/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 254 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 > ~ 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: pow(b, e) -> condLoop(e > 0, b, e, 1) condLoop(FALSE, b, e, r) -> r condLoop(TRUE, b, e, r) -> condMod(e % 2 = 1, b, e, r) condMod(FALSE, b, e, r) -> sqBase(b, e, r) condMod(TRUE, b, e, r) -> sqBase(b, e, r * b) sqBase(b, e, r) -> halfExp(b * b, e, r) halfExp(b, e, r) -> condLoop(e > 0, b, e / 2, r) The set Q consists of the following terms: pow(x0, x1) condLoop(FALSE, x0, x1, x2) condLoop(TRUE, x0, x1, x2) condMod(FALSE, x0, x1, x2) condMod(TRUE, x0, x1, x2) sqBase(x0, x1, x2) halfExp(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 / ~ 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 > ~ 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: pow(b, e) -> condLoop(e > 0, b, e, 1) condLoop(FALSE, b, e, r) -> r condLoop(TRUE, b, e, r) -> condMod(e % 2 = 1, b, e, r) condMod(FALSE, b, e, r) -> sqBase(b, e, r) condMod(TRUE, b, e, r) -> sqBase(b, e, r * b) sqBase(b, e, r) -> halfExp(b * b, e, r) halfExp(b, e, r) -> condLoop(e > 0, b, e / 2, r) The integer pair graph contains the following rules and edges: (0): POW(b[0], e[0]) -> CONDLOOP(e[0] > 0, b[0], e[0], 1) (1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1]) (2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) (3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3]) (4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4]) (5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5]) (0) -> (1), if (e[0] > 0 & b[0] ->^* b[1] & e[0] ->^* e[1] & 1 ->^* r[1]) (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2]) (1) -> (3), if (e[1] % 2 = 1 & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3]) (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4]) (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4]) (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5]) (5) -> (1), if (e[5] > 0 & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1]) The set Q consists of the following terms: pow(x0, x1) condLoop(FALSE, x0, x1, x2) condLoop(TRUE, x0, x1, x2) condMod(FALSE, x0, x1, x2) condMod(TRUE, x0, x1, x2) sqBase(x0, x1, x2) halfExp(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 / ~ 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 > ~ 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 R is empty. The integer pair graph contains the following rules and edges: (0): POW(b[0], e[0]) -> CONDLOOP(e[0] > 0, b[0], e[0], 1) (1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1]) (2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) (3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3]) (4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4]) (5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5]) (0) -> (1), if (e[0] > 0 & b[0] ->^* b[1] & e[0] ->^* e[1] & 1 ->^* r[1]) (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2]) (1) -> (3), if (e[1] % 2 = 1 & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3]) (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4]) (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4]) (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5]) (5) -> (1), if (e[5] > 0 & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1]) The set Q consists of the following terms: pow(x0, x1) condLoop(FALSE, x0, x1, x2) condLoop(TRUE, x0, x1, x2) condMod(FALSE, x0, x1, x2) condMod(TRUE, x0, x1, x2) sqBase(x0, x1, x2) halfExp(x0, x1, x2) ---------------------------------------- (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 > ~ 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 R is empty. The integer pair graph contains the following rules and edges: (3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3]) (5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5]) (4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4]) (2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) (1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1]) (5) -> (1), if (e[5] > 0 & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1]) (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2]) (1) -> (3), if (e[1] % 2 = 1 & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3]) (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4]) (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4]) (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5]) The set Q consists of the following terms: pow(x0, x1) condLoop(FALSE, x0, x1, x2) condLoop(TRUE, x0, x1, x2) condMod(FALSE, x0, x1, x2) condMod(TRUE, x0, x1, x2) sqBase(x0, x1, x2) halfExp(x0, x1, x2) ---------------------------------------- (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@3b3acd33 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 CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) the following chains were created: *We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) which results in the following constraint: (1) (=(%(e[1], 2), 1)=TRUE & b[1]=b[3] & e[1]=e[3] & r[1]=r[3] & b[3]=b[4] & e[3]=e[4] & *(r[3], b[3])=r[4] ==> CONDMOD(TRUE, b[3], e[3], r[3])_>=_NonInfC & CONDMOD(TRUE, b[3], e[3], r[3])_>=_SQBASE(b[3], e[3], *(r[3], b[3])) & (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(%(e[1], 2), 1)=TRUE & <=(%(e[1], 2), 1)=TRUE ==> CONDMOD(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDMOD(TRUE, b[1], e[1], r[1])_>=_SQBASE(b[1], e[1], *(r[1], b[1])) & (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([4] >= 0 & [1] >= 0 & [3] >= 0 ==> (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0) We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint: (6) ([1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & 0 = 0 & [bni_23] = 0 & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) For Pair HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) the following chains were created: *We consider the chain SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]), CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) which results in the following constraint: (1) (*(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5] & >(e[5], 0)=TRUE & b[5]=b[1] & /(e[5], 2)=e[1] & r[5]=r[1] ==> HALFEXP(b[5], e[5], r[5])_>=_NonInfC & HALFEXP(b[5], e[5], r[5])_>=_CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) & (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (>(e[5], 0)=TRUE ==> HALFEXP(*(b[4], b[4]), e[5], r[4])_>=_NonInfC & HALFEXP(*(b[4], b[4]), e[5], r[4])_>=_CONDLOOP(>(e[5], 0), *(b[4], b[4]), /(e[5], 2), r[4]) & (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (e[5] + [-1] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] + e[5] + [-1]max{e[5], [-1]e[5]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (e[5] + [-1] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] + e[5] + [-1]max{e[5], [-1]e[5]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (e[5] + [-1] >= 0 & [2]e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) (e[5] + [-1] >= 0 & [2]e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (e[5] + [-1] >= 0 & [2]e[5] >= 0 & b[4] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) (8) (e[5] + [-1] >= 0 & [2]e[5] >= 0 & b[4] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (10) (e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) For Pair SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) the following chains were created: *We consider the chain CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) which results in the following constraint: (1) (b[2]=b[4] & e[2]=e[4] & r[2]=r[4] & *(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5] ==> SQBASE(b[4], e[4], r[4])_>=_NonInfC & SQBASE(b[4], e[4], r[4])_>=_HALFEXP(*(b[4], b[4]), e[4], r[4]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (SQBASE(b[2], e[2], r[2])_>=_NonInfC & SQBASE(b[2], e[2], r[2])_>=_HALFEXP(*(b[2], b[2]), e[2], r[2]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (b[2] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) (8) (b[2] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) *We consider the chain CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) which results in the following constraint: (1) (b[3]=b[4] & e[3]=e[4] & *(r[3], b[3])=r[4] & *(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5] ==> SQBASE(b[4], e[4], r[4])_>=_NonInfC & SQBASE(b[4], e[4], r[4])_>=_HALFEXP(*(b[4], b[4]), e[4], r[4]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=)) We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: (2) (SQBASE(b[3], e[3], *(r[3], b[3]))_>=_NonInfC & SQBASE(b[3], e[3], *(r[3], b[3]))_>=_HALFEXP(*(b[3], b[3]), e[3], *(r[3], b[3])) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0) We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (6) ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (b[3] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) (8) (b[3] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) For Pair CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) the following chains were created: *We consider the chain CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) which results in the following constraint: (1) (b[2]=b[4] & e[2]=e[4] & r[2]=r[4] ==> CONDMOD(FALSE, b[2], e[2], r[2])_>=_NonInfC & CONDMOD(FALSE, b[2], e[2], r[2])_>=_SQBASE(b[2], e[2], r[2]) & (U^Increasing(SQBASE(b[2], e[2], r[2])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (CONDMOD(FALSE, b[2], e[2], r[2])_>=_NonInfC & CONDMOD(FALSE, b[2], e[2], r[2])_>=_SQBASE(b[2], e[2], r[2]) & (U^Increasing(SQBASE(b[2], e[2], r[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) For Pair CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) the following chains were created: *We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) which results in the following constraint: (1) (=(%(e[1], 2), 1)=FALSE & b[1]=b[2] & e[1]=e[2] & r[1]=r[2] ==> CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: (2) (<(%(e[1], 2), 1)=TRUE ==> CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=)) (3) (>(%(e[1], 2), 1)=TRUE ==> CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (4) ([-1]min{[2], [-2]} >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (5) (max{[2], [-2]} + [-2] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (6) ([-1]min{[2], [-2]} >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (7) (max{[2], [-2]} + [-2] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (8) ([4] >= 0 & [2] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (9) ([4] >= 0 & 0 >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (8) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint: (10) ([1] >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (9) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint: (11) (0 >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0) *We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) which results in the following constraint: (1) (=(%(e[1], 2), 1)=TRUE & b[1]=b[3] & e[1]=e[3] & r[1]=r[3] ==> CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(%(e[1], 2), 1)=TRUE & <=(%(e[1], 2), 1)=TRUE ==> CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([4] >= 0 & [1] >= 0 & [3] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0) We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint: (6) ([1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) *([1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & 0 = 0 & [bni_23] = 0 & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) *HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) *(e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) *(e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0 ==> (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0) *SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) *(b[2] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) *(b[2] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) *(b[3] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) *(b[3] >= 0 ==> (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0) *CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) *((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0) *CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) *([1] >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0) *(0 >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0) *([1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 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(CONDMOD(x_1, x_2, x_3, x_4)) = [-1] + x_3 POL(SQBASE(x_1, x_2, x_3)) = [-1] + x_2 POL(*(x_1, x_2)) = x_1*x_2 POL(HALFEXP(x_1, x_2, x_3)) = [-1] + x_2 POL(CONDLOOP(x_1, x_2, x_3, x_4)) = [-1] + x_3 POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(2) = [2] POL(=(x_1, x_2)) = [-1] POL(1) = [1] Polynomial Interpretations with Context Sensitive Arithemetic Replacement POL(Term^CSAR-Mode @ Context) POL(%(x_1, 2)^1 @ {}) = max{x_2, [-1]x_2} POL(%(x_1, 2)^-1 @ {}) = min{x_2, [-1]x_2} POL(/(x_1, 2)^1 @ {CONDLOOP_4/2}) = max{x_1, [-1]x_1} + [-1] The following pairs are in P_>: HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) The following pairs are in P_bound: HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) The following pairs are in P_>=: CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) At least the following rules have been oriented under context sensitive arithmetic replacement: /^1 -> %^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 > ~ 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 R is empty. The integer pair graph contains the following rules and edges: (3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3]) (4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4]) (2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) (1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1]) (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2]) (1) -> (3), if (e[1] % 2 = 1 & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3]) (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4]) (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4]) The set Q consists of the following terms: pow(x0, x1) condLoop(FALSE, x0, x1, x2) condLoop(TRUE, x0, x1, x2) condMod(FALSE, x0, x1, x2) condMod(TRUE, x0, x1, x2) sqBase(x0, x1, x2) halfExp(x0, x1, x2) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (10) TRUE