4.86/2.15 YES 4.86/2.17 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.86/2.17 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.86/2.17 4.86/2.17 4.86/2.17 Termination of the given ITRS could be proven: 4.86/2.17 4.86/2.17 (0) ITRS 4.86/2.17 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.86/2.17 (2) IDP 4.86/2.17 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.86/2.17 (4) IDP 4.86/2.17 (5) IDPNonInfProof [SOUND, 240 ms] 4.86/2.17 (6) IDP 4.86/2.17 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.86/2.17 (8) TRUE 4.86/2.17 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (0) 4.86/2.17 Obligation: 4.86/2.17 ITRS problem: 4.86/2.17 4.86/2.17 The following function symbols are pre-defined: 4.86/2.17 <<< 4.86/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.86/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.86/2.17 / ~ Div: (Integer, Integer) -> Integer 4.86/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.86/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.86/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.86/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.86/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.86/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.86/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.86/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.86/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.86/2.17 + ~ Add: (Integer, Integer) -> Integer 4.86/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.86/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.86/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.86/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.86/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.86/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.86/2.17 >>> 4.86/2.17 4.86/2.17 The TRS R consists of the following rules: 4.86/2.17 eval(x) -> Cond_eval(x > 0 && !(x = 0) && x % 2 > 0, x) 4.86/2.17 Cond_eval(TRUE, x) -> eval(x - 1) 4.86/2.17 eval(x) -> Cond_eval1(x > 0 && !(x = 0) && x % 2 = 0, x) 4.86/2.17 Cond_eval1(TRUE, x) -> eval(x / 2) 4.86/2.17 The set Q consists of the following terms: 4.86/2.17 eval(x0) 4.86/2.17 Cond_eval(TRUE, x0) 4.86/2.17 Cond_eval1(TRUE, x0) 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (1) ITRStoIDPProof (EQUIVALENT) 4.86/2.17 Added dependency pairs 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (2) 4.86/2.17 Obligation: 4.86/2.17 IDP problem: 4.86/2.17 The following function symbols are pre-defined: 4.86/2.17 <<< 4.86/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.86/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.86/2.17 / ~ Div: (Integer, Integer) -> Integer 4.86/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.86/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.86/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.86/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.86/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.86/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.86/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.86/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.86/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.86/2.17 + ~ Add: (Integer, Integer) -> Integer 4.86/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.86/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.86/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.86/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.86/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.86/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.86/2.17 >>> 4.86/2.17 4.86/2.17 4.86/2.17 The following domains are used: 4.86/2.17 Boolean, Integer 4.86/2.17 4.86/2.17 The ITRS R consists of the following rules: 4.86/2.17 eval(x) -> Cond_eval(x > 0 && !(x = 0) && x % 2 > 0, x) 4.86/2.17 Cond_eval(TRUE, x) -> eval(x - 1) 4.86/2.17 eval(x) -> Cond_eval1(x > 0 && !(x = 0) && x % 2 = 0, x) 4.86/2.17 Cond_eval1(TRUE, x) -> eval(x / 2) 4.86/2.17 4.86/2.17 The integer pair graph contains the following rules and edges: 4.86/2.17 (0): EVAL(x[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0, x[0]) 4.86/2.17 (1): COND_EVAL(TRUE, x[1]) -> EVAL(x[1] - 1) 4.86/2.17 (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) 4.86/2.17 (3): COND_EVAL1(TRUE, x[3]) -> EVAL(x[3] / 2) 4.86/2.17 4.86/2.17 (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0 & x[0] ->^* x[1]) 4.86/2.17 (1) -> (0), if (x[1] - 1 ->^* x[0]) 4.86/2.17 (1) -> (2), if (x[1] - 1 ->^* x[2]) 4.86/2.17 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0 & x[2] ->^* x[3]) 4.86/2.17 (3) -> (0), if (x[3] / 2 ->^* x[0]) 4.86/2.17 (3) -> (2), if (x[3] / 2 ->^* x[2]) 4.86/2.17 4.86/2.17 The set Q consists of the following terms: 4.86/2.17 eval(x0) 4.86/2.17 Cond_eval(TRUE, x0) 4.86/2.17 Cond_eval1(TRUE, x0) 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (3) UsableRulesProof (EQUIVALENT) 4.86/2.17 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. 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (4) 4.86/2.17 Obligation: 4.86/2.17 IDP problem: 4.86/2.17 The following function symbols are pre-defined: 4.86/2.17 <<< 4.86/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.86/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.86/2.17 / ~ Div: (Integer, Integer) -> Integer 4.86/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.86/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.86/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.86/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.86/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.86/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.86/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.86/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.86/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.86/2.17 + ~ Add: (Integer, Integer) -> Integer 4.86/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.86/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.86/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.86/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.86/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.86/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.86/2.17 >>> 4.86/2.17 4.86/2.17 4.86/2.17 The following domains are used: 4.86/2.17 Boolean, Integer 4.86/2.17 4.86/2.17 R is empty. 4.86/2.17 4.86/2.17 The integer pair graph contains the following rules and edges: 4.86/2.17 (0): EVAL(x[0]) -> COND_EVAL(x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0, x[0]) 4.86/2.17 (1): COND_EVAL(TRUE, x[1]) -> EVAL(x[1] - 1) 4.86/2.17 (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) 4.86/2.17 (3): COND_EVAL1(TRUE, x[3]) -> EVAL(x[3] / 2) 4.86/2.17 4.86/2.17 (0) -> (1), if (x[0] > 0 && !(x[0] = 0) && x[0] % 2 > 0 & x[0] ->^* x[1]) 4.86/2.17 (1) -> (0), if (x[1] - 1 ->^* x[0]) 4.86/2.17 (1) -> (2), if (x[1] - 1 ->^* x[2]) 4.86/2.17 (2) -> (3), if (x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0 & x[2] ->^* x[3]) 4.86/2.17 (3) -> (0), if (x[3] / 2 ->^* x[0]) 4.86/2.17 (3) -> (2), if (x[3] / 2 ->^* x[2]) 4.86/2.17 4.86/2.17 The set Q consists of the following terms: 4.86/2.17 eval(x0) 4.86/2.17 Cond_eval(TRUE, x0) 4.86/2.17 Cond_eval1(TRUE, x0) 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (5) IDPNonInfProof (SOUND) 4.86/2.17 Used the following options for this NonInfProof: 4.86/2.17 4.86/2.17 IDPGPoloSolver: 4.86/2.17 Range: [(-1,2)] 4.86/2.17 IsNat: false 4.86/2.17 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@437a2959 4.86/2.17 Constraint Generator: NonInfConstraintGenerator: 4.86/2.17 PathGenerator: MetricPathGenerator: 4.86/2.17 Max Left Steps: 1 4.86/2.17 Max Right Steps: 1 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 The constraints were generated the following way: 4.86/2.17 4.86/2.17 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.86/2.17 4.86/2.17 Note that final constraints are written in bold face. 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 For Pair EVAL(x) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(%(x, 2), 0)), x) the following chains were created: 4.86/2.17 *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) which results in the following constraint: 4.86/2.17 4.86/2.17 (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: 4.86/2.17 4.86/2.17 (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) 4.86/2.17 4.86/2.17 (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) & (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.86/2.17 4.86/2.17 (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.86/2.17 4.86/2.17 (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 For Pair COND_EVAL(TRUE, x) -> EVAL(-(x, 1)) the following chains were created: 4.86/2.17 *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)), EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) which results in the following constraint: 4.86/2.17 4.86/2.17 (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] & -(x[1], 1)=x[0]1 ==> COND_EVAL(TRUE, x[1])_>=_NonInfC & COND_EVAL(TRUE, x[1])_>=_EVAL(-(x[1], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: 4.86/2.17 4.86/2.17 (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.86/2.17 4.86/2.17 (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.86/2.17 4.86/2.17 (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 *We consider the chain EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)), EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) which results in the following constraint: 4.86/2.17 4.86/2.17 (1) (&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0))=TRUE & x[0]=x[1] & -(x[1], 1)=x[2] ==> COND_EVAL(TRUE, x[1])_>=_NonInfC & COND_EVAL(TRUE, x[1])_>=_EVAL(-(x[1], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraints: 4.86/2.17 4.86/2.17 (2) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 (3) (>(%(x[0], 2), 0)=TRUE & >(x[0], 0)=TRUE & <(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0])_>=_NonInfC & COND_EVAL(TRUE, x[0])_>=_EVAL(-(x[0], 1)) & (U^Increasing(EVAL(-(x[1], 1))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (4) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (5) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (6) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (7) (max{[2], [-2]} + [-1] >= 0 & x[0] + [-1] >= 0 & [-1] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.86/2.17 4.86/2.17 (8) (x[0] + [-1] >= 0 & [4] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.86/2.17 4.86/2.17 (9) (x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 For Pair EVAL(x) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), =(%(x, 2), 0)), x) the following chains were created: 4.86/2.17 *We consider the chain EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]), COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) which results in the following constraint: 4.86/2.17 4.86/2.17 (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0))=TRUE & x[2]=x[3] ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints: 4.86/2.17 4.86/2.17 (2) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) 4.86/2.17 4.86/2.17 (3) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE & <(x[2], 0)=TRUE ==> EVAL(x[2])_>=_NonInfC & EVAL(x[2])_>=_COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (4) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (5) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (6) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (7) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.86/2.17 4.86/2.17 (8) (x[2] + [-1] >= 0 & [4] >= 0 & [2] >= 0 & [2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.86/2.17 4.86/2.17 (9) (x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 For Pair COND_EVAL1(TRUE, x) -> EVAL(/(x, 2)) the following chains were created: 4.86/2.17 *We consider the chain EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]), COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) which results in the following constraint: 4.86/2.17 4.86/2.17 (1) (&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0))=TRUE & x[2]=x[3] ==> COND_EVAL1(TRUE, x[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3])_>=_EVAL(/(x[3], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraints: 4.86/2.17 4.86/2.17 (2) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE ==> COND_EVAL1(TRUE, x[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2])_>=_EVAL(/(x[2], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) 4.86/2.17 4.86/2.17 (3) (>(x[2], 0)=TRUE & >=(%(x[2], 2), 0)=TRUE & <=(%(x[2], 2), 0)=TRUE & <(x[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2])_>=_EVAL(/(x[2], 2)) & (U^Increasing(EVAL(/(x[3], 2))), >=)) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (4) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] + [2]x[2] + [-2]max{x[2], [-1]x[2]} >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.86/2.17 4.86/2.17 (5) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] + [2]x[2] + [-2]max{x[2], [-1]x[2]} >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (6) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] + [2]x[2] + [-2]max{x[2], [-1]x[2]} >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.86/2.17 4.86/2.17 (7) (x[2] + [-1] >= 0 & max{[2], [-2]} >= 0 & [-1]min{[2], [-2]} >= 0 & [-1] + [-1]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] + [2]x[2] + [-2]max{x[2], [-1]x[2]} >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.86/2.17 4.86/2.17 (8) (x[2] + [-1] >= 0 & [4] >= 0 & [2] >= 0 & [2] >= 0 & [2]x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 We solved constraint (7) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.86/2.17 4.86/2.17 (9) (x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 To summarize, we get the following constraints P__>=_ for the following pairs. 4.86/2.17 4.86/2.17 *EVAL(x) -> COND_EVAL(&&(&&(>(x, 0), !(=(x, 0))), >(%(x, 2), 0)), x) 4.86/2.17 4.86/2.17 *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 *COND_EVAL(TRUE, x) -> EVAL(-(x, 1)) 4.86/2.17 4.86/2.17 *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 *(x[0] + [-1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1))), >=) & [(-3)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 *EVAL(x) -> COND_EVAL1(&&(&&(>(x, 0), !(=(x, 0))), =(%(x, 2), 0)), x) 4.86/2.17 4.86/2.17 *(x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(2)bni_19]x[2] >= 0 & [(-1)bso_20] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 *COND_EVAL1(TRUE, x) -> EVAL(/(x, 2)) 4.86/2.17 4.86/2.17 *(x[2] + [-1] >= 0 & [1] >= 0 & [1] >= 0 & [1] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(/(x[3], 2))), >=) & [(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x[2] >= 0 & [1 + (-1)bso_25] >= 0) 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 4.86/2.17 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. 4.86/2.17 4.86/2.17 Using the following integer polynomial ordering the resulting constraints can be solved 4.86/2.17 4.86/2.17 Polynomial interpretation over integers[POLO]: 4.86/2.17 4.86/2.17 POL(TRUE) = [2] 4.86/2.17 POL(FALSE) = [1] 4.86/2.17 POL(EVAL(x_1)) = [-1] + [2]x_1 4.86/2.17 POL(COND_EVAL(x_1, x_2)) = [-1] + [2]x_2 + [-1]x_1 4.86/2.17 POL(&&(x_1, x_2)) = [1] 4.86/2.17 POL(>(x_1, x_2)) = [-1] 4.86/2.17 POL(0) = 0 4.86/2.17 POL(!(x_1)) = [-1] 4.86/2.17 POL(=(x_1, x_2)) = [-1] 4.86/2.17 POL(2) = [2] 4.86/2.17 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.86/2.17 POL(1) = [1] 4.86/2.17 POL(COND_EVAL1(x_1, x_2)) = [2]x_2 + [-1]x_1 4.86/2.17 4.86/2.17 Polynomial Interpretations with Context Sensitive Arithemetic Replacement 4.86/2.17 POL(Term^CSAR-Mode @ Context) 4.86/2.17 4.86/2.17 POL(%(x_1, 2)^1 @ {}) = max{x_2, [-1]x_2} 4.86/2.17 POL(%(x_1, 2)^-1 @ {}) = min{x_2, [-1]x_2} 4.86/2.17 POL(/(x_1, 2)^1 @ {EVAL_1/0}) = max{x_1, [-1]x_1} + [-1] 4.86/2.17 4.86/2.17 4.86/2.17 The following pairs are in P_>: 4.86/2.17 4.86/2.17 4.86/2.17 EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) 4.86/2.17 COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) 4.86/2.17 4.86/2.17 4.86/2.17 The following pairs are in P_bound: 4.86/2.17 4.86/2.17 4.86/2.17 EVAL(x[0]) -> COND_EVAL(&&(&&(>(x[0], 0), !(=(x[0], 0))), >(%(x[0], 2), 0)), x[0]) 4.86/2.17 COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) 4.86/2.17 EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) 4.86/2.17 COND_EVAL1(TRUE, x[3]) -> EVAL(/(x[3], 2)) 4.86/2.17 4.86/2.17 4.86/2.17 The following pairs are in P_>=: 4.86/2.17 4.86/2.17 4.86/2.17 COND_EVAL(TRUE, x[1]) -> EVAL(-(x[1], 1)) 4.86/2.17 EVAL(x[2]) -> COND_EVAL1(&&(&&(>(x[2], 0), !(=(x[2], 0))), =(%(x[2], 2), 0)), x[2]) 4.86/2.17 4.86/2.17 4.86/2.17 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.86/2.17 4.86/2.17 TRUE^1 -> &&(TRUE, TRUE)^1 4.86/2.17 &&(TRUE, FALSE)^1 <-> FALSE^1 4.86/2.17 &&(FALSE, TRUE)^1 <-> FALSE^1 4.86/2.17 &&(FALSE, FALSE)^1 <-> FALSE^1 4.86/2.17 /^1 -> 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (6) 4.86/2.17 Obligation: 4.86/2.17 IDP problem: 4.86/2.17 The following function symbols are pre-defined: 4.86/2.17 <<< 4.86/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.86/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.86/2.17 / ~ Div: (Integer, Integer) -> Integer 4.86/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.86/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.86/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.86/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.86/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.86/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.86/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.86/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.86/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.86/2.17 + ~ Add: (Integer, Integer) -> Integer 4.86/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.86/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.86/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.86/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.86/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.86/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.86/2.17 >>> 4.86/2.17 4.86/2.17 4.86/2.17 The following domains are used: 4.86/2.17 Integer, Boolean 4.86/2.17 4.86/2.17 R is empty. 4.86/2.17 4.86/2.17 The integer pair graph contains the following rules and edges: 4.86/2.17 (1): COND_EVAL(TRUE, x[1]) -> EVAL(x[1] - 1) 4.86/2.17 (2): EVAL(x[2]) -> COND_EVAL1(x[2] > 0 && !(x[2] = 0) && x[2] % 2 = 0, x[2]) 4.86/2.17 4.86/2.17 (1) -> (2), if (x[1] - 1 ->^* x[2]) 4.86/2.17 4.86/2.17 The set Q consists of the following terms: 4.86/2.17 eval(x0) 4.86/2.17 Cond_eval(TRUE, x0) 4.86/2.17 Cond_eval1(TRUE, x0) 4.86/2.17 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (7) IDependencyGraphProof (EQUIVALENT) 4.86/2.17 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 4.86/2.17 ---------------------------------------- 4.86/2.17 4.86/2.17 (8) 4.86/2.17 TRUE 5.02/2.22 EOF