5.34/2.43 YES 5.34/2.45 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.34/2.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.34/2.45 5.34/2.45 5.34/2.45 Termination of the given ITRS could be proven: 5.34/2.45 5.34/2.45 (0) ITRS 5.34/2.45 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.34/2.45 (2) IDP 5.34/2.45 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.34/2.45 (4) IDP 5.34/2.45 (5) IDPNonInfProof [SOUND, 242 ms] 5.34/2.45 (6) AND 5.34/2.45 (7) IDP 5.34/2.45 (8) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.34/2.45 (9) IDP 5.34/2.45 (10) IDPNonInfProof [SOUND, 46 ms] 5.34/2.45 (11) IDP 5.34/2.45 (12) PisEmptyProof [EQUIVALENT, 0 ms] 5.34/2.45 (13) YES 5.34/2.45 (14) IDP 5.34/2.45 (15) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.34/2.45 (16) IDP 5.34/2.45 (17) IDPNonInfProof [SOUND, 36 ms] 5.34/2.45 (18) IDP 5.34/2.45 (19) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.34/2.45 (20) TRUE 5.34/2.45 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (0) 5.34/2.45 Obligation: 5.34/2.45 ITRS problem: 5.34/2.45 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 The TRS R consists of the following rules: 5.34/2.45 eval(x, y) -> Cond_eval(x > 0 && y >= x, x, y) 5.34/2.45 Cond_eval(TRUE, x, y) -> eval(x - 1, y) 5.34/2.45 eval(x, y) -> Cond_eval1(x > 0 && x > y, x, y) 5.34/2.45 Cond_eval1(TRUE, x, y) -> eval(y, y) 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (1) ITRStoIDPProof (EQUIVALENT) 5.34/2.45 Added dependency pairs 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (2) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 The ITRS R consists of the following rules: 5.34/2.45 eval(x, y) -> Cond_eval(x > 0 && y >= x, x, y) 5.34/2.45 Cond_eval(TRUE, x, y) -> eval(x - 1, y) 5.34/2.45 eval(x, y) -> Cond_eval1(x > 0 && x > y, x, y) 5.34/2.45 Cond_eval1(TRUE, x, y) -> eval(y, y) 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 5.34/2.45 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && x[2] > y[2], x[2], y[2]) 5.34/2.45 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 (0) -> (1), if (x[0] > 0 && y[0] >= x[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.34/2.45 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.34/2.45 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 5.34/2.45 (2) -> (3), if (x[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.34/2.45 (3) -> (0), if (y[3] ->^* x[0] & y[3] ->^* y[0]) 5.34/2.45 (3) -> (2), if (y[3] ->^* x[2] & y[3] ->^* y[2]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (3) UsableRulesProof (EQUIVALENT) 5.34/2.45 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. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (4) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 5.34/2.45 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && x[2] > y[2], x[2], y[2]) 5.34/2.45 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 (0) -> (1), if (x[0] > 0 && y[0] >= x[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.34/2.45 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.34/2.45 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2]) 5.34/2.45 (2) -> (3), if (x[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.34/2.45 (3) -> (0), if (y[3] ->^* x[0] & y[3] ->^* y[0]) 5.34/2.45 (3) -> (2), if (y[3] ->^* x[2] & y[3] ->^* y[2]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (5) IDPNonInfProof (SOUND) 5.34/2.45 Used the following options for this NonInfProof: 5.34/2.45 5.34/2.45 IDPGPoloSolver: 5.34/2.45 Range: [(-1,2)] 5.34/2.45 IsNat: false 5.34/2.45 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@64fe5e2b 5.34/2.45 Constraint Generator: NonInfConstraintGenerator: 5.34/2.45 PathGenerator: MetricPathGenerator: 5.34/2.45 Max Left Steps: 1 5.34/2.45 Max Right Steps: 1 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 The constraints were generated the following way: 5.34/2.45 5.34/2.45 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.34/2.45 5.34/2.45 Note that final constraints are written in bold face. 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >=(y, x)), x, y) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[0], 0), >=(y[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[0], 0)=TRUE & >=(y[0], x[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[0], 0), >=(y[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[0], 0)=TRUE & >=(y[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[0], 0), >=(y[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[0], 0)=TRUE & >=(y[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair EVAL(x, y) -> COND_EVAL1(&&(>(x, 0), >(x, y)), x, y) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[2], 0), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[2], 0)=TRUE & >(x[2], y[2])=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + [-1]y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + [-1]y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + [-1]y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.34/2.45 5.34/2.45 (6) (x[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 (7) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + [-1]y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(y, y) the following chains were created: 5.34/2.45 *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (y[3]=x[0] & y[3]=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(y[3], y[3]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(y[3], y[3]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (6) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (y[3]=x[2] & y[3]=y[2] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(y[3], y[3]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(y[3], y[3]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (6) ((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 To summarize, we get the following constraints P__>=_ for the following pairs. 5.34/2.45 5.34/2.45 *EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >=(y, x)), x, y) 5.34/2.45 5.34/2.45 *(x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]y[0] + [bni_17]x[0] >= 0 & [(-1)bso_18] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) 5.34/2.45 5.34/2.45 *(x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 *(x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[0] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *EVAL(x, y) -> COND_EVAL1(&&(>(x, 0), >(x, y)), x, y) 5.34/2.45 5.34/2.45 *(x[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 *(x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[2] + [bni_21]x[2] >= 0 & [-1 + (-1)bso_22] + [-1]y[2] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *COND_EVAL1(TRUE, x, y) -> EVAL(y, y) 5.34/2.45 5.34/2.45 *((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 *((U^Increasing(EVAL(y[3], y[3])), >=) & [bni_23] = 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 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. 5.34/2.45 5.34/2.45 Using the following integer polynomial ordering the resulting constraints can be solved 5.34/2.45 5.34/2.45 Polynomial interpretation over integers[POLO]: 5.34/2.45 5.34/2.45 POL(TRUE) = 0 5.34/2.45 POL(FALSE) = [3] 5.34/2.45 POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 5.34/2.45 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 5.34/2.45 POL(&&(x_1, x_2)) = [-1] 5.34/2.45 POL(>(x_1, x_2)) = [-1] 5.34/2.45 POL(0) = 0 5.34/2.45 POL(>=(x_1, x_2)) = [-1] 5.34/2.45 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.34/2.45 POL(1) = [1] 5.34/2.45 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_1 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>: 5.34/2.45 5.34/2.45 5.34/2.45 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_bound: 5.34/2.45 5.34/2.45 5.34/2.45 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>=: 5.34/2.45 5.34/2.45 5.34/2.45 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) 5.34/2.45 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) 5.34/2.45 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 5.34/2.45 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.34/2.45 5.34/2.45 TRUE^1 -> &&(TRUE, TRUE)^1 5.34/2.45 FALSE^1 -> &&(TRUE, FALSE)^1 5.34/2.45 FALSE^1 -> &&(FALSE, TRUE)^1 5.34/2.45 FALSE^1 -> &&(FALSE, FALSE)^1 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (6) 5.34/2.45 Complex Obligation (AND) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (7) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && x[2] > y[2], x[2], y[2]) 5.34/2.45 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 (3) -> (0), if (y[3] ->^* x[0] & y[3] ->^* y[0]) 5.34/2.45 (3) -> (2), if (y[3] ->^* x[2] & y[3] ->^* y[2]) 5.34/2.45 (2) -> (3), if (x[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (8) IDependencyGraphProof (EQUIVALENT) 5.34/2.45 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (9) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 (2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] > 0 && x[2] > y[2], x[2], y[2]) 5.34/2.45 5.34/2.45 (3) -> (2), if (y[3] ->^* x[2] & y[3] ->^* y[2]) 5.34/2.45 (2) -> (3), if (x[2] > 0 && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (10) IDPNonInfProof (SOUND) 5.34/2.45 Used the following options for this NonInfProof: 5.34/2.45 5.34/2.45 IDPGPoloSolver: 5.34/2.45 Range: [(-1,2)] 5.34/2.45 IsNat: false 5.34/2.45 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@64fe5e2b 5.34/2.45 Constraint Generator: NonInfConstraintGenerator: 5.34/2.45 PathGenerator: MetricPathGenerator: 5.34/2.45 Max Left Steps: 1 5.34/2.45 Max Right Steps: 1 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 The constraints were generated the following way: 5.34/2.45 5.34/2.45 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.34/2.45 5.34/2.45 Note that final constraints are written in bold face. 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[2], 0), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & y[3]=x[2]1 & y[3]=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(y[3], y[3]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[2], 0)=TRUE & >(x[2], y[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(y[2], y[2]) & (U^Increasing(EVAL(y[3], y[3])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.34/2.45 5.34/2.45 (6) (x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 (7) (x[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) the following chains were created: 5.34/2.45 *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (y[3]=x[2] & y[3]=y[2] & &&(>(x[2], 0), >(x[2], y[2]))=TRUE & x[2]=x[3]1 & y[2]=y[3]1 ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[2], 0)=TRUE & >(x[2], x[2])=TRUE ==> EVAL(x[2], x[2])_>=_NonInfC & EVAL(x[2], x[2])_>=_COND_EVAL1(&&(>(x[2], 0), >(x[2], x[2])), x[2], x[2]) & (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[2] + [-1] >= 0 & [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]x[2] >= 0 & [2 + (-1)bso_15] + [-1]x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[2] + [-1] >= 0 & [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]x[2] >= 0 & [2 + (-1)bso_15] + [-1]x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[2] + [-1] >= 0 & [-1] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]x[2] >= 0 & [2 + (-1)bso_15] + [-1]x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We solved constraint (5) using rule (IDP_SMT_SPLIT). 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 To summarize, we get the following constraints P__>=_ for the following pairs. 5.34/2.45 5.34/2.45 *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 *(x[2] + [-1] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 *(x[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & y[2] >= 0 ==> (U^Increasing(EVAL(y[3], y[3])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[2] + [bni_12]x[2] >= 0 & [(-1)bso_13] + x[2] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 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. 5.34/2.45 5.34/2.45 Using the following integer polynomial ordering the resulting constraints can be solved 5.34/2.45 5.34/2.45 Polynomial interpretation over integers[POLO]: 5.34/2.45 5.34/2.45 POL(TRUE) = 0 5.34/2.45 POL(FALSE) = 0 5.34/2.45 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 + [2]x_1 5.34/2.45 POL(EVAL(x_1, x_2)) = [-1] + [-1]x_1 5.34/2.45 POL(&&(x_1, x_2)) = [-1] 5.34/2.45 POL(>(x_1, x_2)) = [-1] 5.34/2.45 POL(0) = 0 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>: 5.34/2.45 5.34/2.45 5.34/2.45 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_bound: 5.34/2.45 5.34/2.45 5.34/2.45 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(x[2], 0), >(x[2], y[2])), x[2], y[2]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>=: 5.34/2.45 5.34/2.45 none 5.34/2.45 5.34/2.45 5.34/2.45 There are no usable rules. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (11) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 none 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph is empty. 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (12) PisEmptyProof (EQUIVALENT) 5.34/2.45 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (13) 5.34/2.45 YES 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (14) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 5.34/2.45 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(y[3], y[3]) 5.34/2.45 5.34/2.45 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.34/2.45 (3) -> (0), if (y[3] ->^* x[0] & y[3] ->^* y[0]) 5.34/2.45 (0) -> (1), if (x[0] > 0 && y[0] >= x[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (15) IDependencyGraphProof (EQUIVALENT) 5.34/2.45 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (16) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Integer, Boolean 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1]) 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 5.34/2.45 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0]) 5.34/2.45 (0) -> (1), if (x[0] > 0 && y[0] >= x[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (17) IDPNonInfProof (SOUND) 5.34/2.45 Used the following options for this NonInfProof: 5.34/2.45 5.34/2.45 IDPGPoloSolver: 5.34/2.45 Range: [(-1,2)] 5.34/2.45 IsNat: false 5.34/2.45 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@64fe5e2b 5.34/2.45 Constraint Generator: NonInfConstraintGenerator: 5.34/2.45 PathGenerator: MetricPathGenerator: 5.34/2.45 Max Left Steps: 1 5.34/2.45 Max Right Steps: 1 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 The constraints were generated the following way: 5.34/2.45 5.34/2.45 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.34/2.45 5.34/2.45 Note that final constraints are written in bold face. 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[0], 0), >=(y[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[0], 0)=TRUE & >=(y[0], x[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 For Pair EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) the following chains were created: 5.34/2.45 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint: 5.34/2.45 5.34/2.45 (1) (&&(>(x[0], 0), >=(y[0], x[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.34/2.45 5.34/2.45 (2) (>(x[0], 0)=TRUE & >=(y[0], x[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=)) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.34/2.45 5.34/2.45 (3) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.34/2.45 5.34/2.45 (4) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.34/2.45 5.34/2.45 (5) (x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 To summarize, we get the following constraints P__>=_ for the following pairs. 5.34/2.45 5.34/2.45 *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 5.34/2.45 5.34/2.45 *(x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x[0] >= 0 & [1 + (-1)bso_14] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 *EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) 5.34/2.45 5.34/2.45 *(x[0] + [-1] >= 0 & y[0] + [-1]x[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [(-1)bso_16] >= 0) 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 5.34/2.45 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. 5.34/2.45 5.34/2.45 Using the following integer polynomial ordering the resulting constraints can be solved 5.34/2.45 5.34/2.45 Polynomial interpretation over integers[POLO]: 5.34/2.45 5.34/2.45 POL(TRUE) = 0 5.34/2.45 POL(FALSE) = [1] 5.34/2.45 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1 5.34/2.45 POL(EVAL(x_1, x_2)) = [-1] + x_1 5.34/2.45 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.34/2.45 POL(1) = [1] 5.34/2.45 POL(&&(x_1, x_2)) = 0 5.34/2.45 POL(>(x_1, x_2)) = [-1] 5.34/2.45 POL(0) = 0 5.34/2.45 POL(>=(x_1, x_2)) = [-1] 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>: 5.34/2.45 5.34/2.45 5.34/2.45 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_bound: 5.34/2.45 5.34/2.45 5.34/2.45 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) 5.34/2.45 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) 5.34/2.45 5.34/2.45 5.34/2.45 The following pairs are in P_>=: 5.34/2.45 5.34/2.45 5.34/2.45 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >=(y[0], x[0])), x[0], y[0]) 5.34/2.45 5.34/2.45 5.34/2.45 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.34/2.45 5.34/2.45 TRUE^1 -> &&(TRUE, TRUE)^1 5.34/2.45 FALSE^1 -> &&(TRUE, FALSE)^1 5.34/2.45 FALSE^1 -> &&(FALSE, TRUE)^1 5.34/2.45 FALSE^1 -> &&(FALSE, FALSE)^1 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (18) 5.34/2.45 Obligation: 5.34/2.45 IDP problem: 5.34/2.45 The following function symbols are pre-defined: 5.34/2.45 <<< 5.34/2.45 & ~ Bwand: (Integer, Integer) -> Integer 5.34/2.45 >= ~ Ge: (Integer, Integer) -> Boolean 5.34/2.45 | ~ Bwor: (Integer, Integer) -> Integer 5.34/2.45 / ~ Div: (Integer, Integer) -> Integer 5.34/2.45 != ~ Neq: (Integer, Integer) -> Boolean 5.34/2.45 && ~ Land: (Boolean, Boolean) -> Boolean 5.34/2.45 ! ~ Lnot: (Boolean) -> Boolean 5.34/2.45 = ~ Eq: (Integer, Integer) -> Boolean 5.34/2.45 <= ~ Le: (Integer, Integer) -> Boolean 5.34/2.45 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.34/2.45 % ~ Mod: (Integer, Integer) -> Integer 5.34/2.45 > ~ Gt: (Integer, Integer) -> Boolean 5.34/2.45 + ~ Add: (Integer, Integer) -> Integer 5.34/2.45 -1 ~ UnaryMinus: (Integer) -> Integer 5.34/2.45 < ~ Lt: (Integer, Integer) -> Boolean 5.34/2.45 || ~ Lor: (Boolean, Boolean) -> Boolean 5.34/2.45 - ~ Sub: (Integer, Integer) -> Integer 5.34/2.45 ~ ~ Bwnot: (Integer) -> Integer 5.34/2.45 * ~ Mul: (Integer, Integer) -> Integer 5.34/2.45 >>> 5.34/2.45 5.34/2.45 5.34/2.45 The following domains are used: 5.34/2.45 Boolean, Integer 5.34/2.45 5.34/2.45 R is empty. 5.34/2.45 5.34/2.45 The integer pair graph contains the following rules and edges: 5.34/2.45 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && y[0] >= x[0], x[0], y[0]) 5.34/2.45 5.34/2.45 5.34/2.45 The set Q consists of the following terms: 5.34/2.45 eval(x0, x1) 5.34/2.45 Cond_eval(TRUE, x0, x1) 5.34/2.45 Cond_eval1(TRUE, x0, x1) 5.34/2.45 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (19) IDependencyGraphProof (EQUIVALENT) 5.34/2.45 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.34/2.45 ---------------------------------------- 5.34/2.45 5.34/2.45 (20) 5.34/2.45 TRUE 5.34/2.48 EOF