/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 246 ms] (6) IDP (7) IDPNonInfProof [SOUND, 47 ms] (8) IDP (9) IDependencyGraphProof [EQUIVALENT, 0 ms] (10) 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: f(TRUE, x, y, z) -> g(x > y, x, y, z) g(TRUE, x, y, z) -> f(x > z, x, y + 1, z) g(TRUE, x, y, z) -> f(x > z, x, y, z + 1) The set Q consists of the following terms: f(TRUE, x0, x1, x2) g(TRUE, 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 > ~ 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: f(TRUE, x, y, z) -> g(x > y, x, y, z) g(TRUE, x, y, z) -> f(x > z, x, y + 1, z) g(TRUE, x, y, z) -> f(x > z, x, y, z + 1) The integer pair graph contains the following rules and edges: (0): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) (2): G(TRUE, x[2], y[2], z[2]) -> F(x[2] > z[2], x[2], y[2], z[2] + 1) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (0) -> (2), if (x[0] > y[0] & x[0] ->^* x[2] & y[0] ->^* y[2] & z[0] ->^* z[2]) (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (2) -> (0), if (x[2] > z[2] & x[2] ->^* x[0] & y[2] ->^* y[0] & z[2] + 1 ->^* z[0]) The set Q consists of the following terms: f(TRUE, x0, x1, x2) g(TRUE, 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 > ~ 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): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) (2): G(TRUE, x[2], y[2], z[2]) -> F(x[2] > z[2], x[2], y[2], z[2] + 1) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (0) -> (2), if (x[0] > y[0] & x[0] ->^* x[2] & y[0] ->^* y[2] & z[0] ->^* z[2]) (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (2) -> (0), if (x[2] > z[2] & x[2] ->^* x[0] & y[2] ->^* y[0] & z[2] + 1 ->^* z[0]) The set Q consists of the following terms: f(TRUE, x0, x1, x2) g(TRUE, 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@1141b235 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 F(TRUE, x, y, z) -> G(>(x, y), x, y, z) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[2] & y[0]=y[2] & z[0]=z[2] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) For Pair G(TRUE, x, y, z) -> F(>(x, z), x, +(y, 1), z) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & >(x[1], z[1])=TRUE & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> G(TRUE, x[1], y[1], z[1])_>=_NonInfC & G(TRUE, x[1], y[1], z[1])_>=_F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[1])=TRUE ==> G(TRUE, x[0], y[0], z[1])_>=_NonInfC & G(TRUE, x[0], y[0], z[1])_>=_F(>(x[0], z[1]), x[0], +(y[0], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) (9) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) For Pair G(TRUE, x, y, z) -> F(>(x, z), x, y, +(z, 1)) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[2] & y[0]=y[2] & z[0]=z[2] & >(x[2], z[2])=TRUE & x[2]=x[0]1 & y[2]=y[0]1 & +(z[2], 1)=z[0]1 ==> G(TRUE, x[2], y[2], z[2])_>=_NonInfC & G(TRUE, x[2], y[2], z[2])_>=_F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) & (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[2])=TRUE ==> G(TRUE, x[0], y[0], z[2])_>=_NonInfC & G(TRUE, x[0], y[0], z[2])_>=_F(>(x[0], z[2]), x[0], y[0], +(z[2], 1)) & (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[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 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]y[0] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) (9) (x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *F(TRUE, x, y, z) -> G(>(x, y), x, y, z) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *G(TRUE, x, y, z) -> F(>(x, z), x, +(y, 1), z) *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) *G(TRUE, x, y, z) -> F(>(x, z), x, y, +(z, 1)) *(x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) *(x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 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(F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 POL(G(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 + x_2 POL(1) = [1] The following pairs are in P_>: G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) The following pairs are in P_bound: G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) The following pairs are in P_>=: F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[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): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) The set Q consists of the following terms: f(TRUE, x0, x1, x2) g(TRUE, 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@1141b235 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 F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) For Pair G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) the following chains were created: *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & >(x[1], z[1])=TRUE & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> G(TRUE, x[1], y[1], z[1])_>=_NonInfC & G(TRUE, x[1], y[1], z[1])_>=_F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE & >(x[0], z[1])=TRUE ==> G(TRUE, x[0], y[0], z[1])_>=_NonInfC & G(TRUE, x[0], y[0], z[1])_>=_F(>(x[0], z[1]), x[0], +(y[0], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (8) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) (9) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14 + bni_14] + [bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0) *G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 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(F(x_1, x_2, x_3, x_4)) = [-1]x_3 + x_2 POL(G(x_1, x_2, x_3, x_4)) = [-1] + [-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_>: F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) The following pairs are in P_bound: F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) The following pairs are in P_>=: G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) There are no usable rules. ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean + ~ Add: (Integer, Integer) -> Integer -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) The set Q consists of the following terms: f(TRUE, x0, x1, x2) g(TRUE, x0, x1, x2) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (10) TRUE