4.63/2.62 YES 4.63/2.63 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 4.63/2.63 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.63/2.63 4.63/2.63 4.63/2.63 Termination of the given ITRS could be proven: 4.63/2.63 4.63/2.63 (0) ITRS 4.63/2.63 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.63/2.63 (2) IDP 4.63/2.63 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.63/2.63 (4) IDP 4.63/2.63 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.63/2.63 (6) IDP 4.63/2.63 (7) IDPNonInfProof [SOUND, 278 ms] 4.63/2.63 (8) IDP 4.63/2.63 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.63/2.63 (10) TRUE 4.63/2.63 4.63/2.63 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (0) 4.63/2.63 Obligation: 4.63/2.63 ITRS problem: 4.63/2.63 4.63/2.63 The following function symbols are pre-defined: 4.63/2.63 <<< 4.63/2.63 & ~ Bwand: (Integer, Integer) -> Integer 4.63/2.63 >= ~ Ge: (Integer, Integer) -> Boolean 4.63/2.63 / ~ Div: (Integer, Integer) -> Integer 4.63/2.63 | ~ Bwor: (Integer, Integer) -> Integer 4.63/2.63 != ~ Neq: (Integer, Integer) -> Boolean 4.63/2.63 && ~ Land: (Boolean, Boolean) -> Boolean 4.63/2.63 ! ~ Lnot: (Boolean) -> Boolean 4.63/2.63 = ~ Eq: (Integer, Integer) -> Boolean 4.63/2.63 <= ~ Le: (Integer, Integer) -> Boolean 4.63/2.63 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.63/2.63 % ~ Mod: (Integer, Integer) -> Integer 4.63/2.63 + ~ Add: (Integer, Integer) -> Integer 4.63/2.63 > ~ Gt: (Integer, Integer) -> Boolean 4.63/2.63 -1 ~ UnaryMinus: (Integer) -> Integer 4.63/2.63 < ~ Lt: (Integer, Integer) -> Boolean 4.63/2.63 || ~ Lor: (Boolean, Boolean) -> Boolean 4.63/2.63 - ~ Sub: (Integer, Integer) -> Integer 4.63/2.63 ~ ~ Bwnot: (Integer) -> Integer 4.63/2.63 * ~ Mul: (Integer, Integer) -> Integer 4.63/2.63 >>> 4.63/2.63 4.63/2.63 The TRS R consists of the following rules: 4.63/2.63 log(1, y) -> Cond_log(y >= 2, 1, y) 4.63/2.63 Cond_log(TRUE, 1, y) -> 0 4.63/2.63 log(x, y) -> Cond_log1(x >= 2 && y >= 2, x, y) 4.63/2.63 Cond_log1(TRUE, x, y) -> 1 + log((x - y) / y, y) 4.63/2.63 The set Q consists of the following terms: 4.63/2.63 Cond_log(TRUE, 1, x0) 4.63/2.63 log(x0, x1) 4.63/2.63 Cond_log1(TRUE, x0, x1) 4.63/2.63 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (1) ITRStoIDPProof (EQUIVALENT) 4.63/2.63 Added dependency pairs 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (2) 4.63/2.63 Obligation: 4.63/2.63 IDP problem: 4.63/2.63 The following function symbols are pre-defined: 4.63/2.63 <<< 4.63/2.63 & ~ Bwand: (Integer, Integer) -> Integer 4.63/2.63 >= ~ Ge: (Integer, Integer) -> Boolean 4.63/2.63 / ~ Div: (Integer, Integer) -> Integer 4.63/2.63 | ~ Bwor: (Integer, Integer) -> Integer 4.63/2.63 != ~ Neq: (Integer, Integer) -> Boolean 4.63/2.63 && ~ Land: (Boolean, Boolean) -> Boolean 4.63/2.63 ! ~ Lnot: (Boolean) -> Boolean 4.63/2.63 = ~ Eq: (Integer, Integer) -> Boolean 4.63/2.63 <= ~ Le: (Integer, Integer) -> Boolean 4.63/2.63 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.63/2.63 % ~ Mod: (Integer, Integer) -> Integer 4.63/2.63 + ~ Add: (Integer, Integer) -> Integer 4.63/2.63 > ~ Gt: (Integer, Integer) -> Boolean 4.63/2.63 -1 ~ UnaryMinus: (Integer) -> Integer 4.63/2.63 < ~ Lt: (Integer, Integer) -> Boolean 4.63/2.63 || ~ Lor: (Boolean, Boolean) -> Boolean 4.63/2.63 - ~ Sub: (Integer, Integer) -> Integer 4.63/2.63 ~ ~ Bwnot: (Integer) -> Integer 4.63/2.63 * ~ Mul: (Integer, Integer) -> Integer 4.63/2.63 >>> 4.63/2.63 4.63/2.63 4.63/2.63 The following domains are used: 4.63/2.63 Integer, Boolean 4.63/2.63 4.63/2.63 The ITRS R consists of the following rules: 4.63/2.63 log(1, y) -> Cond_log(y >= 2, 1, y) 4.63/2.63 Cond_log(TRUE, 1, y) -> 0 4.63/2.63 log(x, y) -> Cond_log1(x >= 2 && y >= 2, x, y) 4.63/2.63 Cond_log1(TRUE, x, y) -> 1 + log((x - y) / y, y) 4.63/2.63 4.63/2.63 The integer pair graph contains the following rules and edges: 4.63/2.63 (0): LOG(1, y[0]) -> COND_LOG(y[0] >= 2, 1, y[0]) 4.63/2.63 (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) 4.63/2.63 (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) 4.63/2.63 4.63/2.63 (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) 4.63/2.63 (2) -> (0), if ((x[2] - y[2]) / y[2] ->^* 1 & y[2] ->^* y[0]) 4.63/2.63 (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) 4.63/2.63 4.63/2.63 The set Q consists of the following terms: 4.63/2.63 Cond_log(TRUE, 1, x0) 4.63/2.63 log(x0, x1) 4.63/2.63 Cond_log1(TRUE, x0, x1) 4.63/2.63 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (3) UsableRulesProof (EQUIVALENT) 4.63/2.63 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. 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (4) 4.63/2.63 Obligation: 4.63/2.63 IDP problem: 4.63/2.63 The following function symbols are pre-defined: 4.63/2.63 <<< 4.63/2.63 & ~ Bwand: (Integer, Integer) -> Integer 4.63/2.63 >= ~ Ge: (Integer, Integer) -> Boolean 4.63/2.63 / ~ Div: (Integer, Integer) -> Integer 4.63/2.63 | ~ Bwor: (Integer, Integer) -> Integer 4.63/2.63 != ~ Neq: (Integer, Integer) -> Boolean 4.63/2.63 && ~ Land: (Boolean, Boolean) -> Boolean 4.63/2.63 ! ~ Lnot: (Boolean) -> Boolean 4.63/2.63 = ~ Eq: (Integer, Integer) -> Boolean 4.63/2.63 <= ~ Le: (Integer, Integer) -> Boolean 4.63/2.63 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.63/2.63 % ~ Mod: (Integer, Integer) -> Integer 4.63/2.63 + ~ Add: (Integer, Integer) -> Integer 4.63/2.63 > ~ Gt: (Integer, Integer) -> Boolean 4.63/2.63 -1 ~ UnaryMinus: (Integer) -> Integer 4.63/2.63 < ~ Lt: (Integer, Integer) -> Boolean 4.63/2.63 || ~ Lor: (Boolean, Boolean) -> Boolean 4.63/2.63 - ~ Sub: (Integer, Integer) -> Integer 4.63/2.63 ~ ~ Bwnot: (Integer) -> Integer 4.63/2.63 * ~ Mul: (Integer, Integer) -> Integer 4.63/2.63 >>> 4.63/2.63 4.63/2.63 4.63/2.63 The following domains are used: 4.63/2.63 Integer, Boolean 4.63/2.63 4.63/2.63 R is empty. 4.63/2.63 4.63/2.63 The integer pair graph contains the following rules and edges: 4.63/2.63 (0): LOG(1, y[0]) -> COND_LOG(y[0] >= 2, 1, y[0]) 4.63/2.63 (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) 4.63/2.63 (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) 4.63/2.63 4.63/2.63 (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) 4.63/2.63 (2) -> (0), if ((x[2] - y[2]) / y[2] ->^* 1 & y[2] ->^* y[0]) 4.63/2.63 (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) 4.63/2.63 4.63/2.63 The set Q consists of the following terms: 4.63/2.63 Cond_log(TRUE, 1, x0) 4.63/2.63 log(x0, x1) 4.63/2.63 Cond_log1(TRUE, x0, x1) 4.63/2.63 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (5) IDependencyGraphProof (EQUIVALENT) 4.63/2.63 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.63/2.63 ---------------------------------------- 4.63/2.63 4.63/2.63 (6) 4.63/2.63 Obligation: 4.63/2.63 IDP problem: 4.63/2.63 The following function symbols are pre-defined: 4.63/2.63 <<< 4.63/2.63 & ~ Bwand: (Integer, Integer) -> Integer 4.63/2.63 >= ~ Ge: (Integer, Integer) -> Boolean 4.63/2.63 / ~ Div: (Integer, Integer) -> Integer 4.63/2.63 | ~ Bwor: (Integer, Integer) -> Integer 4.63/2.63 != ~ Neq: (Integer, Integer) -> Boolean 4.63/2.63 && ~ Land: (Boolean, Boolean) -> Boolean 4.63/2.63 ! ~ Lnot: (Boolean) -> Boolean 4.63/2.63 = ~ Eq: (Integer, Integer) -> Boolean 4.63/2.63 <= ~ Le: (Integer, Integer) -> Boolean 4.63/2.63 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.63/2.63 % ~ Mod: (Integer, Integer) -> Integer 4.63/2.63 + ~ Add: (Integer, Integer) -> Integer 4.63/2.63 > ~ Gt: (Integer, Integer) -> Boolean 4.63/2.64 -1 ~ UnaryMinus: (Integer) -> Integer 4.63/2.64 < ~ Lt: (Integer, Integer) -> Boolean 4.63/2.64 || ~ Lor: (Boolean, Boolean) -> Boolean 4.63/2.64 - ~ Sub: (Integer, Integer) -> Integer 4.63/2.64 ~ ~ Bwnot: (Integer) -> Integer 4.63/2.64 * ~ Mul: (Integer, Integer) -> Integer 4.63/2.64 >>> 4.63/2.64 4.63/2.64 4.63/2.64 The following domains are used: 4.63/2.64 Integer, Boolean 4.63/2.64 4.63/2.64 R is empty. 4.63/2.64 4.63/2.64 The integer pair graph contains the following rules and edges: 4.63/2.64 (2): COND_LOG1(TRUE, x[2], y[2]) -> LOG((x[2] - y[2]) / y[2], y[2]) 4.63/2.64 (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) 4.63/2.64 4.63/2.64 (2) -> (1), if ((x[2] - y[2]) / y[2] ->^* x[1] & y[2] ->^* y[1]) 4.63/2.64 (1) -> (2), if (x[1] >= 2 && y[1] >= 2 & x[1] ->^* x[2] & y[1] ->^* y[2]) 4.63/2.64 4.63/2.64 The set Q consists of the following terms: 4.63/2.64 Cond_log(TRUE, 1, x0) 4.63/2.64 log(x0, x1) 4.63/2.64 Cond_log1(TRUE, x0, x1) 4.63/2.64 4.63/2.64 ---------------------------------------- 4.63/2.64 4.63/2.64 (7) IDPNonInfProof (SOUND) 4.63/2.64 Used the following options for this NonInfProof: 4.63/2.64 4.63/2.64 IDPGPoloSolver: 4.63/2.64 Range: [(-1,2)] 4.63/2.64 IsNat: false 4.63/2.64 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3c3eaf57 4.63/2.64 Constraint Generator: NonInfConstraintGenerator: 4.63/2.64 PathGenerator: MetricPathGenerator: 4.63/2.64 Max Left Steps: 1 4.63/2.64 Max Right Steps: 1 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 The constraints were generated the following way: 4.63/2.64 4.63/2.64 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.63/2.64 4.63/2.64 Note that final constraints are written in bold face. 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 For Pair COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) the following chains were created: 4.63/2.64 *We consider the chain LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]), COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]), LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) which results in the following constraint: 4.63/2.64 4.63/2.64 (1) (&&(>=(x[1], 2), >=(y[1], 2))=TRUE & x[1]=x[2] & y[1]=y[2] & /(-(x[2], y[2]), y[2])=x[1]1 & y[2]=y[1]1 ==> COND_LOG1(TRUE, x[2], y[2])_>=_NonInfC & COND_LOG1(TRUE, x[2], y[2])_>=_LOG(/(-(x[2], y[2]), y[2]), y[2]) & (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=)) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.63/2.64 4.63/2.64 (2) (>=(x[1], 2)=TRUE & >=(y[1], 2)=TRUE ==> COND_LOG1(TRUE, x[1], y[1])_>=_NonInfC & COND_LOG1(TRUE, x[1], y[1])_>=_LOG(/(-(x[1], y[1]), y[1]), y[1]) & (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=)) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.63/2.64 4.63/2.64 (3) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] + [-1]max{x[1] + [-1]y[1], [-1]x[1] + y[1]} + min{max{y[1], [-1]y[1]} + [-1], max{x[1] + [-1]y[1], [-1]x[1] + y[1]}} >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.63/2.64 4.63/2.64 (4) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] + [-1]max{x[1] + [-1]y[1], [-1]x[1] + y[1]} + min{max{y[1], [-1]y[1]} + [-1], max{x[1] + [-1]y[1], [-1]x[1] + y[1]}} >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: 4.63/2.64 4.63/2.64 (5) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & x[1] + [-2]y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) 4.63/2.64 4.63/2.64 (6) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & [2]y[1] + [-1] + [-1]x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) 4.63/2.64 4.63/2.64 (7) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [-2]x[1] + [-1] + [2]y[1] >= 0 & [2]y[1] >= 0 & [-1] + x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (5) using rule (IDP_POLY_GCD) which results in the following new constraint: 4.63/2.64 4.63/2.64 (8) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & x[1] + [-2]y[1] >= 0 & x[1] + [-1]y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 For Pair LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) the following chains were created: 4.63/2.64 *We consider the chain LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]), COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) which results in the following constraint: 4.63/2.64 4.63/2.64 (1) (&&(>=(x[1], 2), >=(y[1], 2))=TRUE & x[1]=x[2] & y[1]=y[2] ==> LOG(x[1], y[1])_>=_NonInfC & LOG(x[1], y[1])_>=_COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) & (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=)) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.63/2.64 4.63/2.64 (2) (>=(x[1], 2)=TRUE & >=(y[1], 2)=TRUE ==> LOG(x[1], y[1])_>=_NonInfC & LOG(x[1], y[1])_>=_COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) & (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=)) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.63/2.64 4.63/2.64 (3) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.63/2.64 4.63/2.64 (4) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.63/2.64 4.63/2.64 (5) (x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 To summarize, we get the following constraints P__>=_ for the following pairs. 4.63/2.64 4.63/2.64 *COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) 4.63/2.64 4.63/2.64 *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [2]x[1] + [-2]y[1] >= 0 & [2]y[1] >= 0 & [2]y[1] + [-1] + [-1]x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & [-2]x[1] + [-1] + [2]y[1] >= 0 & [2]y[1] >= 0 & [-1] + x[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [(-1)bso_16] + x[1] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 & x[1] + [-2]y[1] >= 0 & x[1] + [-1]y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(LOG(/(-(x[2], y[2]), y[2]), y[2])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x[1] >= 0 & [-1 + (-1)bso_16] + [2]y[1] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 *LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) 4.63/2.64 4.63/2.64 *(x[1] + [-2] >= 0 & y[1] + [-2] >= 0 ==> (U^Increasing(COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x[1] >= 0 & [(-1)bso_18] >= 0) 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 4.63/2.64 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. 4.63/2.64 4.63/2.64 Using the following integer polynomial ordering the resulting constraints can be solved 4.63/2.64 4.63/2.64 Polynomial interpretation over integers[POLO]: 4.63/2.64 4.63/2.64 POL(TRUE) = 0 4.63/2.64 POL(FALSE) = [1] 4.63/2.64 POL(COND_LOG1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 4.63/2.64 POL(LOG(x_1, x_2)) = [-1] + x_1 4.63/2.64 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.63/2.64 POL(&&(x_1, x_2)) = 0 4.63/2.64 POL(>=(x_1, x_2)) = [-1] 4.63/2.64 POL(2) = [2] 4.63/2.64 4.63/2.64 Polynomial Interpretations with Context Sensitive Arithemetic Replacement 4.63/2.64 POL(Term^CSAR-Mode @ Context) 4.63/2.64 4.63/2.64 POL(/(x_1, y[1])^1 @ {LOG_2/0}) = max{x_1, [-1]x_1} + [-1]min{max{x_2, [-1]x_2} + [-1], max{x_1, [-1]x_1}} 4.63/2.64 4.63/2.64 4.63/2.64 The following pairs are in P_>: 4.63/2.64 4.63/2.64 4.63/2.64 COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) 4.63/2.64 4.63/2.64 4.63/2.64 The following pairs are in P_bound: 4.63/2.64 4.63/2.64 4.63/2.64 COND_LOG1(TRUE, x[2], y[2]) -> LOG(/(-(x[2], y[2]), y[2]), y[2]) 4.63/2.64 LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) 4.63/2.64 4.63/2.64 4.63/2.64 The following pairs are in P_>=: 4.63/2.64 4.63/2.64 4.63/2.64 LOG(x[1], y[1]) -> COND_LOG1(&&(>=(x[1], 2), >=(y[1], 2)), x[1], y[1]) 4.63/2.64 4.63/2.64 4.63/2.64 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.63/2.64 4.63/2.64 /^1 -> 4.63/2.64 TRUE^1 -> &&(TRUE, TRUE)^1 4.63/2.64 FALSE^1 -> &&(TRUE, FALSE)^1 4.63/2.64 FALSE^1 -> &&(FALSE, TRUE)^1 4.63/2.64 FALSE^1 -> &&(FALSE, FALSE)^1 4.63/2.64 4.63/2.64 ---------------------------------------- 4.63/2.64 4.63/2.64 (8) 4.63/2.64 Obligation: 4.63/2.64 IDP problem: 4.63/2.64 The following function symbols are pre-defined: 4.63/2.64 <<< 4.63/2.64 & ~ Bwand: (Integer, Integer) -> Integer 4.63/2.64 >= ~ Ge: (Integer, Integer) -> Boolean 4.63/2.64 / ~ Div: (Integer, Integer) -> Integer 4.63/2.64 | ~ Bwor: (Integer, Integer) -> Integer 4.63/2.64 != ~ Neq: (Integer, Integer) -> Boolean 4.63/2.64 && ~ Land: (Boolean, Boolean) -> Boolean 4.63/2.64 ! ~ Lnot: (Boolean) -> Boolean 4.63/2.64 = ~ Eq: (Integer, Integer) -> Boolean 4.63/2.64 <= ~ Le: (Integer, Integer) -> Boolean 4.63/2.64 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.63/2.64 % ~ Mod: (Integer, Integer) -> Integer 4.63/2.64 + ~ Add: (Integer, Integer) -> Integer 4.63/2.64 > ~ Gt: (Integer, Integer) -> Boolean 4.63/2.64 -1 ~ UnaryMinus: (Integer) -> Integer 4.63/2.64 < ~ Lt: (Integer, Integer) -> Boolean 4.63/2.64 || ~ Lor: (Boolean, Boolean) -> Boolean 4.63/2.64 - ~ Sub: (Integer, Integer) -> Integer 4.63/2.64 ~ ~ Bwnot: (Integer) -> Integer 4.63/2.64 * ~ Mul: (Integer, Integer) -> Integer 4.63/2.64 >>> 4.63/2.64 4.63/2.64 4.63/2.64 The following domains are used: 4.63/2.64 Boolean, Integer 4.63/2.64 4.63/2.64 R is empty. 4.63/2.64 4.63/2.64 The integer pair graph contains the following rules and edges: 4.63/2.64 (1): LOG(x[1], y[1]) -> COND_LOG1(x[1] >= 2 && y[1] >= 2, x[1], y[1]) 4.63/2.64 4.63/2.64 4.63/2.64 The set Q consists of the following terms: 4.63/2.64 Cond_log(TRUE, 1, x0) 4.63/2.64 log(x0, x1) 4.63/2.64 Cond_log1(TRUE, x0, x1) 4.63/2.64 4.63/2.64 ---------------------------------------- 4.63/2.64 4.63/2.64 (9) IDependencyGraphProof (EQUIVALENT) 4.63/2.64 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.63/2.64 ---------------------------------------- 4.63/2.64 4.63/2.64 (10) 4.63/2.64 TRUE 4.63/2.67 EOF