/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 293 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) IDP (9) IDPNonInfProof [SOUND, 29 ms] (10) IDP (11) IDependencyGraphProof [EQUIVALENT, 0 ms] (12) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval_1(i, j) -> Cond_eval_1(i >= 0, i, j) Cond_eval_1(TRUE, i, j) -> eval_2(i, 0) eval_2(i, j) -> Cond_eval_2(j <= i - 1, i, j) Cond_eval_2(TRUE, i, j) -> eval_2(i, j + 1) eval_2(i, j) -> Cond_eval_21(j > i - 1, i, j) Cond_eval_21(TRUE, i, j) -> eval_1(i - 1, j) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer The ITRS R consists of the following rules: eval_1(i, j) -> Cond_eval_1(i >= 0, i, j) Cond_eval_1(TRUE, i, j) -> eval_2(i, 0) eval_2(i, j) -> Cond_eval_2(j <= i - 1, i, j) Cond_eval_2(TRUE, i, j) -> eval_2(i, j + 1) eval_2(i, j) -> Cond_eval_21(j > i - 1, i, j) Cond_eval_21(TRUE, i, j) -> eval_1(i - 1, j) The integer pair graph contains the following rules and edges: (0): EVAL_1(i[0], j[0]) -> COND_EVAL_1(i[0] >= 0, i[0], j[0]) (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) (0) -> (1), if (i[0] >= 0 & i[0] ->^* i[1] & j[0] ->^* j[1]) (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) (5) -> (0), if (i[5] - 1 ->^* i[0] & j[5] ->^* j[0]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL_1(i[0], j[0]) -> COND_EVAL_1(i[0] >= 0, i[0], j[0]) (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) (0) -> (1), if (i[0] >= 0 & i[0] ->^* i[1] & j[0] ->^* j[1]) (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) (5) -> (0), if (i[5] - 1 ->^* i[0] & j[5] ->^* j[0]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@74477c5d Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL_1(i, j) -> COND_EVAL_1(>=(i, 0), i, j) the following chains were created: *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) which results in the following constraint: (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] ==> EVAL_1(i[0], j[0])_>=_NonInfC & EVAL_1(i[0], j[0])_>=_COND_EVAL_1(>=(i[0], 0), i[0], j[0]) & (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(i[0], 0)=TRUE ==> EVAL_1(i[0], j[0])_>=_NonInfC & EVAL_1(i[0], j[0])_>=_COND_EVAL_1(>=(i[0], 0), i[0], j[0]) & (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [(2)bni_24]i[0] >= 0 & [2 + (-1)bso_25] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [(2)bni_24]i[0] >= 0 & [2 + (-1)bso_25] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [bni_24 + (-1)Bound*bni_24] + [(2)bni_24]i[0] >= 0 & [2 + (-1)bso_25] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & 0 = 0 & [bni_24 + (-1)Bound*bni_24] + [(2)bni_24]i[0] >= 0 & [2 + (-1)bso_25] >= 0) For Pair COND_EVAL_1(TRUE, i, j) -> EVAL_2(i, 0) the following chains were created: *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] & i[1]=i[2] & 0=j[2] ==> COND_EVAL_1(TRUE, i[1], j[1])_>=_NonInfC & COND_EVAL_1(TRUE, i[1], j[1])_>=_EVAL_2(i[1], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(i[0], 0)=TRUE ==> COND_EVAL_1(TRUE, i[0], j[0])_>=_NonInfC & COND_EVAL_1(TRUE, i[0], j[0])_>=_EVAL_2(i[0], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0), EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) which results in the following constraint: (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] & i[1]=i[4] & 0=j[4] ==> COND_EVAL_1(TRUE, i[1], j[1])_>=_NonInfC & COND_EVAL_1(TRUE, i[1], j[1])_>=_EVAL_2(i[1], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(i[0], 0)=TRUE ==> COND_EVAL_1(TRUE, i[0], j[0])_>=_NonInfC & COND_EVAL_1(TRUE, i[0], j[0])_>=_EVAL_2(i[0], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) For Pair EVAL_2(i, j) -> COND_EVAL_2(<=(j, -(i, 1)), i, j) the following chains were created: *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) which results in the following constraint: (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (<=(j[2], -(i[2], 1))=TRUE ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (i[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [bni_28 + (-1)Bound*bni_28] + [(2)bni_28]j[2] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [bni_28 + (-1)Bound*bni_28] + [(-2)bni_28]j[2] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [bni_28 + (-1)Bound*bni_28] + [(2)bni_28]j[2] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) For Pair COND_EVAL_2(TRUE, i, j) -> EVAL_2(i, +(j, 1)) the following chains were created: *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[2]1 & +(j[3], 1)=j[2]1 ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(-2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) which results in the following constraint: (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[4] & +(j[3], 1)=j[4] ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(-2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) For Pair EVAL_2(i, j) -> COND_EVAL_21(>(j, -(i, 1)), i, j) the following chains were created: *We consider the chain EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]), COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]) which results in the following constraint: (1) (>(j[4], -(i[4], 1))=TRUE & i[4]=i[5] & j[4]=j[5] ==> EVAL_2(i[4], j[4])_>=_NonInfC & EVAL_2(i[4], j[4])_>=_COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) & (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(j[4], -(i[4], 1))=TRUE ==> EVAL_2(i[4], j[4])_>=_NonInfC & EVAL_2(i[4], j[4])_>=_COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) & (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (j[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) (8) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) For Pair COND_EVAL_21(TRUE, i, j) -> EVAL_1(-(i, 1), j) the following chains were created: *We consider the chain EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]), COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]), EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) which results in the following constraint: (1) (>(j[4], -(i[4], 1))=TRUE & i[4]=i[5] & j[4]=j[5] & -(i[5], 1)=i[0] & j[5]=j[0] ==> COND_EVAL_21(TRUE, i[5], j[5])_>=_NonInfC & COND_EVAL_21(TRUE, i[5], j[5])_>=_EVAL_1(-(i[5], 1), j[5]) & (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(j[4], -(i[4], 1))=TRUE ==> COND_EVAL_21(TRUE, i[4], j[4])_>=_NonInfC & COND_EVAL_21(TRUE, i[4], j[4])_>=_EVAL_1(-(i[4], 1), j[4]) & (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (j[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) (8) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL_1(i, j) -> COND_EVAL_1(>=(i, 0), i, j) *(i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & 0 = 0 & [bni_24 + (-1)Bound*bni_24] + [(2)bni_24]i[0] >= 0 & [2 + (-1)bso_25] >= 0) *COND_EVAL_1(TRUE, i, j) -> EVAL_2(i, 0) *(i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) *(i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) *EVAL_2(i, j) -> COND_EVAL_2(<=(j, -(i, 1)), i, j) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [bni_28 + (-1)Bound*bni_28] + [(-2)bni_28]j[2] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [bni_28 + (-1)Bound*bni_28] + [(2)bni_28]j[2] + [(2)bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) *COND_EVAL_2(TRUE, i, j) -> EVAL_2(i, +(j, 1)) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(-2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [bni_30 + (-1)Bound*bni_30] + [(-2)bni_30]j[2] + [(2)bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) *EVAL_2(i, j) -> COND_EVAL_21(>(j, -(i, 1)), i, j) *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-2)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) *COND_EVAL_21(TRUE, i, j) -> EVAL_1(-(i, 1), j) *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(2)bni_34]i[4] >= 0 & [(-1)bso_35] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL_1(x_1, x_2)) = [1] + [2]x_1 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(>=(x_1, x_2)) = [-1] POL(0) = 0 POL(EVAL_2(x_1, x_2)) = [-1] + [2]x_1 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(<=(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(+(x_1, x_2)) = x_1 + x_2 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + [2]x_2 POL(>(x_1, x_2)) = [-1] The following pairs are in P_>: EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) The following pairs are in P_bound: EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) The following pairs are in P_>=: COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]) There are no usable rules. ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (9) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@74477c5d Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) the following chains were created: *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[2]1 & +(j[3], 1)=j[2]1 ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) For Pair EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) the following chains were created: *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) which results in the following constraint: (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (<=(j[2], -(i[2], 1))=TRUE ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (i[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13 + bni_13] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13 + bni_13] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13 + bni_13] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [(-1)bso_12] >= 0) *EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13 + bni_13] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13 + bni_13] + [bni_13]i[2] >= 0 & [1 + (-1)bso_14] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(EVAL_2(x_1, x_2)) = [-1]x_2 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(<=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 The following pairs are in P_>: EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) The following pairs are in P_bound: COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) The following pairs are in P_>=: COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) There are no usable rules. ---------------------------------------- (10) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) The set Q consists of the following terms: eval_1(x0, x1) Cond_eval_1(TRUE, x0, x1) eval_2(x0, x1) Cond_eval_2(TRUE, x0, x1) Cond_eval_21(TRUE, x0, x1) ---------------------------------------- (11) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (12) TRUE