1102.29/291.54	WORST_CASE(Omega(n^1), ?)
1102.29/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1102.29/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.29/291.54	
1102.29/291.54	(0) CpxTRS
1102.29/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1102.29/291.54	(2) TRS for Loop Detection
1102.29/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1102.29/291.54	(4) BEST
1102.29/291.54	    (5) proven lower bound
1102.29/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1102.29/291.54	        (7) BOUNDS(n^1, INF)
1102.29/291.54	    (8) TRS for Loop Detection
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(0)
1102.29/291.54	Obligation:
1102.29/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	The TRS R consists of the following rules:
1102.29/291.54	
1102.29/291.54	   tower(x) -> f(a, x, s(0))
1102.29/291.54	   f(a, 0, y) -> y
1102.29/291.54	   f(a, s(x), y) -> f(b, y, s(x))
1102.29/291.54	   f(b, y, x) -> f(a, half(x), exp(y))
1102.29/291.54	   exp(0) -> s(0)
1102.29/291.54	   exp(s(x)) -> double(exp(x))
1102.29/291.54	   double(0) -> 0
1102.29/291.54	   double(s(x)) -> s(s(double(x)))
1102.29/291.54	   half(0) -> double(0)
1102.29/291.54	   half(s(0)) -> half(0)
1102.29/291.54	   half(s(s(x))) -> s(half(x))
1102.29/291.54	
1102.29/291.54	S is empty.
1102.29/291.54	Rewrite Strategy: FULL
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1102.29/291.54	Transformed a relative TRS into a decreasing-loop problem.
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(2)
1102.29/291.54	Obligation:
1102.29/291.54	Analyzing the following TRS for decreasing loops:
1102.29/291.54	
1102.29/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	The TRS R consists of the following rules:
1102.29/291.54	
1102.29/291.54	   tower(x) -> f(a, x, s(0))
1102.29/291.54	   f(a, 0, y) -> y
1102.29/291.54	   f(a, s(x), y) -> f(b, y, s(x))
1102.29/291.54	   f(b, y, x) -> f(a, half(x), exp(y))
1102.29/291.54	   exp(0) -> s(0)
1102.29/291.54	   exp(s(x)) -> double(exp(x))
1102.29/291.54	   double(0) -> 0
1102.29/291.54	   double(s(x)) -> s(s(double(x)))
1102.29/291.54	   half(0) -> double(0)
1102.29/291.54	   half(s(0)) -> half(0)
1102.29/291.54	   half(s(s(x))) -> s(half(x))
1102.29/291.54	
1102.29/291.54	S is empty.
1102.29/291.54	Rewrite Strategy: FULL
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
1102.29/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1102.29/291.54	
1102.29/291.54	The rewrite sequence
1102.29/291.54	
1102.29/291.54	exp(s(x)) ->^+ double(exp(x))
1102.29/291.54	
1102.29/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1102.29/291.54	
1102.29/291.54	The pumping substitution is [x / s(x)].
1102.29/291.54	
1102.29/291.54	The result substitution is [ ].
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(4)
1102.29/291.54	Complex Obligation (BEST)
1102.29/291.54	
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(5)
1102.29/291.54	Obligation:
1102.29/291.54	Proved the lower bound n^1 for the following obligation:
1102.29/291.54	
1102.29/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	The TRS R consists of the following rules:
1102.29/291.54	
1102.29/291.54	   tower(x) -> f(a, x, s(0))
1102.29/291.54	   f(a, 0, y) -> y
1102.29/291.54	   f(a, s(x), y) -> f(b, y, s(x))
1102.29/291.54	   f(b, y, x) -> f(a, half(x), exp(y))
1102.29/291.54	   exp(0) -> s(0)
1102.29/291.54	   exp(s(x)) -> double(exp(x))
1102.29/291.54	   double(0) -> 0
1102.29/291.54	   double(s(x)) -> s(s(double(x)))
1102.29/291.54	   half(0) -> double(0)
1102.29/291.54	   half(s(0)) -> half(0)
1102.29/291.54	   half(s(s(x))) -> s(half(x))
1102.29/291.54	
1102.29/291.54	S is empty.
1102.29/291.54	Rewrite Strategy: FULL
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(6) LowerBoundPropagationProof (FINISHED)
1102.29/291.54	Propagated lower bound.
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(7)
1102.29/291.54	BOUNDS(n^1, INF)
1102.29/291.54	
1102.29/291.54	----------------------------------------
1102.29/291.54	
1102.29/291.54	(8)
1102.29/291.54	Obligation:
1102.29/291.54	Analyzing the following TRS for decreasing loops:
1102.29/291.54	
1102.29/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1102.29/291.54	
1102.29/291.54	
1102.29/291.54	The TRS R consists of the following rules:
1102.29/291.54	
1102.29/291.54	   tower(x) -> f(a, x, s(0))
1102.29/291.54	   f(a, 0, y) -> y
1102.29/291.54	   f(a, s(x), y) -> f(b, y, s(x))
1102.29/291.54	   f(b, y, x) -> f(a, half(x), exp(y))
1102.29/291.54	   exp(0) -> s(0)
1102.29/291.54	   exp(s(x)) -> double(exp(x))
1102.29/291.54	   double(0) -> 0
1102.29/291.54	   double(s(x)) -> s(s(double(x)))
1102.29/291.54	   half(0) -> double(0)
1102.29/291.54	   half(s(0)) -> half(0)
1102.29/291.54	   half(s(s(x))) -> s(half(x))
1102.29/291.54	
1102.29/291.54	S is empty.
1102.29/291.54	Rewrite Strategy: FULL
1102.61/291.61	EOF