305.94/291.56	WORST_CASE(Omega(n^1), ?)
305.94/291.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
305.94/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.94/291.57	
305.94/291.57	
305.94/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.94/291.57	
305.94/291.57	(0) CpxTRS
305.94/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
305.94/291.57	(2) TRS for Loop Detection
305.94/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
305.94/291.57	(4) BEST
305.94/291.57	    (5) proven lower bound
305.94/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
305.94/291.57	        (7) BOUNDS(n^1, INF)
305.94/291.57	    (8) TRS for Loop Detection
305.94/291.57	
305.94/291.57	
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(0)
305.94/291.57	Obligation:
305.94/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.94/291.57	
305.94/291.57	
305.94/291.57	The TRS R consists of the following rules:
305.94/291.57	
305.94/291.57	   half(0) -> 0
305.94/291.57	   half(s(0)) -> 0
305.94/291.57	   half(s(s(x))) -> s(half(x))
305.94/291.57	   le(0, y) -> true
305.94/291.57	   le(s(x), 0) -> false
305.94/291.57	   le(s(x), s(y)) -> le(x, y)
305.94/291.57	   inc(0) -> 0
305.94/291.57	   inc(s(x)) -> s(inc(x))
305.94/291.57	   log(x) -> log2(x, 0)
305.94/291.57	   log2(x, y) -> if(le(x, s(0)), x, inc(y))
305.94/291.57	   if(true, x, s(y)) -> y
305.94/291.57	   if(false, x, y) -> log2(half(x), y)
305.94/291.57	
305.94/291.57	S is empty.
305.94/291.57	Rewrite Strategy: FULL
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
305.94/291.57	Transformed a relative TRS into a decreasing-loop problem.
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(2)
305.94/291.57	Obligation:
305.94/291.57	Analyzing the following TRS for decreasing loops:
305.94/291.57	
305.94/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.94/291.57	
305.94/291.57	
305.94/291.57	The TRS R consists of the following rules:
305.94/291.57	
305.94/291.57	   half(0) -> 0
305.94/291.57	   half(s(0)) -> 0
305.94/291.57	   half(s(s(x))) -> s(half(x))
305.94/291.57	   le(0, y) -> true
305.94/291.57	   le(s(x), 0) -> false
305.94/291.57	   le(s(x), s(y)) -> le(x, y)
305.94/291.57	   inc(0) -> 0
305.94/291.57	   inc(s(x)) -> s(inc(x))
305.94/291.57	   log(x) -> log2(x, 0)
305.94/291.57	   log2(x, y) -> if(le(x, s(0)), x, inc(y))
305.94/291.57	   if(true, x, s(y)) -> y
305.94/291.57	   if(false, x, y) -> log2(half(x), y)
305.94/291.57	
305.94/291.57	S is empty.
305.94/291.57	Rewrite Strategy: FULL
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
305.94/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
305.94/291.57	
305.94/291.57	The rewrite sequence
305.94/291.57	
305.94/291.57	half(s(s(x))) ->^+ s(half(x))
305.94/291.57	
305.94/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
305.94/291.57	
305.94/291.57	The pumping substitution is [x / s(s(x))].
305.94/291.57	
305.94/291.57	The result substitution is [ ].
305.94/291.57	
305.94/291.57	
305.94/291.57	
305.94/291.57	
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(4)
305.94/291.57	Complex Obligation (BEST)
305.94/291.57	
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(5)
305.94/291.57	Obligation:
305.94/291.57	Proved the lower bound n^1 for the following obligation:
305.94/291.57	
305.94/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.94/291.57	
305.94/291.57	
305.94/291.57	The TRS R consists of the following rules:
305.94/291.57	
305.94/291.57	   half(0) -> 0
305.94/291.57	   half(s(0)) -> 0
305.94/291.57	   half(s(s(x))) -> s(half(x))
305.94/291.57	   le(0, y) -> true
305.94/291.57	   le(s(x), 0) -> false
305.94/291.57	   le(s(x), s(y)) -> le(x, y)
305.94/291.57	   inc(0) -> 0
305.94/291.57	   inc(s(x)) -> s(inc(x))
305.94/291.57	   log(x) -> log2(x, 0)
305.94/291.57	   log2(x, y) -> if(le(x, s(0)), x, inc(y))
305.94/291.57	   if(true, x, s(y)) -> y
305.94/291.57	   if(false, x, y) -> log2(half(x), y)
305.94/291.57	
305.94/291.57	S is empty.
305.94/291.57	Rewrite Strategy: FULL
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(6) LowerBoundPropagationProof (FINISHED)
305.94/291.57	Propagated lower bound.
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(7)
305.94/291.57	BOUNDS(n^1, INF)
305.94/291.57	
305.94/291.57	----------------------------------------
305.94/291.57	
305.94/291.57	(8)
305.94/291.57	Obligation:
305.94/291.57	Analyzing the following TRS for decreasing loops:
305.94/291.57	
305.94/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.94/291.57	
305.94/291.57	
305.94/291.57	The TRS R consists of the following rules:
305.94/291.57	
305.94/291.57	   half(0) -> 0
305.94/291.57	   half(s(0)) -> 0
305.94/291.57	   half(s(s(x))) -> s(half(x))
305.94/291.57	   le(0, y) -> true
305.94/291.57	   le(s(x), 0) -> false
305.94/291.57	   le(s(x), s(y)) -> le(x, y)
305.94/291.57	   inc(0) -> 0
305.94/291.57	   inc(s(x)) -> s(inc(x))
305.94/291.57	   log(x) -> log2(x, 0)
305.94/291.57	   log2(x, y) -> if(le(x, s(0)), x, inc(y))
305.94/291.57	   if(true, x, s(y)) -> y
305.94/291.57	   if(false, x, y) -> log2(half(x), y)
305.94/291.57	
305.94/291.57	S is empty.
305.94/291.57	Rewrite Strategy: FULL
305.94/291.60	EOF