6.52/2.68	YES
6.83/2.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
6.83/2.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.83/2.70	
6.83/2.70	
6.83/2.70	Termination of the given ITRS could be proven:
6.83/2.70	
6.83/2.70	(0) ITRS
6.83/2.70	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
6.83/2.70	(2) IDP
6.83/2.70	(3) UsableRulesProof [EQUIVALENT, 0 ms]
6.83/2.70	(4) IDP
6.83/2.70	(5) IDPNonInfProof [SOUND, 376 ms]
6.83/2.70	(6) IDP
6.83/2.70	(7) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.83/2.70	(8) IDP
6.83/2.70	(9) IDPNonInfProof [SOUND, 128 ms]
6.83/2.70	(10) IDP
6.83/2.70	(11) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.83/2.70	(12) TRUE
6.83/2.70	
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(0)
6.83/2.70	Obligation:
6.83/2.70	ITRS problem:
6.83/2.70	
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	The TRS R consists of the following rules:
6.83/2.70	eval_1(x, y, z) -> Cond_eval_1(x > z, x, y, z)
6.83/2.70	Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z)
6.83/2.70	eval_2(x, y, z) -> Cond_eval_2(x > z && y > z, x, y, z)
6.83/2.70	Cond_eval_2(TRUE, x, y, z) -> eval_2(x, y - 1, z)
6.83/2.70	eval_2(x, y, z) -> Cond_eval_21(x > z && z >= y, x, y, z)
6.83/2.70	Cond_eval_21(TRUE, x, y, z) -> eval_1(x - 1, y, z)
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(1) ITRStoIDPProof (EQUIVALENT)
6.83/2.70	Added dependency pairs
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(2)
6.83/2.70	Obligation:
6.83/2.70	IDP problem:
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	
6.83/2.70	The following domains are used:
6.83/2.70	   Integer, Boolean
6.83/2.70	
6.83/2.70	The ITRS R consists of the following rules:
6.83/2.70	eval_1(x, y, z) -> Cond_eval_1(x > z, x, y, z)
6.83/2.70	Cond_eval_1(TRUE, x, y, z) -> eval_2(x, y, z)
6.83/2.70	eval_2(x, y, z) -> Cond_eval_2(x > z && y > z, x, y, z)
6.83/2.70	Cond_eval_2(TRUE, x, y, z) -> eval_2(x, y - 1, z)
6.83/2.70	eval_2(x, y, z) -> Cond_eval_21(x > z && z >= y, x, y, z)
6.83/2.70	Cond_eval_21(TRUE, x, y, z) -> eval_1(x - 1, y, z)
6.83/2.70	
6.83/2.70	The integer pair graph contains the following rules and edges:
6.83/2.70	(0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > z[0], x[0], y[0], z[0])
6.83/2.70	(1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	(2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2] && y[2] > z[2], x[2], y[2], z[2])
6.83/2.70	(3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], y[3] - 1, z[3])
6.83/2.70	(4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4] && z[4] >= y[4], x[4], y[4], z[4])
6.83/2.70	(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5] - 1, y[5], z[5])
6.83/2.70	
6.83/2.70	   (0) -> (1), if (x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.83/2.70	   (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2])
6.83/2.70	   (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4])
6.83/2.70	   (2) -> (3), if (x[2] > z[2] && y[2] > z[2]  & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3])
6.83/2.70	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2])
6.83/2.70	   (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4])
6.83/2.70	   (4) -> (5), if (x[4] > z[4] && z[4] >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5])
6.83/2.70	   (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0])
6.83/2.70	
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(3) UsableRulesProof (EQUIVALENT)
6.83/2.70	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.
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(4)
6.83/2.70	Obligation:
6.83/2.70	IDP problem:
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	
6.83/2.70	The following domains are used:
6.83/2.70	   Integer, Boolean
6.83/2.70	
6.83/2.70	R is empty.
6.83/2.70	
6.83/2.70	The integer pair graph contains the following rules and edges:
6.83/2.70	(0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > z[0], x[0], y[0], z[0])
6.83/2.70	(1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	(2): EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(x[2] > z[2] && y[2] > z[2], x[2], y[2], z[2])
6.83/2.70	(3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], y[3] - 1, z[3])
6.83/2.70	(4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4] && z[4] >= y[4], x[4], y[4], z[4])
6.83/2.70	(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5] - 1, y[5], z[5])
6.83/2.70	
6.83/2.70	   (0) -> (1), if (x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.83/2.70	   (1) -> (2), if (x[1] ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2])
6.83/2.70	   (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4])
6.83/2.70	   (2) -> (3), if (x[2] > z[2] && y[2] > z[2]  & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3])
6.83/2.70	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2] & z[3] ->^* z[2])
6.83/2.70	   (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4])
6.83/2.70	   (4) -> (5), if (x[4] > z[4] && z[4] >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5])
6.83/2.70	   (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0])
6.83/2.70	
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(5) IDPNonInfProof (SOUND)
6.83/2.70	Used the following options for this NonInfProof:
6.83/2.70	
6.83/2.70	IDPGPoloSolver:
6.83/2.70	Range: [(-1,2)]
6.83/2.70	IsNat: false
6.83/2.70	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@430b8e6c
6.83/2.70	Constraint Generator: NonInfConstraintGenerator:
6.83/2.70	PathGenerator: MetricPathGenerator:
6.83/2.70	Max Left Steps: 1
6.83/2.70	Max Right Steps: 1
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	The constraints were generated the following way:
6.83/2.70	
6.83/2.70	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.83/2.70	
6.83/2.70	Note that final constraints are written in bold face.
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair EVAL_1(x, y, z) -> COND_EVAL_1(>(x, z), x, y, z) the following chains were created:
6.83/2.70	*We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1]  ==>  EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[0], z[0])=TRUE  ==>  EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] + [bni_29]y[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z) the following chains were created:
6.83/2.70	*We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (x[1]=x[2] & y[1]=y[2] & z[1]=z[2]  ==>  COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (x[1]=x[4] & y[1]=y[4] & z[1]=z[4]  ==>  COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair EVAL_2(x, y, z) -> COND_EVAL_2(&&(>(x, z), >(y, z)), x, y, z) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], -(y[3], 1), z[3]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[2], z[2]), >(y[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3]  ==>  EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[2], z[2])=TRUE & >(y[2], z[2])=TRUE  ==>  EVAL_2(x[2], y[2], z[2])_>=_NonInfC & EVAL_2(x[2], y[2], z[2])_>=_COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(2)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[2] + [bni_33]y[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(2)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[2] + [bni_33]y[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(2)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[2] + [bni_33]y[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(2)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]z[2] + [bni_33]y[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(3)bni_33 + (-1)Bound*bni_33] + [bni_33]z[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(3)bni_33 + (-1)Bound*bni_33] + [bni_33]z[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(3)bni_33 + (-1)Bound*bni_33] + [bni_33]z[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(x, -(y, 1), z) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], -(y[3], 1), z[3]), EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[2], z[2]), >(y[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1 & z[3]=z[2]1  ==>  COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[2], z[2])=TRUE & >(y[2], z[2])=TRUE  ==>  COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_2(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*We consider the chain EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2]), COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], -(y[3], 1), z[3]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[2], z[2]), >(y[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & x[3]=x[4] & -(y[3], 1)=y[4] & z[3]=z[4]  ==>  COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL_2(TRUE, x[3], y[3], z[3])_>=_EVAL_2(x[3], -(y[3], 1), z[3]) & (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[2], z[2])=TRUE & >(y[2], z[2])=TRUE  ==>  COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL_2(TRUE, x[2], y[2], z[2])_>=_EVAL_2(x[2], -(y[2], 1), z[2]) & (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[2] + [-1] + [-1]z[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[2] >= 0 & y[2] + [-1] + [-1]z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]z[2] + [bni_35]y[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[2] >= 0 & z[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair EVAL_2(x, y, z) -> COND_EVAL_21(&&(>(x, z), >=(z, y)), x, y, z) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[4], z[4]), >=(z[4], y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5]  ==>  EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[4], z[4])=TRUE & >=(z[4], y[4])=TRUE  ==>  EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] + [bni_37]y[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] + [bni_37]y[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] + [bni_37]y[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] + [bni_37]y[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[4] >= 0 & z[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[4], z[4]), >=(z[4], y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & -(x[5], 1)=x[0] & y[5]=y[0] & z[5]=z[0]  ==>  COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(-(x[5], 1), y[5], z[5]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[4], z[4])=TRUE & >=(z[4], y[4])=TRUE  ==>  COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_EVAL_1(-(x[4], 1), y[4], z[4]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] + [bni_39]y[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] + [bni_39]y[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] + [bni_39]y[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] + [bni_39]y[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[4] >= 0 & z[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	To summarize, we get the following constraints P__>=_ for the following pairs.
6.83/2.70	
6.83/2.70	*EVAL_1(x, y, z) -> COND_EVAL_1(>(x, z), x, y, z)
6.83/2.70	
6.83/2.70	*(x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_29] = 0 & [(2)bni_29 + (-1)Bound*bni_29] + [bni_29]z[0] >= 0 & [(-1)bso_30] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*COND_EVAL_1(TRUE, x, y, z) -> EVAL_2(x, y, z)
6.83/2.70	
6.83/2.70	*((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_31] = 0 & [(-1)bso_32] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*EVAL_2(x, y, z) -> COND_EVAL_2(&&(>(x, z), >(y, z)), x, y, z)
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(3)bni_33 + (-1)Bound*bni_33] + [bni_33]z[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])), >=) & [(3)bni_33 + (-1)Bound*bni_33] + [bni_33]z[2] >= 0 & [1 + (-1)bso_34] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*COND_EVAL_2(TRUE, x, y, z) -> EVAL_2(x, -(y, 1), z)
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[2] >= 0 & z[2] >= 0 & y[2] >= 0  ==>  (U^Increasing(EVAL_2(x[3], -(y[3], 1), z[3])), >=) & [(2)bni_35 + (-1)Bound*bni_35] + [bni_35]z[2] >= 0 & [(-1)bso_36] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*EVAL_2(x, y, z) -> COND_EVAL_21(&&(>(x, z), >=(z, y)), x, y, z)
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(2)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]z[4] >= 0 & [(-1)bso_38] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*COND_EVAL_21(TRUE, x, y, z) -> EVAL_1(-(x, 1), y, z)
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(2)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]z[4] >= 0 & [(-1)bso_40] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	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. 
6.83/2.70	
6.83/2.70	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.83/2.70	
6.83/2.70	Polynomial interpretation over integers[POLO]:
6.83/2.70	
6.83/2.70	   POL(TRUE) = 0
6.83/2.70	   POL(FALSE) = [3]
6.83/2.70	   POL(EVAL_1(x_1, x_2, x_3)) = [2] + [-1]x_3 + x_2
6.83/2.70	   POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [2] + [-1]x_4 + x_3
6.83/2.70	   POL(>(x_1, x_2)) = [1]
6.83/2.70	   POL(EVAL_2(x_1, x_2, x_3)) = [2] + [-1]x_3 + x_2
6.83/2.70	   POL(COND_EVAL_2(x_1, x_2, x_3, x_4)) = [1] + [-1]x_4 + x_3
6.83/2.70	   POL(&&(x_1, x_2)) = [-1]
6.83/2.70	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
6.83/2.70	   POL(1) = [1]
6.83/2.70	   POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [2] + [-1]x_4 + x_3
6.83/2.70	   POL(>=(x_1, x_2)) = [-1]
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_>:
6.83/2.70	
6.83/2.70	
6.83/2.70	   EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_bound:
6.83/2.70	
6.83/2.70	
6.83/2.70	   EVAL_2(x[2], y[2], z[2]) -> COND_EVAL_2(&&(>(x[2], z[2]), >(y[2], z[2])), x[2], y[2], z[2])
6.83/2.70	   COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], -(y[3], 1), z[3])
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_>=:
6.83/2.70	
6.83/2.70	
6.83/2.70	   EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])
6.83/2.70	   COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	   COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], -(y[3], 1), z[3])
6.83/2.70	   EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])
6.83/2.70	   COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5])
6.83/2.70	
6.83/2.70	
6.83/2.70	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.83/2.70	
6.83/2.70	   TRUE^1 -> &&(TRUE, TRUE)^1
6.83/2.70	   FALSE^1 -> &&(TRUE, FALSE)^1
6.83/2.70	   FALSE^1 -> &&(FALSE, TRUE)^1
6.83/2.70	   FALSE^1 -> &&(FALSE, FALSE)^1
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(6)
6.83/2.70	Obligation:
6.83/2.70	IDP problem:
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	
6.83/2.70	The following domains are used:
6.83/2.70	   Integer, Boolean
6.83/2.70	
6.83/2.70	R is empty.
6.83/2.70	
6.83/2.70	The integer pair graph contains the following rules and edges:
6.83/2.70	(0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > z[0], x[0], y[0], z[0])
6.83/2.70	(1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	(3): COND_EVAL_2(TRUE, x[3], y[3], z[3]) -> EVAL_2(x[3], y[3] - 1, z[3])
6.83/2.70	(4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4] && z[4] >= y[4], x[4], y[4], z[4])
6.83/2.70	(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5] - 1, y[5], z[5])
6.83/2.70	
6.83/2.70	   (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0])
6.83/2.70	   (0) -> (1), if (x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.83/2.70	   (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4])
6.83/2.70	   (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4] & z[3] ->^* z[4])
6.83/2.70	   (4) -> (5), if (x[4] > z[4] && z[4] >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5])
6.83/2.70	
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(7) IDependencyGraphProof (EQUIVALENT)
6.83/2.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(8)
6.83/2.70	Obligation:
6.83/2.70	IDP problem:
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	
6.83/2.70	The following domains are used:
6.83/2.70	   Integer, Boolean
6.83/2.70	
6.83/2.70	R is empty.
6.83/2.70	
6.83/2.70	The integer pair graph contains the following rules and edges:
6.83/2.70	(5): COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(x[5] - 1, y[5], z[5])
6.83/2.70	(4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4] && z[4] >= y[4], x[4], y[4], z[4])
6.83/2.70	(1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	(0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > z[0], x[0], y[0], z[0])
6.83/2.70	
6.83/2.70	   (5) -> (0), if (x[5] - 1 ->^* x[0] & y[5] ->^* y[0] & z[5] ->^* z[0])
6.83/2.70	   (0) -> (1), if (x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.83/2.70	   (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4])
6.83/2.70	   (4) -> (5), if (x[4] > z[4] && z[4] >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5])
6.83/2.70	
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(9) IDPNonInfProof (SOUND)
6.83/2.70	Used the following options for this NonInfProof:
6.83/2.70	
6.83/2.70	IDPGPoloSolver:
6.83/2.70	Range: [(-1,2)]
6.83/2.70	IsNat: false
6.83/2.70	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@430b8e6c
6.83/2.70	Constraint Generator: NonInfConstraintGenerator:
6.83/2.70	PathGenerator: MetricPathGenerator:
6.83/2.70	Max Left Steps: 1
6.83/2.70	Max Right Steps: 1
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	The constraints were generated the following way:
6.83/2.70	
6.83/2.70	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.83/2.70	
6.83/2.70	Note that final constraints are written in bold face.
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5]) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5]), EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[4], z[4]), >=(z[4], y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] & -(x[5], 1)=x[0] & y[5]=y[0] & z[5]=z[0]  ==>  COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL_21(TRUE, x[5], y[5], z[5])_>=_EVAL_1(-(x[5], 1), y[5], z[5]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[4], z[4])=TRUE & >=(z[4], y[4])=TRUE  ==>  COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL_21(TRUE, x[4], y[4], z[4])_>=_EVAL_1(-(x[4], 1), y[4], z[4]) & (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[4] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[4] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]z[4] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[4] >= 0 & z[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) the following chains were created:
6.83/2.70	*We consider the chain EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]), COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (&&(>(x[4], z[4]), >=(z[4], y[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5]  ==>  EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[4], z[4])=TRUE & >=(z[4], y[4])=TRUE  ==>  EVAL_2(x[4], y[4], z[4])_>=_NonInfC & EVAL_2(x[4], y[4], z[4])_>=_COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[4] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[4] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]z[4] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[4] >= 0 & z[4] + [-1]y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[4] >= 0 & z[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) the following chains were created:
6.83/2.70	*We consider the chain COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]), EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (x[1]=x[4] & y[1]=y[4] & z[1]=z[4]  ==>  COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL_1(TRUE, x[1], y[1], z[1])_>=_EVAL_2(x[1], y[1], z[1]) & (U^Increasing(EVAL_2(x[1], y[1], z[1])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_28] = 0 & [(-1)bso_29] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_28] = 0 & [(-1)bso_29] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    ((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_28] = 0 & [(-1)bso_29] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	For Pair EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) the following chains were created:
6.83/2.70	*We consider the chain EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1]) which results in the following constraint:
6.83/2.70	
6.83/2.70	(1)    (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1]  ==>  EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(2)    (>(x[0], z[0])=TRUE  ==>  EVAL_1(x[0], y[0], z[0])_>=_NonInfC & EVAL_1(x[0], y[0], z[0])_>=_COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=))
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(3)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(4)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(5)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(6)    (x[0] + [-1] + [-1]z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]z[0] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.83/2.70	
6.83/2.70	(7)    (x[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.83/2.70	
6.83/2.70	(8)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	(9)    (x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	To summarize, we get the following constraints P__>=_ for the following pairs.
6.83/2.70	
6.83/2.70	*COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5])
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(EVAL_1(-(x[5], 1), y[5], z[5])), >=) & [(-1)Bound*bni_24] + [bni_24]x[4] >= 0 & [1 + (-1)bso_25] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[4] >= 0 & z[4] >= 0 & y[4] >= 0  ==>  (U^Increasing(COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])), >=) & [(-1)Bound*bni_26] + [bni_26]x[4] >= 0 & [(-1)bso_27] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	
6.83/2.70	*((U^Increasing(EVAL_2(x[1], y[1], z[1])), >=) & [bni_28] = 0 & [(-1)bso_29] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	*EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])
6.83/2.70	
6.83/2.70	*(x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	*(x[0] >= 0 & z[0] >= 0  ==>  (U^Increasing(COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_30] + [bni_30]x[0] >= 0 & [(-1)bso_31] >= 0)
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	
6.83/2.70	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. 
6.83/2.70	
6.83/2.70	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.83/2.70	
6.83/2.70	Polynomial interpretation over integers[POLO]:
6.83/2.70	
6.83/2.70	   POL(TRUE) = 0
6.83/2.70	   POL(FALSE) = [3]
6.83/2.70	   POL(COND_EVAL_21(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 + [-1]x_1
6.83/2.70	   POL(EVAL_1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1
6.83/2.70	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
6.83/2.70	   POL(1) = [1]
6.83/2.70	   POL(EVAL_2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1
6.83/2.70	   POL(&&(x_1, x_2)) = 0
6.83/2.70	   POL(>(x_1, x_2)) = [-1]
6.83/2.70	   POL(>=(x_1, x_2)) = [-1]
6.83/2.70	   POL(COND_EVAL_1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_>:
6.83/2.70	
6.83/2.70	
6.83/2.70	   COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5])
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_bound:
6.83/2.70	
6.83/2.70	
6.83/2.70	   COND_EVAL_21(TRUE, x[5], y[5], z[5]) -> EVAL_1(-(x[5], 1), y[5], z[5])
6.83/2.70	   EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])
6.83/2.70	   EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])
6.83/2.70	
6.83/2.70	
6.83/2.70	The following pairs are in P_>=:
6.83/2.70	
6.83/2.70	
6.83/2.70	   EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(&&(>(x[4], z[4]), >=(z[4], y[4])), x[4], y[4], z[4])
6.83/2.70	   COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	   EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(>(x[0], z[0]), x[0], y[0], z[0])
6.83/2.70	
6.83/2.70	
6.83/2.70	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.83/2.70	
6.83/2.70	   TRUE^1 -> &&(TRUE, TRUE)^1
6.83/2.70	   FALSE^1 -> &&(TRUE, FALSE)^1
6.83/2.70	   FALSE^1 -> &&(FALSE, TRUE)^1
6.83/2.70	   FALSE^1 -> &&(FALSE, FALSE)^1
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(10)
6.83/2.70	Obligation:
6.83/2.70	IDP problem:
6.83/2.70	The following function symbols are pre-defined:
6.83/2.70	<<<
6.83/2.70	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.83/2.70	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.83/2.70	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.83/2.70	 / 	~ 	Div: (Integer, Integer) -> Integer
6.83/2.70	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.83/2.70	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.83/2.70	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.83/2.70	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.83/2.70	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.83/2.70	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.83/2.70	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.83/2.70	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.83/2.70	 + 	~ 	Add: (Integer, Integer) -> Integer
6.83/2.70	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.83/2.70	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.83/2.70	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.83/2.70	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.83/2.70	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.83/2.70	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.83/2.70	>>>
6.83/2.70	
6.83/2.70	
6.83/2.70	The following domains are used:
6.83/2.70	   Boolean, Integer
6.83/2.70	
6.83/2.70	R is empty.
6.83/2.70	
6.83/2.70	The integer pair graph contains the following rules and edges:
6.83/2.70	(4): EVAL_2(x[4], y[4], z[4]) -> COND_EVAL_21(x[4] > z[4] && z[4] >= y[4], x[4], y[4], z[4])
6.83/2.70	(1): COND_EVAL_1(TRUE, x[1], y[1], z[1]) -> EVAL_2(x[1], y[1], z[1])
6.83/2.70	(0): EVAL_1(x[0], y[0], z[0]) -> COND_EVAL_1(x[0] > z[0], x[0], y[0], z[0])
6.83/2.70	
6.83/2.70	   (0) -> (1), if (x[0] > z[0]  & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1])
6.83/2.70	   (1) -> (4), if (x[1] ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4])
6.83/2.70	
6.83/2.70	The set Q consists of the following terms:
6.83/2.70	eval_1(x0, x1, x2)
6.83/2.70	Cond_eval_1(TRUE, x0, x1, x2)
6.83/2.70	eval_2(x0, x1, x2)
6.83/2.70	Cond_eval_2(TRUE, x0, x1, x2)
6.83/2.70	Cond_eval_21(TRUE, x0, x1, x2)
6.83/2.70	
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(11) IDependencyGraphProof (EQUIVALENT)
6.83/2.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
6.83/2.70	----------------------------------------
6.83/2.70	
6.83/2.70	(12)
6.83/2.70	TRUE
6.83/2.75	EOF