7.26/2.94 YES 7.59/3.07 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 7.59/3.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.59/3.07 7.59/3.07 7.59/3.07 Termination of the given ITRS could be proven: 7.59/3.07 7.59/3.07 (0) ITRS 7.59/3.07 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 7.59/3.07 (2) IDP 7.59/3.07 (3) UsableRulesProof [EQUIVALENT, 0 ms] 7.59/3.07 (4) IDP 7.59/3.07 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.59/3.07 (6) IDP 7.59/3.07 (7) IDPNonInfProof [SOUND, 249 ms] 7.59/3.07 (8) IDP 7.59/3.07 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.59/3.07 (10) TRUE 7.59/3.07 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (0) 7.59/3.07 Obligation: 7.59/3.07 ITRS problem: 7.59/3.07 7.59/3.07 The following function symbols are pre-defined: 7.59/3.07 <<< 7.59/3.07 & ~ Bwand: (Integer, Integer) -> Integer 7.59/3.07 >= ~ Ge: (Integer, Integer) -> Boolean 7.59/3.07 | ~ Bwor: (Integer, Integer) -> Integer 7.59/3.07 / ~ Div: (Integer, Integer) -> Integer 7.59/3.07 != ~ Neq: (Integer, Integer) -> Boolean 7.59/3.07 && ~ Land: (Boolean, Boolean) -> Boolean 7.59/3.07 ! ~ Lnot: (Boolean) -> Boolean 7.59/3.07 = ~ Eq: (Integer, Integer) -> Boolean 7.59/3.07 <= ~ Le: (Integer, Integer) -> Boolean 7.59/3.07 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.59/3.07 % ~ Mod: (Integer, Integer) -> Integer 7.59/3.07 + ~ Add: (Integer, Integer) -> Integer 7.59/3.07 > ~ Gt: (Integer, Integer) -> Boolean 7.59/3.07 -1 ~ UnaryMinus: (Integer) -> Integer 7.59/3.07 < ~ Lt: (Integer, Integer) -> Boolean 7.59/3.07 || ~ Lor: (Boolean, Boolean) -> Boolean 7.59/3.07 - ~ Sub: (Integer, Integer) -> Integer 7.59/3.07 ~ ~ Bwnot: (Integer) -> Integer 7.59/3.07 * ~ Mul: (Integer, Integer) -> Integer 7.59/3.07 >>> 7.59/3.07 7.59/3.07 The TRS R consists of the following rules: 7.59/3.07 minus(x, y) -> minusNat(y >= 0 && x = y + 1, x, y) 7.59/3.07 minusNat(TRUE, x, y) -> minus(x, round(x)) 7.59/3.07 round(x) -> if(x % 2 = 0, x, x + 1) 7.59/3.07 if(TRUE, u, v) -> u 7.59/3.07 if(FALSE, u, v) -> v 7.59/3.07 The set Q consists of the following terms: 7.59/3.07 minus(x0, x1) 7.59/3.07 minusNat(TRUE, x0, x1) 7.59/3.07 round(x0) 7.59/3.07 if(TRUE, x0, x1) 7.59/3.07 if(FALSE, x0, x1) 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (1) ITRStoIDPProof (EQUIVALENT) 7.59/3.07 Added dependency pairs 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (2) 7.59/3.07 Obligation: 7.59/3.07 IDP problem: 7.59/3.07 The following function symbols are pre-defined: 7.59/3.07 <<< 7.59/3.07 & ~ Bwand: (Integer, Integer) -> Integer 7.59/3.07 >= ~ Ge: (Integer, Integer) -> Boolean 7.59/3.07 | ~ Bwor: (Integer, Integer) -> Integer 7.59/3.07 / ~ Div: (Integer, Integer) -> Integer 7.59/3.07 != ~ Neq: (Integer, Integer) -> Boolean 7.59/3.07 && ~ Land: (Boolean, Boolean) -> Boolean 7.59/3.07 ! ~ Lnot: (Boolean) -> Boolean 7.59/3.07 = ~ Eq: (Integer, Integer) -> Boolean 7.59/3.07 <= ~ Le: (Integer, Integer) -> Boolean 7.59/3.07 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.59/3.07 % ~ Mod: (Integer, Integer) -> Integer 7.59/3.07 + ~ Add: (Integer, Integer) -> Integer 7.59/3.07 > ~ Gt: (Integer, Integer) -> Boolean 7.59/3.07 -1 ~ UnaryMinus: (Integer) -> Integer 7.59/3.07 < ~ Lt: (Integer, Integer) -> Boolean 7.59/3.07 || ~ Lor: (Boolean, Boolean) -> Boolean 7.59/3.07 - ~ Sub: (Integer, Integer) -> Integer 7.59/3.07 ~ ~ Bwnot: (Integer) -> Integer 7.59/3.07 * ~ Mul: (Integer, Integer) -> Integer 7.59/3.07 >>> 7.59/3.07 7.59/3.07 7.59/3.07 The following domains are used: 7.59/3.07 Boolean, Integer 7.59/3.07 7.59/3.07 The ITRS R consists of the following rules: 7.59/3.07 minus(x, y) -> minusNat(y >= 0 && x = y + 1, x, y) 7.59/3.07 minusNat(TRUE, x, y) -> minus(x, round(x)) 7.59/3.07 round(x) -> if(x % 2 = 0, x, x + 1) 7.59/3.07 if(TRUE, u, v) -> u 7.59/3.07 if(FALSE, u, v) -> v 7.59/3.07 7.59/3.07 The integer pair graph contains the following rules and edges: 7.59/3.07 (0): MINUS(x[0], y[0]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) 7.59/3.07 (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 (2): MINUSNAT(TRUE, x[2], y[2]) -> ROUND(x[2]) 7.59/3.07 (3): ROUND(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] + 1) 7.59/3.07 7.59/3.07 (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) 7.59/3.07 (0) -> (2), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[2] & y[0] ->^* y[2]) 7.59/3.07 (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) 7.59/3.07 (2) -> (3), if (x[2] ->^* x[3]) 7.59/3.07 7.59/3.07 The set Q consists of the following terms: 7.59/3.07 minus(x0, x1) 7.59/3.07 minusNat(TRUE, x0, x1) 7.59/3.07 round(x0) 7.59/3.07 if(TRUE, x0, x1) 7.59/3.07 if(FALSE, x0, x1) 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (3) UsableRulesProof (EQUIVALENT) 7.59/3.07 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. 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (4) 7.59/3.07 Obligation: 7.59/3.07 IDP problem: 7.59/3.07 The following function symbols are pre-defined: 7.59/3.07 <<< 7.59/3.07 & ~ Bwand: (Integer, Integer) -> Integer 7.59/3.07 >= ~ Ge: (Integer, Integer) -> Boolean 7.59/3.07 | ~ Bwor: (Integer, Integer) -> Integer 7.59/3.07 / ~ Div: (Integer, Integer) -> Integer 7.59/3.07 != ~ Neq: (Integer, Integer) -> Boolean 7.59/3.07 && ~ Land: (Boolean, Boolean) -> Boolean 7.59/3.07 ! ~ Lnot: (Boolean) -> Boolean 7.59/3.07 = ~ Eq: (Integer, Integer) -> Boolean 7.59/3.07 <= ~ Le: (Integer, Integer) -> Boolean 7.59/3.07 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.59/3.07 % ~ Mod: (Integer, Integer) -> Integer 7.59/3.07 + ~ Add: (Integer, Integer) -> Integer 7.59/3.07 > ~ Gt: (Integer, Integer) -> Boolean 7.59/3.07 -1 ~ UnaryMinus: (Integer) -> Integer 7.59/3.07 < ~ Lt: (Integer, Integer) -> Boolean 7.59/3.07 || ~ Lor: (Boolean, Boolean) -> Boolean 7.59/3.07 - ~ Sub: (Integer, Integer) -> Integer 7.59/3.07 ~ ~ Bwnot: (Integer) -> Integer 7.59/3.07 * ~ Mul: (Integer, Integer) -> Integer 7.59/3.07 >>> 7.59/3.07 7.59/3.07 7.59/3.07 The following domains are used: 7.59/3.07 Integer, Boolean 7.59/3.07 7.59/3.07 The ITRS R consists of the following rules: 7.59/3.07 round(x) -> if(x % 2 = 0, x, x + 1) 7.59/3.07 if(TRUE, u, v) -> u 7.59/3.07 if(FALSE, u, v) -> v 7.59/3.07 7.59/3.07 The integer pair graph contains the following rules and edges: 7.59/3.07 (0): MINUS(x[0], y[0]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) 7.59/3.07 (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 (2): MINUSNAT(TRUE, x[2], y[2]) -> ROUND(x[2]) 7.59/3.07 (3): ROUND(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] + 1) 7.59/3.07 7.59/3.07 (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) 7.59/3.07 (0) -> (2), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[2] & y[0] ->^* y[2]) 7.59/3.07 (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) 7.59/3.07 (2) -> (3), if (x[2] ->^* x[3]) 7.59/3.07 7.59/3.07 The set Q consists of the following terms: 7.59/3.07 minus(x0, x1) 7.59/3.07 minusNat(TRUE, x0, x1) 7.59/3.07 round(x0) 7.59/3.07 if(TRUE, x0, x1) 7.59/3.07 if(FALSE, x0, x1) 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (5) IDependencyGraphProof (EQUIVALENT) 7.59/3.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (6) 7.59/3.07 Obligation: 7.59/3.07 IDP problem: 7.59/3.07 The following function symbols are pre-defined: 7.59/3.07 <<< 7.59/3.07 & ~ Bwand: (Integer, Integer) -> Integer 7.59/3.07 >= ~ Ge: (Integer, Integer) -> Boolean 7.59/3.07 | ~ Bwor: (Integer, Integer) -> Integer 7.59/3.07 / ~ Div: (Integer, Integer) -> Integer 7.59/3.07 != ~ Neq: (Integer, Integer) -> Boolean 7.59/3.07 && ~ Land: (Boolean, Boolean) -> Boolean 7.59/3.07 ! ~ Lnot: (Boolean) -> Boolean 7.59/3.07 = ~ Eq: (Integer, Integer) -> Boolean 7.59/3.07 <= ~ Le: (Integer, Integer) -> Boolean 7.59/3.07 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.59/3.07 % ~ Mod: (Integer, Integer) -> Integer 7.59/3.07 + ~ Add: (Integer, Integer) -> Integer 7.59/3.07 > ~ Gt: (Integer, Integer) -> Boolean 7.59/3.07 -1 ~ UnaryMinus: (Integer) -> Integer 7.59/3.07 < ~ Lt: (Integer, Integer) -> Boolean 7.59/3.07 || ~ Lor: (Boolean, Boolean) -> Boolean 7.59/3.07 - ~ Sub: (Integer, Integer) -> Integer 7.59/3.07 ~ ~ Bwnot: (Integer) -> Integer 7.59/3.07 * ~ Mul: (Integer, Integer) -> Integer 7.59/3.07 >>> 7.59/3.07 7.59/3.07 7.59/3.07 The following domains are used: 7.59/3.07 Integer, Boolean 7.59/3.07 7.59/3.07 The ITRS R consists of the following rules: 7.59/3.07 round(x) -> if(x % 2 = 0, x, x + 1) 7.59/3.07 if(TRUE, u, v) -> u 7.59/3.07 if(FALSE, u, v) -> v 7.59/3.07 7.59/3.07 The integer pair graph contains the following rules and edges: 7.59/3.07 (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 (0): MINUS(x[0], y[0]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) 7.59/3.07 7.59/3.07 (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) 7.59/3.07 (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) 7.59/3.07 7.59/3.07 The set Q consists of the following terms: 7.59/3.07 minus(x0, x1) 7.59/3.07 minusNat(TRUE, x0, x1) 7.59/3.07 round(x0) 7.59/3.07 if(TRUE, x0, x1) 7.59/3.07 if(FALSE, x0, x1) 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (7) IDPNonInfProof (SOUND) 7.59/3.07 Used the following options for this NonInfProof: 7.59/3.07 7.59/3.07 IDPGPoloSolver: 7.59/3.07 Range: [(-1,2)] 7.59/3.07 IsNat: false 7.59/3.07 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@7224f78 7.59/3.07 Constraint Generator: NonInfConstraintGenerator: 7.59/3.07 PathGenerator: MetricPathGenerator: 7.59/3.07 Max Left Steps: 2 7.59/3.07 Max Right Steps: 0 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 The constraints were generated the following way: 7.59/3.07 7.59/3.07 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 7.59/3.07 7.59/3.07 Note that final constraints are written in bold face. 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 For Pair MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) the following chains were created: 7.59/3.07 *We consider the chain MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])), MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]), MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) which results in the following constraint: 7.59/3.07 7.59/3.07 (1) (x[1]=x[0] & round(x[1])=y[0] & &&(>=(y[0], 0), =(x[0], +(y[0], 1)))=TRUE & x[0]=x[1]1 & y[0]=y[1]1 ==> MINUSNAT(TRUE, x[1]1, y[1]1)_>=_NonInfC & MINUSNAT(TRUE, x[1]1, y[1]1)_>=_MINUS(x[1]1, round(x[1]1)) & (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=)) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (1) using rules (III), (VII), (IDP_BOOLEAN), (REWRITING) which results in the following new constraint: 7.59/3.07 7.59/3.07 (2) (=(%(x[1], 2), 0)=x0 & +(x[1], 1)=x1 & if(x0, x[1], x1)=y[0] & >=(y[0], 0)=TRUE & >=(x[1], +(y[0], 1))=TRUE & <=(x[1], +(y[0], 1))=TRUE ==> MINUSNAT(TRUE, x[1], y[0])_>=_NonInfC & MINUSNAT(TRUE, x[1], y[0])_>=_MINUS(x[1], round(x[1])) & (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=)) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.59/3.07 7.59/3.07 (3) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.59/3.07 7.59/3.07 (4) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.59/3.07 7.59/3.07 (5) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.59/3.07 7.59/3.07 (6) ([-1] + x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.59/3.07 7.59/3.07 (7) (y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 For Pair MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) the following chains were created: 7.59/3.07 *We consider the chain MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]), MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) which results in the following constraint: 7.59/3.07 7.59/3.07 (1) (&&(>=(y[0], 0), =(x[0], +(y[0], 1)))=TRUE & x[0]=x[1] & y[0]=y[1] ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) & (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=)) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.59/3.07 7.59/3.07 (2) (>=(y[0], 0)=TRUE & >=(x[0], +(y[0], 1))=TRUE & <=(x[0], +(y[0], 1))=TRUE ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) & (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=)) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.59/3.07 7.59/3.07 (3) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.59/3.07 7.59/3.07 (4) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.59/3.07 7.59/3.07 (5) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 To summarize, we get the following constraints P__>=_ for the following pairs. 7.59/3.07 7.59/3.07 *MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 7.59/3.07 *(y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 *MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) 7.59/3.07 7.59/3.07 *(y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 7.59/3.07 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. 7.59/3.07 7.59/3.07 Using the following integer polynomial ordering the resulting constraints can be solved 7.59/3.07 7.59/3.07 Polynomial interpretation over integers[POLO]: 7.59/3.07 7.59/3.07 POL(TRUE) = 0 7.59/3.07 POL(FALSE) = 0 7.59/3.07 POL(round(x_1)) = x_1 7.59/3.07 POL(if(x_1, x_2, x_3)) = [-1]max{[-1]x_3, [-1]x_2} 7.59/3.07 POL(=(x_1, x_2)) = [-1] 7.59/3.07 POL(2) = [2] 7.59/3.07 POL(0) = 0 7.59/3.07 POL(+(x_1, x_2)) = x_1 + x_2 7.59/3.07 POL(1) = [1] 7.59/3.07 POL(MINUSNAT(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 7.59/3.07 POL(MINUS(x_1, x_2)) = [-1] + [-1]x_2 + x_1 7.59/3.07 POL(&&(x_1, x_2)) = [-1] 7.59/3.07 POL(>=(x_1, x_2)) = [-1] 7.59/3.07 7.59/3.07 7.59/3.07 The following pairs are in P_>: 7.59/3.07 7.59/3.07 7.59/3.07 MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 7.59/3.07 7.59/3.07 The following pairs are in P_bound: 7.59/3.07 7.59/3.07 7.59/3.07 MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) 7.59/3.07 MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) 7.59/3.07 7.59/3.07 7.59/3.07 The following pairs are in P_>=: 7.59/3.07 7.59/3.07 7.59/3.07 MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) 7.59/3.07 7.59/3.07 7.59/3.07 At least the following rules have been oriented under context sensitive arithmetic replacement: 7.59/3.07 7.59/3.07 if(=(%(x, 2), 0), x, +(x, 1))^1 -> round(x)^1 7.59/3.07 u^1 -> if(TRUE, u, v)^1 7.59/3.07 v^1 -> if(FALSE, u, v)^1 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (8) 7.59/3.07 Obligation: 7.59/3.07 IDP problem: 7.59/3.07 The following function symbols are pre-defined: 7.59/3.07 <<< 7.59/3.07 & ~ Bwand: (Integer, Integer) -> Integer 7.59/3.07 >= ~ Ge: (Integer, Integer) -> Boolean 7.59/3.07 | ~ Bwor: (Integer, Integer) -> Integer 7.59/3.07 / ~ Div: (Integer, Integer) -> Integer 7.59/3.07 != ~ Neq: (Integer, Integer) -> Boolean 7.59/3.07 && ~ Land: (Boolean, Boolean) -> Boolean 7.59/3.07 ! ~ Lnot: (Boolean) -> Boolean 7.59/3.07 = ~ Eq: (Integer, Integer) -> Boolean 7.59/3.07 <= ~ Le: (Integer, Integer) -> Boolean 7.59/3.07 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.59/3.07 % ~ Mod: (Integer, Integer) -> Integer 7.59/3.07 + ~ Add: (Integer, Integer) -> Integer 7.59/3.07 > ~ Gt: (Integer, Integer) -> Boolean 7.59/3.07 -1 ~ UnaryMinus: (Integer) -> Integer 7.59/3.07 < ~ Lt: (Integer, Integer) -> Boolean 7.59/3.07 || ~ Lor: (Boolean, Boolean) -> Boolean 7.59/3.07 - ~ Sub: (Integer, Integer) -> Integer 7.59/3.07 ~ ~ Bwnot: (Integer) -> Integer 7.59/3.07 * ~ Mul: (Integer, Integer) -> Integer 7.59/3.07 >>> 7.59/3.07 7.59/3.07 7.59/3.07 The following domains are used: 7.59/3.07 Integer, Boolean 7.59/3.07 7.59/3.07 The ITRS R consists of the following rules: 7.59/3.07 round(x) -> if(x % 2 = 0, x, x + 1) 7.59/3.07 if(TRUE, u, v) -> u 7.59/3.07 if(FALSE, u, v) -> v 7.59/3.07 7.59/3.07 The integer pair graph contains the following rules and edges: 7.59/3.07 (0): MINUS(x[0], y[0]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) 7.59/3.07 7.59/3.07 7.59/3.07 The set Q consists of the following terms: 7.59/3.07 minus(x0, x1) 7.59/3.07 minusNat(TRUE, x0, x1) 7.59/3.07 round(x0) 7.59/3.07 if(TRUE, x0, x1) 7.59/3.07 if(FALSE, x0, x1) 7.59/3.07 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (9) IDependencyGraphProof (EQUIVALENT) 7.59/3.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 7.59/3.07 ---------------------------------------- 7.59/3.07 7.59/3.07 (10) 7.59/3.07 TRUE 7.59/3.10 EOF