/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDependencyGraphProof [EQUIVALENT, 0 ms] (6) IDP (7) IDPNonInfProof [SOUND, 180 ms] (8) AND (9) IDP (10) IDependencyGraphProof [EQUIVALENT, 0 ms] (11) TRUE (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 + ~ 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: div(x, y) -> divNat(x >= 0 && y >= 1, x, y) divNat(TRUE, x, y) -> d(x, y, 0) d(x, y, z) -> dNat(x >= 0 && y >= 1 && z >= 0, x, y, z) dNat(TRUE, x, y, z) -> cond(x >= z, x, y - 1, z) cond(TRUE, x, y, z) -> 1 + d(x, y + 1, y + 1 + z) cond(FALSE, x, y, z) -> 0 The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, 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: Boolean, Integer The ITRS R consists of the following rules: div(x, y) -> divNat(x >= 0 && y >= 1, x, y) divNat(TRUE, x, y) -> d(x, y, 0) d(x, y, z) -> dNat(x >= 0 && y >= 1 && z >= 0, x, y, z) dNat(TRUE, x, y, z) -> cond(x >= z, x, y - 1, z) cond(TRUE, x, y, z) -> 1 + d(x, y + 1, y + 1 + z) cond(FALSE, x, y, z) -> 0 The integer pair graph contains the following rules and edges: (0): DIV(x[0], y[0]) -> DIVNAT(x[0] >= 0 && y[0] >= 1, x[0], y[0]) (1): DIVNAT(TRUE, x[1], y[1]) -> D(x[1], y[1], 0) (2): D(x[2], y[2], z[2]) -> DNAT(x[2] >= 0 && y[2] >= 1 && z[2] >= 0, x[2], y[2], z[2]) (3): DNAT(TRUE, x[3], y[3], z[3]) -> COND(x[3] >= z[3], x[3], y[3] - 1, z[3]) (4): COND(TRUE, x[4], y[4], z[4]) -> D(x[4], y[4] + 1, y[4] + 1 + z[4]) (0) -> (1), if (x[0] >= 0 && y[0] >= 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & 0 ->^* z[2]) (2) -> (3), if (x[2] >= 0 && y[2] >= 1 && z[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (4), if (x[3] >= z[3] & x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (2), if (x[4] ->^* x[2] & y[4] + 1 ->^* y[2] & y[4] + 1 + z[4] ->^* z[2]) The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, 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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): DIV(x[0], y[0]) -> DIVNAT(x[0] >= 0 && y[0] >= 1, x[0], y[0]) (1): DIVNAT(TRUE, x[1], y[1]) -> D(x[1], y[1], 0) (2): D(x[2], y[2], z[2]) -> DNAT(x[2] >= 0 && y[2] >= 1 && z[2] >= 0, x[2], y[2], z[2]) (3): DNAT(TRUE, x[3], y[3], z[3]) -> COND(x[3] >= z[3], x[3], y[3] - 1, z[3]) (4): COND(TRUE, x[4], y[4], z[4]) -> D(x[4], y[4] + 1, y[4] + 1 + z[4]) (0) -> (1), if (x[0] >= 0 && y[0] >= 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & 0 ->^* z[2]) (2) -> (3), if (x[2] >= 0 && y[2] >= 1 && z[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (4), if (x[3] >= z[3] & x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) (4) -> (2), if (x[4] ->^* x[2] & y[4] + 1 ->^* y[2] & y[4] + 1 + z[4] ->^* z[2]) The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, x0, x1, x2) ---------------------------------------- (5) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ 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: (4): COND(TRUE, x[4], y[4], z[4]) -> D(x[4], y[4] + 1, y[4] + 1 + z[4]) (3): DNAT(TRUE, x[3], y[3], z[3]) -> COND(x[3] >= z[3], x[3], y[3] - 1, z[3]) (2): D(x[2], y[2], z[2]) -> DNAT(x[2] >= 0 && y[2] >= 1 && z[2] >= 0, x[2], y[2], z[2]) (4) -> (2), if (x[4] ->^* x[2] & y[4] + 1 ->^* y[2] & y[4] + 1 + z[4] ->^* z[2]) (2) -> (3), if (x[2] >= 0 && y[2] >= 1 && z[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) (3) -> (4), if (x[3] >= z[3] & x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, x0, x1, x2) ---------------------------------------- (7) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@65d2247a 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(TRUE, x[4], y[4], z[4]) -> D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])) the following chains were created: *We consider the chain DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]), COND(TRUE, x[4], y[4], z[4]) -> D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])), D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) which results in the following constraint: (1) (>=(x[3], z[3])=TRUE & x[3]=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] & x[4]=x[2] & +(y[4], 1)=y[2] & +(+(y[4], 1), z[4])=z[2] ==> COND(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND(TRUE, x[4], y[4], z[4])_>=_D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])) & (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>=(x[3], z[3])=TRUE ==> COND(TRUE, x[3], -(y[3], 1), z[3])_>=_NonInfC & COND(TRUE, x[3], -(y[3], 1), z[3])_>=_D(x[3], +(-(y[3], 1), 1), +(+(-(y[3], 1), 1), z[3])) & (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[3] + [-1]z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)Bound*bni_19] + [(-1)bni_19]z[3] + [(-1)bni_19]y[3] + [bni_19]x[3] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[3] + [-1]z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)Bound*bni_19] + [(-1)bni_19]z[3] + [(-1)bni_19]y[3] + [bni_19]x[3] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[3] + [-1]z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)Bound*bni_19] + [(-1)bni_19]z[3] + [(-1)bni_19]y[3] + [bni_19]x[3] >= 0 & [(-1)bso_20] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[3] + [-1]z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [(-1)bni_19]z[3] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[3] >= 0 & z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) (9) (x[3] >= 0 & z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) For Pair DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) the following chains were created: *We consider the chain D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]), DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]), COND(TRUE, x[4], y[4], z[4]) -> D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & >=(x[3], z[3])=TRUE & x[3]=x[4] & -(y[3], 1)=y[4] & z[3]=z[4] ==> DNAT(TRUE, x[3], y[3], z[3])_>=_NonInfC & DNAT(TRUE, x[3], y[3], z[3])_>=_COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) & (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[2], z[2])=TRUE & >=(z[2], 0)=TRUE & >=(x[2], 0)=TRUE & >=(y[2], 1)=TRUE ==> DNAT(TRUE, x[2], y[2], z[2])_>=_NonInfC & DNAT(TRUE, x[2], y[2], z[2])_>=_COND(>=(x[2], z[2]), x[2], -(y[2], 1), z[2]) & (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[2] + [-1]z[2] >= 0 & z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[2] + [-1]z[2] >= 0 & z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[2] + [-1]z[2] >= 0 & z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] >= 0) For Pair D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) the following chains were created: *We consider the chain D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]), DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) which results in the following constraint: (1) (&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> D(x[2], y[2], z[2])_>=_NonInfC & D(x[2], y[2], z[2])_>=_DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) & (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(z[2], 0)=TRUE & >=(x[2], 0)=TRUE & >=(y[2], 1)=TRUE ==> D(x[2], y[2], z[2])_>=_NonInfC & D(x[2], y[2], z[2])_>=_DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) & (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [1 + (-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [1 + (-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [1 + (-1)bso_24] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND(TRUE, x[4], y[4], z[4]) -> D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])) *(x[3] >= 0 & z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) *(x[3] >= 0 & z[3] >= 0 ==> (U^Increasing(D(x[4], +(y[4], 1), +(+(y[4], 1), z[4]))), >=) & [(-1)bni_19] = 0 & [(-1)Bound*bni_19] + [bni_19]x[3] >= 0 & 0 = 0 & [(-1)bso_20] >= 0) *DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) *(x[2] + [-1]z[2] >= 0 & z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] >= 0) *D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) *(z[2] >= 0 & x[2] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [1 + (-1)bso_24] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [3] POL(COND(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + x_2 POL(D(x_1, x_2, x_3)) = [-1]x_3 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(DNAT(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(>=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(&&(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: D(x[2], y[2], z[2]) -> DNAT(&&(&&(>=(x[2], 0), >=(y[2], 1)), >=(z[2], 0)), x[2], y[2], z[2]) The following pairs are in P_bound: DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) The following pairs are in P_>=: COND(TRUE, x[4], y[4], z[4]) -> D(x[4], +(y[4], 1), +(+(y[4], 1), z[4])) DNAT(TRUE, x[3], y[3], z[3]) -> COND(>=(x[3], z[3]), x[3], -(y[3], 1), z[3]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) 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: (4): COND(TRUE, x[4], y[4], z[4]) -> D(x[4], y[4] + 1, y[4] + 1 + z[4]) (3): DNAT(TRUE, x[3], y[3], z[3]) -> COND(x[3] >= z[3], x[3], y[3] - 1, z[3]) (3) -> (4), if (x[3] >= z[3] & x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4]) The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, x0, x1, x2) ---------------------------------------- (10) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (11) TRUE ---------------------------------------- (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 + ~ 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: (4): COND(TRUE, x[4], y[4], z[4]) -> D(x[4], y[4] + 1, y[4] + 1 + z[4]) (2): D(x[2], y[2], z[2]) -> DNAT(x[2] >= 0 && y[2] >= 1 && z[2] >= 0, x[2], y[2], z[2]) (4) -> (2), if (x[4] ->^* x[2] & y[4] + 1 ->^* y[2] & y[4] + 1 + z[4] ->^* z[2]) The set Q consists of the following terms: div(x0, x1) divNat(TRUE, x0, x1) d(x0, x1, x2) dNat(TRUE, x0, x1, x2) cond(TRUE, x0, x1, x2) cond(FALSE, x0, x1, x2) ---------------------------------------- (13) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (14) TRUE