6.06/2.46	YES
6.06/2.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
6.06/2.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.06/2.48	
6.06/2.48	
6.06/2.48	Termination of the given ITRS could be proven:
6.06/2.48	
6.06/2.48	(0) ITRS
6.06/2.48	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
6.06/2.48	(2) IDP
6.06/2.48	(3) UsableRulesProof [EQUIVALENT, 0 ms]
6.06/2.48	(4) IDP
6.06/2.48	(5) IDPNonInfProof [SOUND, 443 ms]
6.06/2.48	(6) AND
6.06/2.48	    (7) IDP
6.06/2.48	        (8) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.48	        (9) IDP
6.06/2.48	        (10) IDPNonInfProof [SOUND, 48 ms]
6.06/2.48	        (11) IDP
6.06/2.48	        (12) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.48	        (13) TRUE
6.06/2.48	    (14) IDP
6.06/2.48	        (15) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.48	        (16) IDP
6.06/2.48	        (17) IDPNonInfProof [SOUND, 42 ms]
6.06/2.48	        (18) IDP
6.06/2.48	        (19) IDependencyGraphProof [EQUIVALENT, 0 ms]
6.06/2.48	        (20) TRUE
6.06/2.48	
6.06/2.48	
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(0)
6.06/2.48	Obligation:
6.06/2.48	ITRS problem:
6.06/2.48	
6.06/2.48	The following function symbols are pre-defined:
6.06/2.48	<<<
6.06/2.48	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.48	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.48	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.48	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.48	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.48	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.48	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.48	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.48	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.48	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.48	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.48	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.48	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.48	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.48	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.48	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.48	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.48	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.48	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.48	>>>
6.06/2.48	
6.06/2.48	The TRS R consists of the following rules:
6.06/2.48	eval(x, y) -> Cond_eval(x + y > 0 && x > 0, x, y)
6.06/2.48	Cond_eval(TRUE, x, y) -> eval(x - 1, y)
6.06/2.48	eval(x, y) -> Cond_eval1(x + y > 0 && 0 >= x && y > 0, x, y)
6.06/2.48	Cond_eval1(TRUE, x, y) -> eval(x, y - 1)
6.06/2.48	eval(x, y) -> Cond_eval2(x + y > 0 && 0 >= x && 0 >= y, x, y)
6.06/2.48	Cond_eval2(TRUE, x, y) -> eval(x, y)
6.06/2.48	The set Q consists of the following terms:
6.06/2.48	eval(x0, x1)
6.06/2.48	Cond_eval(TRUE, x0, x1)
6.06/2.48	Cond_eval1(TRUE, x0, x1)
6.06/2.48	Cond_eval2(TRUE, x0, x1)
6.06/2.48	
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(1) ITRStoIDPProof (EQUIVALENT)
6.06/2.48	Added dependency pairs
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(2)
6.06/2.48	Obligation:
6.06/2.48	IDP problem:
6.06/2.48	The following function symbols are pre-defined:
6.06/2.48	<<<
6.06/2.48	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.48	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.48	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.48	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.48	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.48	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.48	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.48	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.48	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.48	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.48	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.48	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.48	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.48	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.48	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.48	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.48	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.48	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.48	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.48	>>>
6.06/2.48	
6.06/2.48	
6.06/2.48	The following domains are used:
6.06/2.48	   Boolean, Integer
6.06/2.48	
6.06/2.48	The ITRS R consists of the following rules:
6.06/2.48	eval(x, y) -> Cond_eval(x + y > 0 && x > 0, x, y)
6.06/2.48	Cond_eval(TRUE, x, y) -> eval(x - 1, y)
6.06/2.48	eval(x, y) -> Cond_eval1(x + y > 0 && 0 >= x && y > 0, x, y)
6.06/2.48	Cond_eval1(TRUE, x, y) -> eval(x, y - 1)
6.06/2.48	eval(x, y) -> Cond_eval2(x + y > 0 && 0 >= x && 0 >= y, x, y)
6.06/2.48	Cond_eval2(TRUE, x, y) -> eval(x, y)
6.06/2.48	
6.06/2.48	The integer pair graph contains the following rules and edges:
6.06/2.48	(0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > 0, x[0], y[0])
6.06/2.48	(1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1])
6.06/2.48	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.48	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.48	(4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4])
6.06/2.48	(5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5])
6.06/2.48	
6.06/2.48	   (0) -> (1), if (x[0] + y[0] > 0 && x[0] > 0  & x[0] ->^* x[1] & y[0] ->^* y[1])
6.06/2.48	   (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0])
6.06/2.48	   (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2])
6.06/2.48	   (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4])
6.06/2.48	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.48	   (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0])
6.06/2.48	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.48	   (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4])
6.06/2.48	   (4) -> (5), if (x[4] + y[4] > 0 && 0 >= x[4] && 0 >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5])
6.06/2.48	   (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0])
6.06/2.48	   (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2])
6.06/2.48	   (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4])
6.06/2.48	
6.06/2.48	The set Q consists of the following terms:
6.06/2.48	eval(x0, x1)
6.06/2.48	Cond_eval(TRUE, x0, x1)
6.06/2.48	Cond_eval1(TRUE, x0, x1)
6.06/2.48	Cond_eval2(TRUE, x0, x1)
6.06/2.48	
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(3) UsableRulesProof (EQUIVALENT)
6.06/2.48	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.
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(4)
6.06/2.48	Obligation:
6.06/2.48	IDP problem:
6.06/2.48	The following function symbols are pre-defined:
6.06/2.48	<<<
6.06/2.48	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.48	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.48	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.48	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.48	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.48	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.48	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.48	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.48	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.48	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.48	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.48	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.48	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.48	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.48	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.48	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.48	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.48	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.48	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.48	>>>
6.06/2.48	
6.06/2.48	
6.06/2.48	The following domains are used:
6.06/2.48	   Boolean, Integer
6.06/2.48	
6.06/2.48	R is empty.
6.06/2.48	
6.06/2.48	The integer pair graph contains the following rules and edges:
6.06/2.48	(0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > 0, x[0], y[0])
6.06/2.48	(1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1] - 1, y[1])
6.06/2.48	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.48	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.48	(4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] + y[4] > 0 && 0 >= x[4] && 0 >= y[4], x[4], y[4])
6.06/2.48	(5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5])
6.06/2.48	
6.06/2.48	   (0) -> (1), if (x[0] + y[0] > 0 && x[0] > 0  & x[0] ->^* x[1] & y[0] ->^* y[1])
6.06/2.48	   (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0])
6.06/2.48	   (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2])
6.06/2.48	   (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4])
6.06/2.48	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.48	   (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0])
6.06/2.48	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.48	   (3) -> (4), if (x[3] ->^* x[4] & y[3] - 1 ->^* y[4])
6.06/2.48	   (4) -> (5), if (x[4] + y[4] > 0 && 0 >= x[4] && 0 >= y[4]  & x[4] ->^* x[5] & y[4] ->^* y[5])
6.06/2.48	   (5) -> (0), if (x[5] ->^* x[0] & y[5] ->^* y[0])
6.06/2.48	   (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2])
6.06/2.48	   (5) -> (4), if (x[5] ->^* x[4] & y[5] ->^* y[4])
6.06/2.48	
6.06/2.48	The set Q consists of the following terms:
6.06/2.48	eval(x0, x1)
6.06/2.48	Cond_eval(TRUE, x0, x1)
6.06/2.48	Cond_eval1(TRUE, x0, x1)
6.06/2.48	Cond_eval2(TRUE, x0, x1)
6.06/2.48	
6.06/2.48	----------------------------------------
6.06/2.48	
6.06/2.48	(5) IDPNonInfProof (SOUND)
6.06/2.48	Used the following options for this NonInfProof:
6.06/2.48	
6.06/2.48	IDPGPoloSolver:
6.06/2.48	Range: [(-1,2)]
6.06/2.48	IsNat: false
6.06/2.48	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@60b45b4
6.06/2.48	Constraint Generator: NonInfConstraintGenerator:
6.06/2.48	PathGenerator: MetricPathGenerator:
6.06/2.48	Max Left Steps: 1
6.06/2.48	Max Right Steps: 1
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	The constraints were generated the following way:
6.06/2.48	
6.06/2.48	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.06/2.48	
6.06/2.48	Note that final constraints are written in bold face.
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	For Pair EVAL(x, y) -> COND_EVAL(&&(>(+(x, y), 0), >(x, 0)), x, y) the following chains were created:
6.06/2.48	*We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]) which results in the following constraint:
6.06/2.48	
6.06/2.48	(1)    (&&(>(+(x[0], y[0]), 0), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1]  ==>  EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(2)    (>(+(x[0], y[0]), 0)=TRUE & >(x[0], 0)=TRUE  ==>  EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(3)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(4)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(5)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.06/2.48	
6.06/2.48	(6)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.48	
6.06/2.48	(7)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	For Pair COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y) the following chains were created:
6.06/2.48	*We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]) which results in the following constraint:
6.06/2.48	
6.06/2.48	(1)    (&&(>(+(x[0], y[0]), 0), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1  ==>  COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(2)    (>(+(x[0], y[0]), 0)=TRUE & >(x[0], 0)=TRUE  ==>  COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(3)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(4)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(5)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.06/2.48	
6.06/2.48	(6)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	(7)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	*We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) which results in the following constraint:
6.06/2.48	
6.06/2.48	(1)    (&&(>(+(x[0], y[0]), 0), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[2] & y[1]=y[2]  ==>  COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(2)    (>(+(x[0], y[0]), 0)=TRUE & >(x[0], 0)=TRUE  ==>  COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(3)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(4)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.48	
6.06/2.48	(5)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.06/2.48	
6.06/2.48	(6)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	(7)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	
6.06/2.48	*We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1]), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(>(+(x[0], y[0]), 0), >(x[0], 0))=TRUE & x[0]=x[1] & y[0]=y[1] & -(x[1], 1)=x[4] & y[1]=y[4]  ==>  COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(-(x[1], 1), y[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(+(x[0], y[0]), 0)=TRUE & >(x[0], 0)=TRUE  ==>  COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(-(x[0], 1), y[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
6.06/2.49	
6.06/2.49	(6)    (x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	(7)    (x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair EVAL(x, y) -> COND_EVAL1(&&(&&(>(+(x, y), 0), >=(0, x)), >(y, 0)), x, y) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3]  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[2] >= 0 & [(-1)bso_26] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[2] >= 0 & [(-1)bso_26] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x[2] >= 0 & [(-1)bso_26] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1)) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[0] & -(y[3], 1)=y[0]  ==>  COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1  ==>  COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[4] & -(y[3], 1)=y[4]  ==>  COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair EVAL(x, y) -> COND_EVAL2(&&(&&(>(+(x, y), 0), >=(0, x)), >=(0, y)), x, y) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4]))=TRUE & x[4]=x[5] & y[4]=y[5]  ==>  EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>=(0, y[4])=TRUE & >(+(x[4], y[4]), 0)=TRUE & >=(0, x[4])=TRUE  ==>  EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    ([-1]y[4] >= 0 & x[4] + [-1] + y[4] >= 0 & [-1]x[4] >= 0  ==>  (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[4] >= 0 & [-1 + (-1)bso_30] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    ([-1]y[4] >= 0 & x[4] + [-1] + y[4] >= 0 & [-1]x[4] >= 0  ==>  (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[4] >= 0 & [-1 + (-1)bso_30] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    ([-1]y[4] >= 0 & x[4] + [-1] + y[4] >= 0 & [-1]x[4] >= 0  ==>  (U^Increasing(COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]x[4] >= 0 & [-1 + (-1)bso_30] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We solved constraint (5) using rule (IDP_SMT_SPLIT).
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair COND_EVAL2(TRUE, x, y) -> EVAL(x, y) the following chains were created:
6.06/2.49	*We consider the chain COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5]), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (x[5]=x[0] & y[5]=y[0]  ==>  COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*We consider the chain COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5]), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (x[5]=x[2] & y[5]=y[2]  ==>  COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*We consider the chain COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5]), EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (x[5]=x[4] & y[5]=y[4]  ==>  COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rule (IV) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(x[5], y[5]) & (U^Increasing(EVAL(x[5], y[5])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    ((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	To summarize, we get the following constraints P__>=_ for the following pairs.
6.06/2.49	
6.06/2.49	*EVAL(x, y) -> COND_EVAL(&&(>(+(x, y), 0), >(x, 0)), x, y)
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x[0] >= 0 & [(-1)bso_22] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*COND_EVAL(TRUE, x, y) -> EVAL(-(x, 1), y)
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(x[0] + [-1] + [-1]y[0] >= 0 & x[0] + [-1] >= 0 & y[0] >= 0  ==>  (U^Increasing(EVAL(-(x[1], 1), y[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*EVAL(x, y) -> COND_EVAL1(&&(&&(>(+(x, y), 0), >=(0, x)), >(y, 0)), x, y)
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x[2] >= 0 & [(-1)bso_26] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*COND_EVAL1(TRUE, x, y) -> EVAL(x, -(y, 1))
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x[2] >= 0 & [(-1)bso_28] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*EVAL(x, y) -> COND_EVAL2(&&(&&(>(+(x, y), 0), >=(0, x)), >=(0, y)), x, y)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*COND_EVAL2(TRUE, x, y) -> EVAL(x, y)
6.06/2.49	
6.06/2.49	*((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	*((U^Increasing(EVAL(x[5], y[5])), >=) & [bni_31] = 0 & [1 + (-1)bso_32] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	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. 
6.06/2.49	
6.06/2.49	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.06/2.49	
6.06/2.49	Polynomial interpretation over integers[POLO]:
6.06/2.49	
6.06/2.49	   POL(TRUE) = 0
6.06/2.49	   POL(FALSE) = [2]
6.06/2.49	   POL(EVAL(x_1, x_2)) = [-1] + x_1
6.06/2.49	   POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1
6.06/2.49	   POL(&&(x_1, x_2)) = 0
6.06/2.49	   POL(>(x_1, x_2)) = [-1]
6.06/2.49	   POL(+(x_1, x_2)) = x_1 + x_2
6.06/2.49	   POL(0) = 0
6.06/2.49	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
6.06/2.49	   POL(1) = [1]
6.06/2.49	   POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_2 + [-1]x_1
6.06/2.49	   POL(>=(x_1, x_2)) = [-1]
6.06/2.49	   POL(COND_EVAL2(x_1, x_2, x_3)) = x_2 + [2]x_1
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>:
6.06/2.49	
6.06/2.49	
6.06/2.49	   COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1])
6.06/2.49	   EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])
6.06/2.49	   COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5])
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_bound:
6.06/2.49	
6.06/2.49	
6.06/2.49	   EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])
6.06/2.49	   COND_EVAL(TRUE, x[1], y[1]) -> EVAL(-(x[1], 1), y[1])
6.06/2.49	   EVAL(x[4], y[4]) -> COND_EVAL2(&&(&&(>(+(x[4], y[4]), 0), >=(0, x[4])), >=(0, y[4])), x[4], y[4])
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>=:
6.06/2.49	
6.06/2.49	
6.06/2.49	   EVAL(x[0], y[0]) -> COND_EVAL(&&(>(+(x[0], y[0]), 0), >(x[0], 0)), x[0], y[0])
6.06/2.49	   EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	   COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	
6.06/2.49	
6.06/2.49	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.06/2.49	
6.06/2.49	   &&(TRUE, TRUE)^1 <-> TRUE^1
6.06/2.49	   FALSE^1 -> &&(TRUE, FALSE)^1
6.06/2.49	   FALSE^1 -> &&(FALSE, TRUE)^1
6.06/2.49	   FALSE^1 -> &&(FALSE, FALSE)^1
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(6)
6.06/2.49	Complex Obligation (AND)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(7)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Boolean, Integer
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] + y[0] > 0 && x[0] > 0, x[0], y[0])
6.06/2.49	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.49	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.49	
6.06/2.49	   (3) -> (0), if (x[3] ->^* x[0] & y[3] - 1 ->^* y[0])
6.06/2.49	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.49	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(8) IDependencyGraphProof (EQUIVALENT)
6.06/2.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(9)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Integer, Boolean
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.49	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.49	
6.06/2.49	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.49	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(10) IDPNonInfProof (SOUND)
6.06/2.49	Used the following options for this NonInfProof:
6.06/2.49	
6.06/2.49	IDPGPoloSolver:
6.06/2.49	Range: [(-1,2)]
6.06/2.49	IsNat: false
6.06/2.49	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@60b45b4
6.06/2.49	Constraint Generator: NonInfConstraintGenerator:
6.06/2.49	PathGenerator: MetricPathGenerator:
6.06/2.49	Max Left Steps: 1
6.06/2.49	Max Right Steps: 1
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	The constraints were generated the following way:
6.06/2.49	
6.06/2.49	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.06/2.49	
6.06/2.49	Note that final constraints are written in bold face.
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1  ==>  COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [(-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [(-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [(-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]y[2] + [bni_13]x[2] >= 0 & [(-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3]  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	To summarize, we get the following constraints P__>=_ for the following pairs.
6.06/2.49	
6.06/2.49	*COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [bni_13]y[2] + [bni_13]x[2] >= 0 & [(-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]y[2] + [bni_15]x[2] >= 0 & [1 + (-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	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. 
6.06/2.49	
6.06/2.49	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.06/2.49	
6.06/2.49	Polynomial interpretation over integers[POLO]:
6.06/2.49	
6.06/2.49	   POL(TRUE) = [1]
6.06/2.49	   POL(FALSE) = [3]
6.06/2.49	   POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2 + [-1]x_1
6.06/2.49	   POL(EVAL(x_1, x_2)) = [-1] + x_2 + [-1]x_1
6.06/2.49	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
6.06/2.49	   POL(1) = [1]
6.06/2.49	   POL(&&(x_1, x_2)) = [1]
6.06/2.49	   POL(>(x_1, x_2)) = [-1]
6.06/2.49	   POL(+(x_1, x_2)) = x_1 + x_2
6.06/2.49	   POL(0) = 0
6.06/2.49	   POL(>=(x_1, x_2)) = [-1]
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>:
6.06/2.49	
6.06/2.49	
6.06/2.49	   EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_bound:
6.06/2.49	
6.06/2.49	
6.06/2.49	   COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	   EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>=:
6.06/2.49	
6.06/2.49	
6.06/2.49	   COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	
6.06/2.49	
6.06/2.49	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.06/2.49	
6.06/2.49	   &&(TRUE, TRUE)^1 <-> TRUE^1
6.06/2.49	   FALSE^1 -> &&(TRUE, FALSE)^1
6.06/2.49	   FALSE^1 -> &&(FALSE, TRUE)^1
6.06/2.49	   FALSE^1 -> &&(FALSE, FALSE)^1
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(11)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Integer
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.49	
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(12) IDependencyGraphProof (EQUIVALENT)
6.06/2.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(13)
6.06/2.49	TRUE
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(14)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Boolean, Integer
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.49	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.49	(5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5], y[5])
6.06/2.49	
6.06/2.49	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.49	   (5) -> (2), if (x[5] ->^* x[2] & y[5] ->^* y[2])
6.06/2.49	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(15) IDependencyGraphProof (EQUIVALENT)
6.06/2.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(16)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Integer, Boolean
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] - 1)
6.06/2.49	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.49	
6.06/2.49	   (3) -> (2), if (x[3] ->^* x[2] & y[3] - 1 ->^* y[2])
6.06/2.49	   (2) -> (3), if (x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0  & x[2] ->^* x[3] & y[2] ->^* y[3])
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(17) IDPNonInfProof (SOUND)
6.06/2.49	Used the following options for this NonInfProof:
6.06/2.49	
6.06/2.49	IDPGPoloSolver:
6.06/2.49	Range: [(-1,2)]
6.06/2.49	IsNat: false
6.06/2.49	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@60b45b4
6.06/2.49	Constraint Generator: NonInfConstraintGenerator:
6.06/2.49	PathGenerator: MetricPathGenerator:
6.06/2.49	Max Left Steps: 1
6.06/2.49	Max Right Steps: 1
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	The constraints were generated the following way:
6.06/2.49	
6.06/2.49	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
6.06/2.49	
6.06/2.49	Note that final constraints are written in bold face.
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)), EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3] & x[3]=x[2]1 & -(y[3], 1)=y[2]1  ==>  COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], -(y[3], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], -(y[2], 1)) & (U^Increasing(EVAL(x[3], -(y[3], 1))), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] + [(-1)bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] + [bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	For Pair EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) the following chains were created:
6.06/2.49	*We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1)) which results in the following constraint:
6.06/2.49	
6.06/2.49	(1)    (&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0))=TRUE & x[2]=x[3] & y[2]=y[3]  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(2)    (>(y[2], 0)=TRUE & >(+(x[2], y[2]), 0)=TRUE & >=(0, x[2])=TRUE  ==>  EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=))
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(3)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [(-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(4)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [(-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(5)    (y[2] + [-1] >= 0 & x[2] + [-1] + y[2] >= 0 & [-1]x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]y[2] + [(-1)bni_15]x[2] >= 0 & [(-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
6.06/2.49	
6.06/2.49	(6)    (y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	To summarize, we get the following constraints P__>=_ for the following pairs.
6.06/2.49	
6.06/2.49	*COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(EVAL(x[3], -(y[3], 1))), >=) & [(-2)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]y[2] + [bni_13]x[2] >= 0 & [1 + (-1)bso_14] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	*EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	*(y[2] + [-1] >= 0 & [-1]x[2] + [-1] + y[2] >= 0 & x[2] >= 0  ==>  (U^Increasing(COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]y[2] + [bni_15]x[2] >= 0 & [(-1)bso_16] >= 0)
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	
6.06/2.49	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. 
6.06/2.49	
6.06/2.49	Using the following integer polynomial ordering the  resulting constraints can be solved 
6.06/2.49	
6.06/2.49	Polynomial interpretation over integers[POLO]:
6.06/2.49	
6.06/2.49	   POL(TRUE) = [1]
6.06/2.49	   POL(FALSE) = 0
6.06/2.49	   POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [2]x_3 + [-1]x_2 + [-1]x_1
6.06/2.49	   POL(EVAL(x_1, x_2)) = [-1] + [2]x_2 + [-1]x_1
6.06/2.49	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
6.06/2.49	   POL(1) = [1]
6.06/2.49	   POL(&&(x_1, x_2)) = 0
6.06/2.49	   POL(>(x_1, x_2)) = [-1]
6.06/2.49	   POL(+(x_1, x_2)) = x_1 + x_2
6.06/2.49	   POL(0) = 0
6.06/2.49	   POL(>=(x_1, x_2)) = [-1]
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>:
6.06/2.49	
6.06/2.49	
6.06/2.49	   COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_bound:
6.06/2.49	
6.06/2.49	
6.06/2.49	   COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], -(y[3], 1))
6.06/2.49	   EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	
6.06/2.49	The following pairs are in P_>=:
6.06/2.49	
6.06/2.49	
6.06/2.49	   EVAL(x[2], y[2]) -> COND_EVAL1(&&(&&(>(+(x[2], y[2]), 0), >=(0, x[2])), >(y[2], 0)), x[2], y[2])
6.06/2.49	
6.06/2.49	
6.06/2.49	At least the following rules have been oriented under context sensitive arithmetic replacement:
6.06/2.49	
6.06/2.49	   TRUE^1 -> &&(TRUE, TRUE)^1
6.06/2.49	   FALSE^1 -> &&(TRUE, FALSE)^1
6.06/2.49	   FALSE^1 -> &&(FALSE, TRUE)^1
6.06/2.49	   &&(FALSE, FALSE)^1 <-> FALSE^1
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(18)
6.06/2.49	Obligation:
6.06/2.49	IDP problem:
6.06/2.49	The following function symbols are pre-defined:
6.06/2.49	<<<
6.06/2.49	 & 	~ 	Bwand: (Integer, Integer) -> Integer
6.06/2.49	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
6.06/2.49	 | 	~ 	Bwor: (Integer, Integer) -> Integer
6.06/2.49	 / 	~ 	Div: (Integer, Integer) -> Integer
6.06/2.49	 != 	~ 	Neq: (Integer, Integer) -> Boolean
6.06/2.49	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
6.06/2.49	 ! 	~ 	Lnot: (Boolean) -> Boolean
6.06/2.49	 = 	~ 	Eq: (Integer, Integer) -> Boolean
6.06/2.49	 <= 	~ 	Le: (Integer, Integer) -> Boolean
6.06/2.49	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
6.06/2.49	 % 	~ 	Mod: (Integer, Integer) -> Integer
6.06/2.49	 > 	~ 	Gt: (Integer, Integer) -> Boolean
6.06/2.49	 + 	~ 	Add: (Integer, Integer) -> Integer
6.06/2.49	 -1 	~ 	UnaryMinus: (Integer) -> Integer
6.06/2.49	 < 	~ 	Lt: (Integer, Integer) -> Boolean
6.06/2.49	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
6.06/2.49	 - 	~ 	Sub: (Integer, Integer) -> Integer
6.06/2.49	 ~ 	~ 	Bwnot: (Integer) -> Integer
6.06/2.49	 * 	~ 	Mul: (Integer, Integer) -> Integer
6.06/2.49	>>>
6.06/2.49	
6.06/2.49	
6.06/2.49	The following domains are used:
6.06/2.49	   Boolean, Integer
6.06/2.49	
6.06/2.49	R is empty.
6.06/2.49	
6.06/2.49	The integer pair graph contains the following rules and edges:
6.06/2.49	(2): EVAL(x[2], y[2]) -> COND_EVAL1(x[2] + y[2] > 0 && 0 >= x[2] && y[2] > 0, x[2], y[2])
6.06/2.49	
6.06/2.49	
6.06/2.49	The set Q consists of the following terms:
6.06/2.49	eval(x0, x1)
6.06/2.49	Cond_eval(TRUE, x0, x1)
6.06/2.49	Cond_eval1(TRUE, x0, x1)
6.06/2.49	Cond_eval2(TRUE, x0, x1)
6.06/2.49	
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(19) IDependencyGraphProof (EQUIVALENT)
6.06/2.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.06/2.49	----------------------------------------
6.06/2.49	
6.06/2.49	(20)
6.06/2.49	TRUE
6.29/2.51	EOF