301.93/291.49	WORST_CASE(Omega(n^1), ?)
301.93/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
301.93/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.93/291.50	
301.93/291.50	
301.93/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.93/291.50	
301.93/291.50	(0) CpxTRS
301.93/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.93/291.50	(2) TRS for Loop Detection
301.93/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 53 ms]
301.93/291.50	(4) BEST
301.93/291.50	    (5) proven lower bound
301.93/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.93/291.50	        (7) BOUNDS(n^1, INF)
301.93/291.50	    (8) TRS for Loop Detection
301.93/291.50	
301.93/291.50	
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(0)
301.93/291.50	Obligation:
301.93/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.93/291.50	
301.93/291.50	
301.93/291.50	The TRS R consists of the following rules:
301.93/291.50	
301.93/291.50	   is_empty(nil) -> true
301.93/291.50	   is_empty(cons(x, l)) -> false
301.93/291.50	   hd(cons(x, l)) -> x
301.93/291.50	   tl(cons(x, l)) -> l
301.93/291.50	   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
301.93/291.50	   ifappend(l1, l2, true) -> l2
301.93/291.50	   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))
301.93/291.50	
301.93/291.50	S is empty.
301.93/291.50	Rewrite Strategy: FULL
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.93/291.50	Transformed a relative TRS into a decreasing-loop problem.
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(2)
301.93/291.50	Obligation:
301.93/291.50	Analyzing the following TRS for decreasing loops:
301.93/291.50	
301.93/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.93/291.50	
301.93/291.50	
301.93/291.50	The TRS R consists of the following rules:
301.93/291.50	
301.93/291.50	   is_empty(nil) -> true
301.93/291.50	   is_empty(cons(x, l)) -> false
301.93/291.50	   hd(cons(x, l)) -> x
301.93/291.50	   tl(cons(x, l)) -> l
301.93/291.50	   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
301.93/291.50	   ifappend(l1, l2, true) -> l2
301.93/291.50	   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))
301.93/291.50	
301.93/291.50	S is empty.
301.93/291.50	Rewrite Strategy: FULL
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.93/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.93/291.50	
301.93/291.50	The rewrite sequence
301.93/291.50	
301.93/291.50	ifappend(cons(x1_0, cons(x1_1, l2_1)), l2, false) ->^+ cons(hd(cons(x1_0, cons(x1_1, l2_1))), ifappend(cons(x1_1, l2_1), l2, false))
301.93/291.50	
301.93/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
301.93/291.50	
301.93/291.50	The pumping substitution is [l2_1 / cons(x1_1, l2_1)].
301.93/291.50	
301.93/291.50	The result substitution is [x1_0 / x1_1].
301.93/291.50	
301.93/291.50	
301.93/291.50	
301.93/291.50	
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(4)
301.93/291.50	Complex Obligation (BEST)
301.93/291.50	
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(5)
301.93/291.50	Obligation:
301.93/291.50	Proved the lower bound n^1 for the following obligation:
301.93/291.50	
301.93/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.93/291.50	
301.93/291.50	
301.93/291.50	The TRS R consists of the following rules:
301.93/291.50	
301.93/291.50	   is_empty(nil) -> true
301.93/291.50	   is_empty(cons(x, l)) -> false
301.93/291.50	   hd(cons(x, l)) -> x
301.93/291.50	   tl(cons(x, l)) -> l
301.93/291.50	   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
301.93/291.50	   ifappend(l1, l2, true) -> l2
301.93/291.50	   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))
301.93/291.50	
301.93/291.50	S is empty.
301.93/291.50	Rewrite Strategy: FULL
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(6) LowerBoundPropagationProof (FINISHED)
301.93/291.50	Propagated lower bound.
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(7)
301.93/291.50	BOUNDS(n^1, INF)
301.93/291.50	
301.93/291.50	----------------------------------------
301.93/291.50	
301.93/291.50	(8)
301.93/291.50	Obligation:
301.93/291.50	Analyzing the following TRS for decreasing loops:
301.93/291.50	
301.93/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.93/291.50	
301.93/291.50	
301.93/291.50	The TRS R consists of the following rules:
301.93/291.50	
301.93/291.50	   is_empty(nil) -> true
301.93/291.50	   is_empty(cons(x, l)) -> false
301.93/291.50	   hd(cons(x, l)) -> x
301.93/291.50	   tl(cons(x, l)) -> l
301.93/291.50	   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
301.93/291.50	   ifappend(l1, l2, true) -> l2
301.93/291.50	   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))
301.93/291.50	
301.93/291.50	S is empty.
301.93/291.50	Rewrite Strategy: FULL
301.93/291.52	EOF