6.44/2.55	YES
6.44/2.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
6.44/2.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.44/2.57	
6.44/2.57	
6.44/2.57	Termination of the given ITRS could be proven:
6.44/2.57	
6.44/2.57	(0) ITRS
6.44/2.57	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
6.44/2.57	(2) IDP
6.44/2.57	(3) UsableRulesProof [EQUIVALENT, 0 ms]
6.44/2.57	(4) IDP
6.44/2.57	(5) IDPNonInfProof [SOUND, 599 ms]
6.44/2.57	(6) IDP
6.44/2.57	(7) IDPNonInfProof [SOUND, 42 ms]
6.44/2.57	(8) IDP
6.44/2.57	(9) PisEmptyProof [EQUIVALENT, 0 ms]
6.44/2.57	(10) YES
6.44/2.57	
6.44/2.57	
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(0)
6.44/2.57	Obligation:
6.44/2.57	ITRS problem:
6.44/2.57	
6.44/2.57	The following function symbols are pre-defined:
6.44/2.57	<<<
6.44/2.57	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.44/2.57	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.44/2.57	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.44/2.57	 / 	~ 	Div: (Integer, Integer) -> Integer
6.44/2.57	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.44/2.57	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.44/2.57	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.44/2.57	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.44/2.57	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.44/2.57	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.44/2.57	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.44/2.57	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.44/2.57	 + 	~ 	Add: (Integer, Integer) -> Integer
6.44/2.57	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.44/2.57	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.44/2.57	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.44/2.57	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.44/2.57	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.44/2.57	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.44/2.57	>>>
6.44/2.57	
6.44/2.57	The TRS R consists of the following rules:
6.44/2.57	f(TRUE, x, y, z) -> f(x > y && x > z, x, y + 1, z)
6.44/2.57	f(TRUE, x, y, z) -> f(x > y && x > z, x, y, z + 1)
6.44/2.57	The set Q consists of the following terms:
6.44/2.57	f(TRUE, x0, x1, x2)
6.44/2.57	
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(1) ITRStoIDPProof (EQUIVALENT)
6.44/2.57	Added dependency pairs
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(2)
6.44/2.57	Obligation:
6.44/2.57	IDP problem:
6.44/2.57	The following function symbols are pre-defined:
6.44/2.57	<<<
6.44/2.57	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.44/2.57	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.44/2.57	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.44/2.57	 / 	~ 	Div: (Integer, Integer) -> Integer
6.44/2.57	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.44/2.57	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.44/2.57	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.44/2.57	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.44/2.57	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.44/2.57	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.44/2.57	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.44/2.57	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.44/2.57	 + 	~ 	Add: (Integer, Integer) -> Integer
6.44/2.57	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.44/2.57	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.44/2.57	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.44/2.57	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.44/2.57	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.44/2.57	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.44/2.57	>>>
6.44/2.57	
6.44/2.57	
6.44/2.57	The following domains are used:
6.44/2.57	   Boolean, Integer
6.44/2.57	
6.44/2.57	The ITRS R consists of the following rules:
6.44/2.57	f(TRUE, x, y, z) -> f(x > y && x > z, x, y + 1, z)
6.44/2.57	f(TRUE, x, y, z) -> f(x > y && x > z, x, y, z + 1)
6.44/2.57	
6.44/2.57	The integer pair graph contains the following rules and edges:
6.44/2.57	(0): F(TRUE, x[0], y[0], z[0]) -> F(x[0] > y[0] && x[0] > z[0], x[0], y[0] + 1, z[0])
6.44/2.57	(1): F(TRUE, x[1], y[1], z[1]) -> F(x[1] > y[1] && x[1] > z[1], x[1], y[1], z[1] + 1)
6.44/2.57	
6.44/2.57	   (0) -> (0), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[0]' & y[0] + 1 ->^* y[0]' & z[0] ->^* z[0]')
6.44/2.57	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] + 1 ->^* y[1] & z[0] ->^* z[1])
6.44/2.57	   (1) -> (0), if (x[1] > y[1] && x[1] > z[1]  & x[1] ->^* x[0] & y[1] ->^* y[0] & z[1] + 1 ->^* z[0])
6.44/2.57	   (1) -> (1), if (x[1] > y[1] && x[1] > z[1]  & x[1] ->^* x[1]' & y[1] ->^* y[1]' & z[1] + 1 ->^* z[1]')
6.44/2.57	
6.44/2.57	The set Q consists of the following terms:
6.44/2.57	f(TRUE, x0, x1, x2)
6.44/2.57	
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(3) UsableRulesProof (EQUIVALENT)
6.44/2.57	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.
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(4)
6.44/2.57	Obligation:
6.44/2.57	IDP problem:
6.44/2.57	The following function symbols are pre-defined:
6.44/2.57	<<<
6.44/2.57	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.44/2.57	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.44/2.57	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.44/2.57	 / 	~ 	Div: (Integer, Integer) -> Integer
6.44/2.57	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.44/2.57	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.44/2.57	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.44/2.57	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.44/2.57	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.44/2.57	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.44/2.57	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.44/2.57	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.44/2.57	 + 	~ 	Add: (Integer, Integer) -> Integer
6.44/2.57	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.44/2.57	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.44/2.57	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.44/2.57	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.44/2.57	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.44/2.57	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.44/2.57	>>>
6.44/2.57	
6.44/2.57	
6.44/2.57	The following domains are used:
6.44/2.57	   Boolean, Integer
6.44/2.57	
6.44/2.57	R is empty.
6.44/2.57	
6.44/2.57	The integer pair graph contains the following rules and edges:
6.44/2.57	(0): F(TRUE, x[0], y[0], z[0]) -> F(x[0] > y[0] && x[0] > z[0], x[0], y[0] + 1, z[0])
6.44/2.57	(1): F(TRUE, x[1], y[1], z[1]) -> F(x[1] > y[1] && x[1] > z[1], x[1], y[1], z[1] + 1)
6.44/2.57	
6.44/2.57	   (0) -> (0), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[0]' & y[0] + 1 ->^* y[0]' & z[0] ->^* z[0]')
6.44/2.57	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] + 1 ->^* y[1] & z[0] ->^* z[1])
6.44/2.57	   (1) -> (0), if (x[1] > y[1] && x[1] > z[1]  & x[1] ->^* x[0] & y[1] ->^* y[0] & z[1] + 1 ->^* z[0])
6.44/2.57	   (1) -> (1), if (x[1] > y[1] && x[1] > z[1]  & x[1] ->^* x[1]' & y[1] ->^* y[1]' & z[1] + 1 ->^* z[1]')
6.44/2.57	
6.44/2.57	The set Q consists of the following terms:
6.44/2.57	f(TRUE, x0, x1, x2)
6.44/2.57	
6.44/2.57	----------------------------------------
6.44/2.57	
6.44/2.57	(5) IDPNonInfProof (SOUND)
6.44/2.57	Used the following options for this NonInfProof:
6.44/2.57	
6.44/2.57	IDPGPoloSolver:
6.44/2.57	Range: [(-1,2)]
6.44/2.57	IsNat: false
6.44/2.57	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5f64a601
6.44/2.57	Constraint Generator: NonInfConstraintGenerator:
6.44/2.57	PathGenerator: MetricPathGenerator:
6.44/2.57	Max Left Steps: 1
6.44/2.57	Max Right Steps: 1
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	The constraints were generated the following way:
6.44/2.57	
6.44/2.57	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.44/2.57	
6.44/2.57	Note that final constraints are written in bold face.
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	For Pair F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, +(y, 1), z) the following chains were created:
6.44/2.57	*We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 & &&(>(x[0]1, y[0]1), >(x[0]1, z[0]1))=TRUE & x[0]1=x[0]2 & +(y[0]1, 1)=y[0]2 & z[0]1=z[0]2  ==>  F(TRUE, x[0]1, y[0]1, z[0]1)_>=_NonInfC & F(TRUE, x[0]1, y[0]1, z[0]1)_>=_F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE & >(x[0], +(y[0], 1))=TRUE  ==>  F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(+(y[0], 1), 1), z[0]) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [(-1)bni_10]z[0] + [bni_10]y[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0] & y[1]=y[0] & +(z[1], 1)=z[0] & &&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1  ==>  F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE  ==>  F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]y[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 & &&(>(x[0]1, y[0]1), >(x[0]1, z[0]1))=TRUE & x[0]1=x[1] & +(y[0]1, 1)=y[1] & z[0]1=z[1]  ==>  F(TRUE, x[0]1, y[0]1, z[0]1)_>=_NonInfC & F(TRUE, x[0]1, y[0]1, z[0]1)_>=_F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE & >(x[0], +(y[0], 1))=TRUE  ==>  F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(+(y[0], 1), 1), z[0]) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [(-1)bni_10]z[0] + [bni_10]y[0] + [bni_10]x[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0] & y[1]=y[0] & +(z[1], 1)=z[0] & &&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1]1 & +(y[0], 1)=y[1]1 & z[0]=z[1]1  ==>  F(TRUE, x[0], y[0], z[0])_>=_NonInfC & F(TRUE, x[0], y[0], z[0])_>=_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE  ==>  F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], +(y[1], 1), +(z[1], 1)) & (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [(-1)bni_10]z[1] + [bni_10]y[1] + [bni_10]x[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	For Pair F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, y, +(z, 1)) the following chains were created:
6.44/2.57	*We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & +(y[0], 1)=y[1] & z[0]=z[1] & &&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[0]1 & y[1]=y[0]1 & +(z[1], 1)=z[0]1  ==>  F(TRUE, x[1], y[1], z[1])_>=_NonInfC & F(TRUE, x[1], y[1], z[1])_>=_F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE & >(x[0], +(y[0], 1))=TRUE  ==>  F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [(-1)bni_12]z[0] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1 & &&(>(x[1]1, y[1]1), >(x[1]1, z[1]1))=TRUE & x[1]1=x[0] & y[1]1=y[0] & +(z[1]1, 1)=z[0]  ==>  F(TRUE, x[1]1, y[1]1, z[1]1)_>=_NonInfC & F(TRUE, x[1]1, y[1]1, z[1]1)_>=_F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE  ==>  F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], y[1], +(+(z[1], 1), 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]y[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & +(y[0], 1)=y[1] & z[0]=z[1] & &&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1  ==>  F(TRUE, x[1], y[1], z[1])_>=_NonInfC & F(TRUE, x[1], y[1], z[1])_>=_F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE & >(x[0], +(y[0], 1))=TRUE  ==>  F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(y[0], 1), +(z[0], 1)) & (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [(-1)bni_12]z[0] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(7)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.57	
6.44/2.57	(8)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	(9)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	*We consider the chain F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)), F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1)) which results in the following constraint:
6.44/2.57	
6.44/2.57	(1)    (&&(>(x[1], y[1]), >(x[1], z[1]))=TRUE & x[1]=x[1]1 & y[1]=y[1]1 & +(z[1], 1)=z[1]1 & &&(>(x[1]1, y[1]1), >(x[1]1, z[1]1))=TRUE & x[1]1=x[1]2 & y[1]1=y[1]2 & +(z[1]1, 1)=z[1]2  ==>  F(TRUE, x[1]1, y[1]1, z[1]1)_>=_NonInfC & F(TRUE, x[1]1, y[1]1, z[1]1)_>=_F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(2)    (>(x[1], y[1])=TRUE & >(x[1], z[1])=TRUE & >(x[1], +(z[1], 1))=TRUE  ==>  F(TRUE, x[1], y[1], +(z[1], 1))_>=_NonInfC & F(TRUE, x[1], y[1], +(z[1], 1))_>=_F(&&(>(x[1], y[1]), >(x[1], +(z[1], 1))), x[1], y[1], +(+(z[1], 1), 1)) & (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=))
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(3)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(4)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(5)    (x[1] + [-1] + [-1]y[1] >= 0 & x[1] + [-1] + [-1]z[1] >= 0 & x[1] + [-2] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.57	
6.44/2.57	
6.44/2.57	
6.44/2.57	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.57	
6.44/2.57	(6)    (x[1] >= 0 & y[1] + x[1] + [-1]z[1] >= 0 & [-1] + y[1] + x[1] + [-1]z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [(-1)bni_12]z[1] + [bni_12]y[1] + [bni_12]x[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(7)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.58	
6.44/2.58	(8)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	(9)    (x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	To summarize, we get the following constraints P__>=_ for the following pairs.
6.44/2.58	
6.44/2.58	*F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, +(y, 1), z)
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(-1)Bound*bni_10 + bni_10] + [bni_10]z[0] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])), >=) & [(-1)Bound*bni_10] + [bni_10]z[1] >= 0 & [(-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	*F(TRUE, x, y, z) -> F(&&(>(x, y), >(x, z)), x, y, +(z, 1))
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))), >=) & [(-1)Bound*bni_12 + bni_12] + [bni_12]z[0] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[1] >= 0 & z[1] >= 0 & [-1] + z[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(F(&&(>(x[1]1, y[1]1), >(x[1]1, z[1]1)), x[1]1, y[1]1, +(z[1]1, 1))), >=) & [(-1)Bound*bni_12] + [bni_12]z[1] >= 0 & [1 + (-1)bso_13] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	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. 
6.44/2.58	
6.44/2.58	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.44/2.58	
6.44/2.58	Polynomial interpretation over integers[POLO]:
6.44/2.58	
6.44/2.58	   POL(TRUE) = 0
6.44/2.58	   POL(FALSE) = [1]
6.44/2.58	   POL(F(x_1, x_2, x_3, x_4)) = [-1]x_4 + x_2 + [-1]x_1
6.44/2.58	   POL(&&(x_1, x_2)) = 0
6.44/2.58	   POL(>(x_1, x_2)) = [-1]
6.44/2.58	   POL(+(x_1, x_2)) = x_1 + x_2
6.44/2.58	   POL(1) = [1]
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_>:
6.44/2.58	
6.44/2.58	
6.44/2.58	   F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_bound:
6.44/2.58	
6.44/2.58	
6.44/2.58	   F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])
6.44/2.58	   F(TRUE, x[1], y[1], z[1]) -> F(&&(>(x[1], y[1]), >(x[1], z[1])), x[1], y[1], +(z[1], 1))
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_>=:
6.44/2.58	
6.44/2.58	
6.44/2.58	   F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])
6.44/2.58	
6.44/2.58	
6.44/2.58	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.44/2.58	
6.44/2.58	   &&(TRUE, TRUE)^1 <-> TRUE^1
6.44/2.58	   FALSE^1 -> &&(TRUE, FALSE)^1
6.44/2.58	   FALSE^1 -> &&(FALSE, TRUE)^1
6.44/2.58	   FALSE^1 -> &&(FALSE, FALSE)^1
6.44/2.58	
6.44/2.58	----------------------------------------
6.44/2.58	
6.44/2.58	(6)
6.44/2.58	Obligation:
6.44/2.58	IDP problem:
6.44/2.58	The following function symbols are pre-defined:
6.44/2.58	<<<
6.44/2.58	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.44/2.58	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.44/2.58	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.44/2.58	 / 	~ 	Div: (Integer, Integer) -> Integer
6.44/2.58	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.44/2.58	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.44/2.58	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.44/2.58	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.44/2.58	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.44/2.58	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.44/2.58	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.44/2.58	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.44/2.58	 + 	~ 	Add: (Integer, Integer) -> Integer
6.44/2.58	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.44/2.58	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.44/2.58	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.44/2.58	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.44/2.58	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.44/2.58	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.44/2.58	>>>
6.44/2.58	
6.44/2.58	
6.44/2.58	The following domains are used:
6.44/2.58	   Boolean, Integer
6.44/2.58	
6.44/2.58	R is empty.
6.44/2.58	
6.44/2.58	The integer pair graph contains the following rules and edges:
6.44/2.58	(0): F(TRUE, x[0], y[0], z[0]) -> F(x[0] > y[0] && x[0] > z[0], x[0], y[0] + 1, z[0])
6.44/2.58	
6.44/2.58	   (0) -> (0), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[0]' & y[0] + 1 ->^* y[0]' & z[0] ->^* z[0]')
6.44/2.58	
6.44/2.58	The set Q consists of the following terms:
6.44/2.58	f(TRUE, x0, x1, x2)
6.44/2.58	
6.44/2.58	----------------------------------------
6.44/2.58	
6.44/2.58	(7) IDPNonInfProof (SOUND)
6.44/2.58	Used the following options for this NonInfProof:
6.44/2.58	
6.44/2.58	IDPGPoloSolver:
6.44/2.58	Range: [(-1,2)]
6.44/2.58	IsNat: false
6.44/2.58	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5f64a601
6.44/2.58	Constraint Generator: NonInfConstraintGenerator:
6.44/2.58	PathGenerator: MetricPathGenerator:
6.44/2.58	Max Left Steps: 1
6.44/2.58	Max Right Steps: 1
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	The constraints were generated the following way:
6.44/2.58	
6.44/2.58	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.44/2.58	
6.44/2.58	Note that final constraints are written in bold face.
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	For Pair F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) the following chains were created:
6.44/2.58	*We consider the chain F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]), F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0]) which results in the following constraint:
6.44/2.58	
6.44/2.58	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[0]1 & +(y[0], 1)=y[0]1 & z[0]=z[0]1 & &&(>(x[0]1, y[0]1), >(x[0]1, z[0]1))=TRUE & x[0]1=x[0]2 & +(y[0]1, 1)=y[0]2 & z[0]1=z[0]2  ==>  F(TRUE, x[0]1, y[0]1, z[0]1)_>=_NonInfC & F(TRUE, x[0]1, y[0]1, z[0]1)_>=_F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE & >(x[0], +(y[0], 1))=TRUE  ==>  F(TRUE, x[0], +(y[0], 1), z[0])_>=_NonInfC & F(TRUE, x[0], +(y[0], 1), z[0])_>=_F(&&(>(x[0], +(y[0], 1)), >(x[0], z[0])), x[0], +(+(y[0], 1), 1), z[0]) & (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=))
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]y[0] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]y[0] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0 & x[0] + [-2] + [-1]y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]y[0] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.44/2.58	
6.44/2.58	(7)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.44/2.58	
6.44/2.58	(8)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	(9)    (x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	To summarize, we get the following constraints P__>=_ for the following pairs.
6.44/2.58	
6.44/2.58	*F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	*(x[0] >= 0 & z[0] >= 0 & [-1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(&&(>(x[0]1, y[0]1), >(x[0]1, z[0]1)), x[0]1, +(y[0]1, 1), z[0]1)), >=) & [(2)bni_10 + (-1)Bound*bni_10] + [bni_10]x[0] >= 0 & [1 + (-1)bso_11] >= 0)
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	
6.44/2.58	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. 
6.44/2.58	
6.44/2.58	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.44/2.58	
6.44/2.58	Polynomial interpretation over integers[POLO]:
6.44/2.58	
6.44/2.58	   POL(TRUE) = 0
6.44/2.58	   POL(FALSE) = [3]
6.44/2.58	   POL(F(x_1, x_2, x_3, x_4)) = [2] + [-1]x_3 + x_2 + [-1]x_1
6.44/2.58	   POL(&&(x_1, x_2)) = 0
6.44/2.58	   POL(>(x_1, x_2)) = [-1]
6.44/2.58	   POL(+(x_1, x_2)) = x_1 + x_2
6.44/2.58	   POL(1) = [1]
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_>:
6.44/2.58	
6.44/2.58	
6.44/2.58	   F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_bound:
6.44/2.58	
6.44/2.58	
6.44/2.58	   F(TRUE, x[0], y[0], z[0]) -> F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], +(y[0], 1), z[0])
6.44/2.58	
6.44/2.58	
6.44/2.58	The following pairs are in P_>=:
6.44/2.58	
6.44/2.58	none
6.44/2.58	
6.44/2.58	
6.44/2.58	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.44/2.58	
6.44/2.58	   &&(TRUE, TRUE)^1 <-> TRUE^1
6.44/2.58	   FALSE^1 -> &&(TRUE, FALSE)^1
6.44/2.58	   FALSE^1 -> &&(FALSE, TRUE)^1
6.44/2.58	   FALSE^1 -> &&(FALSE, FALSE)^1
6.44/2.58	
6.44/2.58	----------------------------------------
6.44/2.58	
6.44/2.58	(8)
6.44/2.58	Obligation:
6.44/2.58	IDP problem:
6.44/2.58	The following function symbols are pre-defined:
6.44/2.58	<<<
6.44/2.58	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.44/2.58	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.44/2.58	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.44/2.58	 / 	~ 	Div: (Integer, Integer) -> Integer
6.44/2.58	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.44/2.58	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.44/2.58	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.44/2.58	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.44/2.58	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.44/2.58	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.44/2.58	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.44/2.58	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.44/2.58	 + 	~ 	Add: (Integer, Integer) -> Integer
6.44/2.58	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.44/2.58	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.44/2.58	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.44/2.58	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.44/2.58	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.44/2.58	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.44/2.58	>>>
6.44/2.58	
6.44/2.58	
6.44/2.58	The following domains are used:
6.44/2.58	none
6.44/2.58	
6.44/2.58	R is empty.
6.44/2.58	
6.44/2.58	The integer pair graph is empty.
6.44/2.58	
6.44/2.58	The set Q consists of the following terms:
6.44/2.58	f(TRUE, x0, x1, x2)
6.44/2.58	
6.44/2.58	----------------------------------------
6.44/2.58	
6.44/2.58	(9) PisEmptyProof (EQUIVALENT)
6.44/2.58	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.44/2.58	----------------------------------------
6.44/2.58	
6.44/2.58	(10)
6.44/2.58	YES
6.64/2.64	EOF