1122.00/291.58	WORST_CASE(Omega(n^1), ?)
1122.00/291.58	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1122.00/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.00/291.58	
1122.00/291.58	(0) CpxTRS
1122.00/291.58	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1122.00/291.58	(2) TRS for Loop Detection
1122.00/291.58	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1122.00/291.58	(4) BEST
1122.00/291.58	    (5) proven lower bound
1122.00/291.58	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.00/291.58	        (7) BOUNDS(n^1, INF)
1122.00/291.58	    (8) TRS for Loop Detection
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(0)
1122.00/291.58	Obligation:
1122.00/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	The TRS R consists of the following rules:
1122.00/291.58	
1122.00/291.58	   a__f(X) -> a__if(mark(X), c, f(true))
1122.00/291.58	   a__if(true, X, Y) -> mark(X)
1122.00/291.58	   a__if(false, X, Y) -> mark(Y)
1122.00/291.58	   mark(f(X)) -> a__f(mark(X))
1122.00/291.58	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1122.00/291.58	   mark(c) -> c
1122.00/291.58	   mark(true) -> true
1122.00/291.58	   mark(false) -> false
1122.00/291.58	   a__f(X) -> f(X)
1122.00/291.58	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1122.00/291.58	
1122.00/291.58	S is empty.
1122.00/291.58	Rewrite Strategy: INNERMOST
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1122.00/291.58	Transformed a relative TRS into a decreasing-loop problem.
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(2)
1122.00/291.58	Obligation:
1122.00/291.58	Analyzing the following TRS for decreasing loops:
1122.00/291.58	
1122.00/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	The TRS R consists of the following rules:
1122.00/291.58	
1122.00/291.58	   a__f(X) -> a__if(mark(X), c, f(true))
1122.00/291.58	   a__if(true, X, Y) -> mark(X)
1122.00/291.58	   a__if(false, X, Y) -> mark(Y)
1122.00/291.58	   mark(f(X)) -> a__f(mark(X))
1122.00/291.58	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1122.00/291.58	   mark(c) -> c
1122.00/291.58	   mark(true) -> true
1122.00/291.58	   mark(false) -> false
1122.00/291.58	   a__f(X) -> f(X)
1122.00/291.58	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1122.00/291.58	
1122.00/291.58	S is empty.
1122.00/291.58	Rewrite Strategy: INNERMOST
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
1122.00/291.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1122.00/291.58	
1122.00/291.58	The rewrite sequence
1122.00/291.58	
1122.00/291.58	mark(f(X)) ->^+ a__f(mark(X))
1122.00/291.58	
1122.00/291.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1122.00/291.58	
1122.00/291.58	The pumping substitution is [X / f(X)].
1122.00/291.58	
1122.00/291.58	The result substitution is [ ].
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(4)
1122.00/291.58	Complex Obligation (BEST)
1122.00/291.58	
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(5)
1122.00/291.58	Obligation:
1122.00/291.58	Proved the lower bound n^1 for the following obligation:
1122.00/291.58	
1122.00/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	The TRS R consists of the following rules:
1122.00/291.58	
1122.00/291.58	   a__f(X) -> a__if(mark(X), c, f(true))
1122.00/291.58	   a__if(true, X, Y) -> mark(X)
1122.00/291.58	   a__if(false, X, Y) -> mark(Y)
1122.00/291.58	   mark(f(X)) -> a__f(mark(X))
1122.00/291.58	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1122.00/291.58	   mark(c) -> c
1122.00/291.58	   mark(true) -> true
1122.00/291.58	   mark(false) -> false
1122.00/291.58	   a__f(X) -> f(X)
1122.00/291.58	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1122.00/291.58	
1122.00/291.58	S is empty.
1122.00/291.58	Rewrite Strategy: INNERMOST
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(6) LowerBoundPropagationProof (FINISHED)
1122.00/291.58	Propagated lower bound.
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(7)
1122.00/291.58	BOUNDS(n^1, INF)
1122.00/291.58	
1122.00/291.58	----------------------------------------
1122.00/291.58	
1122.00/291.58	(8)
1122.00/291.58	Obligation:
1122.00/291.58	Analyzing the following TRS for decreasing loops:
1122.00/291.58	
1122.00/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1122.00/291.58	
1122.00/291.58	
1122.00/291.58	The TRS R consists of the following rules:
1122.00/291.58	
1122.00/291.58	   a__f(X) -> a__if(mark(X), c, f(true))
1122.00/291.58	   a__if(true, X, Y) -> mark(X)
1122.00/291.58	   a__if(false, X, Y) -> mark(Y)
1122.00/291.58	   mark(f(X)) -> a__f(mark(X))
1122.00/291.58	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1122.00/291.58	   mark(c) -> c
1122.00/291.58	   mark(true) -> true
1122.00/291.58	   mark(false) -> false
1122.00/291.58	   a__f(X) -> f(X)
1122.00/291.58	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1122.00/291.58	
1122.00/291.58	S is empty.
1122.00/291.58	Rewrite Strategy: INNERMOST
1122.33/291.64	EOF