/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 27.5 s] (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 > ~ 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: diff(x, y) -> cond1(x = y, x, y) cond1(TRUE, x, y) -> 0 cond1(FALSE, x, y) -> cond2(x > y, x, y) cond2(TRUE, x, y) -> 1 + diff(x, y + 1) cond2(FALSE, x, y) -> 1 + diff(x + 1, y) The set Q consists of the following terms: diff(x0, x1) cond1(TRUE, x0, x1) cond1(FALSE, x0, x1) cond2(TRUE, x0, x1) cond2(FALSE, 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: Integer The ITRS R consists of the following rules: diff(x, y) -> cond1(x = y, x, y) cond1(TRUE, x, y) -> 0 cond1(FALSE, x, y) -> cond2(x > y, x, y) cond2(TRUE, x, y) -> 1 + diff(x, y + 1) cond2(FALSE, x, y) -> 1 + diff(x + 1, y) The integer pair graph contains the following rules and edges: (0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0]) (1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1]) (2): COND2(TRUE, x[2], y[2]) -> DIFF(x[2], y[2] + 1) (3): COND2(FALSE, x[3], y[3]) -> DIFF(x[3] + 1, y[3]) (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] > y[1] & x[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (3), if (x[1] > y[1] ->^* FALSE & x[1] ->^* x[3] & y[1] ->^* y[3]) (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) (3) -> (0), if (x[3] + 1 ->^* x[0] & y[3] ->^* y[0]) The set Q consists of the following terms: diff(x0, x1) cond1(TRUE, x0, x1) cond1(FALSE, x0, x1) cond2(TRUE, x0, x1) cond2(FALSE, 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: Integer R is empty. The integer pair graph contains the following rules and edges: (0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0]) (1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1]) (2): COND2(TRUE, x[2], y[2]) -> DIFF(x[2], y[2] + 1) (3): COND2(FALSE, x[3], y[3]) -> DIFF(x[3] + 1, y[3]) (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (2), if (x[1] > y[1] & x[1] ->^* x[2] & y[1] ->^* y[2]) (1) -> (3), if (x[1] > y[1] ->^* FALSE & x[1] ->^* x[3] & y[1] ->^* y[3]) (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) (3) -> (0), if (x[3] + 1 ->^* x[0] & y[3] ->^* y[0]) The set Q consists of the following terms: diff(x0, x1) cond1(TRUE, x0, x1) cond1(FALSE, x0, x1) cond2(TRUE, x0, x1) cond2(FALSE, 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.IdpCand1ShapeHeuristic@15e99611 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 2 Max Right Steps: 0 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 DIFF(x, y) -> COND1(=(x, y), x, y) the following chains were created: *We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) which results in the following constraint: (1) (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1] ==> DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (<(x[0], y[0])=TRUE ==> DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=)) (3) (>(x[0], y[0])=TRUE ==> DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) (y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_19] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{[-1]y[0] + x[0], y[0] + [-1]x[0]} >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_19] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{[-1]y[0] + x[0], y[0] + [-1]x[0]} >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_19] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_19] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) (y[0] + [-1] + [-1]x[0] >= 0 & [2]y[0] + [-2]x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]y[0] + [(-1)bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) (x[0] + [-1] + [-1]y[0] >= 0 & [-2]y[0] + [-1] + [2]x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]y[0] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (10) (y[0] >= 0 & [2] + [2]y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (11) (x[0] >= 0 & [1] + [2]x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (12) (y[0] >= 0 & [2] + [2]y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) (13) (y[0] >= 0 & [2] + [2]y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (12) using rule (IDP_POLY_GCD) which results in the following new constraint: (14) (y[0] >= 0 & x[0] >= 0 & [1] + y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (13) using rule (IDP_POLY_GCD) which results in the following new constraint: (15) (y[0] >= 0 & x[0] >= 0 & [1] + y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (16) (x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) (17) (x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) For Pair COND1(FALSE, x, y) -> COND2(>(x, y), x, y) the following chains were created: *We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) which results in the following constraint: (1) (>(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2] ==> COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[1], y[1])=TRUE ==> COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{[-1]y[1] + x[1], y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] + max{[-1]y[1] + x[1], y[1] + [-1]x[1]} + [-1]max{y[1] + [-1]x[1], [-1]y[1] + x[1]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{[-1]y[1] + x[1], y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-1] + [-1]y[1] >= 0 & [-2]y[1] + [2]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[1] >= 0 & [2] + [2]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[1] >= 0 & [2] + [2]x[1] >= 0 & y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) (8) (x[1] >= 0 & [2] + [2]x[1] >= 0 & y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (x[1] >= 0 & y[1] >= 0 & [1] + x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (10) (x[1] >= 0 & y[1] >= 0 & [1] + x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) *We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]) which results in the following constraint: (1) (>(x[1], y[1])=FALSE & x[1]=x[3] & y[1]=y[3] ==> COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[1], y[1])=FALSE ==> COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[1] + [-1]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{[-1]y[1] + x[1], y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] + max{[-1]y[1] + x[1], y[1] + [-1]x[1]} + [-1]max{y[1] + [-1]x[1], [-1]y[1] + x[1]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[1] + [-1]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{[-1]y[1] + x[1], y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: (5) (y[1] + [-1]x[1] >= 0 & [-2]y[1] + [2]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) (6) (y[1] + [-1]x[1] >= 0 & [2]y[1] + [-1] + [-2]x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] + [(-1)bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[1] >= 0 & [-2]y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (y[1] >= 0 & [-1] + [2]y[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (9) (0 >= 0 & 0 >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (10) (0 >= 0 & 0 >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) (11) (0 >= 0 & 0 >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (12) (y[1] >= 0 & [-1] + [2]y[1] >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) (13) (y[1] >= 0 & [-1] + [2]y[1] >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) For Pair COND2(TRUE, x, y) -> DIFF(x, +(y, 1)) the following chains were created: *We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) which results in the following constraint: (1) (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1] & >(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2] ==> COND2(TRUE, x[2], y[2])_>=_NonInfC & COND2(TRUE, x[2], y[2])_>=_DIFF(x[2], +(y[2], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & <(x[0], y[0])=TRUE ==> COND2(TRUE, x[0], y[0])_>=_NonInfC & COND2(TRUE, x[0], y[0])_>=_DIFF(x[0], +(y[0], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_23] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [1] + [-1]x[0], [-1]y[0] + [-1] + x[0]} >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_23] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [1] + [-1]x[0], [-1]y[0] + [-1] + x[0]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_23] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [1] + [-1]x[0], [-1]y[0] + [-1] + x[0]} >= 0) We simplified constraint (5) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: (7) (x[0] + [-1] + [-1]y[0] >= 0 & [-2]y[0] + [-1] + [2]x[0] >= 0 & [2]y[0] + [2] + [-2]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [-1 + (-1)bso_23] + [-2]y[0] + [2]x[0] >= 0) (8) (x[0] + [-1] + [-1]y[0] >= 0 & [-2]y[0] + [-1] + [2]x[0] >= 0 & [-2]y[0] + [-3] + [2]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We solved constraint (6) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (9) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-2]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] + [2]x[0] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (10) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (11) (0 >= 0 & [1] >= 0 & 0 >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (12) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] >= 0 & [1 + (-1)bso_23] >= 0) (13) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] >= 0 & [1 + (-1)bso_23] >= 0) We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (14) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) (15) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) For Pair COND2(FALSE, x, y) -> DIFF(+(x, 1), y) the following chains were created: *We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]) which results in the following constraint: (1) (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1] & >(x[1], y[1])=FALSE & x[1]=x[3] & y[1]=y[3] ==> COND2(FALSE, x[3], y[3])_>=_NonInfC & COND2(FALSE, x[3], y[3])_>=_DIFF(+(x[3], 1), y[3]) & (U^Increasing(DIFF(+(x[3], 1), y[3])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], y[0])=FALSE & <(x[0], y[0])=TRUE ==> COND2(FALSE, x[0], y[0])_>=_NonInfC & COND2(FALSE, x[0], y[0])_>=_DIFF(+(x[0], 1), y[0]) & (U^Increasing(DIFF(+(x[3], 1), y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_25] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [-1] + [-1]x[0], [-1]y[0] + [1] + x[0]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_25] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [-1] + [-1]x[0], [-1]y[0] + [1] + x[0]} >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) (y[0] + [-1]x[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_25] + max{y[0] + [-1]x[0], [-1]y[0] + x[0]} + [-1]max{y[0] + [-1] + [-1]x[0], [-1]y[0] + [1] + x[0]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (7) (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 & [2]y[0] + [-2]x[0] >= 0 & [2]y[0] + [-2] + [-2]x[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [(-1)bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0) We solved constraint (6) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (y[0] >= 0 & [-1] + y[0] >= 0 & [2]y[0] >= 0 & [-2] + [2]y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (y[0] >= 0 & [-1] + y[0] >= 0 & [2]y[0] >= 0 & [-2] + [2]y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) (10) (y[0] >= 0 & [-1] + y[0] >= 0 & [2]y[0] >= 0 & [-2] + [2]y[0] >= 0 & x[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) We simplified constraint (9) using rule (IDP_POLY_GCD) which results in the following new constraint: (11) (y[0] >= 0 & [-1] + y[0] >= 0 & x[0] >= 0 & y[0] >= 0 & [-1] + y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) We simplified constraint (10) using rule (IDP_POLY_GCD) which results in the following new constraint: (12) (y[0] >= 0 & [-1] + y[0] >= 0 & x[0] >= 0 & y[0] >= 0 & [-1] + y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *DIFF(x, y) -> COND1(=(x, y), x, y) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]x[0] >= 0 & [(-1)bso_19] >= 0) *(y[0] >= 0 & x[0] >= 0 & [1] + y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) *(y[0] >= 0 & x[0] >= 0 & [1] + y[0] >= 0 ==> (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0) *COND1(FALSE, x, y) -> COND2(>(x, y), x, y) *(x[1] >= 0 & y[1] >= 0 & [1] + x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) *(x[1] >= 0 & y[1] >= 0 & [1] + x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [(-1)bso_21] >= 0) *(0 >= 0 & 0 >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) *(0 >= 0 & 0 >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) *(y[1] >= 0 & [-1] + [2]y[1] >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) *(y[1] >= 0 & [-1] + [2]y[1] >= 0 & x[1] >= 0 ==> (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0) *COND2(TRUE, x, y) -> DIFF(x, +(y, 1)) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] >= 0 & [1 + (-1)bso_23] >= 0) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] >= 0 & [1 + (-1)bso_23] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_22] + [bni_22]x[0] >= 0 & [1 + (-1)bso_23] >= 0) *COND2(FALSE, x, y) -> DIFF(+(x, 1), y) *(y[0] >= 0 & [-1] + y[0] >= 0 & x[0] >= 0 & y[0] >= 0 & [-1] + y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0) *(y[0] >= 0 & [-1] + y[0] >= 0 & x[0] >= 0 & y[0] >= 0 & [-1] + y[0] >= 0 ==> (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 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(DIFF(x_1, x_2)) = [-1] + max{x_2 + [-1]x_1, [-1]x_2 + x_1} POL(COND1(x_1, x_2, x_3)) = [-1] + max{[-1]x_3 + x_2, x_3 + [-1]x_2} POL(=(x_1, x_2)) = [-1] POL(COND2(x_1, x_2, x_3)) = [-1] + max{x_3 + [-1]x_2, [-1]x_3 + x_2} POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] The following pairs are in P_>: COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]) The following pairs are in P_bound: DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]) The following pairs are in P_>=: DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) 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 > ~ 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: (0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0]) (1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1]) (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1]) The set Q consists of the following terms: diff(x0, x1) cond1(TRUE, x0, x1) cond1(FALSE, x0, x1) cond2(TRUE, x0, x1) cond2(FALSE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (8) TRUE