3.29/1.98	WORST_CASE(Omega(n^1), O(n^1))
3.29/1.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.29/1.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.29/1.98	
3.29/1.98	
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.98	
3.29/1.98	(0) CpxTRS
3.29/1.98	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.29/1.98	(2) CpxTRS
3.29/1.98	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.29/1.98	    (4) BOUNDS(1, n^1)
3.29/1.98	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.29/1.98	(6) TRS for Loop Detection
3.29/1.98	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.29/1.98	    (8) BEST
3.29/1.98	        (9) proven lower bound
3.29/1.98	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.29/1.98	            (11) BOUNDS(n^1, INF)
3.29/1.98	        (12) TRS for Loop Detection
3.29/1.98	
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(0)
3.29/1.98	Obligation:
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.98	
3.29/1.98	
3.29/1.98	The TRS R consists of the following rules:
3.29/1.98	
3.29/1.98	   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
3.29/1.98	   revapp(Nil, rest) -> rest
3.29/1.98	   goal(xs, ys) -> revapp(xs, ys)
3.29/1.98	
3.29/1.98	S is empty.
3.29/1.98	Rewrite Strategy: INNERMOST
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.29/1.98	transformed relative TRS to TRS
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(2)
3.29/1.98	Obligation:
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.29/1.98	
3.29/1.98	
3.29/1.98	The TRS R consists of the following rules:
3.29/1.98	
3.29/1.98	   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
3.29/1.98	   revapp(Nil, rest) -> rest
3.29/1.98	   goal(xs, ys) -> revapp(xs, ys)
3.29/1.98	
3.29/1.98	S is empty.
3.29/1.98	Rewrite Strategy: INNERMOST
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.29/1.98	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. 
3.29/1.98	
3.29/1.98	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.29/1.98	final states : [1, 2]
3.29/1.98	transitions: 
3.29/1.98	Cons0(0, 0) -> 0
3.29/1.98	Nil0() -> 0
3.29/1.98	revapp0(0, 0) -> 1
3.29/1.98	goal0(0, 0) -> 2
3.29/1.98	Cons1(0, 0) -> 3
3.29/1.98	revapp1(0, 3) -> 1
3.29/1.98	revapp1(0, 0) -> 2
3.29/1.98	Cons1(0, 3) -> 3
3.29/1.98	revapp1(0, 3) -> 2
3.29/1.98	0 -> 1
3.29/1.98	0 -> 2
3.29/1.98	3 -> 1
3.29/1.98	3 -> 2
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(4)
3.29/1.98	BOUNDS(1, n^1)
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.29/1.98	Transformed a relative TRS into a decreasing-loop problem.
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(6)
3.29/1.98	Obligation:
3.29/1.98	Analyzing the following TRS for decreasing loops:
3.29/1.98	
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.98	
3.29/1.98	
3.29/1.98	The TRS R consists of the following rules:
3.29/1.98	
3.29/1.98	   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
3.29/1.98	   revapp(Nil, rest) -> rest
3.29/1.98	   goal(xs, ys) -> revapp(xs, ys)
3.29/1.98	
3.29/1.98	S is empty.
3.29/1.98	Rewrite Strategy: INNERMOST
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.29/1.98	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.29/1.98	
3.29/1.98	The rewrite sequence
3.29/1.98	
3.29/1.98	revapp(Cons(x, xs), rest) ->^+ revapp(xs, Cons(x, rest))
3.29/1.98	
3.29/1.98	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.29/1.98	
3.29/1.98	The pumping substitution is [xs / Cons(x, xs)].
3.29/1.98	
3.29/1.98	The result substitution is [rest / Cons(x, rest)].
3.29/1.98	
3.29/1.98	
3.29/1.98	
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(8)
3.29/1.98	Complex Obligation (BEST)
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(9)
3.29/1.98	Obligation:
3.29/1.98	Proved the lower bound n^1 for the following obligation:
3.29/1.98	
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.98	
3.29/1.98	
3.29/1.98	The TRS R consists of the following rules:
3.29/1.98	
3.29/1.98	   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
3.29/1.98	   revapp(Nil, rest) -> rest
3.29/1.98	   goal(xs, ys) -> revapp(xs, ys)
3.29/1.98	
3.29/1.98	S is empty.
3.29/1.98	Rewrite Strategy: INNERMOST
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(10) LowerBoundPropagationProof (FINISHED)
3.29/1.98	Propagated lower bound.
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(11)
3.29/1.98	BOUNDS(n^1, INF)
3.29/1.98	
3.29/1.98	----------------------------------------
3.29/1.98	
3.29/1.98	(12)
3.29/1.98	Obligation:
3.29/1.98	Analyzing the following TRS for decreasing loops:
3.29/1.98	
3.29/1.98	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.29/1.98	
3.29/1.98	
3.29/1.98	The TRS R consists of the following rules:
3.29/1.98	
3.29/1.98	   revapp(Cons(x, xs), rest) -> revapp(xs, Cons(x, rest))
3.29/1.98	   revapp(Nil, rest) -> rest
3.29/1.98	   goal(xs, ys) -> revapp(xs, ys)
3.29/1.98	
3.29/1.98	S is empty.
3.29/1.98	Rewrite Strategy: INNERMOST
3.63/2.02	EOF