1121.90/291.70	WORST_CASE(Omega(n^1), ?)
1121.90/291.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1121.90/291.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.90/291.71	
1121.90/291.71	(0) CpxTRS
1121.90/291.71	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1121.90/291.71	(2) TRS for Loop Detection
1121.90/291.71	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1121.90/291.71	(4) BEST
1121.90/291.71	    (5) proven lower bound
1121.90/291.71	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1121.90/291.71	        (7) BOUNDS(n^1, INF)
1121.90/291.71	    (8) TRS for Loop Detection
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(0)
1121.90/291.71	Obligation:
1121.90/291.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	The TRS R consists of the following rules:
1121.90/291.71	
1121.90/291.71	   a__f(X) -> a__if(mark(X), c, f(true))
1121.90/291.71	   a__if(true, X, Y) -> mark(X)
1121.90/291.71	   a__if(false, X, Y) -> mark(Y)
1121.90/291.71	   mark(f(X)) -> a__f(mark(X))
1121.90/291.71	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1121.90/291.71	   mark(c) -> c
1121.90/291.71	   mark(true) -> true
1121.90/291.71	   mark(false) -> false
1121.90/291.71	   a__f(X) -> f(X)
1121.90/291.71	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1121.90/291.71	
1121.90/291.71	S is empty.
1121.90/291.71	Rewrite Strategy: FULL
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1121.90/291.71	Transformed a relative TRS into a decreasing-loop problem.
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(2)
1121.90/291.71	Obligation:
1121.90/291.71	Analyzing the following TRS for decreasing loops:
1121.90/291.71	
1121.90/291.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	The TRS R consists of the following rules:
1121.90/291.71	
1121.90/291.71	   a__f(X) -> a__if(mark(X), c, f(true))
1121.90/291.71	   a__if(true, X, Y) -> mark(X)
1121.90/291.71	   a__if(false, X, Y) -> mark(Y)
1121.90/291.71	   mark(f(X)) -> a__f(mark(X))
1121.90/291.71	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1121.90/291.71	   mark(c) -> c
1121.90/291.71	   mark(true) -> true
1121.90/291.71	   mark(false) -> false
1121.90/291.71	   a__f(X) -> f(X)
1121.90/291.71	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1121.90/291.71	
1121.90/291.71	S is empty.
1121.90/291.71	Rewrite Strategy: FULL
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(3) DecreasingLoopProof (LOWER BOUND(ID))
1121.90/291.71	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1121.90/291.71	
1121.90/291.71	The rewrite sequence
1121.90/291.71	
1121.90/291.71	mark(f(X)) ->^+ a__f(mark(X))
1121.90/291.71	
1121.90/291.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1121.90/291.71	
1121.90/291.71	The pumping substitution is [X / f(X)].
1121.90/291.71	
1121.90/291.71	The result substitution is [ ].
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(4)
1121.90/291.71	Complex Obligation (BEST)
1121.90/291.71	
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(5)
1121.90/291.71	Obligation:
1121.90/291.71	Proved the lower bound n^1 for the following obligation:
1121.90/291.71	
1121.90/291.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	The TRS R consists of the following rules:
1121.90/291.71	
1121.90/291.71	   a__f(X) -> a__if(mark(X), c, f(true))
1121.90/291.71	   a__if(true, X, Y) -> mark(X)
1121.90/291.71	   a__if(false, X, Y) -> mark(Y)
1121.90/291.71	   mark(f(X)) -> a__f(mark(X))
1121.90/291.71	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1121.90/291.71	   mark(c) -> c
1121.90/291.71	   mark(true) -> true
1121.90/291.71	   mark(false) -> false
1121.90/291.71	   a__f(X) -> f(X)
1121.90/291.71	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1121.90/291.71	
1121.90/291.71	S is empty.
1121.90/291.71	Rewrite Strategy: FULL
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(6) LowerBoundPropagationProof (FINISHED)
1121.90/291.71	Propagated lower bound.
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(7)
1121.90/291.71	BOUNDS(n^1, INF)
1121.90/291.71	
1121.90/291.71	----------------------------------------
1121.90/291.71	
1121.90/291.71	(8)
1121.90/291.71	Obligation:
1121.90/291.71	Analyzing the following TRS for decreasing loops:
1121.90/291.71	
1121.90/291.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1121.90/291.71	
1121.90/291.71	
1121.90/291.71	The TRS R consists of the following rules:
1121.90/291.71	
1121.90/291.71	   a__f(X) -> a__if(mark(X), c, f(true))
1121.90/291.71	   a__if(true, X, Y) -> mark(X)
1121.90/291.71	   a__if(false, X, Y) -> mark(Y)
1121.90/291.71	   mark(f(X)) -> a__f(mark(X))
1121.90/291.71	   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
1121.90/291.71	   mark(c) -> c
1121.90/291.71	   mark(true) -> true
1121.90/291.71	   mark(false) -> false
1121.90/291.71	   a__f(X) -> f(X)
1121.90/291.71	   a__if(X1, X2, X3) -> if(X1, X2, X3)
1121.90/291.71	
1121.90/291.71	S is empty.
1121.90/291.71	Rewrite Strategy: FULL
1122.21/291.78	EOF