/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 247 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 / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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: log(1, y) -> Cond_log(y >= 2, 1, y) Cond_log(TRUE, 1, y) -> 0 log(x, y) -> Cond_log1(x >= 2 && y >= 2, x, y) Cond_log1(TRUE, x, y) -> 1 + log((x - y) / y, y) The set Q consists of the following terms: Cond_log(TRUE, 1, x0) log(x0, x1) Cond_log1(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 / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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, Boolean The ITRS R consists of the following rules: log(1, y) -> Cond_log(y >= 2, 1, y) Cond_log(TRUE, 1, y) -> 0 log(x, y) -> Cond_log1(x >= 2 && y >= 2, x, y) Cond_log1(TRUE, x, y) -> 1 + log((x - y) / y, y) The integer pair graph contains the following rules and edges: (0): LOG(1, y[0]) -> COND_LOG(y[0] >= 2, 1, y[0]) (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (0), if ((x[2] - y[2]) / y[2] ->^* 1 & y[2] ->^* y[0]) (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) The set Q consists of the following terms: Cond_log(TRUE, 1, x0) log(x0, x1) Cond_log1(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 / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): LOG(1, y[0]) -> COND_LOG(y[0] >= 2, 1, y[0]) (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (0), if ((x[2] - y[2]) / y[2] ->^* 1 & y[2] ->^* y[0]) (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) The set Q consists of the following terms: Cond_log(TRUE, 1, x0) log(x0, x1) Cond_log1(TRUE, x0, x1) ---------------------------------------- (5) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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, Boolean R is empty. The integer pair graph contains the following rules and edges: (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) The set Q consists of the following terms: Cond_log(TRUE, 1, x0) log(x0, x1) Cond_log1(TRUE, x0, x1) ---------------------------------------- (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@7cc7bb1a 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_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) the following chains were created: *We consider the chain LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]), COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]), LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) which results in the following constraint: (1) (&&(>=(x[1], 2), >=(y[1], 2))=TRUE & x[1]=x[2] & y[1]=y[2] & /(-(x[2], y[2]), y[2])=x[1]1 & y[2]=y[1]1 ==> COND_LOG1(TRUE, x[2], y[2])_>=_NonInfC & COND_LOG1(TRUE, x[2], y[2])_>=_LOG(/(-(x[2], y[2]), y[2]), y[2]) & (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[1], 2)=TRUE & >=(y[1], 2)=TRUE ==> COND_LOG1(TRUE, x[1], y[1])_>=_NonInfC & COND_LOG1(TRUE, x[1], y[1])_>=_LOG(/(-(x[1], y[1]), y[1]), y[1]) & (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] + [-1]max{x[1] + [-1]y[1], [-1]x[1] + y[1]} + min{max{y[1], [-1]y[1]} + [-1], max{x[1] + [-1]y[1], [-1]x[1] + y[1]}} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] + [-1]max{x[1] + [-1]y[1], [-1]x[1] + y[1]} + min{max{y[1], [-1]y[1]} + [-1], max{x[1] + [-1]y[1], [-1]x[1] + y[1]}} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: (5) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & x[1] + [-2]y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) (6) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & [2]y[1] + [-1] + [-1]x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) (7) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [-2]x[1] + [-1] + [2]y[1] >= 0 & [2]y[1] >= 0 & [-1] + x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) We simplified constraint (5) using rule (IDP_POLY_GCD) which results in the following new constraint: (8) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & x[1] + [-2]y[1] >= 0 & x[1] + [-1]y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) For Pair LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) the following chains were created: *We consider the chain LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]), COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) which results in the following constraint: (1) (&&(>=(x[1], 2), >=(y[1], 2))=TRUE & x[1]=x[2] & y[1]=y[2] ==> LOG(x[1], y[1])_>=_NonInfC & LOG(x[1], y[1])_>=_COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) & (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[1], 2)=TRUE & >=(y[1], 2)=TRUE ==> LOG(x[1], y[1])_>=_NonInfC & LOG(x[1], y[1])_>=_COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) & (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & [2]y[1] + [-1] + [-1]x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [-2]x[1] + [-1] + [2]y[1] >= 0 & [2]y[1] >= 0 & [-1] + x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & x[1] + [-2]y[1] >= 0 & x[1] + [-1]y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) *LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 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(COND_LOG1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 POL(LOG(x_1, x_2)) = [-1] + x_1 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(&&(x_1, x_2)) = 0 POL(>=(x_1, x_2)) = [-1] POL(2) = [2] Polynomial Interpretations with Context Sensitive Arithemetic Replacement POL(Term^CSAR-Mode @ Context) POL(/(x_1, y[1])^1 @ {LOG_2/0}) = max{x_1, [-1]x_1} + [-1]min{max{x_2, [-1]x_2} + [-1], max{x_1, [-1]x_1}} The following pairs are in P_>: COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) The following pairs are in P_bound: COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) The following pairs are in P_>=: LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) At least the following rules have been oriented under context sensitive arithmetic replacement: /^1 -> TRUE^1 -> &&(TRUE, TRUE)^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean / ~ Div: (Integer, Integer) -> Integer | ~ Bwor: (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: (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) The set Q consists of the following terms: Cond_log(TRUE, 1, x0) log(x0, x1) Cond_log1(TRUE, x0, x1) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (10) TRUE