/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 136 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) 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: b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38) Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38) b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1) b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && (1 < sv14_14 && (TRUE && (0 < sv24_38 && 1 < sv23_37))), sv14_14, sv23_37, sv24_38) The set Q consists of the following terms: b10(x0, x1, x2) Cond_b14(TRUE, x0, x1, x2) b15(x0, x1, x2) b14(x0, x1, x2) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean The ITRS R consists of the following rules: b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38) Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38) b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1) b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && (1 < sv14_14 && (TRUE && (0 < sv24_38 && 1 < sv23_37))), sv14_14, sv23_37, sv24_38) The integer pair graph contains the following rules and edges: (0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) (1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) (2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1) (3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))), sv14_14[3], sv23_37[3], sv24_38[3]) (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3]) (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2]) (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0]) (3) -> (1), if (sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))) & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1]) The set Q consists of the following terms: b10(x0, x1, x2) Cond_b14(TRUE, x0, x1, x2) b15(x0, x1, x2) b14(x0, x1, x2) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) (1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) (2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1) (3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))), sv14_14[3], sv23_37[3], sv24_38[3]) (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3]) (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2]) (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0]) (3) -> (1), if (sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))) & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1]) The set Q consists of the following terms: b10(x0, x1, x2) Cond_b14(TRUE, x0, x1, x2) b15(x0, x1, x2) b14(x0, x1, x2) ---------------------------------------- (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@589a4faa 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 B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38) the following chains were created: *We consider the chain B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]), B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) which results in the following constraint: (1) (sv14_14[0]=sv14_14[3] & sv23_37[0]=sv23_37[3] & sv24_38[0]=sv24_38[3] ==> B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_NonInfC & B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_B14(sv14_14[0], sv23_37[0], sv24_38[0]) & (U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_NonInfC & B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_B14(sv14_14[0], sv23_37[0], sv24_38[0]) & (U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) For Pair COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38) the following chains were created: *We consider the chain COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)) which results in the following constraint: (1) (sv14_14[1]=sv14_14[2] & sv23_37[1]=sv23_37[2] & sv24_38[1]=sv24_38[2] ==> COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_B15(sv14_14[1], sv23_37[1], sv24_38[1]) & (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_B15(sv14_14[1], sv23_37[1], sv24_38[1]) & (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0) For Pair B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1)) the following chains were created: *We consider the chain COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)), B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) which results in the following constraint: (1) (sv14_14[1]=sv14_14[2] & sv23_37[1]=sv23_37[2] & sv24_38[1]=sv24_38[2] & sv14_14[2]=sv14_14[0] & -(sv23_37[2], sv14_14[2])=sv23_37[0] & +(sv24_38[2], 1)=sv24_38[0] ==> B15(sv14_14[2], sv23_37[2], sv24_38[2])_>=_NonInfC & B15(sv14_14[2], sv23_37[2], sv24_38[2])_>=_B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)) & (U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (B15(sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & B15(sv14_14[1], sv23_37[1], sv24_38[1])_>=_B10(sv14_14[1], -(sv23_37[1], sv14_14[1]), +(sv24_38[1], 1)) & (U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & 0 = 0 & [(-1)bso_28] >= 0) For Pair B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), &&(<(1, sv14_14), &&(TRUE, &&(<(0, sv24_38), <(1, sv23_37))))), sv14_14, sv23_37, sv24_38) the following chains were created: *We consider the chain B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]), COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) which results in the following constraint: (1) (&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3])))))=TRUE & sv14_14[3]=sv14_14[1] & sv23_37[3]=sv23_37[1] & sv24_38[3]=sv24_38[1] ==> B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_NonInfC & B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=)) We simplified constraint (1) using rules (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(sv23_37[3], sv14_14[3])=TRUE & <(1, sv14_14[3])=TRUE & <(0, sv24_38[3])=TRUE & <(1, sv23_37[3])=TRUE ==> B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_NonInfC & B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0 ==> (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-1 + (-1)bso_30] + sv14_14[3] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0 ==> (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-1 + (-1)bso_30] + sv14_14[3] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0 ==> (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-1 + (-1)bso_30] + sv14_14[3] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38) *((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) *COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38) *((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0) *B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1)) *((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & 0 = 0 & [(-1)bso_28] >= 0) *B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), &&(<(1, sv14_14), &&(TRUE, &&(<(0, sv24_38), <(1, sv23_37))))), sv14_14, sv23_37, sv24_38) *(sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0 ==> (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-1 + (-1)bso_30] + sv14_14[3] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [3] POL(B10(x_1, x_2, x_3)) = [-1] + x_2 + [2]x_1 POL(B14(x_1, x_2, x_3)) = [-1] + x_2 + [2]x_1 POL(COND_B14(x_1, x_2, x_3, x_4)) = x_3 + x_2 + [-1]x_1 POL(B15(x_1, x_2, x_3)) = [-1] + x_2 + x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(&&(x_1, x_2)) = 0 POL(>=(x_1, x_2)) = [-1] POL(<(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) The following pairs are in P_bound: B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) The following pairs are in P_>=: B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)) 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 + ~ 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): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) (1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) (2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1) (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0]) (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2]) The set Q consists of the following terms: b10(x0, x1, x2) Cond_b14(TRUE, x0, x1, x2) b15(x0, x1, x2) b14(x0, x1, x2) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (8) TRUE