6.33/2.59 YES 6.33/2.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 6.33/2.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.33/2.61 6.33/2.61 6.33/2.61 Termination of the given ITRS could be proven: 6.33/2.61 6.33/2.61 (0) ITRS 6.33/2.61 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 6.33/2.61 (2) IDP 6.33/2.61 (3) UsableRulesProof [EQUIVALENT, 0 ms] 6.33/2.61 (4) IDP 6.33/2.61 (5) IDPNonInfProof [SOUND, 379 ms] 6.33/2.61 (6) IDP 6.33/2.61 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.33/2.61 (8) IDP 6.33/2.61 (9) IDPNonInfProof [SOUND, 93 ms] 6.33/2.61 (10) IDP 6.33/2.61 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 6.33/2.61 (12) TRUE 6.33/2.61 6.33/2.61 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (0) 6.33/2.61 Obligation: 6.33/2.61 ITRS problem: 6.33/2.61 6.33/2.61 The following function symbols are pre-defined: 6.33/2.61 <<< 6.33/2.61 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.61 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.61 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.61 / ~ Div: (Integer, Integer) -> Integer 6.33/2.61 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.61 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.61 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.61 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.61 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.61 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.61 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.61 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.61 + ~ Add: (Integer, Integer) -> Integer 6.33/2.61 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.61 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.61 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.61 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.61 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.61 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.61 >>> 6.33/2.61 6.33/2.61 The TRS R consists of the following rules: 6.33/2.61 eval(x, y) -> Cond_eval(x + y > 0 && x > y, x, y) 6.33/2.61 Cond_eval(TRUE, x, y) -> eval(x - 1, y) 6.33/2.61 eval(x, y) -> Cond_eval1(x + y > 0 && y >= x && x = y, x, y) 6.33/2.61 Cond_eval1(TRUE, x, y) -> eval(x - 1, y) 6.33/2.61 eval(x, y) -> Cond_eval2(x + y > 0 && y >= x && y > x, x, y) 6.33/2.61 Cond_eval2(TRUE, x, y) -> eval(x, y - 1) 6.33/2.61 eval(x, y) -> Cond_eval3(x + y > 0 && y >= x && x > y, x, y) 6.33/2.61 Cond_eval3(TRUE, x, y) -> eval(x, y - 1) 6.33/2.61 The set Q consists of the following terms: 6.33/2.61 eval(x0, x1) 6.33/2.61 Cond_eval(TRUE, x0, x1) 6.33/2.61 Cond_eval1(TRUE, x0, x1) 6.33/2.61 Cond_eval2(TRUE, x0, x1) 6.33/2.61 Cond_eval3(TRUE, x0, x1) 6.33/2.61 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (1) ITRStoIDPProof (EQUIVALENT) 6.33/2.61 Added dependency pairs 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (2) 6.33/2.61 Obligation: 6.33/2.61 IDP problem: 6.33/2.61 The following function symbols are pre-defined: 6.33/2.61 <<< 6.33/2.61 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.61 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.61 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.61 / ~ Div: (Integer, Integer) -> Integer 6.33/2.61 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.61 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.61 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.61 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.61 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.61 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.61 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.61 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.61 + ~ Add: (Integer, Integer) -> Integer 6.33/2.61 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.61 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.61 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.61 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.61 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.61 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.61 >>> 6.33/2.61 6.33/2.61 6.33/2.61 The following domains are used: 6.33/2.61 Boolean, Integer 6.33/2.61 6.33/2.61 The ITRS R consists of the following rules: 6.33/2.61 eval(x, y) -> Cond_eval(x + y > 0 && x > y, x, y) 6.33/2.61 Cond_eval(TRUE, x, y) -> eval(x - 1, y) 6.33/2.61 eval(x, y) -> Cond_eval1(x + y > 0 && y >= x && x = y, x, y) 6.33/2.61 Cond_eval1(TRUE, x, y) -> eval(x - 1, y) 6.33/2.61 eval(x, y) -> Cond_eval2(x + y > 0 && y >= x && y > x, x, y) 6.33/2.61 Cond_eval2(TRUE, x, y) -> eval(x, y - 1) 6.33/2.61 eval(x, y) -> Cond_eval3(x + y > 0 && y >= x && x > y, x, y) 6.33/2.61 Cond_eval3(TRUE, x, y) -> eval(x, y - 1) 6.33/2.61 6.33/2.61 The integer pair graph contains the following rules and edges: 6.33/2.61 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.33/2.61 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 6.33/2.61 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && y[2] >= x[2] && x[2] = y[2], x[2], y[2]) 6.33/2.61 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 6.33/2.61 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4], x[4], y[4]) 6.33/2.61 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) 6.33/2.61 (6): EVAL(x[6], y[6]) -> COND_EVAL3(x[6] + y[6] > 0 && y[6] >= x[6] && x[6] > y[6], x[6], y[6]) 6.33/2.61 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - 1) 6.33/2.61 6.33/2.61 (0) -> (1), if (x[0] + y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.33/2.61 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 6.33/2.61 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 6.33/2.61 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4]) 6.33/2.61 (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) 6.33/2.61 (2) -> (3), if (x[2] + y[2] > 0 && y[2] >= x[2] && x[2] = y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.33/2.61 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) 6.33/2.61 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 6.33/2.61 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4]) 6.33/2.61 (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) 6.33/2.61 (4) -> (5), if (x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.33/2.61 (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0]) 6.33/2.61 (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2]) 6.33/2.61 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) 6.33/2.61 (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6]) 6.33/2.61 (6) -> (7), if (x[6] + y[6] > 0 && y[6] >= x[6] && x[6] > y[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.33/2.61 (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0]) 6.33/2.61 (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2]) 6.33/2.61 (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4]) 6.33/2.61 (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6]) 6.33/2.61 6.33/2.61 The set Q consists of the following terms: 6.33/2.61 eval(x0, x1) 6.33/2.61 Cond_eval(TRUE, x0, x1) 6.33/2.61 Cond_eval1(TRUE, x0, x1) 6.33/2.61 Cond_eval2(TRUE, x0, x1) 6.33/2.61 Cond_eval3(TRUE, x0, x1) 6.33/2.61 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (3) UsableRulesProof (EQUIVALENT) 6.33/2.61 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. 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (4) 6.33/2.61 Obligation: 6.33/2.61 IDP problem: 6.33/2.61 The following function symbols are pre-defined: 6.33/2.61 <<< 6.33/2.61 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.61 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.61 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.61 / ~ Div: (Integer, Integer) -> Integer 6.33/2.61 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.61 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.61 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.61 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.61 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.61 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.61 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.61 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.61 + ~ Add: (Integer, Integer) -> Integer 6.33/2.61 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.61 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.61 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.61 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.61 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.61 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.61 >>> 6.33/2.61 6.33/2.61 6.33/2.61 The following domains are used: 6.33/2.61 Boolean, Integer 6.33/2.61 6.33/2.61 R is empty. 6.33/2.61 6.33/2.61 The integer pair graph contains the following rules and edges: 6.33/2.61 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.33/2.61 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 6.33/2.61 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && y[2] >= x[2] && x[2] = y[2], x[2], y[2]) 6.33/2.61 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 6.33/2.61 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4], x[4], y[4]) 6.33/2.61 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) 6.33/2.61 (6): EVAL(x[6], y[6]) -> COND_EVAL3(x[6] + y[6] > 0 && y[6] >= x[6] && x[6] > y[6], x[6], y[6]) 6.33/2.61 (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], y[7] - 1) 6.33/2.61 6.33/2.61 (0) -> (1), if (x[0] + y[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 6.33/2.61 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 6.33/2.61 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 6.33/2.61 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4]) 6.33/2.61 (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6]) 6.33/2.61 (2) -> (3), if (x[2] + y[2] > 0 && y[2] >= x[2] && x[2] = y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 6.33/2.61 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) 6.33/2.61 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 6.33/2.61 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4]) 6.33/2.61 (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6]) 6.33/2.61 (4) -> (5), if (x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.33/2.61 (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0]) 6.33/2.61 (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2]) 6.33/2.61 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) 6.33/2.61 (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6]) 6.33/2.61 (6) -> (7), if (x[6] + y[6] > 0 && y[6] >= x[6] && x[6] > y[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) 6.33/2.61 (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0]) 6.33/2.61 (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2]) 6.33/2.61 (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4]) 6.33/2.61 (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6]) 6.33/2.61 6.33/2.61 The set Q consists of the following terms: 6.33/2.61 eval(x0, x1) 6.33/2.61 Cond_eval(TRUE, x0, x1) 6.33/2.61 Cond_eval1(TRUE, x0, x1) 6.33/2.61 Cond_eval2(TRUE, x0, x1) 6.33/2.61 Cond_eval3(TRUE, x0, x1) 6.33/2.61 6.33/2.61 ---------------------------------------- 6.33/2.61 6.33/2.61 (5) IDPNonInfProof (SOUND) 6.33/2.61 Used the following options for this NonInfProof: 6.33/2.61 6.33/2.61 IDPGPoloSolver: 6.33/2.61 Range: [(-1,2)] 6.33/2.61 IsNat: false 6.33/2.61 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5933351 6.33/2.61 Constraint Generator: NonInfConstraintGenerator: 6.33/2.61 PathGenerator: MetricPathGenerator: 6.33/2.61 Max Left Steps: 1 6.33/2.61 Max Right Steps: 1 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 The constraints were generated the following way: 6.33/2.61 6.33/2.61 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.33/2.61 6.33/2.61 Note that final constraints are written in bold face. 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 For Pair EVAL(x, y) -> COND_EVAL(&&(>(+(x, y), 0), >(x, y)), x, y) the following chains were created: 6.33/2.61 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: 6.33/2.61 6.33/2.61 (1) (&&(>(+(x[0], y[0]), 0), >(x[0], y[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]), 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.61 6.33/2.61 (2) (>(+(x[0], y[0]), 0)=TRUE & >(x[0], y[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.61 6.33/2.61 (3) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.61 6.33/2.61 (4) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.61 6.33/2.61 (5) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.61 6.33/2.61 (6) (x[0] >= 0 & [-2]y[0] + x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)Bound*bni_26] + [(-1)bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.61 6.33/2.61 (7) (x[0] >= 0 & [-2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)Bound*bni_26] + [(-1)bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 (8) (x[0] >= 0 & [2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)Bound*bni_26] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: 6.33/2.61 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: 6.33/2.61 6.33/2.61 (1) (&&(>(+(x[0], y[0]), 0), >(x[0], y[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], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.61 6.33/2.61 (2) (>(+(x[0], y[0]), 0)=TRUE & >(x[0], y[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.61 6.33/2.61 (3) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.61 6.33/2.61 (4) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.61 6.33/2.61 (5) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.61 6.33/2.61 (6) (x[0] >= 0 & [-2]y[0] + x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.61 6.33/2.61 (7) (x[0] >= 0 & [2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_28] + [bni_28]y[0] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 (8) (x[0] >= 0 & [-2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(>(+(x, y), 0), >=(y, x)), =(x, y)), x, y) the following chains were created: 6.33/2.61 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: 6.33/2.61 6.33/2.61 (1) (&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(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(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.61 6.33/2.61 (2) (>(+(x[2], y[2]), 0)=TRUE & >=(y[2], x[2])=TRUE & >=(x[2], y[2])=TRUE & <=(x[2], y[2])=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.61 6.33/2.61 (3) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] + [-1]y[2] + x[2] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.61 6.33/2.61 (4) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] + [-1]y[2] + x[2] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.61 6.33/2.61 (5) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x[2] >= 0 & [(-1)bso_31] + [-1]y[2] + x[2] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.61 6.33/2.61 (6) (x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 & [1] + [-2]y[2] + x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] + [-2]y[2] + x[2] >= 0) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: 6.33/2.61 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) which results in the following constraint: 6.33/2.61 6.33/2.61 (1) (&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(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], 1), y[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) 6.33/2.61 6.33/2.61 6.33/2.61 6.33/2.61 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.61 6.33/2.61 (2) (>(+(x[2], y[2]), 0)=TRUE & >=(y[2], x[2])=TRUE & >=(x[2], y[2])=TRUE & <=(x[2], y[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(-(x[2], 1), y[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[2] >= 0 & [1 + (-1)bso_33] + y[2] + [-1]x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[2] >= 0 & [1 + (-1)bso_33] + y[2] + [-1]x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (x[2] + [-1] + y[2] >= 0 & y[2] + [-1]x[2] >= 0 & x[2] + [-1]y[2] >= 0 & y[2] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[2] >= 0 & [1 + (-1)bso_33] + y[2] + [-1]x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.62 6.33/2.62 (6) (x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 & [1] + [-2]y[2] + x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[2] >= 0 & [(-1)bso_33] + [2]y[2] + [-1]x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair EVAL(x, y) -> COND_EVAL2(&&(&&(>(+(x, y), 0), >=(y, x)), >(y, x)), x, y) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(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]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(y[4], x[4])=TRUE & >(+(x[4], y[4]), 0)=TRUE & >=(y[4], x[4])=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.62 6.33/2.62 (6) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.62 6.33/2.62 (7) (y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 (8) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(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], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(y[4], x[4])=TRUE & >(+(x[4], y[4]), 0)=TRUE & >=(y[4], x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(x[4], -(y[4], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.62 6.33/2.62 (6) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.62 6.33/2.62 (7) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 (8) (y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair EVAL(x, y) -> COND_EVAL3(&&(&&(>(+(x, y), 0), >=(y, x)), >(x, y)), x, y) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(x[6], y[6])=TRUE & >(+(x[6], y[6]), 0)=TRUE & >=(y[6], x[6])=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [bni_38]x[6] >= 0 & [-1 + (-1)bso_39] + y[6] + [2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [bni_38]x[6] >= 0 & [-1 + (-1)bso_39] + y[6] + [2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6])), >=) & [(-1)bni_38 + (-1)Bound*bni_38] + [bni_38]x[6] >= 0 & [-1 + (-1)bso_39] + y[6] + [2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We solved constraint (5) using rule (IDP_SMT_SPLIT). 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[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], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(x[6], y[6])=TRUE & >(+(x[6], y[6]), 0)=TRUE & >=(y[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(x[6], -(y[6], 1)) & (U^Increasing(EVAL(x[7], -(y[7], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]y[6] + [(-1)bni_40]x[6] >= 0 & [(-1)bso_41] + [-1]y[6] + [-2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]y[6] + [(-1)bni_40]x[6] >= 0 & [(-1)bso_41] + [-1]y[6] + [-2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (x[6] + [-1] + [-1]y[6] >= 0 & x[6] + [-1] + y[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1))), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]y[6] + [(-1)bni_40]x[6] >= 0 & [(-1)bso_41] + [-1]y[6] + [-2]x[6] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We solved constraint (5) using rule (IDP_SMT_SPLIT). 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 To summarize, we get the following constraints P__>=_ for the following pairs. 6.33/2.62 6.33/2.62 *EVAL(x, y) -> COND_EVAL(&&(>(+(x, y), 0), >(x, y)), x, y) 6.33/2.62 6.33/2.62 *(x[0] >= 0 & [-2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)Bound*bni_26] + [(-1)bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(x[0] >= 0 & [2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)Bound*bni_26] + [bni_26]y[0] + [bni_26]x[0] >= 0 & [(-1)bso_27] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) 6.33/2.62 6.33/2.62 *(x[0] >= 0 & [2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_28] + [bni_28]y[0] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(x[0] >= 0 & [-2]y[0] + x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]y[0] + [bni_28]x[0] >= 0 & [1 + (-1)bso_29] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *EVAL(x, y) -> COND_EVAL1(&&(&&(>(+(x, y), 0), >=(y, x)), =(x, y)), x, y) 6.33/2.62 6.33/2.62 *(x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 & [1] + [-2]y[2] + x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2])), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]y[2] + [bni_30]x[2] >= 0 & [1 + (-1)bso_31] + [-2]y[2] + x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) 6.33/2.62 6.33/2.62 *(x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 & [1] + [-2]y[2] + x[2] >= 0 & [2]y[2] + [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]y[2] >= 0 & [(-1)bso_33] + [2]y[2] + [-1]x[2] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *EVAL(x, y) -> COND_EVAL2(&&(&&(>(+(x, y), 0), >=(y, x)), >(y, x)), x, y) 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]x[4] >= 0 & [(-1)bso_35] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *COND_EVAL2(TRUE, x, y) -> EVAL(x, -(y, 1)) 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]x[4] >= 0 & [(-1)bso_37] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *EVAL(x, y) -> COND_EVAL3(&&(&&(>(+(x, y), 0), >=(y, x)), >(x, y)), x, y) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *COND_EVAL3(TRUE, x, y) -> EVAL(x, -(y, 1)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 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. 6.33/2.62 6.33/2.62 Using the following integer polynomial ordering the resulting constraints can be solved 6.33/2.62 6.33/2.62 Polynomial interpretation over integers[POLO]: 6.33/2.62 6.33/2.62 POL(TRUE) = 0 6.33/2.62 POL(FALSE) = [1] 6.33/2.62 POL(EVAL(x_1, x_2)) = [-1] + x_1 6.33/2.62 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_2 6.33/2.62 POL(&&(x_1, x_2)) = [-1] 6.33/2.62 POL(>(x_1, x_2)) = [-1] 6.33/2.62 POL(+(x_1, x_2)) = x_1 + x_2 6.33/2.62 POL(0) = 0 6.33/2.62 POL(-(x_1, x_2)) = x_1 + [-1]x_2 6.33/2.62 POL(1) = [1] 6.33/2.62 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 6.33/2.62 POL(>=(x_1, x_2)) = [-1] 6.33/2.62 POL(=(x_1, x_2)) = [-1] 6.33/2.62 POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + x_2 6.33/2.62 POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_>: 6.33/2.62 6.33/2.62 6.33/2.62 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 6.33/2.62 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 6.33/2.62 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]) 6.33/2.62 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_bound: 6.33/2.62 6.33/2.62 6.33/2.62 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0]) 6.33/2.62 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 6.33/2.62 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]) 6.33/2.62 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 6.33/2.62 EVAL(x[6], y[6]) -> COND_EVAL3(&&(&&(>(+(x[6], y[6]), 0), >=(y[6], x[6])), >(x[6], y[6])), x[6], y[6]) 6.33/2.62 COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7], -(y[7], 1)) 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_>=: 6.33/2.62 6.33/2.62 6.33/2.62 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], y[0])), x[0], y[0]) 6.33/2.62 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(y[2], x[2])), =(x[2], y[2])), x[2], y[2]) 6.33/2.62 EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) 6.33/2.62 COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) 6.33/2.62 6.33/2.62 6.33/2.62 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.33/2.62 6.33/2.62 TRUE^1 -> &&(TRUE, TRUE)^1 6.33/2.62 FALSE^1 -> &&(TRUE, FALSE)^1 6.33/2.62 FALSE^1 -> &&(FALSE, TRUE)^1 6.33/2.62 FALSE^1 -> &&(FALSE, FALSE)^1 6.33/2.62 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (6) 6.33/2.62 Obligation: 6.33/2.62 IDP problem: 6.33/2.62 The following function symbols are pre-defined: 6.33/2.62 <<< 6.33/2.62 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.62 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.62 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.62 / ~ Div: (Integer, Integer) -> Integer 6.33/2.62 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.62 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.62 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.62 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.62 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.62 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.62 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.62 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.62 + ~ Add: (Integer, Integer) -> Integer 6.33/2.62 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.62 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.62 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.62 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.62 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.62 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.62 >>> 6.33/2.62 6.33/2.62 6.33/2.62 The following domains are used: 6.33/2.62 Boolean, Integer 6.33/2.62 6.33/2.62 R is empty. 6.33/2.62 6.33/2.62 The integer pair graph contains the following rules and edges: 6.33/2.62 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > y[0], x[0], y[0]) 6.33/2.62 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && y[2] >= x[2] && x[2] = y[2], x[2], y[2]) 6.33/2.62 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4], x[4], y[4]) 6.33/2.62 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) 6.33/2.62 6.33/2.62 (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0]) 6.33/2.62 (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2]) 6.33/2.62 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) 6.33/2.62 (4) -> (5), if (x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.33/2.62 6.33/2.62 The set Q consists of the following terms: 6.33/2.62 eval(x0, x1) 6.33/2.62 Cond_eval(TRUE, x0, x1) 6.33/2.62 Cond_eval1(TRUE, x0, x1) 6.33/2.62 Cond_eval2(TRUE, x0, x1) 6.33/2.62 Cond_eval3(TRUE, x0, x1) 6.33/2.62 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (7) IDependencyGraphProof (EQUIVALENT) 6.33/2.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (8) 6.33/2.62 Obligation: 6.33/2.62 IDP problem: 6.33/2.62 The following function symbols are pre-defined: 6.33/2.62 <<< 6.33/2.62 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.62 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.62 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.62 / ~ Div: (Integer, Integer) -> Integer 6.33/2.62 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.62 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.62 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.62 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.62 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.62 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.62 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.62 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.62 + ~ Add: (Integer, Integer) -> Integer 6.33/2.62 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.62 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.62 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.62 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.62 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.62 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.62 >>> 6.33/2.62 6.33/2.62 6.33/2.62 The following domains are used: 6.33/2.62 Integer, Boolean 6.33/2.62 6.33/2.62 R is empty. 6.33/2.62 6.33/2.62 The integer pair graph contains the following rules and edges: 6.33/2.62 (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5] - 1) 6.33/2.62 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4], x[4], y[4]) 6.33/2.62 6.33/2.62 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4]) 6.33/2.62 (4) -> (5), if (x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) 6.33/2.62 6.33/2.62 The set Q consists of the following terms: 6.33/2.62 eval(x0, x1) 6.33/2.62 Cond_eval(TRUE, x0, x1) 6.33/2.62 Cond_eval1(TRUE, x0, x1) 6.33/2.62 Cond_eval2(TRUE, x0, x1) 6.33/2.62 Cond_eval3(TRUE, x0, x1) 6.33/2.62 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (9) IDPNonInfProof (SOUND) 6.33/2.62 Used the following options for this NonInfProof: 6.33/2.62 6.33/2.62 IDPGPoloSolver: 6.33/2.62 Range: [(-1,2)] 6.33/2.62 IsNat: false 6.33/2.62 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5933351 6.33/2.62 Constraint Generator: NonInfConstraintGenerator: 6.33/2.62 PathGenerator: MetricPathGenerator: 6.33/2.62 Max Left Steps: 1 6.33/2.62 Max Right Steps: 1 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 The constraints were generated the following way: 6.33/2.62 6.33/2.62 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 6.33/2.62 6.33/2.62 Note that final constraints are written in bold face. 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & x[5]=x[4]1 & -(y[5], 1)=y[4]1 ==> COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], -(y[5], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(y[4], x[4])=TRUE & >(+(x[4], y[4]), 0)=TRUE & >=(y[4], x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(x[4], -(y[4], 1)) & (U^Increasing(EVAL(x[5], -(y[5], 1))), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[4] + [(-1)bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[4] + [(-1)bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[4] + [(-1)bni_13]x[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.62 6.33/2.62 (6) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)Bound*bni_13] + [bni_13]y[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.62 6.33/2.62 (7) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)Bound*bni_13] + [bni_13]y[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 (8) (y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)Bound*bni_13] + [bni_13]y[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 For Pair EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) the following chains were created: 6.33/2.62 *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) which results in the following constraint: 6.33/2.62 6.33/2.62 (1) (&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(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]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 6.33/2.62 6.33/2.62 (2) (>(y[4], x[4])=TRUE & >(+(x[4], y[4]), 0)=TRUE & >=(y[4], x[4])=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=)) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 6.33/2.62 6.33/2.62 (3) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[4] + [(-1)bni_15]x[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 6.33/2.62 6.33/2.62 (4) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[4] + [(-1)bni_15]x[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 6.33/2.62 6.33/2.62 (5) (y[4] + [-1] + [-1]x[4] >= 0 & x[4] + [-1] + y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[4] + [(-1)bni_15]x[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 6.33/2.62 6.33/2.62 (6) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)Bound*bni_15] + [bni_15]y[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 6.33/2.62 6.33/2.62 (7) (y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)Bound*bni_15] + [bni_15]y[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 (8) (y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)Bound*bni_15] + [bni_15]y[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 To summarize, we get the following constraints P__>=_ for the following pairs. 6.33/2.62 6.33/2.62 *COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)Bound*bni_13] + [bni_13]y[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1))), >=) & [(-1)Bound*bni_13] + [bni_13]y[4] >= 0 & [1 + (-1)bso_14] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 *EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [-2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)Bound*bni_15] + [bni_15]y[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 *(y[4] >= 0 & [2]x[4] + y[4] >= 0 & [1] + y[4] >= 0 & x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4])), >=) & [(-1)Bound*bni_15] + [bni_15]y[4] >= 0 & [(-1)bso_16] >= 0) 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 6.33/2.62 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. 6.33/2.62 6.33/2.62 Using the following integer polynomial ordering the resulting constraints can be solved 6.33/2.62 6.33/2.62 Polynomial interpretation over integers[POLO]: 6.33/2.62 6.33/2.62 POL(TRUE) = [2] 6.33/2.62 POL(FALSE) = 0 6.33/2.62 POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 6.33/2.62 POL(EVAL(x_1, x_2)) = [-1] + x_2 + [-1]x_1 6.33/2.62 POL(-(x_1, x_2)) = x_1 + [-1]x_2 6.33/2.62 POL(1) = [1] 6.33/2.62 POL(&&(x_1, x_2)) = [-1] 6.33/2.62 POL(>(x_1, x_2)) = [-1] 6.33/2.62 POL(+(x_1, x_2)) = x_1 + x_2 6.33/2.62 POL(0) = 0 6.33/2.62 POL(>=(x_1, x_2)) = [-1] 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_>: 6.33/2.62 6.33/2.62 6.33/2.62 COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_bound: 6.33/2.62 6.33/2.62 6.33/2.62 COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], -(y[5], 1)) 6.33/2.62 EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) 6.33/2.62 6.33/2.62 6.33/2.62 The following pairs are in P_>=: 6.33/2.62 6.33/2.62 6.33/2.62 EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(y[4], x[4])), >(y[4], x[4])), x[4], y[4]) 6.33/2.62 6.33/2.62 6.33/2.62 At least the following rules have been oriented under context sensitive arithmetic replacement: 6.33/2.62 6.33/2.62 TRUE^1 -> &&(TRUE, TRUE)^1 6.33/2.62 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (10) 6.33/2.62 Obligation: 6.33/2.62 IDP problem: 6.33/2.62 The following function symbols are pre-defined: 6.33/2.62 <<< 6.33/2.62 & ~ Bwand: (Integer, Integer) -> Integer 6.33/2.62 >= ~ Ge: (Integer, Integer) -> Boolean 6.33/2.62 | ~ Bwor: (Integer, Integer) -> Integer 6.33/2.62 / ~ Div: (Integer, Integer) -> Integer 6.33/2.62 != ~ Neq: (Integer, Integer) -> Boolean 6.33/2.62 && ~ Land: (Boolean, Boolean) -> Boolean 6.33/2.62 ! ~ Lnot: (Boolean) -> Boolean 6.33/2.62 = ~ Eq: (Integer, Integer) -> Boolean 6.33/2.62 <= ~ Le: (Integer, Integer) -> Boolean 6.33/2.62 ^ ~ Bwxor: (Integer, Integer) -> Integer 6.33/2.62 % ~ Mod: (Integer, Integer) -> Integer 6.33/2.62 > ~ Gt: (Integer, Integer) -> Boolean 6.33/2.62 + ~ Add: (Integer, Integer) -> Integer 6.33/2.62 -1 ~ UnaryMinus: (Integer) -> Integer 6.33/2.62 < ~ Lt: (Integer, Integer) -> Boolean 6.33/2.62 || ~ Lor: (Boolean, Boolean) -> Boolean 6.33/2.62 - ~ Sub: (Integer, Integer) -> Integer 6.33/2.62 ~ ~ Bwnot: (Integer) -> Integer 6.33/2.62 * ~ Mul: (Integer, Integer) -> Integer 6.33/2.62 >>> 6.33/2.62 6.33/2.62 6.33/2.62 The following domains are used: 6.33/2.62 Boolean, Integer 6.33/2.62 6.33/2.62 R is empty. 6.33/2.62 6.33/2.62 The integer pair graph contains the following rules and edges: 6.33/2.62 (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && y[4] >= x[4] && y[4] > x[4], x[4], y[4]) 6.33/2.62 6.33/2.62 6.33/2.62 The set Q consists of the following terms: 6.33/2.62 eval(x0, x1) 6.33/2.62 Cond_eval(TRUE, x0, x1) 6.33/2.62 Cond_eval1(TRUE, x0, x1) 6.33/2.62 Cond_eval2(TRUE, x0, x1) 6.33/2.62 Cond_eval3(TRUE, x0, x1) 6.33/2.62 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (11) IDependencyGraphProof (EQUIVALENT) 6.33/2.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 6.33/2.62 ---------------------------------------- 6.33/2.62 6.33/2.62 (12) 6.33/2.62 TRUE 6.51/2.65 EOF