8.71/3.05	WORST_CASE(Omega(n^1), O(n^1))
8.71/3.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.71/3.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.71/3.06	
8.71/3.06	
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.71/3.06	
8.71/3.06	(0) CpxTRS
8.71/3.06	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
8.71/3.06	(2) CpxTRS
8.71/3.06	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
8.71/3.06	    (4) BOUNDS(1, n^1)
8.71/3.06	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
8.71/3.06	(6) TRS for Loop Detection
8.71/3.06	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
8.71/3.06	    (8) BEST
8.71/3.06	        (9) proven lower bound
8.71/3.06	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
8.71/3.06	            (11) BOUNDS(n^1, INF)
8.71/3.06	        (12) TRS for Loop Detection
8.71/3.06	
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(0)
8.71/3.06	Obligation:
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.71/3.06	
8.71/3.06	
8.71/3.06	The TRS R consists of the following rules:
8.71/3.06	
8.71/3.06	   h(f(x), y) -> f(g(x, y))
8.71/3.06	   g(x, y) -> h(x, y)
8.71/3.06	
8.71/3.06	S is empty.
8.71/3.06	Rewrite Strategy: FULL
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(1) RelTrsToTrsProof (UPPER BOUND(ID))
8.71/3.06	transformed relative TRS to TRS
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(2)
8.71/3.06	Obligation:
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
8.71/3.06	
8.71/3.06	
8.71/3.06	The TRS R consists of the following rules:
8.71/3.06	
8.71/3.06	   h(f(x), y) -> f(g(x, y))
8.71/3.06	   g(x, y) -> h(x, y)
8.71/3.06	
8.71/3.06	S is empty.
8.71/3.06	Rewrite Strategy: FULL
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(3) CpxTrsMatchBoundsTAProof (FINISHED)
8.71/3.06	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 2. 
8.71/3.06	
8.71/3.06	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
8.71/3.06	final states : [1, 2]
8.71/3.06	transitions: 
8.71/3.06	f0(0) -> 0
8.71/3.06	h0(0, 0) -> 1
8.71/3.06	g0(0, 0) -> 2
8.71/3.06	g1(0, 0) -> 3
8.71/3.06	f1(3) -> 1
8.71/3.06	h1(0, 0) -> 2
8.71/3.06	f1(3) -> 2
8.71/3.06	h2(0, 0) -> 3
8.71/3.06	f1(3) -> 3
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(4)
8.71/3.06	BOUNDS(1, n^1)
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
8.71/3.06	Transformed a relative TRS into a decreasing-loop problem.
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(6)
8.71/3.06	Obligation:
8.71/3.06	Analyzing the following TRS for decreasing loops:
8.71/3.06	
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.71/3.06	
8.71/3.06	
8.71/3.06	The TRS R consists of the following rules:
8.71/3.06	
8.71/3.06	   h(f(x), y) -> f(g(x, y))
8.71/3.06	   g(x, y) -> h(x, y)
8.71/3.06	
8.71/3.06	S is empty.
8.71/3.06	Rewrite Strategy: FULL
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(7) DecreasingLoopProof (LOWER BOUND(ID))
8.71/3.06	The following loop(s) give(s) rise to the lower bound Omega(n^1):
8.71/3.06	
8.71/3.06	The rewrite sequence
8.71/3.06	
8.71/3.06	h(f(x), y) ->^+ f(h(x, y))
8.71/3.06	
8.71/3.06	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
8.71/3.06	
8.71/3.06	The pumping substitution is [x / f(x)].
8.71/3.06	
8.71/3.06	The result substitution is [ ].
8.71/3.06	
8.71/3.06	
8.71/3.06	
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(8)
8.71/3.06	Complex Obligation (BEST)
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(9)
8.71/3.06	Obligation:
8.71/3.06	Proved the lower bound n^1 for the following obligation:
8.71/3.06	
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.71/3.06	
8.71/3.06	
8.71/3.06	The TRS R consists of the following rules:
8.71/3.06	
8.71/3.06	   h(f(x), y) -> f(g(x, y))
8.71/3.06	   g(x, y) -> h(x, y)
8.71/3.06	
8.71/3.06	S is empty.
8.71/3.06	Rewrite Strategy: FULL
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(10) LowerBoundPropagationProof (FINISHED)
8.71/3.06	Propagated lower bound.
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(11)
8.71/3.06	BOUNDS(n^1, INF)
8.71/3.06	
8.71/3.06	----------------------------------------
8.71/3.06	
8.71/3.06	(12)
8.71/3.06	Obligation:
8.71/3.06	Analyzing the following TRS for decreasing loops:
8.71/3.06	
8.71/3.06	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.71/3.06	
8.71/3.06	
8.71/3.06	The TRS R consists of the following rules:
8.71/3.06	
8.71/3.06	   h(f(x), y) -> f(g(x, y))
8.71/3.06	   g(x, y) -> h(x, y)
8.71/3.06	
8.71/3.06	S is empty.
8.71/3.06	Rewrite Strategy: FULL
9.04/3.22	EOF