5.92/2.49	YES
6.21/2.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
6.21/2.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.21/2.51	
6.21/2.51	
6.21/2.51	Termination of the given ITRS could be proven:
6.21/2.51	
6.21/2.51	(0) ITRS
6.21/2.51	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
6.21/2.51	(2) IDP
6.21/2.51	(3) UsableRulesProof [EQUIVALENT, 0 ms]
6.21/2.51	(4) IDP
6.21/2.51	(5) IDPNonInfProof [SOUND, 409 ms]
6.21/2.51	(6) IDP
6.21/2.51	(7) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.21/2.51	(8) IDP
6.21/2.51	(9) IDPNonInfProof [SOUND, 61 ms]
6.21/2.51	(10) IDP
6.21/2.51	(11) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.21/2.51	(12) TRUE
6.21/2.51	
6.21/2.51	
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(0)
6.21/2.51	Obligation:
6.21/2.51	ITRS problem:
6.21/2.51	
6.21/2.51	The following function symbols are pre-defined:
6.21/2.51	<<<
6.21/2.51	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.51	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.51	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.51	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.51	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.51	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.51	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.51	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.51	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.51	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.51	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.51	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.51	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.51	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.51	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.51	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.51	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.51	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.51	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.51	>>>
6.21/2.51	
6.21/2.51	The TRS R consists of the following rules:
6.21/2.51	f(x, y, z) -> Cond_f(x > y && x > z, x, y, z)
6.21/2.51	Cond_f(TRUE, x, y, z) -> f(x, y + 1, z)
6.21/2.51	f(x, y, z) -> Cond_f1(x > y && x > z, x, y, z)
6.21/2.51	Cond_f1(TRUE, x, y, z) -> f(x, y, z + 1)
6.21/2.51	The set Q consists of the following terms:
6.21/2.51	f(x0, x1, x2)
6.21/2.51	Cond_f(TRUE, x0, x1, x2)
6.21/2.51	Cond_f1(TRUE, x0, x1, x2)
6.21/2.51	
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(1) ITRStoIDPProof (EQUIVALENT)
6.21/2.51	Added dependency pairs
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(2)
6.21/2.51	Obligation:
6.21/2.51	IDP problem:
6.21/2.51	The following function symbols are pre-defined:
6.21/2.51	<<<
6.21/2.51	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.51	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.51	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.51	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.51	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.51	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.51	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.51	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.51	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.51	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.51	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.51	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.51	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.51	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.51	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.51	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.51	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.51	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.51	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.51	>>>
6.21/2.51	
6.21/2.51	
6.21/2.51	The following domains are used:
6.21/2.51	   Boolean, Integer
6.21/2.51	
6.21/2.51	The ITRS R consists of the following rules:
6.21/2.51	f(x, y, z) -> Cond_f(x > y && x > z, x, y, z)
6.21/2.51	Cond_f(TRUE, x, y, z) -> f(x, y + 1, z)
6.21/2.51	f(x, y, z) -> Cond_f1(x > y && x > z, x, y, z)
6.21/2.51	Cond_f1(TRUE, x, y, z) -> f(x, y, z + 1)
6.21/2.51	
6.21/2.51	The integer pair graph contains the following rules and edges:
6.21/2.51	(0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0])
6.21/2.51	(1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1])
6.21/2.51	(2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2])
6.21/2.51	(3): COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], z[3] + 1)
6.21/2.51	
6.21/2.51	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.21/2.51	   (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0])
6.21/2.51	   (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2])
6.21/2.51	   (2) -> (3), if (x[2] > y[2] && x[2] > z[2]  & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3])
6.21/2.51	   (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0])
6.21/2.51	   (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2])
6.21/2.51	
6.21/2.51	The set Q consists of the following terms:
6.21/2.51	f(x0, x1, x2)
6.21/2.51	Cond_f(TRUE, x0, x1, x2)
6.21/2.51	Cond_f1(TRUE, x0, x1, x2)
6.21/2.51	
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(3) UsableRulesProof (EQUIVALENT)
6.21/2.51	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.
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(4)
6.21/2.51	Obligation:
6.21/2.51	IDP problem:
6.21/2.51	The following function symbols are pre-defined:
6.21/2.51	<<<
6.21/2.51	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.51	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.51	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.51	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.51	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.51	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.51	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.51	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.51	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.51	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.51	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.51	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.51	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.51	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.51	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.51	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.51	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.51	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.51	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.51	>>>
6.21/2.51	
6.21/2.51	
6.21/2.51	The following domains are used:
6.21/2.51	   Boolean, Integer
6.21/2.51	
6.21/2.51	R is empty.
6.21/2.51	
6.21/2.51	The integer pair graph contains the following rules and edges:
6.21/2.51	(0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0])
6.21/2.51	(1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1])
6.21/2.51	(2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2])
6.21/2.51	(3): COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], z[3] + 1)
6.21/2.51	
6.21/2.51	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.21/2.51	   (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0])
6.21/2.51	   (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2])
6.21/2.51	   (2) -> (3), if (x[2] > y[2] && x[2] > z[2]  & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3])
6.21/2.51	   (3) -> (0), if (x[3] ->^* x[0] & y[3] ->^* y[0] & z[3] + 1 ->^* z[0])
6.21/2.51	   (3) -> (2), if (x[3] ->^* x[2] & y[3] ->^* y[2] & z[3] + 1 ->^* z[2])
6.21/2.51	
6.21/2.51	The set Q consists of the following terms:
6.21/2.51	f(x0, x1, x2)
6.21/2.51	Cond_f(TRUE, x0, x1, x2)
6.21/2.51	Cond_f1(TRUE, x0, x1, x2)
6.21/2.51	
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(5) IDPNonInfProof (SOUND)
6.21/2.51	Used the following options for this NonInfProof:
6.21/2.51	
6.21/2.51	IDPGPoloSolver:
6.21/2.51	Range: [(-1,2)]
6.21/2.51	IsNat: false
6.21/2.51	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6b1f24ae
6.21/2.51	Constraint Generator: NonInfConstraintGenerator:
6.21/2.51	PathGenerator: MetricPathGenerator:
6.21/2.51	Max Left Steps: 1
6.21/2.51	Max Right Steps: 1
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	The constraints were generated the following way:
6.21/2.51	
6.21/2.51	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.21/2.51	
6.21/2.51	Note that final constraints are written in bold face.
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	For Pair F(x, y, z) -> COND_F(&&(>(x, y), >(x, z)), x, y, z) the following chains were created:
6.21/2.51	*We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1]  ==>  F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE  ==>  F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [(-1)bni_19]z[0] + [bni_19]y[0] + [bni_19]x[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	For Pair COND_F(TRUE, x, y, z) -> F(x, +(y, 1), z) the following chains were created:
6.21/2.51	*We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1  ==>  COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE  ==>  COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	*We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[2] & +(y[1], 1)=y[2] & z[1]=z[2]  ==>  COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE  ==>  COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [(-1)bni_21]z[0] + [bni_21]y[0] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	For Pair F(x, y, z) -> COND_F1(&&(>(x, y), >(x, z)), x, y, z) the following chains were created:
6.21/2.51	*We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[2], y[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3]  ==>  F(x[2], y[2], z[2])_>=_NonInfC & F(x[2], y[2], z[2])_>=_COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE  ==>  F(x[2], y[2], z[2])_>=_NonInfC & F(x[2], y[2], z[2])_>=_COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [(-1)bni_23]z[2] + [bni_23]y[2] + [bni_23]x[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	For Pair COND_F1(TRUE, x, y, z) -> F(x, y, +(z, 1)) the following chains were created:
6.21/2.51	*We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[2], y[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[0] & y[3]=y[0] & +(z[3], 1)=z[0]  ==>  COND_F1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_F1(TRUE, x[3], y[3], z[3])_>=_F(x[3], y[3], +(z[3], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE  ==>  COND_F1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_F1(TRUE, x[2], y[2], z[2])_>=_F(x[2], y[2], +(z[2], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	*We consider the chain F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1)), F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
6.21/2.51	
6.21/2.51	(1)    (&&(>(x[2], y[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & y[3]=y[2]1 & +(z[3], 1)=z[2]1  ==>  COND_F1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_F1(TRUE, x[3], y[3], z[3])_>=_F(x[3], y[3], +(z[3], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(2)    (>(x[2], y[2])=TRUE & >(x[2], z[2])=TRUE  ==>  COND_F1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_F1(TRUE, x[2], y[2], z[2])_>=_F(x[2], y[2], +(z[2], 1)) & (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=))
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(3)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(4)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(5)    (x[2] + [-1] + [-1]y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(6)    (x[2] >= 0 & y[2] + x[2] + [-1]z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [(-1)bni_25]z[2] + [bni_25]y[2] + [bni_25]x[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.51	
6.21/2.51	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.51	
6.21/2.51	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	To summarize, we get the following constraints P__>=_ for the following pairs.
6.21/2.51	
6.21/2.51	*F(x, y, z) -> COND_F(&&(>(x, y), >(x, z)), x, y, z)
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)Bound*bni_19] + [bni_19]z[0] >= 0 & [(-1)bso_20] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	*COND_F(TRUE, x, y, z) -> F(x, +(y, 1), z)
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)Bound*bni_21] + [bni_21]z[0] >= 0 & [(-1)bso_22] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	*F(x, y, z) -> COND_F1(&&(>(x, y), >(x, z)), x, y, z)
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_23] + [bni_23]z[2] >= 0 & [(-1)bso_24] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	*COND_F1(TRUE, x, y, z) -> F(x, y, +(z, 1))
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(F(x[3], y[3], +(z[3], 1))), >=) & [(-1)Bound*bni_25] + [bni_25]z[2] >= 0 & [1 + (-1)bso_26] >= 0)
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	
6.21/2.51	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. 
6.21/2.51	
6.21/2.51	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.21/2.51	
6.21/2.51	Polynomial interpretation over integers[POLO]:
6.21/2.51	
6.21/2.51	   POL(TRUE) = 0
6.21/2.51	   POL(FALSE) = 0
6.21/2.51	   POL(F(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1
6.21/2.51	   POL(COND_F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2
6.21/2.51	   POL(&&(x_1, x_2)) = [-1]
6.21/2.51	   POL(>(x_1, x_2)) = [-1]
6.21/2.51	   POL(+(x_1, x_2)) = x_1 + x_2
6.21/2.51	   POL(1) = [1]
6.21/2.51	   POL(COND_F1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2
6.21/2.51	
6.21/2.51	
6.21/2.51	The following pairs are in P_>:
6.21/2.51	
6.21/2.51	
6.21/2.51	   COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1))
6.21/2.51	
6.21/2.51	
6.21/2.51	The following pairs are in P_bound:
6.21/2.51	
6.21/2.51	
6.21/2.51	   F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])
6.21/2.51	   COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1])
6.21/2.51	   F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])
6.21/2.51	   COND_F1(TRUE, x[3], y[3], z[3]) -> F(x[3], y[3], +(z[3], 1))
6.21/2.51	
6.21/2.51	
6.21/2.51	The following pairs are in P_>=:
6.21/2.51	
6.21/2.51	
6.21/2.51	   F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])
6.21/2.51	   COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1])
6.21/2.51	   F(x[2], y[2], z[2]) -> COND_F1(&&(>(x[2], y[2]), >(x[2], z[2])), x[2], y[2], z[2])
6.21/2.51	
6.21/2.51	
6.21/2.51	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.21/2.51	
6.21/2.51	   TRUE^1 -> &&(TRUE, TRUE)^1
6.21/2.51	   FALSE^1 -> &&(TRUE, FALSE)^1
6.21/2.51	   FALSE^1 -> &&(FALSE, TRUE)^1
6.21/2.51	   FALSE^1 -> &&(FALSE, FALSE)^1
6.21/2.51	
6.21/2.51	----------------------------------------
6.21/2.51	
6.21/2.51	(6)
6.21/2.51	Obligation:
6.21/2.51	IDP problem:
6.21/2.52	The following function symbols are pre-defined:
6.21/2.52	<<<
6.21/2.52	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.52	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.52	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.52	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.52	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.52	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.52	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.52	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.52	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.52	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.52	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.52	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.52	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.52	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.52	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.52	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.52	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.52	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.52	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.52	>>>
6.21/2.52	
6.21/2.52	
6.21/2.52	The following domains are used:
6.21/2.52	   Boolean, Integer
6.21/2.52	
6.21/2.52	R is empty.
6.21/2.52	
6.21/2.52	The integer pair graph contains the following rules and edges:
6.21/2.52	(0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0])
6.21/2.52	(1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1])
6.21/2.52	(2): F(x[2], y[2], z[2]) -> COND_F1(x[2] > y[2] && x[2] > z[2], x[2], y[2], z[2])
6.21/2.52	
6.21/2.52	   (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0])
6.21/2.52	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.21/2.52	   (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2] & z[1] ->^* z[2])
6.21/2.52	
6.21/2.52	The set Q consists of the following terms:
6.21/2.52	f(x0, x1, x2)
6.21/2.52	Cond_f(TRUE, x0, x1, x2)
6.21/2.52	Cond_f1(TRUE, x0, x1, x2)
6.21/2.52	
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(7) IDependencyGraphProof (EQUIVALENT)
6.21/2.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(8)
6.21/2.52	Obligation:
6.21/2.52	IDP problem:
6.21/2.52	The following function symbols are pre-defined:
6.21/2.52	<<<
6.21/2.52	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.52	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.52	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.52	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.52	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.52	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.52	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.52	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.52	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.52	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.52	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.52	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.52	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.52	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.52	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.52	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.52	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.52	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.52	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.52	>>>
6.21/2.52	
6.21/2.52	
6.21/2.52	The following domains are used:
6.21/2.52	   Integer, Boolean
6.21/2.52	
6.21/2.52	R is empty.
6.21/2.52	
6.21/2.52	The integer pair graph contains the following rules and edges:
6.21/2.52	(1): COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], y[1] + 1, z[1])
6.21/2.52	(0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0])
6.21/2.52	
6.21/2.52	   (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0] & z[1] ->^* z[0])
6.21/2.52	   (0) -> (1), if (x[0] > y[0] && x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.21/2.52	
6.21/2.52	The set Q consists of the following terms:
6.21/2.52	f(x0, x1, x2)
6.21/2.52	Cond_f(TRUE, x0, x1, x2)
6.21/2.52	Cond_f1(TRUE, x0, x1, x2)
6.21/2.52	
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(9) IDPNonInfProof (SOUND)
6.21/2.52	Used the following options for this NonInfProof:
6.21/2.52	
6.21/2.52	IDPGPoloSolver:
6.21/2.52	Range: [(-1,2)]
6.21/2.52	IsNat: false
6.21/2.52	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@6b1f24ae
6.21/2.52	Constraint Generator: NonInfConstraintGenerator:
6.21/2.52	PathGenerator: MetricPathGenerator:
6.21/2.52	Max Left Steps: 1
6.21/2.52	Max Right Steps: 1
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	The constraints were generated the following way:
6.21/2.52	
6.21/2.52	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.21/2.52	
6.21/2.52	Note that final constraints are written in bold face.
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	For Pair COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) the following chains were created:
6.21/2.52	*We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]), F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) which results in the following constraint:
6.21/2.52	
6.21/2.52	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & x[1]=x[0]1 & +(y[1], 1)=y[0]1 & z[1]=z[0]1  ==>  COND_F(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_F(TRUE, x[1], y[1], z[1])_>=_F(x[1], +(y[1], 1), z[1]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE  ==>  COND_F(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_F(TRUE, x[0], y[0], z[0])_>=_F(x[0], +(y[0], 1), z[0]) & (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=))
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [(-1)bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]z[0] + [bni_14]y[0] + [(2)bni_14]x[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(7)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.52	
6.21/2.52	(8)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	(9)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	For Pair F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) the following chains were created:
6.21/2.52	*We consider the chain F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]), COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1]) which results in the following constraint:
6.21/2.52	
6.21/2.52	(1)    (&&(>(x[0], y[0]), >(x[0], z[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1]  ==>  F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=))
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(2)    (>(x[0], y[0])=TRUE & >(x[0], z[0])=TRUE  ==>  F(x[0], y[0], z[0])_>=_NonInfC & F(x[0], y[0], z[0])_>=_COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0]) & (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=))
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(4)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(5)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [(-1)bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(6)    (x[0] >= 0 & y[0] + x[0] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]z[0] + [bni_16]y[0] + [(2)bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.21/2.52	
6.21/2.52	(7)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.21/2.52	
6.21/2.52	(8)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	(9)    (x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	To summarize, we get the following constraints P__>=_ for the following pairs.
6.21/2.52	
6.21/2.52	*COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1])
6.21/2.52	
6.21/2.52	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(F(x[1], +(y[1], 1), z[1])), >=) & [bni_14 + (-1)Bound*bni_14] + [bni_14]x[0] + [bni_14]z[0] >= 0 & [1 + (-1)bso_15] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	*F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])
6.21/2.52	
6.21/2.52	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	*(x[0] >= 0 & z[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])), >=) & [bni_16 + (-1)Bound*bni_16] + [bni_16]x[0] + [bni_16]z[0] >= 0 & [(-1)bso_17] >= 0)
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	
6.21/2.52	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. 
6.21/2.52	
6.21/2.52	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.21/2.52	
6.21/2.52	Polynomial interpretation over integers[POLO]:
6.21/2.52	
6.21/2.52	   POL(TRUE) = [1]
6.21/2.52	   POL(FALSE) = [3]
6.21/2.52	   POL(COND_F(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [2]x_2
6.21/2.52	   POL(F(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [2]x_1
6.21/2.52	   POL(+(x_1, x_2)) = x_1 + x_2
6.21/2.52	   POL(1) = [1]
6.21/2.52	   POL(&&(x_1, x_2)) = [-1]
6.21/2.52	   POL(>(x_1, x_2)) = [-1]
6.21/2.52	
6.21/2.52	
6.21/2.52	The following pairs are in P_>:
6.21/2.52	
6.21/2.52	
6.21/2.52	   COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1])
6.21/2.52	
6.21/2.52	
6.21/2.52	The following pairs are in P_bound:
6.21/2.52	
6.21/2.52	
6.21/2.52	   COND_F(TRUE, x[1], y[1], z[1]) -> F(x[1], +(y[1], 1), z[1])
6.21/2.52	   F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])
6.21/2.52	
6.21/2.52	
6.21/2.52	The following pairs are in P_>=:
6.21/2.52	
6.21/2.52	
6.21/2.52	   F(x[0], y[0], z[0]) -> COND_F(&&(>(x[0], y[0]), >(x[0], z[0])), x[0], y[0], z[0])
6.21/2.52	
6.21/2.52	
6.21/2.52	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.21/2.52	
6.21/2.52	   TRUE^1 -> &&(TRUE, TRUE)^1
6.21/2.52	   FALSE^1 -> &&(TRUE, FALSE)^1
6.21/2.52	   FALSE^1 -> &&(FALSE, TRUE)^1
6.21/2.52	
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(10)
6.21/2.52	Obligation:
6.21/2.52	IDP problem:
6.21/2.52	The following function symbols are pre-defined:
6.21/2.52	<<<
6.21/2.52	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.21/2.52	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.21/2.52	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.21/2.52	 / 	~ 	Div: (Integer, Integer) -> Integer
6.21/2.52	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.21/2.52	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.21/2.52	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.21/2.52	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.21/2.52	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.21/2.52	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.21/2.52	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.21/2.52	 + 	~ 	Add: (Integer, Integer) -> Integer
6.21/2.52	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.21/2.52	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.21/2.52	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.21/2.52	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.21/2.52	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.21/2.52	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.21/2.52	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.21/2.52	>>>
6.21/2.52	
6.21/2.52	
6.21/2.52	The following domains are used:
6.21/2.52	   Boolean, Integer
6.21/2.52	
6.21/2.52	R is empty.
6.21/2.52	
6.21/2.52	The integer pair graph contains the following rules and edges:
6.21/2.52	(0): F(x[0], y[0], z[0]) -> COND_F(x[0] > y[0] && x[0] > z[0], x[0], y[0], z[0])
6.21/2.52	
6.21/2.52	
6.21/2.52	The set Q consists of the following terms:
6.21/2.52	f(x0, x1, x2)
6.21/2.52	Cond_f(TRUE, x0, x1, x2)
6.21/2.52	Cond_f1(TRUE, x0, x1, x2)
6.21/2.52	
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(11) IDependencyGraphProof (EQUIVALENT)
6.21/2.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.21/2.52	----------------------------------------
6.21/2.52	
6.21/2.52	(12)
6.21/2.52	TRUE
6.21/2.58	EOF