4.12/2.00 YES 4.12/2.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.12/2.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.12/2.01 4.12/2.01 4.12/2.01 Termination of the given ITRS could be proven: 4.12/2.01 4.12/2.01 (0) ITRS 4.12/2.01 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.12/2.01 (2) IDP 4.12/2.01 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.12/2.01 (4) IDP 4.12/2.01 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.12/2.01 (6) AND 4.12/2.01 (7) IDP 4.12/2.01 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.12/2.01 (9) IDP 4.12/2.01 (10) IDPNonInfProof [SOUND, 130 ms] 4.12/2.01 (11) IDP 4.12/2.01 (12) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.12/2.01 (13) TRUE 4.12/2.01 (14) IDP 4.12/2.01 (15) IDPNonInfProof [SOUND, 56 ms] 4.12/2.01 (16) IDP 4.12/2.01 (17) PisEmptyProof [EQUIVALENT, 0 ms] 4.12/2.01 (18) YES 4.12/2.01 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (0) 4.12/2.01 Obligation: 4.12/2.01 ITRS problem: 4.12/2.01 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 The TRS R consists of the following rules: 4.12/2.01 cu(TRUE, x) -> cu(x < exp(10, 2), x + 1) 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (1) ITRStoIDPProof (EQUIVALENT) 4.12/2.01 Added dependency pairs 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (2) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 The ITRS R consists of the following rules: 4.12/2.01 cu(TRUE, x) -> cu(x < exp(10, 2), x + 1) 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (0): CU(TRUE, x[0]) -> CU(x[0] < exp(10, 2), x[0] + 1) 4.12/2.01 (1): CU(TRUE, x[1]) -> EXP(10, 2) 4.12/2.01 (2): EXP(x[2], y[2]) -> IF(y[2] > 0, x[2], y[2]) 4.12/2.01 (3): IF(TRUE, x[3], y[3]) -> EXP(x[3], y[3] - 1) 4.12/2.01 4.12/2.01 (0) -> (0), if (x[0] < exp(10, 2) & x[0] + 1 ->^* x[0]') 4.12/2.01 (0) -> (1), if (x[0] < exp(10, 2) & x[0] + 1 ->^* x[1]) 4.12/2.01 (1) -> (2), if (10 ->^* x[2] & 2 ->^* y[2]) 4.12/2.01 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.12/2.01 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (3) UsableRulesProof (EQUIVALENT) 4.12/2.01 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. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (4) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 The ITRS R consists of the following rules: 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (0): CU(TRUE, x[0]) -> CU(x[0] < exp(10, 2), x[0] + 1) 4.12/2.01 (1): CU(TRUE, x[1]) -> EXP(10, 2) 4.12/2.01 (2): EXP(x[2], y[2]) -> IF(y[2] > 0, x[2], y[2]) 4.12/2.01 (3): IF(TRUE, x[3], y[3]) -> EXP(x[3], y[3] - 1) 4.12/2.01 4.12/2.01 (0) -> (0), if (x[0] < exp(10, 2) & x[0] + 1 ->^* x[0]') 4.12/2.01 (0) -> (1), if (x[0] < exp(10, 2) & x[0] + 1 ->^* x[1]) 4.12/2.01 (1) -> (2), if (10 ->^* x[2] & 2 ->^* y[2]) 4.12/2.01 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.12/2.01 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (5) IDependencyGraphProof (EQUIVALENT) 4.12/2.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (6) 4.12/2.01 Complex Obligation (AND) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (7) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 The ITRS R consists of the following rules: 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (3): IF(TRUE, x[3], y[3]) -> EXP(x[3], y[3] - 1) 4.12/2.01 (2): EXP(x[2], y[2]) -> IF(y[2] > 0, x[2], y[2]) 4.12/2.01 4.12/2.01 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.12/2.01 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (8) UsableRulesProof (EQUIVALENT) 4.12/2.01 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. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (9) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 R is empty. 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (3): IF(TRUE, x[3], y[3]) -> EXP(x[3], y[3] - 1) 4.12/2.01 (2): EXP(x[2], y[2]) -> IF(y[2] > 0, x[2], y[2]) 4.12/2.01 4.12/2.01 (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2]) 4.12/2.01 (2) -> (3), if (y[2] > 0 & x[2] ->^* x[3] & y[2] ->^* y[3]) 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (10) IDPNonInfProof (SOUND) 4.12/2.01 Used the following options for this NonInfProof: 4.12/2.01 4.12/2.01 IDPGPoloSolver: 4.12/2.01 Range: [(-1,2)] 4.12/2.01 IsNat: false 4.12/2.01 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@d575d34 4.12/2.01 Constraint Generator: NonInfConstraintGenerator: 4.12/2.01 PathGenerator: MetricPathGenerator: 4.12/2.01 Max Left Steps: 1 4.12/2.01 Max Right Steps: 1 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 The constraints were generated the following way: 4.12/2.01 4.12/2.01 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.12/2.01 4.12/2.01 Note that final constraints are written in bold face. 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 For Pair IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)) the following chains were created: 4.12/2.01 *We consider the chain EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]), IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)), EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]) which results in the following constraint: 4.12/2.01 4.12/2.01 (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 ==> IF(TRUE, x[3], y[3])_>=_NonInfC & IF(TRUE, x[3], y[3])_>=_EXP(x[3], -(y[3], 1)) & (U^Increasing(EXP(x[3], -(y[3], 1))), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.12/2.01 4.12/2.01 (2) (>(y[2], 0)=TRUE ==> IF(TRUE, x[2], y[2])_>=_NonInfC & IF(TRUE, x[2], y[2])_>=_EXP(x[2], -(y[2], 1)) & (U^Increasing(EXP(x[3], -(y[3], 1))), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.12/2.01 4.12/2.01 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(EXP(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.12/2.01 4.12/2.01 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(EXP(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.12/2.01 4.12/2.01 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(EXP(x[3], -(y[3], 1))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.12/2.01 4.12/2.01 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(EXP(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 For Pair EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]) the following chains were created: 4.12/2.01 *We consider the chain EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]), IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)) which results in the following constraint: 4.12/2.01 4.12/2.01 (1) (>(y[2], 0)=TRUE & x[2]=x[3] & y[2]=y[3] ==> EXP(x[2], y[2])_>=_NonInfC & EXP(x[2], y[2])_>=_IF(>(y[2], 0), x[2], y[2]) & (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (1) using rule (IV) which results in the following new constraint: 4.12/2.01 4.12/2.01 (2) (>(y[2], 0)=TRUE ==> EXP(x[2], y[2])_>=_NonInfC & EXP(x[2], y[2])_>=_IF(>(y[2], 0), x[2], y[2]) & (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.12/2.01 4.12/2.01 (3) (y[2] + [-1] >= 0 ==> (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.12/2.01 4.12/2.01 (4) (y[2] + [-1] >= 0 ==> (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.12/2.01 4.12/2.01 (5) (y[2] + [-1] >= 0 ==> (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=) & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.12/2.01 4.12/2.01 (6) (y[2] + [-1] >= 0 ==> (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 To summarize, we get the following constraints P__>=_ for the following pairs. 4.12/2.01 4.12/2.01 *IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)) 4.12/2.01 4.12/2.01 *(y[2] + [-1] >= 0 ==> (U^Increasing(EXP(x[3], -(y[3], 1))), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]y[2] >= 0 & [(-1)bso_12] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 *EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]) 4.12/2.01 4.12/2.01 *(y[2] + [-1] >= 0 ==> (U^Increasing(IF(>(y[2], 0), x[2], y[2])), >=) & 0 = 0 & [bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] >= 0 & [2 + (-1)bso_14] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 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. 4.12/2.01 4.12/2.01 Using the following integer polynomial ordering the resulting constraints can be solved 4.12/2.01 4.12/2.01 Polynomial interpretation over integers[POLO]: 4.12/2.01 4.12/2.01 POL(TRUE) = 0 4.12/2.01 POL(FALSE) = 0 4.12/2.01 POL(IF(x_1, x_2, x_3)) = [-1] + [2]x_3 4.12/2.01 POL(EXP(x_1, x_2)) = [1] + [2]x_2 4.12/2.01 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.12/2.01 POL(1) = [1] 4.12/2.01 POL(>(x_1, x_2)) = [2] 4.12/2.01 POL(0) = 0 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_>: 4.12/2.01 4.12/2.01 4.12/2.01 EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]) 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_bound: 4.12/2.01 4.12/2.01 4.12/2.01 IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)) 4.12/2.01 EXP(x[2], y[2]) -> IF(>(y[2], 0), x[2], y[2]) 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_>=: 4.12/2.01 4.12/2.01 4.12/2.01 IF(TRUE, x[3], y[3]) -> EXP(x[3], -(y[3], 1)) 4.12/2.01 4.12/2.01 4.12/2.01 There are no usable rules. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (11) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 R is empty. 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (3): IF(TRUE, x[3], y[3]) -> EXP(x[3], y[3] - 1) 4.12/2.01 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (12) IDependencyGraphProof (EQUIVALENT) 4.12/2.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (13) 4.12/2.01 TRUE 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (14) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 The ITRS R consists of the following rules: 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 4.12/2.01 The integer pair graph contains the following rules and edges: 4.12/2.01 (0): CU(TRUE, x[0]) -> CU(x[0] < exp(10, 2), x[0] + 1) 4.12/2.01 4.12/2.01 (0) -> (0), if (x[0] < exp(10, 2) & x[0] + 1 ->^* x[0]') 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (15) IDPNonInfProof (SOUND) 4.12/2.01 Used the following options for this NonInfProof: 4.12/2.01 4.12/2.01 IDPGPoloSolver: 4.12/2.01 Range: [(-1,2)] 4.12/2.01 IsNat: false 4.12/2.01 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@d575d34 4.12/2.01 Constraint Generator: NonInfConstraintGenerator: 4.12/2.01 PathGenerator: MetricPathGenerator: 4.12/2.01 Max Left Steps: 1 4.12/2.01 Max Right Steps: 1 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 The constraints were generated the following way: 4.12/2.01 4.12/2.01 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.12/2.01 4.12/2.01 Note that final constraints are written in bold face. 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 For Pair CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)) the following chains were created: 4.12/2.01 *We consider the chain CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)), CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)), CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)) which results in the following constraint: 4.12/2.01 4.12/2.01 (1) (<(x[0], exp(10, 2))=TRUE & +(x[0], 1)=x[0]1 & <(x[0]1, exp(10, 2))=TRUE & +(x[0]1, 1)=x[0]2 ==> CU(TRUE, x[0]1)_>=_NonInfC & CU(TRUE, x[0]1)_>=_CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1)) & (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (1) using rules (III), (IV), (IDP_CONSTANT_FOLD), (REWRITING) which results in the following new constraint: 4.12/2.01 4.12/2.01 (2) (<(x[0], 100)=TRUE & <(+(x[0], 1), 100)=TRUE ==> CU(TRUE, +(x[0], 1))_>=_NonInfC & CU(TRUE, +(x[0], 1))_>=_CU(<(+(x[0], 1), exp(10, 2)), +(+(x[0], 1), 1)) & (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=)) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.12/2.01 4.12/2.01 (3) ([99] + [-1]x[0] >= 0 & [98] + [-1]x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.12/2.01 4.12/2.01 (4) ([99] + [-1]x[0] >= 0 & [98] + [-1]x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.12/2.01 4.12/2.01 (5) ([99] + [-1]x[0] >= 0 & [98] + [-1]x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.12/2.01 4.12/2.01 (6) ([99] + x[0] >= 0 & [98] + x[0] >= 0 & x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 (7) ([99] + [-1]x[0] >= 0 & [98] + [-1]x[0] >= 0 & x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 To summarize, we get the following constraints P__>=_ for the following pairs. 4.12/2.01 4.12/2.01 *CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)) 4.12/2.01 4.12/2.01 *([99] + x[0] >= 0 & [98] + x[0] >= 0 & x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 *([99] + [-1]x[0] >= 0 & [98] + [-1]x[0] >= 0 & x[0] >= 0 ==> (U^Increasing(CU(<(x[0]1, exp(10, 2)), +(x[0]1, 1))), >=) & [bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x[0] >= 0 & [1 + (-1)bso_16] >= 0) 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 4.12/2.01 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. 4.12/2.01 4.12/2.01 Using the following integer polynomial ordering the resulting constraints can be solved 4.12/2.01 4.12/2.01 Polynomial interpretation over integers[POLO]: 4.12/2.01 4.12/2.01 POL(TRUE) = [1] 4.12/2.01 POL(FALSE) = 0 4.12/2.01 POL(exp(x_1, x_2)) = [-1] + [-1]x_2 + [-1]x_1 4.12/2.01 POL(if(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 4.12/2.01 POL(>(x_1, x_2)) = [-1] 4.12/2.01 POL(0) = 0 4.12/2.01 POL(*(x_1, x_2)) = x_1*x_2 4.12/2.01 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.12/2.01 POL(1) = [1] 4.12/2.01 POL(CU(x_1, x_2)) = [2] + [-1]x_2 4.12/2.01 POL(<(x_1, x_2)) = [-1] 4.12/2.01 POL(10) = [10] 4.12/2.01 POL(2) = [2] 4.12/2.01 POL(+(x_1, x_2)) = x_1 + x_2 4.12/2.01 POL(100) = [100] 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_>: 4.12/2.01 4.12/2.01 4.12/2.01 CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)) 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_bound: 4.12/2.01 4.12/2.01 4.12/2.01 CU(TRUE, x[0]) -> CU(<(x[0], exp(10, 2)), +(x[0], 1)) 4.12/2.01 4.12/2.01 4.12/2.01 The following pairs are in P_>=: 4.12/2.01 4.12/2.01 none 4.12/2.01 4.12/2.01 4.12/2.01 There are no usable rules. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (16) 4.12/2.01 Obligation: 4.12/2.01 IDP problem: 4.12/2.01 The following function symbols are pre-defined: 4.12/2.01 <<< 4.12/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.12/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.12/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.12/2.01 / ~ Div: (Integer, Integer) -> Integer 4.12/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.12/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.12/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.12/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.12/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.12/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.12/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.12/2.01 + ~ Add: (Integer, Integer) -> Integer 4.12/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.12/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.12/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.12/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.12/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.12/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.12/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.12/2.01 >>> 4.12/2.01 4.12/2.01 4.12/2.01 The following domains are used: 4.12/2.01 Integer 4.12/2.01 4.12/2.01 The ITRS R consists of the following rules: 4.12/2.01 exp(x, y) -> if(y > 0, x, y) 4.12/2.01 if(TRUE, x, y) -> x * exp(x, y - 1) 4.12/2.01 if(FALSE, x, y) -> 1 4.12/2.01 4.12/2.01 The integer pair graph is empty. 4.12/2.01 4.12/2.01 The set Q consists of the following terms: 4.12/2.01 cu(TRUE, x0) 4.12/2.01 exp(x0, x1) 4.12/2.01 if(TRUE, x0, x1) 4.12/2.01 if(FALSE, x0, x1) 4.12/2.01 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (17) PisEmptyProof (EQUIVALENT) 4.12/2.01 The TRS P is empty. Hence, there is no (P,Q,R) chain. 4.12/2.01 ---------------------------------------- 4.12/2.01 4.12/2.01 (18) 4.12/2.01 YES 4.12/2.03 EOF