134.27/38.89	YES
134.27/38.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
134.27/38.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
134.27/38.91	
134.27/38.91	
134.27/38.91	Termination of the given ITRS could be proven:
134.27/38.91	
134.27/38.91	(0) ITRS
134.27/38.91	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
134.27/38.91	(2) IDP
134.27/38.91	(3) UsableRulesProof [EQUIVALENT, 0 ms]
134.27/38.91	(4) IDP
134.27/38.91	(5) IDPNonInfProof [SOUND, 26.9 s]
134.27/38.91	(6) IDP
134.27/38.91	(7) IDPNonInfProof [SOUND, 35 ms]
134.27/38.91	(8) IDP
134.27/38.91	(9) IDependencyGraphProof [EQUIVALENT, 0 ms]
134.27/38.91	(10) TRUE
134.27/38.91	
134.27/38.91	
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(0)
134.27/38.91	Obligation:
134.27/38.91	ITRS problem:
134.27/38.91	
134.27/38.91	The following function symbols are pre-defined:
134.27/38.91	<<<
134.27/38.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
134.27/38.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
134.27/38.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
134.27/38.91	 / 	~ 	Div: (Integer, Integer) -> Integer
134.27/38.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
134.27/38.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
134.27/38.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
134.27/38.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
134.27/38.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
134.27/38.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
134.27/38.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
134.27/38.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
134.27/38.91	 + 	~ 	Add: (Integer, Integer) -> Integer
134.27/38.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
134.27/38.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
134.27/38.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
134.27/38.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
134.27/38.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
134.27/38.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
134.27/38.91	>>>
134.27/38.91	
134.27/38.91	The TRS R consists of the following rules:
134.27/38.91	diff(x, y) -> cond1(x = y, x, y)
134.27/38.91	cond1(TRUE, x, y) -> 0
134.27/38.91	cond1(FALSE, x, y) -> cond2(x > y, x, y)
134.27/38.91	cond2(TRUE, x, y) -> 1 + diff(x, y + 1)
134.27/38.91	cond2(FALSE, x, y) -> 1 + diff(x + 1, y)
134.27/38.91	The set Q consists of the following terms:
134.27/38.91	diff(x0, x1)
134.27/38.91	cond1(TRUE, x0, x1)
134.27/38.91	cond1(FALSE, x0, x1)
134.27/38.91	cond2(TRUE, x0, x1)
134.27/38.91	cond2(FALSE, x0, x1)
134.27/38.91	
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(1) ITRStoIDPProof (EQUIVALENT)
134.27/38.91	Added dependency pairs
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(2)
134.27/38.91	Obligation:
134.27/38.91	IDP problem:
134.27/38.91	The following function symbols are pre-defined:
134.27/38.91	<<<
134.27/38.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
134.27/38.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
134.27/38.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
134.27/38.91	 / 	~ 	Div: (Integer, Integer) -> Integer
134.27/38.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
134.27/38.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
134.27/38.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
134.27/38.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
134.27/38.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
134.27/38.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
134.27/38.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
134.27/38.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
134.27/38.91	 + 	~ 	Add: (Integer, Integer) -> Integer
134.27/38.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
134.27/38.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
134.27/38.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
134.27/38.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
134.27/38.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
134.27/38.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
134.27/38.91	>>>
134.27/38.91	
134.27/38.91	
134.27/38.91	The following domains are used:
134.27/38.91	   Integer
134.27/38.91	
134.27/38.91	The ITRS R consists of the following rules:
134.27/38.91	diff(x, y) -> cond1(x = y, x, y)
134.27/38.91	cond1(TRUE, x, y) -> 0
134.27/38.91	cond1(FALSE, x, y) -> cond2(x > y, x, y)
134.27/38.91	cond2(TRUE, x, y) -> 1 + diff(x, y + 1)
134.27/38.91	cond2(FALSE, x, y) -> 1 + diff(x + 1, y)
134.27/38.91	
134.27/38.91	The integer pair graph contains the following rules and edges:
134.27/38.91	(0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0])
134.27/38.91	(1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1])
134.27/38.91	(2): COND2(TRUE, x[2], y[2]) -> DIFF(x[2], y[2] + 1)
134.27/38.91	(3): COND2(FALSE, x[3], y[3]) -> DIFF(x[3] + 1, y[3])
134.27/38.91	
134.27/38.91	   (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1])
134.27/38.91	   (1) -> (2), if (x[1] > y[1]  & x[1] ->^* x[2] & y[1] ->^* y[2])
134.27/38.91	   (1) -> (3), if (x[1] > y[1] ->^* FALSE & x[1] ->^* x[3] & y[1] ->^* y[3])
134.27/38.91	   (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0])
134.27/38.91	   (3) -> (0), if (x[3] + 1 ->^* x[0] & y[3] ->^* y[0])
134.27/38.91	
134.27/38.91	The set Q consists of the following terms:
134.27/38.91	diff(x0, x1)
134.27/38.91	cond1(TRUE, x0, x1)
134.27/38.91	cond1(FALSE, x0, x1)
134.27/38.91	cond2(TRUE, x0, x1)
134.27/38.91	cond2(FALSE, x0, x1)
134.27/38.91	
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(3) UsableRulesProof (EQUIVALENT)
134.27/38.91	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.
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(4)
134.27/38.91	Obligation:
134.27/38.91	IDP problem:
134.27/38.91	The following function symbols are pre-defined:
134.27/38.91	<<<
134.27/38.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
134.27/38.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
134.27/38.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
134.27/38.91	 / 	~ 	Div: (Integer, Integer) -> Integer
134.27/38.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
134.27/38.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
134.27/38.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
134.27/38.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
134.27/38.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
134.27/38.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
134.27/38.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
134.27/38.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
134.27/38.91	 + 	~ 	Add: (Integer, Integer) -> Integer
134.27/38.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
134.27/38.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
134.27/38.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
134.27/38.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
134.27/38.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
134.27/38.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
134.27/38.91	>>>
134.27/38.91	
134.27/38.91	
134.27/38.91	The following domains are used:
134.27/38.91	   Integer
134.27/38.91	
134.27/38.91	R is empty.
134.27/38.91	
134.27/38.91	The integer pair graph contains the following rules and edges:
134.27/38.91	(0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0])
134.27/38.91	(1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1])
134.27/38.91	(2): COND2(TRUE, x[2], y[2]) -> DIFF(x[2], y[2] + 1)
134.27/38.91	(3): COND2(FALSE, x[3], y[3]) -> DIFF(x[3] + 1, y[3])
134.27/38.91	
134.27/38.91	   (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1])
134.27/38.91	   (1) -> (2), if (x[1] > y[1]  & x[1] ->^* x[2] & y[1] ->^* y[2])
134.27/38.91	   (1) -> (3), if (x[1] > y[1] ->^* FALSE & x[1] ->^* x[3] & y[1] ->^* y[3])
134.27/38.91	   (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0])
134.27/38.91	   (3) -> (0), if (x[3] + 1 ->^* x[0] & y[3] ->^* y[0])
134.27/38.91	
134.27/38.91	The set Q consists of the following terms:
134.27/38.91	diff(x0, x1)
134.27/38.91	cond1(TRUE, x0, x1)
134.27/38.91	cond1(FALSE, x0, x1)
134.27/38.91	cond2(TRUE, x0, x1)
134.27/38.91	cond2(FALSE, x0, x1)
134.27/38.91	
134.27/38.91	----------------------------------------
134.27/38.91	
134.27/38.91	(5) IDPNonInfProof (SOUND)
134.27/38.91	Used the following options for this NonInfProof:
134.27/38.91	
134.27/38.91	IDPGPoloSolver:
134.27/38.91	Range: [(-1,2)]
134.27/38.91	IsNat: false
134.27/38.91	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@600d1117
134.27/38.91	Constraint Generator: NonInfConstraintGenerator:
134.27/38.91	PathGenerator: MetricPathGenerator:
134.27/38.91	Max Left Steps: 2
134.27/38.92	Max Right Steps: 1
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	The constraints were generated the following way:
134.27/38.92	
134.27/38.92	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
134.27/38.92	
134.27/38.92	Note that final constraints are written in bold face.
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair DIFF(x, y) -> COND1(=(x, y), x, y) the following chains were created:
134.27/38.92	*We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1]  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(2)    (<(x[0], y[0])=TRUE  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	(3)    (>(x[0], y[0])=TRUE  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(7)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(8)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]y[0] + [(-1)bni_18]x[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(9)    (x[0] + [-1] + [-1]y[0] >= 0 & [-1]y[0] + x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(10)    (y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(11)    (x[0] >= 0 & [1] + x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(12)    (y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	(13)    (y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(14)    (x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	(15)    (x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair COND1(FALSE, x, y) -> COND2(>(x, y), x, y) the following chains were created:
134.27/38.92	*We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (>(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2]  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rule (IV) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[1], y[1])=TRUE  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{0, y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{0, y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[1] + [-1] + [-1]y[1] >= 0 & [-1]y[1] + x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (x[1] >= 0 & [1] + x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(7)    (x[1] >= 0 & [1] + x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	(8)    (x[1] >= 0 & [1] + x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (>(x[1], y[1])=FALSE & x[1]=x[3] & y[1]=y[3]  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rule (IV) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[1], y[1])=FALSE  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{0, y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{0, y[1] + [-1]x[1]} >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(5)    (y[1] + [-1]x[1] >= 0 & [-1]y[1] + x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	(6)    (y[1] + [-1]x[1] >= 0 & y[1] + [-1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] + [(-1)bni_20]x[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(7)    (y[1] >= 0 & [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(8)    (y[1] >= 0 & [-1] + y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(9)    (0 >= 0 & 0 >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(10)    (0 >= 0 & 0 >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	(11)    (0 >= 0 & 0 >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(12)    (y[1] >= 0 & [-1] + y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	(13)    (y[1] >= 0 & [-1] + y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair COND2(TRUE, x, y) -> DIFF(x, +(y, 1)) the following chains were created:
134.27/38.92	*We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)), DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1] & >(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2] & x[2]=x[0]1 & +(y[2], 1)=y[0]1  ==>  COND2(TRUE, x[2], y[2])_>=_NonInfC & COND2(TRUE, x[2], y[2])_>=_DIFF(x[2], +(y[2], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[0], y[0])=TRUE & <(x[0], y[0])=TRUE  ==>  COND2(TRUE, x[0], y[0])_>=_NonInfC & COND2(TRUE, x[0], y[0])_>=_DIFF(x[0], +(y[0], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_23] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_23] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_23] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(7)    (x[0] + [-1] + [-1]y[0] >= 0 & [-1]y[0] + x[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We solved constraint (6) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(8)    (x[0] >= 0 & [1] + x[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(9)    (x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	(10)    (x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair COND2(FALSE, x, y) -> DIFF(+(x, 1), y) the following chains were created:
134.27/38.92	*We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3]), DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1] & >(x[1], y[1])=FALSE & x[1]=x[3] & y[1]=y[3] & +(x[3], 1)=x[0]1 & y[3]=y[0]1  ==>  COND2(FALSE, x[3], y[3])_>=_NonInfC & COND2(FALSE, x[3], y[3])_>=_DIFF(+(x[3], 1), y[3]) & (U^Increasing(DIFF(+(x[3], 1), y[3])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[0], y[0])=FALSE & <(x[0], y[0])=TRUE  ==>  COND2(FALSE, x[0], y[0])_>=_NonInfC & COND2(FALSE, x[0], y[0])_>=_DIFF(+(x[0], 1), y[0]) & (U^Increasing(DIFF(+(x[3], 1), y[3])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_25] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [-1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_25] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [-1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (y[0] + [-1]x[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]max{0, y[0] + [-1]x[0]} >= 0 & [(-1)bso_25] + max{0, y[0] + [-1]x[0]} + [-1]max{0, y[0] + [-1] + [-1]x[0]} >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(7)    (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 & [-1]y[0] + [1] + x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [(-1)bni_24]x[0] >= 0 & [(-1)bso_25] + y[0] + [-1]x[0] >= 0)
134.27/38.92	
134.27/38.92	(8)    (y[0] + [-1]x[0] >= 0 & y[0] + [-1] + [-1]x[0] >= 0 & y[0] + [-2] + [-1]x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] + [(-1)bni_24]x[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We solved constraint (6) using rule (POLY_REMOVE_MIN_MAX).We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(9)    (y[0] >= 0 & [-1] + y[0] >= 0 & [1] + [-1]y[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [(-1)bso_25] + y[0] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(10)    (y[0] >= 0 & [-1] + y[0] >= 0 & [-2] + y[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(11)    (y[0] >= 0 & [-1] + y[0] >= 0 & [1] + [-1]y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [(-1)bso_25] + y[0] >= 0)
134.27/38.92	
134.27/38.92	(12)    (y[0] >= 0 & [-1] + y[0] >= 0 & [1] + [-1]y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [(-1)bso_25] + y[0] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(13)    (y[0] >= 0 & [-1] + y[0] >= 0 & [-2] + y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	(14)    (y[0] >= 0 & [-1] + y[0] >= 0 & [-2] + y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	To summarize, we get the following constraints P__>=_ for the following pairs.
134.27/38.92	
134.27/38.92	*DIFF(x, y) -> COND1(=(x, y), x, y)
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_18] + [bni_18]y[0] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*COND1(FALSE, x, y) -> COND2(>(x, y), x, y)
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & [1] + x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & [1] + x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(0 >= 0 & 0 >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(0 >= 0 & 0 >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[1] >= 0 & [-1] + y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[1] >= 0 & [-1] + y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [(-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*COND2(TRUE, x, y) -> DIFF(x, +(y, 1))
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & [1] + x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_22 + (-1)Bound*bni_22] >= 0 & [(-1)bso_23] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*COND2(FALSE, x, y) -> DIFF(+(x, 1), y)
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & [-1] + y[0] >= 0 & [1] + [-1]y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [(-1)bso_25] + y[0] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & [-1] + y[0] >= 0 & [1] + [-1]y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [(-1)bso_25] + y[0] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & [-1] + y[0] >= 0 & [-2] + y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & [-1] + y[0] >= 0 & [-2] + y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(DIFF(+(x[3], 1), y[3])), >=) & [(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]y[0] >= 0 & [1 + (-1)bso_25] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	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. 
134.27/38.92	
134.27/38.92	Using the following integer polynomial ordering the  resulting constraints can be solved 
134.27/38.92	
134.27/38.92	Polynomial interpretation over integers[POLO]:
134.27/38.92	
134.27/38.92	   POL(TRUE) = 0
134.27/38.92	   POL(FALSE) = 0
134.27/38.92	   POL(DIFF(x_1, x_2)) = [-1] + max{0, x_2 + [-1]x_1}
134.27/38.92	   POL(COND1(x_1, x_2, x_3)) = [-1] + max{0, x_3 + [-1]x_2}
134.27/38.92	   POL(=(x_1, x_2)) = [-1]
134.27/38.92	   POL(COND2(x_1, x_2, x_3)) = [-1] + max{0, x_3 + [-1]x_2}
134.27/38.92	   POL(>(x_1, x_2)) = [-1]
134.27/38.92	   POL(+(x_1, x_2)) = x_1 + x_2
134.27/38.92	   POL(1) = [1]
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_>:
134.27/38.92	
134.27/38.92	
134.27/38.92	   COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3])
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_bound:
134.27/38.92	
134.27/38.92	
134.27/38.92	   DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0])
134.27/38.92	   COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1])
134.27/38.92	   COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1))
134.27/38.92	   COND2(FALSE, x[3], y[3]) -> DIFF(+(x[3], 1), y[3])
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_>=:
134.27/38.92	
134.27/38.92	
134.27/38.92	   DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0])
134.27/38.92	   COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1])
134.27/38.92	   COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1))
134.27/38.92	
134.27/38.92	
134.27/38.92	There are no usable rules.
134.27/38.92	----------------------------------------
134.27/38.92	
134.27/38.92	(6)
134.27/38.92	Obligation:
134.27/38.92	IDP problem:
134.27/38.92	The following function symbols are pre-defined:
134.27/38.92	<<<
134.27/38.92	 & 	~ 	Bwand: (Integer, Integer) -> Integer
134.27/38.92	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
134.27/38.92	 | 	~ 	Bwor: (Integer, Integer) -> Integer
134.27/38.92	 / 	~ 	Div: (Integer, Integer) -> Integer
134.27/38.92	 != 	~ 	Neq: (Integer, Integer) -> Boolean
134.27/38.92	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
134.27/38.92	 ! 	~ 	Lnot: (Boolean) -> Boolean
134.27/38.92	 = 	~ 	Eq: (Integer, Integer) -> Boolean
134.27/38.92	 <= 	~ 	Le: (Integer, Integer) -> Boolean
134.27/38.92	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
134.27/38.92	 % 	~ 	Mod: (Integer, Integer) -> Integer
134.27/38.92	 > 	~ 	Gt: (Integer, Integer) -> Boolean
134.27/38.92	 + 	~ 	Add: (Integer, Integer) -> Integer
134.27/38.92	 -1 	~ 	UnaryMinus: (Integer) -> Integer
134.27/38.92	 < 	~ 	Lt: (Integer, Integer) -> Boolean
134.27/38.92	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
134.27/38.92	 - 	~ 	Sub: (Integer, Integer) -> Integer
134.27/38.92	 ~ 	~ 	Bwnot: (Integer) -> Integer
134.27/38.92	 * 	~ 	Mul: (Integer, Integer) -> Integer
134.27/38.92	>>>
134.27/38.92	
134.27/38.92	
134.27/38.92	The following domains are used:
134.27/38.92	   Integer
134.27/38.92	
134.27/38.92	R is empty.
134.27/38.92	
134.27/38.92	The integer pair graph contains the following rules and edges:
134.27/38.92	(0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0])
134.27/38.92	(1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1])
134.27/38.92	(2): COND2(TRUE, x[2], y[2]) -> DIFF(x[2], y[2] + 1)
134.27/38.92	
134.27/38.92	   (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0])
134.27/38.92	   (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1])
134.27/38.92	   (1) -> (2), if (x[1] > y[1]  & x[1] ->^* x[2] & y[1] ->^* y[2])
134.27/38.92	
134.27/38.92	The set Q consists of the following terms:
134.27/38.92	diff(x0, x1)
134.27/38.92	cond1(TRUE, x0, x1)
134.27/38.92	cond1(FALSE, x0, x1)
134.27/38.92	cond2(TRUE, x0, x1)
134.27/38.92	cond2(FALSE, x0, x1)
134.27/38.92	
134.27/38.92	----------------------------------------
134.27/38.92	
134.27/38.92	(7) IDPNonInfProof (SOUND)
134.27/38.92	Used the following options for this NonInfProof:
134.27/38.92	
134.27/38.92	IDPGPoloSolver:
134.27/38.92	Range: [(-1,2)]
134.27/38.92	IsNat: false
134.27/38.92	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7507ddd9
134.27/38.92	Constraint Generator: NonInfConstraintGenerator:
134.27/38.92	PathGenerator: MetricPathGenerator:
134.27/38.92	Max Left Steps: 1
134.27/38.92	Max Right Steps: 1
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	The constraints were generated the following way:
134.27/38.92	
134.27/38.92	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
134.27/38.92	
134.27/38.92	Note that final constraints are written in bold face.
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) the following chains were created:
134.27/38.92	*We consider the chain DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]), COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (=(x[0], y[0])=FALSE & x[0]=x[1] & y[0]=y[1]  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(2)    (<(x[0], y[0])=TRUE  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	(3)    (>(x[0], y[0])=TRUE  ==>  DIFF(x[0], y[0])_>=_NonInfC & DIFF(x[0], y[0])_>=_COND1(=(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(7)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(8)    (y[0] + [-1] + [-1]x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(9)    (x[0] + [-1] + [-1]y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(10)    (y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(11)    (x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(12)    (y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	(13)    (y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(14)    (x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	(15)    (x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]) the following chains were created:
134.27/38.92	*We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (>(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2]  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rule (IV) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[1], y[1])=TRUE  ==>  COND1(FALSE, x[1], y[1])_>=_NonInfC & COND1(FALSE, x[1], y[1])_>=_COND2(>(x[1], y[1]), x[1], y[1]) & (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]y[1] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]y[1] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]y[1] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (x[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_18] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(7)    (x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_18] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	(8)    (x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_18] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	For Pair COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)) the following chains were created:
134.27/38.92	*We consider the chain COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1]), COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1)), DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0]) which results in the following constraint:
134.27/38.92	
134.27/38.92	(1)    (>(x[1], y[1])=TRUE & x[1]=x[2] & y[1]=y[2] & x[2]=x[0] & +(y[2], 1)=y[0]  ==>  COND2(TRUE, x[2], y[2])_>=_NonInfC & COND2(TRUE, x[2], y[2])_>=_DIFF(x[2], +(y[2], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(2)    (>(x[1], y[1])=TRUE  ==>  COND2(TRUE, x[1], y[1])_>=_NonInfC & COND2(TRUE, x[1], y[1])_>=_DIFF(x[1], +(y[1], 1)) & (U^Increasing(DIFF(x[2], +(y[2], 1))), >=))
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(3)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(4)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(5)    (x[1] + [-1] + [-1]y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[1] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
134.27/38.92	
134.27/38.92	(6)    (x[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
134.27/38.92	
134.27/38.92	(7)    (x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	(8)    (x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	To summarize, we get the following constraints P__>=_ for the following pairs.
134.27/38.92	
134.27/38.92	*DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0])
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(y[0] >= 0 & x[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[0] >= 0 & y[0] >= 0  ==>  (U^Increasing(COND1(=(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [(-1)bso_17] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1])
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_18] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(COND2(>(x[1], y[1]), x[1], y[1])), >=) & [(-1)Bound*bni_18] + [bni_18]x[1] >= 0 & [(-1)bso_19] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	*COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1))
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	*(x[1] >= 0 & y[1] >= 0  ==>  (U^Increasing(DIFF(x[2], +(y[2], 1))), >=) & [(-1)Bound*bni_20] + [bni_20]x[1] >= 0 & [1 + (-1)bso_21] >= 0)
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	
134.27/38.92	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. 
134.27/38.92	
134.27/38.92	Using the following integer polynomial ordering the  resulting constraints can be solved 
134.27/38.92	
134.27/38.92	Polynomial interpretation over integers[POLO]:
134.27/38.92	
134.27/38.92	   POL(TRUE) = 0
134.27/38.92	   POL(FALSE) = 0
134.27/38.92	   POL(DIFF(x_1, x_2)) = [-1] + [-1]x_2 + x_1
134.27/38.92	   POL(COND1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2
134.27/38.92	   POL(=(x_1, x_2)) = [-1]
134.27/38.92	   POL(COND2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2
134.27/38.92	   POL(>(x_1, x_2)) = [-1]
134.27/38.92	   POL(+(x_1, x_2)) = x_1 + x_2
134.27/38.92	   POL(1) = [1]
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_>:
134.27/38.92	
134.27/38.92	
134.27/38.92	   COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1))
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_bound:
134.27/38.92	
134.27/38.92	
134.27/38.92	   COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1])
134.27/38.92	   COND2(TRUE, x[2], y[2]) -> DIFF(x[2], +(y[2], 1))
134.27/38.92	
134.27/38.92	
134.27/38.92	The following pairs are in P_>=:
134.27/38.92	
134.27/38.92	
134.27/38.92	   DIFF(x[0], y[0]) -> COND1(=(x[0], y[0]), x[0], y[0])
134.27/38.92	   COND1(FALSE, x[1], y[1]) -> COND2(>(x[1], y[1]), x[1], y[1])
134.27/38.92	
134.27/38.92	
134.27/38.92	There are no usable rules.
134.27/38.92	----------------------------------------
134.27/38.92	
134.27/38.92	(8)
134.27/38.92	Obligation:
134.27/38.92	IDP problem:
134.27/38.92	The following function symbols are pre-defined:
134.27/38.92	<<<
134.27/38.92	 & 	~ 	Bwand: (Integer, Integer) -> Integer
134.27/38.92	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
134.27/38.92	 | 	~ 	Bwor: (Integer, Integer) -> Integer
134.27/38.92	 / 	~ 	Div: (Integer, Integer) -> Integer
134.27/38.92	 != 	~ 	Neq: (Integer, Integer) -> Boolean
134.27/38.92	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
134.27/38.92	 ! 	~ 	Lnot: (Boolean) -> Boolean
134.27/38.92	 = 	~ 	Eq: (Integer, Integer) -> Boolean
134.27/38.92	 <= 	~ 	Le: (Integer, Integer) -> Boolean
134.27/38.92	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
134.27/38.92	 % 	~ 	Mod: (Integer, Integer) -> Integer
134.27/38.92	 > 	~ 	Gt: (Integer, Integer) -> Boolean
134.27/38.92	 + 	~ 	Add: (Integer, Integer) -> Integer
134.27/38.92	 -1 	~ 	UnaryMinus: (Integer) -> Integer
134.27/38.92	 < 	~ 	Lt: (Integer, Integer) -> Boolean
134.27/38.92	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
134.27/38.92	 - 	~ 	Sub: (Integer, Integer) -> Integer
134.27/38.92	 ~ 	~ 	Bwnot: (Integer) -> Integer
134.27/38.92	 * 	~ 	Mul: (Integer, Integer) -> Integer
134.27/38.92	>>>
134.27/38.92	
134.27/38.92	
134.27/38.92	The following domains are used:
134.27/38.92	   Integer
134.27/38.92	
134.27/38.92	R is empty.
134.27/38.92	
134.27/38.92	The integer pair graph contains the following rules and edges:
134.27/38.92	(0): DIFF(x[0], y[0]) -> COND1(x[0] = y[0], x[0], y[0])
134.27/38.92	(1): COND1(FALSE, x[1], y[1]) -> COND2(x[1] > y[1], x[1], y[1])
134.27/38.92	
134.27/38.92	   (0) -> (1), if (x[0] = y[0] ->^* FALSE & x[0] ->^* x[1] & y[0] ->^* y[1])
134.27/38.92	
134.27/38.92	The set Q consists of the following terms:
134.27/38.92	diff(x0, x1)
134.27/38.92	cond1(TRUE, x0, x1)
134.27/38.92	cond1(FALSE, x0, x1)
134.27/38.92	cond2(TRUE, x0, x1)
134.27/38.92	cond2(FALSE, x0, x1)
134.27/38.92	
134.27/38.92	----------------------------------------
134.27/38.92	
134.27/38.92	(9) IDependencyGraphProof (EQUIVALENT)
134.27/38.92	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
134.27/38.92	----------------------------------------
134.27/38.92	
134.27/38.92	(10)
134.27/38.92	TRUE
134.54/38.96	EOF