4.04/1.89	YES
4.04/1.91	proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs
4.04/1.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.04/1.91	
4.04/1.91	
4.04/1.91	Termination of the given ITRS could be proven:
4.04/1.91	
4.04/1.91	(0) ITRS
4.04/1.91	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
4.04/1.91	(2) IDP
4.04/1.91	(3) UsableRulesProof [EQUIVALENT, 0 ms]
4.04/1.91	(4) IDP
4.04/1.91	(5) IDPNonInfProof [SOUND, 149 ms]
4.04/1.91	(6) AND
4.04/1.91	    (7) IDP
4.04/1.91	        (8) IDependencyGraphProof [EQUIVALENT, 0 ms]
4.04/1.91	        (9) TRUE
4.04/1.91	    (10) IDP
4.04/1.91	        (11) IDependencyGraphProof [EQUIVALENT, 0 ms]
4.04/1.91	        (12) TRUE
4.04/1.91	
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(0)
4.04/1.91	Obligation:
4.04/1.91	ITRS problem:
4.04/1.91	
4.04/1.91	The following function symbols are pre-defined:
4.04/1.91	<<<
4.04/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.04/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.04/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.04/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
4.04/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.04/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.04/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.04/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.04/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.04/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.04/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.04/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
4.04/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.04/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.04/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.04/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.04/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.04/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.04/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.04/1.91	>>>
4.04/1.91	
4.04/1.91	The TRS R consists of the following rules:
4.04/1.91	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
4.04/1.91	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
4.04/1.91	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
4.04/1.91	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && (1 < sv14_14 && (TRUE && (0 < sv24_38 && 1 < sv23_37))), sv14_14, sv23_37, sv24_38)
4.04/1.91	The set Q consists of the following terms:
4.04/1.91	b10(x0, x1, x2)
4.04/1.91	Cond_b14(TRUE, x0, x1, x2)
4.04/1.91	b15(x0, x1, x2)
4.04/1.91	b14(x0, x1, x2)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(1) ITRStoIDPProof (EQUIVALENT)
4.04/1.91	Added dependency pairs
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(2)
4.04/1.91	Obligation:
4.04/1.91	IDP problem:
4.04/1.91	The following function symbols are pre-defined:
4.04/1.91	<<<
4.04/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.04/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.04/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.04/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
4.04/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.04/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.04/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.04/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.04/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.04/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.04/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.04/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
4.04/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.04/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.04/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.04/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.04/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.04/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.04/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.04/1.91	>>>
4.04/1.91	
4.04/1.91	
4.04/1.91	The following domains are used:
4.04/1.91	   Integer, Boolean
4.04/1.91	
4.04/1.91	The ITRS R consists of the following rules:
4.04/1.91	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
4.04/1.91	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
4.04/1.91	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
4.04/1.91	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && (1 < sv14_14 && (TRUE && (0 < sv24_38 && 1 < sv23_37))), sv14_14, sv23_37, sv24_38)
4.04/1.91	
4.04/1.91	The integer pair graph contains the following rules and edges:
4.04/1.91	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.04/1.91	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.04/1.91	(2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1)
4.04/1.91	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))), sv14_14[3], sv23_37[3], sv24_38[3])
4.04/1.91	
4.04/1.91	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
4.04/1.91	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
4.04/1.91	   (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0])
4.04/1.91	   (3) -> (1), if (sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3])))  & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1])
4.04/1.91	
4.04/1.91	The set Q consists of the following terms:
4.04/1.91	b10(x0, x1, x2)
4.04/1.91	Cond_b14(TRUE, x0, x1, x2)
4.04/1.91	b15(x0, x1, x2)
4.04/1.91	b14(x0, x1, x2)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(3) UsableRulesProof (EQUIVALENT)
4.04/1.91	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.
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(4)
4.04/1.91	Obligation:
4.04/1.91	IDP problem:
4.04/1.91	The following function symbols are pre-defined:
4.04/1.91	<<<
4.04/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.04/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.04/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.04/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
4.04/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.04/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.04/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.04/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.04/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.04/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.04/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.04/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
4.04/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.04/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.04/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.04/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.04/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.04/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.04/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.04/1.91	>>>
4.04/1.91	
4.04/1.91	
4.04/1.91	The following domains are used:
4.04/1.91	   Integer, Boolean
4.04/1.91	
4.04/1.91	R is empty.
4.04/1.91	
4.04/1.91	The integer pair graph contains the following rules and edges:
4.04/1.91	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.04/1.91	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.04/1.91	(2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1)
4.04/1.91	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))), sv14_14[3], sv23_37[3], sv24_38[3])
4.04/1.91	
4.04/1.91	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
4.04/1.91	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
4.04/1.91	   (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0])
4.04/1.91	   (3) -> (1), if (sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3])))  & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1])
4.04/1.91	
4.04/1.91	The set Q consists of the following terms:
4.04/1.91	b10(x0, x1, x2)
4.04/1.91	Cond_b14(TRUE, x0, x1, x2)
4.04/1.91	b15(x0, x1, x2)
4.04/1.91	b14(x0, x1, x2)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(5) IDPNonInfProof (SOUND)
4.04/1.91	Used the following options for this NonInfProof:
4.04/1.91	
4.04/1.91	IDPGPoloSolver:
4.04/1.91	Range: [(-1,2)]
4.04/1.91	IsNat: false
4.04/1.91	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7881af86
4.04/1.91	Constraint Generator: NonInfConstraintGenerator:
4.04/1.91	PathGenerator: MetricPathGenerator:
4.04/1.91	Max Left Steps: 1
4.04/1.91	Max Right Steps: 1
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	The constraints were generated the following way:
4.04/1.91	
4.04/1.91	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
4.04/1.91	
4.04/1.91	Note that final constraints are written in bold face.
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	For Pair B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38) the following chains were created:
4.04/1.91	*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[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) which results in the following constraint:
4.04/1.91	
4.04/1.91	(1)    (sv14_14[0]=sv14_14[3] & sv23_37[0]=sv23_37[3] & sv24_38[0]=sv24_38[3]  ==>  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])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (1) using rule (IV) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(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])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(3)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(4)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(5)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	For Pair COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38) the following chains were created:
4.04/1.91	*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[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)) which results in the following constraint:
4.04/1.91	
4.04/1.91	(1)    (sv14_14[1]=sv14_14[2] & sv23_37[1]=sv23_37[2] & sv24_38[1]=sv24_38[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])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (1) using rule (IV) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(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])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(3)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(4)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(5)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	For Pair B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1)) the following chains were created:
4.04/1.91	*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[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 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:
4.04/1.91	
4.04/1.91	(1)    (sv14_14[1]=sv14_14[2] & sv23_37[1]=sv23_37[2] & sv24_38[1]=sv24_38[2] & sv14_14[2]=sv14_14[0] & -(sv23_37[2], sv14_14[2])=sv23_37[0] & +(sv24_38[2], 1)=sv24_38[0]  ==>  B15(sv14_14[2], sv23_37[2], sv24_38[2])_>=_NonInfC & B15(sv14_14[2], sv23_37[2], sv24_38[2])_>=_B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1)) & (U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(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[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(3)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(4)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(5)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(6)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & 0 = 0 & [(-1)bso_28] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	For Pair B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), &&(<(1, sv14_14), &&(TRUE, &&(<(0, sv24_38), <(1, sv23_37))))), sv14_14, sv23_37, sv24_38) the following chains were created:
4.04/1.91	*We consider the chain B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]), 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:
4.04/1.91	
4.04/1.91	(1)    (&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3])))))=TRUE & sv14_14[3]=sv14_14[1] & sv23_37[3]=sv23_37[1] & sv24_38[3]=sv24_38[1]  ==>  B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_NonInfC & B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (1) using rules (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(2)    (>=(sv23_37[3], sv14_14[3])=TRUE & <(1, sv14_14[3])=TRUE & <(0, sv24_38[3])=TRUE & <(1, sv23_37[3])=TRUE  ==>  B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_NonInfC & B14(sv14_14[3], sv23_37[3], sv24_38[3])_>=_COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=))
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(3)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(4)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.04/1.91	
4.04/1.91	(5)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	To summarize, we get the following constraints P__>=_ for the following pairs.
4.04/1.91	
4.04/1.91	*B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38)
4.04/1.91	
4.04/1.91	*((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	*COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38)
4.04/1.91	
4.04/1.91	*((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	*B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1))
4.04/1.91	
4.04/1.91	*((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & 0 = 0 & [(-1)bso_28] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	*B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), &&(<(1, sv14_14), &&(TRUE, &&(<(0, sv24_38), <(1, sv23_37))))), sv14_14, sv23_37, sv24_38)
4.04/1.91	
4.04/1.91	*(sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0 & sv24_38[3] + [-1] >= 0 & sv23_37[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	
4.04/1.91	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. 
4.04/1.91	
4.04/1.91	Using the following integer polynomial ordering the  resulting constraints can be solved 
4.04/1.91	
4.04/1.91	Polynomial interpretation over integers[POLO]:
4.04/1.91	
4.04/1.91	   POL(TRUE) = 0
4.04/1.91	   POL(FALSE) = [2]
4.04/1.91	   POL(B10(x_1, x_2, x_3)) = [-1] + x_2
4.04/1.91	   POL(B14(x_1, x_2, x_3)) = [-1] + x_2
4.04/1.91	   POL(COND_B14(x_1, x_2, x_3, x_4)) = x_3 + [-1]x_2 + [-1]x_1
4.04/1.91	   POL(B15(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1
4.04/1.91	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
4.04/1.91	   POL(+(x_1, x_2)) = x_1 + x_2
4.04/1.91	   POL(1) = [1]
4.04/1.91	   POL(&&(x_1, x_2)) = [-1]
4.04/1.91	   POL(>=(x_1, x_2)) = [-1]
4.04/1.91	   POL(<(x_1, x_2)) = [-1]
4.04/1.91	   POL(0) = 0
4.04/1.91	
4.04/1.91	
4.04/1.91	The following pairs are in P_>:
4.04/1.91	
4.04/1.91	
4.04/1.91	   COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.04/1.91	
4.04/1.91	
4.04/1.91	The following pairs are in P_bound:
4.04/1.91	
4.04/1.91	
4.04/1.91	   B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])
4.04/1.91	
4.04/1.91	
4.04/1.91	The following pairs are in P_>=:
4.04/1.91	
4.04/1.91	
4.04/1.91	   B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.04/1.91	   B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))
4.04/1.91	   B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), &&(<(1, sv14_14[3]), &&(TRUE, &&(<(0, sv24_38[3]), <(1, sv23_37[3]))))), sv14_14[3], sv23_37[3], sv24_38[3])
4.04/1.91	
4.04/1.91	
4.04/1.91	At least the following rules have been oriented under context sensitive arithmetic replacement:
4.04/1.91	
4.04/1.91	   TRUE^1 -> &&(TRUE, TRUE)^1
4.04/1.91	   FALSE^1 -> &&(TRUE, FALSE)^1
4.04/1.91	   FALSE^1 -> &&(FALSE, TRUE)^1
4.04/1.91	   FALSE^1 -> &&(FALSE, FALSE)^1
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(6)
4.04/1.91	Complex Obligation (AND)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(7)
4.04/1.91	Obligation:
4.04/1.91	IDP problem:
4.04/1.91	The following function symbols are pre-defined:
4.04/1.91	<<<
4.04/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.04/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.04/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.04/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
4.04/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.04/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.04/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.04/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.04/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.04/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.04/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.04/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
4.04/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.04/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.04/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.04/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.04/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.04/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.04/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.04/1.91	>>>
4.04/1.91	
4.04/1.91	
4.04/1.91	The following domains are used:
4.04/1.91	   Integer, Boolean
4.04/1.91	
4.04/1.91	R is empty.
4.04/1.91	
4.04/1.91	The integer pair graph contains the following rules and edges:
4.04/1.91	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.04/1.91	(2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1)
4.04/1.91	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && (1 < sv14_14[3] && (TRUE && (0 < sv24_38[3] && 1 < sv23_37[3]))), sv14_14[3], sv23_37[3], sv24_38[3])
4.04/1.91	
4.04/1.91	   (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0])
4.04/1.91	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
4.04/1.91	
4.04/1.91	The set Q consists of the following terms:
4.04/1.91	b10(x0, x1, x2)
4.04/1.91	Cond_b14(TRUE, x0, x1, x2)
4.04/1.91	b15(x0, x1, x2)
4.04/1.91	b14(x0, x1, x2)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(8) IDependencyGraphProof (EQUIVALENT)
4.04/1.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(9)
4.04/1.91	TRUE
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(10)
4.04/1.91	Obligation:
4.04/1.91	IDP problem:
4.04/1.91	The following function symbols are pre-defined:
4.04/1.91	<<<
4.04/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.04/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.04/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.04/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
4.04/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.04/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.04/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.04/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.04/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.04/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.04/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.04/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
4.04/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.04/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.04/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.04/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.04/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.04/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.04/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.04/1.91	>>>
4.04/1.91	
4.04/1.91	
4.04/1.91	The following domains are used:
4.04/1.91	   Integer
4.04/1.91	
4.04/1.91	R is empty.
4.04/1.91	
4.04/1.91	The integer pair graph contains the following rules and edges:
4.04/1.91	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
4.04/1.91	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
4.04/1.91	(2): B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], sv23_37[2] - sv14_14[2], sv24_38[2] + 1)
4.04/1.91	
4.04/1.91	   (2) -> (0), if (sv14_14[2] ->^* sv14_14[0] & sv23_37[2] - sv14_14[2] ->^* sv23_37[0] & sv24_38[2] + 1 ->^* sv24_38[0])
4.04/1.91	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
4.04/1.91	
4.04/1.91	The set Q consists of the following terms:
4.04/1.91	b10(x0, x1, x2)
4.04/1.91	Cond_b14(TRUE, x0, x1, x2)
4.04/1.91	b15(x0, x1, x2)
4.04/1.91	b14(x0, x1, x2)
4.04/1.91	
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(11) IDependencyGraphProof (EQUIVALENT)
4.04/1.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
4.04/1.91	----------------------------------------
4.04/1.91	
4.04/1.91	(12)
4.04/1.91	TRUE
4.04/1.92	EOF