/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 136 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: random(x) -> Cond_random(x >= 0, x) Cond_random(TRUE, x) -> rand(x, 0) rand(x, y) -> Cond_rand(x = 0, x, y) Cond_rand(TRUE, x, y) -> y rand(x, y) -> Cond_rand1(x > 0, x, y) Cond_rand1(TRUE, x, y) -> rand(x - 1, id_inc(y)) id_inc(x) -> x id_inc(x) -> x + 1 The set Q consists of the following terms: random(x0) Cond_random(TRUE, x0) rand(x0, x1) Cond_rand(TRUE, x0, x1) Cond_rand1(TRUE, x0, x1) id_inc(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: random(x) -> Cond_random(x >= 0, x) Cond_random(TRUE, x) -> rand(x, 0) rand(x, y) -> Cond_rand(x = 0, x, y) Cond_rand(TRUE, x, y) -> y rand(x, y) -> Cond_rand1(x > 0, x, y) Cond_rand1(TRUE, x, y) -> rand(x - 1, id_inc(y)) id_inc(x) -> x id_inc(x) -> x + 1 The integer pair graph contains the following rules and edges: (0): RANDOM(x[0]) -> COND_RANDOM(x[0] >= 0, x[0]) (1): COND_RANDOM(TRUE, x[1]) -> RAND(x[1], 0) (2): RAND(x[2], y[2]) -> COND_RAND(x[2] = 0, x[2], y[2]) (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) (5): COND_RAND1(TRUE, x[5], y[5]) -> ID_INC(y[5]) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) (1) -> (2), if (x[1] ->^* x[2] & 0 ->^* y[2]) (1) -> (3), if (x[1] ->^* x[3] & 0 ->^* y[3]) (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) (3) -> (5), if (x[3] > 0 & x[3] ->^* x[5] & y[3] ->^* y[5]) (4) -> (2), if (x[4] - 1 ->^* x[2] & id_inc(y[4]) ->^* y[2]) (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) The set Q consists of the following terms: random(x0) Cond_random(TRUE, x0) rand(x0, x1) Cond_rand(TRUE, x0, x1) Cond_rand1(TRUE, x0, x1) id_inc(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 The ITRS R consists of the following rules: id_inc(x) -> x id_inc(x) -> x + 1 The integer pair graph contains the following rules and edges: (0): RANDOM(x[0]) -> COND_RANDOM(x[0] >= 0, x[0]) (1): COND_RANDOM(TRUE, x[1]) -> RAND(x[1], 0) (2): RAND(x[2], y[2]) -> COND_RAND(x[2] = 0, x[2], y[2]) (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) (5): COND_RAND1(TRUE, x[5], y[5]) -> ID_INC(y[5]) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) (1) -> (2), if (x[1] ->^* x[2] & 0 ->^* y[2]) (1) -> (3), if (x[1] ->^* x[3] & 0 ->^* y[3]) (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) (3) -> (5), if (x[3] > 0 & x[3] ->^* x[5] & y[3] ->^* y[5]) (4) -> (2), if (x[4] - 1 ->^* x[2] & id_inc(y[4]) ->^* y[2]) (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) The set Q consists of the following terms: random(x0) Cond_random(TRUE, x0) rand(x0, x1) Cond_rand(TRUE, x0, x1) Cond_rand1(TRUE, x0, x1) id_inc(x0) ---------------------------------------- (5) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 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 > ~ 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: id_inc(x) -> x id_inc(x) -> x + 1 The integer pair graph contains the following rules and edges: (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) The set Q consists of the following terms: random(x0) Cond_random(TRUE, x0) rand(x0, x1) Cond_rand(TRUE, x0, x1) Cond_rand1(TRUE, x0, x1) id_inc(x0) ---------------------------------------- (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@139a354b 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_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) the following chains were created: *We consider the chain RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]), COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])), RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) which results in the following constraint: (1) (>(x[3], 0)=TRUE & x[3]=x[4] & y[3]=y[4] & -(x[4], 1)=x[3]1 & id_inc(y[4])=y[3]1 ==> COND_RAND1(TRUE, x[4], y[4])_>=_NonInfC & COND_RAND1(TRUE, x[4], y[4])_>=_RAND(-(x[4], 1), id_inc(y[4])) & (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(x[3], 0)=TRUE ==> COND_RAND1(TRUE, x[3], y[3])_>=_NonInfC & COND_RAND1(TRUE, x[3], y[3])_>=_RAND(-(x[3], 1), id_inc(y[3])) & (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[3] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[3] >= 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[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[3] >= 0 & [1 + (-1)bso_14] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[3] >= 0 & 0 = 0 & [1 + (-1)bso_14] >= 0) For Pair RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) the following chains were created: *We consider the chain RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]), COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) which results in the following constraint: (1) (>(x[3], 0)=TRUE & x[3]=x[4] & y[3]=y[4] ==> RAND(x[3], y[3])_>=_NonInfC & RAND(x[3], y[3])_>=_COND_RAND1(>(x[3], 0), x[3], y[3]) & (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[3], 0)=TRUE ==> RAND(x[3], y[3])_>=_NonInfC & RAND(x[3], y[3])_>=_COND_RAND1(>(x[3], 0), x[3], y[3]) & (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[3] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[3] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[3] >= 0 & [(-1)bso_16] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & 0 = 0 & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[3] >= 0 & [(-1)bso_16] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) *(x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[3] >= 0 & 0 = 0 & [1 + (-1)bso_14] >= 0) *RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) *(x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & 0 = 0 & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[3] >= 0 & [(-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) = [1] POL(FALSE) = 0 POL(id_inc(x_1)) = [-1] + [-1]x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_RAND1(x_1, x_2, x_3)) = [-1] + x_2 POL(RAND(x_1, x_2)) = [-1] + x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(>(x_1, x_2)) = 0 POL(0) = 0 The following pairs are in P_>: COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) The following pairs are in P_bound: COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) The following pairs are in P_>=: RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) 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 The ITRS R consists of the following rules: id_inc(x) -> x id_inc(x) -> x + 1 The integer pair graph contains the following rules and edges: (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) The set Q consists of the following terms: random(x0) Cond_random(TRUE, x0) rand(x0, x1) Cond_rand(TRUE, x0, x1) Cond_rand1(TRUE, x0, x1) id_inc(x0) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (10) TRUE