4.10/2.02	YES
4.10/2.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
4.10/2.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.10/2.03	
4.10/2.03	
4.10/2.03	Termination of the given ITRS could be proven:
4.10/2.03	
4.10/2.03	(0) ITRS
4.10/2.03	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
4.10/2.03	(2) IDP
4.10/2.03	(3) UsableRulesProof [EQUIVALENT, 0 ms]
4.10/2.03	(4) IDP
4.10/2.03	(5) IDPNonInfProof [SOUND, 203 ms]
4.10/2.03	(6) IDP
4.10/2.03	(7) IDependencyGraphProof [EQUIVALENT, 0 ms]
4.10/2.03	(8) TRUE
4.10/2.03	
4.10/2.03	
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(0)
4.10/2.03	Obligation:
4.10/2.03	ITRS problem:
4.10/2.03	
4.10/2.03	The following function symbols are pre-defined:
4.10/2.03	<<<
4.10/2.03	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.10/2.03	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.10/2.03	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.10/2.03	 / 	~ 	Div: (Integer, Integer) -> Integer
4.10/2.03	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.10/2.03	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.10/2.03	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.10/2.03	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.10/2.03	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.10/2.03	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.10/2.03	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.10/2.03	 + 	~ 	Add: (Integer, Integer) -> Integer
4.10/2.03	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.10/2.03	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.10/2.03	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.10/2.03	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.10/2.03	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.10/2.03	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.10/2.03	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.10/2.03	>>>
4.10/2.03	
4.10/2.03	The TRS R consists of the following rules:
4.10/2.03	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
4.10/2.03	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
4.10/2.03	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && 1 < sv14_14 && 0 < sv24_38 && 1 < sv23_37, sv14_14, sv23_37, sv24_38)
4.10/2.03	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
4.10/2.03	The set Q consists of the following terms:
4.10/2.03	b10(x0, x1, x2)
4.10/2.03	Cond_b14(TRUE, x0, x1, x2)
4.10/2.03	b14(x0, x1, x2)
4.10/2.03	b15(x0, x1, x2)
4.10/2.03	
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(1) ITRStoIDPProof (EQUIVALENT)
4.10/2.03	Added dependency pairs
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(2)
4.10/2.03	Obligation:
4.10/2.03	IDP problem:
4.10/2.03	The following function symbols are pre-defined:
4.10/2.03	<<<
4.10/2.03	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.10/2.03	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.10/2.03	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.10/2.03	 / 	~ 	Div: (Integer, Integer) -> Integer
4.10/2.03	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.10/2.03	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.10/2.03	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.10/2.03	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.10/2.03	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.10/2.03	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.10/2.03	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.10/2.03	 + 	~ 	Add: (Integer, Integer) -> Integer
4.10/2.03	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.10/2.03	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.10/2.03	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.10/2.03	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.10/2.03	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.10/2.03	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.10/2.03	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.10/2.03	>>>
4.10/2.03	
4.10/2.03	
4.10/2.03	The following domains are used:
4.10/2.03	   Boolean, Integer
4.10/2.03	
4.10/2.03	The ITRS R consists of the following rules:
4.10/2.03	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
4.10/2.03	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
4.10/2.03	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && 1 < sv14_14 && 0 < sv24_38 && 1 < sv23_37, sv14_14, sv23_37, sv24_38)
4.10/2.03	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
4.10/2.03	
4.10/2.03	The integer pair graph contains the following rules and edges:
4.10/2.03	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.10/2.03	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.10/2.03	(2): B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(sv23_37[2] >= sv14_14[2] && 1 < sv14_14[2] && 0 < sv24_38[2] && 1 < sv23_37[2], sv14_14[2], sv23_37[2], sv24_38[2])
4.10/2.03	(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], sv23_37[3] - sv14_14[3], sv24_38[3] + 1)
4.10/2.03	
4.10/2.03	   (0) -> (2), if (sv14_14[0] ->^* sv14_14[2] & sv23_37[0] ->^* sv23_37[2] & sv24_38[0] ->^* sv24_38[2])
4.10/2.03	   (1) -> (3), if (sv14_14[1] ->^* sv14_14[3] & sv23_37[1] ->^* sv23_37[3] & sv24_38[1] ->^* sv24_38[3])
4.10/2.03	   (2) -> (1), if (sv23_37[2] >= sv14_14[2] && 1 < sv14_14[2] && 0 < sv24_38[2] && 1 < sv23_37[2]  & sv14_14[2] ->^* sv14_14[1] & sv23_37[2] ->^* sv23_37[1] & sv24_38[2] ->^* sv24_38[1])
4.10/2.03	   (3) -> (0), if (sv14_14[3] ->^* sv14_14[0] & sv23_37[3] - sv14_14[3] ->^* sv23_37[0] & sv24_38[3] + 1 ->^* sv24_38[0])
4.10/2.03	
4.10/2.03	The set Q consists of the following terms:
4.10/2.03	b10(x0, x1, x2)
4.10/2.03	Cond_b14(TRUE, x0, x1, x2)
4.10/2.03	b14(x0, x1, x2)
4.10/2.03	b15(x0, x1, x2)
4.10/2.03	
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(3) UsableRulesProof (EQUIVALENT)
4.10/2.03	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.
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(4)
4.10/2.03	Obligation:
4.10/2.03	IDP problem:
4.10/2.03	The following function symbols are pre-defined:
4.10/2.03	<<<
4.10/2.03	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.10/2.03	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.10/2.03	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.10/2.03	 / 	~ 	Div: (Integer, Integer) -> Integer
4.10/2.03	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.10/2.03	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.10/2.03	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.10/2.03	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.10/2.03	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.10/2.03	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.10/2.03	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.10/2.03	 + 	~ 	Add: (Integer, Integer) -> Integer
4.10/2.03	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.10/2.03	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.10/2.03	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.10/2.03	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.10/2.03	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.10/2.03	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.10/2.03	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.10/2.03	>>>
4.10/2.03	
4.10/2.03	
4.10/2.03	The following domains are used:
4.10/2.03	   Boolean, Integer
4.10/2.03	
4.10/2.03	R is empty.
4.10/2.03	
4.10/2.03	The integer pair graph contains the following rules and edges:
4.10/2.03	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.10/2.03	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.10/2.03	(2): B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(sv23_37[2] >= sv14_14[2] && 1 < sv14_14[2] && 0 < sv24_38[2] && 1 < sv23_37[2], sv14_14[2], sv23_37[2], sv24_38[2])
4.10/2.03	(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], sv23_37[3] - sv14_14[3], sv24_38[3] + 1)
4.10/2.03	
4.10/2.03	   (0) -> (2), if (sv14_14[0] ->^* sv14_14[2] & sv23_37[0] ->^* sv23_37[2] & sv24_38[0] ->^* sv24_38[2])
4.10/2.03	   (1) -> (3), if (sv14_14[1] ->^* sv14_14[3] & sv23_37[1] ->^* sv23_37[3] & sv24_38[1] ->^* sv24_38[3])
4.10/2.03	   (2) -> (1), if (sv23_37[2] >= sv14_14[2] && 1 < sv14_14[2] && 0 < sv24_38[2] && 1 < sv23_37[2]  & sv14_14[2] ->^* sv14_14[1] & sv23_37[2] ->^* sv23_37[1] & sv24_38[2] ->^* sv24_38[1])
4.10/2.03	   (3) -> (0), if (sv14_14[3] ->^* sv14_14[0] & sv23_37[3] - sv14_14[3] ->^* sv23_37[0] & sv24_38[3] + 1 ->^* sv24_38[0])
4.10/2.03	
4.10/2.03	The set Q consists of the following terms:
4.10/2.03	b10(x0, x1, x2)
4.10/2.03	Cond_b14(TRUE, x0, x1, x2)
4.10/2.03	b14(x0, x1, x2)
4.10/2.03	b15(x0, x1, x2)
4.10/2.03	
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(5) IDPNonInfProof (SOUND)
4.10/2.03	Used the following options for this NonInfProof:
4.10/2.03	
4.10/2.03	IDPGPoloSolver:
4.10/2.03	Range: [(-1,2)]
4.10/2.03	IsNat: false
4.10/2.03	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6e4e9201
4.10/2.03	Constraint Generator: NonInfConstraintGenerator:
4.10/2.03	PathGenerator: MetricPathGenerator:
4.10/2.03	Max Left Steps: 1
4.10/2.03	Max Right Steps: 1
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	The constraints were generated the following way:
4.10/2.03	
4.10/2.03	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
4.10/2.03	
4.10/2.03	Note that final constraints are written in bold face.
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	For Pair B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38) the following chains were created:
4.10/2.03	*We consider the chain B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]), B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2]) which results in the following constraint:
4.10/2.03	
4.10/2.03	(1)    (sv14_14[0]=sv14_14[2] & sv23_37[0]=sv23_37[2] & sv24_38[0]=sv24_38[2]  ==>  B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_NonInfC & B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_B14(sv14_14[0], sv23_37[0], sv24_38[0]) & (U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (1) using rule (IV) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(2)    (B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_NonInfC & B10(sv14_14[0], sv23_37[0], sv24_38[0])_>=_B14(sv14_14[0], sv23_37[0], sv24_38[0]) & (U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(3)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(4)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(5)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	For Pair COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38) the following chains were created:
4.10/2.03	*We consider the chain COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1)) which results in the following constraint:
4.10/2.03	
4.10/2.03	(1)    (sv14_14[1]=sv14_14[3] & sv23_37[1]=sv23_37[3] & sv24_38[1]=sv24_38[3]  ==>  COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_B15(sv14_14[1], sv23_37[1], sv24_38[1]) & (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (1) using rule (IV) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(2)    (COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])_>=_B15(sv14_14[1], sv23_37[1], sv24_38[1]) & (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(3)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(4)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(5)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	For Pair B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(&&(&&(>=(sv23_37, sv14_14), <(1, sv14_14)), <(0, sv24_38)), <(1, sv23_37)), sv14_14, sv23_37, sv24_38) the following chains were created:
4.10/2.03	*We consider the chain B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2]), COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]) which results in the following constraint:
4.10/2.03	
4.10/2.03	(1)    (&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2]))=TRUE & sv14_14[2]=sv14_14[1] & sv23_37[2]=sv23_37[1] & sv24_38[2]=sv24_38[1]  ==>  B14(sv14_14[2], sv23_37[2], sv24_38[2])_>=_NonInfC & B14(sv14_14[2], sv23_37[2], sv24_38[2])_>=_COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2]) & (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(2)    (<(1, sv23_37[2])=TRUE & <(0, sv24_38[2])=TRUE & >=(sv23_37[2], sv14_14[2])=TRUE & <(1, sv14_14[2])=TRUE  ==>  B14(sv14_14[2], sv23_37[2], sv24_38[2])_>=_NonInfC & B14(sv14_14[2], sv23_37[2], sv24_38[2])_>=_COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2]) & (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(3)    (sv23_37[2] + [-2] >= 0 & sv24_38[2] + [-1] >= 0 & sv23_37[2] + [-1]sv14_14[2] >= 0 & sv14_14[2] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]sv23_37[2] >= 0 & [-1 + (-1)bso_28] + sv14_14[2] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(4)    (sv23_37[2] + [-2] >= 0 & sv24_38[2] + [-1] >= 0 & sv23_37[2] + [-1]sv14_14[2] >= 0 & sv14_14[2] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]sv23_37[2] >= 0 & [-1 + (-1)bso_28] + sv14_14[2] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(5)    (sv23_37[2] + [-2] >= 0 & sv24_38[2] + [-1] >= 0 & sv23_37[2] + [-1]sv14_14[2] >= 0 & sv14_14[2] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]sv23_37[2] >= 0 & [-1 + (-1)bso_28] + sv14_14[2] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	For Pair B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1)) the following chains were created:
4.10/2.03	*We consider the chain COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1)), B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0]) which results in the following constraint:
4.10/2.03	
4.10/2.03	(1)    (sv14_14[1]=sv14_14[3] & sv23_37[1]=sv23_37[3] & sv24_38[1]=sv24_38[3] & sv14_14[3]=sv14_14[0] & -(sv23_37[3], sv14_14[3])=sv23_37[0] & +(sv24_38[3], 1)=sv24_38[0]  ==>  B15(sv14_14[3], sv23_37[3], sv24_38[3])_>=_NonInfC & B15(sv14_14[3], sv23_37[3], sv24_38[3])_>=_B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1)) & (U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(2)    (B15(sv14_14[1], sv23_37[1], sv24_38[1])_>=_NonInfC & B15(sv14_14[1], sv23_37[1], sv24_38[1])_>=_B10(sv14_14[1], -(sv23_37[1], sv14_14[1]), +(sv24_38[1], 1)) & (U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=))
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(3)    ((U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=) & [bni_29] = 0 & [(-1)bso_30] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(4)    ((U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=) & [bni_29] = 0 & [(-1)bso_30] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(5)    ((U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=) & [bni_29] = 0 & [(-1)bso_30] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
4.10/2.03	
4.10/2.03	(6)    ((U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=) & [bni_29] = 0 & 0 = 0 & [(-1)bso_30] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	To summarize, we get the following constraints P__>=_ for the following pairs.
4.10/2.03	
4.10/2.03	*B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38)
4.10/2.03	
4.10/2.03	*((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	*COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38)
4.10/2.03	
4.10/2.03	*((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [(-1)bso_26] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	*B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(&&(&&(>=(sv23_37, sv14_14), <(1, sv14_14)), <(0, sv24_38)), <(1, sv23_37)), sv14_14, sv23_37, sv24_38)
4.10/2.03	
4.10/2.03	*(sv23_37[2] + [-2] >= 0 & sv24_38[2] + [-1] >= 0 & sv23_37[2] + [-1]sv14_14[2] >= 0 & sv14_14[2] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]sv23_37[2] >= 0 & [-1 + (-1)bso_28] + sv14_14[2] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	*B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1))
4.10/2.03	
4.10/2.03	*((U^Increasing(B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))), >=) & [bni_29] = 0 & 0 = 0 & [(-1)bso_30] >= 0)
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	
4.10/2.03	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. 
4.10/2.03	
4.10/2.03	Using the following integer polynomial ordering the  resulting constraints can be solved 
4.10/2.03	
4.10/2.03	Polynomial interpretation over integers[POLO]:
4.10/2.03	
4.10/2.03	   POL(TRUE) = 0
4.10/2.03	   POL(FALSE) = [1]
4.10/2.03	   POL(B10(x_1, x_2, x_3)) = [-1] + x_2
4.10/2.03	   POL(B14(x_1, x_2, x_3)) = [-1] + x_2
4.10/2.03	   POL(COND_B14(x_1, x_2, x_3, x_4)) = [-1] + x_3 + [-1]x_2 + [-1]x_1
4.10/2.03	   POL(B15(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1
4.10/2.03	   POL(&&(x_1, x_2)) = [-1]
4.10/2.03	   POL(>=(x_1, x_2)) = [-1]
4.10/2.03	   POL(<(x_1, x_2)) = [-1]
4.10/2.03	   POL(1) = [1]
4.10/2.03	   POL(0) = 0
4.10/2.03	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
4.10/2.03	   POL(+(x_1, x_2)) = x_1 + x_2
4.10/2.03	
4.10/2.03	
4.10/2.03	The following pairs are in P_>:
4.10/2.03	
4.10/2.03	
4.10/2.03	   B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])
4.10/2.03	
4.10/2.03	
4.10/2.03	The following pairs are in P_bound:
4.10/2.03	
4.10/2.03	
4.10/2.03	   B14(sv14_14[2], sv23_37[2], sv24_38[2]) -> COND_B14(&&(&&(&&(>=(sv23_37[2], sv14_14[2]), <(1, sv14_14[2])), <(0, sv24_38[2])), <(1, sv23_37[2])), sv14_14[2], sv23_37[2], sv24_38[2])
4.10/2.03	
4.10/2.03	
4.10/2.03	The following pairs are in P_>=:
4.10/2.03	
4.10/2.03	
4.10/2.03	   B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.10/2.03	   COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.10/2.03	   B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], -(sv23_37[3], sv14_14[3]), +(sv24_38[3], 1))
4.10/2.03	
4.10/2.03	
4.10/2.03	At least the following rules have been oriented under context sensitive arithmetic replacement:
4.10/2.03	
4.10/2.03	   TRUE^1 -> &&(TRUE, TRUE)^1
4.10/2.03	   FALSE^1 -> &&(TRUE, FALSE)^1
4.10/2.03	   FALSE^1 -> &&(FALSE, TRUE)^1
4.10/2.03	   FALSE^1 -> &&(FALSE, FALSE)^1
4.10/2.03	
4.10/2.03	----------------------------------------
4.10/2.03	
4.10/2.03	(6)
4.10/2.03	Obligation:
4.10/2.03	IDP problem:
4.10/2.03	The following function symbols are pre-defined:
4.10/2.03	<<<
4.10/2.03	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.10/2.03	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.10/2.03	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.10/2.04	 / 	~ 	Div: (Integer, Integer) -> Integer
4.10/2.04	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.10/2.04	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.10/2.04	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.10/2.04	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.10/2.04	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.10/2.04	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.10/2.04	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.10/2.04	 + 	~ 	Add: (Integer, Integer) -> Integer
4.10/2.04	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.10/2.04	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.10/2.04	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.10/2.04	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.10/2.04	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.10/2.04	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.10/2.04	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.10/2.04	>>>
4.10/2.04	
4.10/2.04	
4.10/2.04	The following domains are used:
4.10/2.04	   Integer
4.10/2.04	
4.10/2.04	R is empty.
4.10/2.04	
4.10/2.04	The integer pair graph contains the following rules and edges:
4.10/2.04	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.10/2.04	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.10/2.04	(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) -> B10(sv14_14[3], sv23_37[3] - sv14_14[3], sv24_38[3] + 1)
4.10/2.04	
4.10/2.04	   (3) -> (0), if (sv14_14[3] ->^* sv14_14[0] & sv23_37[3] - sv14_14[3] ->^* sv23_37[0] & sv24_38[3] + 1 ->^* sv24_38[0])
4.10/2.04	   (1) -> (3), if (sv14_14[1] ->^* sv14_14[3] & sv23_37[1] ->^* sv23_37[3] & sv24_38[1] ->^* sv24_38[3])
4.10/2.04	
4.10/2.04	The set Q consists of the following terms:
4.10/2.04	b10(x0, x1, x2)
4.10/2.04	Cond_b14(TRUE, x0, x1, x2)
4.10/2.04	b14(x0, x1, x2)
4.10/2.04	b15(x0, x1, x2)
4.10/2.04	
4.10/2.04	----------------------------------------
4.10/2.04	
4.10/2.04	(7) IDependencyGraphProof (EQUIVALENT)
4.10/2.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
4.10/2.04	----------------------------------------
4.10/2.04	
4.10/2.04	(8)
4.10/2.04	TRUE
4.10/2.05	EOF