5.27/2.20 YES 5.27/2.22 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 5.27/2.22 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.27/2.22 5.27/2.22 5.27/2.22 Termination of the given ITRS could be proven: 5.27/2.22 5.27/2.22 (0) ITRS 5.27/2.22 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.27/2.22 (2) IDP 5.27/2.22 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.27/2.22 (4) IDP 5.27/2.22 (5) IDPNonInfProof [SOUND, 310 ms] 5.27/2.22 (6) IDP 5.27/2.22 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.27/2.22 (8) IDP 5.27/2.22 (9) IDPNonInfProof [SOUND, 29 ms] 5.27/2.22 (10) IDP 5.27/2.22 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.27/2.22 (12) TRUE 5.27/2.22 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (0) 5.27/2.22 Obligation: 5.27/2.22 ITRS problem: 5.27/2.22 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 The TRS R consists of the following rules: 5.27/2.22 eval(x, y) -> Cond_eval(x > 0 && y > 0, x, y, z) 5.27/2.22 Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) 5.27/2.22 eval(x, y) -> Cond_eval1(x > 0 && y > 0, x, y) 5.27/2.22 Cond_eval1(TRUE, x, y) -> eval(x, y - 1) 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (1) ITRStoIDPProof (EQUIVALENT) 5.27/2.22 Added dependency pairs 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (2) 5.27/2.22 Obligation: 5.27/2.22 IDP problem: 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 5.27/2.22 The following domains are used: 5.27/2.22 Boolean, Integer 5.27/2.22 5.27/2.22 The ITRS R consists of the following rules: 5.27/2.22 eval(x, y) -> Cond_eval(x > 0 && y > 0, x, y, z) 5.27/2.22 Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) 5.27/2.22 eval(x, y) -> Cond_eval1(x > 0 && y > 0, x, y) 5.27/2.22 Cond_eval1(TRUE, x, y) -> eval(x, y - 1) 5.27/2.22 5.27/2.22 The integer pair graph contains the following rules and edges: 5.27/2.22 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] > 0, x[0], y[0], z[0]) 5.27/2.22 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) 5.27/2.22 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.27/2.22 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 5.27/2.22 5.27/2.22 (0) -> (1), if (x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.27/2.22 (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) 5.27/2.22 (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) 5.27/2.22 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.27/2.22 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 5.27/2.22 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.27/2.22 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (3) UsableRulesProof (EQUIVALENT) 5.27/2.22 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. 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (4) 5.27/2.22 Obligation: 5.27/2.22 IDP problem: 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 5.27/2.22 The following domains are used: 5.27/2.22 Boolean, Integer 5.27/2.22 5.27/2.22 R is empty. 5.27/2.22 5.27/2.22 The integer pair graph contains the following rules and edges: 5.27/2.22 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] > 0, x[0], y[0], z[0]) 5.27/2.22 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) 5.27/2.22 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.27/2.22 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 5.27/2.22 5.27/2.22 (0) -> (1), if (x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.27/2.22 (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) 5.27/2.22 (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) 5.27/2.22 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.27/2.22 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 5.27/2.22 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.27/2.22 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (5) IDPNonInfProof (SOUND) 5.27/2.22 Used the following options for this NonInfProof: 5.27/2.22 5.27/2.22 IDPGPoloSolver: 5.27/2.22 Range: [(-1,2)] 5.27/2.22 IsNat: false 5.27/2.22 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7bf91d93 5.27/2.22 Constraint Generator: NonInfConstraintGenerator: 5.27/2.22 PathGenerator: MetricPathGenerator: 5.27/2.22 Max Left Steps: 1 5.27/2.22 Max Right Steps: 1 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 The constraints were generated the following way: 5.27/2.22 5.27/2.22 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.27/2.22 5.27/2.22 Note that final constraints are written in bold face. 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >(y, 0)), x, y, z) the following chains were created: 5.27/2.22 *We consider the chain COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (-(x[1], 1)=x[0] & z[1]=y[0] & &&(>(x[0], 0), >(y[0], 0))=TRUE & x[0]=x[1]1 & y[0]=y[1]1 & z[0]=z[1]1 ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(-(x[1], 1), 0)=TRUE & >(y[0], 0)=TRUE ==> EVAL(-(x[1], 1), y[0])_>=_NonInfC & EVAL(-(x[1], 1), y[0])_>=_COND_EVAL(&&(>(-(x[1], 1), 0), >(y[0], 0)), -(x[1], 1), y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[1] + [-2] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[1] + [-2] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[1] + [-2] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (6) (x[1] + [-2] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & 0 = 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (x[3]=x[0] & -(y[3], 1)=y[0] & &&(>(x[0], 0), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[0], 0)=TRUE & >(-(y[3], 1), 0)=TRUE ==> EVAL(x[0], -(y[3], 1))_>=_NonInfC & EVAL(x[0], -(y[3], 1))_>=_COND_EVAL(&&(>(x[0], 0), >(-(y[3], 1), 0)), x[0], -(y[3], 1), z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[0] + [-1] >= 0 & y[3] + [-2] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[0] + [-1] >= 0 & y[3] + [-2] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[0] + [-1] >= 0 & y[3] + [-2] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (6) (x[0] + [-1] >= 0 & y[3] + [-2] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & 0 = 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) the following chains were created: 5.27/2.22 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[0], 0), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & z[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[0], 0)=TRUE & >(y[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (6) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[0], 0), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & z[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[0], 0)=TRUE & >(y[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (6) (x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair EVAL(x, y) -> COND_EVAL1(&&(>(x, 0), >(y, 0)), x, y) the following chains were created: 5.27/2.22 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: 5.27/2.22 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[0] & -(y[3], 1)=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 To summarize, we get the following constraints P__>=_ for the following pairs. 5.27/2.22 5.27/2.22 *EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >(y, 0)), x, y, z) 5.27/2.22 5.27/2.22 *(x[1] + [-2] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & 0 = 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 *(x[0] + [-1] >= 0 & y[3] + [-2] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[0] >= 0 & 0 = 0 & [(-1)bso_18] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) 5.27/2.22 5.27/2.22 *(x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 *(x[0] + [-1] >= 0 & y[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *EVAL(x, y) -> COND_EVAL1(&&(>(x, 0), >(y, 0)), x, y) 5.27/2.22 5.27/2.22 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) 5.27/2.22 5.27/2.22 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 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. 5.27/2.22 5.27/2.22 Using the following integer polynomial ordering the resulting constraints can be solved 5.27/2.22 5.27/2.22 Polynomial interpretation over integers[POLO]: 5.27/2.22 5.27/2.22 POL(TRUE) = 0 5.27/2.22 POL(FALSE) = [1] 5.27/2.22 POL(EVAL(x_1, x_2)) = [-1] + x_1 5.27/2.22 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_2 5.27/2.22 POL(&&(x_1, x_2)) = [-1] 5.27/2.22 POL(>(x_1, x_2)) = [-1] 5.27/2.22 POL(0) = 0 5.27/2.22 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.27/2.22 POL(1) = [1] 5.27/2.22 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_2 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_>: 5.27/2.22 5.27/2.22 5.27/2.22 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_bound: 5.27/2.22 5.27/2.22 5.27/2.22 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) 5.27/2.22 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) 5.27/2.22 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.27/2.22 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_>=: 5.27/2.22 5.27/2.22 5.27/2.22 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(y[0], 0)), x[0], y[0], z[0]) 5.27/2.22 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.27/2.22 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 5.27/2.22 5.27/2.22 5.27/2.22 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.27/2.22 5.27/2.22 TRUE^1 -> &&(TRUE, TRUE)^1 5.27/2.22 FALSE^1 -> &&(TRUE, FALSE)^1 5.27/2.22 FALSE^1 -> &&(FALSE, TRUE)^1 5.27/2.22 FALSE^1 -> &&(FALSE, FALSE)^1 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (6) 5.27/2.22 Obligation: 5.27/2.22 IDP problem: 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 5.27/2.22 The following domains are used: 5.27/2.22 Boolean, Integer 5.27/2.22 5.27/2.22 R is empty. 5.27/2.22 5.27/2.22 The integer pair graph contains the following rules and edges: 5.27/2.22 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] > 0, x[0], y[0], z[0]) 5.27/2.22 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.27/2.22 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 5.27/2.22 5.27/2.22 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 5.27/2.22 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.27/2.22 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.27/2.22 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (7) IDependencyGraphProof (EQUIVALENT) 5.27/2.22 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (8) 5.27/2.22 Obligation: 5.27/2.22 IDP problem: 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 5.27/2.22 The following domains are used: 5.27/2.22 Integer, Boolean 5.27/2.22 5.27/2.22 R is empty. 5.27/2.22 5.27/2.22 The integer pair graph contains the following rules and edges: 5.27/2.22 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 5.27/2.22 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.27/2.22 5.27/2.22 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.27/2.22 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.27/2.22 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (9) IDPNonInfProof (SOUND) 5.27/2.22 Used the following options for this NonInfProof: 5.27/2.22 5.27/2.22 IDPGPoloSolver: 5.27/2.22 Range: [(-1,2)] 5.27/2.22 IsNat: false 5.27/2.22 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7bf91d93 5.27/2.22 Constraint Generator: NonInfConstraintGenerator: 5.27/2.22 PathGenerator: MetricPathGenerator: 5.27/2.22 Max Left Steps: 1 5.27/2.22 Max Right Steps: 1 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 The constraints were generated the following way: 5.27/2.22 5.27/2.22 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.27/2.22 5.27/2.22 Note that final constraints are written in bold face. 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) the following chains were created: 5.27/2.22 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] >= 0 & [1 + (-1)bso_13] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] >= 0 & [1 + (-1)bso_13] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] >= 0 & [1 + (-1)bso_13] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) the following chains were created: 5.27/2.22 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: 5.27/2.22 5.27/2.22 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.27/2.22 5.27/2.22 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.27/2.22 5.27/2.22 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] >= 0 & [(-1)bso_15] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.27/2.22 5.27/2.22 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] >= 0 & [(-1)bso_15] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.27/2.22 5.27/2.22 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] >= 0 & [(-1)bso_15] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 To summarize, we get the following constraints P__>=_ for the following pairs. 5.27/2.22 5.27/2.22 *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 5.27/2.22 5.27/2.22 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] >= 0 & [1 + (-1)bso_13] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 *EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.27/2.22 5.27/2.22 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[2] >= 0 & [(-1)bso_15] >= 0) 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 5.27/2.22 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. 5.27/2.22 5.27/2.22 Using the following integer polynomial ordering the resulting constraints can be solved 5.27/2.22 5.27/2.22 Polynomial interpretation over integers[POLO]: 5.27/2.22 5.27/2.22 POL(TRUE) = 0 5.27/2.22 POL(FALSE) = [3] 5.27/2.22 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_1 5.27/2.22 POL(EVAL(x_1, x_2)) = [-1] + x_2 5.27/2.22 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.27/2.22 POL(1) = [1] 5.27/2.22 POL(&&(x_1, x_2)) = 0 5.27/2.22 POL(>(x_1, x_2)) = [-1] 5.27/2.22 POL(0) = 0 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_>: 5.27/2.22 5.27/2.22 5.27/2.22 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_bound: 5.27/2.22 5.27/2.22 5.27/2.22 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 5.27/2.22 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.27/2.22 5.27/2.22 5.27/2.22 The following pairs are in P_>=: 5.27/2.22 5.27/2.22 5.27/2.22 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.27/2.22 5.27/2.22 5.27/2.22 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.27/2.22 5.27/2.22 TRUE^1 -> &&(TRUE, TRUE)^1 5.27/2.22 FALSE^1 -> &&(TRUE, FALSE)^1 5.27/2.22 FALSE^1 -> &&(FALSE, TRUE)^1 5.27/2.22 FALSE^1 -> &&(FALSE, FALSE)^1 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (10) 5.27/2.22 Obligation: 5.27/2.22 IDP problem: 5.27/2.22 The following function symbols are pre-defined: 5.27/2.22 <<< 5.27/2.22 & ~ Bwand: (Integer, Integer) -> Integer 5.27/2.22 >= ~ Ge: (Integer, Integer) -> Boolean 5.27/2.22 | ~ Bwor: (Integer, Integer) -> Integer 5.27/2.22 / ~ Div: (Integer, Integer) -> Integer 5.27/2.22 != ~ Neq: (Integer, Integer) -> Boolean 5.27/2.22 && ~ Land: (Boolean, Boolean) -> Boolean 5.27/2.22 ! ~ Lnot: (Boolean) -> Boolean 5.27/2.22 = ~ Eq: (Integer, Integer) -> Boolean 5.27/2.22 <= ~ Le: (Integer, Integer) -> Boolean 5.27/2.22 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.27/2.22 % ~ Mod: (Integer, Integer) -> Integer 5.27/2.22 > ~ Gt: (Integer, Integer) -> Boolean 5.27/2.22 + ~ Add: (Integer, Integer) -> Integer 5.27/2.22 -1 ~ UnaryMinus: (Integer) -> Integer 5.27/2.22 < ~ Lt: (Integer, Integer) -> Boolean 5.27/2.22 || ~ Lor: (Boolean, Boolean) -> Boolean 5.27/2.22 - ~ Sub: (Integer, Integer) -> Integer 5.27/2.22 ~ ~ Bwnot: (Integer) -> Integer 5.27/2.22 * ~ Mul: (Integer, Integer) -> Integer 5.27/2.22 >>> 5.27/2.22 5.27/2.22 5.27/2.22 The following domains are used: 5.27/2.22 Boolean, Integer 5.27/2.22 5.27/2.22 R is empty. 5.27/2.22 5.27/2.22 The integer pair graph contains the following rules and edges: 5.27/2.22 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.27/2.22 5.27/2.22 5.27/2.22 The set Q consists of the following terms: 5.27/2.22 eval(x0, x1) 5.27/2.22 Cond_eval(TRUE, x0, x1, x2) 5.27/2.22 Cond_eval1(TRUE, x0, x1) 5.27/2.22 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (11) IDependencyGraphProof (EQUIVALENT) 5.27/2.22 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.27/2.22 ---------------------------------------- 5.27/2.22 5.27/2.22 (12) 5.27/2.22 TRUE 5.27/2.26 EOF