/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 279 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 > ~ 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: gcd(x, 0) -> x gcd(0, y) -> y gcd(x, y) -> Cond_gcd(x >= y && y > 0, x, y) Cond_gcd(TRUE, x, y) -> gcd(x - y, y) gcd(x, y) -> Cond_gcd1(y > x && x > 0, x, y) Cond_gcd1(TRUE, x, y) -> gcd(y - x, x) The set Q consists of the following terms: gcd(x0, x1) Cond_gcd(TRUE, x0, x1) Cond_gcd1(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 > ~ 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: Boolean, Integer The ITRS R consists of the following rules: gcd(x, 0) -> x gcd(0, y) -> y gcd(x, y) -> Cond_gcd(x >= y && y > 0, x, y) Cond_gcd(TRUE, x, y) -> gcd(x - y, y) gcd(x, y) -> Cond_gcd1(y > x && x > 0, x, y) Cond_gcd1(TRUE, x, y) -> gcd(y - x, x) The integer pair graph contains the following rules and edges: (0): GCD(x[0], y[0]) -> COND_GCD(x[0] >= y[0] && y[0] > 0, x[0], y[0]) (1): COND_GCD(TRUE, x[1], y[1]) -> GCD(x[1] - y[1], y[1]) (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) (3): COND_GCD1(TRUE, x[3], y[3]) -> GCD(y[3] - x[3], x[3]) (0) -> (1), if (x[0] >= y[0] && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (y[2] > x[2] && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (y[3] - x[3] ->^* x[0] & x[3] ->^* y[0]) (3) -> (2), if (y[3] - x[3] ->^* x[2] & x[3] ->^* y[2]) The set Q consists of the following terms: gcd(x0, x1) Cond_gcd(TRUE, x0, x1) Cond_gcd1(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 > ~ 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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): GCD(x[0], y[0]) -> COND_GCD(x[0] >= y[0] && y[0] > 0, x[0], y[0]) (1): COND_GCD(TRUE, x[1], y[1]) -> GCD(x[1] - y[1], y[1]) (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) (3): COND_GCD1(TRUE, x[3], y[3]) -> GCD(y[3] - x[3], x[3]) (0) -> (1), if (x[0] >= y[0] && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) (2) -> (3), if (y[2] > x[2] && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (y[3] - x[3] ->^* x[0] & x[3] ->^* y[0]) (3) -> (2), if (y[3] - x[3] ->^* x[2] & x[3] ->^* y[2]) The set Q consists of the following terms: gcd(x0, x1) Cond_gcd(TRUE, x0, x1) Cond_gcd1(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@78f4cd2e 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 GCD(x, y) -> COND_GCD(&&(>=(x, y), >(y, 0)), x, y) the following chains were created: *We consider the chain GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]), COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]) which results in the following constraint: (1) (&&(>=(x[0], y[0]), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> GCD(x[0], y[0])_>=_NonInfC & GCD(x[0], y[0])_>=_COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[0], y[0])=TRUE & >(y[0], 0)=TRUE ==> GCD(x[0], y[0])_>=_NonInfC & GCD(x[0], y[0])_>=_COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]y[0] + [(2)bni_17]x[0] >= 0 & [-1 + (-1)bso_18] + y[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]y[0] + [(2)bni_17]x[0] >= 0 & [-1 + (-1)bso_18] + y[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]y[0] + [(2)bni_17]x[0] >= 0 & [-1 + (-1)bso_18] + y[0] >= 0) For Pair COND_GCD(TRUE, x, y) -> GCD(-(x, y), y) the following chains were created: *We consider the chain GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]), COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]), GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) which results in the following constraint: (1) (&&(>=(x[0], y[0]), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], y[1])=x[0]1 & y[1]=y[0]1 ==> COND_GCD(TRUE, x[1], y[1])_>=_NonInfC & COND_GCD(TRUE, x[1], y[1])_>=_GCD(-(x[1], y[1]), y[1]) & (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[0], y[0])=TRUE & >(y[0], 0)=TRUE ==> COND_GCD(TRUE, x[0], y[0])_>=_NonInfC & COND_GCD(TRUE, x[0], y[0])_>=_GCD(-(x[0], y[0]), y[0]) & (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) *We consider the chain GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]), COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]), GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>=(x[0], y[0]), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], y[1])=x[2] & y[1]=y[2] ==> COND_GCD(TRUE, x[1], y[1])_>=_NonInfC & COND_GCD(TRUE, x[1], y[1])_>=_GCD(-(x[1], y[1]), y[1]) & (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(x[0], y[0])=TRUE & >(y[0], 0)=TRUE ==> COND_GCD(TRUE, x[0], y[0])_>=_NonInfC & COND_GCD(TRUE, x[0], y[0])_>=_GCD(-(x[0], y[0]), y[0]) & (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) For Pair GCD(x, y) -> COND_GCD1(&&(>(y, x), >(x, 0)), x, y) the following chains were created: *We consider the chain GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]), COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]) which results in the following constraint: (1) (&&(>(y[2], x[2]), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> GCD(x[2], y[2])_>=_NonInfC & GCD(x[2], y[2])_>=_COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> GCD(x[2], y[2])_>=_NonInfC & GCD(x[2], y[2])_>=_COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) & (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]y[2] + [(2)bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]y[2] + [(2)bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]y[2] + [(2)bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) For Pair COND_GCD1(TRUE, x, y) -> GCD(-(y, x), x) the following chains were created: *We consider the chain GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]), COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]), GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) which results in the following constraint: (1) (&&(>(y[2], x[2]), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(y[3], x[3])=x[0] & x[3]=y[0] ==> COND_GCD1(TRUE, x[3], y[3])_>=_NonInfC & COND_GCD1(TRUE, x[3], y[3])_>=_GCD(-(y[3], x[3]), x[3]) & (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> COND_GCD1(TRUE, x[2], y[2])_>=_NonInfC & COND_GCD1(TRUE, x[2], y[2])_>=_GCD(-(y[2], x[2]), x[2]) & (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) *We consider the chain GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]), COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]), GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) which results in the following constraint: (1) (&&(>(y[2], x[2]), >(x[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & -(y[3], x[3])=x[2]1 & x[3]=y[2]1 ==> COND_GCD1(TRUE, x[3], y[3])_>=_NonInfC & COND_GCD1(TRUE, x[3], y[3])_>=_GCD(-(y[3], x[3]), x[3]) & (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> COND_GCD1(TRUE, x[2], y[2])_>=_NonInfC & COND_GCD1(TRUE, x[2], y[2])_>=_GCD(-(y[2], x[2]), x[2]) & (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *GCD(x, y) -> COND_GCD(&&(>=(x, y), >(y, 0)), x, y) *(x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]y[0] + [(2)bni_17]x[0] >= 0 & [-1 + (-1)bso_18] + y[0] >= 0) *COND_GCD(TRUE, x, y) -> GCD(-(x, y), y) *(x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) *(x[0] + [-1]y[0] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(GCD(-(x[1], y[1]), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [(2)bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] >= 0) *GCD(x, y) -> COND_GCD1(&&(>(y, x), >(x, 0)), x, y) *(y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]y[2] + [(2)bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) *COND_GCD1(TRUE, x, y) -> GCD(-(y, x), x) *(y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 0) *(y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(GCD(-(y[3], x[3]), x[3])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]y[2] + [(2)bni_23]x[2] >= 0 & [(-1)bso_24] + [2]x[2] >= 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) = [3] POL(GCD(x_1, x_2)) = [-1] + [2]x_2 + [2]x_1 POL(COND_GCD(x_1, x_2, x_3)) = [-1] + x_3 + [2]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [-1] POL(>(x_1, x_2)) = [-1] POL(0) = 0 POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(COND_GCD1(x_1, x_2, x_3)) = [-1] + [2]x_3 + [2]x_2 The following pairs are in P_>: COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]) COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]) The following pairs are in P_bound: GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]) GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]) The following pairs are in P_>=: GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) 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 > ~ 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: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): GCD(x[0], y[0]) -> COND_GCD(x[0] >= y[0] && y[0] > 0, x[0], y[0]) (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) The set Q consists of the following terms: gcd(x0, x1) Cond_gcd(TRUE, x0, x1) Cond_gcd1(TRUE, x0, x1) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (8) TRUE