2]) (3) -> (4), if (x[3] - y[3] ->^* x[4] & y[3] ->^* y[4]) (3) -> (6), if (x[3] - y[3] ->^* x[6] & y[3] ->^* y[6]) (4) -> (5), if (x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - x[5] ->^* y[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - x[5] ->^* y[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - x[5] ->^* y[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - x[5] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - x[7] ->^* y[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - x[7] ->^* y[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(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(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - y[3], y[3]) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4], x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - x[5]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (4), if (x[1] - y[1] ->^* x[4] & y[1] ->^* y[4]) (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) (2) -> (3), if (y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] - y[3] ->^* x[0] & y[3] ->^* y[0]) (3) -> (2), if (x[3] - y[3] ->^* x[2] & y[3] ->^* y[2]) (3) -> (4), if (x[3] - y[3] ->^* x[4] & y[3] ->^* y[4]) (3) -> (6), if (x[3] - y[3] ->^* x[6] & y[3] ->^* y[6]) (4) -> (5), if (x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] ->^* x[0] & y[5] - x[5] ->^* y[0]) (5) -> (2), if (x[5] ->^* x[2] & y[5] - x[5] ->^* y[2]) (5) -> (4), if (x[5] ->^* x[4] & y[5] - x[5] ->^* y[4]) (5) -> (6), if (x[5] ->^* x[6] & y[5] - x[5] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) (7) -> (2), if (x[7] ->^* x[2] & y[7] - x[7] ->^* y[2]) (7) -> (4), if (x[7] ->^* x[4] & y[7] - x[7] ->^* y[4]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(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@315f1419 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x, y) -> COND_EVAL(&&(&&(>(x, y), >(x, 0)), >(y, 0)), x, y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(2)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(2)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(2)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] >= 0) For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, y), y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], y[1]), y[1]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], y[0]), y[0]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(2)bni_27 + (-1)Bound*bni_27] >= 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(2)bni_27 + (-1)Bound*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) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(2)bni_27 + (-1)Bound*bni_27] >= 0 & [(-1)bso_28] >= 0) For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >(x, y)), x, y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) which results in the following constraint: (1) (&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[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 & >(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[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] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(2)bni_29 + (-1)Bound*bni_29] >= 0 & [3 + (-1)bso_30] + y[2] + x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(2)bni_29 + (-1)Bound*bni_29] >= 0 & [3 + (-1)bso_30] + y[2] + x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(2)bni_29 + (-1)Bound*bni_29] >= 0 & [3 + (-1)bso_30] + y[2] + x[2] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL1(TRUE, x, y) -> EVAL(-(x, y), y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) which results in the following constraint: (1) (&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], y[3]), y[3]) & (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[2], y[2])=TRUE & >(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], y[2]), y[2]) & (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [-3 + (-1)bso_32] + [-1]y[2] + [-1]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [-3 + (-1)bso_32] + [-1]y[2] + [-1]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [-3 + (-1)bso_32] + [-1]y[2] + [-1]x[2] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y) -> COND_EVAL2(&&(&&(&&(>(x, y), >(x, 0)), >(y, 0)), >=(y, x)), x, y) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) which results in the following constraint: (1) (&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(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 & >(y[4], 0)=TRUE & >(x[4], y[4])=TRUE & >(x[4], 0)=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(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]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(2)bni_33 + (-1)Bound*bni_33] >= 0 & [3 + (-1)bso_34] + y[4] + x[4] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(2)bni_33 + (-1)Bound*bni_33] >= 0 & [3 + (-1)bso_34] + y[4] + x[4] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(2)bni_33 + (-1)Bound*bni_33] >= 0 & [3 + (-1)bso_34] + y[4] + x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, x)) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) which results in the following constraint: (1) (&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], -(y[5], x[5])) & (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[4], x[4])=TRUE & >(y[4], 0)=TRUE & >(x[4], y[4])=TRUE & >(x[4], 0)=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(x[4], -(y[4], x[4])) & (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [-3 + (-1)bso_36] + [-1]y[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]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [-3 + (-1)bso_36] + [-1]y[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]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [-3 + (-1)bso_36] + [-1]y[4] + [-1]x[4] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y) -> COND_EVAL3(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >=(y, x)), x, y) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(2)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(2)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(2)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) For Pair COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, x)) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], x[7])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], x[6])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(2)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(2)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(2)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(&&(&&(>(x, y), >(x, 0)), >(y, 0)), x, y) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(2)bni_25 + (-1)Bound*bni_25] >= 0 & [(-1)bso_26] >= 0) *COND_EVAL(TRUE, x, y) -> EVAL(-(x, y), y) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(2)bni_27 + (-1)Bound*bni_27] >= 0 & [(-1)bso_28] >= 0) *EVAL(x, y) -> COND_EVAL1(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >(x, y)), x, y) *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, y), y) *EVAL(x, y) -> COND_EVAL2(&&(&&(&&(>(x, y), >(x, 0)), >(y, 0)), >=(y, x)), x, y) *COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, x)) *EVAL(x, y) -> COND_EVAL3(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >=(y, x)), x, y) *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(2)bni_37 + (-1)Bound*bni_37] >= 0 & [(-1)bso_38] >= 0) *COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, x)) *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(2)bni_39 + (-1)Bound*bni_39] >= 0 & [(-1)bso_40] >= 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) = [2] POL(EVAL(x_1, x_2)) = [2] POL(COND_EVAL(x_1, x_2, x_3)) = [2] + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] POL(COND_EVAL3(x_1, x_2, x_3)) = [2] + [-1]x_1 The following pairs are in P_>: EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 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 ---------------------------------------- (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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (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@315f1419 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], y[1])=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], y[1]), y[1]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], y[0]), y[0]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], y[1])=x[6] & y[1]=y[6] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], y[1]), y[1]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], y[0]), y[0]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) For Pair EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[6] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[6] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[6] >= 0 & [(-1)bso_22] >= 0) For Pair COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) which results in the following constraint: (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[0] & -(y[7], x[7])=y[0] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], x[7])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], x[6])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])), EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) which results in the following constraint: (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[6]1 & -(y[7], x[7])=y[6]1 ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], x[7])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], x[6])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]y[0] >= 0 & [(-1)bso_18] >= 0) *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] >= 0 & [(-1)bso_20] >= 0) *EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[6] >= 0 & [(-1)bso_22] >= 0) *COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 0) *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[6] >= 0 & [(-1)bso_24] + x[6] >= 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(x_1, x_2)) = [-1] + x_2 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_3 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + x_3 POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 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 ---------------------------------------- (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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (11) 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@315f1419 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(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], y[1])=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], y[1]), y[1]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], y[0]), y[0]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) For Pair EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] + y[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] + y[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] + y[0] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) *EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] + y[0] >= 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(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 + [-1]x_1 POL(EVAL(x_1, x_2)) = [-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 The following pairs are in P_>: EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) The following pairs are in P_bound: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) The following pairs are in P_>=: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) 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 ---------------------------------------- (12) 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(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (13) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (14) TRUE