3.08/1.58	WORST_CASE(Omega(n^1), O(n^1))
3.08/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.08/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.08/1.59	
3.08/1.59	
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.08/1.59	
3.08/1.59	(0) CpxTRS
3.08/1.59	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.08/1.59	(2) CpxTRS
3.08/1.59	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.08/1.59	    (4) BOUNDS(1, n^1)
3.08/1.59	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.08/1.59	(6) TRS for Loop Detection
3.08/1.59	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.08/1.59	    (8) BEST
3.08/1.59	        (9) proven lower bound
3.08/1.59	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.08/1.59	            (11) BOUNDS(n^1, INF)
3.08/1.59	        (12) TRS for Loop Detection
3.08/1.59	
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(0)
3.08/1.59	Obligation:
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.08/1.59	
3.08/1.59	
3.08/1.59	The TRS R consists of the following rules:
3.08/1.59	
3.08/1.59	   g(f(x), y) -> f(h(x, y))
3.08/1.59	   h(x, y) -> g(x, f(y))
3.08/1.59	
3.08/1.59	S is empty.
3.08/1.59	Rewrite Strategy: FULL
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.08/1.59	transformed relative TRS to TRS
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(2)
3.08/1.59	Obligation:
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.08/1.59	
3.08/1.59	
3.08/1.59	The TRS R consists of the following rules:
3.08/1.59	
3.08/1.59	   g(f(x), y) -> f(h(x, y))
3.08/1.59	   h(x, y) -> g(x, f(y))
3.08/1.59	
3.08/1.59	S is empty.
3.08/1.59	Rewrite Strategy: FULL
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.08/1.59	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. 
3.08/1.59	
3.08/1.59	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.08/1.59	final states : [1, 2]
3.08/1.59	transitions: 
3.08/1.59	f0(0) -> 0
3.08/1.59	g0(0, 0) -> 1
3.08/1.59	h0(0, 0) -> 2
3.08/1.59	h1(0, 0) -> 3
3.08/1.59	f1(3) -> 1
3.08/1.59	f1(0) -> 4
3.08/1.59	g1(0, 4) -> 2
3.08/1.59	h1(0, 4) -> 3
3.08/1.59	f1(3) -> 2
3.08/1.59	f2(0) -> 5
3.08/1.59	g2(0, 5) -> 3
3.08/1.59	f2(4) -> 5
3.08/1.59	h1(0, 5) -> 3
3.08/1.59	f1(3) -> 3
3.08/1.59	f2(5) -> 5
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(4)
3.08/1.59	BOUNDS(1, n^1)
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.08/1.59	Transformed a relative TRS into a decreasing-loop problem.
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(6)
3.08/1.59	Obligation:
3.08/1.59	Analyzing the following TRS for decreasing loops:
3.08/1.59	
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.08/1.59	
3.08/1.59	
3.08/1.59	The TRS R consists of the following rules:
3.08/1.59	
3.08/1.59	   g(f(x), y) -> f(h(x, y))
3.08/1.59	   h(x, y) -> g(x, f(y))
3.08/1.59	
3.08/1.59	S is empty.
3.08/1.59	Rewrite Strategy: FULL
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.08/1.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.08/1.59	
3.08/1.59	The rewrite sequence
3.08/1.59	
3.08/1.59	h(f(x1_0), y) ->^+ f(h(x1_0, f(y)))
3.08/1.59	
3.08/1.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.08/1.59	
3.08/1.59	The pumping substitution is [x1_0 / f(x1_0)].
3.08/1.59	
3.08/1.59	The result substitution is [y / f(y)].
3.08/1.59	
3.08/1.59	
3.08/1.59	
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(8)
3.08/1.59	Complex Obligation (BEST)
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(9)
3.08/1.59	Obligation:
3.08/1.59	Proved the lower bound n^1 for the following obligation:
3.08/1.59	
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.08/1.59	
3.08/1.59	
3.08/1.59	The TRS R consists of the following rules:
3.08/1.59	
3.08/1.59	   g(f(x), y) -> f(h(x, y))
3.08/1.59	   h(x, y) -> g(x, f(y))
3.08/1.59	
3.08/1.59	S is empty.
3.08/1.59	Rewrite Strategy: FULL
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(10) LowerBoundPropagationProof (FINISHED)
3.08/1.59	Propagated lower bound.
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(11)
3.08/1.59	BOUNDS(n^1, INF)
3.08/1.59	
3.08/1.59	----------------------------------------
3.08/1.59	
3.08/1.59	(12)
3.08/1.59	Obligation:
3.08/1.59	Analyzing the following TRS for decreasing loops:
3.08/1.59	
3.08/1.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.08/1.59	
3.08/1.59	
3.08/1.59	The TRS R consists of the following rules:
3.08/1.59	
3.08/1.59	   g(f(x), y) -> f(h(x, y))
3.08/1.59	   h(x, y) -> g(x, f(y))
3.08/1.59	
3.08/1.59	S is empty.
3.08/1.59	Rewrite Strategy: FULL
3.38/1.77	EOF