4.18/1.84	WORST_CASE(NON_POLY, ?)
4.18/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.18/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.18/1.84	
4.18/1.84	
4.18/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.18/1.84	
4.18/1.84	(0) CpxTRS
4.18/1.84	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.18/1.84	(2) TRS for Loop Detection
4.18/1.84	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.18/1.84	(4) BEST
4.18/1.84	    (5) proven lower bound
4.18/1.84	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
4.18/1.84	        (7) BOUNDS(n^1, INF)
4.18/1.84	    (8) TRS for Loop Detection
4.18/1.84	        (9) DecreasingLoopProof [FINISHED, 123 ms]
4.18/1.84	        (10) BOUNDS(EXP, INF)
4.18/1.84	
4.18/1.84	
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(0)
4.18/1.84	Obligation:
4.18/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.18/1.84	
4.18/1.84	
4.18/1.84	The TRS R consists of the following rules:
4.18/1.84	
4.18/1.84	   minus_active(0, y) -> 0
4.18/1.84	   mark(0) -> 0
4.18/1.84	   minus_active(s(x), s(y)) -> minus_active(x, y)
4.18/1.84	   mark(s(x)) -> s(mark(x))
4.18/1.84	   ge_active(x, 0) -> true
4.18/1.84	   mark(minus(x, y)) -> minus_active(x, y)
4.18/1.84	   ge_active(0, s(y)) -> false
4.18/1.84	   mark(ge(x, y)) -> ge_active(x, y)
4.18/1.84	   ge_active(s(x), s(y)) -> ge_active(x, y)
4.18/1.84	   mark(div(x, y)) -> div_active(mark(x), y)
4.18/1.84	   div_active(0, s(y)) -> 0
4.18/1.84	   mark(if(x, y, z)) -> if_active(mark(x), y, z)
4.18/1.84	   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
4.18/1.84	   if_active(true, x, y) -> mark(x)
4.18/1.84	   minus_active(x, y) -> minus(x, y)
4.18/1.84	   if_active(false, x, y) -> mark(y)
4.18/1.84	   ge_active(x, y) -> ge(x, y)
4.18/1.84	   if_active(x, y, z) -> if(x, y, z)
4.18/1.84	   div_active(x, y) -> div(x, y)
4.18/1.84	
4.18/1.84	S is empty.
4.18/1.84	Rewrite Strategy: FULL
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.18/1.84	Transformed a relative TRS into a decreasing-loop problem.
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(2)
4.18/1.84	Obligation:
4.18/1.84	Analyzing the following TRS for decreasing loops:
4.18/1.84	
4.18/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.18/1.84	
4.18/1.84	
4.18/1.84	The TRS R consists of the following rules:
4.18/1.84	
4.18/1.84	   minus_active(0, y) -> 0
4.18/1.84	   mark(0) -> 0
4.18/1.84	   minus_active(s(x), s(y)) -> minus_active(x, y)
4.18/1.84	   mark(s(x)) -> s(mark(x))
4.18/1.84	   ge_active(x, 0) -> true
4.18/1.84	   mark(minus(x, y)) -> minus_active(x, y)
4.18/1.84	   ge_active(0, s(y)) -> false
4.18/1.84	   mark(ge(x, y)) -> ge_active(x, y)
4.18/1.84	   ge_active(s(x), s(y)) -> ge_active(x, y)
4.18/1.84	   mark(div(x, y)) -> div_active(mark(x), y)
4.18/1.84	   div_active(0, s(y)) -> 0
4.18/1.84	   mark(if(x, y, z)) -> if_active(mark(x), y, z)
4.18/1.84	   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
4.18/1.84	   if_active(true, x, y) -> mark(x)
4.18/1.84	   minus_active(x, y) -> minus(x, y)
4.18/1.84	   if_active(false, x, y) -> mark(y)
4.18/1.84	   ge_active(x, y) -> ge(x, y)
4.18/1.84	   if_active(x, y, z) -> if(x, y, z)
4.18/1.84	   div_active(x, y) -> div(x, y)
4.18/1.84	
4.18/1.84	S is empty.
4.18/1.84	Rewrite Strategy: FULL
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(3) DecreasingLoopProof (LOWER BOUND(ID))
4.18/1.84	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.18/1.84	
4.18/1.84	The rewrite sequence
4.18/1.84	
4.18/1.84	mark(s(x)) ->^+ s(mark(x))
4.18/1.84	
4.18/1.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
4.18/1.84	
4.18/1.84	The pumping substitution is [x / s(x)].
4.18/1.84	
4.18/1.84	The result substitution is [ ].
4.18/1.84	
4.18/1.84	
4.18/1.84	
4.18/1.84	
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(4)
4.18/1.84	Complex Obligation (BEST)
4.18/1.84	
4.18/1.84	----------------------------------------
4.18/1.84	
4.18/1.84	(5)
4.18/1.84	Obligation:
4.18/1.84	Proved the lower bound n^1 for the following obligation:
4.18/1.84	
4.18/1.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.18/1.84	
4.18/1.84	
4.18/1.84	The TRS R consists of the following rules:
4.18/1.84	
4.18/1.84	   minus_active(0, y) -> 0
4.18/1.84	   mark(0) -> 0
4.18/1.84	   minus_active(s(x), s(y)) -> minus_active(x, y)
4.18/1.84	   mark(s(x)) -> s(mark(x))
4.18/1.84	   ge_active(x, 0) -> true
4.18/1.84	   mark(minus(x, y)) -> minus_active(x, y)
4.18/1.84	   ge_active(0, s(y)) -> false
4.18/1.84	   mark(ge(x, y)) -> ge_active(x, y)
4.18/1.84	   ge_active(s(x), s(y)) -> ge_active(x, y)
4.18/1.84	   mark(div(x, y)) -> div_active(mark(x), y)
4.18/1.84	   div_active(0, s(y)) -> 0
4.18/1.84	   mark(if(x, y, z)) -> if_active(mark(x), y, z)
4.18/1.84	   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
4.18/1.84	   if_active(true, x, y) -> mark(x)
4.18/1.84	   minus_active(x, y) -> minus(x, y)
4.18/1.84	   if_active(false, x, y) -> mark(y)
4.18/1.84	   ge_active(x, y) -> ge(x, y)
4.18/1.84	   if_active(x, y, z) -> if(x, y, z)
4.18/1.85	   div_active(x, y) -> div(x, y)
4.18/1.85	
4.18/1.85	S is empty.
4.18/1.85	Rewrite Strategy: FULL
4.18/1.85	----------------------------------------
4.18/1.85	
4.18/1.85	(6) LowerBoundPropagationProof (FINISHED)
4.18/1.85	Propagated lower bound.
4.18/1.85	----------------------------------------
4.18/1.85	
4.18/1.85	(7)
4.18/1.85	BOUNDS(n^1, INF)
4.18/1.85	
4.18/1.85	----------------------------------------
4.18/1.85	
4.18/1.85	(8)
4.18/1.85	Obligation:
4.18/1.85	Analyzing the following TRS for decreasing loops:
4.18/1.85	
4.18/1.85	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
4.18/1.85	
4.18/1.85	
4.18/1.85	The TRS R consists of the following rules:
4.18/1.85	
4.18/1.85	   minus_active(0, y) -> 0
4.18/1.85	   mark(0) -> 0
4.18/1.85	   minus_active(s(x), s(y)) -> minus_active(x, y)
4.18/1.85	   mark(s(x)) -> s(mark(x))
4.18/1.85	   ge_active(x, 0) -> true
4.18/1.85	   mark(minus(x, y)) -> minus_active(x, y)
4.18/1.85	   ge_active(0, s(y)) -> false
4.18/1.85	   mark(ge(x, y)) -> ge_active(x, y)
4.18/1.85	   ge_active(s(x), s(y)) -> ge_active(x, y)
4.18/1.85	   mark(div(x, y)) -> div_active(mark(x), y)
4.18/1.85	   div_active(0, s(y)) -> 0
4.18/1.85	   mark(if(x, y, z)) -> if_active(mark(x), y, z)
4.18/1.85	   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
4.18/1.85	   if_active(true, x, y) -> mark(x)
4.18/1.85	   minus_active(x, y) -> minus(x, y)
4.18/1.85	   if_active(false, x, y) -> mark(y)
4.18/1.85	   ge_active(x, y) -> ge(x, y)
4.18/1.85	   if_active(x, y, z) -> if(x, y, z)
4.18/1.85	   div_active(x, y) -> div(x, y)
4.18/1.85	
4.18/1.85	S is empty.
4.18/1.85	Rewrite Strategy: FULL
4.18/1.85	----------------------------------------
4.18/1.85	
4.18/1.85	(9) DecreasingLoopProof (FINISHED)
4.18/1.85	The following loop(s) give(s) rise to the lower bound EXP:
4.18/1.85	
4.18/1.85	The rewrite sequence
4.18/1.85	
4.18/1.85	mark(div(s(x1_0), s(y2_1))) ->^+ if_active(ge_active(mark(x1_0), y2_1), s(div(minus(mark(x1_0), y2_1), s(y2_1))), 0)
4.18/1.85	
4.18/1.85	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
4.18/1.85	
4.18/1.85	The pumping substitution is [x1_0 / div(s(x1_0), s(y2_1))].
4.18/1.85	
4.18/1.85	The result substitution is [ ].
4.18/1.85	
4.18/1.85	
4.18/1.85	
4.18/1.85	The rewrite sequence
4.18/1.85	
4.18/1.85	mark(div(s(x1_0), s(y2_1))) ->^+ if_active(ge_active(mark(x1_0), y2_1), s(div(minus(mark(x1_0), y2_1), s(y2_1))), 0)
4.18/1.85	
4.18/1.85	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0,0].
4.18/1.85	
4.18/1.85	The pumping substitution is [x1_0 / div(s(x1_0), s(y2_1))].
4.18/1.85	
4.18/1.85	The result substitution is [ ].
4.18/1.85	
4.18/1.85	
4.18/1.85	
4.18/1.85	
4.18/1.85	----------------------------------------
4.18/1.85	
4.18/1.85	(10)
4.18/1.85	BOUNDS(EXP, INF)
4.45/1.88	EOF