1092.33/291.53	WORST_CASE(Omega(n^1), ?)
1092.33/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1092.33/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1092.33/291.53	
1092.33/291.53	(0) CpxTRS
1092.33/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1092.33/291.53	(2) TRS for Loop Detection
1092.33/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1092.33/291.53	(4) BEST
1092.33/291.53	    (5) proven lower bound
1092.33/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1092.33/291.53	        (7) BOUNDS(n^1, INF)
1092.33/291.53	    (8) TRS for Loop Detection
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(0)
1092.33/291.53	Obligation:
1092.33/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	The TRS R consists of the following rules:
1092.33/291.53	
1092.33/291.53	   top(sent(x)) -> top(check(rest(x)))
1092.33/291.53	   rest(nil) -> sent(nil)
1092.33/291.53	   rest(cons(x, y)) -> sent(y)
1092.33/291.53	   check(sent(x)) -> sent(check(x))
1092.33/291.53	   check(rest(x)) -> rest(check(x))
1092.33/291.53	   check(cons(x, y)) -> cons(check(x), y)
1092.33/291.53	   check(cons(x, y)) -> cons(x, check(y))
1092.33/291.53	   check(cons(x, y)) -> cons(x, y)
1092.33/291.53	
1092.33/291.53	S is empty.
1092.33/291.53	Rewrite Strategy: FULL
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1092.33/291.53	Transformed a relative TRS into a decreasing-loop problem.
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(2)
1092.33/291.53	Obligation:
1092.33/291.53	Analyzing the following TRS for decreasing loops:
1092.33/291.53	
1092.33/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	The TRS R consists of the following rules:
1092.33/291.53	
1092.33/291.53	   top(sent(x)) -> top(check(rest(x)))
1092.33/291.53	   rest(nil) -> sent(nil)
1092.33/291.53	   rest(cons(x, y)) -> sent(y)
1092.33/291.53	   check(sent(x)) -> sent(check(x))
1092.33/291.53	   check(rest(x)) -> rest(check(x))
1092.33/291.53	   check(cons(x, y)) -> cons(check(x), y)
1092.33/291.53	   check(cons(x, y)) -> cons(x, check(y))
1092.33/291.53	   check(cons(x, y)) -> cons(x, y)
1092.33/291.53	
1092.33/291.53	S is empty.
1092.33/291.53	Rewrite Strategy: FULL
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
1092.33/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1092.33/291.53	
1092.33/291.53	The rewrite sequence
1092.33/291.53	
1092.33/291.53	check(cons(x, y)) ->^+ cons(check(x), y)
1092.33/291.53	
1092.33/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1092.33/291.53	
1092.33/291.53	The pumping substitution is [x / cons(x, y)].
1092.33/291.53	
1092.33/291.53	The result substitution is [ ].
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(4)
1092.33/291.53	Complex Obligation (BEST)
1092.33/291.53	
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(5)
1092.33/291.53	Obligation:
1092.33/291.53	Proved the lower bound n^1 for the following obligation:
1092.33/291.53	
1092.33/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	The TRS R consists of the following rules:
1092.33/291.53	
1092.33/291.53	   top(sent(x)) -> top(check(rest(x)))
1092.33/291.53	   rest(nil) -> sent(nil)
1092.33/291.53	   rest(cons(x, y)) -> sent(y)
1092.33/291.53	   check(sent(x)) -> sent(check(x))
1092.33/291.53	   check(rest(x)) -> rest(check(x))
1092.33/291.53	   check(cons(x, y)) -> cons(check(x), y)
1092.33/291.53	   check(cons(x, y)) -> cons(x, check(y))
1092.33/291.53	   check(cons(x, y)) -> cons(x, y)
1092.33/291.53	
1092.33/291.53	S is empty.
1092.33/291.53	Rewrite Strategy: FULL
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(6) LowerBoundPropagationProof (FINISHED)
1092.33/291.53	Propagated lower bound.
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(7)
1092.33/291.53	BOUNDS(n^1, INF)
1092.33/291.53	
1092.33/291.53	----------------------------------------
1092.33/291.53	
1092.33/291.53	(8)
1092.33/291.53	Obligation:
1092.33/291.53	Analyzing the following TRS for decreasing loops:
1092.33/291.53	
1092.33/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1092.33/291.53	
1092.33/291.53	
1092.33/291.53	The TRS R consists of the following rules:
1092.33/291.53	
1092.33/291.53	   top(sent(x)) -> top(check(rest(x)))
1092.33/291.53	   rest(nil) -> sent(nil)
1092.33/291.53	   rest(cons(x, y)) -> sent(y)
1092.33/291.53	   check(sent(x)) -> sent(check(x))
1092.33/291.53	   check(rest(x)) -> rest(check(x))
1092.33/291.53	   check(cons(x, y)) -> cons(check(x), y)
1092.33/291.53	   check(cons(x, y)) -> cons(x, check(y))
1092.33/291.53	   check(cons(x, y)) -> cons(x, y)
1092.33/291.53	
1092.33/291.53	S is empty.
1092.33/291.53	Rewrite Strategy: FULL
1093.30/291.84	EOF