1036.59/291.48	WORST_CASE(Omega(n^1), ?)
1036.59/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1036.59/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.59/291.49	
1036.59/291.49	(0) CpxTRS
1036.59/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1036.59/291.49	(2) TRS for Loop Detection
1036.59/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1036.59/291.49	(4) BEST
1036.59/291.49	    (5) proven lower bound
1036.59/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1036.59/291.49	        (7) BOUNDS(n^1, INF)
1036.59/291.49	    (8) TRS for Loop Detection
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(0)
1036.59/291.49	Obligation:
1036.59/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	The TRS R consists of the following rules:
1036.59/291.49	
1036.59/291.49	   minus(x, 0) -> x
1036.59/291.49	   minus(s(x), s(y)) -> minus(x, y)
1036.59/291.49	   f(0) -> s(0)
1036.59/291.49	   f(s(x)) -> minus(s(x), g(f(x)))
1036.59/291.49	   g(0) -> 0
1036.59/291.49	   g(s(x)) -> minus(s(x), f(g(x)))
1036.59/291.49	
1036.59/291.49	S is empty.
1036.59/291.49	Rewrite Strategy: INNERMOST
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1036.59/291.49	Transformed a relative TRS into a decreasing-loop problem.
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(2)
1036.59/291.49	Obligation:
1036.59/291.49	Analyzing the following TRS for decreasing loops:
1036.59/291.49	
1036.59/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	The TRS R consists of the following rules:
1036.59/291.49	
1036.59/291.49	   minus(x, 0) -> x
1036.59/291.49	   minus(s(x), s(y)) -> minus(x, y)
1036.59/291.49	   f(0) -> s(0)
1036.59/291.49	   f(s(x)) -> minus(s(x), g(f(x)))
1036.59/291.49	   g(0) -> 0
1036.59/291.49	   g(s(x)) -> minus(s(x), f(g(x)))
1036.59/291.49	
1036.59/291.49	S is empty.
1036.59/291.49	Rewrite Strategy: INNERMOST
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
1036.59/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1036.59/291.49	
1036.59/291.49	The rewrite sequence
1036.59/291.49	
1036.59/291.49	g(s(x)) ->^+ minus(s(x), f(g(x)))
1036.59/291.49	
1036.59/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
1036.59/291.49	
1036.59/291.49	The pumping substitution is [x / s(x)].
1036.59/291.49	
1036.59/291.49	The result substitution is [ ].
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(4)
1036.59/291.49	Complex Obligation (BEST)
1036.59/291.49	
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(5)
1036.59/291.49	Obligation:
1036.59/291.49	Proved the lower bound n^1 for the following obligation:
1036.59/291.49	
1036.59/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	The TRS R consists of the following rules:
1036.59/291.49	
1036.59/291.49	   minus(x, 0) -> x
1036.59/291.49	   minus(s(x), s(y)) -> minus(x, y)
1036.59/291.49	   f(0) -> s(0)
1036.59/291.49	   f(s(x)) -> minus(s(x), g(f(x)))
1036.59/291.49	   g(0) -> 0
1036.59/291.49	   g(s(x)) -> minus(s(x), f(g(x)))
1036.59/291.49	
1036.59/291.49	S is empty.
1036.59/291.49	Rewrite Strategy: INNERMOST
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(6) LowerBoundPropagationProof (FINISHED)
1036.59/291.49	Propagated lower bound.
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(7)
1036.59/291.49	BOUNDS(n^1, INF)
1036.59/291.49	
1036.59/291.49	----------------------------------------
1036.59/291.49	
1036.59/291.49	(8)
1036.59/291.49	Obligation:
1036.59/291.49	Analyzing the following TRS for decreasing loops:
1036.59/291.49	
1036.59/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.59/291.49	
1036.59/291.49	
1036.59/291.49	The TRS R consists of the following rules:
1036.59/291.49	
1036.59/291.49	   minus(x, 0) -> x
1036.59/291.49	   minus(s(x), s(y)) -> minus(x, y)
1036.59/291.49	   f(0) -> s(0)
1036.59/291.49	   f(s(x)) -> minus(s(x), g(f(x)))
1036.59/291.49	   g(0) -> 0
1036.59/291.49	   g(s(x)) -> minus(s(x), f(g(x)))
1036.59/291.49	
1036.59/291.49	S is empty.
1036.59/291.49	Rewrite Strategy: INNERMOST
1036.87/291.55	EOF