5.32/2.25 YES 5.32/2.27 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.32/2.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.32/2.27 5.32/2.27 5.32/2.27 Termination of the given ITRS could be proven: 5.32/2.27 5.32/2.27 (0) ITRS 5.32/2.27 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.32/2.27 (2) IDP 5.32/2.27 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.27 (4) IDP 5.32/2.27 (5) IDPNonInfProof [SOUND, 249 ms] 5.32/2.27 (6) IDP 5.32/2.27 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.32/2.27 (8) IDP 5.32/2.27 (9) IDPNonInfProof [SOUND, 61 ms] 5.32/2.27 (10) IDP 5.32/2.27 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.32/2.27 (12) TRUE 5.32/2.27 5.32/2.27 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (0) 5.32/2.27 Obligation: 5.32/2.27 ITRS problem: 5.32/2.27 5.32/2.27 The following function symbols are pre-defined: 5.32/2.27 <<< 5.32/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.27 / ~ Div: (Integer, Integer) -> Integer 5.32/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.27 + ~ Add: (Integer, Integer) -> Integer 5.32/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.27 >>> 5.32/2.27 5.32/2.27 The TRS R consists of the following rules: 5.32/2.27 eval(x, y) -> Cond_eval(x > y && x > 0 && y > 0, x, y) 5.32/2.27 Cond_eval(TRUE, x, y) -> eval(x - y, y) 5.32/2.27 eval(x, y) -> Cond_eval1(y > x && x > 0 && y > 0 && x > y, x, y) 5.32/2.27 Cond_eval1(TRUE, x, y) -> eval(x - y, y) 5.32/2.27 eval(x, y) -> Cond_eval2(x > y && x > 0 && y > 0 && y >= x, x, y) 5.32/2.27 Cond_eval2(TRUE, x, y) -> eval(x, y - x) 5.32/2.27 eval(x, y) -> Cond_eval3(y > x && x > 0 && y > 0 && y >= x, x, y) 5.32/2.27 Cond_eval3(TRUE, x, y) -> eval(x, y - x) 5.32/2.27 The set Q consists of the following terms: 5.32/2.27 eval(x0, x1) 5.32/2.27 Cond_eval(TRUE, x0, x1) 5.32/2.27 Cond_eval1(TRUE, x0, x1) 5.32/2.27 Cond_eval2(TRUE, x0, x1) 5.32/2.27 Cond_eval3(TRUE, x0, x1) 5.32/2.27 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (1) ITRStoIDPProof (EQUIVALENT) 5.32/2.27 Added dependency pairs 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (2) 5.32/2.27 Obligation: 5.32/2.27 IDP problem: 5.32/2.27 The following function symbols are pre-defined: 5.32/2.27 <<< 5.32/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.27 / ~ Div: (Integer, Integer) -> Integer 5.32/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.27 + ~ Add: (Integer, Integer) -> Integer 5.32/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.27 >>> 5.32/2.27 5.32/2.27 5.32/2.27 The following domains are used: 5.32/2.27 Boolean, Integer 5.32/2.27 5.32/2.27 The ITRS R consists of the following rules: 5.32/2.27 eval(x, y) -> Cond_eval(x > y && x > 0 && y > 0, x, y) 5.32/2.27 Cond_eval(TRUE, x, y) -> eval(x - y, y) 5.32/2.27 eval(x, y) -> Cond_eval1(y > x && x > 0 && y > 0 && x > y, x, y) 5.32/2.27 Cond_eval1(TRUE, x, y) -> eval(x - y, y) 5.32/2.27 eval(x, y) -> Cond_eval2(x > y && x > 0 && y > 0 && y >= x, x, y) 5.32/2.27 Cond_eval2(TRUE, x, y) -> eval(x, y - x) 5.32/2.27 eval(x, y) -> Cond_eval3(y > x && x > 0 && y > 0 && y >= x, x, y) 5.32/2.27 Cond_eval3(TRUE, x, y) -> eval(x, y - x) 5.32/2.27 5.32/2.27 The integer pair graph contains the following rules and edges: 5.32/2.27 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) 5.32/2.27 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) 5.32/2.27 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2], x[2], y[2]) 5.32/2.27 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - y[3], y[3]) 5.32/2.27 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4], x[4], y[4]) 5.32/2.27 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - x[5]) 5.32/2.27 (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) 5.32/2.27 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) 5.32/2.27 5.32/2.27 (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.32/2.27 (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) 5.32/2.27 (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) 5.32/2.27 (1) -> (4), if (x[1] - y[1] ->^* x[4] & y[1] ->^* y[4]) 5.32/2.27 (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) 5.32/2.27 (2) -> (3), if (y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.32/2.27 (3) -> (0), if (x[3] - y[3] ->^* x[0] & y[3] ->^* y[0]) 5.32/2.27 (3) -> (2), if (x[3] - y[3] ->^* x[2] & y[3] ->^* y[2]) 5.32/2.27 (3) -> (4), if (x[3] - y[3] ->^* x[4] & y[3] ->^* y[4]) 5.32/2.27 (3) -> (6), if (x[3] - y[3] ->^* x[6] & y[3] ->^* y[6]) 5.32/2.27 (4) -> (5), if (x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.32/2.27 (5) -> (0), if (x[5] ->^* x[0] & y[5] - x[5] ->^* y[0]) 5.32/2.27 (5) -> (2), if (x[5] ->^* x[2] & y[5] - x[5] ->^* y[2]) 5.32/2.27 (5) -> (4), if (x[5] ->^* x[4] & y[5] - x[5] ->^* y[4]) 5.32/2.27 (5) -> (6), if (x[5] ->^* x[6] & y[5] - x[5] ->^* y[6]) 5.32/2.27 (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 5.32/2.27 (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) 5.32/2.27 (7) -> (2), if (x[7] ->^* x[2] & y[7] - x[7] ->^* y[2]) 5.32/2.27 (7) -> (4), if (x[7] ->^* x[4] & y[7] - x[7] ->^* y[4]) 5.32/2.27 (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) 5.32/2.27 5.32/2.27 The set Q consists of the following terms: 5.32/2.27 eval(x0, x1) 5.32/2.27 Cond_eval(TRUE, x0, x1) 5.32/2.27 Cond_eval1(TRUE, x0, x1) 5.32/2.27 Cond_eval2(TRUE, x0, x1) 5.32/2.27 Cond_eval3(TRUE, x0, x1) 5.32/2.27 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (3) UsableRulesProof (EQUIVALENT) 5.32/2.27 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. 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (4) 5.32/2.27 Obligation: 5.32/2.27 IDP problem: 5.32/2.27 The following function symbols are pre-defined: 5.32/2.27 <<< 5.32/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.27 / ~ Div: (Integer, Integer) -> Integer 5.32/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.27 + ~ Add: (Integer, Integer) -> Integer 5.32/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.27 >>> 5.32/2.27 5.32/2.27 5.32/2.27 The following domains are used: 5.32/2.27 Boolean, Integer 5.32/2.27 5.32/2.27 R is empty. 5.32/2.27 5.32/2.27 The integer pair graph contains the following rules and edges: 5.32/2.27 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) 5.32/2.27 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - y[1], y[1]) 5.32/2.27 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2], x[2], y[2]) 5.32/2.27 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - y[3], y[3]) 5.32/2.27 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4], x[4], y[4]) 5.32/2.27 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - x[5]) 5.32/2.27 (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) 5.32/2.27 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) 5.32/2.27 5.32/2.27 (0) -> (1), if (x[0] > y[0] && x[0] > 0 && y[0] > 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.32/2.27 (1) -> (0), if (x[1] - y[1] ->^* x[0] & y[1] ->^* y[0]) 5.32/2.27 (1) -> (2), if (x[1] - y[1] ->^* x[2] & y[1] ->^* y[2]) 5.32/2.27 (1) -> (4), if (x[1] - y[1] ->^* x[4] & y[1] ->^* y[4]) 5.32/2.27 (1) -> (6), if (x[1] - y[1] ->^* x[6] & y[1] ->^* y[6]) 5.32/2.27 (2) -> (3), if (y[2] > x[2] && x[2] > 0 && y[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.32/2.27 (3) -> (0), if (x[3] - y[3] ->^* x[0] & y[3] ->^* y[0]) 5.32/2.27 (3) -> (2), if (x[3] - y[3] ->^* x[2] & y[3] ->^* y[2]) 5.32/2.27 (3) -> (4), if (x[3] - y[3] ->^* x[4] & y[3] ->^* y[4]) 5.32/2.27 (3) -> (6), if (x[3] - y[3] ->^* x[6] & y[3] ->^* y[6]) 5.32/2.27 (4) -> (5), if (x[4] > y[4] && x[4] > 0 && y[4] > 0 && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 5.32/2.27 (5) -> (0), if (x[5] ->^* x[0] & y[5] - x[5] ->^* y[0]) 5.32/2.27 (5) -> (2), if (x[5] ->^* x[2] & y[5] - x[5] ->^* y[2]) 5.32/2.27 (5) -> (4), if (x[5] ->^* x[4] & y[5] - x[5] ->^* y[4]) 5.32/2.27 (5) -> (6), if (x[5] ->^* x[6] & y[5] - x[5] ->^* y[6]) 5.32/2.27 (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 5.32/2.27 (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) 5.32/2.27 (7) -> (2), if (x[7] ->^* x[2] & y[7] - x[7] ->^* y[2]) 5.32/2.27 (7) -> (4), if (x[7] ->^* x[4] & y[7] - x[7] ->^* y[4]) 5.32/2.27 (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) 5.32/2.27 5.32/2.27 The set Q consists of the following terms: 5.32/2.27 eval(x0, x1) 5.32/2.27 Cond_eval(TRUE, x0, x1) 5.32/2.27 Cond_eval1(TRUE, x0, x1) 5.32/2.27 Cond_eval2(TRUE, x0, x1) 5.32/2.27 Cond_eval3(TRUE, x0, x1) 5.32/2.27 5.32/2.27 ---------------------------------------- 5.32/2.27 5.32/2.27 (5) IDPNonInfProof (SOUND) 5.32/2.27 Used the following options for this NonInfProof: 5.32/2.27 5.32/2.27 IDPGPoloSolver: 5.32/2.27 Range: [(-1,2)] 5.32/2.27 IsNat: false 5.32/2.27 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@37d1e291 5.32/2.27 Constraint Generator: NonInfConstraintGenerator: 5.32/2.27 PathGenerator: MetricPathGenerator: 5.32/2.27 Max Left Steps: 1 5.32/2.27 Max Right Steps: 1 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 The constraints were generated the following way: 5.32/2.27 5.32/2.27 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.32/2.27 5.32/2.27 Note that final constraints are written in bold face. 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair EVAL(x, y) -> COND_EVAL(&&(&&(>(x, y), >(x, 0)), >(y, 0)), x, y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, y), y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], y[1]), y[1]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>(y[0], 0)=TRUE & >(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], y[0]), y[0]) & (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >(x, y)), x, y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>(x[2], y[2])=TRUE & >(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [-1 + (-1)bso_30] + y[2] + [2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [-1 + (-1)bso_30] + y[2] + [2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[2] >= 0 & [-1 + (-1)bso_30] + y[2] + [2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We solved constraint (5) using rule (IDP_SMT_SPLIT). 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(-(x, y), y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(-(x[3], y[3]), y[3]) & (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>(x[2], y[2])=TRUE & >(y[2], 0)=TRUE & >(y[2], x[2])=TRUE & >(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], y[2]), y[2]) & (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [(-1)bso_32] + [-2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [(-1)bso_32] + [-2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (x[2] + [-1] + [-1]y[2] >= 0 & y[2] + [-1] >= 0 & y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], y[3]), y[3])), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [(-1)bso_32] + [-2]x[2] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We solved constraint (5) using rule (IDP_SMT_SPLIT). 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair EVAL(x, y) -> COND_EVAL2(&&(&&(&&(>(x, y), >(x, 0)), >(y, 0)), >=(y, x)), x, y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>=(y[4], x[4])=TRUE & >(y[4], 0)=TRUE & >(x[4], y[4])=TRUE & >(x[4], 0)=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [-1 + (-1)bso_34] + y[4] + [2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [-1 + (-1)bso_34] + y[4] + [2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x[4] >= 0 & [-1 + (-1)bso_34] + y[4] + [2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We solved constraint (5) using rule (IDP_SMT_SPLIT). 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, x)) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], -(y[5], x[5])) & (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>=(y[4], x[4])=TRUE & >(y[4], 0)=TRUE & >(x[4], y[4])=TRUE & >(x[4], 0)=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(x[4], -(y[4], x[4])) & (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [(-1)bso_36] + [-1]y[4] + [-2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [(-1)bso_36] + [-1]y[4] + [-2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (y[4] + [-1]x[4] >= 0 & y[4] + [-1] >= 0 & x[4] + [-1] + [-1]y[4] >= 0 & x[4] + [-1] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], x[5]))), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [(-1)bso_36] + [-1]y[4] + [-2]x[4] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We solved constraint (5) using rule (IDP_SMT_SPLIT). 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair EVAL(x, y) -> COND_EVAL3(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >=(y, x)), x, y) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: 5.32/2.27 5.32/2.27 (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.27 5.32/2.27 (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.27 5.32/2.27 (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.27 5.32/2.27 (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.27 5.32/2.27 (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 5.32/2.27 For Pair COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, x)) the following chains were created: 5.32/2.27 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: 5.32/2.28 5.32/2.28 (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], x[7])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.28 5.32/2.28 (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], x[6])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.28 5.32/2.28 (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x[6] >= 0 & [(-1)bso_40] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.28 5.32/2.28 (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x[6] >= 0 & [(-1)bso_40] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.28 5.32/2.28 (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x[6] >= 0 & [(-1)bso_40] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 To summarize, we get the following constraints P__>=_ for the following pairs. 5.32/2.28 5.32/2.28 *EVAL(x, y) -> COND_EVAL(&&(&&(>(x, y), >(x, 0)), >(y, 0)), x, y) 5.32/2.28 5.32/2.28 *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *COND_EVAL(TRUE, x, y) -> EVAL(-(x, y), y) 5.32/2.28 5.32/2.28 *(y[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], y[1]), y[1])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[0] >= 0 & [(-1)bso_28] + y[0] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *EVAL(x, y) -> COND_EVAL1(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >(x, y)), x, y) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, y), y) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *EVAL(x, y) -> COND_EVAL2(&&(&&(&&(>(x, y), >(x, 0)), >(y, 0)), >=(y, x)), x, y) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, x)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *EVAL(x, y) -> COND_EVAL3(&&(&&(&&(>(y, x), >(x, 0)), >(y, 0)), >=(y, x)), x, y) 5.32/2.28 5.32/2.28 *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, x)) 5.32/2.28 5.32/2.28 *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x[6] >= 0 & [(-1)bso_40] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 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. 5.32/2.28 5.32/2.28 Using the following integer polynomial ordering the resulting constraints can be solved 5.32/2.28 5.32/2.28 Polynomial interpretation over integers[POLO]: 5.32/2.28 5.32/2.28 POL(TRUE) = 0 5.32/2.28 POL(FALSE) = [1] 5.32/2.28 POL(EVAL(x_1, x_2)) = [-1] + x_1 5.32/2.28 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_2 5.32/2.28 POL(&&(x_1, x_2)) = [-1] 5.32/2.28 POL(>(x_1, x_2)) = [-1] 5.32/2.28 POL(0) = 0 5.32/2.28 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.32/2.28 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 5.32/2.28 POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 5.32/2.28 POL(>=(x_1, x_2)) = [-1] 5.32/2.28 POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + x_2 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_>: 5.32/2.28 5.32/2.28 5.32/2.28 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) 5.32/2.28 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) 5.32/2.28 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) 5.32/2.28 EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) 5.32/2.28 COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_bound: 5.32/2.28 5.32/2.28 5.32/2.28 EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) 5.32/2.28 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], y[1]), y[1]) 5.32/2.28 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(&&(>(y[2], x[2]), >(x[2], 0)), >(y[2], 0)), >(x[2], y[2])), x[2], y[2]) 5.32/2.28 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], y[3]), y[3]) 5.32/2.28 EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(&&(>(x[4], y[4]), >(x[4], 0)), >(y[4], 0)), >=(y[4], x[4])), x[4], y[4]) 5.32/2.28 COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], x[5])) 5.32/2.28 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 5.32/2.28 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_>=: 5.32/2.28 5.32/2.28 5.32/2.28 EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], y[0]), >(x[0], 0)), >(y[0], 0)), x[0], y[0]) 5.32/2.28 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 5.32/2.28 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 5.32/2.28 5.32/2.28 5.32/2.28 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.32/2.28 5.32/2.28 TRUE^1 -> &&(TRUE, TRUE)^1 5.32/2.28 FALSE^1 -> &&(TRUE, FALSE)^1 5.32/2.28 FALSE^1 -> &&(FALSE, TRUE)^1 5.32/2.28 FALSE^1 -> &&(FALSE, FALSE)^1 5.32/2.28 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (6) 5.32/2.28 Obligation: 5.32/2.28 IDP problem: 5.32/2.28 The following function symbols are pre-defined: 5.32/2.28 <<< 5.32/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.28 / ~ Div: (Integer, Integer) -> Integer 5.32/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.28 + ~ Add: (Integer, Integer) -> Integer 5.32/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.28 >>> 5.32/2.28 5.32/2.28 5.32/2.28 The following domains are used: 5.32/2.28 Boolean, Integer 5.32/2.28 5.32/2.28 R is empty. 5.32/2.28 5.32/2.28 The integer pair graph contains the following rules and edges: 5.32/2.28 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0] && x[0] > 0 && y[0] > 0, x[0], y[0]) 5.32/2.28 (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) 5.32/2.28 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) 5.32/2.28 5.32/2.28 (7) -> (0), if (x[7] ->^* x[0] & y[7] - x[7] ->^* y[0]) 5.32/2.28 (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) 5.32/2.28 (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 5.32/2.28 5.32/2.28 The set Q consists of the following terms: 5.32/2.28 eval(x0, x1) 5.32/2.28 Cond_eval(TRUE, x0, x1) 5.32/2.28 Cond_eval1(TRUE, x0, x1) 5.32/2.28 Cond_eval2(TRUE, x0, x1) 5.32/2.28 Cond_eval3(TRUE, x0, x1) 5.32/2.28 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (7) IDependencyGraphProof (EQUIVALENT) 5.32/2.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (8) 5.32/2.28 Obligation: 5.32/2.28 IDP problem: 5.32/2.28 The following function symbols are pre-defined: 5.32/2.28 <<< 5.32/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.28 / ~ Div: (Integer, Integer) -> Integer 5.32/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.28 + ~ Add: (Integer, Integer) -> Integer 5.32/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.28 >>> 5.32/2.28 5.32/2.28 5.32/2.28 The following domains are used: 5.32/2.28 Integer, Boolean 5.32/2.28 5.32/2.28 R is empty. 5.32/2.28 5.32/2.28 The integer pair graph contains the following rules and edges: 5.32/2.28 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - x[7]) 5.32/2.28 (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) 5.32/2.28 5.32/2.28 (7) -> (6), if (x[7] ->^* x[6] & y[7] - x[7] ->^* y[6]) 5.32/2.28 (6) -> (7), if (y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 5.32/2.28 5.32/2.28 The set Q consists of the following terms: 5.32/2.28 eval(x0, x1) 5.32/2.28 Cond_eval(TRUE, x0, x1) 5.32/2.28 Cond_eval1(TRUE, x0, x1) 5.32/2.28 Cond_eval2(TRUE, x0, x1) 5.32/2.28 Cond_eval3(TRUE, x0, x1) 5.32/2.28 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (9) IDPNonInfProof (SOUND) 5.32/2.28 Used the following options for this NonInfProof: 5.32/2.28 5.32/2.28 IDPGPoloSolver: 5.32/2.28 Range: [(-1,2)] 5.32/2.28 IsNat: false 5.32/2.28 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@37d1e291 5.32/2.28 Constraint Generator: NonInfConstraintGenerator: 5.32/2.28 PathGenerator: MetricPathGenerator: 5.32/2.28 Max Left Steps: 1 5.32/2.28 Max Right Steps: 1 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 The constraints were generated the following way: 5.32/2.28 5.32/2.28 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.32/2.28 5.32/2.28 Note that final constraints are written in bold face. 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 For Pair COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) the following chains were created: 5.32/2.28 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])), EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) which results in the following constraint: 5.32/2.28 5.32/2.28 (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & x[7]=x[6]1 & -(y[7], x[7])=y[6]1 ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(x[7], -(y[7], x[7])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.28 5.32/2.28 (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], x[6])) & (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.28 5.32/2.28 (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[6] + [bni_13]x[6] >= 0 & [(-1)bso_14] + x[6] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.28 5.32/2.28 (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[6] + [bni_13]x[6] >= 0 & [(-1)bso_14] + x[6] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.28 5.32/2.28 (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[6] + [bni_13]x[6] >= 0 & [(-1)bso_14] + x[6] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 For Pair EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) the following chains were created: 5.32/2.28 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) which results in the following constraint: 5.32/2.28 5.32/2.28 (1) (&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.32/2.28 5.32/2.28 (2) (>=(y[6], x[6])=TRUE & >(y[6], 0)=TRUE & >(y[6], x[6])=TRUE & >(x[6], 0)=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=)) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.32/2.28 5.32/2.28 (3) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[6] + [bni_15]x[6] >= 0 & [(-1)bso_16] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.32/2.28 5.32/2.28 (4) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[6] + [bni_15]x[6] >= 0 & [(-1)bso_16] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.32/2.28 5.32/2.28 (5) (y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[6] + [bni_15]x[6] >= 0 & [(-1)bso_16] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 To summarize, we get the following constraints P__>=_ for the following pairs. 5.32/2.28 5.32/2.28 *COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 5.32/2.28 5.32/2.28 *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], x[7]))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[6] + [bni_13]x[6] >= 0 & [(-1)bso_14] + x[6] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 *EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 5.32/2.28 5.32/2.28 *(y[6] + [-1]x[6] >= 0 & y[6] + [-1] >= 0 & y[6] + [-1] + [-1]x[6] >= 0 & x[6] + [-1] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[6] + [bni_15]x[6] >= 0 & [(-1)bso_16] >= 0) 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 5.32/2.28 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. 5.32/2.28 5.32/2.28 Using the following integer polynomial ordering the resulting constraints can be solved 5.32/2.28 5.32/2.28 Polynomial interpretation over integers[POLO]: 5.32/2.28 5.32/2.28 POL(TRUE) = 0 5.32/2.28 POL(FALSE) = [1] 5.32/2.28 POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + x_3 + x_2 + [-1]x_1 5.32/2.28 POL(EVAL(x_1, x_2)) = [-1] + x_2 + x_1 5.32/2.28 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.32/2.28 POL(&&(x_1, x_2)) = 0 5.32/2.28 POL(>(x_1, x_2)) = [-1] 5.32/2.28 POL(0) = 0 5.32/2.28 POL(>=(x_1, x_2)) = [-1] 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_>: 5.32/2.28 5.32/2.28 5.32/2.28 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_bound: 5.32/2.28 5.32/2.28 5.32/2.28 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], x[7])) 5.32/2.28 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 5.32/2.28 5.32/2.28 5.32/2.28 The following pairs are in P_>=: 5.32/2.28 5.32/2.28 5.32/2.28 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(&&(>(y[6], x[6]), >(x[6], 0)), >(y[6], 0)), >=(y[6], x[6])), x[6], y[6]) 5.32/2.28 5.32/2.28 5.32/2.28 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.32/2.28 5.32/2.28 &&(TRUE, TRUE)^1 <-> TRUE^1 5.32/2.28 FALSE^1 -> &&(TRUE, FALSE)^1 5.32/2.28 FALSE^1 -> &&(FALSE, TRUE)^1 5.32/2.28 FALSE^1 -> &&(FALSE, FALSE)^1 5.32/2.28 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (10) 5.32/2.28 Obligation: 5.32/2.28 IDP problem: 5.32/2.28 The following function symbols are pre-defined: 5.32/2.28 <<< 5.32/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.32/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.32/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.32/2.28 / ~ Div: (Integer, Integer) -> Integer 5.32/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.32/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.32/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.32/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.32/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.32/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.32/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.32/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.32/2.28 + ~ Add: (Integer, Integer) -> Integer 5.32/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.32/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.32/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.32/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.32/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.32/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.32/2.28 >>> 5.32/2.28 5.32/2.28 5.32/2.28 The following domains are used: 5.32/2.28 Boolean, Integer 5.32/2.28 5.32/2.28 R is empty. 5.32/2.28 5.32/2.28 The integer pair graph contains the following rules and edges: 5.32/2.28 (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && x[6] > 0 && y[6] > 0 && y[6] >= x[6], x[6], y[6]) 5.32/2.28 5.32/2.28 5.32/2.28 The set Q consists of the following terms: 5.32/2.28 eval(x0, x1) 5.32/2.28 Cond_eval(TRUE, x0, x1) 5.32/2.28 Cond_eval1(TRUE, x0, x1) 5.32/2.28 Cond_eval2(TRUE, x0, x1) 5.32/2.28 Cond_eval3(TRUE, x0, x1) 5.32/2.28 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (11) IDependencyGraphProof (EQUIVALENT) 5.32/2.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.32/2.28 ---------------------------------------- 5.32/2.28 5.32/2.28 (12) 5.32/2.28 TRUE 5.44/2.30 EOF