1076.93/291.64	WORST_CASE(Omega(n^1), ?)
1079.45/292.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1079.45/292.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1079.45/292.32	
1079.45/292.32	(0) CpxTRS
1079.45/292.32	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1079.45/292.32	(2) TRS for Loop Detection
1079.45/292.32	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1079.45/292.32	(4) BEST
1079.45/292.32	    (5) proven lower bound
1079.45/292.32	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1079.45/292.32	        (7) BOUNDS(n^1, INF)
1079.45/292.32	    (8) TRS for Loop Detection
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(0)
1079.45/292.32	Obligation:
1079.45/292.32	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	The TRS R consists of the following rules:
1079.45/292.32	
1079.45/292.32	   f(x, a(b(y))) -> f(c(d(x)), y)
1079.45/292.32	   f(c(x), y) -> f(x, a(y))
1079.45/292.32	   f(d(x), y) -> f(x, b(y))
1079.45/292.32	
1079.45/292.32	S is empty.
1079.45/292.32	Rewrite Strategy: INNERMOST
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1079.45/292.32	Transformed a relative TRS into a decreasing-loop problem.
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(2)
1079.45/292.32	Obligation:
1079.45/292.32	Analyzing the following TRS for decreasing loops:
1079.45/292.32	
1079.45/292.32	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	The TRS R consists of the following rules:
1079.45/292.32	
1079.45/292.32	   f(x, a(b(y))) -> f(c(d(x)), y)
1079.45/292.32	   f(c(x), y) -> f(x, a(y))
1079.45/292.32	   f(d(x), y) -> f(x, b(y))
1079.45/292.32	
1079.45/292.32	S is empty.
1079.45/292.32	Rewrite Strategy: INNERMOST
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(3) DecreasingLoopProof (LOWER BOUND(ID))
1079.45/292.32	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1079.45/292.32	
1079.45/292.32	The rewrite sequence
1079.45/292.32	
1079.45/292.32	f(c(x), y) ->^+ f(x, a(y))
1079.45/292.32	
1079.45/292.32	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1079.45/292.32	
1079.45/292.32	The pumping substitution is [x / c(x)].
1079.45/292.32	
1079.45/292.32	The result substitution is [y / a(y)].
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(4)
1079.45/292.32	Complex Obligation (BEST)
1079.45/292.32	
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(5)
1079.45/292.32	Obligation:
1079.45/292.32	Proved the lower bound n^1 for the following obligation:
1079.45/292.32	
1079.45/292.32	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	The TRS R consists of the following rules:
1079.45/292.32	
1079.45/292.32	   f(x, a(b(y))) -> f(c(d(x)), y)
1079.45/292.32	   f(c(x), y) -> f(x, a(y))
1079.45/292.32	   f(d(x), y) -> f(x, b(y))
1079.45/292.32	
1079.45/292.32	S is empty.
1079.45/292.32	Rewrite Strategy: INNERMOST
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(6) LowerBoundPropagationProof (FINISHED)
1079.45/292.32	Propagated lower bound.
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(7)
1079.45/292.32	BOUNDS(n^1, INF)
1079.45/292.32	
1079.45/292.32	----------------------------------------
1079.45/292.32	
1079.45/292.32	(8)
1079.45/292.32	Obligation:
1079.45/292.32	Analyzing the following TRS for decreasing loops:
1079.45/292.32	
1079.45/292.32	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1079.45/292.32	
1079.45/292.32	
1079.45/292.32	The TRS R consists of the following rules:
1079.45/292.32	
1079.45/292.32	   f(x, a(b(y))) -> f(c(d(x)), y)
1079.45/292.32	   f(c(x), y) -> f(x, a(y))
1079.45/292.32	   f(d(x), y) -> f(x, b(y))
1079.45/292.32	
1079.45/292.32	S is empty.
1079.45/292.32	Rewrite Strategy: INNERMOST
1079.73/292.38	EOF