8.79/3.00	WORST_CASE(Omega(n^1), O(n^1))
8.91/3.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.91/3.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.91/3.00	
8.91/3.00	
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.91/3.00	
8.91/3.00	(0) CpxTRS
8.91/3.00	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
8.91/3.00	(2) CpxTRS
8.91/3.00	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
8.91/3.00	    (4) CpxTRS
8.91/3.00	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 16 ms]
8.91/3.00	    (6) BOUNDS(1, n^1)
8.91/3.00	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
8.91/3.00	(8) TRS for Loop Detection
8.91/3.00	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
8.91/3.00	    (10) BEST
8.91/3.00	        (11) proven lower bound
8.91/3.00	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
8.91/3.00	            (13) BOUNDS(n^1, INF)
8.91/3.00	        (14) TRS for Loop Detection
8.91/3.00	
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(0)
8.91/3.00	Obligation:
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(+(x, y), z) -> +(x, +(y, z))
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	   +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
8.91/3.00	The TRS does not nest defined symbols.
8.91/3.00	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
8.91/3.00	+(+(x, y), z) -> +(x, +(y, z))
8.91/3.00	+(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(2)
8.91/3.00	Obligation:
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(3) RelTrsToTrsProof (UPPER BOUND(ID))
8.91/3.00	transformed relative TRS to TRS
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(4)
8.91/3.00	Obligation:
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(5) CpxTrsMatchBoundsTAProof (FINISHED)
8.91/3.00	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. 
8.91/3.00	
8.91/3.00	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
8.91/3.00	final states : [1]
8.91/3.00	transitions: 
8.91/3.00	f0(0) -> 0
8.91/3.00	+0(0, 0) -> 1
8.91/3.00	+1(0, 0) -> 2
8.91/3.00	f1(2) -> 1
8.91/3.00	f1(2) -> 2
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(6)
8.91/3.00	BOUNDS(1, n^1)
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
8.91/3.00	Transformed a relative TRS into a decreasing-loop problem.
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(8)
8.91/3.00	Obligation:
8.91/3.00	Analyzing the following TRS for decreasing loops:
8.91/3.00	
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(+(x, y), z) -> +(x, +(y, z))
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	   +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(9) DecreasingLoopProof (LOWER BOUND(ID))
8.91/3.00	The following loop(s) give(s) rise to the lower bound Omega(n^1):
8.91/3.00	
8.91/3.00	The rewrite sequence
8.91/3.00	
8.91/3.00	+(f(x), f(y)) ->^+ f(+(x, y))
8.91/3.00	
8.91/3.00	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
8.91/3.00	
8.91/3.00	The pumping substitution is [x / f(x), y / f(y)].
8.91/3.00	
8.91/3.00	The result substitution is [ ].
8.91/3.00	
8.91/3.00	
8.91/3.00	
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(10)
8.91/3.00	Complex Obligation (BEST)
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(11)
8.91/3.00	Obligation:
8.91/3.00	Proved the lower bound n^1 for the following obligation:
8.91/3.00	
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(+(x, y), z) -> +(x, +(y, z))
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	   +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(12) LowerBoundPropagationProof (FINISHED)
8.91/3.00	Propagated lower bound.
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(13)
8.91/3.00	BOUNDS(n^1, INF)
8.91/3.00	
8.91/3.00	----------------------------------------
8.91/3.00	
8.91/3.00	(14)
8.91/3.00	Obligation:
8.91/3.00	Analyzing the following TRS for decreasing loops:
8.91/3.00	
8.91/3.00	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.91/3.00	
8.91/3.00	
8.91/3.00	The TRS R consists of the following rules:
8.91/3.00	
8.91/3.00	   +(+(x, y), z) -> +(x, +(y, z))
8.91/3.00	   +(f(x), f(y)) -> f(+(x, y))
8.91/3.00	   +(f(x), +(f(y), z)) -> +(f(+(x, y)), z)
8.91/3.00	
8.91/3.00	S is empty.
8.91/3.00	Rewrite Strategy: FULL
8.95/3.04	EOF