/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 164 ms] (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 + ~ 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) -> Cond_eval(x > 0 && x > y, x, y) Cond_eval(TRUE, x, y) -> eval(x, x + y) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, 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 + ~ 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: Boolean, Integer The ITRS R consists of the following rules: eval(x, y) -> Cond_eval(x > 0 && x > y, x, y) Cond_eval(TRUE, x, y) -> eval(x, x + y) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], x[1] + y[1]) (0) -> (1), if (x[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] ->^* x[0] & x[1] + y[1] ->^* y[0]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, 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 + ~ 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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], x[1] + y[1]) (0) -> (1), if (x[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] ->^* x[0] & x[1] + y[1] ->^* y[0]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, 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.IdpDefaultShapeHeuristic@6a289fe2 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) -> COND_EVAL(&&(>(x, 0), >(x, y)), x, y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])) which results in the following constraint: (1) (&&(>(x[0], 0), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(x[0], y[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 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] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) For Pair COND_EVAL(TRUE, x, y) -> EVAL(x, +(x, y)) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: (1) (&&(>(x[0], 0), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & x[1]=x[0]1 & +(x[1], y[1])=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(x[1], +(x[1], y[1])) & (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[0], 0)=TRUE & >(x[0], y[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(x[0], +(x[0], y[0])) & (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 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] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (6) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 0) (7) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >(x, y)), x, y) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] >= 0) *COND_EVAL(TRUE, x, y) -> EVAL(x, +(x, y)) *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 0) *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] + x[0] >= 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) = [1] POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(&&(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(+(x_1, x_2)) = x_1 + x_2 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && x[0] > y[0], x[0], y[0]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (8) TRUE