7.56/2.77	WORST_CASE(Omega(n^1), O(n^1))
7.56/2.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
7.56/2.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.56/2.78	
7.56/2.78	
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.56/2.78	
7.56/2.78	(0) CpxTRS
7.56/2.78	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
7.56/2.78	(2) CpxTRS
7.56/2.78	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.56/2.78	    (4) CpxTRS
7.56/2.78	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
7.56/2.78	    (6) BOUNDS(1, n^1)
7.56/2.78	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.56/2.78	(8) TRS for Loop Detection
7.56/2.78	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.56/2.78	    (10) BEST
7.56/2.78	        (11) proven lower bound
7.56/2.78	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
7.56/2.78	            (13) BOUNDS(n^1, INF)
7.56/2.78	        (14) TRS for Loop Detection
7.56/2.78	
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(0)
7.56/2.78	Obligation:
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(f(0, x), 1) -> f(g(f(x, x)), x)
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
7.56/2.78	The TRS does not nest defined symbols.
7.56/2.78	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
7.56/2.78	f(f(0, x), 1) -> f(g(f(x, x)), x)
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(2)
7.56/2.78	Obligation:
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(3) RelTrsToTrsProof (UPPER BOUND(ID))
7.56/2.78	transformed relative TRS to TRS
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(4)
7.56/2.78	Obligation:
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(5) CpxTrsMatchBoundsTAProof (FINISHED)
7.56/2.78	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. 
7.56/2.78	
7.56/2.78	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
7.56/2.78	final states : [1]
7.56/2.78	transitions: 
7.56/2.78	g0(0) -> 0
7.56/2.78	f0(0, 0) -> 1
7.56/2.78	f1(0, 0) -> 2
7.56/2.78	g1(2) -> 1
7.56/2.78	g1(2) -> 2
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(6)
7.56/2.78	BOUNDS(1, n^1)
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.56/2.78	Transformed a relative TRS into a decreasing-loop problem.
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(8)
7.56/2.78	Obligation:
7.56/2.78	Analyzing the following TRS for decreasing loops:
7.56/2.78	
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(f(0, x), 1) -> f(g(f(x, x)), x)
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(9) DecreasingLoopProof (LOWER BOUND(ID))
7.56/2.78	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.56/2.78	
7.56/2.78	The rewrite sequence
7.56/2.78	
7.56/2.78	f(g(x), y) ->^+ g(f(x, y))
7.56/2.78	
7.56/2.78	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
7.56/2.78	
7.56/2.78	The pumping substitution is [x / g(x)].
7.56/2.78	
7.56/2.78	The result substitution is [ ].
7.56/2.78	
7.56/2.78	
7.56/2.78	
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(10)
7.56/2.78	Complex Obligation (BEST)
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(11)
7.56/2.78	Obligation:
7.56/2.78	Proved the lower bound n^1 for the following obligation:
7.56/2.78	
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(f(0, x), 1) -> f(g(f(x, x)), x)
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(12) LowerBoundPropagationProof (FINISHED)
7.56/2.78	Propagated lower bound.
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(13)
7.56/2.78	BOUNDS(n^1, INF)
7.56/2.78	
7.56/2.78	----------------------------------------
7.56/2.78	
7.56/2.78	(14)
7.56/2.78	Obligation:
7.56/2.78	Analyzing the following TRS for decreasing loops:
7.56/2.78	
7.56/2.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.56/2.78	
7.56/2.78	
7.56/2.78	The TRS R consists of the following rules:
7.56/2.78	
7.56/2.78	   f(f(0, x), 1) -> f(g(f(x, x)), x)
7.56/2.78	   f(g(x), y) -> g(f(x, y))
7.56/2.78	
7.56/2.78	S is empty.
7.56/2.78	Rewrite Strategy: FULL
7.76/3.12	EOF