4.30/1.97 YES 4.30/1.98 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.30/1.98 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.30/1.98 4.30/1.98 4.30/1.98 Termination of the given ITRS could be proven: 4.30/1.98 4.30/1.98 (0) ITRS 4.30/1.98 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.30/1.98 (2) IDP 4.30/1.98 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.30/1.98 (4) IDP 4.30/1.98 (5) IDPNonInfProof [SOUND, 218 ms] 4.30/1.98 (6) IDP 4.30/1.98 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.30/1.98 (8) TRUE 4.30/1.98 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (0) 4.30/1.98 Obligation: 4.30/1.98 ITRS problem: 4.30/1.98 4.30/1.98 The following function symbols are pre-defined: 4.30/1.98 <<< 4.30/1.98 & ~ Bwand: (Integer, Integer) -> Integer 4.30/1.98 >= ~ Ge: (Integer, Integer) -> Boolean 4.30/1.98 | ~ Bwor: (Integer, Integer) -> Integer 4.30/1.98 / ~ Div: (Integer, Integer) -> Integer 4.30/1.98 != ~ Neq: (Integer, Integer) -> Boolean 4.30/1.98 && ~ Land: (Boolean, Boolean) -> Boolean 4.30/1.98 ! ~ Lnot: (Boolean) -> Boolean 4.30/1.98 = ~ Eq: (Integer, Integer) -> Boolean 4.30/1.98 <= ~ Le: (Integer, Integer) -> Boolean 4.30/1.98 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.30/1.98 % ~ Mod: (Integer, Integer) -> Integer 4.30/1.98 + ~ Add: (Integer, Integer) -> Integer 4.30/1.98 > ~ Gt: (Integer, Integer) -> Boolean 4.30/1.98 -1 ~ UnaryMinus: (Integer) -> Integer 4.30/1.98 < ~ Lt: (Integer, Integer) -> Boolean 4.30/1.98 || ~ Lor: (Boolean, Boolean) -> Boolean 4.30/1.98 - ~ Sub: (Integer, Integer) -> Integer 4.30/1.98 ~ ~ Bwnot: (Integer) -> Integer 4.30/1.98 * ~ Mul: (Integer, Integer) -> Integer 4.30/1.98 >>> 4.30/1.98 4.30/1.98 The TRS R consists of the following rules: 4.30/1.98 eval(x, y, z) -> Cond_eval(x > y && x > z, x, y, z) 4.30/1.98 Cond_eval(TRUE, x, y, z) -> eval(x, y + 1, z + 1) 4.30/1.98 The set Q consists of the following terms: 4.30/1.98 eval(x0, x1, x2) 4.30/1.98 Cond_eval(TRUE, x0, x1, x2) 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (1) ITRStoIDPProof (EQUIVALENT) 4.30/1.98 Added dependency pairs 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (2) 4.30/1.98 Obligation: 4.30/1.98 IDP problem: 4.30/1.98 The following function symbols are pre-defined: 4.30/1.98 <<< 4.30/1.98 & ~ Bwand: (Integer, Integer) -> Integer 4.30/1.98 >= ~ Ge: (Integer, Integer) -> Boolean 4.30/1.98 | ~ Bwor: (Integer, Integer) -> Integer 4.30/1.98 / ~ Div: (Integer, Integer) -> Integer 4.30/1.98 != ~ Neq: (Integer, Integer) -> Boolean 4.30/1.98 && ~ Land: (Boolean, Boolean) -> Boolean 4.30/1.98 ! ~ Lnot: (Boolean) -> Boolean 4.30/1.98 = ~ Eq: (Integer, Integer) -> Boolean 4.30/1.98 <= ~ Le: (Integer, Integer) -> Boolean 4.30/1.98 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.30/1.98 % ~ Mod: (Integer, Integer) -> Integer 4.30/1.98 + ~ Add: (Integer, Integer) -> Integer 4.30/1.98 > ~ Gt: (Integer, Integer) -> Boolean 4.30/1.98 -1 ~ UnaryMinus: (Integer) -> Integer 4.30/1.98 < ~ Lt: (Integer, Integer) -> Boolean 4.30/1.98 || ~ Lor: (Boolean, Boolean) -> Boolean 4.30/1.98 - ~ Sub: (Integer, Integer) -> Integer 4.30/1.98 ~ ~ Bwnot: (Integer) -> Integer 4.30/1.98 * ~ Mul: (Integer, Integer) -> Integer 4.30/1.98 >>> 4.30/1.98 4.30/1.98 4.30/1.98 The following domains are used: 4.30/1.98 Boolean, Integer 4.30/1.98 4.30/1.98 The ITRS R consists of the following rules: 4.30/1.98 eval(x, y, z) -> Cond_eval(x > y && x > z, x, y, z) 4.30/1.98 Cond_eval(TRUE, x, y, z) -> eval(x, y + 1, z + 1) 4.30/1.98 4.30/1.98 The integer pair graph contains the following rules and edges: 4.30/1.98 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) 4.30/1.98 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], y[1] + 1, z[1] + 1) 4.30/1.98 4.30/1.98 (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 4.30/1.98 (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] + 1 ->^* z[0]) 4.30/1.98 4.30/1.98 The set Q consists of the following terms: 4.30/1.98 eval(x0, x1, x2) 4.30/1.98 Cond_eval(TRUE, x0, x1, x2) 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (3) UsableRulesProof (EQUIVALENT) 4.30/1.98 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. 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (4) 4.30/1.98 Obligation: 4.30/1.98 IDP problem: 4.30/1.98 The following function symbols are pre-defined: 4.30/1.98 <<< 4.30/1.98 & ~ Bwand: (Integer, Integer) -> Integer 4.30/1.98 >= ~ Ge: (Integer, Integer) -> Boolean 4.30/1.98 | ~ Bwor: (Integer, Integer) -> Integer 4.30/1.98 / ~ Div: (Integer, Integer) -> Integer 4.30/1.98 != ~ Neq: (Integer, Integer) -> Boolean 4.30/1.98 && ~ Land: (Boolean, Boolean) -> Boolean 4.30/1.98 ! ~ Lnot: (Boolean) -> Boolean 4.30/1.98 = ~ Eq: (Integer, Integer) -> Boolean 4.30/1.98 <= ~ Le: (Integer, Integer) -> Boolean 4.30/1.98 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.30/1.98 % ~ Mod: (Integer, Integer) -> Integer 4.30/1.98 + ~ Add: (Integer, Integer) -> Integer 4.30/1.98 > ~ Gt: (Integer, Integer) -> Boolean 4.30/1.98 -1 ~ UnaryMinus: (Integer) -> Integer 4.30/1.98 < ~ Lt: (Integer, Integer) -> Boolean 4.30/1.98 || ~ Lor: (Boolean, Boolean) -> Boolean 4.30/1.98 - ~ Sub: (Integer, Integer) -> Integer 4.30/1.98 ~ ~ Bwnot: (Integer) -> Integer 4.30/1.98 * ~ Mul: (Integer, Integer) -> Integer 4.30/1.98 >>> 4.30/1.98 4.30/1.98 4.30/1.98 The following domains are used: 4.30/1.98 Boolean, Integer 4.30/1.98 4.30/1.98 R is empty. 4.30/1.98 4.30/1.98 The integer pair graph contains the following rules and edges: 4.30/1.98 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) 4.30/1.98 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], y[1] + 1, z[1] + 1) 4.30/1.98 4.30/1.98 (0) -> (1), if (x[0] > y[0] && x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 4.30/1.98 (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] + 1 ->^* z[0]) 4.30/1.98 4.30/1.98 The set Q consists of the following terms: 4.30/1.98 eval(x0, x1, x2) 4.30/1.98 Cond_eval(TRUE, x0, x1, x2) 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (5) IDPNonInfProof (SOUND) 4.30/1.98 Used the following options for this NonInfProof: 4.30/1.98 4.30/1.98 IDPGPoloSolver: 4.30/1.98 Range: [(-1,2)] 4.30/1.98 IsNat: false 4.30/1.98 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@64693039 4.30/1.98 Constraint Generator: NonInfConstraintGenerator: 4.30/1.98 PathGenerator: MetricPathGenerator: 4.30/1.98 Max Left Steps: 1 4.30/1.98 Max Right Steps: 1 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 The constraints were generated the following way: 4.30/1.98 4.30/1.98 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.30/1.98 4.30/1.98 Note that final constraints are written in bold face. 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 For Pair EVAL(x, y, z) -> COND_EVAL(&&(>(x, y), >(x, z)), x, y, z) the following chains were created: 4.30/1.98 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)) which results in the following constraint: 4.30/1.98 4.30/1.98 (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.30/1.98 4.30/1.98 (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=)) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.30/1.98 4.30/1.98 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.30/1.98 4.30/1.98 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.30/1.98 4.30/1.98 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.30/1.98 4.30/1.98 (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.30/1.98 4.30/1.98 (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.30/1.98 4.30/1.98 (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(x, +(y, 1), +(z, 1)) the following chains were created: 4.30/1.98 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)), EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint: 4.30/1.98 4.30/1.98 (1) (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & +(z[1], 1)=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=)) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.30/1.98 4.30/1.98 (2) (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=)) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.30/1.98 4.30/1.98 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.30/1.98 4.30/1.98 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.30/1.98 4.30/1.98 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.30/1.98 4.30/1.98 (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.30/1.98 4.30/1.98 (7) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.30/1.98 4.30/1.98 (8) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 (9) (x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 To summarize, we get the following constraints P__>=_ for the following pairs. 4.30/1.98 4.30/1.98 *EVAL(x, y, z) -> COND_EVAL(&&(>(x, y), >(x, z)), x, y, z) 4.30/1.98 4.30/1.98 *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [(-1)bso_15] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 *COND_EVAL(TRUE, x, y, z) -> EVAL(x, +(y, 1), +(z, 1)) 4.30/1.98 4.30/1.98 *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 *(x[0] >= 0 & z[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1), +(z[1], 1))), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 4.30/1.98 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. 4.30/1.98 4.30/1.98 Using the following integer polynomial ordering the resulting constraints can be solved 4.30/1.98 4.30/1.98 Polynomial interpretation over integers[POLO]: 4.30/1.98 4.30/1.98 POL(TRUE) = [1] 4.30/1.98 POL(FALSE) = [2] 4.30/1.98 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [2]x_1 4.30/1.98 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1]x_4 + [-1]x_3 + [2]x_2 + [-1]x_1 4.30/1.98 POL(&&(x_1, x_2)) = [1] 4.30/1.98 POL(>(x_1, x_2)) = [-1] 4.30/1.98 POL(+(x_1, x_2)) = x_1 + x_2 4.30/1.98 POL(1) = [1] 4.30/1.98 4.30/1.98 4.30/1.98 The following pairs are in P_>: 4.30/1.98 4.30/1.98 4.30/1.98 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)) 4.30/1.98 4.30/1.98 4.30/1.98 The following pairs are in P_bound: 4.30/1.98 4.30/1.98 4.30/1.98 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 4.30/1.98 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1], +(y[1], 1), +(z[1], 1)) 4.30/1.98 4.30/1.98 4.30/1.98 The following pairs are in P_>=: 4.30/1.98 4.30/1.98 4.30/1.98 EVAL(x[0], y[0], z[0]) -> COND_EVAL(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) 4.30/1.98 4.30/1.98 4.30/1.98 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.30/1.98 4.30/1.98 TRUE^1 -> &&(TRUE, TRUE)^1 4.30/1.98 FALSE^1 -> &&(TRUE, FALSE)^1 4.30/1.98 FALSE^1 -> &&(FALSE, TRUE)^1 4.30/1.98 FALSE^1 -> &&(FALSE, FALSE)^1 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (6) 4.30/1.98 Obligation: 4.30/1.98 IDP problem: 4.30/1.98 The following function symbols are pre-defined: 4.30/1.98 <<< 4.30/1.98 & ~ Bwand: (Integer, Integer) -> Integer 4.30/1.98 >= ~ Ge: (Integer, Integer) -> Boolean 4.30/1.98 | ~ Bwor: (Integer, Integer) -> Integer 4.30/1.98 / ~ Div: (Integer, Integer) -> Integer 4.30/1.98 != ~ Neq: (Integer, Integer) -> Boolean 4.30/1.98 && ~ Land: (Boolean, Boolean) -> Boolean 4.30/1.98 ! ~ Lnot: (Boolean) -> Boolean 4.30/1.98 = ~ Eq: (Integer, Integer) -> Boolean 4.30/1.98 <= ~ Le: (Integer, Integer) -> Boolean 4.30/1.98 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.30/1.98 % ~ Mod: (Integer, Integer) -> Integer 4.30/1.98 + ~ Add: (Integer, Integer) -> Integer 4.30/1.98 > ~ Gt: (Integer, Integer) -> Boolean 4.30/1.98 -1 ~ UnaryMinus: (Integer) -> Integer 4.30/1.98 < ~ Lt: (Integer, Integer) -> Boolean 4.30/1.98 || ~ Lor: (Boolean, Boolean) -> Boolean 4.30/1.98 - ~ Sub: (Integer, Integer) -> Integer 4.30/1.98 ~ ~ Bwnot: (Integer) -> Integer 4.30/1.98 * ~ Mul: (Integer, Integer) -> Integer 4.30/1.98 >>> 4.30/1.98 4.30/1.98 4.30/1.98 The following domains are used: 4.30/1.98 Boolean, Integer 4.30/1.98 4.30/1.98 R is empty. 4.30/1.98 4.30/1.98 The integer pair graph contains the following rules and edges: 4.30/1.98 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0]) 4.30/1.98 4.30/1.98 4.30/1.98 The set Q consists of the following terms: 4.30/1.98 eval(x0, x1, x2) 4.30/1.98 Cond_eval(TRUE, x0, x1, x2) 4.30/1.98 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (7) IDependencyGraphProof (EQUIVALENT) 4.30/1.98 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.30/1.98 ---------------------------------------- 4.30/1.98 4.30/1.98 (8) 4.30/1.98 TRUE 4.54/2.02 EOF