3.60/1.70	WORST_CASE(Omega(n^1), O(n^1))
3.60/1.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.60/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.71	
3.60/1.71	
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.71	
3.60/1.71	(0) CpxTRS
3.60/1.71	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.60/1.71	(2) CpxTRS
3.60/1.71	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 47 ms]
3.60/1.71	    (4) BOUNDS(1, n^1)
3.60/1.71	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.60/1.71	(6) TRS for Loop Detection
3.60/1.71	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.60/1.71	    (8) BEST
3.60/1.71	        (9) proven lower bound
3.60/1.71	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.60/1.71	            (11) BOUNDS(n^1, INF)
3.60/1.71	        (12) TRS for Loop Detection
3.60/1.71	
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(0)
3.60/1.71	Obligation:
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.71	
3.60/1.71	
3.60/1.71	The TRS R consists of the following rules:
3.60/1.71	
3.60/1.71	   revApp#2(Nil, x16) -> x16
3.60/1.71	   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
3.60/1.71	   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
3.60/1.71	   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)
3.60/1.71	
3.60/1.71	S is empty.
3.60/1.71	Rewrite Strategy: INNERMOST
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.60/1.71	transformed relative TRS to TRS
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(2)
3.60/1.71	Obligation:
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.60/1.71	
3.60/1.71	
3.60/1.71	The TRS R consists of the following rules:
3.60/1.71	
3.60/1.71	   revApp#2(Nil, x16) -> x16
3.60/1.71	   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
3.60/1.71	   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
3.60/1.71	   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)
3.60/1.71	
3.60/1.71	S is empty.
3.60/1.71	Rewrite Strategy: INNERMOST
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.60/1.71	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 
3.60/1.71	
3.60/1.71	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.60/1.71	final states : [1, 2, 3]
3.60/1.71	transitions: 
3.60/1.71	Nil0() -> 0
3.60/1.71	Cons0(0, 0) -> 0
3.60/1.71	Leaf0(0) -> 0
3.60/1.71	Node0(0, 0) -> 0
3.60/1.71	revApp#20(0, 0) -> 1
3.60/1.71	dfsAcc#30(0, 0) -> 2
3.60/1.71	main0(0) -> 3
3.60/1.71	Cons1(0, 0) -> 4
3.60/1.71	revApp#21(0, 4) -> 1
3.60/1.71	Cons1(0, 0) -> 2
3.60/1.71	dfsAcc#31(0, 0) -> 5
3.60/1.71	dfsAcc#31(0, 5) -> 2
3.60/1.71	Nil1() -> 7
3.60/1.71	dfsAcc#31(0, 7) -> 6
3.60/1.71	Nil1() -> 8
3.60/1.71	revApp#21(6, 8) -> 3
3.60/1.71	Cons1(0, 4) -> 4
3.60/1.71	Cons1(0, 5) -> 2
3.60/1.71	Cons1(0, 0) -> 5
3.60/1.71	Cons1(0, 7) -> 6
3.60/1.71	dfsAcc#31(0, 5) -> 5
3.60/1.71	dfsAcc#31(0, 7) -> 5
3.60/1.71	dfsAcc#31(0, 5) -> 6
3.60/1.71	Cons2(0, 8) -> 9
3.60/1.71	revApp#22(7, 9) -> 3
3.60/1.71	Cons1(0, 5) -> 5
3.60/1.71	Cons1(0, 7) -> 5
3.60/1.71	Cons1(0, 5) -> 6
3.60/1.71	revApp#22(5, 9) -> 3
3.60/1.71	Cons2(0, 9) -> 9
3.60/1.71	revApp#22(0, 9) -> 3
3.60/1.71	Cons1(0, 9) -> 4
3.60/1.71	revApp#21(0, 4) -> 3
3.60/1.71	0 -> 1
3.60/1.71	4 -> 1
3.60/1.71	4 -> 3
3.60/1.71	9 -> 3
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(4)
3.60/1.71	BOUNDS(1, n^1)
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.60/1.71	Transformed a relative TRS into a decreasing-loop problem.
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(6)
3.60/1.71	Obligation:
3.60/1.71	Analyzing the following TRS for decreasing loops:
3.60/1.71	
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.71	
3.60/1.71	
3.60/1.71	The TRS R consists of the following rules:
3.60/1.71	
3.60/1.71	   revApp#2(Nil, x16) -> x16
3.60/1.71	   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
3.60/1.71	   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
3.60/1.71	   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)
3.60/1.71	
3.60/1.71	S is empty.
3.60/1.71	Rewrite Strategy: INNERMOST
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.60/1.71	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.60/1.71	
3.60/1.71	The rewrite sequence
3.60/1.71	
3.60/1.71	dfsAcc#3(Node(x6, x4), x2) ->^+ dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	
3.60/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.60/1.71	
3.60/1.71	The pumping substitution is [x4 / Node(x6, x4)].
3.60/1.71	
3.60/1.71	The result substitution is [x2 / dfsAcc#3(x6, x2)].
3.60/1.71	
3.60/1.71	
3.60/1.71	
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(8)
3.60/1.71	Complex Obligation (BEST)
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(9)
3.60/1.71	Obligation:
3.60/1.71	Proved the lower bound n^1 for the following obligation:
3.60/1.71	
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.71	
3.60/1.71	
3.60/1.71	The TRS R consists of the following rules:
3.60/1.71	
3.60/1.71	   revApp#2(Nil, x16) -> x16
3.60/1.71	   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
3.60/1.71	   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
3.60/1.71	   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)
3.60/1.71	
3.60/1.71	S is empty.
3.60/1.71	Rewrite Strategy: INNERMOST
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(10) LowerBoundPropagationProof (FINISHED)
3.60/1.71	Propagated lower bound.
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(11)
3.60/1.71	BOUNDS(n^1, INF)
3.60/1.71	
3.60/1.71	----------------------------------------
3.60/1.71	
3.60/1.71	(12)
3.60/1.71	Obligation:
3.60/1.71	Analyzing the following TRS for decreasing loops:
3.60/1.71	
3.60/1.71	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.71	
3.60/1.71	
3.60/1.71	The TRS R consists of the following rules:
3.60/1.71	
3.60/1.71	   revApp#2(Nil, x16) -> x16
3.60/1.71	   revApp#2(Cons(x6, x4), x2) -> revApp#2(x4, Cons(x6, x2))
3.60/1.71	   dfsAcc#3(Leaf(x8), x16) -> Cons(x8, x16)
3.60/1.71	   dfsAcc#3(Node(x6, x4), x2) -> dfsAcc#3(x4, dfsAcc#3(x6, x2))
3.60/1.71	   main(x1) -> revApp#2(dfsAcc#3(x1, Nil), Nil)
3.60/1.71	
3.60/1.71	S is empty.
3.60/1.71	Rewrite Strategy: INNERMOST
3.69/1.75	EOF