4.70/2.58	YES
4.70/2.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
4.70/2.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.70/2.60	
4.70/2.60	
4.70/2.60	Termination of the given ITRS could be proven:
4.70/2.60	
4.70/2.60	(0) ITRS
4.70/2.60	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
4.70/2.60	(2) IDP
4.70/2.60	(3) UsableRulesProof [EQUIVALENT, 0 ms]
4.70/2.60	(4) IDP
4.70/2.60	(5) IDependencyGraphProof [EQUIVALENT, 0 ms]
4.70/2.60	(6) IDP
4.70/2.60	(7) IDPNonInfProof [SOUND, 245 ms]
4.70/2.60	(8) IDP
4.70/2.60	(9) IDependencyGraphProof [EQUIVALENT, 0 ms]
4.70/2.60	(10) TRUE
4.70/2.60	
4.70/2.60	
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(0)
4.70/2.60	Obligation:
4.70/2.60	ITRS problem:
4.70/2.60	
4.70/2.60	The following function symbols are pre-defined:
4.70/2.60	<<<
4.70/2.60	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.70/2.60	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.70/2.60	 / 	~ 	Div: (Integer, Integer) -> Integer
4.70/2.60	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.70/2.60	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.70/2.60	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.70/2.60	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.70/2.60	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.70/2.60	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.70/2.60	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.70/2.60	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.70/2.60	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.70/2.60	 + 	~ 	Add: (Integer, Integer) -> Integer
4.70/2.60	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.70/2.60	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.70/2.60	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.70/2.60	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.70/2.60	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.70/2.60	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.70/2.60	>>>
4.70/2.60	
4.70/2.60	The TRS R consists of the following rules:
4.70/2.60	pow(b, e) -> condLoop(e > 0, b, e, 1)
4.70/2.60	condLoop(FALSE, b, e, r) -> r
4.70/2.60	condLoop(TRUE, b, e, r) -> condMod(e % 2 = 1, b, e, r)
4.70/2.60	condMod(FALSE, b, e, r) -> sqBase(b, e, r)
4.70/2.60	condMod(TRUE, b, e, r) -> sqBase(b, e, r * b)
4.70/2.60	sqBase(b, e, r) -> halfExp(b * b, e, r)
4.70/2.60	halfExp(b, e, r) -> condLoop(e > 0, b, e / 2, r)
4.70/2.60	The set Q consists of the following terms:
4.70/2.60	pow(x0, x1)
4.70/2.60	condLoop(FALSE, x0, x1, x2)
4.70/2.60	condLoop(TRUE, x0, x1, x2)
4.70/2.60	condMod(FALSE, x0, x1, x2)
4.70/2.60	condMod(TRUE, x0, x1, x2)
4.70/2.60	sqBase(x0, x1, x2)
4.70/2.60	halfExp(x0, x1, x2)
4.70/2.60	
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(1) ITRStoIDPProof (EQUIVALENT)
4.70/2.60	Added dependency pairs
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(2)
4.70/2.60	Obligation:
4.70/2.60	IDP problem:
4.70/2.60	The following function symbols are pre-defined:
4.70/2.60	<<<
4.70/2.60	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.70/2.60	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.70/2.60	 / 	~ 	Div: (Integer, Integer) -> Integer
4.70/2.60	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.70/2.60	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.70/2.60	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.70/2.60	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.70/2.60	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.70/2.60	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.70/2.60	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.70/2.60	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.70/2.60	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.70/2.60	 + 	~ 	Add: (Integer, Integer) -> Integer
4.70/2.60	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.70/2.60	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.70/2.60	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.70/2.60	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.70/2.60	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.70/2.60	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.70/2.60	>>>
4.70/2.60	
4.70/2.60	
4.70/2.60	The following domains are used:
4.70/2.60	   Integer
4.70/2.60	
4.70/2.60	The ITRS R consists of the following rules:
4.70/2.60	pow(b, e) -> condLoop(e > 0, b, e, 1)
4.70/2.60	condLoop(FALSE, b, e, r) -> r
4.70/2.60	condLoop(TRUE, b, e, r) -> condMod(e % 2 = 1, b, e, r)
4.70/2.60	condMod(FALSE, b, e, r) -> sqBase(b, e, r)
4.70/2.60	condMod(TRUE, b, e, r) -> sqBase(b, e, r * b)
4.70/2.60	sqBase(b, e, r) -> halfExp(b * b, e, r)
4.70/2.60	halfExp(b, e, r) -> condLoop(e > 0, b, e / 2, r)
4.70/2.60	
4.70/2.60	The integer pair graph contains the following rules and edges:
4.70/2.60	(0): POW(b[0], e[0]) -> CONDLOOP(e[0] > 0, b[0], e[0], 1)
4.70/2.60	(1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1])
4.70/2.60	(2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.70/2.60	(3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3])
4.70/2.60	(4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4])
4.70/2.60	(5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5])
4.70/2.60	
4.70/2.60	   (0) -> (1), if (e[0] > 0  & b[0] ->^* b[1] & e[0] ->^* e[1] & 1 ->^* r[1])
4.70/2.60	   (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2])
4.70/2.60	   (1) -> (3), if (e[1] % 2 = 1  & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3])
4.70/2.60	   (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4])
4.70/2.60	   (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4])
4.70/2.60	   (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5])
4.70/2.60	   (5) -> (1), if (e[5] > 0  & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1])
4.70/2.60	
4.70/2.60	The set Q consists of the following terms:
4.70/2.60	pow(x0, x1)
4.70/2.60	condLoop(FALSE, x0, x1, x2)
4.70/2.60	condLoop(TRUE, x0, x1, x2)
4.70/2.60	condMod(FALSE, x0, x1, x2)
4.70/2.60	condMod(TRUE, x0, x1, x2)
4.70/2.60	sqBase(x0, x1, x2)
4.70/2.60	halfExp(x0, x1, x2)
4.70/2.60	
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(3) UsableRulesProof (EQUIVALENT)
4.70/2.60	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.
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(4)
4.70/2.60	Obligation:
4.70/2.60	IDP problem:
4.70/2.60	The following function symbols are pre-defined:
4.70/2.60	<<<
4.70/2.60	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.70/2.60	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.70/2.60	 / 	~ 	Div: (Integer, Integer) -> Integer
4.70/2.60	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.70/2.60	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.70/2.60	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.70/2.60	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.70/2.60	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.70/2.60	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.70/2.60	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.70/2.60	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.70/2.60	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.70/2.60	 + 	~ 	Add: (Integer, Integer) -> Integer
4.70/2.60	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.70/2.60	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.70/2.60	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.70/2.60	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.70/2.60	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.70/2.60	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.70/2.60	>>>
4.70/2.60	
4.70/2.60	
4.70/2.60	The following domains are used:
4.70/2.60	   Integer
4.70/2.60	
4.70/2.60	R is empty.
4.70/2.60	
4.70/2.60	The integer pair graph contains the following rules and edges:
4.70/2.60	(0): POW(b[0], e[0]) -> CONDLOOP(e[0] > 0, b[0], e[0], 1)
4.70/2.60	(1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1])
4.70/2.60	(2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.70/2.60	(3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3])
4.70/2.60	(4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4])
4.70/2.60	(5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5])
4.70/2.60	
4.70/2.60	   (0) -> (1), if (e[0] > 0  & b[0] ->^* b[1] & e[0] ->^* e[1] & 1 ->^* r[1])
4.70/2.60	   (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2])
4.70/2.60	   (1) -> (3), if (e[1] % 2 = 1  & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3])
4.70/2.60	   (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4])
4.70/2.60	   (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4])
4.70/2.60	   (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5])
4.70/2.60	   (5) -> (1), if (e[5] > 0  & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1])
4.70/2.60	
4.70/2.60	The set Q consists of the following terms:
4.70/2.60	pow(x0, x1)
4.70/2.60	condLoop(FALSE, x0, x1, x2)
4.70/2.60	condLoop(TRUE, x0, x1, x2)
4.70/2.60	condMod(FALSE, x0, x1, x2)
4.70/2.60	condMod(TRUE, x0, x1, x2)
4.70/2.60	sqBase(x0, x1, x2)
4.70/2.60	halfExp(x0, x1, x2)
4.70/2.60	
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(5) IDependencyGraphProof (EQUIVALENT)
4.70/2.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(6)
4.70/2.60	Obligation:
4.70/2.60	IDP problem:
4.70/2.60	The following function symbols are pre-defined:
4.70/2.60	<<<
4.70/2.60	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.70/2.60	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.70/2.60	 / 	~ 	Div: (Integer, Integer) -> Integer
4.70/2.60	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.70/2.60	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.70/2.60	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.70/2.60	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.70/2.60	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.70/2.60	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.70/2.60	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.70/2.60	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.70/2.60	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.70/2.60	 + 	~ 	Add: (Integer, Integer) -> Integer
4.70/2.60	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.70/2.60	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.70/2.60	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.70/2.60	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.70/2.60	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.70/2.60	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.70/2.60	>>>
4.70/2.60	
4.70/2.60	
4.70/2.60	The following domains are used:
4.70/2.60	   Integer
4.70/2.60	
4.70/2.60	R is empty.
4.70/2.60	
4.70/2.60	The integer pair graph contains the following rules and edges:
4.70/2.60	(3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3])
4.70/2.60	(5): HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(e[5] > 0, b[5], e[5] / 2, r[5])
4.70/2.60	(4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4])
4.70/2.60	(2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.70/2.60	(1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1])
4.70/2.60	
4.70/2.60	   (5) -> (1), if (e[5] > 0  & b[5] ->^* b[1] & e[5] / 2 ->^* e[1] & r[5] ->^* r[1])
4.70/2.60	   (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2])
4.70/2.60	   (1) -> (3), if (e[1] % 2 = 1  & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3])
4.70/2.60	   (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4])
4.70/2.60	   (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4])
4.70/2.60	   (4) -> (5), if (b[4] * b[4] ->^* b[5] & e[4] ->^* e[5] & r[4] ->^* r[5])
4.70/2.60	
4.70/2.60	The set Q consists of the following terms:
4.70/2.60	pow(x0, x1)
4.70/2.60	condLoop(FALSE, x0, x1, x2)
4.70/2.60	condLoop(TRUE, x0, x1, x2)
4.70/2.60	condMod(FALSE, x0, x1, x2)
4.70/2.60	condMod(TRUE, x0, x1, x2)
4.70/2.60	sqBase(x0, x1, x2)
4.70/2.60	halfExp(x0, x1, x2)
4.70/2.60	
4.70/2.60	----------------------------------------
4.70/2.60	
4.70/2.60	(7) IDPNonInfProof (SOUND)
4.70/2.60	Used the following options for this NonInfProof:
4.70/2.60	
4.70/2.60	IDPGPoloSolver:
4.70/2.60	Range: [(-1,2)]
4.70/2.60	IsNat: false
4.70/2.60	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@30b72e9a
4.70/2.60	Constraint Generator: NonInfConstraintGenerator:
4.70/2.60	PathGenerator: MetricPathGenerator:
4.70/2.60	Max Left Steps: 1
4.70/2.60	Max Right Steps: 1
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	The constraints were generated the following way:
4.70/2.60	
4.70/2.60	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
4.70/2.60	
4.70/2.60	Note that final constraints are written in bold face.
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	For Pair CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) the following chains were created:
4.70/2.60	*We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) which results in the following constraint:
4.70/2.60	
4.70/2.60	(1)    (=(%(e[1], 2), 1)=TRUE & b[1]=b[3] & e[1]=e[3] & r[1]=r[3] & b[3]=b[4] & e[3]=e[4] & *(r[3], b[3])=r[4]  ==>  CONDMOD(TRUE, b[3], e[3], r[3])_>=_NonInfC & CONDMOD(TRUE, b[3], e[3], r[3])_>=_SQBASE(b[3], e[3], *(r[3], b[3])) & (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(2)    (>=(%(e[1], 2), 1)=TRUE & <=(%(e[1], 2), 1)=TRUE  ==>  CONDMOD(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDMOD(TRUE, b[1], e[1], r[1])_>=_SQBASE(b[1], e[1], *(r[1], b[1])) & (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(3)    (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(4)    (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(5)    ([4] >= 0 & [1] >= 0 & [3] >= 0  ==>  (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]e[1] >= 0 & [(-1)bso_24] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(6)    ([1] >= 0 & [1] >= 0 & [1] >= 0  ==>  (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & 0 = 0 & [bni_23] = 0 & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	For Pair HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) the following chains were created:
4.70/2.60	*We consider the chain SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]), CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) which results in the following constraint:
4.70/2.60	
4.70/2.60	(1)    (*(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5] & >(e[5], 0)=TRUE & b[5]=b[1] & /(e[5], 2)=e[1] & r[5]=r[1]  ==>  HALFEXP(b[5], e[5], r[5])_>=_NonInfC & HALFEXP(b[5], e[5], r[5])_>=_CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) & (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(2)    (>(e[5], 0)=TRUE  ==>  HALFEXP(*(b[4], b[4]), e[5], r[4])_>=_NonInfC & HALFEXP(*(b[4], b[4]), e[5], r[4])_>=_CONDLOOP(>(e[5], 0), *(b[4], b[4]), /(e[5], 2), r[4]) & (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(3)    (e[5] + [-1] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] + e[5] + [-1]max{e[5], [-1]e[5]} >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(4)    (e[5] + [-1] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] + e[5] + [-1]max{e[5], [-1]e[5]} >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(5)    (e[5] + [-1] >= 0 & [2]e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(6)    (e[5] + [-1] >= 0 & [2]e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
4.70/2.60	
4.70/2.60	(7)    (e[5] + [-1] >= 0 & [2]e[5] >= 0 & b[4] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	(8)    (e[5] + [-1] >= 0 & [2]e[5] >= 0 & b[4] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(9)    (e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(10)    (e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	For Pair SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) the following chains were created:
4.70/2.60	*We consider the chain CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) which results in the following constraint:
4.70/2.60	
4.70/2.60	(1)    (b[2]=b[4] & e[2]=e[4] & r[2]=r[4] & *(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5]  ==>  SQBASE(b[4], e[4], r[4])_>=_NonInfC & SQBASE(b[4], e[4], r[4])_>=_HALFEXP(*(b[4], b[4]), e[4], r[4]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(2)    (SQBASE(b[2], e[2], r[2])_>=_NonInfC & SQBASE(b[2], e[2], r[2])_>=_HALFEXP(*(b[2], b[2]), e[2], r[2]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(3)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(4)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(5)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(6)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
4.70/2.60	
4.70/2.60	(7)    (b[2] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	(8)    (b[2] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	*We consider the chain CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]), HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5]) which results in the following constraint:
4.70/2.60	
4.70/2.60	(1)    (b[3]=b[4] & e[3]=e[4] & *(r[3], b[3])=r[4] & *(b[4], b[4])=b[5] & e[4]=e[5] & r[4]=r[5]  ==>  SQBASE(b[4], e[4], r[4])_>=_NonInfC & SQBASE(b[4], e[4], r[4])_>=_HALFEXP(*(b[4], b[4]), e[4], r[4]) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(2)    (SQBASE(b[3], e[3], *(r[3], b[3]))_>=_NonInfC & SQBASE(b[3], e[3], *(r[3], b[3]))_>=_HALFEXP(*(b[3], b[3]), e[3], *(r[3], b[3])) & (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(3)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(4)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(5)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(6)    ((U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
4.70/2.60	
4.70/2.60	(7)    (b[3] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	(8)    (b[3] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	For Pair CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) the following chains were created:
4.70/2.60	*We consider the chain CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]), SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4]) which results in the following constraint:
4.70/2.60	
4.70/2.60	(1)    (b[2]=b[4] & e[2]=e[4] & r[2]=r[4]  ==>  CONDMOD(FALSE, b[2], e[2], r[2])_>=_NonInfC & CONDMOD(FALSE, b[2], e[2], r[2])_>=_SQBASE(b[2], e[2], r[2]) & (U^Increasing(SQBASE(b[2], e[2], r[2])), >=))
4.70/2.60	
4.70/2.60	
4.70/2.60	
4.70/2.60	We simplified constraint (1) using rule (IV) which results in the following new constraint:
4.70/2.60	
4.70/2.60	(2)    (CONDMOD(FALSE, b[2], e[2], r[2])_>=_NonInfC & CONDMOD(FALSE, b[2], e[2], r[2])_>=_SQBASE(b[2], e[2], r[2]) & (U^Increasing(SQBASE(b[2], e[2], r[2])), >=))
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(3)    ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(4)    ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(5)    ((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	For Pair CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) the following chains were created:
4.92/2.60	*We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2]) which results in the following constraint:
4.92/2.60	
4.92/2.60	(1)    (=(%(e[1], 2), 1)=FALSE & b[1]=b[2] & e[1]=e[2] & r[1]=r[2]  ==>  CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=))
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
4.92/2.60	
4.92/2.60	(2)    (<(%(e[1], 2), 1)=TRUE  ==>  CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=))
4.92/2.60	
4.92/2.60	(3)    (>(%(e[1], 2), 1)=TRUE  ==>  CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=))
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(4)    ([-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(5)    (max{[2], [-2]} + [-2] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(6)    ([-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(7)    (max{[2], [-2]} + [-2] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(8)    ([4] >= 0 & [2] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(9)    ([4] >= 0 & 0 >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (8) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(10)    ([1] >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (9) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(11)    (0 >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	*We consider the chain CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]), CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3])) which results in the following constraint:
4.92/2.60	
4.92/2.60	(1)    (=(%(e[1], 2), 1)=TRUE & b[1]=b[3] & e[1]=e[3] & r[1]=r[3]  ==>  CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=))
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(2)    (>=(%(e[1], 2), 1)=TRUE & <=(%(e[1], 2), 1)=TRUE  ==>  CONDLOOP(TRUE, b[1], e[1], r[1])_>=_NonInfC & CONDLOOP(TRUE, b[1], e[1], r[1])_>=_CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1]) & (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=))
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(3)    (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(4)    (max{[2], [-2]} + [-1] >= 0 & [1] + [-1]min{[2], [-2]} >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(5)    ([4] >= 0 & [1] >= 0 & [3] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & [(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]e[1] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
4.92/2.60	
4.92/2.60	(6)    ([1] >= 0 & [1] >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	To summarize, we get the following constraints P__>=_ for the following pairs.
4.92/2.60	
4.92/2.60	*CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3]))
4.92/2.60	
4.92/2.60	*([1] >= 0 & [1] >= 0 & [1] >= 0  ==>  (U^Increasing(SQBASE(b[3], e[3], *(r[3], b[3]))), >=) & 0 = 0 & [bni_23] = 0 & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] >= 0 & 0 = 0 & 0 = 0 & [(-1)bso_24] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	*HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])
4.92/2.60	
4.92/2.60	*(e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*(e[5] + [-1] >= 0 & b[4] >= 0 & e[5] >= 0  ==>  (U^Increasing(CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]e[5] >= 0 & [1 + (-1)bso_29] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	*SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4])
4.92/2.60	
4.92/2.60	*(b[2] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*(b[2] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*(b[3] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*(b[3] >= 0  ==>  (U^Increasing(HALFEXP(*(b[4], b[4]), e[4], r[4])), >=) & [bni_30] = 0 & 0 = 0 & [(-1)bso_31] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	*CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.92/2.60	
4.92/2.60	*((U^Increasing(SQBASE(b[2], e[2], r[2])), >=) & [bni_32] = 0 & [(-1)bso_33] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	*CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])
4.92/2.60	
4.92/2.60	*([1] >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*(0 >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	*([1] >= 0 & [1] >= 0 & [1] >= 0  ==>  (U^Increasing(CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])), >=) & 0 = 0 & [bni_34] = 0 & 0 = 0 & [(-1)bni_34 + (-1)Bound*bni_34] >= 0 & [(-1)bso_35] >= 0)
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	
4.92/2.60	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. 
4.92/2.60	
4.92/2.60	Using the following integer polynomial ordering the  resulting constraints can be solved 
4.92/2.60	
4.92/2.60	Polynomial interpretation over integers[POLO]:
4.92/2.60	
4.92/2.60	   POL(TRUE) = 0
4.92/2.60	   POL(FALSE) = 0
4.92/2.60	   POL(CONDMOD(x_1, x_2, x_3, x_4)) = [-1] + x_3
4.92/2.60	   POL(SQBASE(x_1, x_2, x_3)) = [-1] + x_2
4.92/2.60	   POL(*(x_1, x_2)) = x_1*x_2
4.92/2.60	   POL(HALFEXP(x_1, x_2, x_3)) = [-1] + x_2
4.92/2.60	   POL(CONDLOOP(x_1, x_2, x_3, x_4)) = [-1] + x_3
4.92/2.60	   POL(>(x_1, x_2)) = [-1]
4.92/2.60	   POL(0) = 0
4.92/2.60	   POL(2) = [2]
4.92/2.60	   POL(=(x_1, x_2)) = [-1]
4.92/2.60	   POL(1) = [1]
4.92/2.60	   
4.92/2.60	   Polynomial Interpretations with Context Sensitive Arithemetic Replacement
4.92/2.60	   POL(Term^CSAR-Mode @ Context)
4.92/2.60	   
4.92/2.60	   POL(%(x_1, 2)^1 @ {}) = max{x_2, [-1]x_2}
4.92/2.60	   POL(%(x_1, 2)^-1 @ {}) = min{x_2, [-1]x_2}
4.92/2.60	   POL(/(x_1, 2)^1 @ {CONDLOOP_4/2}) = max{x_1, [-1]x_1} + [-1]
4.92/2.60	
4.92/2.60	
4.92/2.60	The following pairs are in P_>:
4.92/2.60	
4.92/2.60	
4.92/2.60	   HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])
4.92/2.60	
4.92/2.60	
4.92/2.60	The following pairs are in P_bound:
4.92/2.60	
4.92/2.60	
4.92/2.60	   HALFEXP(b[5], e[5], r[5]) -> CONDLOOP(>(e[5], 0), b[5], /(e[5], 2), r[5])
4.92/2.60	
4.92/2.60	
4.92/2.60	The following pairs are in P_>=:
4.92/2.60	
4.92/2.60	
4.92/2.60	   CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], *(r[3], b[3]))
4.92/2.60	   SQBASE(b[4], e[4], r[4]) -> HALFEXP(*(b[4], b[4]), e[4], r[4])
4.92/2.60	   CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.92/2.60	   CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(=(%(e[1], 2), 1), b[1], e[1], r[1])
4.92/2.60	
4.92/2.60	
4.92/2.60	At least the following rules have been oriented under context sensitive arithmetic replacement:
4.92/2.60	
4.92/2.60	   /^1 ->
4.92/2.61	   %^1 ->
4.92/2.61	
4.92/2.61	----------------------------------------
4.92/2.61	
4.92/2.61	(8)
4.92/2.61	Obligation:
4.92/2.61	IDP problem:
4.92/2.61	The following function symbols are pre-defined:
4.92/2.61	<<<
4.92/2.61	 & 	~ 	Bwand: (Integer, Integer) -> Integer
4.92/2.61	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
4.92/2.61	 / 	~ 	Div: (Integer, Integer) -> Integer
4.92/2.61	 | 	~ 	Bwor: (Integer, Integer) -> Integer
4.92/2.61	 != 	~ 	Neq: (Integer, Integer) -> Boolean
4.92/2.61	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
4.92/2.61	 ! 	~ 	Lnot: (Boolean) -> Boolean
4.92/2.61	 = 	~ 	Eq: (Integer, Integer) -> Boolean
4.92/2.61	 <= 	~ 	Le: (Integer, Integer) -> Boolean
4.92/2.61	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
4.92/2.61	 % 	~ 	Mod: (Integer, Integer) -> Integer
4.92/2.61	 > 	~ 	Gt: (Integer, Integer) -> Boolean
4.92/2.61	 + 	~ 	Add: (Integer, Integer) -> Integer
4.92/2.61	 -1 	~ 	UnaryMinus: (Integer) -> Integer
4.92/2.61	 < 	~ 	Lt: (Integer, Integer) -> Boolean
4.92/2.61	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
4.92/2.61	 - 	~ 	Sub: (Integer, Integer) -> Integer
4.92/2.61	 ~ 	~ 	Bwnot: (Integer) -> Integer
4.92/2.61	 * 	~ 	Mul: (Integer, Integer) -> Integer
4.92/2.61	>>>
4.92/2.61	
4.92/2.61	
4.92/2.61	The following domains are used:
4.92/2.61	   Integer
4.92/2.61	
4.92/2.61	R is empty.
4.92/2.61	
4.92/2.61	The integer pair graph contains the following rules and edges:
4.92/2.61	(3): CONDMOD(TRUE, b[3], e[3], r[3]) -> SQBASE(b[3], e[3], r[3] * b[3])
4.92/2.61	(4): SQBASE(b[4], e[4], r[4]) -> HALFEXP(b[4] * b[4], e[4], r[4])
4.92/2.61	(2): CONDMOD(FALSE, b[2], e[2], r[2]) -> SQBASE(b[2], e[2], r[2])
4.92/2.61	(1): CONDLOOP(TRUE, b[1], e[1], r[1]) -> CONDMOD(e[1] % 2 = 1, b[1], e[1], r[1])
4.92/2.61	
4.92/2.61	   (1) -> (2), if (e[1] % 2 = 1 ->^* FALSE & b[1] ->^* b[2] & e[1] ->^* e[2] & r[1] ->^* r[2])
4.92/2.61	   (1) -> (3), if (e[1] % 2 = 1  & b[1] ->^* b[3] & e[1] ->^* e[3] & r[1] ->^* r[3])
4.92/2.61	   (2) -> (4), if (b[2] ->^* b[4] & e[2] ->^* e[4] & r[2] ->^* r[4])
4.92/2.61	   (3) -> (4), if (b[3] ->^* b[4] & e[3] ->^* e[4] & r[3] * b[3] ->^* r[4])
4.92/2.61	
4.92/2.61	The set Q consists of the following terms:
4.92/2.61	pow(x0, x1)
4.92/2.61	condLoop(FALSE, x0, x1, x2)
4.92/2.61	condLoop(TRUE, x0, x1, x2)
4.92/2.61	condMod(FALSE, x0, x1, x2)
4.92/2.61	condMod(TRUE, x0, x1, x2)
4.92/2.61	sqBase(x0, x1, x2)
4.92/2.61	halfExp(x0, x1, x2)
4.92/2.61	
4.92/2.61	----------------------------------------
4.92/2.61	
4.92/2.61	(9) IDependencyGraphProof (EQUIVALENT)
4.92/2.61	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
4.92/2.61	----------------------------------------
4.92/2.61	
4.92/2.61	(10)
4.92/2.61	TRUE
4.98/2.67	EOF