3.43/1.63	WORST_CASE(Omega(n^1), O(n^1))
3.43/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.64	
3.43/1.64	
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	(0) CpxTRS
3.43/1.64	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.43/1.64	(2) CpxTRS
3.43/1.64	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.43/1.64	    (4) BOUNDS(1, n^1)
3.43/1.64	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.43/1.64	(6) TRS for Loop Detection
3.43/1.64	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.43/1.64	    (8) BEST
3.43/1.64	        (9) proven lower bound
3.43/1.64	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.43/1.64	            (11) BOUNDS(n^1, INF)
3.43/1.64	        (12) TRS for Loop Detection
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(0)
3.43/1.64	Obligation:
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   foldl#3(x2, Nil) -> x2
3.43/1.64	   foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6)
3.43/1.64	   main(x1) -> foldl#3(Nil, x1)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: INNERMOST
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.43/1.64	transformed relative TRS to TRS
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(2)
3.43/1.64	Obligation:
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   foldl#3(x2, Nil) -> x2
3.43/1.64	   foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6)
3.43/1.64	   main(x1) -> foldl#3(Nil, x1)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: INNERMOST
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.43/1.64	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 1. 
3.43/1.64	
3.43/1.64	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.43/1.64	final states : [1, 2]
3.43/1.64	transitions: 
3.43/1.64	Nil0() -> 0
3.43/1.64	Cons0(0, 0) -> 0
3.43/1.64	foldl#30(0, 0) -> 1
3.43/1.64	main0(0) -> 2
3.43/1.64	Cons1(0, 0) -> 3
3.43/1.64	foldl#31(3, 0) -> 1
3.43/1.64	Nil1() -> 4
3.43/1.64	foldl#31(4, 0) -> 2
3.43/1.64	Cons1(0, 3) -> 3
3.43/1.64	Cons1(0, 4) -> 3
3.43/1.64	foldl#31(3, 0) -> 2
3.43/1.64	0 -> 1
3.43/1.64	3 -> 1
3.43/1.64	3 -> 2
3.43/1.64	4 -> 2
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(4)
3.43/1.64	BOUNDS(1, n^1)
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.43/1.64	Transformed a relative TRS into a decreasing-loop problem.
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(6)
3.43/1.64	Obligation:
3.43/1.64	Analyzing the following TRS for decreasing loops:
3.43/1.64	
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   foldl#3(x2, Nil) -> x2
3.43/1.64	   foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6)
3.43/1.64	   main(x1) -> foldl#3(Nil, x1)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: INNERMOST
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.43/1.64	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.43/1.64	
3.43/1.64	The rewrite sequence
3.43/1.64	
3.43/1.64	foldl#3(x16, Cons(x24, x6)) ->^+ foldl#3(Cons(x24, x16), x6)
3.43/1.64	
3.43/1.64	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.43/1.64	
3.43/1.64	The pumping substitution is [x6 / Cons(x24, x6)].
3.43/1.64	
3.43/1.64	The result substitution is [x16 / Cons(x24, x16)].
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(8)
3.43/1.64	Complex Obligation (BEST)
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(9)
3.43/1.64	Obligation:
3.43/1.64	Proved the lower bound n^1 for the following obligation:
3.43/1.64	
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   foldl#3(x2, Nil) -> x2
3.43/1.64	   foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6)
3.43/1.64	   main(x1) -> foldl#3(Nil, x1)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: INNERMOST
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(10) LowerBoundPropagationProof (FINISHED)
3.43/1.64	Propagated lower bound.
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(11)
3.43/1.64	BOUNDS(n^1, INF)
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(12)
3.43/1.64	Obligation:
3.43/1.64	Analyzing the following TRS for decreasing loops:
3.43/1.64	
3.43/1.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.43/1.64	
3.43/1.64	
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   foldl#3(x2, Nil) -> x2
3.43/1.64	   foldl#3(x16, Cons(x24, x6)) -> foldl#3(Cons(x24, x16), x6)
3.43/1.64	   main(x1) -> foldl#3(Nil, x1)
3.43/1.64	
3.43/1.64	S is empty.
3.43/1.64	Rewrite Strategy: INNERMOST
3.43/1.67	EOF