5.43/2.32 YES 5.43/2.34 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 5.43/2.34 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.43/2.34 5.43/2.34 5.43/2.34 Termination of the given ITRS could be proven: 5.43/2.34 5.43/2.34 (0) ITRS 5.43/2.34 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.43/2.34 (2) IDP 5.43/2.34 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.43/2.34 (4) IDP 5.43/2.34 (5) IDPNonInfProof [SOUND, 207 ms] 5.43/2.34 (6) AND 5.43/2.34 (7) IDP 5.43/2.34 (8) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.43/2.34 (9) IDP 5.43/2.34 (10) IDPNonInfProof [SOUND, 58 ms] 5.43/2.34 (11) IDP 5.43/2.34 (12) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.43/2.34 (13) TRUE 5.43/2.34 (14) IDP 5.43/2.34 (15) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.43/2.34 (16) IDP 5.43/2.34 (17) IDPNonInfProof [SOUND, 20 ms] 5.43/2.34 (18) IDP 5.43/2.34 (19) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.43/2.34 (20) TRUE 5.43/2.34 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (0) 5.43/2.34 Obligation: 5.43/2.34 ITRS problem: 5.43/2.34 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 The TRS R consists of the following rules: 5.43/2.34 eval_1(x, y) -> Cond_eval_1(x = y && x > 0, x, y) 5.43/2.34 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 5.43/2.34 eval_2(x, y) -> Cond_eval_2(y > 0, x, y) 5.43/2.34 Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y - 1) 5.43/2.34 eval_2(x, y) -> Cond_eval_21(0 >= y, x, y) 5.43/2.34 Cond_eval_21(TRUE, x, y) -> eval_1(x, y) 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (1) ITRStoIDPProof (EQUIVALENT) 5.43/2.34 Added dependency pairs 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (2) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Boolean, Integer 5.43/2.34 5.43/2.34 The ITRS R consists of the following rules: 5.43/2.34 eval_1(x, y) -> Cond_eval_1(x = y && x > 0, x, y) 5.43/2.34 Cond_eval_1(TRUE, x, y) -> eval_2(x, y) 5.43/2.34 eval_2(x, y) -> Cond_eval_2(y > 0, x, y) 5.43/2.34 Cond_eval_2(TRUE, x, y) -> eval_2(x - 1, y - 1) 5.43/2.34 eval_2(x, y) -> Cond_eval_21(0 >= y, x, y) 5.43/2.34 Cond_eval_21(TRUE, x, y) -> eval_1(x, y) 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) 5.43/2.34 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.43/2.34 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.43/2.34 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) 5.43/2.34 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (3) UsableRulesProof (EQUIVALENT) 5.43/2.34 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. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (4) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Boolean, Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) 5.43/2.34 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.43/2.34 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.43/2.34 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) 5.43/2.34 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (5) IDPNonInfProof (SOUND) 5.43/2.34 Used the following options for this NonInfProof: 5.43/2.34 5.43/2.34 IDPGPoloSolver: 5.43/2.34 Range: [(-1,2)] 5.43/2.34 IsNat: false 5.43/2.34 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@68e91735 5.43/2.34 Constraint Generator: NonInfConstraintGenerator: 5.43/2.34 PathGenerator: MetricPathGenerator: 5.43/2.34 Max Left Steps: 1 5.43/2.34 Max Right Steps: 1 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 The constraints were generated the following way: 5.43/2.34 5.43/2.34 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.43/2.34 5.43/2.34 Note that final constraints are written in bold face. 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_1(x, y) -> COND_EVAL_1(&&(=(x, y), >(x, 0)), x, y) the following chains were created: 5.43/2.34 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(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: 5.43/2.34 5.43/2.34 (1) (&&(=(x[0], y[0]), >(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], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(x[0], 0)=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [-1 + (-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [-1 + (-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [-1 + (-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) the following chains were created: 5.43/2.34 *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(>(y[2], 0), x[2], y[2]) which results in the following constraint: 5.43/2.34 5.43/2.34 (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])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (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])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *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(>=(0, y[4]), x[4], y[4]) which results in the following constraint: 5.43/2.34 5.43/2.34 (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])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (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])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_2(x, y) -> COND_EVAL_2(>(y, 0), x, y) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>(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(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(y[2], 0)=TRUE ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), -(y, 1)) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_2(x, y) -> COND_EVAL_21(>=(0, y), x, y) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>=(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(>=(0, y[4]), x[4], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>=(0, y[4])=TRUE ==> EVAL_2(x[4], y[4])_>=_NonInfC & EVAL_2(x[4], y[4])_>=_COND_EVAL_21(>=(0, y[4]), x[4], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-1]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-1]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [(-1)bso_34] + [-1]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] >= 0 & [(-1)bso_34] + [-1]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.43/2.34 5.43/2.34 (7) (y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] >= 0 & [(-1)bso_34] + y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) the following chains were created: 5.43/2.34 *We consider the chain COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (x[5]=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], y[5]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (COND_EVAL_21(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5])_>=_EVAL_1(x[5], y[5]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 To summarize, we get the following constraints P__>=_ for the following pairs. 5.43/2.34 5.43/2.34 *EVAL_1(x, y) -> COND_EVAL_1(&&(=(x, y), >(x, 0)), x, y) 5.43/2.34 5.43/2.34 *(x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]y[0] + [bni_25]x[0] >= 0 & [-1 + (-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *COND_EVAL_1(TRUE, x, y) -> EVAL_2(x, y) 5.43/2.34 5.43/2.34 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 *((U^Increasing(EVAL_2(x[1], y[1])), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *EVAL_2(x, y) -> COND_EVAL_2(>(y, 0), x, y) 5.43/2.34 5.43/2.34 *(y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [bni_29] = 0 & [(-1)bni_29 + (-1)Bound*bni_29] >= 0 & [(-1)bso_30] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *COND_EVAL_2(TRUE, x, y) -> EVAL_2(-(x, 1), -(y, 1)) 5.43/2.34 5.43/2.34 *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [bni_31] = 0 & [(-1)bni_31 + (-1)Bound*bni_31] >= 0 & [1 + (-1)bso_32] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *EVAL_2(x, y) -> COND_EVAL_21(>=(0, y), x, y) 5.43/2.34 5.43/2.34 *(y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [bni_33] = 0 & [(-1)bni_33 + (-1)Bound*bni_33] >= 0 & [(-1)bso_34] + y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *COND_EVAL_21(TRUE, x, y) -> EVAL_1(x, y) 5.43/2.34 5.43/2.34 *((U^Increasing(EVAL_1(x[5], y[5])), >=) & [bni_35] = 0 & [(-1)bso_36] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 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. 5.43/2.34 5.43/2.34 Using the following integer polynomial ordering the resulting constraints can be solved 5.43/2.34 5.43/2.34 Polynomial interpretation over integers[POLO]: 5.43/2.34 5.43/2.34 POL(TRUE) = 0 5.43/2.34 POL(FALSE) = [3] 5.43/2.34 POL(EVAL_1(x_1, x_2)) = [-1] + x_2 + x_1 5.43/2.34 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 5.43/2.34 POL(&&(x_1, x_2)) = [-1] 5.43/2.34 POL(=(x_1, x_2)) = [-1] 5.43/2.34 POL(>(x_1, x_2)) = [-1] 5.43/2.34 POL(0) = 0 5.43/2.34 POL(EVAL_2(x_1, x_2)) = [-1] + x_1 5.43/2.34 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_2 5.43/2.34 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.43/2.34 POL(1) = [1] 5.43/2.34 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_3 + x_2 5.43/2.34 POL(>=(x_1, x_2)) = [2] 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>: 5.43/2.34 5.43/2.34 5.43/2.34 COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_bound: 5.43/2.34 5.43/2.34 5.43/2.34 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>=: 5.43/2.34 5.43/2.34 5.43/2.34 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) 5.43/2.34 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) 5.43/2.34 EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) 5.43/2.34 COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 5.43/2.34 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.43/2.34 5.43/2.34 TRUE^1 -> &&(TRUE, TRUE)^1 5.43/2.34 FALSE^1 -> &&(TRUE, FALSE)^1 5.43/2.34 FALSE^1 -> &&(FALSE, TRUE)^1 5.43/2.34 FALSE^1 -> &&(FALSE, FALSE)^1 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (6) 5.43/2.34 Complex Obligation (AND) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (7) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Boolean, Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0]) 5.43/2.34 (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.43/2.34 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (8) IDependencyGraphProof (EQUIVALENT) 5.43/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (9) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Integer, Boolean 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 (0): EVAL_1(x[0], y[0]) -> COND_EVAL_1(x[0] = y[0] && x[0] > 0, x[0], y[0]) 5.43/2.34 5.43/2.34 (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0]) 5.43/2.34 (0) -> (1), if (x[0] = y[0] && x[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (10) IDPNonInfProof (SOUND) 5.43/2.34 Used the following options for this NonInfProof: 5.43/2.34 5.43/2.34 IDPGPoloSolver: 5.43/2.34 Range: [(-1,2)] 5.43/2.34 IsNat: false 5.43/2.34 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@68e91735 5.43/2.34 Constraint Generator: NonInfConstraintGenerator: 5.43/2.34 PathGenerator: MetricPathGenerator: 5.43/2.34 Max Left Steps: 1 5.43/2.34 Max Right Steps: 1 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 The constraints were generated the following way: 5.43/2.34 5.43/2.34 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.43/2.34 5.43/2.34 Note that final constraints are written in bold face. 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>=(0, y[4])=TRUE & x[4]=x[5] & y[4]=y[5] & x[5]=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], y[5]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>=(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], y[4]) & (U^Increasing(EVAL_1(x[5], y[5])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ([-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ([-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ([-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[4] + [(-1)bni_21]x[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) ([-1]y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.43/2.34 5.43/2.34 (7) (y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) the following chains were created: 5.43/2.34 *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(>=(0, y[4]), x[4], y[4]), COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (x[1]=x[4] & y[1]=y[4] & >=(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(>=(0, y[4]), x[4], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>=(0, y[4])=TRUE ==> EVAL_2(x[1], y[4])_>=_NonInfC & EVAL_2(x[1], y[4])_>=_COND_EVAL_21(>=(0, y[4]), x[1], y[4]) & (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[4] + [(-1)bni_23]x[1] >= 0 & [(-1)bso_24] + [-2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[4] + [(-1)bni_23]x[1] >= 0 & [(-1)bso_24] + [-2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[4] + [(-1)bni_23]x[1] >= 0 & [(-1)bso_24] + [-2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) ([-1]y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[4] >= 0 & [(-1)bso_24] + [-2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.43/2.34 5.43/2.34 (7) (y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[4] >= 0 & [(-1)bso_24] + [2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) the following chains were created: 5.43/2.34 *We consider the chain EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]), COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]), EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (&&(=(x[0], y[0]), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[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])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(x[0], 0)=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> COND_EVAL_1(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL_1(TRUE, x[0], y[0])_>=_EVAL_2(x[0], y[0]) & (U^Increasing(EVAL_2(x[1], y[1])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) the following chains were created: 5.43/2.34 *We consider the chain COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]), EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(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: 5.43/2.34 5.43/2.34 (1) (x[5]=x[0] & y[5]=y[0] & &&(=(x[0], y[0]), >(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], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(x[0], 0)=TRUE & >=(x[0], y[0])=TRUE & <=(x[0], y[0])=TRUE ==> EVAL_1(x[0], y[0])_>=_NonInfC & EVAL_1(x[0], y[0])_>=_COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 To summarize, we get the following constraints P__>=_ for the following pairs. 5.43/2.34 5.43/2.34 *COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 *(y[4] >= 0 ==> (U^Increasing(EVAL_1(x[5], y[5])), >=) & [(-1)bni_21] = 0 & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[4] >= 0 & [(-1)bso_22] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) 5.43/2.34 5.43/2.34 *(y[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>=(0, y[4]), x[4], y[4])), >=) & [(-1)bni_23] = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]y[4] >= 0 & [(-1)bso_24] + [2]y[4] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 5.43/2.34 *(x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL_2(x[1], y[1])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) 5.43/2.34 5.43/2.34 *(x[0] + [-1] >= 0 & x[0] + [-1]y[0] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 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. 5.43/2.34 5.43/2.34 Using the following integer polynomial ordering the resulting constraints can be solved 5.43/2.34 5.43/2.34 Polynomial interpretation over integers[POLO]: 5.43/2.34 5.43/2.34 POL(TRUE) = 0 5.43/2.34 POL(FALSE) = [1] 5.43/2.34 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 5.43/2.34 POL(EVAL_1(x_1, x_2)) = [-1] + x_2 + [-1]x_1 5.43/2.34 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 + [-1]x_1 5.43/2.34 POL(>=(x_1, x_2)) = [-1] 5.43/2.34 POL(0) = 0 5.43/2.34 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_2 5.43/2.34 POL(&&(x_1, x_2)) = [-1] 5.43/2.34 POL(=(x_1, x_2)) = [-1] 5.43/2.34 POL(>(x_1, x_2)) = [-1] 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>: 5.43/2.34 5.43/2.34 5.43/2.34 COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_bound: 5.43/2.34 5.43/2.34 5.43/2.34 EVAL_1(x[0], y[0]) -> COND_EVAL_1(&&(=(x[0], y[0]), >(x[0], 0)), x[0], y[0]) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>=: 5.43/2.34 5.43/2.34 5.43/2.34 COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 EVAL_2(x[4], y[4]) -> COND_EVAL_21(>=(0, y[4]), x[4], y[4]) 5.43/2.34 5.43/2.34 5.43/2.34 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.43/2.34 5.43/2.34 FALSE^1 -> &&(TRUE, FALSE)^1 5.43/2.34 FALSE^1 -> &&(FALSE, TRUE)^1 5.43/2.34 FALSE^1 -> &&(FALSE, FALSE)^1 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (11) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (12) IDependencyGraphProof (EQUIVALENT) 5.43/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (13) 5.43/2.34 TRUE 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (14) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (1): COND_EVAL_1(TRUE, x[1], y[1]) -> EVAL_2(x[1], y[1]) 5.43/2.34 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) 5.43/2.34 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) 5.43/2.34 (4): EVAL_2(x[4], y[4]) -> COND_EVAL_21(0 >= y[4], x[4], y[4]) 5.43/2.34 (5): COND_EVAL_21(TRUE, x[5], y[5]) -> EVAL_1(x[5], y[5]) 5.43/2.34 5.43/2.34 (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2]) 5.43/2.34 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) 5.43/2.34 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.43/2.34 (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4]) 5.43/2.34 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] - 1 ->^* y[4]) 5.43/2.34 (4) -> (5), if (0 >= y[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (15) IDependencyGraphProof (EQUIVALENT) 5.43/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (16) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) 5.43/2.34 (2): EVAL_2(x[2], y[2]) -> COND_EVAL_2(y[2] > 0, x[2], y[2]) 5.43/2.34 5.43/2.34 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] - 1 ->^* y[2]) 5.43/2.34 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (17) IDPNonInfProof (SOUND) 5.43/2.34 Used the following options for this NonInfProof: 5.43/2.34 5.43/2.34 IDPGPoloSolver: 5.43/2.34 Range: [(-1,2)] 5.43/2.34 IsNat: false 5.43/2.34 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@68e91735 5.43/2.34 Constraint Generator: NonInfConstraintGenerator: 5.43/2.34 PathGenerator: MetricPathGenerator: 5.43/2.34 Max Left Steps: 1 5.43/2.34 Max Right Steps: 1 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 The constraints were generated the following way: 5.43/2.34 5.43/2.34 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.43/2.34 5.43/2.34 Note that final constraints are written in bold face. 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)), EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=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], 1), -(y[3], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(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], 1), -(y[2], 1)) & (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & 0 = 0 & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 For Pair EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) the following chains were created: 5.43/2.34 *We consider the chain EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]), COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) which results in the following constraint: 5.43/2.34 5.43/2.34 (1) (>(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(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.43/2.34 5.43/2.34 (2) (>(y[2], 0)=TRUE ==> EVAL_2(x[2], y[2])_>=_NonInfC & EVAL_2(x[2], y[2])_>=_COND_EVAL_2(>(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=)) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.43/2.34 5.43/2.34 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.43/2.34 5.43/2.34 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.43/2.34 5.43/2.34 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 To summarize, we get the following constraints P__>=_ for the following pairs. 5.43/2.34 5.43/2.34 *COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) 5.43/2.34 5.43/2.34 *(y[2] + [-1] >= 0 ==> (U^Increasing(EVAL_2(-(x[3], 1), -(y[3], 1))), >=) & 0 = 0 & [(-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 *EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) 5.43/2.34 5.43/2.34 *(y[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL_2(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 5.43/2.34 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. 5.43/2.34 5.43/2.34 Using the following integer polynomial ordering the resulting constraints can be solved 5.43/2.34 5.43/2.34 Polynomial interpretation over integers[POLO]: 5.43/2.34 5.43/2.34 POL(TRUE) = 0 5.43/2.34 POL(FALSE) = 0 5.43/2.34 POL(COND_EVAL_2(x_1, x_2, x_3)) = [2]x_3 5.43/2.34 POL(EVAL_2(x_1, x_2)) = [2] + [2]x_2 5.43/2.34 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.43/2.34 POL(1) = [1] 5.43/2.34 POL(>(x_1, x_2)) = [-1] 5.43/2.34 POL(0) = 0 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>: 5.43/2.34 5.43/2.34 5.43/2.34 EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_bound: 5.43/2.34 5.43/2.34 5.43/2.34 COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) 5.43/2.34 EVAL_2(x[2], y[2]) -> COND_EVAL_2(>(y[2], 0), x[2], y[2]) 5.43/2.34 5.43/2.34 5.43/2.34 The following pairs are in P_>=: 5.43/2.34 5.43/2.34 5.43/2.34 COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(-(x[3], 1), -(y[3], 1)) 5.43/2.34 5.43/2.34 5.43/2.34 There are no usable rules. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (18) 5.43/2.34 Obligation: 5.43/2.34 IDP problem: 5.43/2.34 The following function symbols are pre-defined: 5.43/2.34 <<< 5.43/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.43/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.43/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.43/2.34 / ~ Div: (Integer, Integer) -> Integer 5.43/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.43/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.43/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.43/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.43/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.43/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.43/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.43/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.43/2.34 + ~ Add: (Integer, Integer) -> Integer 5.43/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.43/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.43/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.43/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.43/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.43/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.43/2.34 >>> 5.43/2.34 5.43/2.34 5.43/2.34 The following domains are used: 5.43/2.34 Integer 5.43/2.34 5.43/2.34 R is empty. 5.43/2.34 5.43/2.34 The integer pair graph contains the following rules and edges: 5.43/2.34 (3): COND_EVAL_2(TRUE, x[3], y[3]) -> EVAL_2(x[3] - 1, y[3] - 1) 5.43/2.34 5.43/2.34 5.43/2.34 The set Q consists of the following terms: 5.43/2.34 eval_1(x0, x1) 5.43/2.34 Cond_eval_1(TRUE, x0, x1) 5.43/2.34 eval_2(x0, x1) 5.43/2.34 Cond_eval_2(TRUE, x0, x1) 5.43/2.34 Cond_eval_21(TRUE, x0, x1) 5.43/2.34 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (19) IDependencyGraphProof (EQUIVALENT) 5.43/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.43/2.34 ---------------------------------------- 5.43/2.34 5.43/2.34 (20) 5.43/2.34 TRUE 5.71/2.37 EOF