3.78/1.90	YES
3.78/1.91	proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs
3.78/1.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.78/1.91	
3.78/1.91	
3.78/1.91	Termination of the given ITRS could be proven:
3.78/1.91	
3.78/1.91	(0) ITRS
3.78/1.91	(1) ITRStoIDPProof [EQUIVALENT, 0 ms]
3.78/1.91	(2) IDP
3.78/1.91	(3) UsableRulesProof [EQUIVALENT, 0 ms]
3.78/1.91	(4) IDP
3.78/1.91	(5) IDPNonInfProof [SOUND, 143 ms]
3.78/1.91	(6) IDP
3.78/1.91	(7) IDependencyGraphProof [EQUIVALENT, 0 ms]
3.78/1.91	(8) TRUE
3.78/1.91	
3.78/1.91	
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(0)
3.78/1.91	Obligation:
3.78/1.91	ITRS problem:
3.78/1.91	
3.78/1.91	The following function symbols are pre-defined:
3.78/1.91	<<<
3.78/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.78/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.78/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.78/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
3.78/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.78/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.78/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.78/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.78/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.78/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.78/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.78/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.78/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
3.78/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.78/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.78/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.78/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.78/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.78/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.78/1.91	>>>
3.78/1.91	
3.78/1.91	The TRS R consists of the following rules:
3.78/1.91	pow(x, y) -> if(y > 0, x, y)
3.78/1.91	if(TRUE, x, y) -> x * pow(x, y - 1)
3.78/1.91	if(FALSE, x, y) -> 1
3.78/1.91	The set Q consists of the following terms:
3.78/1.91	pow(x0, x1)
3.78/1.91	if(TRUE, x0, x1)
3.78/1.91	if(FALSE, x0, x1)
3.78/1.91	
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(1) ITRStoIDPProof (EQUIVALENT)
3.78/1.91	Added dependency pairs
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(2)
3.78/1.91	Obligation:
3.78/1.91	IDP problem:
3.78/1.91	The following function symbols are pre-defined:
3.78/1.91	<<<
3.78/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.78/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.78/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.78/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
3.78/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.78/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.78/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.78/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.78/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.78/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.78/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.78/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.78/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
3.78/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.78/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.78/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.78/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.78/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.78/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.78/1.91	>>>
3.78/1.91	
3.78/1.91	
3.78/1.91	The following domains are used:
3.78/1.91	   Integer
3.78/1.91	
3.78/1.91	The ITRS R consists of the following rules:
3.78/1.91	pow(x, y) -> if(y > 0, x, y)
3.78/1.91	if(TRUE, x, y) -> x * pow(x, y - 1)
3.78/1.91	if(FALSE, x, y) -> 1
3.78/1.91	
3.78/1.91	The integer pair graph contains the following rules and edges:
3.78/1.91	(0): POW(x[0], y[0]) -> IF(y[0] > 0, x[0], y[0])
3.78/1.91	(1): IF(TRUE, x[1], y[1]) -> POW(x[1], y[1] - 1)
3.78/1.91	
3.78/1.91	   (0) -> (1), if (y[0] > 0  & x[0] ->^* x[1] & y[0] ->^* y[1])
3.78/1.91	   (1) -> (0), if (x[1] ->^* x[0] & y[1] - 1 ->^* y[0])
3.78/1.91	
3.78/1.91	The set Q consists of the following terms:
3.78/1.91	pow(x0, x1)
3.78/1.91	if(TRUE, x0, x1)
3.78/1.91	if(FALSE, x0, x1)
3.78/1.91	
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(3) UsableRulesProof (EQUIVALENT)
3.78/1.91	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.
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(4)
3.78/1.91	Obligation:
3.78/1.91	IDP problem:
3.78/1.91	The following function symbols are pre-defined:
3.78/1.91	<<<
3.78/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.78/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.78/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.78/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
3.78/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.78/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.78/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.78/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.78/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.78/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.78/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.78/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.78/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
3.78/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.78/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.78/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.78/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.78/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.78/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.78/1.91	>>>
3.78/1.91	
3.78/1.91	
3.78/1.91	The following domains are used:
3.78/1.91	   Integer
3.78/1.91	
3.78/1.91	R is empty.
3.78/1.91	
3.78/1.91	The integer pair graph contains the following rules and edges:
3.78/1.91	(0): POW(x[0], y[0]) -> IF(y[0] > 0, x[0], y[0])
3.78/1.91	(1): IF(TRUE, x[1], y[1]) -> POW(x[1], y[1] - 1)
3.78/1.91	
3.78/1.91	   (0) -> (1), if (y[0] > 0  & x[0] ->^* x[1] & y[0] ->^* y[1])
3.78/1.91	   (1) -> (0), if (x[1] ->^* x[0] & y[1] - 1 ->^* y[0])
3.78/1.91	
3.78/1.91	The set Q consists of the following terms:
3.78/1.91	pow(x0, x1)
3.78/1.91	if(TRUE, x0, x1)
3.78/1.91	if(FALSE, x0, x1)
3.78/1.91	
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(5) IDPNonInfProof (SOUND)
3.78/1.91	Used the following options for this NonInfProof:
3.78/1.91	
3.78/1.91	IDPGPoloSolver:
3.78/1.91	Range: [(-1,2)]
3.78/1.91	IsNat: false
3.78/1.91	Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4c874750
3.78/1.91	Constraint Generator: NonInfConstraintGenerator:
3.78/1.91	PathGenerator: MetricPathGenerator:
3.78/1.91	Max Left Steps: 1
3.78/1.91	Max Right Steps: 1
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	The constraints were generated the following way:
3.78/1.91	
3.78/1.91	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
3.78/1.91	
3.78/1.91	Note that final constraints are written in bold face.
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	For Pair POW(x, y) -> IF(>(y, 0), x, y) the following chains were created:
3.78/1.91	*We consider the chain POW(x[0], y[0]) -> IF(>(y[0], 0), x[0], y[0]), IF(TRUE, x[1], y[1]) -> POW(x[1], -(y[1], 1)) which results in the following constraint:
3.78/1.91	
3.78/1.91	(1)    (>(y[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1]  ==>  POW(x[0], y[0])_>=_NonInfC & POW(x[0], y[0])_>=_IF(>(y[0], 0), x[0], y[0]) & (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=))
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (1) using rule (IV) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(2)    (>(y[0], 0)=TRUE  ==>  POW(x[0], y[0])_>=_NonInfC & POW(x[0], y[0])_>=_IF(>(y[0], 0), x[0], y[0]) & (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=))
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(3)    (y[0] + [-1] >= 0  ==>  (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[0] >= 0 & [(-1)bso_12] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(4)    (y[0] + [-1] >= 0  ==>  (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[0] >= 0 & [(-1)bso_12] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(5)    (y[0] + [-1] >= 0  ==>  (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=) & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[0] >= 0 & [(-1)bso_12] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(6)    (y[0] + [-1] >= 0  ==>  (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[0] >= 0 & [(-1)bso_12] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	For Pair IF(TRUE, x, y) -> POW(x, -(y, 1)) the following chains were created:
3.78/1.91	*We consider the chain POW(x[0], y[0]) -> IF(>(y[0], 0), x[0], y[0]), IF(TRUE, x[1], y[1]) -> POW(x[1], -(y[1], 1)), POW(x[0], y[0]) -> IF(>(y[0], 0), x[0], y[0]) which results in the following constraint:
3.78/1.91	
3.78/1.91	(1)    (>(y[0], 0)=TRUE & x[0]=x[1] & y[0]=y[1] & x[1]=x[0]1 & -(y[1], 1)=y[0]1  ==>  IF(TRUE, x[1], y[1])_>=_NonInfC & IF(TRUE, x[1], y[1])_>=_POW(x[1], -(y[1], 1)) & (U^Increasing(POW(x[1], -(y[1], 1))), >=))
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(2)    (>(y[0], 0)=TRUE  ==>  IF(TRUE, x[0], y[0])_>=_NonInfC & IF(TRUE, x[0], y[0])_>=_POW(x[0], -(y[0], 1)) & (U^Increasing(POW(x[1], -(y[1], 1))), >=))
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(3)    (y[0] + [-1] >= 0  ==>  (U^Increasing(POW(x[1], -(y[1], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[0] >= 0 & [1 + (-1)bso_14] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(4)    (y[0] + [-1] >= 0  ==>  (U^Increasing(POW(x[1], -(y[1], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[0] >= 0 & [1 + (-1)bso_14] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(5)    (y[0] + [-1] >= 0  ==>  (U^Increasing(POW(x[1], -(y[1], 1))), >=) & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[0] >= 0 & [1 + (-1)bso_14] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
3.78/1.91	
3.78/1.91	(6)    (y[0] + [-1] >= 0  ==>  (U^Increasing(POW(x[1], -(y[1], 1))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[0] >= 0 & [1 + (-1)bso_14] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	To summarize, we get the following constraints P__>=_ for the following pairs.
3.78/1.91	
3.78/1.91	*POW(x, y) -> IF(>(y, 0), x, y)
3.78/1.91	
3.78/1.91	*(y[0] + [-1] >= 0  ==>  (U^Increasing(IF(>(y[0], 0), x[0], y[0])), >=) & 0 = 0 & [(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]y[0] >= 0 & [(-1)bso_12] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	*IF(TRUE, x, y) -> POW(x, -(y, 1))
3.78/1.91	
3.78/1.91	*(y[0] + [-1] >= 0  ==>  (U^Increasing(POW(x[1], -(y[1], 1))), >=) & 0 = 0 & [(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]y[0] >= 0 & [1 + (-1)bso_14] >= 0)
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	
3.78/1.91	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. 
3.78/1.91	
3.78/1.91	Using the following integer polynomial ordering the  resulting constraints can be solved 
3.78/1.91	
3.78/1.91	Polynomial interpretation over integers[POLO]:
3.78/1.91	
3.78/1.91	   POL(TRUE) = 0
3.78/1.91	   POL(FALSE) = 0
3.78/1.91	   POL(POW(x_1, x_2)) = [-1] + x_2
3.78/1.91	   POL(IF(x_1, x_2, x_3)) = [-1] + x_3
3.78/1.91	   POL(>(x_1, x_2)) = [-1]
3.78/1.91	   POL(0) = 0
3.78/1.91	   POL(-(x_1, x_2)) = x_1 + [-1]x_2
3.78/1.91	   POL(1) = [1]
3.78/1.91	
3.78/1.91	
3.78/1.91	The following pairs are in P_>:
3.78/1.91	
3.78/1.91	
3.78/1.91	   IF(TRUE, x[1], y[1]) -> POW(x[1], -(y[1], 1))
3.78/1.91	
3.78/1.91	
3.78/1.91	The following pairs are in P_bound:
3.78/1.91	
3.78/1.91	
3.78/1.91	   POW(x[0], y[0]) -> IF(>(y[0], 0), x[0], y[0])
3.78/1.91	   IF(TRUE, x[1], y[1]) -> POW(x[1], -(y[1], 1))
3.78/1.91	
3.78/1.91	
3.78/1.91	The following pairs are in P_>=:
3.78/1.91	
3.78/1.91	
3.78/1.91	   POW(x[0], y[0]) -> IF(>(y[0], 0), x[0], y[0])
3.78/1.91	
3.78/1.91	
3.78/1.91	There are no usable rules.
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(6)
3.78/1.91	Obligation:
3.78/1.91	IDP problem:
3.78/1.91	The following function symbols are pre-defined:
3.78/1.91	<<<
3.78/1.91	 & 	~ 	Bwand: (Integer, Integer) -> Integer
3.78/1.91	 >= 	~ 	Ge: (Integer, Integer) -> Boolean
3.78/1.91	 | 	~ 	Bwor: (Integer, Integer) -> Integer
3.78/1.91	 / 	~ 	Div: (Integer, Integer) -> Integer
3.78/1.91	 != 	~ 	Neq: (Integer, Integer) -> Boolean
3.78/1.91	 && 	~ 	Land: (Boolean, Boolean) -> Boolean
3.78/1.91	 ! 	~ 	Lnot: (Boolean) -> Boolean
3.78/1.91	 = 	~ 	Eq: (Integer, Integer) -> Boolean
3.78/1.91	 <= 	~ 	Le: (Integer, Integer) -> Boolean
3.78/1.91	 ^ 	~ 	Bwxor: (Integer, Integer) -> Integer
3.78/1.91	 % 	~ 	Mod: (Integer, Integer) -> Integer
3.78/1.91	 > 	~ 	Gt: (Integer, Integer) -> Boolean
3.78/1.91	 + 	~ 	Add: (Integer, Integer) -> Integer
3.78/1.91	 -1 	~ 	UnaryMinus: (Integer) -> Integer
3.78/1.91	 < 	~ 	Lt: (Integer, Integer) -> Boolean
3.78/1.91	 || 	~ 	Lor: (Boolean, Boolean) -> Boolean
3.78/1.91	 - 	~ 	Sub: (Integer, Integer) -> Integer
3.78/1.91	 ~ 	~ 	Bwnot: (Integer) -> Integer
3.78/1.91	 * 	~ 	Mul: (Integer, Integer) -> Integer
3.78/1.91	>>>
3.78/1.91	
3.78/1.91	
3.78/1.91	The following domains are used:
3.78/1.91	   Integer
3.78/1.91	
3.78/1.91	R is empty.
3.78/1.91	
3.78/1.91	The integer pair graph contains the following rules and edges:
3.78/1.91	(0): POW(x[0], y[0]) -> IF(y[0] > 0, x[0], y[0])
3.78/1.91	
3.78/1.91	
3.78/1.91	The set Q consists of the following terms:
3.78/1.91	pow(x0, x1)
3.78/1.91	if(TRUE, x0, x1)
3.78/1.91	if(FALSE, x0, x1)
3.78/1.91	
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(7) IDependencyGraphProof (EQUIVALENT)
3.78/1.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
3.78/1.91	----------------------------------------
3.78/1.91	
3.78/1.91	(8)
3.78/1.91	TRUE
3.78/1.93	EOF