5.50/3.14 YES 5.50/3.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 5.50/3.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.50/3.16 5.50/3.16 5.50/3.16 Termination of the given ITRS could be proven: 5.50/3.16 5.50/3.16 (0) ITRS 5.50/3.16 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.50/3.16 (2) IDP 5.50/3.16 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.50/3.16 (4) IDP 5.50/3.16 (5) IDPNonInfProof [SOUND, 282 ms] 5.50/3.16 (6) IDP 5.50/3.16 (7) IDPNonInfProof [SOUND, 79 ms] 5.50/3.16 (8) IDP 5.50/3.16 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.50/3.16 (10) TRUE 5.50/3.16 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (0) 5.50/3.16 Obligation: 5.50/3.16 ITRS problem: 5.50/3.16 5.50/3.16 The following function symbols are pre-defined: 5.50/3.16 <<< 5.50/3.16 & ~ Bwand: (Integer, Integer) -> Integer 5.50/3.16 >= ~ Ge: (Integer, Integer) -> Boolean 5.50/3.16 | ~ Bwor: (Integer, Integer) -> Integer 5.50/3.16 / ~ Div: (Integer, Integer) -> Integer 5.50/3.16 != ~ Neq: (Integer, Integer) -> Boolean 5.50/3.16 && ~ Land: (Boolean, Boolean) -> Boolean 5.50/3.16 ! ~ Lnot: (Boolean) -> Boolean 5.50/3.16 = ~ Eq: (Integer, Integer) -> Boolean 5.50/3.16 <= ~ Le: (Integer, Integer) -> Boolean 5.50/3.16 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.50/3.16 % ~ Mod: (Integer, Integer) -> Integer 5.50/3.16 > ~ Gt: (Integer, Integer) -> Boolean 5.50/3.16 + ~ Add: (Integer, Integer) -> Integer 5.50/3.16 -1 ~ UnaryMinus: (Integer) -> Integer 5.50/3.16 < ~ Lt: (Integer, Integer) -> Boolean 5.50/3.16 || ~ Lor: (Boolean, Boolean) -> Boolean 5.50/3.16 - ~ Sub: (Integer, Integer) -> Integer 5.50/3.16 ~ ~ Bwnot: (Integer) -> Integer 5.50/3.16 * ~ Mul: (Integer, Integer) -> Integer 5.50/3.16 >>> 5.50/3.16 5.50/3.16 The TRS R consists of the following rules: 5.50/3.16 f(TRUE, x, y, z) -> g(x > y, x, y, z) 5.50/3.16 g(TRUE, x, y, z) -> f(x > z, x, y + 1, z) 5.50/3.16 g(TRUE, x, y, z) -> f(x > z, x, y, z + 1) 5.50/3.16 The set Q consists of the following terms: 5.50/3.16 f(TRUE, x0, x1, x2) 5.50/3.16 g(TRUE, x0, x1, x2) 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (1) ITRStoIDPProof (EQUIVALENT) 5.50/3.16 Added dependency pairs 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (2) 5.50/3.16 Obligation: 5.50/3.16 IDP problem: 5.50/3.16 The following function symbols are pre-defined: 5.50/3.16 <<< 5.50/3.16 & ~ Bwand: (Integer, Integer) -> Integer 5.50/3.16 >= ~ Ge: (Integer, Integer) -> Boolean 5.50/3.16 | ~ Bwor: (Integer, Integer) -> Integer 5.50/3.16 / ~ Div: (Integer, Integer) -> Integer 5.50/3.16 != ~ Neq: (Integer, Integer) -> Boolean 5.50/3.16 && ~ Land: (Boolean, Boolean) -> Boolean 5.50/3.16 ! ~ Lnot: (Boolean) -> Boolean 5.50/3.16 = ~ Eq: (Integer, Integer) -> Boolean 5.50/3.16 <= ~ Le: (Integer, Integer) -> Boolean 5.50/3.16 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.50/3.16 % ~ Mod: (Integer, Integer) -> Integer 5.50/3.16 > ~ Gt: (Integer, Integer) -> Boolean 5.50/3.16 + ~ Add: (Integer, Integer) -> Integer 5.50/3.16 -1 ~ UnaryMinus: (Integer) -> Integer 5.50/3.16 < ~ Lt: (Integer, Integer) -> Boolean 5.50/3.16 || ~ Lor: (Boolean, Boolean) -> Boolean 5.50/3.16 - ~ Sub: (Integer, Integer) -> Integer 5.50/3.16 ~ ~ Bwnot: (Integer) -> Integer 5.50/3.16 * ~ Mul: (Integer, Integer) -> Integer 5.50/3.16 >>> 5.50/3.16 5.50/3.16 5.50/3.16 The following domains are used: 5.50/3.16 Integer 5.50/3.16 5.50/3.16 The ITRS R consists of the following rules: 5.50/3.16 f(TRUE, x, y, z) -> g(x > y, x, y, z) 5.50/3.16 g(TRUE, x, y, z) -> f(x > z, x, y + 1, z) 5.50/3.16 g(TRUE, x, y, z) -> f(x > z, x, y, z + 1) 5.50/3.16 5.50/3.16 The integer pair graph contains the following rules and edges: 5.50/3.16 (0): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) 5.50/3.16 (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) 5.50/3.16 (2): G(TRUE, x[2], y[2], z[2]) -> F(x[2] > z[2], x[2], y[2], z[2] + 1) 5.50/3.16 5.50/3.16 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.50/3.16 (0) -> (2), if (x[0] > y[0] & x[0] ->^* x[2] & y[0] ->^* y[2] & z[0] ->^* z[2]) 5.50/3.16 (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) 5.50/3.16 (2) -> (0), if (x[2] > z[2] & x[2] ->^* x[0] & y[2] ->^* y[0] & z[2] + 1 ->^* z[0]) 5.50/3.16 5.50/3.16 The set Q consists of the following terms: 5.50/3.16 f(TRUE, x0, x1, x2) 5.50/3.16 g(TRUE, x0, x1, x2) 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (3) UsableRulesProof (EQUIVALENT) 5.50/3.16 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. 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (4) 5.50/3.16 Obligation: 5.50/3.16 IDP problem: 5.50/3.16 The following function symbols are pre-defined: 5.50/3.16 <<< 5.50/3.16 & ~ Bwand: (Integer, Integer) -> Integer 5.50/3.16 >= ~ Ge: (Integer, Integer) -> Boolean 5.50/3.16 | ~ Bwor: (Integer, Integer) -> Integer 5.50/3.16 / ~ Div: (Integer, Integer) -> Integer 5.50/3.16 != ~ Neq: (Integer, Integer) -> Boolean 5.50/3.16 && ~ Land: (Boolean, Boolean) -> Boolean 5.50/3.16 ! ~ Lnot: (Boolean) -> Boolean 5.50/3.16 = ~ Eq: (Integer, Integer) -> Boolean 5.50/3.16 <= ~ Le: (Integer, Integer) -> Boolean 5.50/3.16 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.50/3.16 % ~ Mod: (Integer, Integer) -> Integer 5.50/3.16 > ~ Gt: (Integer, Integer) -> Boolean 5.50/3.16 + ~ Add: (Integer, Integer) -> Integer 5.50/3.16 -1 ~ UnaryMinus: (Integer) -> Integer 5.50/3.16 < ~ Lt: (Integer, Integer) -> Boolean 5.50/3.16 || ~ Lor: (Boolean, Boolean) -> Boolean 5.50/3.16 - ~ Sub: (Integer, Integer) -> Integer 5.50/3.16 ~ ~ Bwnot: (Integer) -> Integer 5.50/3.16 * ~ Mul: (Integer, Integer) -> Integer 5.50/3.16 >>> 5.50/3.16 5.50/3.16 5.50/3.16 The following domains are used: 5.50/3.16 Integer 5.50/3.16 5.50/3.16 R is empty. 5.50/3.16 5.50/3.16 The integer pair graph contains the following rules and edges: 5.50/3.16 (0): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) 5.50/3.16 (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) 5.50/3.16 (2): G(TRUE, x[2], y[2], z[2]) -> F(x[2] > z[2], x[2], y[2], z[2] + 1) 5.50/3.16 5.50/3.16 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.50/3.16 (0) -> (2), if (x[0] > y[0] & x[0] ->^* x[2] & y[0] ->^* y[2] & z[0] ->^* z[2]) 5.50/3.16 (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) 5.50/3.16 (2) -> (0), if (x[2] > z[2] & x[2] ->^* x[0] & y[2] ->^* y[0] & z[2] + 1 ->^* z[0]) 5.50/3.16 5.50/3.16 The set Q consists of the following terms: 5.50/3.16 f(TRUE, x0, x1, x2) 5.50/3.16 g(TRUE, x0, x1, x2) 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (5) IDPNonInfProof (SOUND) 5.50/3.16 Used the following options for this NonInfProof: 5.50/3.16 5.50/3.16 IDPGPoloSolver: 5.50/3.16 Range: [(-1,2)] 5.50/3.16 IsNat: false 5.50/3.16 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3f0e4678 5.50/3.16 Constraint Generator: NonInfConstraintGenerator: 5.50/3.16 PathGenerator: MetricPathGenerator: 5.50/3.16 Max Left Steps: 1 5.50/3.16 Max Right Steps: 1 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 The constraints were generated the following way: 5.50/3.16 5.50/3.16 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.50/3.16 5.50/3.16 Note that final constraints are written in bold face. 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 For Pair F(TRUE, x, y, z) -> G(>(x, y), x, y, z) the following chains were created: 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[2] & y[0]=y[2] & z[0]=z[2] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 For Pair G(TRUE, x, y, z) -> F(>(x, z), x, +(y, 1), z) the following chains were created: 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & >(x[1], z[1])=TRUE & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> G(TRUE, x[1], y[1], z[1])_>=_NonInfC & G(TRUE, x[1], y[1], z[1])_>=_F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE & >(x[0], z[1])=TRUE ==> G(TRUE, x[0], y[0], z[1])_>=_NonInfC & G(TRUE, x[0], y[0], z[1])_>=_F(>(x[0], z[1]), x[0], +(y[0], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [(-1)bni_16]z[1] + [bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 & z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 For Pair G(TRUE, x, y, z) -> F(>(x, z), x, y, +(z, 1)) the following chains were created: 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[2] & y[0]=y[2] & z[0]=z[2] & >(x[2], z[2])=TRUE & x[2]=x[0]1 & y[2]=y[0]1 & +(z[2], 1)=z[0]1 ==> G(TRUE, x[2], y[2], z[2])_>=_NonInfC & G(TRUE, x[2], y[2], z[2])_>=_F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) & (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE & >(x[0], z[2])=TRUE ==> G(TRUE, x[0], y[0], z[2])_>=_NonInfC & G(TRUE, x[0], y[0], z[2])_>=_F(>(x[0], z[2]), x[0], y[0], +(z[2], 1)) & (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [(-1)bni_18]z[2] + [bni_18]y[0] + [bni_18]x[0] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 & z[2] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 To summarize, we get the following constraints P__>=_ for the following pairs. 5.50/3.16 5.50/3.16 *F(TRUE, x, y, z) -> G(>(x, y), x, y, z) 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14] = 0 & [(-1)Bound*bni_14] + [bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 *G(TRUE, x, y, z) -> F(>(x, z), x, +(y, 1), z) 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]z[1] >= 0 & [(-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 *G(TRUE, x, y, z) -> F(>(x, z), x, y, +(z, 1)) 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[2] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[2], z[2]), x[2], y[2], +(z[2], 1))), >=) & [(-1)Bound*bni_18] + [bni_18]z[2] >= 0 & [1 + (-1)bso_19] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 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. 5.50/3.16 5.50/3.16 Using the following integer polynomial ordering the resulting constraints can be solved 5.50/3.16 5.50/3.16 Polynomial interpretation over integers[POLO]: 5.50/3.16 5.50/3.16 POL(TRUE) = 0 5.50/3.16 POL(FALSE) = 0 5.50/3.16 POL(F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.50/3.16 POL(G(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 5.50/3.16 POL(>(x_1, x_2)) = [-1] 5.50/3.16 POL(+(x_1, x_2)) = x_1 + x_2 5.50/3.16 POL(1) = [1] 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_>: 5.50/3.16 5.50/3.16 5.50/3.16 G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_bound: 5.50/3.16 5.50/3.16 5.50/3.16 G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) 5.50/3.16 G(TRUE, x[2], y[2], z[2]) -> F(>(x[2], z[2]), x[2], y[2], +(z[2], 1)) 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_>=: 5.50/3.16 5.50/3.16 5.50/3.16 F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) 5.50/3.16 G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) 5.50/3.16 5.50/3.16 5.50/3.16 There are no usable rules. 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (6) 5.50/3.16 Obligation: 5.50/3.16 IDP problem: 5.50/3.16 The following function symbols are pre-defined: 5.50/3.16 <<< 5.50/3.16 & ~ Bwand: (Integer, Integer) -> Integer 5.50/3.16 >= ~ Ge: (Integer, Integer) -> Boolean 5.50/3.16 | ~ Bwor: (Integer, Integer) -> Integer 5.50/3.16 / ~ Div: (Integer, Integer) -> Integer 5.50/3.16 != ~ Neq: (Integer, Integer) -> Boolean 5.50/3.16 && ~ Land: (Boolean, Boolean) -> Boolean 5.50/3.16 ! ~ Lnot: (Boolean) -> Boolean 5.50/3.16 = ~ Eq: (Integer, Integer) -> Boolean 5.50/3.16 <= ~ Le: (Integer, Integer) -> Boolean 5.50/3.16 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.50/3.16 % ~ Mod: (Integer, Integer) -> Integer 5.50/3.16 > ~ Gt: (Integer, Integer) -> Boolean 5.50/3.16 + ~ Add: (Integer, Integer) -> Integer 5.50/3.16 -1 ~ UnaryMinus: (Integer) -> Integer 5.50/3.16 < ~ Lt: (Integer, Integer) -> Boolean 5.50/3.16 || ~ Lor: (Boolean, Boolean) -> Boolean 5.50/3.16 - ~ Sub: (Integer, Integer) -> Integer 5.50/3.16 ~ ~ Bwnot: (Integer) -> Integer 5.50/3.16 * ~ Mul: (Integer, Integer) -> Integer 5.50/3.16 >>> 5.50/3.16 5.50/3.16 5.50/3.16 The following domains are used: 5.50/3.16 Integer 5.50/3.16 5.50/3.16 R is empty. 5.50/3.16 5.50/3.16 The integer pair graph contains the following rules and edges: 5.50/3.16 (0): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) 5.50/3.16 (1): G(TRUE, x[1], y[1], z[1]) -> F(x[1] > z[1], x[1], y[1] + 1, z[1]) 5.50/3.16 5.50/3.16 (1) -> (0), if (x[1] > z[1] & x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0]) 5.50/3.16 (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 5.50/3.16 5.50/3.16 The set Q consists of the following terms: 5.50/3.16 f(TRUE, x0, x1, x2) 5.50/3.16 g(TRUE, x0, x1, x2) 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (7) IDPNonInfProof (SOUND) 5.50/3.16 Used the following options for this NonInfProof: 5.50/3.16 5.50/3.16 IDPGPoloSolver: 5.50/3.16 Range: [(-1,2)] 5.50/3.16 IsNat: false 5.50/3.16 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3f0e4678 5.50/3.16 Constraint Generator: NonInfConstraintGenerator: 5.50/3.16 PathGenerator: MetricPathGenerator: 5.50/3.16 Max Left Steps: 1 5.50/3.16 Max Right Steps: 1 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 The constraints were generated the following way: 5.50/3.16 5.50/3.16 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.50/3.16 5.50/3.16 Note that final constraints are written in bold face. 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 For Pair F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) the following chains were created: 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rule (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE ==> F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_G(>(x[0], y[0]), x[0], y[0], z[0]) & (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 For Pair G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) the following chains were created: 5.50/3.16 *We consider the chain F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]), G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]), F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) which results in the following constraint: 5.50/3.16 5.50/3.16 (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & >(x[1], z[1])=TRUE & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1 ==> G(TRUE, x[1], y[1], z[1])_>=_NonInfC & G(TRUE, x[1], y[1], z[1])_>=_F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 5.50/3.16 5.50/3.16 (2) (>(x[0], y[0])=TRUE & >(x[0], z[1])=TRUE ==> G(TRUE, x[0], y[0], z[1])_>=_NonInfC & G(TRUE, x[0], y[0], z[1])_>=_F(>(x[0], z[1]), x[0], +(y[0], 1), z[1]) & (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=)) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.50/3.16 5.50/3.16 (3) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.50/3.16 5.50/3.16 (4) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.50/3.16 5.50/3.16 (5) (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (6) (x[0] >= 0 & y[0] + x[0] + [-1]z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.50/3.16 5.50/3.16 (7) (x[0] >= 0 & z[1] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.50/3.16 5.50/3.16 (8) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 (9) (x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 To summarize, we get the following constraints P__>=_ for the following pairs. 5.50/3.16 5.50/3.16 *F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(G(>(x[0], y[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 *G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 *(x[0] >= 0 & z[1] >= 0 & y[0] >= 0 ==> (U^Increasing(F(>(x[1], z[1]), x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 5.50/3.16 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. 5.50/3.16 5.50/3.16 Using the following integer polynomial ordering the resulting constraints can be solved 5.50/3.16 5.50/3.16 Polynomial interpretation over integers[POLO]: 5.50/3.16 5.50/3.16 POL(TRUE) = 0 5.50/3.16 POL(FALSE) = 0 5.50/3.16 POL(F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 5.50/3.16 POL(G(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_3 + x_2 5.50/3.16 POL(>(x_1, x_2)) = [-1] 5.50/3.16 POL(+(x_1, x_2)) = x_1 + x_2 5.50/3.16 POL(1) = [1] 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_>: 5.50/3.16 5.50/3.16 5.50/3.16 G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_bound: 5.50/3.16 5.50/3.16 5.50/3.16 F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) 5.50/3.16 G(TRUE, x[1], y[1], z[1]) -> F(>(x[1], z[1]), x[1], +(y[1], 1), z[1]) 5.50/3.16 5.50/3.16 5.50/3.16 The following pairs are in P_>=: 5.50/3.16 5.50/3.16 5.50/3.16 F(TRUE, x[0], y[0], z[0]) -> G(>(x[0], y[0]), x[0], y[0], z[0]) 5.50/3.16 5.50/3.16 5.50/3.16 There are no usable rules. 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (8) 5.50/3.16 Obligation: 5.50/3.16 IDP problem: 5.50/3.16 The following function symbols are pre-defined: 5.50/3.16 <<< 5.50/3.16 & ~ Bwand: (Integer, Integer) -> Integer 5.50/3.16 >= ~ Ge: (Integer, Integer) -> Boolean 5.50/3.16 | ~ Bwor: (Integer, Integer) -> Integer 5.50/3.16 / ~ Div: (Integer, Integer) -> Integer 5.50/3.16 != ~ Neq: (Integer, Integer) -> Boolean 5.50/3.16 && ~ Land: (Boolean, Boolean) -> Boolean 5.50/3.16 ! ~ Lnot: (Boolean) -> Boolean 5.50/3.16 = ~ Eq: (Integer, Integer) -> Boolean 5.50/3.16 <= ~ Le: (Integer, Integer) -> Boolean 5.50/3.16 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.50/3.16 % ~ Mod: (Integer, Integer) -> Integer 5.50/3.16 > ~ Gt: (Integer, Integer) -> Boolean 5.50/3.16 + ~ Add: (Integer, Integer) -> Integer 5.50/3.16 -1 ~ UnaryMinus: (Integer) -> Integer 5.50/3.16 < ~ Lt: (Integer, Integer) -> Boolean 5.50/3.16 || ~ Lor: (Boolean, Boolean) -> Boolean 5.50/3.16 - ~ Sub: (Integer, Integer) -> Integer 5.50/3.16 ~ ~ Bwnot: (Integer) -> Integer 5.50/3.16 * ~ Mul: (Integer, Integer) -> Integer 5.50/3.16 >>> 5.50/3.16 5.50/3.16 5.50/3.16 The following domains are used: 5.50/3.16 Integer 5.50/3.16 5.50/3.16 R is empty. 5.50/3.16 5.50/3.16 The integer pair graph contains the following rules and edges: 5.50/3.16 (0): F(TRUE, x[0], y[0], z[0]) -> G(x[0] > y[0], x[0], y[0], z[0]) 5.50/3.16 5.50/3.16 5.50/3.16 The set Q consists of the following terms: 5.50/3.16 f(TRUE, x0, x1, x2) 5.50/3.16 g(TRUE, x0, x1, x2) 5.50/3.16 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (9) IDependencyGraphProof (EQUIVALENT) 5.50/3.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.50/3.16 ---------------------------------------- 5.50/3.16 5.50/3.16 (10) 5.50/3.16 TRUE 5.67/3.18 EOF