3.94/1.88	YES
3.94/1.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
3.94/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.94/1.89	
3.94/1.89	
3.94/1.89	Termination of the given ITRS could be proven:
3.94/1.89	
3.94/1.89	(0) ITRS
3.94/1.89	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
3.94/1.89	(2) IDP
3.94/1.89	(3) UsableRulesProof [EQUIVALENT, 0 ms]
3.94/1.89	(4) IDP
3.94/1.89	(5) IDPNonInfProof [SOUND, 126 ms]
3.94/1.89	(6) AND
3.94/1.89	    (7) IDP
3.94/1.89	        (8) IDependencyGraphProof [EQUIVALENT, 0 ms]
3.94/1.89	        (9) TRUE
3.94/1.89	    (10) IDP
3.94/1.89	        (11) IDependencyGraphProof [EQUIVALENT, 0 ms]
3.94/1.89	        (12) TRUE
3.94/1.89	
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(0)
3.94/1.89	Obligation:
3.94/1.89	ITRS problem:
3.94/1.89	
3.94/1.89	The following function symbols are pre-defined:
3.94/1.89	<<<
3.94/1.89	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.94/1.89	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.94/1.89	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.94/1.89	 / 	~ 	Div: (Integer, Integer) -> Integer
3.94/1.89	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.94/1.89	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.94/1.89	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.94/1.89	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.94/1.89	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.94/1.89	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.94/1.89	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.94/1.89	 + 	~ 	Add: (Integer, Integer) -> Integer
3.94/1.89	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.94/1.89	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.94/1.89	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.94/1.89	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.94/1.89	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.94/1.89	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.94/1.89	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.94/1.89	>>>
3.94/1.89	
3.94/1.89	The TRS R consists of the following rules:
3.94/1.89	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
3.94/1.89	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
3.94/1.89	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
3.94/1.89	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && 1 < sv14_14, sv14_14, sv23_37, sv24_38)
3.94/1.89	The set Q consists of the following terms:
3.94/1.89	b10(x0, x1, x2)
3.94/1.89	Cond_b14(TRUE, x0, x1, x2)
3.94/1.89	b15(x0, x1, x2)
3.94/1.89	b14(x0, x1, x2)
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(1) ITRStoIDPProof (EQUIVALENT)
3.94/1.89	Added dependency pairs
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(2)
3.94/1.89	Obligation:
3.94/1.89	IDP problem:
3.94/1.89	The following function symbols are pre-defined:
3.94/1.89	<<<
3.94/1.89	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.94/1.89	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.94/1.89	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.94/1.89	 / 	~ 	Div: (Integer, Integer) -> Integer
3.94/1.89	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.94/1.89	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.94/1.89	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.94/1.89	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.94/1.89	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.94/1.89	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.94/1.89	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.94/1.89	 + 	~ 	Add: (Integer, Integer) -> Integer
3.94/1.89	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.94/1.89	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.94/1.89	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.94/1.89	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.94/1.89	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.94/1.89	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.94/1.89	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.94/1.89	>>>
3.94/1.89	
3.94/1.89	
3.94/1.89	The following domains are used:
3.94/1.89	   Integer, Boolean
3.94/1.89	
3.94/1.89	The ITRS R consists of the following rules:
3.94/1.89	b10(sv14_14, sv23_37, sv24_38) -> b14(sv14_14, sv23_37, sv24_38)
3.94/1.89	Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) -> b15(sv14_14, sv23_37, sv24_38)
3.94/1.89	b15(sv14_14, sv23_37, sv24_38) -> b10(sv14_14, sv23_37 - sv14_14, sv24_38 + 1)
3.94/1.89	b14(sv14_14, sv23_37, sv24_38) -> Cond_b14(sv23_37 >= sv14_14 && 1 < sv14_14, sv14_14, sv23_37, sv24_38)
3.94/1.89	
3.94/1.89	The integer pair graph contains the following rules and edges:
3.94/1.89	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
3.94/1.89	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
3.94/1.89	(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)
3.94/1.89	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && 1 < sv14_14[3], sv14_14[3], sv23_37[3], sv24_38[3])
3.94/1.89	
3.94/1.89	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
3.94/1.89	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
3.94/1.89	   (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])
3.94/1.89	   (3) -> (1), if (sv23_37[3] >= sv14_14[3] && 1 < sv14_14[3]  & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1])
3.94/1.89	
3.94/1.89	The set Q consists of the following terms:
3.94/1.89	b10(x0, x1, x2)
3.94/1.89	Cond_b14(TRUE, x0, x1, x2)
3.94/1.89	b15(x0, x1, x2)
3.94/1.89	b14(x0, x1, x2)
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(3) UsableRulesProof (EQUIVALENT)
3.94/1.89	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.
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(4)
3.94/1.89	Obligation:
3.94/1.89	IDP problem:
3.94/1.89	The following function symbols are pre-defined:
3.94/1.89	<<<
3.94/1.89	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.94/1.89	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.94/1.89	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.94/1.89	 / 	~ 	Div: (Integer, Integer) -> Integer
3.94/1.89	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.94/1.89	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.94/1.89	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.94/1.89	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.94/1.89	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.94/1.89	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.94/1.89	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.94/1.89	 + 	~ 	Add: (Integer, Integer) -> Integer
3.94/1.89	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.94/1.89	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.94/1.89	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.94/1.89	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.94/1.89	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.94/1.89	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.94/1.89	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.94/1.89	>>>
3.94/1.89	
3.94/1.89	
3.94/1.89	The following domains are used:
3.94/1.89	   Integer, Boolean
3.94/1.89	
3.94/1.89	R is empty.
3.94/1.89	
3.94/1.89	The integer pair graph contains the following rules and edges:
3.94/1.89	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
3.94/1.89	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
3.94/1.89	(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)
3.94/1.89	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && 1 < sv14_14[3], sv14_14[3], sv23_37[3], sv24_38[3])
3.94/1.89	
3.94/1.89	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
3.94/1.89	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
3.94/1.89	   (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])
3.94/1.89	   (3) -> (1), if (sv23_37[3] >= sv14_14[3] && 1 < sv14_14[3]  & sv14_14[3] ->^* sv14_14[1] & sv23_37[3] ->^* sv23_37[1] & sv24_38[3] ->^* sv24_38[1])
3.94/1.89	
3.94/1.89	The set Q consists of the following terms:
3.94/1.89	b10(x0, x1, x2)
3.94/1.89	Cond_b14(TRUE, x0, x1, x2)
3.94/1.89	b15(x0, x1, x2)
3.94/1.89	b14(x0, x1, x2)
3.94/1.89	
3.94/1.89	----------------------------------------
3.94/1.89	
3.94/1.89	(5) IDPNonInfProof (SOUND)
3.94/1.89	Used the following options for this NonInfProof:
3.94/1.89	
3.94/1.89	IDPGPoloSolver:
3.94/1.89	Range: [(-1,2)]
3.94/1.89	IsNat: false
3.94/1.89	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7294dff5
3.94/1.89	Constraint Generator: NonInfConstraintGenerator:
3.94/1.89	PathGenerator: MetricPathGenerator:
3.94/1.89	Max Left Steps: 1
3.94/1.89	Max Right Steps: 1
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	The constraints were generated the following way:
3.94/1.89	
3.94/1.89	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
3.94/1.89	
3.94/1.89	Note that final constraints are written in bold face.
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	For Pair B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38) the following chains were created:
3.94/1.89	*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])), sv14_14[3], sv23_37[3], sv24_38[3]) which results in the following constraint:
3.94/1.89	
3.94/1.89	(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])), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (1) using rule (IV) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(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])), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(3)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(4)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(5)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	For Pair COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38) the following chains were created:
3.94/1.89	*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:
3.94/1.89	
3.94/1.89	(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])), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (1) using rule (IV) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(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])), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(3)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(4)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(5)    ((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	For Pair B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1)) the following chains were created:
3.94/1.89	*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:
3.94/1.89	
3.94/1.89	(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))), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(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))), >=))
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(3)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(4)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(5)    ((U^Increasing(B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))), >=) & [bni_27] = 0 & [(-1)bso_28] >= 0)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
3.94/1.89	
3.94/1.89	(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)
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	
3.94/1.89	For Pair B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), <(1, sv14_14)), sv14_14, sv23_37, sv24_38) the following chains were created:
3.94/1.89	*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])), 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:
3.94/1.90	
3.94/1.90	(1)    (&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[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])), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=))
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
3.94/1.90	
3.94/1.90	(2)    (>=(sv23_37[3], sv14_14[3])=TRUE & <(1, sv14_14[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])), sv14_14[3], sv23_37[3], sv24_38[3]) & (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=))
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.94/1.90	
3.94/1.90	(3)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.94/1.90	
3.94/1.90	(4)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.94/1.90	
3.94/1.90	(5)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
3.94/1.90	
3.94/1.90	(6)    (sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & 0 = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	To summarize, we get the following constraints P__>=_ for the following pairs.
3.94/1.90	
3.94/1.90	*B10(sv14_14, sv23_37, sv24_38) -> B14(sv14_14, sv23_37, sv24_38)
3.94/1.90	
3.94/1.90	*((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), >=) & [bni_23] = 0 & [(-1)bso_24] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	*COND_B14(TRUE, sv14_14, sv23_37, sv24_38) -> B15(sv14_14, sv23_37, sv24_38)
3.94/1.90	
3.94/1.90	*((U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), >=) & [bni_25] = 0 & [1 + (-1)bso_26] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	*B15(sv14_14, sv23_37, sv24_38) -> B10(sv14_14, -(sv23_37, sv14_14), +(sv24_38, 1))
3.94/1.90	
3.94/1.90	*((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)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	*B14(sv14_14, sv23_37, sv24_38) -> COND_B14(&&(>=(sv23_37, sv14_14), <(1, sv14_14)), sv14_14, sv23_37, sv24_38)
3.94/1.90	
3.94/1.90	*(sv23_37[3] + [-1]sv14_14[3] >= 0 & sv14_14[3] + [-2] >= 0  ==>  (U^Increasing(COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])), >=) & 0 = 0 & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]sv23_37[3] + [(2)bni_29]sv14_14[3] >= 0 & [-2 + (-1)bso_30] + sv14_14[3] >= 0)
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	
3.94/1.90	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. 
3.94/1.90	
3.94/1.90	Using the following integer polynomial ordering the  resulting constraints can be solved 
3.94/1.90	
3.94/1.90	Polynomial interpretation over integers[POLO]:
3.94/1.90	
3.94/1.90	   POL(TRUE) = 0
3.94/1.90	   POL(FALSE) = [1]
3.94/1.90	   POL(B10(x_1, x_2, x_3)) = [-1] + x_2 + [2]x_1
3.94/1.90	   POL(B14(x_1, x_2, x_3)) = [-1] + x_2 + [2]x_1
3.94/1.90	   POL(COND_B14(x_1, x_2, x_3, x_4)) = x_3 + x_2 + [-1]x_1
3.94/1.90	   POL(B15(x_1, x_2, x_3)) = [-1] + x_2 + x_1
3.94/1.90	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
3.94/1.90	   POL(+(x_1, x_2)) = x_1 + x_2
3.94/1.90	   POL(1) = [1]
3.94/1.90	   POL(&&(x_1, x_2)) = [-1]
3.94/1.90	   POL(>=(x_1, x_2)) = [-1]
3.94/1.90	   POL(<(x_1, x_2)) = [-1]
3.94/1.90	
3.94/1.90	
3.94/1.90	The following pairs are in P_>:
3.94/1.90	
3.94/1.90	
3.94/1.90	   COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
3.94/1.90	
3.94/1.90	
3.94/1.90	The following pairs are in P_bound:
3.94/1.90	
3.94/1.90	
3.94/1.90	   B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])
3.94/1.90	
3.94/1.90	
3.94/1.90	The following pairs are in P_>=:
3.94/1.90	
3.94/1.90	
3.94/1.90	   B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
3.94/1.90	   B15(sv14_14[2], sv23_37[2], sv24_38[2]) -> B10(sv14_14[2], -(sv23_37[2], sv14_14[2]), +(sv24_38[2], 1))
3.94/1.90	   B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(&&(>=(sv23_37[3], sv14_14[3]), <(1, sv14_14[3])), sv14_14[3], sv23_37[3], sv24_38[3])
3.94/1.90	
3.94/1.90	
3.94/1.90	At least the following rules have been oriented under context sensitive arithmetic replacement:
3.94/1.90	
3.94/1.90	   TRUE^1 -> &&(TRUE, TRUE)^1
3.94/1.90	   FALSE^1 -> &&(TRUE, FALSE)^1
3.94/1.90	   FALSE^1 -> &&(FALSE, TRUE)^1
3.94/1.90	   FALSE^1 -> &&(FALSE, FALSE)^1
3.94/1.90	
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(6)
3.94/1.90	Complex Obligation (AND)
3.94/1.90	
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(7)
3.94/1.90	Obligation:
3.94/1.90	IDP problem:
3.94/1.90	The following function symbols are pre-defined:
3.94/1.90	<<<
3.94/1.90	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.94/1.90	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.94/1.90	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.94/1.90	 / 	~ 	Div: (Integer, Integer) -> Integer
3.94/1.90	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.94/1.90	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.94/1.90	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.94/1.90	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.94/1.90	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.94/1.90	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.94/1.90	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.94/1.90	 + 	~ 	Add: (Integer, Integer) -> Integer
3.94/1.90	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.94/1.90	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.94/1.90	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.94/1.90	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.94/1.90	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.94/1.90	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.94/1.90	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.94/1.90	>>>
3.94/1.90	
3.94/1.90	
3.94/1.90	The following domains are used:
3.94/1.90	   Integer, Boolean
3.94/1.90	
3.94/1.90	R is empty.
3.94/1.90	
3.94/1.90	The integer pair graph contains the following rules and edges:
3.94/1.90	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
3.94/1.90	(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)
3.94/1.90	(3): B14(sv14_14[3], sv23_37[3], sv24_38[3]) -> COND_B14(sv23_37[3] >= sv14_14[3] && 1 < sv14_14[3], sv14_14[3], sv23_37[3], sv24_38[3])
3.94/1.90	
3.94/1.90	   (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])
3.94/1.90	   (0) -> (3), if (sv14_14[0] ->^* sv14_14[3] & sv23_37[0] ->^* sv23_37[3] & sv24_38[0] ->^* sv24_38[3])
3.94/1.90	
3.94/1.90	The set Q consists of the following terms:
3.94/1.90	b10(x0, x1, x2)
3.94/1.90	Cond_b14(TRUE, x0, x1, x2)
3.94/1.90	b15(x0, x1, x2)
3.94/1.90	b14(x0, x1, x2)
3.94/1.90	
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(8) IDependencyGraphProof (EQUIVALENT)
3.94/1.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(9)
3.94/1.90	TRUE
3.94/1.90	
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(10)
3.94/1.90	Obligation:
3.94/1.90	IDP problem:
3.94/1.90	The following function symbols are pre-defined:
3.94/1.90	<<<
3.94/1.90	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.94/1.90	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.94/1.90	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.94/1.90	 / 	~ 	Div: (Integer, Integer) -> Integer
3.94/1.90	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.94/1.90	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.94/1.90	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.94/1.90	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.94/1.90	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.94/1.90	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.94/1.90	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.94/1.90	 + 	~ 	Add: (Integer, Integer) -> Integer
3.94/1.90	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.94/1.90	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.94/1.90	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.94/1.90	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.94/1.90	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.94/1.90	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.94/1.90	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.94/1.90	>>>
3.94/1.90	
3.94/1.90	
3.94/1.90	The following domains are used:
3.94/1.90	   Integer
3.94/1.90	
3.94/1.90	R is empty.
3.94/1.90	
3.94/1.90	The integer pair graph contains the following rules and edges:
3.94/1.90	(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) -> B14(sv14_14[0], sv23_37[0], sv24_38[0])
3.94/1.90	(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) -> B15(sv14_14[1], sv23_37[1], sv24_38[1])
3.94/1.90	(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)
3.94/1.90	
3.94/1.90	   (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])
3.94/1.90	   (1) -> (2), if (sv14_14[1] ->^* sv14_14[2] & sv23_37[1] ->^* sv23_37[2] & sv24_38[1] ->^* sv24_38[2])
3.94/1.90	
3.94/1.90	The set Q consists of the following terms:
3.94/1.90	b10(x0, x1, x2)
3.94/1.90	Cond_b14(TRUE, x0, x1, x2)
3.94/1.90	b15(x0, x1, x2)
3.94/1.90	b14(x0, x1, x2)
3.94/1.90	
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(11) IDependencyGraphProof (EQUIVALENT)
3.94/1.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
3.94/1.90	----------------------------------------
3.94/1.90	
3.94/1.90	(12)
3.94/1.90	TRUE
4.05/1.91	EOF