5.57/2.32 YES 5.61/2.34 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.61/2.34 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.61/2.34 5.61/2.34 5.61/2.34 Termination of the given ITRS could be proven: 5.61/2.34 5.61/2.34 (0) ITRS 5.61/2.34 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.61/2.34 (2) IDP 5.61/2.34 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.61/2.34 (4) IDP 5.61/2.34 (5) IDPNonInfProof [SOUND, 247 ms] 5.61/2.34 (6) AND 5.61/2.34 (7) IDP 5.61/2.34 (8) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.34 (9) IDP 5.61/2.34 (10) IDPNonInfProof [SOUND, 41 ms] 5.61/2.34 (11) IDP 5.61/2.34 (12) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.34 (13) TRUE 5.61/2.34 (14) IDP 5.61/2.34 (15) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.34 (16) IDP 5.61/2.34 (17) IDPNonInfProof [SOUND, 43 ms] 5.61/2.34 (18) IDP 5.61/2.34 (19) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.34 (20) TRUE 5.61/2.34 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (0) 5.61/2.34 Obligation: 5.61/2.34 ITRS problem: 5.61/2.34 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 The TRS R consists of the following rules: 5.61/2.34 eval_1(i, j) -> Cond_eval_1(i >= 0, i, j) 5.61/2.34 Cond_eval_1(TRUE, i, j) -> eval_2(i, 0) 5.61/2.34 eval_2(i, j) -> Cond_eval_2(j <= i - 1, i, j) 5.61/2.34 Cond_eval_2(TRUE, i, j) -> eval_2(i, j + 1) 5.61/2.34 eval_2(i, j) -> Cond_eval_21(j > i - 1, i, j) 5.61/2.34 Cond_eval_21(TRUE, i, j) -> eval_1(i - 1, j) 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.34 Cond_eval_21(TRUE, x0, x1) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (1) ITRStoIDPProof (EQUIVALENT) 5.61/2.34 Added dependency pairs 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (2) 5.61/2.34 Obligation: 5.61/2.34 IDP problem: 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 5.61/2.34 The following domains are used: 5.61/2.34 Integer 5.61/2.34 5.61/2.34 The ITRS R consists of the following rules: 5.61/2.34 eval_1(i, j) -> Cond_eval_1(i >= 0, i, j) 5.61/2.34 Cond_eval_1(TRUE, i, j) -> eval_2(i, 0) 5.61/2.34 eval_2(i, j) -> Cond_eval_2(j <= i - 1, i, j) 5.61/2.34 Cond_eval_2(TRUE, i, j) -> eval_2(i, j + 1) 5.61/2.34 eval_2(i, j) -> Cond_eval_21(j > i - 1, i, j) 5.61/2.34 Cond_eval_21(TRUE, i, j) -> eval_1(i - 1, j) 5.61/2.34 5.61/2.34 The integer pair graph contains the following rules and edges: 5.61/2.34 (0): EVAL_1(i[0], j[0]) -> COND_EVAL_1(i[0] >= 0, i[0], j[0]) 5.61/2.34 (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) 5.61/2.34 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.34 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.34 (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) 5.61/2.34 (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) 5.61/2.34 5.61/2.34 (0) -> (1), if (i[0] >= 0 & i[0] ->^* i[1] & j[0] ->^* j[1]) 5.61/2.34 (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) 5.61/2.34 (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) 5.61/2.34 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.34 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.34 (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) 5.61/2.34 (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) 5.61/2.34 (5) -> (0), if (i[5] - 1 ->^* i[0] & j[5] ->^* j[0]) 5.61/2.34 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.34 Cond_eval_21(TRUE, x0, x1) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (3) UsableRulesProof (EQUIVALENT) 5.61/2.34 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. 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (4) 5.61/2.34 Obligation: 5.61/2.34 IDP problem: 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 5.61/2.34 The following domains are used: 5.61/2.34 Integer 5.61/2.34 5.61/2.34 R is empty. 5.61/2.34 5.61/2.34 The integer pair graph contains the following rules and edges: 5.61/2.34 (0): EVAL_1(i[0], j[0]) -> COND_EVAL_1(i[0] >= 0, i[0], j[0]) 5.61/2.34 (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) 5.61/2.34 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.34 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.34 (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) 5.61/2.34 (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) 5.61/2.34 5.61/2.34 (0) -> (1), if (i[0] >= 0 & i[0] ->^* i[1] & j[0] ->^* j[1]) 5.61/2.34 (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) 5.61/2.34 (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) 5.61/2.34 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.34 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.34 (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) 5.61/2.34 (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) 5.61/2.34 (5) -> (0), if (i[5] - 1 ->^* i[0] & j[5] ->^* j[0]) 5.61/2.34 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.34 Cond_eval_21(TRUE, x0, x1) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (5) IDPNonInfProof (SOUND) 5.61/2.34 Used the following options for this NonInfProof: 5.61/2.34 5.61/2.34 IDPGPoloSolver: 5.61/2.34 Range: [(-1,2)] 5.61/2.34 IsNat: false 5.61/2.34 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@525fc0aa 5.61/2.34 Constraint Generator: NonInfConstraintGenerator: 5.61/2.34 PathGenerator: MetricPathGenerator: 5.61/2.34 Max Left Steps: 1 5.61/2.34 Max Right Steps: 1 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 The constraints were generated the following way: 5.61/2.34 5.61/2.34 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.61/2.34 5.61/2.34 Note that final constraints are written in bold face. 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair EVAL_1(i, j) -> COND_EVAL_1(>=(i, 0), i, j) the following chains were created: 5.61/2.34 *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] ==> EVAL_1(i[0], j[0])_>=_NonInfC & EVAL_1(i[0], j[0])_>=_COND_EVAL_1(>=(i[0], 0), i[0], j[0]) & (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (>=(i[0], 0)=TRUE ==> EVAL_1(i[0], j[0])_>=_NonInfC & EVAL_1(i[0], j[0])_>=_COND_EVAL_1(>=(i[0], 0), i[0], j[0]) & (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i[0] >= 0 & [(-1)bso_25] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i[0] >= 0 & [(-1)bso_25] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i[0] >= 0 & [(-1)bso_25] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & 0 = 0 & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i[0] >= 0 & [(-1)bso_25] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair COND_EVAL_1(TRUE, i, j) -> EVAL_2(i, 0) the following chains were created: 5.61/2.34 *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] & i[1]=i[2] & 0=j[2] ==> COND_EVAL_1(TRUE, i[1], j[1])_>=_NonInfC & COND_EVAL_1(TRUE, i[1], j[1])_>=_EVAL_2(i[1], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (>=(i[0], 0)=TRUE ==> COND_EVAL_1(TRUE, i[0], j[0])_>=_NonInfC & COND_EVAL_1(TRUE, i[0], j[0])_>=_EVAL_2(i[0], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *We consider the chain EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]), COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0), EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (>=(i[0], 0)=TRUE & i[0]=i[1] & j[0]=j[1] & i[1]=i[4] & 0=j[4] ==> COND_EVAL_1(TRUE, i[1], j[1])_>=_NonInfC & COND_EVAL_1(TRUE, i[1], j[1])_>=_EVAL_2(i[1], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (>=(i[0], 0)=TRUE ==> COND_EVAL_1(TRUE, i[0], j[0])_>=_NonInfC & COND_EVAL_1(TRUE, i[0], j[0])_>=_EVAL_2(i[0], 0) & (U^Increasing(EVAL_2(i[1], 0)), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair EVAL_2(i, j) -> COND_EVAL_2(<=(j, -(i, 1)), i, j) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (<=(j[2], -(i[2], 1))=TRUE ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_28] + [bni_28]j[2] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_28] + [bni_28]j[2] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]j[2] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair COND_EVAL_2(TRUE, i, j) -> EVAL_2(i, +(j, 1)) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[2]1 & +(j[3], 1)=j[2]1 ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[4] & +(j[3], 1)=j[4] ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair EVAL_2(i, j) -> COND_EVAL_21(>(j, -(i, 1)), i, j) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]), COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (>(j[4], -(i[4], 1))=TRUE & i[4]=i[5] & j[4]=j[5] ==> EVAL_2(i[4], j[4])_>=_NonInfC & EVAL_2(i[4], j[4])_>=_COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) & (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (>(j[4], -(i[4], 1))=TRUE ==> EVAL_2(i[4], j[4])_>=_NonInfC & EVAL_2(i[4], j[4])_>=_COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) & (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (j[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 (8) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair COND_EVAL_21(TRUE, i, j) -> EVAL_1(-(i, 1), j) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]), COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]), EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (>(j[4], -(i[4], 1))=TRUE & i[4]=i[5] & j[4]=j[5] & -(i[5], 1)=i[0] & j[5]=j[0] ==> COND_EVAL_21(TRUE, i[5], j[5])_>=_NonInfC & COND_EVAL_21(TRUE, i[5], j[5])_>=_EVAL_1(-(i[5], 1), j[5]) & (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (>(j[4], -(i[4], 1))=TRUE ==> COND_EVAL_21(TRUE, i[4], j[4])_>=_NonInfC & COND_EVAL_21(TRUE, i[4], j[4])_>=_EVAL_1(-(i[4], 1), j[4]) & (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (j[4] + [-1]i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (j[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 (8) (j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 To summarize, we get the following constraints P__>=_ for the following pairs. 5.61/2.34 5.61/2.34 *EVAL_1(i, j) -> COND_EVAL_1(>=(i, 0), i, j) 5.61/2.34 5.61/2.34 *(i[0] >= 0 ==> (U^Increasing(COND_EVAL_1(>=(i[0], 0), i[0], j[0])), >=) & 0 = 0 & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i[0] >= 0 & [(-1)bso_25] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *COND_EVAL_1(TRUE, i, j) -> EVAL_2(i, 0) 5.61/2.34 5.61/2.34 *(i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[0] >= 0 ==> (U^Increasing(EVAL_2(i[1], 0)), >=) & 0 = 0 & [(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]i[0] >= 0 & 0 = 0 & [(-1)bso_27] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *EVAL_2(i, j) -> COND_EVAL_2(<=(j, -(i, 1)), i, j) 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_28] + [bni_28]j[2] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_28] + [(-1)bni_28]j[2] + [bni_28]i[2] >= 0 & [(-1)bso_29] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *COND_EVAL_2(TRUE, i, j) -> EVAL_2(i, +(j, 1)) 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [(-1)bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_30] + [bni_30]j[2] + [bni_30]i[2] >= 0 & [(-1)bso_31] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *EVAL_2(i, j) -> COND_EVAL_21(>(j, -(i, 1)), i, j) 5.61/2.34 5.61/2.34 *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4])), >=) & [(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i[4] >= 0 & [(-1)bso_33] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *COND_EVAL_21(TRUE, i, j) -> EVAL_1(-(i, 1), j) 5.61/2.34 5.61/2.34 *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(j[4] >= 0 & i[4] >= 0 ==> (U^Increasing(EVAL_1(-(i[5], 1), j[5])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i[4] >= 0 & [1 + (-1)bso_35] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 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. 5.61/2.34 5.61/2.34 Using the following integer polynomial ordering the resulting constraints can be solved 5.61/2.34 5.61/2.34 Polynomial interpretation over integers[POLO]: 5.61/2.34 5.61/2.34 POL(TRUE) = 0 5.61/2.34 POL(FALSE) = 0 5.61/2.34 POL(EVAL_1(x_1, x_2)) = [-1] + x_1 5.61/2.34 POL(COND_EVAL_1(x_1, x_2, x_3)) = [-1] + x_2 5.61/2.34 POL(>=(x_1, x_2)) = [-1] 5.61/2.34 POL(0) = 0 5.61/2.34 POL(EVAL_2(x_1, x_2)) = [-1] + x_1 5.61/2.34 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + x_2 5.61/2.34 POL(<=(x_1, x_2)) = [-1] 5.61/2.34 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.61/2.34 POL(1) = [1] 5.61/2.34 POL(+(x_1, x_2)) = x_1 + x_2 5.61/2.34 POL(COND_EVAL_21(x_1, x_2, x_3)) = [-1] + x_2 5.61/2.34 POL(>(x_1, x_2)) = [-1] 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_>: 5.61/2.34 5.61/2.34 5.61/2.34 COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(-(i[5], 1), j[5]) 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_bound: 5.61/2.34 5.61/2.34 5.61/2.34 EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) 5.61/2.34 COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_>=: 5.61/2.34 5.61/2.34 5.61/2.34 EVAL_1(i[0], j[0]) -> COND_EVAL_1(>=(i[0], 0), i[0], j[0]) 5.61/2.34 COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) 5.61/2.34 EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.34 COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.34 EVAL_2(i[4], j[4]) -> COND_EVAL_21(>(j[4], -(i[4], 1)), i[4], j[4]) 5.61/2.34 5.61/2.34 5.61/2.34 There are no usable rules. 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (6) 5.61/2.34 Complex Obligation (AND) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (7) 5.61/2.34 Obligation: 5.61/2.34 IDP problem: 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 5.61/2.34 The following domains are used: 5.61/2.34 Integer 5.61/2.34 5.61/2.34 R is empty. 5.61/2.34 5.61/2.34 The integer pair graph contains the following rules and edges: 5.61/2.34 (0): EVAL_1(i[0], j[0]) -> COND_EVAL_1(i[0] >= 0, i[0], j[0]) 5.61/2.34 (1): COND_EVAL_1(TRUE, i[1], j[1]) -> EVAL_2(i[1], 0) 5.61/2.34 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.34 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.34 (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) 5.61/2.34 5.61/2.34 (0) -> (1), if (i[0] >= 0 & i[0] ->^* i[1] & j[0] ->^* j[1]) 5.61/2.34 (1) -> (2), if (i[1] ->^* i[2] & 0 ->^* j[2]) 5.61/2.34 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.34 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.34 (1) -> (4), if (i[1] ->^* i[4] & 0 ->^* j[4]) 5.61/2.34 (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) 5.61/2.34 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.34 Cond_eval_21(TRUE, x0, x1) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (8) IDependencyGraphProof (EQUIVALENT) 5.61/2.34 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (9) 5.61/2.34 Obligation: 5.61/2.34 IDP problem: 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 5.61/2.34 The following domains are used: 5.61/2.34 Integer 5.61/2.34 5.61/2.34 R is empty. 5.61/2.34 5.61/2.34 The integer pair graph contains the following rules and edges: 5.61/2.34 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.34 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.34 5.61/2.34 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.34 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.34 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.34 Cond_eval_21(TRUE, x0, x1) 5.61/2.34 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (10) IDPNonInfProof (SOUND) 5.61/2.34 Used the following options for this NonInfProof: 5.61/2.34 5.61/2.34 IDPGPoloSolver: 5.61/2.34 Range: [(-1,2)] 5.61/2.34 IsNat: false 5.61/2.34 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@525fc0aa 5.61/2.34 Constraint Generator: NonInfConstraintGenerator: 5.61/2.34 PathGenerator: MetricPathGenerator: 5.61/2.34 Max Left Steps: 1 5.61/2.34 Max Right Steps: 1 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 The constraints were generated the following way: 5.61/2.34 5.61/2.34 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.61/2.34 5.61/2.34 Note that final constraints are written in bold face. 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[2]1 & +(j[3], 1)=j[2]1 ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 For Pair EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) the following chains were created: 5.61/2.34 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) which results in the following constraint: 5.61/2.34 5.61/2.34 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.61/2.34 5.61/2.34 (2) (<=(j[2], -(i[2], 1))=TRUE ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.34 5.61/2.34 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.34 5.61/2.34 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.34 5.61/2.34 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.34 5.61/2.34 (6) (i[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.34 5.61/2.34 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 To summarize, we get the following constraints P__>=_ for the following pairs. 5.61/2.34 5.61/2.34 *COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 *EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 5.61/2.34 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. 5.61/2.34 5.61/2.34 Using the following integer polynomial ordering the resulting constraints can be solved 5.61/2.34 5.61/2.34 Polynomial interpretation over integers[POLO]: 5.61/2.34 5.61/2.34 POL(TRUE) = 0 5.61/2.34 POL(FALSE) = 0 5.61/2.34 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 5.61/2.34 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 + x_1 5.61/2.34 POL(+(x_1, x_2)) = x_1 + x_2 5.61/2.34 POL(1) = [1] 5.61/2.34 POL(<=(x_1, x_2)) = [-1] 5.61/2.34 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_>: 5.61/2.34 5.61/2.34 5.61/2.34 COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_bound: 5.61/2.34 5.61/2.34 5.61/2.34 COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.34 EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.34 5.61/2.34 5.61/2.34 The following pairs are in P_>=: 5.61/2.34 5.61/2.34 5.61/2.34 EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.34 5.61/2.34 5.61/2.34 There are no usable rules. 5.61/2.34 ---------------------------------------- 5.61/2.34 5.61/2.34 (11) 5.61/2.34 Obligation: 5.61/2.34 IDP problem: 5.61/2.34 The following function symbols are pre-defined: 5.61/2.34 <<< 5.61/2.34 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.34 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.34 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.34 / ~ Div: (Integer, Integer) -> Integer 5.61/2.34 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.34 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.34 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.34 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.34 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.34 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.34 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.34 + ~ Add: (Integer, Integer) -> Integer 5.61/2.34 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.34 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.34 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.34 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.34 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.34 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.34 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.34 >>> 5.61/2.34 5.61/2.34 5.61/2.34 The following domains are used: 5.61/2.34 Integer 5.61/2.34 5.61/2.34 R is empty. 5.61/2.34 5.61/2.34 The integer pair graph contains the following rules and edges: 5.61/2.34 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.34 5.61/2.34 5.61/2.34 The set Q consists of the following terms: 5.61/2.34 eval_1(x0, x1) 5.61/2.34 Cond_eval_1(TRUE, x0, x1) 5.61/2.34 eval_2(x0, x1) 5.61/2.34 Cond_eval_2(TRUE, x0, x1) 5.61/2.35 Cond_eval_21(TRUE, x0, x1) 5.61/2.35 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (12) IDependencyGraphProof (EQUIVALENT) 5.61/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (13) 5.61/2.35 TRUE 5.61/2.35 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (14) 5.61/2.35 Obligation: 5.61/2.35 IDP problem: 5.61/2.35 The following function symbols are pre-defined: 5.61/2.35 <<< 5.61/2.35 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.35 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.35 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.35 / ~ Div: (Integer, Integer) -> Integer 5.61/2.35 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.35 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.35 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.35 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.35 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.35 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.35 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.35 + ~ Add: (Integer, Integer) -> Integer 5.61/2.35 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.35 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.35 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.35 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.35 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.35 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.35 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.35 >>> 5.61/2.35 5.61/2.35 5.61/2.35 The following domains are used: 5.61/2.35 Integer 5.61/2.35 5.61/2.35 R is empty. 5.61/2.35 5.61/2.35 The integer pair graph contains the following rules and edges: 5.61/2.35 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.35 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.35 (4): EVAL_2(i[4], j[4]) -> COND_EVAL_21(j[4] > i[4] - 1, i[4], j[4]) 5.61/2.35 (5): COND_EVAL_21(TRUE, i[5], j[5]) -> EVAL_1(i[5] - 1, j[5]) 5.61/2.35 5.61/2.35 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.35 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.35 (3) -> (4), if (i[3] ->^* i[4] & j[3] + 1 ->^* j[4]) 5.61/2.35 (4) -> (5), if (j[4] > i[4] - 1 & i[4] ->^* i[5] & j[4] ->^* j[5]) 5.61/2.35 5.61/2.35 The set Q consists of the following terms: 5.61/2.35 eval_1(x0, x1) 5.61/2.35 Cond_eval_1(TRUE, x0, x1) 5.61/2.35 eval_2(x0, x1) 5.61/2.35 Cond_eval_2(TRUE, x0, x1) 5.61/2.35 Cond_eval_21(TRUE, x0, x1) 5.61/2.35 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (15) IDependencyGraphProof (EQUIVALENT) 5.61/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (16) 5.61/2.35 Obligation: 5.61/2.35 IDP problem: 5.61/2.35 The following function symbols are pre-defined: 5.61/2.35 <<< 5.61/2.35 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.35 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.35 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.35 / ~ Div: (Integer, Integer) -> Integer 5.61/2.35 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.35 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.35 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.35 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.35 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.35 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.35 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.35 + ~ Add: (Integer, Integer) -> Integer 5.61/2.35 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.35 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.35 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.35 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.35 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.35 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.35 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.35 >>> 5.61/2.35 5.61/2.35 5.61/2.35 The following domains are used: 5.61/2.35 Integer 5.61/2.35 5.61/2.35 R is empty. 5.61/2.35 5.61/2.35 The integer pair graph contains the following rules and edges: 5.61/2.35 (3): COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], j[3] + 1) 5.61/2.35 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.35 5.61/2.35 (3) -> (2), if (i[3] ->^* i[2] & j[3] + 1 ->^* j[2]) 5.61/2.35 (2) -> (3), if (j[2] <= i[2] - 1 & i[2] ->^* i[3] & j[2] ->^* j[3]) 5.61/2.35 5.61/2.35 The set Q consists of the following terms: 5.61/2.35 eval_1(x0, x1) 5.61/2.35 Cond_eval_1(TRUE, x0, x1) 5.61/2.35 eval_2(x0, x1) 5.61/2.35 Cond_eval_2(TRUE, x0, x1) 5.61/2.35 Cond_eval_21(TRUE, x0, x1) 5.61/2.35 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (17) IDPNonInfProof (SOUND) 5.61/2.35 Used the following options for this NonInfProof: 5.61/2.35 5.61/2.35 IDPGPoloSolver: 5.61/2.35 Range: [(-1,2)] 5.61/2.35 IsNat: false 5.61/2.35 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@525fc0aa 5.61/2.35 Constraint Generator: NonInfConstraintGenerator: 5.61/2.35 PathGenerator: MetricPathGenerator: 5.61/2.35 Max Left Steps: 1 5.61/2.35 Max Right Steps: 1 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 The constraints were generated the following way: 5.61/2.35 5.61/2.35 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.61/2.35 5.61/2.35 Note that final constraints are written in bold face. 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 For Pair COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) the following chains were created: 5.61/2.35 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)), EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) which results in the following constraint: 5.61/2.35 5.61/2.35 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] & i[3]=i[2]1 & +(j[3], 1)=j[2]1 ==> COND_EVAL_2(TRUE, i[3], j[3])_>=_NonInfC & COND_EVAL_2(TRUE, i[3], j[3])_>=_EVAL_2(i[3], +(j[3], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.61/2.35 5.61/2.35 (2) (<=(j[2], -(i[2], 1))=TRUE ==> COND_EVAL_2(TRUE, i[2], j[2])_>=_NonInfC & COND_EVAL_2(TRUE, i[2], j[2])_>=_EVAL_2(i[2], +(j[2], 1)) & (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=)) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.35 5.61/2.35 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.35 5.61/2.35 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.35 5.61/2.35 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]j[2] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.35 5.61/2.35 (6) (i[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.35 5.61/2.35 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 For Pair EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) the following chains were created: 5.61/2.35 *We consider the chain EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]), COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) which results in the following constraint: 5.61/2.35 5.61/2.35 (1) (<=(j[2], -(i[2], 1))=TRUE & i[2]=i[3] & j[2]=j[3] ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.61/2.35 5.61/2.35 (2) (<=(j[2], -(i[2], 1))=TRUE ==> EVAL_2(i[2], j[2])_>=_NonInfC & EVAL_2(i[2], j[2])_>=_COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) & (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=)) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.61/2.35 5.61/2.35 (3) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.61/2.35 5.61/2.35 (4) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.61/2.35 5.61/2.35 (5) (i[2] + [-1] + [-1]j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]j[2] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.61/2.35 5.61/2.35 (6) (i[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.61/2.35 5.61/2.35 (7) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 (8) (i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 To summarize, we get the following constraints P__>=_ for the following pairs. 5.61/2.35 5.61/2.35 *COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.35 5.61/2.35 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(EVAL_2(i[3], +(j[3], 1))), >=) & [(-1)Bound*bni_11] + [bni_11]i[2] >= 0 & [1 + (-1)bso_12] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 *EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.35 5.61/2.35 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 *(i[2] >= 0 & j[2] >= 0 ==> (U^Increasing(COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2])), >=) & [(-1)Bound*bni_13] + [bni_13]i[2] >= 0 & [(-1)bso_14] >= 0) 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 5.61/2.35 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. 5.61/2.35 5.61/2.35 Using the following integer polynomial ordering the resulting constraints can be solved 5.61/2.35 5.61/2.35 Polynomial interpretation over integers[POLO]: 5.61/2.35 5.61/2.35 POL(TRUE) = 0 5.61/2.35 POL(FALSE) = 0 5.61/2.35 POL(COND_EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 5.61/2.35 POL(EVAL_2(x_1, x_2)) = [-1] + [-1]x_2 + x_1 5.61/2.35 POL(+(x_1, x_2)) = x_1 + x_2 5.61/2.35 POL(1) = [1] 5.61/2.35 POL(<=(x_1, x_2)) = [2] 5.61/2.35 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.61/2.35 5.61/2.35 5.61/2.35 The following pairs are in P_>: 5.61/2.35 5.61/2.35 5.61/2.35 COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.35 5.61/2.35 5.61/2.35 The following pairs are in P_bound: 5.61/2.35 5.61/2.35 5.61/2.35 COND_EVAL_2(TRUE, i[3], j[3]) -> EVAL_2(i[3], +(j[3], 1)) 5.61/2.35 EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.35 5.61/2.35 5.61/2.35 The following pairs are in P_>=: 5.61/2.35 5.61/2.35 5.61/2.35 EVAL_2(i[2], j[2]) -> COND_EVAL_2(<=(j[2], -(i[2], 1)), i[2], j[2]) 5.61/2.35 5.61/2.35 5.61/2.35 There are no usable rules. 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (18) 5.61/2.35 Obligation: 5.61/2.35 IDP problem: 5.61/2.35 The following function symbols are pre-defined: 5.61/2.35 <<< 5.61/2.35 & ~ Bwand: (Integer, Integer) -> Integer 5.61/2.35 >= ~ Ge: (Integer, Integer) -> Boolean 5.61/2.35 | ~ Bwor: (Integer, Integer) -> Integer 5.61/2.35 / ~ Div: (Integer, Integer) -> Integer 5.61/2.35 != ~ Neq: (Integer, Integer) -> Boolean 5.61/2.35 && ~ Land: (Boolean, Boolean) -> Boolean 5.61/2.35 ! ~ Lnot: (Boolean) -> Boolean 5.61/2.35 = ~ Eq: (Integer, Integer) -> Boolean 5.61/2.35 <= ~ Le: (Integer, Integer) -> Boolean 5.61/2.35 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.61/2.35 % ~ Mod: (Integer, Integer) -> Integer 5.61/2.35 + ~ Add: (Integer, Integer) -> Integer 5.61/2.35 > ~ Gt: (Integer, Integer) -> Boolean 5.61/2.35 -1 ~ UnaryMinus: (Integer) -> Integer 5.61/2.35 < ~ Lt: (Integer, Integer) -> Boolean 5.61/2.35 || ~ Lor: (Boolean, Boolean) -> Boolean 5.61/2.35 - ~ Sub: (Integer, Integer) -> Integer 5.61/2.35 ~ ~ Bwnot: (Integer) -> Integer 5.61/2.35 * ~ Mul: (Integer, Integer) -> Integer 5.61/2.35 >>> 5.61/2.35 5.61/2.35 5.61/2.35 The following domains are used: 5.61/2.35 Integer 5.61/2.35 5.61/2.35 R is empty. 5.61/2.35 5.61/2.35 The integer pair graph contains the following rules and edges: 5.61/2.35 (2): EVAL_2(i[2], j[2]) -> COND_EVAL_2(j[2] <= i[2] - 1, i[2], j[2]) 5.61/2.35 5.61/2.35 5.61/2.35 The set Q consists of the following terms: 5.61/2.35 eval_1(x0, x1) 5.61/2.35 Cond_eval_1(TRUE, x0, x1) 5.61/2.35 eval_2(x0, x1) 5.61/2.35 Cond_eval_2(TRUE, x0, x1) 5.61/2.35 Cond_eval_21(TRUE, x0, x1) 5.61/2.35 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (19) IDependencyGraphProof (EQUIVALENT) 5.61/2.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.61/2.35 ---------------------------------------- 5.61/2.35 5.61/2.35 (20) 5.61/2.35 TRUE 5.61/2.38 EOF