318.14/291.49	WORST_CASE(Omega(n^1), ?)
318.14/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
318.14/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
318.14/291.50	
318.14/291.50	
318.14/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.50	
318.14/291.50	(0) CpxTRS
318.14/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
318.14/291.50	(2) TRS for Loop Detection
318.14/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
318.14/291.50	(4) BEST
318.14/291.50	    (5) proven lower bound
318.14/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
318.14/291.50	        (7) BOUNDS(n^1, INF)
318.14/291.50	    (8) TRS for Loop Detection
318.14/291.50	
318.14/291.50	
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(0)
318.14/291.50	Obligation:
318.14/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.50	
318.14/291.50	
318.14/291.50	The TRS R consists of the following rules:
318.14/291.50	
318.14/291.50	   half(0) -> 0
318.14/291.50	   half(s(0)) -> 0
318.14/291.50	   half(s(s(x))) -> s(half(x))
318.14/291.50	   inc(0) -> 0
318.14/291.50	   inc(s(x)) -> s(inc(x))
318.14/291.50	   zero(0) -> true
318.14/291.50	   zero(s(x)) -> false
318.14/291.50	   p(0) -> 0
318.14/291.50	   p(s(x)) -> x
318.14/291.50	   bits(x) -> bitIter(x, 0)
318.14/291.50	   bitIter(x, y) -> if(zero(x), x, inc(y))
318.14/291.50	   if(true, x, y) -> p(y)
318.14/291.50	   if(false, x, y) -> bitIter(half(x), y)
318.14/291.50	
318.14/291.50	S is empty.
318.14/291.50	Rewrite Strategy: FULL
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
318.14/291.50	Transformed a relative TRS into a decreasing-loop problem.
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(2)
318.14/291.50	Obligation:
318.14/291.50	Analyzing the following TRS for decreasing loops:
318.14/291.50	
318.14/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.50	
318.14/291.50	
318.14/291.50	The TRS R consists of the following rules:
318.14/291.50	
318.14/291.50	   half(0) -> 0
318.14/291.50	   half(s(0)) -> 0
318.14/291.50	   half(s(s(x))) -> s(half(x))
318.14/291.50	   inc(0) -> 0
318.14/291.50	   inc(s(x)) -> s(inc(x))
318.14/291.50	   zero(0) -> true
318.14/291.50	   zero(s(x)) -> false
318.14/291.50	   p(0) -> 0
318.14/291.50	   p(s(x)) -> x
318.14/291.50	   bits(x) -> bitIter(x, 0)
318.14/291.50	   bitIter(x, y) -> if(zero(x), x, inc(y))
318.14/291.50	   if(true, x, y) -> p(y)
318.14/291.50	   if(false, x, y) -> bitIter(half(x), y)
318.14/291.50	
318.14/291.50	S is empty.
318.14/291.50	Rewrite Strategy: FULL
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
318.14/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
318.14/291.50	
318.14/291.50	The rewrite sequence
318.14/291.50	
318.14/291.50	half(s(s(x))) ->^+ s(half(x))
318.14/291.50	
318.14/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
318.14/291.50	
318.14/291.50	The pumping substitution is [x / s(s(x))].
318.14/291.50	
318.14/291.50	The result substitution is [ ].
318.14/291.50	
318.14/291.50	
318.14/291.50	
318.14/291.50	
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(4)
318.14/291.50	Complex Obligation (BEST)
318.14/291.50	
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(5)
318.14/291.50	Obligation:
318.14/291.50	Proved the lower bound n^1 for the following obligation:
318.14/291.50	
318.14/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.50	
318.14/291.50	
318.14/291.50	The TRS R consists of the following rules:
318.14/291.50	
318.14/291.50	   half(0) -> 0
318.14/291.50	   half(s(0)) -> 0
318.14/291.50	   half(s(s(x))) -> s(half(x))
318.14/291.50	   inc(0) -> 0
318.14/291.50	   inc(s(x)) -> s(inc(x))
318.14/291.50	   zero(0) -> true
318.14/291.50	   zero(s(x)) -> false
318.14/291.50	   p(0) -> 0
318.14/291.50	   p(s(x)) -> x
318.14/291.50	   bits(x) -> bitIter(x, 0)
318.14/291.50	   bitIter(x, y) -> if(zero(x), x, inc(y))
318.14/291.50	   if(true, x, y) -> p(y)
318.14/291.50	   if(false, x, y) -> bitIter(half(x), y)
318.14/291.50	
318.14/291.50	S is empty.
318.14/291.50	Rewrite Strategy: FULL
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(6) LowerBoundPropagationProof (FINISHED)
318.14/291.50	Propagated lower bound.
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(7)
318.14/291.50	BOUNDS(n^1, INF)
318.14/291.50	
318.14/291.50	----------------------------------------
318.14/291.50	
318.14/291.50	(8)
318.14/291.50	Obligation:
318.14/291.50	Analyzing the following TRS for decreasing loops:
318.14/291.50	
318.14/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.50	
318.14/291.50	
318.14/291.50	The TRS R consists of the following rules:
318.14/291.50	
318.14/291.50	   half(0) -> 0
318.14/291.50	   half(s(0)) -> 0
318.14/291.50	   half(s(s(x))) -> s(half(x))
318.14/291.50	   inc(0) -> 0
318.14/291.50	   inc(s(x)) -> s(inc(x))
318.14/291.50	   zero(0) -> true
318.14/291.50	   zero(s(x)) -> false
318.14/291.50	   p(0) -> 0
318.14/291.50	   p(s(x)) -> x
318.14/291.50	   bits(x) -> bitIter(x, 0)
318.14/291.50	   bitIter(x, y) -> if(zero(x), x, inc(y))
318.14/291.50	   if(true, x, y) -> p(y)
318.14/291.50	   if(false, x, y) -> bitIter(half(x), y)
318.14/291.50	
318.14/291.50	S is empty.
318.14/291.50	Rewrite Strategy: FULL
318.25/291.52	EOF