4.59/2.16 YES 4.59/2.17 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.59/2.17 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.59/2.17 4.59/2.17 4.59/2.17 Termination of the given ITRS could be proven: 4.59/2.17 4.59/2.17 (0) ITRS 4.59/2.17 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.59/2.17 (2) IDP 4.59/2.17 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.59/2.17 (4) IDP 4.59/2.17 (5) IDPNonInfProof [SOUND, 269 ms] 4.59/2.17 (6) IDP 4.59/2.17 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.59/2.17 (8) TRUE 4.59/2.17 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (0) 4.59/2.17 Obligation: 4.59/2.17 ITRS problem: 4.59/2.17 4.59/2.17 The following function symbols are pre-defined: 4.59/2.17 <<< 4.59/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.59/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.59/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.59/2.17 / ~ Div: (Integer, Integer) -> Integer 4.59/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.59/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.59/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.59/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.59/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.59/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.59/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.59/2.17 + ~ Add: (Integer, Integer) -> Integer 4.59/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.59/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.59/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.59/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.59/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.59/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.59/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.59/2.17 >>> 4.59/2.17 4.59/2.17 The TRS R consists of the following rules: 4.59/2.17 eval(i, j, k) -> Cond_eval(i <= 100 && j <= k, i, j, k) 4.59/2.17 Cond_eval(TRUE, i, j, k) -> eval(j, i + 1, k - 1) 4.59/2.17 The set Q consists of the following terms: 4.59/2.17 eval(x0, x1, x2) 4.59/2.17 Cond_eval(TRUE, x0, x1, x2) 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (1) ITRStoIDPProof (EQUIVALENT) 4.59/2.17 Added dependency pairs 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (2) 4.59/2.17 Obligation: 4.59/2.17 IDP problem: 4.59/2.17 The following function symbols are pre-defined: 4.59/2.17 <<< 4.59/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.59/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.59/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.59/2.17 / ~ Div: (Integer, Integer) -> Integer 4.59/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.59/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.59/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.59/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.59/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.59/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.59/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.59/2.17 + ~ Add: (Integer, Integer) -> Integer 4.59/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.59/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.59/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.59/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.59/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.59/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.59/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.59/2.17 >>> 4.59/2.17 4.59/2.17 4.59/2.17 The following domains are used: 4.59/2.17 Boolean, Integer 4.59/2.17 4.59/2.17 The ITRS R consists of the following rules: 4.59/2.17 eval(i, j, k) -> Cond_eval(i <= 100 && j <= k, i, j, k) 4.59/2.17 Cond_eval(TRUE, i, j, k) -> eval(j, i + 1, k - 1) 4.59/2.17 4.59/2.17 The integer pair graph contains the following rules and edges: 4.59/2.17 (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) 4.59/2.17 (1): COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], i[1] + 1, k[1] - 1) 4.59/2.17 4.59/2.17 (0) -> (1), if (i[0] <= 100 && j[0] <= k[0] & i[0] ->^* i[1] & j[0] ->^* j[1] & k[0] ->^* k[1]) 4.59/2.17 (1) -> (0), if (j[1] ->^* i[0] & i[1] + 1 ->^* j[0] & k[1] - 1 ->^* k[0]) 4.59/2.17 4.59/2.17 The set Q consists of the following terms: 4.59/2.17 eval(x0, x1, x2) 4.59/2.17 Cond_eval(TRUE, x0, x1, x2) 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (3) UsableRulesProof (EQUIVALENT) 4.59/2.17 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. 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (4) 4.59/2.17 Obligation: 4.59/2.17 IDP problem: 4.59/2.17 The following function symbols are pre-defined: 4.59/2.17 <<< 4.59/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.59/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.59/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.59/2.17 / ~ Div: (Integer, Integer) -> Integer 4.59/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.59/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.59/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.59/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.59/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.59/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.59/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.59/2.17 + ~ Add: (Integer, Integer) -> Integer 4.59/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.59/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.59/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.59/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.59/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.59/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.59/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.59/2.17 >>> 4.59/2.17 4.59/2.17 4.59/2.17 The following domains are used: 4.59/2.17 Boolean, Integer 4.59/2.17 4.59/2.17 R is empty. 4.59/2.17 4.59/2.17 The integer pair graph contains the following rules and edges: 4.59/2.17 (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) 4.59/2.17 (1): COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], i[1] + 1, k[1] - 1) 4.59/2.17 4.59/2.17 (0) -> (1), if (i[0] <= 100 && j[0] <= k[0] & i[0] ->^* i[1] & j[0] ->^* j[1] & k[0] ->^* k[1]) 4.59/2.17 (1) -> (0), if (j[1] ->^* i[0] & i[1] + 1 ->^* j[0] & k[1] - 1 ->^* k[0]) 4.59/2.17 4.59/2.17 The set Q consists of the following terms: 4.59/2.17 eval(x0, x1, x2) 4.59/2.17 Cond_eval(TRUE, x0, x1, x2) 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (5) IDPNonInfProof (SOUND) 4.59/2.17 Used the following options for this NonInfProof: 4.59/2.17 4.59/2.17 IDPGPoloSolver: 4.59/2.17 Range: [(-1,2)] 4.59/2.17 IsNat: false 4.59/2.17 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@28bf23fb 4.59/2.17 Constraint Generator: NonInfConstraintGenerator: 4.59/2.17 PathGenerator: MetricPathGenerator: 4.59/2.17 Max Left Steps: 1 4.59/2.17 Max Right Steps: 1 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 The constraints were generated the following way: 4.59/2.17 4.59/2.17 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.59/2.17 4.59/2.17 Note that final constraints are written in bold face. 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 For Pair EVAL(i, j, k) -> COND_EVAL(&&(<=(i, 100), <=(j, k)), i, j, k) the following chains were created: 4.59/2.17 *We consider the chain EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]), COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) which results in the following constraint: 4.59/2.17 4.59/2.17 (1) (&&(<=(i[0], 100), <=(j[0], k[0]))=TRUE & i[0]=i[1] & j[0]=j[1] & k[0]=k[1] ==> EVAL(i[0], j[0], k[0])_>=_NonInfC & EVAL(i[0], j[0], k[0])_>=_COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) & (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=)) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.59/2.17 4.59/2.17 (2) (<=(i[0], 100)=TRUE & <=(j[0], k[0])=TRUE ==> EVAL(i[0], j[0], k[0])_>=_NonInfC & EVAL(i[0], j[0], k[0])_>=_COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) & (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=)) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.59/2.17 4.59/2.17 (3) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.59/2.17 4.59/2.17 (4) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.59/2.17 4.59/2.17 (5) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.59/2.17 4.59/2.17 (6) ([100] + [-1]i[0] >= 0 & k[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (7) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 (8) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (9) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 (10) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (11) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 (12) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 For Pair COND_EVAL(TRUE, i, j, k) -> EVAL(j, +(i, 1), -(k, 1)) the following chains were created: 4.59/2.17 *We consider the chain EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]), COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)), EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) which results in the following constraint: 4.59/2.17 4.59/2.17 (1) (&&(<=(i[0], 100), <=(j[0], k[0]))=TRUE & i[0]=i[1] & j[0]=j[1] & k[0]=k[1] & j[1]=i[0]1 & +(i[1], 1)=j[0]1 & -(k[1], 1)=k[0]1 ==> COND_EVAL(TRUE, i[1], j[1], k[1])_>=_NonInfC & COND_EVAL(TRUE, i[1], j[1], k[1])_>=_EVAL(j[1], +(i[1], 1), -(k[1], 1)) & (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=)) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.59/2.17 4.59/2.17 (2) (<=(i[0], 100)=TRUE & <=(j[0], k[0])=TRUE ==> COND_EVAL(TRUE, i[0], j[0], k[0])_>=_NonInfC & COND_EVAL(TRUE, i[0], j[0], k[0])_>=_EVAL(j[0], +(i[0], 1), -(k[0], 1)) & (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=)) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.59/2.17 4.59/2.17 (3) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.59/2.17 4.59/2.17 (4) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.59/2.17 4.59/2.17 (5) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.59/2.17 4.59/2.17 (6) ([100] + [-1]i[0] >= 0 & k[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (7) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 (8) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (9) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 (10) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.59/2.17 4.59/2.17 (11) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 (12) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 To summarize, we get the following constraints P__>=_ for the following pairs. 4.59/2.17 4.59/2.17 *EVAL(i, j, k) -> COND_EVAL(&&(<=(i, 100), <=(j, k)), i, j, k) 4.59/2.17 4.59/2.17 *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 *COND_EVAL(TRUE, i, j, k) -> EVAL(j, +(i, 1), -(k, 1)) 4.59/2.17 4.59/2.17 *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [2 + (-1)bso_17] >= 0) 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 4.59/2.17 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. 4.59/2.17 4.59/2.17 Using the following integer polynomial ordering the resulting constraints can be solved 4.59/2.17 4.59/2.17 Polynomial interpretation over integers[POLO]: 4.59/2.17 4.59/2.17 POL(TRUE) = 0 4.59/2.17 POL(FALSE) = [3] 4.59/2.17 POL(EVAL(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 + [-1]x_1 4.59/2.17 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_4 + [-1]x_3 + [-1]x_2 4.59/2.17 POL(&&(x_1, x_2)) = [-1] 4.59/2.17 POL(<=(x_1, x_2)) = [-1] 4.59/2.17 POL(100) = [100] 4.59/2.17 POL(+(x_1, x_2)) = x_1 + x_2 4.59/2.17 POL(1) = [1] 4.59/2.17 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.59/2.17 4.59/2.17 4.59/2.17 The following pairs are in P_>: 4.59/2.17 4.59/2.17 4.59/2.17 COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) 4.59/2.17 4.59/2.17 4.59/2.17 The following pairs are in P_bound: 4.59/2.17 4.59/2.17 4.59/2.17 EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) 4.59/2.17 COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) 4.59/2.17 4.59/2.17 4.59/2.17 The following pairs are in P_>=: 4.59/2.17 4.59/2.17 4.59/2.17 EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) 4.59/2.17 4.59/2.17 4.59/2.17 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.59/2.17 4.59/2.17 FALSE^1 -> &&(TRUE, FALSE)^1 4.59/2.17 FALSE^1 -> &&(FALSE, TRUE)^1 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (6) 4.59/2.17 Obligation: 4.59/2.17 IDP problem: 4.59/2.17 The following function symbols are pre-defined: 4.59/2.17 <<< 4.59/2.17 & ~ Bwand: (Integer, Integer) -> Integer 4.59/2.17 >= ~ Ge: (Integer, Integer) -> Boolean 4.59/2.17 | ~ Bwor: (Integer, Integer) -> Integer 4.59/2.17 / ~ Div: (Integer, Integer) -> Integer 4.59/2.17 != ~ Neq: (Integer, Integer) -> Boolean 4.59/2.17 && ~ Land: (Boolean, Boolean) -> Boolean 4.59/2.17 ! ~ Lnot: (Boolean) -> Boolean 4.59/2.17 = ~ Eq: (Integer, Integer) -> Boolean 4.59/2.17 <= ~ Le: (Integer, Integer) -> Boolean 4.59/2.17 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.59/2.17 % ~ Mod: (Integer, Integer) -> Integer 4.59/2.17 + ~ Add: (Integer, Integer) -> Integer 4.59/2.17 > ~ Gt: (Integer, Integer) -> Boolean 4.59/2.17 -1 ~ UnaryMinus: (Integer) -> Integer 4.59/2.17 < ~ Lt: (Integer, Integer) -> Boolean 4.59/2.17 || ~ Lor: (Boolean, Boolean) -> Boolean 4.59/2.17 - ~ Sub: (Integer, Integer) -> Integer 4.59/2.17 ~ ~ Bwnot: (Integer) -> Integer 4.59/2.17 * ~ Mul: (Integer, Integer) -> Integer 4.59/2.17 >>> 4.59/2.17 4.59/2.17 4.59/2.17 The following domains are used: 4.59/2.17 Boolean, Integer 4.59/2.17 4.59/2.17 R is empty. 4.59/2.17 4.59/2.17 The integer pair graph contains the following rules and edges: 4.59/2.17 (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) 4.59/2.17 4.59/2.17 4.59/2.17 The set Q consists of the following terms: 4.59/2.17 eval(x0, x1, x2) 4.59/2.17 Cond_eval(TRUE, x0, x1, x2) 4.59/2.17 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (7) IDependencyGraphProof (EQUIVALENT) 4.59/2.17 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.59/2.17 ---------------------------------------- 4.59/2.17 4.59/2.17 (8) 4.59/2.17 TRUE 4.59/2.21 EOF