4.66/2.07 YES 4.66/2.09 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.66/2.09 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.66/2.09 4.66/2.09 4.66/2.09 Termination of the given ITRS could be proven: 4.66/2.09 4.66/2.09 (0) ITRS 4.66/2.09 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.66/2.09 (2) IDP 4.66/2.09 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.66/2.09 (4) IDP 4.66/2.09 (5) IDPNonInfProof [SOUND, 214 ms] 4.66/2.09 (6) IDP 4.66/2.09 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.66/2.09 (8) IDP 4.66/2.09 (9) IDPNonInfProof [SOUND, 23 ms] 4.66/2.09 (10) IDP 4.66/2.09 (11) PisEmptyProof [EQUIVALENT, 0 ms] 4.66/2.09 (12) YES 4.66/2.09 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (0) 4.66/2.09 Obligation: 4.66/2.09 ITRS problem: 4.66/2.09 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 The TRS R consists of the following rules: 4.66/2.09 eval(x, y) -> Cond_eval(x >= 0, x, y, z) 4.66/2.09 Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) 4.66/2.09 eval(x, y) -> Cond_eval1(y >= 0, x, y) 4.66/2.09 Cond_eval1(TRUE, x, y) -> eval(x, y - 1) 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (1) ITRStoIDPProof (EQUIVALENT) 4.66/2.09 Added dependency pairs 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (2) 4.66/2.09 Obligation: 4.66/2.09 IDP problem: 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 4.66/2.09 The following domains are used: 4.66/2.09 Integer 4.66/2.09 4.66/2.09 The ITRS R consists of the following rules: 4.66/2.09 eval(x, y) -> Cond_eval(x >= 0, x, y, z) 4.66/2.09 Cond_eval(TRUE, x, y, z) -> eval(x - 1, z) 4.66/2.09 eval(x, y) -> Cond_eval1(y >= 0, x, y) 4.66/2.09 Cond_eval1(TRUE, x, y) -> eval(x, y - 1) 4.66/2.09 4.66/2.09 The integer pair graph contains the following rules and edges: 4.66/2.09 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) 4.66/2.09 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) 4.66/2.09 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) 4.66/2.09 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 4.66/2.09 4.66/2.09 (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 4.66/2.09 (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) 4.66/2.09 (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) 4.66/2.09 (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.66/2.09 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 4.66/2.09 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.66/2.09 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (3) UsableRulesProof (EQUIVALENT) 4.66/2.09 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. 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (4) 4.66/2.09 Obligation: 4.66/2.09 IDP problem: 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 4.66/2.09 The following domains are used: 4.66/2.09 Integer 4.66/2.09 4.66/2.09 R is empty. 4.66/2.09 4.66/2.09 The integer pair graph contains the following rules and edges: 4.66/2.09 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) 4.66/2.09 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, z[1]) 4.66/2.09 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) 4.66/2.09 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 4.66/2.09 4.66/2.09 (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 4.66/2.09 (1) -> (0), if (x[1] - 1 ->^* x[0] & z[1] ->^* y[0]) 4.66/2.09 (1) -> (2), if (x[1] - 1 ->^* x[2] & z[1] ->^* y[2]) 4.66/2.09 (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.66/2.09 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 4.66/2.09 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.66/2.09 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (5) IDPNonInfProof (SOUND) 4.66/2.09 Used the following options for this NonInfProof: 4.66/2.09 4.66/2.09 IDPGPoloSolver: 4.66/2.09 Range: [(-1,2)] 4.66/2.09 IsNat: false 4.66/2.09 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7bf91d93 4.66/2.09 Constraint Generator: NonInfConstraintGenerator: 4.66/2.09 PathGenerator: MetricPathGenerator: 4.66/2.09 Max Left Steps: 1 4.66/2.09 Max Right Steps: 1 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 The constraints were generated the following way: 4.66/2.09 4.66/2.09 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.66/2.09 4.66/2.09 Note that final constraints are written in bold face. 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair EVAL(x, y) -> COND_EVAL(>=(x, 0), x, y, z) the following chains were created: 4.66/2.09 *We consider the chain COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (-(x[1], 1)=x[0] & z[1]=y[0] & >=(x[0], 0)=TRUE & x[0]=x[1]1 & y[0]=y[1]1 & z[0]=z[1]1 ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(-(x[1], 1), 0)=TRUE ==> EVAL(-(x[1], 1), z[1])_>=_NonInfC & EVAL(-(x[1], 1), z[1])_>=_COND_EVAL(>=(-(x[1], 1), 0), -(x[1], 1), z[1], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *We consider the chain COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (x[3]=x[0] & -(y[3], 1)=y[0] & >=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(x[0], 0)=TRUE ==> EVAL(x[0], -(y[3], 1))_>=_NonInfC & EVAL(x[0], -(y[3], 1))_>=_COND_EVAL(>=(x[0], 0), x[0], -(y[3], 1), z[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) the following chains were created: 4.66/2.09 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & z[1]=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(x[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & z[1]=y[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), z[1]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(x[0], 0)=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), z[0]) & (U^Increasing(EVAL(-(x[1], 1), z[1])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair EVAL(x, y) -> COND_EVAL1(>=(y, 0), x, y) the following chains were created: 4.66/2.09 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rule (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(2)bni_20]x[2] >= 0 & [(-1)bso_21] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(2)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created: 4.66/2.09 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[0] & -(y[3], 1)=y[0] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x[2] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 To summarize, we get the following constraints P__>=_ for the following pairs. 4.66/2.09 4.66/2.09 *EVAL(x, y) -> COND_EVAL(>=(x, 0), x, y, z) 4.66/2.09 4.66/2.09 *(x[1] + [-1] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[1] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 *(x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x[0] >= 0 & 0 = 0 & [(-1)bso_17] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), z) 4.66/2.09 4.66/2.09 *(x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 *(x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), z[1])), >=) & 0 = 0 & 0 = 0 & [(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x[0] >= 0 & 0 = 0 & [2 + (-1)bso_19] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *EVAL(x, y) -> COND_EVAL1(>=(y, 0), x, y) 4.66/2.09 4.66/2.09 *(y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(2)bni_20] = 0 & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) 4.66/2.09 4.66/2.09 *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(2)bni_22] = 0 & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 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. 4.66/2.09 4.66/2.09 Using the following integer polynomial ordering the resulting constraints can be solved 4.66/2.09 4.66/2.09 Polynomial interpretation over integers[POLO]: 4.66/2.09 4.66/2.09 POL(TRUE) = 0 4.66/2.09 POL(FALSE) = 0 4.66/2.09 POL(EVAL(x_1, x_2)) = [-1] + [2]x_1 4.66/2.09 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [2]x_2 4.66/2.09 POL(>=(x_1, x_2)) = [-1] 4.66/2.09 POL(0) = 0 4.66/2.09 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.66/2.09 POL(1) = [1] 4.66/2.09 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [2]x_2 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_>: 4.66/2.09 4.66/2.09 4.66/2.09 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_bound: 4.66/2.09 4.66/2.09 4.66/2.09 EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) 4.66/2.09 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), z[1]) 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_>=: 4.66/2.09 4.66/2.09 4.66/2.09 EVAL(x[0], y[0]) -> COND_EVAL(>=(x[0], 0), x[0], y[0], z[0]) 4.66/2.09 EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) 4.66/2.09 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 4.66/2.09 4.66/2.09 4.66/2.09 There are no usable rules. 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (6) 4.66/2.09 Obligation: 4.66/2.09 IDP problem: 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 4.66/2.09 The following domains are used: 4.66/2.09 Integer 4.66/2.09 4.66/2.09 R is empty. 4.66/2.09 4.66/2.09 The integer pair graph contains the following rules and edges: 4.66/2.09 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] >= 0, x[0], y[0], z[0]) 4.66/2.09 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) 4.66/2.09 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 4.66/2.09 4.66/2.09 (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0]) 4.66/2.09 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.66/2.09 (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.66/2.09 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (7) IDependencyGraphProof (EQUIVALENT) 4.66/2.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (8) 4.66/2.09 Obligation: 4.66/2.09 IDP problem: 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 4.66/2.09 The following domains are used: 4.66/2.09 Integer 4.66/2.09 4.66/2.09 R is empty. 4.66/2.09 4.66/2.09 The integer pair graph contains the following rules and edges: 4.66/2.09 (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1) 4.66/2.09 (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] >= 0, x[2], y[2]) 4.66/2.09 4.66/2.09 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.66/2.09 (2) -> (3), if (y[2] >= 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.66/2.09 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (9) IDPNonInfProof (SOUND) 4.66/2.09 Used the following options for this NonInfProof: 4.66/2.09 4.66/2.09 IDPGPoloSolver: 4.66/2.09 Range: [(-1,2)] 4.66/2.09 IsNat: false 4.66/2.09 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7bf91d93 4.66/2.09 Constraint Generator: NonInfConstraintGenerator: 4.66/2.09 PathGenerator: MetricPathGenerator: 4.66/2.09 Max Left Steps: 1 4.66/2.09 Max Right Steps: 1 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 The constraints were generated the following way: 4.66/2.09 4.66/2.09 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.66/2.09 4.66/2.09 Note that final constraints are written in bold face. 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) the following chains were created: 4.66/2.09 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(y[2], 0)=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 For Pair EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) the following chains were created: 4.66/2.09 *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint: 4.66/2.09 4.66/2.09 (1) (>=(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (1) using rule (IV) which results in the following new constraint: 4.66/2.09 4.66/2.09 (2) (>=(y[2], 0)=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(>=(y[2], 0), x[2], y[2]) & (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=)) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (3) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.66/2.09 4.66/2.09 (4) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.66/2.09 4.66/2.09 (5) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.66/2.09 4.66/2.09 (6) (y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 To summarize, we get the following constraints P__>=_ for the following pairs. 4.66/2.09 4.66/2.09 *COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 4.66/2.09 4.66/2.09 *(y[2] >= 0 ==> (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [1 + (-1)bso_12] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 *EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) 4.66/2.09 4.66/2.09 *(y[2] >= 0 ==> (U^Increasing(COND_EVAL1(>=(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [(-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [1 + (-1)bso_14] >= 0) 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 4.66/2.09 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. 4.66/2.09 4.66/2.09 Using the following integer polynomial ordering the resulting constraints can be solved 4.66/2.09 4.66/2.09 Polynomial interpretation over integers[POLO]: 4.66/2.09 4.66/2.09 POL(TRUE) = 0 4.66/2.09 POL(FALSE) = 0 4.66/2.09 POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [2]x_3 4.66/2.09 POL(EVAL(x_1, x_2)) = [2]x_2 4.66/2.09 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.66/2.09 POL(1) = [1] 4.66/2.09 POL(>=(x_1, x_2)) = [-1] 4.66/2.09 POL(0) = 0 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_>: 4.66/2.09 4.66/2.09 4.66/2.09 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 4.66/2.09 EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_bound: 4.66/2.09 4.66/2.09 4.66/2.09 COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) 4.66/2.09 EVAL(x[2], y[2]) -> COND_EVAL1(>=(y[2], 0), x[2], y[2]) 4.66/2.09 4.66/2.09 4.66/2.09 The following pairs are in P_>=: 4.66/2.09 4.66/2.09 none 4.66/2.09 4.66/2.09 4.66/2.09 There are no usable rules. 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (10) 4.66/2.09 Obligation: 4.66/2.09 IDP problem: 4.66/2.09 The following function symbols are pre-defined: 4.66/2.09 <<< 4.66/2.09 & ~ Bwand: (Integer, Integer) -> Integer 4.66/2.09 >= ~ Ge: (Integer, Integer) -> Boolean 4.66/2.09 | ~ Bwor: (Integer, Integer) -> Integer 4.66/2.09 / ~ Div: (Integer, Integer) -> Integer 4.66/2.09 != ~ Neq: (Integer, Integer) -> Boolean 4.66/2.09 && ~ Land: (Boolean, Boolean) -> Boolean 4.66/2.09 ! ~ Lnot: (Boolean) -> Boolean 4.66/2.09 = ~ Eq: (Integer, Integer) -> Boolean 4.66/2.09 <= ~ Le: (Integer, Integer) -> Boolean 4.66/2.09 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.66/2.09 % ~ Mod: (Integer, Integer) -> Integer 4.66/2.09 + ~ Add: (Integer, Integer) -> Integer 4.66/2.09 > ~ Gt: (Integer, Integer) -> Boolean 4.66/2.09 -1 ~ UnaryMinus: (Integer) -> Integer 4.66/2.09 < ~ Lt: (Integer, Integer) -> Boolean 4.66/2.09 || ~ Lor: (Boolean, Boolean) -> Boolean 4.66/2.09 - ~ Sub: (Integer, Integer) -> Integer 4.66/2.09 ~ ~ Bwnot: (Integer) -> Integer 4.66/2.09 * ~ Mul: (Integer, Integer) -> Integer 4.66/2.09 >>> 4.66/2.09 4.66/2.09 4.66/2.09 The following domains are used: 4.66/2.09 none 4.66/2.09 4.66/2.09 R is empty. 4.66/2.09 4.66/2.09 The integer pair graph is empty. 4.66/2.09 4.66/2.09 The set Q consists of the following terms: 4.66/2.09 eval(x0, x1) 4.66/2.09 Cond_eval(TRUE, x0, x1, x2) 4.66/2.09 Cond_eval1(TRUE, x0, x1) 4.66/2.09 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (11) PisEmptyProof (EQUIVALENT) 4.66/2.09 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.66/2.09 ---------------------------------------- 4.66/2.09 4.66/2.09 (12) 4.66/2.09 YES 4.95/2.14 EOF