7.94/3.10 YES 8.21/3.11 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 8.21/3.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.21/3.11 8.21/3.11 8.21/3.11 Termination of the given ITRS could be proven: 8.21/3.11 8.21/3.11 (0) ITRS 8.21/3.11 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 8.21/3.11 (2) IDP 8.21/3.11 (3) UsableRulesProof [EQUIVALENT, 0 ms] 8.21/3.11 (4) IDP 8.21/3.11 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 8.21/3.11 (6) IDP 8.21/3.11 (7) IDPNonInfProof [SOUND, 237 ms] 8.21/3.11 (8) IDP 8.21/3.11 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 8.21/3.11 (10) TRUE 8.21/3.11 8.21/3.11 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (0) 8.21/3.11 Obligation: 8.21/3.11 ITRS problem: 8.21/3.11 8.21/3.11 The following function symbols are pre-defined: 8.21/3.11 <<< 8.21/3.11 & ~ Bwand: (Integer, Integer) -> Integer 8.21/3.11 >= ~ Ge: (Integer, Integer) -> Boolean 8.21/3.11 | ~ Bwor: (Integer, Integer) -> Integer 8.21/3.11 / ~ Div: (Integer, Integer) -> Integer 8.21/3.11 != ~ Neq: (Integer, Integer) -> Boolean 8.21/3.11 && ~ Land: (Boolean, Boolean) -> Boolean 8.21/3.11 ! ~ Lnot: (Boolean) -> Boolean 8.21/3.11 = ~ Eq: (Integer, Integer) -> Boolean 8.21/3.11 <= ~ Le: (Integer, Integer) -> Boolean 8.21/3.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 8.21/3.11 % ~ Mod: (Integer, Integer) -> Integer 8.21/3.11 > ~ Gt: (Integer, Integer) -> Boolean 8.21/3.11 + ~ Add: (Integer, Integer) -> Integer 8.21/3.11 -1 ~ UnaryMinus: (Integer) -> Integer 8.21/3.11 < ~ Lt: (Integer, Integer) -> Boolean 8.21/3.11 || ~ Lor: (Boolean, Boolean) -> Boolean 8.21/3.11 - ~ Sub: (Integer, Integer) -> Integer 8.21/3.11 ~ ~ Bwnot: (Integer) -> Integer 8.21/3.11 * ~ Mul: (Integer, Integer) -> Integer 8.21/3.11 >>> 8.21/3.11 8.21/3.11 The TRS R consists of the following rules: 8.21/3.11 f(TRUE, x, y) -> fNat(x >= 0 && y >= 0, x, y) 8.21/3.11 fNat(TRUE, x, y) -> f(x > y, trunc(x), y + 1) 8.21/3.11 trunc(x) -> if(x % 2 = 0, x, x - 1) 8.21/3.11 if(TRUE, u, v) -> u 8.21/3.11 if(FALSE, u, v) -> v 8.21/3.11 The set Q consists of the following terms: 8.21/3.11 f(TRUE, x0, x1) 8.21/3.11 fNat(TRUE, x0, x1) 8.21/3.11 trunc(x0) 8.21/3.11 if(TRUE, x0, x1) 8.21/3.11 if(FALSE, x0, x1) 8.21/3.11 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (1) ITRStoIDPProof (EQUIVALENT) 8.21/3.11 Added dependency pairs 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (2) 8.21/3.11 Obligation: 8.21/3.11 IDP problem: 8.21/3.11 The following function symbols are pre-defined: 8.21/3.11 <<< 8.21/3.11 & ~ Bwand: (Integer, Integer) -> Integer 8.21/3.11 >= ~ Ge: (Integer, Integer) -> Boolean 8.21/3.11 | ~ Bwor: (Integer, Integer) -> Integer 8.21/3.11 / ~ Div: (Integer, Integer) -> Integer 8.21/3.11 != ~ Neq: (Integer, Integer) -> Boolean 8.21/3.11 && ~ Land: (Boolean, Boolean) -> Boolean 8.21/3.11 ! ~ Lnot: (Boolean) -> Boolean 8.21/3.11 = ~ Eq: (Integer, Integer) -> Boolean 8.21/3.11 <= ~ Le: (Integer, Integer) -> Boolean 8.21/3.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 8.21/3.11 % ~ Mod: (Integer, Integer) -> Integer 8.21/3.11 > ~ Gt: (Integer, Integer) -> Boolean 8.21/3.11 + ~ Add: (Integer, Integer) -> Integer 8.21/3.11 -1 ~ UnaryMinus: (Integer) -> Integer 8.21/3.11 < ~ Lt: (Integer, Integer) -> Boolean 8.21/3.11 || ~ Lor: (Boolean, Boolean) -> Boolean 8.21/3.11 - ~ Sub: (Integer, Integer) -> Integer 8.21/3.11 ~ ~ Bwnot: (Integer) -> Integer 8.21/3.11 * ~ Mul: (Integer, Integer) -> Integer 8.21/3.11 >>> 8.21/3.11 8.21/3.11 8.21/3.11 The following domains are used: 8.21/3.11 Boolean, Integer 8.21/3.11 8.21/3.11 The ITRS R consists of the following rules: 8.21/3.11 f(TRUE, x, y) -> fNat(x >= 0 && y >= 0, x, y) 8.21/3.11 fNat(TRUE, x, y) -> f(x > y, trunc(x), y + 1) 8.21/3.11 trunc(x) -> if(x % 2 = 0, x, x - 1) 8.21/3.11 if(TRUE, u, v) -> u 8.21/3.11 if(FALSE, u, v) -> v 8.21/3.11 8.21/3.11 The integer pair graph contains the following rules and edges: 8.21/3.11 (0): F(TRUE, x[0], y[0]) -> FNAT(x[0] >= 0 && y[0] >= 0, x[0], y[0]) 8.21/3.11 (1): FNAT(TRUE, x[1], y[1]) -> F(x[1] > y[1], trunc(x[1]), y[1] + 1) 8.21/3.11 (2): FNAT(TRUE, x[2], y[2]) -> TRUNC(x[2]) 8.21/3.11 (3): TRUNC(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] - 1) 8.21/3.11 8.21/3.11 (0) -> (1), if (x[0] >= 0 && y[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 8.21/3.11 (0) -> (2), if (x[0] >= 0 && y[0] >= 0 & x[0] ->^* x[2] & y[0] ->^* y[2]) 8.21/3.11 (1) -> (0), if (x[1] > y[1] & trunc(x[1]) ->^* x[0] & y[1] + 1 ->^* y[0]) 8.21/3.11 (2) -> (3), if (x[2] ->^* x[3]) 8.21/3.11 8.21/3.11 The set Q consists of the following terms: 8.21/3.11 f(TRUE, x0, x1) 8.21/3.11 fNat(TRUE, x0, x1) 8.21/3.11 trunc(x0) 8.21/3.11 if(TRUE, x0, x1) 8.21/3.11 if(FALSE, x0, x1) 8.21/3.11 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (3) UsableRulesProof (EQUIVALENT) 8.21/3.11 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. 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (4) 8.21/3.11 Obligation: 8.21/3.11 IDP problem: 8.21/3.11 The following function symbols are pre-defined: 8.21/3.11 <<< 8.21/3.11 & ~ Bwand: (Integer, Integer) -> Integer 8.21/3.11 >= ~ Ge: (Integer, Integer) -> Boolean 8.21/3.11 | ~ Bwor: (Integer, Integer) -> Integer 8.21/3.11 / ~ Div: (Integer, Integer) -> Integer 8.21/3.11 != ~ Neq: (Integer, Integer) -> Boolean 8.21/3.11 && ~ Land: (Boolean, Boolean) -> Boolean 8.21/3.11 ! ~ Lnot: (Boolean) -> Boolean 8.21/3.11 = ~ Eq: (Integer, Integer) -> Boolean 8.21/3.11 <= ~ Le: (Integer, Integer) -> Boolean 8.21/3.11 ^ ~ Bwxor: (Integer, Integer) -> Integer 8.21/3.11 % ~ Mod: (Integer, Integer) -> Integer 8.21/3.11 > ~ Gt: (Integer, Integer) -> Boolean 8.21/3.11 + ~ Add: (Integer, Integer) -> Integer 8.21/3.11 -1 ~ UnaryMinus: (Integer) -> Integer 8.21/3.11 < ~ Lt: (Integer, Integer) -> Boolean 8.21/3.11 || ~ Lor: (Boolean, Boolean) -> Boolean 8.21/3.11 - ~ Sub: (Integer, Integer) -> Integer 8.21/3.11 ~ ~ Bwnot: (Integer) -> Integer 8.21/3.11 * ~ Mul: (Integer, Integer) -> Integer 8.21/3.11 >>> 8.21/3.11 8.21/3.11 8.21/3.11 The following domains are used: 8.21/3.11 Integer, Boolean 8.21/3.11 8.21/3.11 The ITRS R consists of the following rules: 8.21/3.11 trunc(x) -> if(x % 2 = 0, x, x - 1) 8.21/3.11 if(TRUE, u, v) -> u 8.21/3.11 if(FALSE, u, v) -> v 8.21/3.11 8.21/3.11 The integer pair graph contains the following rules and edges: 8.21/3.11 (0): F(TRUE, x[0], y[0]) -> FNAT(x[0] >= 0 && y[0] >= 0, x[0], y[0]) 8.21/3.11 (1): FNAT(TRUE, x[1], y[1]) -> F(x[1] > y[1], trunc(x[1]), y[1] + 1) 8.21/3.11 (2): FNAT(TRUE, x[2], y[2]) -> TRUNC(x[2]) 8.21/3.11 (3): TRUNC(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] - 1) 8.21/3.11 8.21/3.11 (0) -> (1), if (x[0] >= 0 && y[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 8.21/3.11 (0) -> (2), if (x[0] >= 0 && y[0] >= 0 & x[0] ->^* x[2] & y[0] ->^* y[2]) 8.21/3.11 (1) -> (0), if (x[1] > y[1] & trunc(x[1]) ->^* x[0] & y[1] + 1 ->^* y[0]) 8.21/3.11 (2) -> (3), if (x[2] ->^* x[3]) 8.21/3.11 8.21/3.11 The set Q consists of the following terms: 8.21/3.11 f(TRUE, x0, x1) 8.21/3.11 fNat(TRUE, x0, x1) 8.21/3.11 trunc(x0) 8.21/3.11 if(TRUE, x0, x1) 8.21/3.11 if(FALSE, x0, x1) 8.21/3.11 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (5) IDependencyGraphProof (EQUIVALENT) 8.21/3.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 8.21/3.11 ---------------------------------------- 8.21/3.11 8.21/3.11 (6) 8.21/3.11 Obligation: 8.21/3.11 IDP problem: 8.21/3.11 The following function symbols are pre-defined: 8.21/3.11 <<< 8.21/3.11 & ~ Bwand: (Integer, Integer) -> Integer 8.21/3.11 >= ~ Ge: (Integer, Integer) -> Boolean 8.21/3.11 | ~ Bwor: (Integer, Integer) -> Integer 8.21/3.12 / ~ Div: (Integer, Integer) -> Integer 8.21/3.12 != ~ Neq: (Integer, Integer) -> Boolean 8.21/3.12 && ~ Land: (Boolean, Boolean) -> Boolean 8.21/3.12 ! ~ Lnot: (Boolean) -> Boolean 8.21/3.12 = ~ Eq: (Integer, Integer) -> Boolean 8.21/3.12 <= ~ Le: (Integer, Integer) -> Boolean 8.21/3.12 ^ ~ Bwxor: (Integer, Integer) -> Integer 8.21/3.12 % ~ Mod: (Integer, Integer) -> Integer 8.21/3.12 > ~ Gt: (Integer, Integer) -> Boolean 8.21/3.12 + ~ Add: (Integer, Integer) -> Integer 8.21/3.12 -1 ~ UnaryMinus: (Integer) -> Integer 8.21/3.12 < ~ Lt: (Integer, Integer) -> Boolean 8.21/3.12 || ~ Lor: (Boolean, Boolean) -> Boolean 8.21/3.12 - ~ Sub: (Integer, Integer) -> Integer 8.21/3.12 ~ ~ Bwnot: (Integer) -> Integer 8.21/3.12 * ~ Mul: (Integer, Integer) -> Integer 8.21/3.12 >>> 8.21/3.12 8.21/3.12 8.21/3.12 The following domains are used: 8.21/3.12 Integer, Boolean 8.21/3.12 8.21/3.12 The ITRS R consists of the following rules: 8.21/3.12 trunc(x) -> if(x % 2 = 0, x, x - 1) 8.21/3.12 if(TRUE, u, v) -> u 8.21/3.12 if(FALSE, u, v) -> v 8.21/3.12 8.21/3.12 The integer pair graph contains the following rules and edges: 8.21/3.12 (1): FNAT(TRUE, x[1], y[1]) -> F(x[1] > y[1], trunc(x[1]), y[1] + 1) 8.21/3.12 (0): F(TRUE, x[0], y[0]) -> FNAT(x[0] >= 0 && y[0] >= 0, x[0], y[0]) 8.21/3.12 8.21/3.12 (1) -> (0), if (x[1] > y[1] & trunc(x[1]) ->^* x[0] & y[1] + 1 ->^* y[0]) 8.21/3.12 (0) -> (1), if (x[0] >= 0 && y[0] >= 0 & x[0] ->^* x[1] & y[0] ->^* y[1]) 8.21/3.12 8.21/3.12 The set Q consists of the following terms: 8.21/3.12 f(TRUE, x0, x1) 8.21/3.12 fNat(TRUE, x0, x1) 8.21/3.12 trunc(x0) 8.21/3.12 if(TRUE, x0, x1) 8.21/3.12 if(FALSE, x0, x1) 8.21/3.12 8.21/3.12 ---------------------------------------- 8.21/3.12 8.21/3.12 (7) IDPNonInfProof (SOUND) 8.21/3.12 Used the following options for this NonInfProof: 8.21/3.12 8.21/3.12 IDPGPoloSolver: 8.21/3.12 Range: [(-1,2)] 8.21/3.12 IsNat: false 8.21/3.12 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@66ed5744 8.21/3.12 Constraint Generator: NonInfConstraintGenerator: 8.21/3.12 PathGenerator: MetricPathGenerator: 8.21/3.12 Max Left Steps: 2 8.21/3.12 Max Right Steps: 1 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 The constraints were generated the following way: 8.21/3.12 8.21/3.12 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 8.21/3.12 8.21/3.12 Note that final constraints are written in bold face. 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 For Pair FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)) the following chains were created: 8.21/3.12 *We consider the chain FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)), F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]), FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)), F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) which results in the following constraint: 8.21/3.12 8.21/3.12 (1) (>(x[1], y[1])=TRUE & trunc(x[1])=x[0] & +(y[1], 1)=y[0] & &&(>=(x[0], 0), >=(y[0], 0))=TRUE & x[0]=x[1]1 & y[0]=y[1]1 & >(x[1]1, y[1]1)=TRUE & trunc(x[1]1)=x[0]1 & +(y[1]1, 1)=y[0]1 ==> FNAT(TRUE, x[1]1, y[1]1)_>=_NonInfC & FNAT(TRUE, x[1]1, y[1]1)_>=_F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1)) & (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=)) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN), (REWRITING) which results in the following new constraint: 8.21/3.12 8.21/3.12 (2) (>(x[1], y[1])=TRUE & >(x[0], +(y[1], 1))=TRUE & if(=(%(x[1], 2), 0), x[1], -(x[1], 1))=x[0] & >=(x[0], 0)=TRUE & >=(+(y[1], 1), 0)=TRUE ==> FNAT(TRUE, x[0], +(y[1], 1))_>=_NonInfC & FNAT(TRUE, x[0], +(y[1], 1))_>=_F(>(x[0], +(y[1], 1)), trunc(x[0]), +(+(y[1], 1), 1)) & (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=)) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 8.21/3.12 8.21/3.12 (3) (x[1] + [-1] + [-1]y[1] >= 0 & x[0] + [-2] + [-1]y[1] >= 0 & x[0] >= 0 & y[1] + [1] >= 0 ==> (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=) & [(-2)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[1] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 8.21/3.12 8.21/3.12 (4) (x[1] + [-1] + [-1]y[1] >= 0 & x[0] + [-2] + [-1]y[1] >= 0 & x[0] >= 0 & y[1] + [1] >= 0 ==> (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=) & [(-2)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[1] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 8.21/3.12 8.21/3.12 (5) (x[1] + [-1] + [-1]y[1] >= 0 & x[0] + [-2] + [-1]y[1] >= 0 & x[0] >= 0 & y[1] + [1] >= 0 ==> (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=) & [(-2)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]y[1] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 8.21/3.12 8.21/3.12 (6) (y[1] >= 0 & x[0] + [-1] + [-1]x[1] + y[1] >= 0 & x[0] >= 0 & x[1] + [-1]y[1] >= 0 ==> (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x[1] + [bni_19]y[1] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 For Pair F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) the following chains were created: 8.21/3.12 *We consider the chain F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]), FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)) which results in the following constraint: 8.21/3.12 8.21/3.12 (1) (&&(>=(x[0], 0), >=(y[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] ==> F(TRUE, x[0], y[0])_>=_NonInfC & F(TRUE, x[0], y[0])_>=_FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) & (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=)) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 8.21/3.12 8.21/3.12 (2) (>=(x[0], 0)=TRUE & >=(y[0], 0)=TRUE ==> F(TRUE, x[0], y[0])_>=_NonInfC & F(TRUE, x[0], y[0])_>=_FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) & (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=)) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 8.21/3.12 8.21/3.12 (3) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 8.21/3.12 8.21/3.12 (4) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 8.21/3.12 8.21/3.12 (5) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 To summarize, we get the following constraints P__>=_ for the following pairs. 8.21/3.12 8.21/3.12 *FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)) 8.21/3.12 8.21/3.12 *(y[1] >= 0 & x[0] + [-1] + [-1]x[1] + y[1] >= 0 & x[0] >= 0 & x[1] + [-1]y[1] >= 0 ==> (U^Increasing(F(>(x[1]1, y[1]1), trunc(x[1]1), +(y[1]1, 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x[1] + [bni_19]y[1] + [bni_19]x[0] >= 0 & [1 + (-1)bso_20] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 *F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) 8.21/3.12 8.21/3.12 *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0) 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 8.21/3.12 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. 8.21/3.12 8.21/3.12 Using the following integer polynomial ordering the resulting constraints can be solved 8.21/3.12 8.21/3.12 Polynomial interpretation over integers[POLO]: 8.21/3.12 8.21/3.12 POL(TRUE) = 0 8.21/3.12 POL(FALSE) = 0 8.21/3.12 POL(trunc(x_1)) = x_1 8.21/3.12 POL(if(x_1, x_2, x_3)) = max{x_3, x_2} 8.21/3.12 POL(=(x_1, x_2)) = [-1] 8.21/3.12 POL(2) = [2] 8.21/3.12 POL(0) = 0 8.21/3.12 POL(-(x_1, x_2)) = x_1 + [-1]x_2 8.21/3.12 POL(1) = [1] 8.21/3.12 POL(FNAT(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 8.21/3.12 POL(F(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 8.21/3.12 POL(>(x_1, x_2)) = [-1] 8.21/3.12 POL(+(x_1, x_2)) = x_1 + x_2 8.21/3.12 POL(&&(x_1, x_2)) = [-1] 8.21/3.12 POL(>=(x_1, x_2)) = [-1] 8.21/3.12 8.21/3.12 8.21/3.12 The following pairs are in P_>: 8.21/3.12 8.21/3.12 8.21/3.12 FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)) 8.21/3.12 8.21/3.12 8.21/3.12 The following pairs are in P_bound: 8.21/3.12 8.21/3.12 8.21/3.12 FNAT(TRUE, x[1], y[1]) -> F(>(x[1], y[1]), trunc(x[1]), +(y[1], 1)) 8.21/3.12 8.21/3.12 8.21/3.12 The following pairs are in P_>=: 8.21/3.12 8.21/3.12 8.21/3.12 F(TRUE, x[0], y[0]) -> FNAT(&&(>=(x[0], 0), >=(y[0], 0)), x[0], y[0]) 8.21/3.12 8.21/3.12 8.21/3.12 At least the following rules have been oriented under context sensitive arithmetic replacement: 8.21/3.12 8.21/3.12 trunc(x)^1 -> if(=(%(x, 2), 0), x, -(x, 1))^1 8.21/3.12 if(TRUE, u, v)^1 -> u^1 8.21/3.12 if(FALSE, u, v)^1 -> v^1 8.21/3.12 8.21/3.12 ---------------------------------------- 8.21/3.12 8.21/3.12 (8) 8.21/3.12 Obligation: 8.21/3.12 IDP problem: 8.21/3.12 The following function symbols are pre-defined: 8.21/3.12 <<< 8.21/3.12 & ~ Bwand: (Integer, Integer) -> Integer 8.21/3.12 >= ~ Ge: (Integer, Integer) -> Boolean 8.21/3.12 | ~ Bwor: (Integer, Integer) -> Integer 8.21/3.12 / ~ Div: (Integer, Integer) -> Integer 8.21/3.12 != ~ Neq: (Integer, Integer) -> Boolean 8.21/3.12 && ~ Land: (Boolean, Boolean) -> Boolean 8.21/3.12 ! ~ Lnot: (Boolean) -> Boolean 8.21/3.12 = ~ Eq: (Integer, Integer) -> Boolean 8.21/3.12 <= ~ Le: (Integer, Integer) -> Boolean 8.21/3.12 ^ ~ Bwxor: (Integer, Integer) -> Integer 8.21/3.12 % ~ Mod: (Integer, Integer) -> Integer 8.21/3.12 > ~ Gt: (Integer, Integer) -> Boolean 8.21/3.12 + ~ Add: (Integer, Integer) -> Integer 8.21/3.12 -1 ~ UnaryMinus: (Integer) -> Integer 8.21/3.12 < ~ Lt: (Integer, Integer) -> Boolean 8.21/3.12 || ~ Lor: (Boolean, Boolean) -> Boolean 8.21/3.12 - ~ Sub: (Integer, Integer) -> Integer 8.21/3.12 ~ ~ Bwnot: (Integer) -> Integer 8.21/3.12 * ~ Mul: (Integer, Integer) -> Integer 8.21/3.12 >>> 8.21/3.12 8.21/3.12 8.21/3.12 The following domains are used: 8.21/3.12 Integer, Boolean 8.21/3.12 8.21/3.12 The ITRS R consists of the following rules: 8.21/3.12 trunc(x) -> if(x % 2 = 0, x, x - 1) 8.21/3.12 if(TRUE, u, v) -> u 8.21/3.12 if(FALSE, u, v) -> v 8.21/3.12 8.21/3.12 The integer pair graph contains the following rules and edges: 8.21/3.12 (0): F(TRUE, x[0], y[0]) -> FNAT(x[0] >= 0 && y[0] >= 0, x[0], y[0]) 8.21/3.12 8.21/3.12 8.21/3.12 The set Q consists of the following terms: 8.21/3.12 f(TRUE, x0, x1) 8.21/3.12 fNat(TRUE, x0, x1) 8.21/3.12 trunc(x0) 8.21/3.12 if(TRUE, x0, x1) 8.21/3.12 if(FALSE, x0, x1) 8.21/3.12 8.21/3.12 ---------------------------------------- 8.21/3.12 8.21/3.12 (9) IDependencyGraphProof (EQUIVALENT) 8.21/3.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 8.21/3.12 ---------------------------------------- 8.21/3.12 8.21/3.12 (10) 8.21/3.12 TRUE 8.28/3.17 EOF