884.71/291.54	WORST_CASE(Omega(n^1), ?)
884.71/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
884.71/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
884.71/291.54	
884.71/291.54	
884.71/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
884.71/291.54	
884.71/291.54	(0) CpxTRS
884.71/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
884.71/291.54	(2) TRS for Loop Detection
884.71/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
884.71/291.54	(4) BEST
884.71/291.54	    (5) proven lower bound
884.71/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
884.71/291.54	        (7) BOUNDS(n^1, INF)
884.71/291.54	    (8) TRS for Loop Detection
884.71/291.54	
884.71/291.54	
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(0)
884.71/291.54	Obligation:
884.71/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
884.71/291.54	
884.71/291.54	
884.71/291.54	The TRS R consists of the following rules:
884.71/291.54	
884.71/291.54	   g(x, 0) -> 0
884.71/291.54	   g(d, s(x)) -> s(s(g(d, x)))
884.71/291.54	   g(h, s(0)) -> 0
884.71/291.54	   g(h, s(s(x))) -> s(g(h, x))
884.71/291.54	   double(x) -> g(d, x)
884.71/291.54	   half(x) -> g(h, x)
884.71/291.54	   f(s(x), y) -> f(half(s(x)), double(y))
884.71/291.54	   f(s(0), y) -> y
884.71/291.54	   id(x) -> f(x, s(0))
884.71/291.54	
884.71/291.54	S is empty.
884.71/291.54	Rewrite Strategy: FULL
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
884.71/291.54	Transformed a relative TRS into a decreasing-loop problem.
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(2)
884.71/291.54	Obligation:
884.71/291.54	Analyzing the following TRS for decreasing loops:
884.71/291.54	
884.71/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
884.71/291.54	
884.71/291.54	
884.71/291.54	The TRS R consists of the following rules:
884.71/291.54	
884.71/291.54	   g(x, 0) -> 0
884.71/291.54	   g(d, s(x)) -> s(s(g(d, x)))
884.71/291.54	   g(h, s(0)) -> 0
884.71/291.54	   g(h, s(s(x))) -> s(g(h, x))
884.71/291.54	   double(x) -> g(d, x)
884.71/291.54	   half(x) -> g(h, x)
884.71/291.54	   f(s(x), y) -> f(half(s(x)), double(y))
884.71/291.54	   f(s(0), y) -> y
884.71/291.54	   id(x) -> f(x, s(0))
884.71/291.54	
884.71/291.54	S is empty.
884.71/291.54	Rewrite Strategy: FULL
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
884.71/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
884.71/291.54	
884.71/291.54	The rewrite sequence
884.71/291.54	
884.71/291.54	g(h, s(s(x))) ->^+ s(g(h, x))
884.71/291.54	
884.71/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
884.71/291.54	
884.71/291.54	The pumping substitution is [x / s(s(x))].
884.71/291.54	
884.71/291.54	The result substitution is [ ].
884.71/291.54	
884.71/291.54	
884.71/291.54	
884.71/291.54	
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(4)
884.71/291.54	Complex Obligation (BEST)
884.71/291.54	
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(5)
884.71/291.54	Obligation:
884.71/291.54	Proved the lower bound n^1 for the following obligation:
884.71/291.54	
884.71/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
884.71/291.54	
884.71/291.54	
884.71/291.54	The TRS R consists of the following rules:
884.71/291.54	
884.71/291.54	   g(x, 0) -> 0
884.71/291.54	   g(d, s(x)) -> s(s(g(d, x)))
884.71/291.54	   g(h, s(0)) -> 0
884.71/291.54	   g(h, s(s(x))) -> s(g(h, x))
884.71/291.54	   double(x) -> g(d, x)
884.71/291.54	   half(x) -> g(h, x)
884.71/291.54	   f(s(x), y) -> f(half(s(x)), double(y))
884.71/291.54	   f(s(0), y) -> y
884.71/291.54	   id(x) -> f(x, s(0))
884.71/291.54	
884.71/291.54	S is empty.
884.71/291.54	Rewrite Strategy: FULL
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(6) LowerBoundPropagationProof (FINISHED)
884.71/291.54	Propagated lower bound.
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(7)
884.71/291.54	BOUNDS(n^1, INF)
884.71/291.54	
884.71/291.54	----------------------------------------
884.71/291.54	
884.71/291.54	(8)
884.71/291.54	Obligation:
884.71/291.54	Analyzing the following TRS for decreasing loops:
884.71/291.54	
884.71/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
884.71/291.54	
884.71/291.54	
884.71/291.54	The TRS R consists of the following rules:
884.71/291.54	
884.71/291.54	   g(x, 0) -> 0
884.71/291.54	   g(d, s(x)) -> s(s(g(d, x)))
884.71/291.54	   g(h, s(0)) -> 0
884.71/291.54	   g(h, s(s(x))) -> s(g(h, x))
884.71/291.54	   double(x) -> g(d, x)
884.71/291.54	   half(x) -> g(h, x)
884.71/291.54	   f(s(x), y) -> f(half(s(x)), double(y))
884.71/291.54	   f(s(0), y) -> y
884.71/291.54	   id(x) -> f(x, s(0))
884.71/291.54	
884.71/291.54	S is empty.
884.71/291.54	Rewrite Strategy: FULL
884.83/291.59	EOF