5.20/2.25 YES 5.20/2.27 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.20/2.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.20/2.27 5.20/2.27 5.20/2.27 Termination of the given ITRS could be proven: 5.20/2.27 5.20/2.27 (0) ITRS 5.20/2.27 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.20/2.27 (2) IDP 5.20/2.27 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.20/2.27 (4) IDP 5.20/2.27 (5) IDPNonInfProof [SOUND, 285 ms] 5.20/2.27 (6) IDP 5.20/2.27 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.20/2.27 (8) IDP 5.20/2.27 (9) IDPNonInfProof [SOUND, 36 ms] 5.20/2.27 (10) IDP 5.20/2.27 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.20/2.27 (12) TRUE 5.20/2.27 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (0) 5.20/2.27 Obligation: 5.20/2.27 ITRS problem: 5.20/2.27 5.20/2.27 The following function symbols are pre-defined: 5.20/2.27 <<< 5.20/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.27 / ~ Div: (Integer, Integer) -> Integer 5.20/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.27 + ~ Add: (Integer, Integer) -> Integer 5.20/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.27 >>> 5.20/2.27 5.20/2.27 The TRS R consists of the following rules: 5.20/2.27 eval(x, y) -> Cond_eval(x > 0 && !(x = 0) && x > y, x, y) 5.20/2.27 Cond_eval(TRUE, x, y) -> eval(y, y) 5.20/2.27 eval(x, y) -> Cond_eval1(x > 0 && !(x = 0) && x <= y, x, y) 5.20/2.27 Cond_eval1(TRUE, x, y) -> eval(x - 1, y) 5.20/2.27 The set Q consists of the following terms: 5.20/2.27 eval(x0, x1) 5.20/2.27 Cond_eval(TRUE, x0, x1) 5.20/2.27 Cond_eval1(TRUE, x0, x1) 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (1) ITRStoIDPProof (EQUIVALENT) 5.20/2.27 Added dependency pairs 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (2) 5.20/2.27 Obligation: 5.20/2.27 IDP problem: 5.20/2.27 The following function symbols are pre-defined: 5.20/2.27 <<< 5.20/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.27 / ~ Div: (Integer, Integer) -> Integer 5.20/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.27 + ~ Add: (Integer, Integer) -> Integer 5.20/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.27 >>> 5.20/2.27 5.20/2.27 5.20/2.27 The following domains are used: 5.20/2.27 Boolean, Integer 5.20/2.27 5.20/2.27 The ITRS R consists of the following rules: 5.20/2.27 eval(x, y) -> Cond_eval(x > 0 && !(x = 0) && x > y, x, y) 5.20/2.27 Cond_eval(TRUE, x, y) -> eval(y, y) 5.20/2.27 eval(x, y) -> Cond_eval1(x > 0 && !(x = 0) && x <= y, x, y) 5.20/2.27 Cond_eval1(TRUE, x, y) -> eval(x - 1, y) 5.20/2.27 5.20/2.27 The integer pair graph contains the following rules and edges: 5.20/2.27 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] > y[0], x[0], y[0]) 5.20/2.27 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) 5.20/2.27 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) 5.20/2.27 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 5.20/2.27 5.20/2.27 (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.20/2.27 (1) -> (0), if (y[1] ->^* x[0] & y[1] ->^* y[0]) 5.20/2.27 (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) 5.20/2.27 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.20/2.27 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) 5.20/2.27 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 5.20/2.27 5.20/2.27 The set Q consists of the following terms: 5.20/2.27 eval(x0, x1) 5.20/2.27 Cond_eval(TRUE, x0, x1) 5.20/2.27 Cond_eval1(TRUE, x0, x1) 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (3) UsableRulesProof (EQUIVALENT) 5.20/2.27 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. 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (4) 5.20/2.27 Obligation: 5.20/2.27 IDP problem: 5.20/2.27 The following function symbols are pre-defined: 5.20/2.27 <<< 5.20/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.27 / ~ Div: (Integer, Integer) -> Integer 5.20/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.27 + ~ Add: (Integer, Integer) -> Integer 5.20/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.27 >>> 5.20/2.27 5.20/2.27 5.20/2.27 The following domains are used: 5.20/2.27 Boolean, Integer 5.20/2.27 5.20/2.27 R is empty. 5.20/2.27 5.20/2.27 The integer pair graph contains the following rules and edges: 5.20/2.27 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] > y[0], x[0], y[0]) 5.20/2.27 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) 5.20/2.27 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) 5.20/2.27 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 5.20/2.27 5.20/2.27 (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.20/2.27 (1) -> (0), if (y[1] ->^* x[0] & y[1] ->^* y[0]) 5.20/2.27 (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) 5.20/2.27 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.20/2.27 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0]) 5.20/2.27 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 5.20/2.27 5.20/2.27 The set Q consists of the following terms: 5.20/2.27 eval(x0, x1) 5.20/2.27 Cond_eval(TRUE, x0, x1) 5.20/2.27 Cond_eval1(TRUE, x0, x1) 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (5) IDPNonInfProof (SOUND) 5.20/2.27 Used the following options for this NonInfProof: 5.20/2.27 5.20/2.27 IDPGPoloSolver: 5.20/2.27 Range: [(-1,2)] 5.20/2.27 IsNat: false 5.20/2.27 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@2bdd06f5 5.20/2.27 Constraint Generator: NonInfConstraintGenerator: 5.20/2.27 PathGenerator: MetricPathGenerator: 5.20/2.27 Max Left Steps: 1 5.20/2.27 Max Right Steps: 1 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 The constraints were generated the following way: 5.20/2.27 5.20/2.27 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.20/2.27 5.20/2.27 Note that final constraints are written in bold face. 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 For Pair EVAL(x, y) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(x, y)), x, y) the following chains were created: 5.20/2.27 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) which results in the following constraint: 5.20/2.27 5.20/2.27 (1) (&&(&&(>(x[0], 0), !(=(x[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], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.27 5.20/2.27 (2) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) 5.20/2.27 5.20/2.27 (3) (>(x[0], y[0])=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (7) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (8) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (9) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.20/2.27 5.20/2.27 (10) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 (11) (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 For Pair COND_EVAL(TRUE, x, y) -> EVAL(y, y) the following chains were created: 5.20/2.27 *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 5.20/2.27 5.20/2.27 (1) (y[1]=x[0] & y[1]=y[0] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.20/2.27 5.20/2.27 (2) (COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (3) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.20/2.27 5.20/2.27 (1) (y[1]=x[2] & y[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.20/2.27 5.20/2.27 (2) (COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(y[1], y[1]) & (U^Increasing(EVAL(y[1], y[1])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (3) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) ((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), <=(x, y)), x, y) the following chains were created: 5.20/2.27 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(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: 5.20/2.27 5.20/2.27 (1) (&&(&&(>(x[2], 0), !(=(x[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(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.27 5.20/2.27 (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.27 5.20/2.27 (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: 5.20/2.27 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 5.20/2.27 5.20/2.27 (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[0] & y[3]=y[0] ==> 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])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.27 5.20/2.27 (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=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])), >=)) 5.20/2.27 5.20/2.27 (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=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])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.27 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.20/2.27 5.20/2.27 (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 ==> 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])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.27 5.20/2.27 (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=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])), >=)) 5.20/2.27 5.20/2.27 (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=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])), >=)) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.27 5.20/2.27 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.27 5.20/2.27 (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.27 5.20/2.27 (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 To summarize, we get the following constraints P__>=_ for the following pairs. 5.20/2.27 5.20/2.27 *EVAL(x, y) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(x, y)), x, y) 5.20/2.27 5.20/2.27 *(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + [-1]y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 *(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] + y[0] + x[0] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 *COND_EVAL(TRUE, x, y) -> EVAL(y, y) 5.20/2.27 5.20/2.27 *((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 *((U^Increasing(EVAL(y[1], y[1])), >=) & [bni_21] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_22] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 *EVAL(x, y) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), <=(x, y)), x, y) 5.20/2.27 5.20/2.27 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 *COND_EVAL1(TRUE, x, y) -> EVAL(-(x, 1), y) 5.20/2.27 5.20/2.27 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0) 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 5.20/2.27 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. 5.20/2.27 5.20/2.27 Using the following integer polynomial ordering the resulting constraints can be solved 5.20/2.27 5.20/2.27 Polynomial interpretation over integers[POLO]: 5.20/2.27 5.20/2.27 POL(TRUE) = 0 5.20/2.27 POL(FALSE) = 0 5.20/2.27 POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 5.20/2.27 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] 5.20/2.27 POL(&&(x_1, x_2)) = [-1] 5.20/2.27 POL(>(x_1, x_2)) = [-1] 5.20/2.27 POL(0) = 0 5.20/2.27 POL(!(x_1)) = [-1] 5.20/2.27 POL(=(x_1, x_2)) = [-1] 5.20/2.27 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 5.20/2.27 POL(<=(x_1, x_2)) = [-1] 5.20/2.27 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.20/2.27 POL(1) = [1] 5.20/2.27 5.20/2.27 5.20/2.27 The following pairs are in P_>: 5.20/2.27 5.20/2.27 5.20/2.27 EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) 5.20/2.27 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 5.20/2.27 5.20/2.27 5.20/2.27 The following pairs are in P_bound: 5.20/2.27 5.20/2.27 5.20/2.27 EVAL(x[0], y[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(x[0], y[0])), x[0], y[0]) 5.20/2.27 5.20/2.27 5.20/2.27 The following pairs are in P_>=: 5.20/2.27 5.20/2.27 5.20/2.27 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) 5.20/2.27 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) 5.20/2.27 5.20/2.27 5.20/2.27 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.20/2.27 5.20/2.27 TRUE^1 -> &&(TRUE, TRUE)^1 5.20/2.27 FALSE^1 -> &&(TRUE, FALSE)^1 5.20/2.27 FALSE^1 -> &&(FALSE, TRUE)^1 5.20/2.27 FALSE^1 -> &&(FALSE, FALSE)^1 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (6) 5.20/2.27 Obligation: 5.20/2.27 IDP problem: 5.20/2.27 The following function symbols are pre-defined: 5.20/2.27 <<< 5.20/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.27 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.27 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.27 / ~ Div: (Integer, Integer) -> Integer 5.20/2.27 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.27 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.27 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.27 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.27 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.27 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.27 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.27 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.27 + ~ Add: (Integer, Integer) -> Integer 5.20/2.27 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.27 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.27 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.27 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.27 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.27 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.27 >>> 5.20/2.27 5.20/2.27 5.20/2.27 The following domains are used: 5.20/2.27 Boolean, Integer 5.20/2.27 5.20/2.27 R is empty. 5.20/2.27 5.20/2.27 The integer pair graph contains the following rules and edges: 5.20/2.27 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(y[1], y[1]) 5.20/2.27 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) 5.20/2.27 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 5.20/2.27 5.20/2.27 (1) -> (2), if (y[1] ->^* x[2] & y[1] ->^* y[2]) 5.20/2.27 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 5.20/2.27 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.20/2.27 5.20/2.27 The set Q consists of the following terms: 5.20/2.27 eval(x0, x1) 5.20/2.27 Cond_eval(TRUE, x0, x1) 5.20/2.27 Cond_eval1(TRUE, x0, x1) 5.20/2.27 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (7) IDependencyGraphProof (EQUIVALENT) 5.20/2.27 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.20/2.27 ---------------------------------------- 5.20/2.27 5.20/2.27 (8) 5.20/2.27 Obligation: 5.20/2.27 IDP problem: 5.20/2.27 The following function symbols are pre-defined: 5.20/2.27 <<< 5.20/2.27 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.28 / ~ Div: (Integer, Integer) -> Integer 5.20/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.28 + ~ Add: (Integer, Integer) -> Integer 5.20/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.28 >>> 5.20/2.28 5.20/2.28 5.20/2.28 The following domains are used: 5.20/2.28 Integer, Boolean 5.20/2.28 5.20/2.28 R is empty. 5.20/2.28 5.20/2.28 The integer pair graph contains the following rules and edges: 5.20/2.28 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3] - 1, y[3]) 5.20/2.28 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) 5.20/2.28 5.20/2.28 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2]) 5.20/2.28 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] <= y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.20/2.28 5.20/2.28 The set Q consists of the following terms: 5.20/2.28 eval(x0, x1) 5.20/2.28 Cond_eval(TRUE, x0, x1) 5.20/2.28 Cond_eval1(TRUE, x0, x1) 5.20/2.28 5.20/2.28 ---------------------------------------- 5.20/2.28 5.20/2.28 (9) IDPNonInfProof (SOUND) 5.20/2.28 Used the following options for this NonInfProof: 5.20/2.28 5.20/2.28 IDPGPoloSolver: 5.20/2.28 Range: [(-1,2)] 5.20/2.28 IsNat: false 5.20/2.28 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@2bdd06f5 5.20/2.28 Constraint Generator: NonInfConstraintGenerator: 5.20/2.28 PathGenerator: MetricPathGenerator: 5.20/2.28 Max Left Steps: 1 5.20/2.28 Max Right Steps: 1 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 The constraints were generated the following way: 5.20/2.28 5.20/2.28 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.20/2.28 5.20/2.28 Note that final constraints are written in bold face. 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) the following chains were created: 5.20/2.28 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.20/2.28 5.20/2.28 (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 ==> 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])), >=)) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.28 5.20/2.28 (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=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])), >=)) 5.20/2.28 5.20/2.28 (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=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])), >=)) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.28 5.20/2.28 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.28 5.20/2.28 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.28 5.20/2.28 (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.28 5.20/2.28 (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.28 5.20/2.28 (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.28 5.20/2.28 (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) the following chains were created: 5.20/2.28 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(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: 5.20/2.28 5.20/2.28 (1) (&&(&&(>(x[2], 0), !(=(x[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(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: 5.20/2.28 5.20/2.28 (2) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.28 5.20/2.28 (3) (<=(x[2], y[2])=TRUE & >(x[2], 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=)) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.28 5.20/2.28 (4) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.20/2.28 5.20/2.28 (5) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.28 5.20/2.28 (6) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.20/2.28 5.20/2.28 (7) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.28 5.20/2.28 (8) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.20/2.28 5.20/2.28 (9) (y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 We solved constraint (9) using rule (IDP_SMT_SPLIT). 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 To summarize, we get the following constraints P__>=_ for the following pairs. 5.20/2.28 5.20/2.28 *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 5.20/2.28 5.20/2.28 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 *EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) 5.20/2.28 5.20/2.28 *(y[2] + [-1]x[2] >= 0 & x[2] + [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[2] >= 0 & [(-1)bso_18] >= 0) 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 5.20/2.28 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. 5.20/2.28 5.20/2.28 Using the following integer polynomial ordering the resulting constraints can be solved 5.20/2.28 5.20/2.28 Polynomial interpretation over integers[POLO]: 5.20/2.28 5.20/2.28 POL(TRUE) = 0 5.20/2.28 POL(FALSE) = [1] 5.20/2.28 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 5.20/2.28 POL(EVAL(x_1, x_2)) = [-1] + x_1 5.20/2.28 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.20/2.28 POL(1) = [1] 5.20/2.28 POL(&&(x_1, x_2)) = 0 5.20/2.28 POL(>(x_1, x_2)) = [-1] 5.20/2.28 POL(0) = 0 5.20/2.28 POL(!(x_1)) = [-1] 5.20/2.28 POL(=(x_1, x_2)) = [-1] 5.20/2.28 POL(<=(x_1, x_2)) = [-1] 5.20/2.28 5.20/2.28 5.20/2.28 The following pairs are in P_>: 5.20/2.28 5.20/2.28 5.20/2.28 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 5.20/2.28 5.20/2.28 5.20/2.28 The following pairs are in P_bound: 5.20/2.28 5.20/2.28 5.20/2.28 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(-(x[3], 1), y[3]) 5.20/2.28 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) 5.20/2.28 5.20/2.28 5.20/2.28 The following pairs are in P_>=: 5.20/2.28 5.20/2.28 5.20/2.28 EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), <=(x[2], y[2])), x[2], y[2]) 5.20/2.28 5.20/2.28 5.20/2.28 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.20/2.28 5.20/2.28 &&(TRUE, TRUE)^1 <-> TRUE^1 5.20/2.28 FALSE^1 -> &&(TRUE, FALSE)^1 5.20/2.28 FALSE^1 -> &&(FALSE, TRUE)^1 5.20/2.28 FALSE^1 -> &&(FALSE, FALSE)^1 5.20/2.28 5.20/2.28 ---------------------------------------- 5.20/2.28 5.20/2.28 (10) 5.20/2.28 Obligation: 5.20/2.28 IDP problem: 5.20/2.28 The following function symbols are pre-defined: 5.20/2.28 <<< 5.20/2.28 & ~ Bwand: (Integer, Integer) -> Integer 5.20/2.28 >= ~ Ge: (Integer, Integer) -> Boolean 5.20/2.28 | ~ Bwor: (Integer, Integer) -> Integer 5.20/2.28 / ~ Div: (Integer, Integer) -> Integer 5.20/2.28 != ~ Neq: (Integer, Integer) -> Boolean 5.20/2.28 && ~ Land: (Boolean, Boolean) -> Boolean 5.20/2.28 ! ~ Lnot: (Boolean) -> Boolean 5.20/2.28 = ~ Eq: (Integer, Integer) -> Boolean 5.20/2.28 <= ~ Le: (Integer, Integer) -> Boolean 5.20/2.28 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.20/2.28 % ~ Mod: (Integer, Integer) -> Integer 5.20/2.28 > ~ Gt: (Integer, Integer) -> Boolean 5.20/2.28 + ~ Add: (Integer, Integer) -> Integer 5.20/2.28 -1 ~ UnaryMinus: (Integer) -> Integer 5.20/2.28 < ~ Lt: (Integer, Integer) -> Boolean 5.20/2.28 || ~ Lor: (Boolean, Boolean) -> Boolean 5.20/2.28 - ~ Sub: (Integer, Integer) -> Integer 5.20/2.28 ~ ~ Bwnot: (Integer) -> Integer 5.20/2.28 * ~ Mul: (Integer, Integer) -> Integer 5.20/2.28 >>> 5.20/2.28 5.20/2.28 5.20/2.28 The following domains are used: 5.20/2.28 Boolean, Integer 5.20/2.28 5.20/2.28 R is empty. 5.20/2.28 5.20/2.28 The integer pair graph contains the following rules and edges: 5.20/2.28 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] <= y[2], x[2], y[2]) 5.20/2.28 5.20/2.28 5.20/2.28 The set Q consists of the following terms: 5.20/2.28 eval(x0, x1) 5.20/2.28 Cond_eval(TRUE, x0, x1) 5.20/2.28 Cond_eval1(TRUE, x0, x1) 5.20/2.28 5.20/2.28 ---------------------------------------- 5.20/2.28 5.20/2.28 (11) IDependencyGraphProof (EQUIVALENT) 5.20/2.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.20/2.28 ---------------------------------------- 5.20/2.28 5.20/2.28 (12) 5.20/2.28 TRUE 5.20/2.30 EOF