5.12/2.26 YES 5.12/2.28 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.12/2.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.12/2.28 5.12/2.28 5.12/2.28 Termination of the given ITRS could be proven: 5.12/2.28 5.12/2.28 (0) ITRS 5.12/2.28 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.12/2.28 (2) IDP 5.12/2.28 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.12/2.28 (4) IDP 5.12/2.28 (5) IDPNonInfProof [SOUND, 287 ms] 5.12/2.28 (6) AND 5.12/2.28 (7) IDP 5.12/2.28 (8) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.28 (9) TRUE 5.12/2.28 (10) IDP 5.12/2.28 (11) IDPNonInfProof [SOUND, 69 ms] 5.12/2.28 (12) IDP 5.12/2.28 (13) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.28 (14) TRUE 5.12/2.28 5.12/2.28 5.12/2.28 ---------------------------------------- 5.12/2.28 5.12/2.28 (0) 5.12/2.28 Obligation: 5.12/2.28 ITRS problem: 5.12/2.28 5.12/2.28 The following function symbols are pre-defined: 5.12/2.28 <<< 5.12/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.28 / ~ Div: (Integer, Integer) -> Integer 5.12/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.28 + ~ Add: (Integer, Integer) -> Integer 5.12/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.28 >>> 5.12/2.28 5.12/2.28 The TRS R consists of the following rules: 5.12/2.28 eval_1(x, y) -> Cond_eval_1(x > 0, x, y) 5.12/2.28 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 5.12/2.28 eval_2(x, y) -> Cond_eval_2(x > 0 && y > 0, x, y) 5.12/2.28 Cond_eval_2(TRUE, x, y) -> eval_2(x, y - 1) 5.12/2.28 eval_2(x, y) -> Cond_eval_21(x > 0 && 0 >= y, x, y) 5.12/2.28 Cond_eval_21(TRUE, x, y) -> eval_1(x - 1, y) 5.12/2.28 The set Q consists of the following terms: 5.12/2.28 eval_1(x0, x1) 5.12/2.28 Cond_eval_1(TRUE, x0, x1) 5.12/2.28 eval_2(x0, x1) 5.12/2.28 Cond_eval_2(TRUE, x0, x1) 5.12/2.28 Cond_eval_21(TRUE, x0, x1) 5.12/2.28 5.12/2.28 ---------------------------------------- 5.12/2.28 5.12/2.28 (1) ITRStoIDPProof (EQUIVALENT) 5.12/2.28 Added dependency pairs 5.12/2.28 ---------------------------------------- 5.12/2.28 5.12/2.28 (2) 5.12/2.28 Obligation: 5.12/2.28 IDP problem: 5.12/2.28 The following function symbols are pre-defined: 5.12/2.28 <<< 5.12/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.28 / ~ Div: (Integer, Integer) -> Integer 5.12/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.28 + ~ Add: (Integer, Integer) -> Integer 5.12/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.28 >>> 5.12/2.28 5.12/2.28 5.12/2.28 The following domains are used: 5.12/2.28 Integer, Boolean 5.12/2.28 5.12/2.28 The ITRS R consists of the following rules: 5.12/2.28 eval_1(x, y) -> Cond_eval_1(x > 0, x, y) 5.12/2.28 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 5.12/2.28 eval_2(x, y) -> Cond_eval_2(x > 0 && y > 0, x, y) 5.12/2.28 Cond_eval_2(TRUE, x, y) -> eval_2(x, y - 1) 5.12/2.28 eval_2(x, y) -> Cond_eval_21(x > 0 && 0 >= y, x, y) 5.12/2.28 Cond_eval_21(TRUE, x, y) -> eval_1(x - 1, y) 5.12/2.28 5.12/2.28 The integer pair graph contains the following rules and edges: 5.12/2.28 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0, x[0], y[0]) 5.12/2.28 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.28 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.12/2.28 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] - 1) 5.12/2.28 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] > 0 && 0 >= y[4], x[4], y[4]) 5.12/2.28 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 1, y[5]) 5.12/2.28 5.12/2.28 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.12/2.28 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.12/2.28 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.12/2.28 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.12/2.28 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.12/2.29 (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4]) 5.12/2.29 (4) -> (5), if (x[4] > 0 && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.12/2.29 (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0]) 5.12/2.29 5.12/2.29 The set Q consists of the following terms: 5.12/2.29 eval_1(x0, x1) 5.12/2.29 Cond_eval_1(TRUE, x0, x1) 5.12/2.29 eval_2(x0, x1) 5.12/2.29 Cond_eval_2(TRUE, x0, x1) 5.12/2.29 Cond_eval_21(TRUE, x0, x1) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (3) UsableRulesProof (EQUIVALENT) 5.12/2.29 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. 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (4) 5.12/2.29 Obligation: 5.12/2.29 IDP problem: 5.12/2.29 The following function symbols are pre-defined: 5.12/2.29 <<< 5.12/2.29 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.29 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.29 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.29 / ~ Div: (Integer, Integer) -> Integer 5.12/2.29 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.29 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.29 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.29 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.29 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.29 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.29 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.29 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.29 + ~ Add: (Integer, Integer) -> Integer 5.12/2.29 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.29 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.29 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.29 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.29 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.29 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.29 >>> 5.12/2.29 5.12/2.29 5.12/2.29 The following domains are used: 5.12/2.29 Integer, Boolean 5.12/2.29 5.12/2.29 R is empty. 5.12/2.29 5.12/2.29 The integer pair graph contains the following rules and edges: 5.12/2.29 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0, x[0], y[0]) 5.12/2.29 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.12/2.29 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], y[3] - 1) 5.12/2.29 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] > 0 && 0 >= y[4], x[4], y[4]) 5.12/2.29 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 1, y[5]) 5.12/2.29 5.12/2.29 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.12/2.29 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.12/2.29 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.12/2.29 (2) -> (3), if (x[2] > 0 && y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.12/2.29 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 5.12/2.29 (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4]) 5.12/2.29 (4) -> (5), if (x[4] > 0 && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.12/2.29 (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0]) 5.12/2.29 5.12/2.29 The set Q consists of the following terms: 5.12/2.29 eval_1(x0, x1) 5.12/2.29 Cond_eval_1(TRUE, x0, x1) 5.12/2.29 eval_2(x0, x1) 5.12/2.29 Cond_eval_2(TRUE, x0, x1) 5.12/2.29 Cond_eval_21(TRUE, x0, x1) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (5) IDPNonInfProof (SOUND) 5.12/2.29 Used the following options for this NonInfProof: 5.12/2.29 5.12/2.29 IDPGPoloSolver: 5.12/2.29 Range: [(-1,2)] 5.12/2.29 IsNat: false 5.12/2.29 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@18f02bc2 5.12/2.29 Constraint Generator: NonInfConstraintGenerator: 5.12/2.29 PathGenerator: MetricPathGenerator: 5.12/2.29 Max Left Steps: 1 5.12/2.29 Max Right Steps: 1 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 The constraints were generated the following way: 5.12/2.29 5.12/2.29 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.12/2.29 5.12/2.29 Note that final constraints are written in bold face. 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair EVAL_1(x, y) -> COND_EVAL_1(>(x, 0), x, y) the following chains were created: 5.12/2.29 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[0], 0)=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [bni_24] = 0 & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) the following chains were created: 5.12/2.29 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (x[1]=x[2] & y[1]=y[2] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair EVAL_2(x, y) -> COND_EVAL_2(&&(>(x, 0), >(y, 0)), x, y) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], -(y[3], 1)) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[2] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[2] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[2] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(x, -(y, 1)) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], -(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) which results in the following constraint: 5.12/2.29 5.12/2.29 (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_EVAL_2(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3])_>=_EVAL_2(x[3], -(y[3], 1)) & (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2])_>=_EVAL_2(x[2], -(y[2], 1)) & (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], -(y[3], 1)), EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[2], 0), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[4] & -(y[3], 1)=y[4] ==> COND_EVAL_2(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3])_>=_EVAL_2(x[3], -(y[3], 1)) & (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[2], 0)=TRUE & >(y[2], 0)=TRUE ==> COND_EVAL_2(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2])_>=_EVAL_2(x[2], -(y[2], 1)) & (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair EVAL_2(x, y) -> COND_EVAL_21(&&(>(x, 0), >=(0, y)), x, y) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[4], 0), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[4], 0)=TRUE & >=(0, y[4])=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[4] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[4] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[4] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]y[4] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(-(x, 1), y) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[4], 0), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[0] & y[5]=y[0] ==> COND_EVAL_21(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5])_>=_EVAL_1(-(x[5], 1), y[5]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[4], 0)=TRUE & >=(0, y[4])=TRUE ==> COND_EVAL_21(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4])_>=_EVAL_1(-(x[4], 1), y[4]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]y[4] + [bni_34]x[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]y[4] + [bni_34]x[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]y[4] + [bni_34]x[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]y[4] + [bni_34]x[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 To summarize, we get the following constraints P__>=_ for the following pairs. 5.12/2.29 5.12/2.29 *EVAL_1(x, y) -> COND_EVAL_1(>(x, 0), x, y) 5.12/2.29 5.12/2.29 *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [bni_24] = 0 & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x[0] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) 5.12/2.29 5.12/2.29 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_26] = 0 & [(-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *EVAL_2(x, y) -> COND_EVAL_2(&&(>(x, 0), >(y, 0)), x, y) 5.12/2.29 5.12/2.29 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]y[2] + [bni_28]x[2] >= 0 & [(-1)bso_29] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *COND_EVAL_2(TRUE, x, y) -> EVAL_2(x, -(y, 1)) 5.12/2.29 5.12/2.29 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 *(x[2] + [-1] >= 0 & y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(x[3], -(y[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *EVAL_2(x, y) -> COND_EVAL_21(&&(>(x, 0), >=(0, y)), x, y) 5.12/2.29 5.12/2.29 *(x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]y[4] + [bni_32]x[4] >= 0 & [(-1)bso_33] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *COND_EVAL_21(TRUE, x, y) -> EVAL_1(-(x, 1), y) 5.12/2.29 5.12/2.29 *(x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]y[4] + [bni_34]x[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 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. 5.12/2.29 5.12/2.29 Using the following integer polynomial ordering the resulting constraints can be solved 5.12/2.29 5.12/2.29 Polynomial interpretation over integers[POLO]: 5.12/2.29 5.12/2.29 POL(TRUE) = 0 5.12/2.29 POL(FALSE) = [1] 5.12/2.29 POL(EVAL_1(x_1, x_2)) = [-1] + x_2 + x_1 5.12/2.29 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + x_3 + x_2 5.12/2.29 POL(>(x_1, x_2)) = [-1] 5.12/2.29 POL(0) = 0 5.12/2.29 POL(EVAL_2(x_1, x_2)) = [-1] + x_2 + x_1 5.12/2.29 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_3 + x_2 5.12/2.29 POL(&&(x_1, x_2)) = [-1] 5.12/2.29 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.12/2.29 POL(1) = [1] 5.12/2.29 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_3 + x_2 5.12/2.29 POL(>=(x_1, x_2)) = [-1] 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_>: 5.12/2.29 5.12/2.29 5.12/2.29 COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], -(y[3], 1)) 5.12/2.29 COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_bound: 5.12/2.29 5.12/2.29 5.12/2.29 EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.12/2.29 COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3], -(y[3], 1)) 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_>=: 5.12/2.29 5.12/2.29 5.12/2.29 EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) 5.12/2.29 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 EVAL_2(x[2], y[2]) -> COND_EVAL_2(&&(>(x[2], 0), >(y[2], 0)), x[2], y[2]) 5.12/2.29 EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) 5.12/2.29 5.12/2.29 5.12/2.29 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.12/2.29 5.12/2.29 TRUE^1 -> &&(TRUE, TRUE)^1 5.12/2.29 FALSE^1 -> &&(TRUE, FALSE)^1 5.12/2.29 FALSE^1 -> &&(FALSE, TRUE)^1 5.12/2.29 FALSE^1 -> &&(FALSE, FALSE)^1 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (6) 5.12/2.29 Complex Obligation (AND) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (7) 5.12/2.29 Obligation: 5.12/2.29 IDP problem: 5.12/2.29 The following function symbols are pre-defined: 5.12/2.29 <<< 5.12/2.29 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.29 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.29 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.29 / ~ Div: (Integer, Integer) -> Integer 5.12/2.29 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.29 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.29 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.29 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.29 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.29 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.29 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.29 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.29 + ~ Add: (Integer, Integer) -> Integer 5.12/2.29 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.29 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.29 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.29 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.29 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.29 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.29 >>> 5.12/2.29 5.12/2.29 5.12/2.29 The following domains are used: 5.12/2.29 Integer, Boolean 5.12/2.29 5.12/2.29 R is empty. 5.12/2.29 5.12/2.29 The integer pair graph contains the following rules and edges: 5.12/2.29 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0, x[0], y[0]) 5.12/2.29 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(x[2] > 0 && y[2] > 0, x[2], y[2]) 5.12/2.29 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] > 0 && 0 >= y[4], x[4], y[4]) 5.12/2.29 5.12/2.29 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.12/2.29 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.12/2.29 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.12/2.29 5.12/2.29 The set Q consists of the following terms: 5.12/2.29 eval_1(x0, x1) 5.12/2.29 Cond_eval_1(TRUE, x0, x1) 5.12/2.29 eval_2(x0, x1) 5.12/2.29 Cond_eval_2(TRUE, x0, x1) 5.12/2.29 Cond_eval_21(TRUE, x0, x1) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (8) IDependencyGraphProof (EQUIVALENT) 5.12/2.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (9) 5.12/2.29 TRUE 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (10) 5.12/2.29 Obligation: 5.12/2.29 IDP problem: 5.12/2.29 The following function symbols are pre-defined: 5.12/2.29 <<< 5.12/2.29 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.29 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.29 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.29 / ~ Div: (Integer, Integer) -> Integer 5.12/2.29 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.29 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.29 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.29 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.29 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.29 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.29 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.29 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.29 + ~ Add: (Integer, Integer) -> Integer 5.12/2.29 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.29 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.29 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.29 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.29 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.29 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.29 >>> 5.12/2.29 5.12/2.29 5.12/2.29 The following domains are used: 5.12/2.29 Integer, Boolean 5.12/2.29 5.12/2.29 R is empty. 5.12/2.29 5.12/2.29 The integer pair graph contains the following rules and edges: 5.12/2.29 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0, x[0], y[0]) 5.12/2.29 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] > 0 && 0 >= y[4], x[4], y[4]) 5.12/2.29 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5] - 1, y[5]) 5.12/2.29 5.12/2.29 (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0]) 5.12/2.29 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.12/2.29 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.12/2.29 (4) -> (5), if (x[4] > 0 && 0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.12/2.29 5.12/2.29 The set Q consists of the following terms: 5.12/2.29 eval_1(x0, x1) 5.12/2.29 Cond_eval_1(TRUE, x0, x1) 5.12/2.29 eval_2(x0, x1) 5.12/2.29 Cond_eval_2(TRUE, x0, x1) 5.12/2.29 Cond_eval_21(TRUE, x0, x1) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (11) IDPNonInfProof (SOUND) 5.12/2.29 Used the following options for this NonInfProof: 5.12/2.29 5.12/2.29 IDPGPoloSolver: 5.12/2.29 Range: [(-1,2)] 5.12/2.29 IsNat: false 5.12/2.29 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@18f02bc2 5.12/2.29 Constraint Generator: NonInfConstraintGenerator: 5.12/2.29 PathGenerator: MetricPathGenerator: 5.12/2.29 Max Left Steps: 1 5.12/2.29 Max Right Steps: 1 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 The constraints were generated the following way: 5.12/2.29 5.12/2.29 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.12/2.29 5.12/2.29 Note that final constraints are written in bold face. 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) the following chains were created: 5.12/2.29 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (>(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[0], 0)=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(>(x[0], 0), x[0], y[0]) & (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_20] = 0 & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) the following chains were created: 5.12/2.29 *We consider the chain COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (x[1]=x[4] & y[1]=y[4] ==> COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (COND_EVAL_1(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1])_>=_EVAL_2(x[1], y[1]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_22] = 0 & [(-1)bso_23] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_22] = 0 & [(-1)bso_23] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_22] = 0 & [(-1)bso_23] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[4], 0), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[4], 0)=TRUE & >=(0, y[4])=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[4] + [bni_24]x[4] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[4] + [bni_24]x[4] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]y[4] + [bni_24]x[4] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(2)bni_24 + (-1)Bound*bni_24] + [bni_24]y[4] + [bni_24]x[4] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 For Pair COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) the following chains were created: 5.12/2.29 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) which results in the following constraint: 5.12/2.29 5.12/2.29 (1) (&&(>(x[4], 0), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & -(x[5], 1)=x[0] & y[5]=y[0] ==> COND_EVAL_21(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5])_>=_EVAL_1(-(x[5], 1), y[5]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.12/2.29 5.12/2.29 (2) (>(x[4], 0)=TRUE & >=(0, y[4])=TRUE ==> COND_EVAL_21(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4])_>=_EVAL_1(-(x[4], 1), y[4]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=)) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.12/2.29 5.12/2.29 (3) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[4] + [bni_26]x[4] >= 0 & [1 + (-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.12/2.29 5.12/2.29 (4) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[4] + [bni_26]x[4] >= 0 & [1 + (-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.12/2.29 5.12/2.29 (5) (x[4] + [-1] >= 0 & [-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]y[4] + [bni_26]x[4] >= 0 & [1 + (-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.12/2.29 5.12/2.29 (6) (x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(2)bni_26 + (-1)Bound*bni_26] + [bni_26]y[4] + [bni_26]x[4] >= 0 & [1 + (-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 To summarize, we get the following constraints P__>=_ for the following pairs. 5.12/2.29 5.12/2.29 *EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) 5.12/2.29 5.12/2.29 *(x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_1(>(x[0], 0), x[0], y[0])), >=) & [(-1)bni_20] = 0 & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x[0] >= 0 & [(-1)bso_21] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 5.12/2.29 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_22] = 0 & [(-1)bso_23] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) 5.12/2.29 5.12/2.29 *(x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4])), >=) & [(2)bni_24 + (-1)Bound*bni_24] + [bni_24]y[4] + [bni_24]x[4] >= 0 & [(-1)bso_25] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 *COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) 5.12/2.29 5.12/2.29 *(x[4] + [-1] >= 0 & y[4] >= 0 ==> (U^Increasing(EVAL_1(-(x[5], 1), y[5])), >=) & [(2)bni_26 + (-1)Bound*bni_26] + [bni_26]y[4] + [bni_26]x[4] >= 0 & [1 + (-1)bso_27] >= 0) 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 5.12/2.29 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. 5.12/2.29 5.12/2.29 Using the following integer polynomial ordering the resulting constraints can be solved 5.12/2.29 5.12/2.29 Polynomial interpretation over integers[POLO]: 5.12/2.29 5.12/2.29 POL(TRUE) = [3] 5.12/2.29 POL(FALSE) = [2] 5.12/2.29 POL(EVAL_1(x_1, x_2)) = [2] + [-1]x_2 + x_1 5.12/2.29 POL(COND_EVAL_1(x_1, x_2, x_3)) = [2] + [-1]x_3 + x_2 5.12/2.29 POL(>(x_1, x_2)) = [-1] 5.12/2.29 POL(0) = 0 5.12/2.29 POL(EVAL_2(x_1, x_2)) = [2] + [-1]x_2 + x_1 5.12/2.29 POL(COND_EVAL_21(x_1, x_2, x_3)) = [2] + [-1]x_3 + x_2 5.12/2.29 POL(&&(x_1, x_2)) = [-1] 5.12/2.29 POL(>=(x_1, x_2)) = [-1] 5.12/2.29 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.12/2.29 POL(1) = [1] 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_>: 5.12/2.29 5.12/2.29 5.12/2.29 COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_bound: 5.12/2.29 5.12/2.29 5.12/2.29 EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) 5.12/2.29 COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(-(x[5], 1), y[5]) 5.12/2.29 5.12/2.29 5.12/2.29 The following pairs are in P_>=: 5.12/2.29 5.12/2.29 5.12/2.29 EVAL_1(x[0], y[0]) -> COND_EVAL_1(>(x[0], 0), x[0], y[0]) 5.12/2.29 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 EVAL_2(x[4], y[4]) -> COND_EVAL_21(&&(>(x[4], 0), >=(0, y[4])), x[4], y[4]) 5.12/2.29 5.12/2.29 5.12/2.29 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.12/2.29 5.12/2.29 TRUE^1 -> &&(TRUE, TRUE)^1 5.12/2.29 FALSE^1 -> &&(FALSE, FALSE)^1 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (12) 5.12/2.29 Obligation: 5.12/2.29 IDP problem: 5.12/2.29 The following function symbols are pre-defined: 5.12/2.29 <<< 5.12/2.29 & ~ Bwand: (Integer, Integer) -> Integer 5.12/2.29 >= ~ Ge: (Integer, Integer) -> Boolean 5.12/2.29 | ~ Bwor: (Integer, Integer) -> Integer 5.12/2.29 / ~ Div: (Integer, Integer) -> Integer 5.12/2.29 != ~ Neq: (Integer, Integer) -> Boolean 5.12/2.29 && ~ Land: (Boolean, Boolean) -> Boolean 5.12/2.29 ! ~ Lnot: (Boolean) -> Boolean 5.12/2.29 = ~ Eq: (Integer, Integer) -> Boolean 5.12/2.29 <= ~ Le: (Integer, Integer) -> Boolean 5.12/2.29 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.12/2.29 % ~ Mod: (Integer, Integer) -> Integer 5.12/2.29 > ~ Gt: (Integer, Integer) -> Boolean 5.12/2.29 + ~ Add: (Integer, Integer) -> Integer 5.12/2.29 -1 ~ UnaryMinus: (Integer) -> Integer 5.12/2.29 < ~ Lt: (Integer, Integer) -> Boolean 5.12/2.29 || ~ Lor: (Boolean, Boolean) -> Boolean 5.12/2.29 - ~ Sub: (Integer, Integer) -> Integer 5.12/2.29 ~ ~ Bwnot: (Integer) -> Integer 5.12/2.29 * ~ Mul: (Integer, Integer) -> Integer 5.12/2.29 >>> 5.12/2.29 5.12/2.29 5.12/2.29 The following domains are used: 5.12/2.29 Integer, Boolean 5.12/2.29 5.12/2.29 R is empty. 5.12/2.29 5.12/2.29 The integer pair graph contains the following rules and edges: 5.12/2.29 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] > 0, x[0], y[0]) 5.12/2.29 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.12/2.29 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(x[4] > 0 && 0 >= y[4], x[4], y[4]) 5.12/2.29 5.12/2.29 (0) -> (1), if (x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.12/2.29 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.12/2.29 5.12/2.29 The set Q consists of the following terms: 5.12/2.29 eval_1(x0, x1) 5.12/2.29 Cond_eval_1(TRUE, x0, x1) 5.12/2.29 eval_2(x0, x1) 5.12/2.29 Cond_eval_2(TRUE, x0, x1) 5.12/2.29 Cond_eval_21(TRUE, x0, x1) 5.12/2.29 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (13) IDependencyGraphProof (EQUIVALENT) 5.12/2.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.12/2.29 ---------------------------------------- 5.12/2.29 5.12/2.29 (14) 5.12/2.29 TRUE 5.21/2.33 EOF