3.62/1.82	YES
3.62/1.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs
3.62/1.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.62/1.83	
3.62/1.83	
3.62/1.83	Termination of the given ITRS could be proven:
3.62/1.83	
3.62/1.83	(0) ITRS
3.62/1.83	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
3.62/1.83	(2) IDP
3.62/1.83	(3) IDependencyGraphProof [EQUIVALENT, 0 ms]
3.62/1.83	(4) IDP
3.62/1.83	(5) UsableRulesProof [EQUIVALENT, 0 ms]
3.62/1.83	(6) IDP
3.62/1.83	(7) IDPNonInfProof [SOUND, 157 ms]
3.62/1.83	(8) IDP
3.62/1.83	(9) IDependencyGraphProof [EQUIVALENT, 0 ms]
3.62/1.83	(10) TRUE
3.62/1.83	
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(0)
3.62/1.83	Obligation:
3.62/1.83	ITRS problem:
3.62/1.83	
3.62/1.83	The following function symbols are pre-defined:
3.62/1.83	<<<
3.62/1.83	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.62/1.83	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.62/1.83	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.62/1.83	 / 	~ 	Div: (Integer, Integer) -> Integer
3.62/1.83	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.62/1.83	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.62/1.83	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.62/1.83	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.62/1.83	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.62/1.83	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.62/1.83	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.62/1.83	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.62/1.83	 + 	~ 	Add: (Integer, Integer) -> Integer
3.62/1.83	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.62/1.83	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.62/1.83	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.62/1.83	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.62/1.83	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.62/1.83	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.62/1.83	>>>
3.62/1.83	
3.62/1.83	The TRS R consists of the following rules:
3.62/1.83	sumto(x, y) -> Cond_sumto(x > y, x, y)
3.62/1.83	Cond_sumto(TRUE, x, y) -> wrap(0)
3.62/1.83	sumto(x, y) -> Cond_sumto1(y >= x, x, y)
3.62/1.83	Cond_sumto1(TRUE, x, y) -> if(sumto(x + 1, y), x, y)
3.62/1.83	if(wrap(z), x, y) -> wrap(x + z)
3.62/1.83	The set Q consists of the following terms:
3.62/1.83	sumto(x0, x1)
3.62/1.83	Cond_sumto(TRUE, x0, x1)
3.62/1.83	Cond_sumto1(TRUE, x0, x1)
3.62/1.83	if(wrap(x0), x1, x2)
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(1) ITRStoIDPProof (EQUIVALENT)
3.62/1.83	Added dependency pairs
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(2)
3.62/1.83	Obligation:
3.62/1.83	IDP problem:
3.62/1.83	The following function symbols are pre-defined:
3.62/1.83	<<<
3.62/1.83	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.62/1.83	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.62/1.83	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.62/1.83	 / 	~ 	Div: (Integer, Integer) -> Integer
3.62/1.83	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.62/1.83	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.62/1.83	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.62/1.83	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.62/1.83	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.62/1.83	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.62/1.83	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.62/1.83	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.62/1.83	 + 	~ 	Add: (Integer, Integer) -> Integer
3.62/1.83	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.62/1.83	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.62/1.83	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.62/1.83	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.62/1.83	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.62/1.83	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.62/1.83	>>>
3.62/1.83	
3.62/1.83	
3.62/1.83	The following domains are used:
3.62/1.83	   Integer
3.62/1.83	
3.62/1.83	The ITRS R consists of the following rules:
3.62/1.83	sumto(x, y) -> Cond_sumto(x > y, x, y)
3.62/1.83	Cond_sumto(TRUE, x, y) -> wrap(0)
3.62/1.83	sumto(x, y) -> Cond_sumto1(y >= x, x, y)
3.62/1.83	Cond_sumto1(TRUE, x, y) -> if(sumto(x + 1, y), x, y)
3.62/1.83	if(wrap(z), x, y) -> wrap(x + z)
3.62/1.83	
3.62/1.83	The integer pair graph contains the following rules and edges:
3.62/1.83	(0): SUMTO(x[0], y[0]) -> COND_SUMTO(x[0] > y[0], x[0], y[0])
3.62/1.83	(1): SUMTO(x[1], y[1]) -> COND_SUMTO1(y[1] >= x[1], x[1], y[1])
3.62/1.83	(2): COND_SUMTO1(TRUE, x[2], y[2]) -> IF(sumto(x[2] + 1, y[2]), x[2], y[2])
3.62/1.83	(3): COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(x[3] + 1, y[3])
3.62/1.83	
3.62/1.83	   (1) -> (2), if (y[1] >= x[1]  & x[1] ->^* x[2] & y[1] ->^* y[2])
3.62/1.83	   (1) -> (3), if (y[1] >= x[1]  & x[1] ->^* x[3] & y[1] ->^* y[3])
3.62/1.83	   (3) -> (0), if (x[3] + 1 ->^* x[0] & y[3] ->^* y[0])
3.62/1.83	   (3) -> (1), if (x[3] + 1 ->^* x[1] & y[3] ->^* y[1])
3.62/1.83	
3.62/1.83	The set Q consists of the following terms:
3.62/1.83	sumto(x0, x1)
3.62/1.83	Cond_sumto(TRUE, x0, x1)
3.62/1.83	Cond_sumto1(TRUE, x0, x1)
3.62/1.83	if(wrap(x0), x1, x2)
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(3) IDependencyGraphProof (EQUIVALENT)
3.62/1.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(4)
3.62/1.83	Obligation:
3.62/1.83	IDP problem:
3.62/1.83	The following function symbols are pre-defined:
3.62/1.83	<<<
3.62/1.83	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.62/1.83	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.62/1.83	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.62/1.83	 / 	~ 	Div: (Integer, Integer) -> Integer
3.62/1.83	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.62/1.83	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.62/1.83	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.62/1.83	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.62/1.83	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.62/1.83	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.62/1.83	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.62/1.83	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.62/1.83	 + 	~ 	Add: (Integer, Integer) -> Integer
3.62/1.83	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.62/1.83	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.62/1.83	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.62/1.83	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.62/1.83	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.62/1.83	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.62/1.83	>>>
3.62/1.83	
3.62/1.83	
3.62/1.83	The following domains are used:
3.62/1.83	   Integer
3.62/1.83	
3.62/1.83	The ITRS R consists of the following rules:
3.62/1.83	sumto(x, y) -> Cond_sumto(x > y, x, y)
3.62/1.83	Cond_sumto(TRUE, x, y) -> wrap(0)
3.62/1.83	sumto(x, y) -> Cond_sumto1(y >= x, x, y)
3.62/1.83	Cond_sumto1(TRUE, x, y) -> if(sumto(x + 1, y), x, y)
3.62/1.83	if(wrap(z), x, y) -> wrap(x + z)
3.62/1.83	
3.62/1.83	The integer pair graph contains the following rules and edges:
3.62/1.83	(3): COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(x[3] + 1, y[3])
3.62/1.83	(1): SUMTO(x[1], y[1]) -> COND_SUMTO1(y[1] >= x[1], x[1], y[1])
3.62/1.83	
3.62/1.83	   (3) -> (1), if (x[3] + 1 ->^* x[1] & y[3] ->^* y[1])
3.62/1.83	   (1) -> (3), if (y[1] >= x[1]  & x[1] ->^* x[3] & y[1] ->^* y[3])
3.62/1.83	
3.62/1.83	The set Q consists of the following terms:
3.62/1.83	sumto(x0, x1)
3.62/1.83	Cond_sumto(TRUE, x0, x1)
3.62/1.83	Cond_sumto1(TRUE, x0, x1)
3.62/1.83	if(wrap(x0), x1, x2)
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(5) UsableRulesProof (EQUIVALENT)
3.62/1.83	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.
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(6)
3.62/1.83	Obligation:
3.62/1.83	IDP problem:
3.62/1.83	The following function symbols are pre-defined:
3.62/1.83	<<<
3.62/1.83	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.62/1.83	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.62/1.83	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.62/1.83	 / 	~ 	Div: (Integer, Integer) -> Integer
3.62/1.83	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.62/1.83	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.62/1.83	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.62/1.83	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.62/1.83	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.62/1.83	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.62/1.83	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.62/1.83	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.62/1.83	 + 	~ 	Add: (Integer, Integer) -> Integer
3.62/1.83	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.62/1.83	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.62/1.83	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.62/1.83	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.62/1.83	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.62/1.83	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.62/1.83	>>>
3.62/1.83	
3.62/1.83	
3.62/1.83	The following domains are used:
3.62/1.83	   Integer
3.62/1.83	
3.62/1.83	R is empty.
3.62/1.83	
3.62/1.83	The integer pair graph contains the following rules and edges:
3.62/1.83	(3): COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(x[3] + 1, y[3])
3.62/1.83	(1): SUMTO(x[1], y[1]) -> COND_SUMTO1(y[1] >= x[1], x[1], y[1])
3.62/1.83	
3.62/1.83	   (3) -> (1), if (x[3] + 1 ->^* x[1] & y[3] ->^* y[1])
3.62/1.83	   (1) -> (3), if (y[1] >= x[1]  & x[1] ->^* x[3] & y[1] ->^* y[3])
3.62/1.83	
3.62/1.83	The set Q consists of the following terms:
3.62/1.83	sumto(x0, x1)
3.62/1.83	Cond_sumto(TRUE, x0, x1)
3.62/1.83	Cond_sumto1(TRUE, x0, x1)
3.62/1.83	if(wrap(x0), x1, x2)
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(7) IDPNonInfProof (SOUND)
3.62/1.83	Used the following options for this NonInfProof:
3.62/1.83	
3.62/1.83	IDPGPoloSolver:
3.62/1.83	Range: [(-1,2)]
3.62/1.83	IsNat: false
3.62/1.83	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7316c6a
3.62/1.83	Constraint Generator: NonInfConstraintGenerator:
3.62/1.83	PathGenerator: MetricPathGenerator:
3.62/1.83	Max Left Steps: 1
3.62/1.83	Max Right Steps: 1
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	The constraints were generated the following way:
3.62/1.83	
3.62/1.83	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
3.62/1.83	
3.62/1.83	Note that final constraints are written in bold face.
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	For Pair COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3]) the following chains were created:
3.62/1.83	*We consider the chain SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]), COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3]), SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]) which results in the following constraint:
3.62/1.83	
3.62/1.83	(1)    (>=(y[1], x[1])=TRUE & x[1]=x[3] & y[1]=y[3] & +(x[3], 1)=x[1]1 & y[3]=y[1]1  ==>  COND_SUMTO1(TRUE, x[3], y[3])_>=_NonInfC & COND_SUMTO1(TRUE, x[3], y[3])_>=_SUMTO(+(x[3], 1), y[3]) & (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=))
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(2)    (>=(y[1], x[1])=TRUE  ==>  COND_SUMTO1(TRUE, x[1], y[1])_>=_NonInfC & COND_SUMTO1(TRUE, x[1], y[1])_>=_SUMTO(+(x[1], 1), y[1]) & (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=))
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(3)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] + [(-1)bni_11]x[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(4)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] + [(-1)bni_11]x[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(5)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] + [(-1)bni_11]x[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(6)    (y[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
3.62/1.83	
3.62/1.83	(7)    (y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	(8)    (y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	For Pair SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]) the following chains were created:
3.62/1.83	*We consider the chain SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]), COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3]) which results in the following constraint:
3.62/1.83	
3.62/1.83	(1)    (>=(y[1], x[1])=TRUE & x[1]=x[3] & y[1]=y[3]  ==>  SUMTO(x[1], y[1])_>=_NonInfC & SUMTO(x[1], y[1])_>=_COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]) & (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=))
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (1) using rule (IV) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(2)    (>=(y[1], x[1])=TRUE  ==>  SUMTO(x[1], y[1])_>=_NonInfC & SUMTO(x[1], y[1])_>=_COND_SUMTO1(>=(y[1], x[1]), x[1], y[1]) & (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=))
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(3)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] + [(-1)bni_13]x[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(4)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] + [(-1)bni_13]x[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(5)    (y[1] + [-1]x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] + [(-1)bni_13]x[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
3.62/1.83	
3.62/1.83	(6)    (y[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
3.62/1.83	
3.62/1.83	(7)    (y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	(8)    (y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	To summarize, we get the following constraints P__>=_ for the following pairs.
3.62/1.83	
3.62/1.83	*COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3])
3.62/1.83	
3.62/1.83	*(y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	*(y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(SUMTO(+(x[3], 1), y[3])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[1] >= 0 & [(-1)bso_12] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	*SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])
3.62/1.83	
3.62/1.83	*(y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	*(y[1] >= 0 & x[1] >= 0  ==>  (U^Increasing(COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])), >=) & [(-1)Bound*bni_13] + [bni_13]y[1] >= 0 & [1 + (-1)bso_14] >= 0)
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	
3.62/1.83	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. 
3.62/1.83	
3.62/1.83	Using the following integer polynomial ordering the  resulting constraints can be solved 
3.62/1.83	
3.62/1.83	Polynomial interpretation over integers[POLO]:
3.62/1.83	
3.62/1.83	   POL(TRUE) = 0
3.62/1.83	   POL(FALSE) = 0
3.62/1.83	   POL(COND_SUMTO1(x_1, x_2, x_3)) = [-1] + x_3 + [-1]x_2
3.62/1.83	   POL(SUMTO(x_1, x_2)) = x_2 + [-1]x_1
3.62/1.83	   POL(+(x_1, x_2)) = x_1 + x_2
3.62/1.83	   POL(1) = [1]
3.62/1.83	   POL(>=(x_1, x_2)) = [-1]
3.62/1.83	
3.62/1.83	
3.62/1.83	The following pairs are in P_>:
3.62/1.83	
3.62/1.83	
3.62/1.83	   SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])
3.62/1.83	
3.62/1.83	
3.62/1.83	The following pairs are in P_bound:
3.62/1.83	
3.62/1.83	
3.62/1.83	   COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3])
3.62/1.83	   SUMTO(x[1], y[1]) -> COND_SUMTO1(>=(y[1], x[1]), x[1], y[1])
3.62/1.83	
3.62/1.83	
3.62/1.83	The following pairs are in P_>=:
3.62/1.83	
3.62/1.83	
3.62/1.83	   COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(+(x[3], 1), y[3])
3.62/1.83	
3.62/1.83	
3.62/1.83	There are no usable rules.
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(8)
3.62/1.83	Obligation:
3.62/1.83	IDP problem:
3.62/1.83	The following function symbols are pre-defined:
3.62/1.83	<<<
3.62/1.83	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.62/1.83	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.62/1.83	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.62/1.83	 / 	~ 	Div: (Integer, Integer) -> Integer
3.62/1.83	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.62/1.83	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.62/1.83	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.62/1.83	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.62/1.83	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.62/1.83	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.62/1.83	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.62/1.83	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.62/1.83	 + 	~ 	Add: (Integer, Integer) -> Integer
3.62/1.83	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.62/1.83	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.62/1.83	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.62/1.83	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.62/1.83	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.62/1.83	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.62/1.83	>>>
3.62/1.83	
3.62/1.83	
3.62/1.83	The following domains are used:
3.62/1.83	   Integer
3.62/1.83	
3.62/1.83	R is empty.
3.62/1.83	
3.62/1.83	The integer pair graph contains the following rules and edges:
3.62/1.83	(3): COND_SUMTO1(TRUE, x[3], y[3]) -> SUMTO(x[3] + 1, y[3])
3.62/1.83	
3.62/1.83	
3.62/1.83	The set Q consists of the following terms:
3.62/1.83	sumto(x0, x1)
3.62/1.83	Cond_sumto(TRUE, x0, x1)
3.62/1.83	Cond_sumto1(TRUE, x0, x1)
3.62/1.83	if(wrap(x0), x1, x2)
3.62/1.83	
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(9) IDependencyGraphProof (EQUIVALENT)
3.62/1.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
3.62/1.83	----------------------------------------
3.62/1.83	
3.62/1.83	(10)
3.62/1.83	TRUE
3.79/1.85	EOF