4.59/2.09 YES 4.80/2.11 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.80/2.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.80/2.11 4.80/2.11 4.80/2.11 Termination of the given ITRS could be proven: 4.80/2.11 4.80/2.11 (0) ITRS 4.80/2.11 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.80/2.11 (2) IDP 4.80/2.11 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.80/2.11 (4) IDP 4.80/2.11 (5) IDPNonInfProof [SOUND, 273 ms] 4.80/2.11 (6) IDP 4.80/2.11 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.80/2.11 (8) TRUE 4.80/2.11 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (0) 4.80/2.11 Obligation: 4.80/2.11 ITRS problem: 4.80/2.11 4.80/2.11 The following function symbols are pre-defined: 4.80/2.11 <<< 4.80/2.11 & ~ Bwand: (Integer, Integer) -> Integer 4.80/2.11 >= ~ Ge: (Integer, Integer) -> Boolean 4.80/2.11 | ~ Bwor: (Integer, Integer) -> Integer 4.80/2.11 / ~ Div: (Integer, Integer) -> Integer 4.80/2.11 != ~ Neq: (Integer, Integer) -> Boolean 4.80/2.11 && ~ Land: (Boolean, Boolean) -> Boolean 4.80/2.11 ! ~ Lnot: (Boolean) -> Boolean 4.80/2.11 = ~ Eq: (Integer, Integer) -> Boolean 4.80/2.11 <= ~ Le: (Integer, Integer) -> Boolean 4.80/2.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.80/2.11 % ~ Mod: (Integer, Integer) -> Integer 4.80/2.11 > ~ Gt: (Integer, Integer) -> Boolean 4.80/2.11 + ~ Add: (Integer, Integer) -> Integer 4.80/2.11 -1 ~ UnaryMinus: (Integer) -> Integer 4.80/2.11 < ~ Lt: (Integer, Integer) -> Boolean 4.80/2.11 || ~ Lor: (Boolean, Boolean) -> Boolean 4.80/2.11 - ~ Sub: (Integer, Integer) -> Integer 4.80/2.11 ~ ~ Bwnot: (Integer) -> Integer 4.80/2.11 * ~ Mul: (Integer, Integer) -> Integer 4.80/2.11 >>> 4.80/2.11 4.80/2.11 The TRS R consists of the following rules: 4.80/2.11 gcd(x, 0) -> x 4.80/2.11 gcd(0, y) -> y 4.80/2.11 gcd(x, y) -> Cond_gcd(x >= y && y > 0, x, y) 4.80/2.11 Cond_gcd(TRUE, x, y) -> gcd(x - y, y) 4.80/2.11 gcd(x, y) -> Cond_gcd1(y > x && x > 0, x, y) 4.80/2.11 Cond_gcd1(TRUE, x, y) -> gcd(y - x, x) 4.80/2.11 The set Q consists of the following terms: 4.80/2.11 gcd(x0, x1) 4.80/2.11 Cond_gcd(TRUE, x0, x1) 4.80/2.11 Cond_gcd1(TRUE, x0, x1) 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (1) ITRStoIDPProof (EQUIVALENT) 4.80/2.11 Added dependency pairs 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (2) 4.80/2.11 Obligation: 4.80/2.11 IDP problem: 4.80/2.11 The following function symbols are pre-defined: 4.80/2.11 <<< 4.80/2.11 & ~ Bwand: (Integer, Integer) -> Integer 4.80/2.11 >= ~ Ge: (Integer, Integer) -> Boolean 4.80/2.11 | ~ Bwor: (Integer, Integer) -> Integer 4.80/2.11 / ~ Div: (Integer, Integer) -> Integer 4.80/2.11 != ~ Neq: (Integer, Integer) -> Boolean 4.80/2.11 && ~ Land: (Boolean, Boolean) -> Boolean 4.80/2.11 ! ~ Lnot: (Boolean) -> Boolean 4.80/2.11 = ~ Eq: (Integer, Integer) -> Boolean 4.80/2.11 <= ~ Le: (Integer, Integer) -> Boolean 4.80/2.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.80/2.11 % ~ Mod: (Integer, Integer) -> Integer 4.80/2.11 > ~ Gt: (Integer, Integer) -> Boolean 4.80/2.11 + ~ Add: (Integer, Integer) -> Integer 4.80/2.11 -1 ~ UnaryMinus: (Integer) -> Integer 4.80/2.11 < ~ Lt: (Integer, Integer) -> Boolean 4.80/2.11 || ~ Lor: (Boolean, Boolean) -> Boolean 4.80/2.11 - ~ Sub: (Integer, Integer) -> Integer 4.80/2.11 ~ ~ Bwnot: (Integer) -> Integer 4.80/2.11 * ~ Mul: (Integer, Integer) -> Integer 4.80/2.11 >>> 4.80/2.11 4.80/2.11 4.80/2.11 The following domains are used: 4.80/2.11 Boolean, Integer 4.80/2.11 4.80/2.11 The ITRS R consists of the following rules: 4.80/2.11 gcd(x, 0) -> x 4.80/2.11 gcd(0, y) -> y 4.80/2.11 gcd(x, y) -> Cond_gcd(x >= y && y > 0, x, y) 4.80/2.11 Cond_gcd(TRUE, x, y) -> gcd(x - y, y) 4.80/2.11 gcd(x, y) -> Cond_gcd1(y > x && x > 0, x, y) 4.80/2.11 Cond_gcd1(TRUE, x, y) -> gcd(y - x, x) 4.80/2.11 4.80/2.11 The integer pair graph contains the following rules and edges: 4.80/2.11 (0): GCD(x[0], y[0]) -> COND_GCD(x[0] >= y[0] && y[0] > 0, x[0], y[0]) 4.80/2.11 (1): COND_GCD(TRUE, x[1], y[1]) -> GCD(x[1] - y[1], y[1]) 4.80/2.11 (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) 4.80/2.11 (3): COND_GCD1(TRUE, x[3], y[3]) -> GCD(y[3] - x[3], x[3]) 4.80/2.11 4.80/2.11 (0) -> (1), if (x[0] >= y[0] && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 4.80/2.11 (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) 4.80/2.11 (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) 4.80/2.11 (2) -> (3), if (y[2] > x[2] && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.80/2.11 (3) -> (0), if (y[3] - x[3] ->^* x[0] & x[3] ->^* y[0]) 4.80/2.11 (3) -> (2), if (y[3] - x[3] ->^* x[2] & x[3] ->^* y[2]) 4.80/2.11 4.80/2.11 The set Q consists of the following terms: 4.80/2.11 gcd(x0, x1) 4.80/2.11 Cond_gcd(TRUE, x0, x1) 4.80/2.11 Cond_gcd1(TRUE, x0, x1) 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (3) UsableRulesProof (EQUIVALENT) 4.80/2.11 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. 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (4) 4.80/2.11 Obligation: 4.80/2.11 IDP problem: 4.80/2.11 The following function symbols are pre-defined: 4.80/2.11 <<< 4.80/2.11 & ~ Bwand: (Integer, Integer) -> Integer 4.80/2.11 >= ~ Ge: (Integer, Integer) -> Boolean 4.80/2.11 | ~ Bwor: (Integer, Integer) -> Integer 4.80/2.11 / ~ Div: (Integer, Integer) -> Integer 4.80/2.11 != ~ Neq: (Integer, Integer) -> Boolean 4.80/2.11 && ~ Land: (Boolean, Boolean) -> Boolean 4.80/2.11 ! ~ Lnot: (Boolean) -> Boolean 4.80/2.11 = ~ Eq: (Integer, Integer) -> Boolean 4.80/2.11 <= ~ Le: (Integer, Integer) -> Boolean 4.80/2.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.80/2.11 % ~ Mod: (Integer, Integer) -> Integer 4.80/2.11 > ~ Gt: (Integer, Integer) -> Boolean 4.80/2.11 + ~ Add: (Integer, Integer) -> Integer 4.80/2.11 -1 ~ UnaryMinus: (Integer) -> Integer 4.80/2.11 < ~ Lt: (Integer, Integer) -> Boolean 4.80/2.11 || ~ Lor: (Boolean, Boolean) -> Boolean 4.80/2.11 - ~ Sub: (Integer, Integer) -> Integer 4.80/2.11 ~ ~ Bwnot: (Integer) -> Integer 4.80/2.11 * ~ Mul: (Integer, Integer) -> Integer 4.80/2.11 >>> 4.80/2.11 4.80/2.11 4.80/2.11 The following domains are used: 4.80/2.11 Boolean, Integer 4.80/2.11 4.80/2.11 R is empty. 4.80/2.11 4.80/2.11 The integer pair graph contains the following rules and edges: 4.80/2.11 (0): GCD(x[0], y[0]) -> COND_GCD(x[0] >= y[0] && y[0] > 0, x[0], y[0]) 4.80/2.11 (1): COND_GCD(TRUE, x[1], y[1]) -> GCD(x[1] - y[1], y[1]) 4.80/2.11 (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) 4.80/2.11 (3): COND_GCD1(TRUE, x[3], y[3]) -> GCD(y[3] - x[3], x[3]) 4.80/2.11 4.80/2.11 (0) -> (1), if (x[0] >= y[0] && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 4.80/2.11 (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) 4.80/2.11 (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) 4.80/2.11 (2) -> (3), if (y[2] > x[2] && x[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.80/2.11 (3) -> (0), if (y[3] - x[3] ->^* x[0] & x[3] ->^* y[0]) 4.80/2.11 (3) -> (2), if (y[3] - x[3] ->^* x[2] & x[3] ->^* y[2]) 4.80/2.11 4.80/2.11 The set Q consists of the following terms: 4.80/2.11 gcd(x0, x1) 4.80/2.11 Cond_gcd(TRUE, x0, x1) 4.80/2.11 Cond_gcd1(TRUE, x0, x1) 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (5) IDPNonInfProof (SOUND) 4.80/2.11 Used the following options for this NonInfProof: 4.80/2.11 4.80/2.11 IDPGPoloSolver: 4.80/2.11 Range: [(-1,2)] 4.80/2.11 IsNat: false 4.80/2.11 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5b1d1d9f 4.80/2.11 Constraint Generator: NonInfConstraintGenerator: 4.80/2.11 PathGenerator: MetricPathGenerator: 4.80/2.11 Max Left Steps: 1 4.80/2.11 Max Right Steps: 1 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 The constraints were generated the following way: 4.80/2.11 4.80/2.11 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.80/2.11 4.80/2.11 Note that final constraints are written in bold face. 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 For Pair GCD(x, y) -> COND_GCD(&&(>=(x, y), >(y, 0)), x, y) the following chains were created: 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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)bso_18] + y[0] >= 0) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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)bso_18] + y[0] >= 0) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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)bso_18] + y[0] >= 0) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 For Pair COND_GCD(TRUE, x, y) -> GCD(-(x, y), y) the following chains were created: 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 For Pair GCD(x, y) -> COND_GCD1(&&(>(y, x), >(x, 0)), x, y) the following chains were created: 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 For Pair COND_GCD1(TRUE, x, y) -> GCD(-(y, x), x) the following chains were created: 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 *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: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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])), >=)) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.80/2.11 4.80/2.11 (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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 To summarize, we get the following constraints P__>=_ for the following pairs. 4.80/2.11 4.80/2.11 *GCD(x, y) -> COND_GCD(&&(>=(x, y), >(y, 0)), x, y) 4.80/2.11 4.80/2.11 *(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)bso_18] + y[0] >= 0) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 *COND_GCD(TRUE, x, y) -> GCD(-(x, y), y) 4.80/2.11 4.80/2.11 *(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) 4.80/2.11 4.80/2.11 4.80/2.11 *(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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 *GCD(x, y) -> COND_GCD1(&&(>(y, x), >(x, 0)), x, y) 4.80/2.11 4.80/2.11 *(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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 *COND_GCD1(TRUE, x, y) -> GCD(-(y, x), x) 4.80/2.11 4.80/2.11 *(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) 4.80/2.11 4.80/2.11 4.80/2.11 *(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) 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 4.80/2.11 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. 4.80/2.11 4.80/2.11 Using the following integer polynomial ordering the resulting constraints can be solved 4.80/2.11 4.80/2.11 Polynomial interpretation over integers[POLO]: 4.80/2.11 4.80/2.11 POL(TRUE) = 0 4.80/2.11 POL(FALSE) = [1] 4.80/2.11 POL(GCD(x_1, x_2)) = [-1] + [2]x_2 + [2]x_1 4.80/2.11 POL(COND_GCD(x_1, x_2, x_3)) = [-1] + x_3 + [2]x_2 4.80/2.11 POL(&&(x_1, x_2)) = 0 4.80/2.11 POL(>=(x_1, x_2)) = [-1] 4.80/2.11 POL(>(x_1, x_2)) = [-1] 4.80/2.11 POL(0) = 0 4.80/2.11 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.80/2.11 POL(COND_GCD1(x_1, x_2, x_3)) = [-1] + [2]x_3 + [2]x_2 + [-1]x_1 4.80/2.11 4.80/2.11 4.80/2.11 The following pairs are in P_>: 4.80/2.11 4.80/2.11 4.80/2.11 GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) 4.80/2.11 COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]) 4.80/2.11 COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]) 4.80/2.11 4.80/2.11 4.80/2.11 The following pairs are in P_bound: 4.80/2.11 4.80/2.11 4.80/2.11 GCD(x[0], y[0]) -> COND_GCD(&&(>=(x[0], y[0]), >(y[0], 0)), x[0], y[0]) 4.80/2.11 COND_GCD(TRUE, x[1], y[1]) -> GCD(-(x[1], y[1]), y[1]) 4.80/2.11 GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) 4.80/2.11 COND_GCD1(TRUE, x[3], y[3]) -> GCD(-(y[3], x[3]), x[3]) 4.80/2.11 4.80/2.11 4.80/2.11 The following pairs are in P_>=: 4.80/2.11 4.80/2.11 4.80/2.11 GCD(x[2], y[2]) -> COND_GCD1(&&(>(y[2], x[2]), >(x[2], 0)), x[2], y[2]) 4.80/2.11 4.80/2.11 4.80/2.11 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.80/2.11 4.80/2.11 &&(TRUE, TRUE)^1 <-> TRUE^1 4.80/2.11 FALSE^1 -> &&(TRUE, FALSE)^1 4.80/2.11 FALSE^1 -> &&(FALSE, TRUE)^1 4.80/2.11 FALSE^1 -> &&(FALSE, FALSE)^1 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (6) 4.80/2.11 Obligation: 4.80/2.11 IDP problem: 4.80/2.11 The following function symbols are pre-defined: 4.80/2.11 <<< 4.80/2.11 & ~ Bwand: (Integer, Integer) -> Integer 4.80/2.11 >= ~ Ge: (Integer, Integer) -> Boolean 4.80/2.11 | ~ Bwor: (Integer, Integer) -> Integer 4.80/2.11 / ~ Div: (Integer, Integer) -> Integer 4.80/2.11 != ~ Neq: (Integer, Integer) -> Boolean 4.80/2.11 && ~ Land: (Boolean, Boolean) -> Boolean 4.80/2.11 ! ~ Lnot: (Boolean) -> Boolean 4.80/2.11 = ~ Eq: (Integer, Integer) -> Boolean 4.80/2.11 <= ~ Le: (Integer, Integer) -> Boolean 4.80/2.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.80/2.11 % ~ Mod: (Integer, Integer) -> Integer 4.80/2.11 > ~ Gt: (Integer, Integer) -> Boolean 4.80/2.11 + ~ Add: (Integer, Integer) -> Integer 4.80/2.11 -1 ~ UnaryMinus: (Integer) -> Integer 4.80/2.11 < ~ Lt: (Integer, Integer) -> Boolean 4.80/2.11 || ~ Lor: (Boolean, Boolean) -> Boolean 4.80/2.11 - ~ Sub: (Integer, Integer) -> Integer 4.80/2.11 ~ ~ Bwnot: (Integer) -> Integer 4.80/2.11 * ~ Mul: (Integer, Integer) -> Integer 4.80/2.11 >>> 4.80/2.11 4.80/2.11 4.80/2.11 The following domains are used: 4.80/2.11 Boolean, Integer 4.80/2.11 4.80/2.11 R is empty. 4.80/2.11 4.80/2.11 The integer pair graph contains the following rules and edges: 4.80/2.11 (2): GCD(x[2], y[2]) -> COND_GCD1(y[2] > x[2] && x[2] > 0, x[2], y[2]) 4.80/2.11 4.80/2.11 4.80/2.11 The set Q consists of the following terms: 4.80/2.11 gcd(x0, x1) 4.80/2.11 Cond_gcd(TRUE, x0, x1) 4.80/2.11 Cond_gcd1(TRUE, x0, x1) 4.80/2.11 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (7) IDependencyGraphProof (EQUIVALENT) 4.80/2.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.80/2.11 ---------------------------------------- 4.80/2.11 4.80/2.11 (8) 4.80/2.11 TRUE 4.83/2.17 EOF