3.35/1.68	WORST_CASE(Omega(n^1), O(n^1))
3.35/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.35/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.69	
3.35/1.69	
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.35/1.69	
3.35/1.69	(0) CpxTRS
3.35/1.69	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.35/1.69	(2) CpxTRS
3.35/1.69	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 38 ms]
3.35/1.69	    (4) BOUNDS(1, n^1)
3.35/1.69	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.35/1.69	(6) TRS for Loop Detection
3.35/1.69	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.35/1.69	    (8) BEST
3.35/1.69	        (9) proven lower bound
3.35/1.69	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.35/1.69	            (11) BOUNDS(n^1, INF)
3.35/1.69	        (12) TRS for Loop Detection
3.35/1.69	
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(0)
3.35/1.69	Obligation:
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.35/1.69	
3.35/1.69	
3.35/1.69	The TRS R consists of the following rules:
3.35/1.69	
3.35/1.69	   merge(x, nil) -> x
3.35/1.69	   merge(nil, y) -> y
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
3.35/1.69	
3.35/1.69	S is empty.
3.35/1.69	Rewrite Strategy: FULL
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.35/1.69	transformed relative TRS to TRS
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(2)
3.35/1.69	Obligation:
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.35/1.69	
3.35/1.69	
3.35/1.69	The TRS R consists of the following rules:
3.35/1.69	
3.35/1.69	   merge(x, nil) -> x
3.35/1.69	   merge(nil, y) -> y
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
3.35/1.69	
3.35/1.69	S is empty.
3.35/1.69	Rewrite Strategy: FULL
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.35/1.69	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. 
3.35/1.69	
3.35/1.69	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.35/1.69	final states : [1]
3.35/1.69	transitions: 
3.35/1.69	nil0() -> 0
3.35/1.69	++0(0, 0) -> 0
3.35/1.69	u0() -> 0
3.35/1.69	v0() -> 0
3.35/1.69	merge0(0, 0) -> 1
3.35/1.69	u1() -> 4
3.35/1.69	v1() -> 5
3.35/1.69	++1(4, 5) -> 3
3.35/1.69	merge1(0, 3) -> 2
3.35/1.69	++1(0, 2) -> 1
3.35/1.69	++1(0, 0) -> 7
3.35/1.69	v1() -> 8
3.35/1.69	merge1(7, 8) -> 6
3.35/1.69	++1(4, 6) -> 1
3.35/1.69	++1(0, 2) -> 2
3.35/1.69	++1(4, 6) -> 2
3.35/1.69	0 -> 1
3.35/1.69	3 -> 2
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(4)
3.35/1.69	BOUNDS(1, n^1)
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.35/1.69	Transformed a relative TRS into a decreasing-loop problem.
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(6)
3.35/1.69	Obligation:
3.35/1.69	Analyzing the following TRS for decreasing loops:
3.35/1.69	
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.35/1.69	
3.35/1.69	
3.35/1.69	The TRS R consists of the following rules:
3.35/1.69	
3.35/1.69	   merge(x, nil) -> x
3.35/1.69	   merge(nil, y) -> y
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
3.35/1.69	
3.35/1.69	S is empty.
3.35/1.69	Rewrite Strategy: FULL
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.35/1.69	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.35/1.69	
3.35/1.69	The rewrite sequence
3.35/1.69	
3.35/1.69	merge(++(x, y), ++(u, v)) ->^+ ++(x, merge(y, ++(u, v)))
3.35/1.69	
3.35/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.35/1.69	
3.35/1.69	The pumping substitution is [y / ++(x, y)].
3.35/1.69	
3.35/1.69	The result substitution is [ ].
3.35/1.69	
3.35/1.69	
3.35/1.69	
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(8)
3.35/1.69	Complex Obligation (BEST)
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(9)
3.35/1.69	Obligation:
3.35/1.69	Proved the lower bound n^1 for the following obligation:
3.35/1.69	
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.35/1.69	
3.35/1.69	
3.35/1.69	The TRS R consists of the following rules:
3.35/1.69	
3.35/1.69	   merge(x, nil) -> x
3.35/1.69	   merge(nil, y) -> y
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
3.35/1.69	
3.35/1.69	S is empty.
3.35/1.69	Rewrite Strategy: FULL
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(10) LowerBoundPropagationProof (FINISHED)
3.35/1.69	Propagated lower bound.
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(11)
3.35/1.69	BOUNDS(n^1, INF)
3.35/1.69	
3.35/1.69	----------------------------------------
3.35/1.69	
3.35/1.69	(12)
3.35/1.69	Obligation:
3.35/1.69	Analyzing the following TRS for decreasing loops:
3.35/1.69	
3.35/1.69	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.35/1.69	
3.35/1.69	
3.35/1.69	The TRS R consists of the following rules:
3.35/1.69	
3.35/1.69	   merge(x, nil) -> x
3.35/1.69	   merge(nil, y) -> y
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
3.35/1.69	   merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))
3.35/1.69	
3.35/1.69	S is empty.
3.35/1.69	Rewrite Strategy: FULL
3.60/1.72	EOF