/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 100 ms] (6) IDP (7) PisEmptyProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (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: cd(TRUE, x) -> cd(x > 0, x - 1) The set Q consists of the following terms: cd(TRUE, x0) ---------------------------------------- (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: cd(TRUE, x) -> cd(x > 0, x - 1) The integer pair graph contains the following rules and edges: (0): CD(TRUE, x[0]) -> CD(x[0] > 0, x[0] - 1) (0) -> (0), if (x[0] > 0 & x[0] - 1 ->^* x[0]') The set Q consists of the following terms: cd(TRUE, x0) ---------------------------------------- (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): CD(TRUE, x[0]) -> CD(x[0] > 0, x[0] - 1) (0) -> (0), if (x[0] > 0 & x[0] - 1 ->^* x[0]') The set Q consists of the following terms: cd(TRUE, x0) ---------------------------------------- (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@6012b267 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 CD(TRUE, x) -> CD(>(x, 0), -(x, 1)) the following chains were created: *We consider the chain CD(TRUE, x[0]) -> CD(>(x[0], 0), -(x[0], 1)), CD(TRUE, x[0]) -> CD(>(x[0], 0), -(x[0], 1)), CD(TRUE, x[0]) -> CD(>(x[0], 0), -(x[0], 1)) which results in the following constraint: (1) (>(x[0], 0)=TRUE & -(x[0], 1)=x[0]1 & >(x[0]1, 0)=TRUE & -(x[0]1, 1)=x[0]2 ==> CD(TRUE, x[0]1)_>=_NonInfC & CD(TRUE, x[0]1)_>=_CD(>(x[0]1, 0), -(x[0]1, 1)) & (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(-(x[0], 1), 0)=TRUE ==> CD(TRUE, -(x[0], 1))_>=_NonInfC & CD(TRUE, -(x[0], 1))_>=_CD(>(-(x[0], 1), 0), -(-(x[0], 1), 1)) & (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 ==> (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=) & [(-1)Bound*bni_7] + [(2)bni_7]x[0] >= 0 & [2 + (-1)bso_8] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 ==> (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=) & [(-1)Bound*bni_7] + [(2)bni_7]x[0] >= 0 & [2 + (-1)bso_8] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] >= 0 & x[0] + [-2] >= 0 ==> (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=) & [(-1)Bound*bni_7] + [(2)bni_7]x[0] >= 0 & [2 + (-1)bso_8] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *CD(TRUE, x) -> CD(>(x, 0), -(x, 1)) *(x[0] + [-1] >= 0 & x[0] + [-2] >= 0 ==> (U^Increasing(CD(>(x[0]1, 0), -(x[0]1, 1))), >=) & [(-1)Bound*bni_7] + [(2)bni_7]x[0] >= 0 & [2 + (-1)bso_8] >= 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) = [1] POL(FALSE) = 0 POL(CD(x_1, x_2)) = [2] + [2]x_2 POL(>(x_1, x_2)) = [1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] The following pairs are in P_>: CD(TRUE, x[0]) -> CD(>(x[0], 0), -(x[0], 1)) The following pairs are in P_bound: CD(TRUE, x[0]) -> CD(>(x[0], 0), -(x[0], 1)) The following pairs are in P_>=: none 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: none R is empty. The integer pair graph is empty. The set Q consists of the following terms: cd(TRUE, x0) ---------------------------------------- (7) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (8) YES