19.00/6.85	WORST_CASE(Omega(n^1), O(n^1))
19.00/6.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.00/6.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.00/6.86	
19.00/6.86	
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.00/6.86	
19.00/6.86	(0) CpxTRS
19.00/6.86	(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
19.00/6.86	(2) CpxTRS
19.00/6.86	    (3) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
19.00/6.86	    (4) CpxTRS
19.00/6.86	    (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
19.00/6.86	    (6) CpxTRS
19.00/6.86	    (7) CpxTrsMatchBoundsTAProof [FINISHED, 3 ms]
19.00/6.86	    (8) BOUNDS(1, n^1)
19.00/6.86	(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
19.00/6.86	(10) TRS for Loop Detection
19.00/6.86	    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
19.00/6.86	    (12) BEST
19.00/6.86	        (13) proven lower bound
19.00/6.86	            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
19.00/6.86	            (15) BOUNDS(n^1, INF)
19.00/6.86	        (16) TRS for Loop Detection
19.00/6.86	
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(0)
19.00/6.86	Obligation:
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   rev(rev(x)) -> x
19.00/6.86	   rev(++(x, y)) -> ++(rev(y), rev(x))
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   ++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(1) DependencyGraphProof (UPPER BOUND(ID))
19.00/6.86	The following rules are not reachable from basic terms in the dependency graph and can be removed:
19.00/6.86	
19.00/6.86	rev(rev(x)) -> x
19.00/6.86	rev(++(x, y)) -> ++(rev(y), rev(x))
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(2)
19.00/6.86	Obligation:
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   ++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(3) NestedDefinedSymbolProof (UPPER BOUND(ID))
19.00/6.86	The TRS does not nest defined symbols.
19.00/6.86	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
19.00/6.86	++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(4)
19.00/6.86	Obligation:
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(5) RelTrsToTrsProof (UPPER BOUND(ID))
19.00/6.86	transformed relative TRS to TRS
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(6)
19.00/6.86	Obligation:
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(7) CpxTrsMatchBoundsTAProof (FINISHED)
19.00/6.86	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. 
19.00/6.86	
19.00/6.86	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
19.00/6.86	final states : [1, 2, 3]
19.00/6.86	transitions: 
19.00/6.86	nil0() -> 0
19.00/6.86	.0(0, 0) -> 0
19.00/6.86	rev0(0) -> 1
19.00/6.86	++0(0, 0) -> 2
19.00/6.86	make0(0) -> 3
19.00/6.86	nil1() -> 1
19.00/6.86	++1(0, 0) -> 4
19.00/6.86	.1(0, 4) -> 2
19.00/6.86	nil1() -> 5
19.00/6.86	.1(0, 5) -> 3
19.00/6.86	.1(0, 4) -> 4
19.00/6.86	0 -> 2
19.00/6.86	0 -> 4
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(8)
19.00/6.86	BOUNDS(1, n^1)
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
19.00/6.86	Transformed a relative TRS into a decreasing-loop problem.
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(10)
19.00/6.86	Obligation:
19.00/6.86	Analyzing the following TRS for decreasing loops:
19.00/6.86	
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   rev(rev(x)) -> x
19.00/6.86	   rev(++(x, y)) -> ++(rev(y), rev(x))
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   ++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(11) DecreasingLoopProof (LOWER BOUND(ID))
19.00/6.86	The following loop(s) give(s) rise to the lower bound Omega(n^1):
19.00/6.86	
19.00/6.86	The rewrite sequence
19.00/6.86	
19.00/6.86	++(.(x, y), z) ->^+ .(x, ++(y, z))
19.00/6.86	
19.00/6.86	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
19.00/6.86	
19.00/6.86	The pumping substitution is [y / .(x, y)].
19.00/6.86	
19.00/6.86	The result substitution is [ ].
19.00/6.86	
19.00/6.86	
19.00/6.86	
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(12)
19.00/6.86	Complex Obligation (BEST)
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(13)
19.00/6.86	Obligation:
19.00/6.86	Proved the lower bound n^1 for the following obligation:
19.00/6.86	
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   rev(rev(x)) -> x
19.00/6.86	   rev(++(x, y)) -> ++(rev(y), rev(x))
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   ++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(14) LowerBoundPropagationProof (FINISHED)
19.00/6.86	Propagated lower bound.
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(15)
19.00/6.86	BOUNDS(n^1, INF)
19.00/6.86	
19.00/6.86	----------------------------------------
19.00/6.86	
19.00/6.86	(16)
19.00/6.86	Obligation:
19.00/6.86	Analyzing the following TRS for decreasing loops:
19.00/6.86	
19.00/6.86	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.00/6.86	
19.00/6.86	
19.00/6.86	The TRS R consists of the following rules:
19.00/6.86	
19.00/6.86	   rev(nil) -> nil
19.00/6.86	   rev(rev(x)) -> x
19.00/6.86	   rev(++(x, y)) -> ++(rev(y), rev(x))
19.00/6.86	   ++(nil, y) -> y
19.00/6.86	   ++(x, nil) -> x
19.00/6.86	   ++(.(x, y), z) -> .(x, ++(y, z))
19.00/6.86	   ++(x, ++(y, z)) -> ++(++(x, y), z)
19.00/6.86	   make(x) -> .(x, nil)
19.00/6.86	
19.00/6.86	S is empty.
19.00/6.86	Rewrite Strategy: FULL
19.12/6.89	EOF