3.59/1.65	WORST_CASE(Omega(n^1), O(n^1))
3.59/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.59/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.66	
3.59/1.66	
3.59/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.59/1.66	
3.59/1.66	(0) CpxTRS
3.59/1.66	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.59/1.66	(2) CpxTRS
3.59/1.66	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 34 ms]
3.59/1.66	    (4) BOUNDS(1, n^1)
3.59/1.66	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.59/1.66	(6) TRS for Loop Detection
3.59/1.66	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.59/1.66	    (8) BEST
3.59/1.66	        (9) proven lower bound
3.59/1.66	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.59/1.66	            (11) BOUNDS(n^1, INF)
3.59/1.66	        (12) TRS for Loop Detection
3.59/1.66	
3.59/1.66	
3.59/1.66	----------------------------------------
3.59/1.66	
3.59/1.66	(0)
3.59/1.66	Obligation:
3.59/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.59/1.66	
3.59/1.66	
3.59/1.66	The TRS R consists of the following rules:
3.59/1.66	
3.59/1.66	   f(nil) -> nil
3.59/1.66	   f(.(nil, y)) -> .(nil, f(y))
3.59/1.66	   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
3.59/1.66	   g(nil) -> nil
3.59/1.66	   g(.(x, nil)) -> .(g(x), nil)
3.59/1.67	   g(.(x, .(y, z))) -> g(.(.(x, y), z))
3.59/1.67	
3.59/1.67	S is empty.
3.59/1.67	Rewrite Strategy: FULL
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.59/1.67	transformed relative TRS to TRS
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(2)
3.59/1.67	Obligation:
3.59/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.59/1.67	
3.59/1.67	
3.59/1.67	The TRS R consists of the following rules:
3.59/1.67	
3.59/1.67	   f(nil) -> nil
3.59/1.67	   f(.(nil, y)) -> .(nil, f(y))
3.59/1.67	   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
3.59/1.67	   g(nil) -> nil
3.59/1.67	   g(.(x, nil)) -> .(g(x), nil)
3.59/1.67	   g(.(x, .(y, z))) -> g(.(.(x, y), z))
3.59/1.67	
3.59/1.67	S is empty.
3.59/1.67	Rewrite Strategy: FULL
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.59/1.67	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. 
3.59/1.67	
3.59/1.67	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.59/1.67	final states : [1, 2]
3.59/1.67	transitions: 
3.59/1.67	nil0() -> 0
3.59/1.67	.0(0, 0) -> 0
3.59/1.67	f0(0) -> 1
3.59/1.67	g0(0) -> 2
3.59/1.67	nil1() -> 1
3.59/1.67	nil1() -> 3
3.59/1.67	f1(0) -> 4
3.59/1.67	.1(3, 4) -> 1
3.59/1.67	.1(0, 0) -> 6
3.59/1.67	.1(0, 6) -> 5
3.59/1.67	f1(5) -> 1
3.59/1.67	nil1() -> 2
3.59/1.67	g1(0) -> 7
3.59/1.67	nil1() -> 8
3.59/1.67	.1(7, 8) -> 2
3.59/1.67	.1(0, 0) -> 10
3.59/1.67	.1(10, 0) -> 9
3.59/1.67	g1(9) -> 2
3.59/1.67	nil1() -> 4
3.59/1.67	.1(3, 4) -> 4
3.59/1.67	f1(6) -> 4
3.59/1.67	f1(5) -> 4
3.59/1.67	.1(0, 6) -> 6
3.59/1.67	nil1() -> 7
3.59/1.67	.1(7, 8) -> 7
3.59/1.67	g1(10) -> 7
3.59/1.67	g1(9) -> 7
3.59/1.67	.1(10, 0) -> 10
3.59/1.67	
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(4)
3.59/1.67	BOUNDS(1, n^1)
3.59/1.67	
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.59/1.67	Transformed a relative TRS into a decreasing-loop problem.
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(6)
3.59/1.67	Obligation:
3.59/1.67	Analyzing the following TRS for decreasing loops:
3.59/1.67	
3.59/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.59/1.67	
3.59/1.67	
3.59/1.67	The TRS R consists of the following rules:
3.59/1.67	
3.59/1.67	   f(nil) -> nil
3.59/1.67	   f(.(nil, y)) -> .(nil, f(y))
3.59/1.67	   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
3.59/1.67	   g(nil) -> nil
3.59/1.67	   g(.(x, nil)) -> .(g(x), nil)
3.59/1.67	   g(.(x, .(y, z))) -> g(.(.(x, y), z))
3.59/1.67	
3.59/1.67	S is empty.
3.59/1.67	Rewrite Strategy: FULL
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.59/1.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.59/1.67	
3.59/1.67	The rewrite sequence
3.59/1.67	
3.59/1.67	g(.(x, nil)) ->^+ .(g(x), nil)
3.59/1.67	
3.59/1.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.59/1.67	
3.59/1.67	The pumping substitution is [x / .(x, nil)].
3.59/1.67	
3.59/1.67	The result substitution is [ ].
3.59/1.67	
3.59/1.67	
3.59/1.67	
3.59/1.67	
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(8)
3.59/1.67	Complex Obligation (BEST)
3.59/1.67	
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(9)
3.59/1.67	Obligation:
3.59/1.67	Proved the lower bound n^1 for the following obligation:
3.59/1.67	
3.59/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.59/1.67	
3.59/1.67	
3.59/1.67	The TRS R consists of the following rules:
3.59/1.67	
3.59/1.67	   f(nil) -> nil
3.59/1.67	   f(.(nil, y)) -> .(nil, f(y))
3.59/1.67	   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
3.59/1.67	   g(nil) -> nil
3.59/1.67	   g(.(x, nil)) -> .(g(x), nil)
3.59/1.67	   g(.(x, .(y, z))) -> g(.(.(x, y), z))
3.59/1.67	
3.59/1.67	S is empty.
3.59/1.67	Rewrite Strategy: FULL
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(10) LowerBoundPropagationProof (FINISHED)
3.59/1.67	Propagated lower bound.
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(11)
3.59/1.67	BOUNDS(n^1, INF)
3.59/1.67	
3.59/1.67	----------------------------------------
3.59/1.67	
3.59/1.67	(12)
3.59/1.67	Obligation:
3.59/1.67	Analyzing the following TRS for decreasing loops:
3.59/1.67	
3.59/1.67	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.59/1.67	
3.59/1.67	
3.59/1.67	The TRS R consists of the following rules:
3.59/1.67	
3.59/1.67	   f(nil) -> nil
3.59/1.67	   f(.(nil, y)) -> .(nil, f(y))
3.59/1.67	   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
3.59/1.67	   g(nil) -> nil
3.59/1.67	   g(.(x, nil)) -> .(g(x), nil)
3.59/1.67	   g(.(x, .(y, z))) -> g(.(.(x, y), z))
3.59/1.67	
3.59/1.67	S is empty.
3.59/1.67	Rewrite Strategy: FULL
3.73/1.75	EOF