/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 220 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 + ~ 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: eval(x, y, z) -> Cond_eval(x > y + z, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x, y + 1, z + 1) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(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 + ~ 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 The ITRS R consists of the following rules: eval(x, y, z) -> Cond_eval(x > y + z, x, y, z) Cond_eval(TRUE, x, y, z) -> eval(x, y + 1, z + 1) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > y[0] + z[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], y[1] + 1, z[1] + 1) (0) -> (1), if (x[0] > y[0] + z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] + 1 ->^* z[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(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 + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > y[0] + z[0], x[0], y[0], z[0]) (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], y[1] + 1, z[1] + 1) (0) -> (1), if (x[0] > y[0] + z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] + 1 ->^* z[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(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@52f68a54 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 EVAL(x, y, z) -> COND_EVAL(>(x, +(y, z)), x, y, z) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)) which results in the following constraint: (1) (>(x[0], +(y[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[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], z[0]))=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[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] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[0] + [(-1)bni_13]y[0] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[0] + [(-1)bni_13]y[0] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 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] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]z[0] + [(-1)bni_13]y[0] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) (10) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (11) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) (12) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(x, +(y, 1), +(z, 1)) the following chains were created: *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: (1) (>(x[0], +(y[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & +(z[1], 1)=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[0], +(y[0], z[0]))=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15] + [(-1)bni_15]z[0] + [(-1)bni_15]y[0] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15] + [(-1)bni_15]z[0] + [(-1)bni_15]y[0] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 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] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15] + [(-1)bni_15]z[0] + [(-1)bni_15]y[0] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) (10) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (11) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) (12) (x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y, z) -> COND_EVAL(>(x, +(y, z)), x, y, z) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) *COND_EVAL(TRUE, x, y, z) -> EVAL(x, +(y, 1), +(z, 1)) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) *(x[0] >= 0 & y[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)Bound*bni_15 + bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 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(EVAL(x_1, x_2, x_3)) = [1] + [-1]x_3 + [-1]x_2 + x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1]x_4 + [-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_>: EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)) The following pairs are in P_bound: EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], +(y[0], z[0])), x[0], y[0], z[0]) COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 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 + ~ 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: none R is empty. The integer pair graph is empty. The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) ---------------------------------------- (7) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (8) YES