1106.09/294.43	WORST_CASE(Omega(n^1), ?)
1108.28/295.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1108.28/295.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1108.28/295.04	
1108.28/295.04	(0) CpxTRS
1108.28/295.04	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1108.28/295.04	(2) TRS for Loop Detection
1108.28/295.04	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1108.28/295.04	(4) BEST
1108.28/295.04	    (5) proven lower bound
1108.28/295.04	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1108.28/295.04	        (7) BOUNDS(n^1, INF)
1108.28/295.04	    (8) TRS for Loop Detection
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(0)
1108.28/295.04	Obligation:
1108.28/295.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	The TRS R consists of the following rules:
1108.28/295.04	
1108.28/295.04	   f(x, a(b(y))) -> f(c(d(x)), y)
1108.28/295.04	   f(c(x), y) -> f(x, a(y))
1108.28/295.04	   f(d(x), y) -> f(x, b(y))
1108.28/295.04	
1108.28/295.04	S is empty.
1108.28/295.04	Rewrite Strategy: FULL
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1108.28/295.04	Transformed a relative TRS into a decreasing-loop problem.
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(2)
1108.28/295.04	Obligation:
1108.28/295.04	Analyzing the following TRS for decreasing loops:
1108.28/295.04	
1108.28/295.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	The TRS R consists of the following rules:
1108.28/295.04	
1108.28/295.04	   f(x, a(b(y))) -> f(c(d(x)), y)
1108.28/295.04	   f(c(x), y) -> f(x, a(y))
1108.28/295.04	   f(d(x), y) -> f(x, b(y))
1108.28/295.04	
1108.28/295.04	S is empty.
1108.28/295.04	Rewrite Strategy: FULL
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(3) DecreasingLoopProof (LOWER BOUND(ID))
1108.28/295.04	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1108.28/295.04	
1108.28/295.04	The rewrite sequence
1108.28/295.04	
1108.28/295.04	f(c(x), y) ->^+ f(x, a(y))
1108.28/295.04	
1108.28/295.04	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1108.28/295.04	
1108.28/295.04	The pumping substitution is [x / c(x)].
1108.28/295.04	
1108.28/295.04	The result substitution is [y / a(y)].
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(4)
1108.28/295.04	Complex Obligation (BEST)
1108.28/295.04	
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(5)
1108.28/295.04	Obligation:
1108.28/295.04	Proved the lower bound n^1 for the following obligation:
1108.28/295.04	
1108.28/295.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	The TRS R consists of the following rules:
1108.28/295.04	
1108.28/295.04	   f(x, a(b(y))) -> f(c(d(x)), y)
1108.28/295.04	   f(c(x), y) -> f(x, a(y))
1108.28/295.04	   f(d(x), y) -> f(x, b(y))
1108.28/295.04	
1108.28/295.04	S is empty.
1108.28/295.04	Rewrite Strategy: FULL
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(6) LowerBoundPropagationProof (FINISHED)
1108.28/295.04	Propagated lower bound.
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(7)
1108.28/295.04	BOUNDS(n^1, INF)
1108.28/295.04	
1108.28/295.04	----------------------------------------
1108.28/295.04	
1108.28/295.04	(8)
1108.28/295.04	Obligation:
1108.28/295.04	Analyzing the following TRS for decreasing loops:
1108.28/295.04	
1108.28/295.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1108.28/295.04	
1108.28/295.04	
1108.28/295.04	The TRS R consists of the following rules:
1108.28/295.04	
1108.28/295.04	   f(x, a(b(y))) -> f(c(d(x)), y)
1108.28/295.04	   f(c(x), y) -> f(x, a(y))
1108.28/295.04	   f(d(x), y) -> f(x, b(y))
1108.28/295.04	
1108.28/295.04	S is empty.
1108.28/295.04	Rewrite Strategy: FULL
1108.69/295.10	EOF