/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 82 ms] (8) AND (9) IDP (10) IDependencyGraphProof [EQUIVALENT, 0 ms] (11) TRUE (12) IDP (13) IDependencyGraphProof [EQUIVALENT, 0 ms] (14) 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: sqrt(x) -> sqrtAcc(x, 0) sqrtAcc(x, y) -> condAcc(y * y >= x || y < 0, x, y) condAcc(TRUE, x, y) -> y condAcc(FALSE, x, y) -> sqrtAcc(x, y + 1) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, 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: sqrt(x) -> sqrtAcc(x, 0) sqrtAcc(x, y) -> condAcc(y * y >= x || y < 0, x, y) condAcc(TRUE, x, y) -> y condAcc(FALSE, x, y) -> sqrtAcc(x, y + 1) The integer pair graph contains the following rules and edges: (0): SQRT(x[0]) -> SQRTACC(x[0], 0) (1): SQRTACC(x[1], y[1]) -> CONDACC(y[1] * y[1] >= x[1] || y[1] < 0, x[1], y[1]) (2): CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], y[2] + 1) (0) -> (1), if (x[0] ->^* x[1] & 0 ->^* y[1]) (1) -> (2), if (y[1] * y[1] >= x[1] || y[1] < 0 ->^* FALSE & x[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, 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): SQRT(x[0]) -> SQRTACC(x[0], 0) (1): SQRTACC(x[1], y[1]) -> CONDACC(y[1] * y[1] >= x[1] || y[1] < 0, x[1], y[1]) (2): CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], y[2] + 1) (0) -> (1), if (x[0] ->^* x[1] & 0 ->^* y[1]) (1) -> (2), if (y[1] * y[1] >= x[1] || y[1] < 0 ->^* FALSE & x[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, 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 | ~ 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, Boolean R is empty. The integer pair graph contains the following rules and edges: (2): CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], y[2] + 1) (1): SQRTACC(x[1], y[1]) -> CONDACC(y[1] * y[1] >= x[1] || y[1] < 0, x[1], y[1]) (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) (1) -> (2), if (y[1] * y[1] >= x[1] || y[1] < 0 ->^* FALSE & x[1] ->^* x[2] & y[1] ->^* y[2]) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, 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.IdpCand1ShapeHeuristic@51acc52e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 2 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 CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)) the following chains were created: *We consider the chain CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)), SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]), CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)), SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) which results in the following constraint: (1) (x[2]=x[1] & +(y[2], 1)=y[1] & ||(>=(*(y[1], y[1]), x[1]), <(y[1], 0))=FALSE & x[1]=x[2]1 & y[1]=y[2]1 & x[2]1=x[1]1 & +(y[2]1, 1)=y[1]1 ==> CONDACC(FALSE, x[2]1, y[2]1)_>=_NonInfC & CONDACC(FALSE, x[2]1, y[2]1)_>=_SQRTACC(x[2]1, +(y[2]1, 1)) & (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(*(+(y[2], 1), +(y[2], 1)), x[1])=FALSE & <(+(y[2], 1), 0)=FALSE ==> CONDACC(FALSE, x[1], +(y[2], 1))_>=_NonInfC & CONDACC(FALSE, x[1], +(y[2], 1))_>=_SQRTACC(x[1], +(+(y[2], 1), 1)) & (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-2] + [-1]y[2]^2 + [-2]y[2] >= 0 & y[2] + [1] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] + [(-1)bni_12]y[2] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-2] + [-1]y[2]^2 + [-2]y[2] >= 0 & y[2] + [1] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] + [(-1)bni_12]y[2] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [1] >= 0 & x[1] + [-2] + [-1]y[2]^2 + [-2]y[2] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-2)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] + [(-1)bni_12]y[2] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[2] + [1] >= 0 & x[1] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]y[2]^2 + [bni_12]y[2] + [bni_12]x[1] >= 0 & [(-1)bso_13] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) ([-1]y[2] + [1] >= 0 & x[1] >= 0 & y[2] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]y[2]^2 + [(-1)bni_12]y[2] + [bni_12]x[1] >= 0 & [(-1)bso_13] >= 0) (8) (y[2] + [1] >= 0 & x[1] >= 0 & y[2] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]y[2]^2 + [bni_12]y[2] + [bni_12]x[1] >= 0 & [(-1)bso_13] >= 0) For Pair SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) the following chains were created: *We consider the chain SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]), CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)) which results in the following constraint: (1) (||(>=(*(y[1], y[1]), x[1]), <(y[1], 0))=FALSE & x[1]=x[2] & y[1]=y[2] ==> SQRTACC(x[1], y[1])_>=_NonInfC & SQRTACC(x[1], y[1])_>=_CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) & (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(*(y[1], y[1]), x[1])=FALSE & <(y[1], 0)=FALSE ==> SQRTACC(x[1], y[1])_>=_NonInfC & SQRTACC(x[1], y[1])_>=_CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) & (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[1] + [-1] + [-1]y[1]^2 >= 0 & y[1] >= 0 ==> (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[1] + [(-1)bni_14]y[1] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[1] + [-1] + [-1]y[1]^2 >= 0 & y[1] >= 0 ==> (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[1] + [(-1)bni_14]y[1] >= 0 & [1 + (-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[1] >= 0 & x[1] + [-1] + [-1]y[1]^2 >= 0 ==> (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[1] + [(-1)bni_14]y[1] >= 0 & [1 + (-1)bso_15] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)) *([-1]y[2] + [1] >= 0 & x[1] >= 0 & y[2] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]y[2]^2 + [(-1)bni_12]y[2] + [bni_12]x[1] >= 0 & [(-1)bso_13] >= 0) *(y[2] + [1] >= 0 & x[1] >= 0 & y[2] >= 0 ==> (U^Increasing(SQRTACC(x[2]1, +(y[2]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]y[2]^2 + [bni_12]y[2] + [bni_12]x[1] >= 0 & [(-1)bso_13] >= 0) *SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) *(y[1] >= 0 & x[1] + [-1] + [-1]y[1]^2 >= 0 ==> (U^Increasing(CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1])), >=) & [(-1)Bound*bni_14] + [bni_14]x[1] + [(-1)bni_14]y[1] >= 0 & [1 + (-1)bso_15] >= 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(CONDACC(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_3 POL(SQRTACC(x_1, x_2)) = x_1 + [-1]x_2 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(||(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [-1] POL(*(x_1, x_2)) = x_1*x_2 POL(<(x_1, x_2)) = [-1] POL(0) = 0 The following pairs are in P_>: SQRTACC(x[1], y[1]) -> CONDACC(||(>=(*(y[1], y[1]), x[1]), <(y[1], 0)), x[1], y[1]) The following pairs are in P_bound: CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)) The following pairs are in P_>=: CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], +(y[2], 1)) There are no usable rules. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) 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: (2): CONDACC(FALSE, x[2], y[2]) -> SQRTACC(x[2], y[2] + 1) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, x0, x1) ---------------------------------------- (10) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (11) TRUE ---------------------------------------- (12) 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: (1): SQRTACC(x[1], y[1]) -> CONDACC(y[1] * y[1] >= x[1] || y[1] < 0, x[1], y[1]) The set Q consists of the following terms: sqrt(x0) sqrtAcc(x0, x1) condAcc(TRUE, x0, x1) condAcc(FALSE, x0, x1) ---------------------------------------- (13) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (14) TRUE