3.28/1.62	WORST_CASE(Omega(n^1), O(n^1))
3.28/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.28/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.28/1.63	
3.28/1.63	
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.28/1.63	
3.28/1.63	(0) CpxTRS
3.28/1.63	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.28/1.63	(2) CpxTRS
3.28/1.63	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.28/1.63	    (4) BOUNDS(1, n^1)
3.28/1.63	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.28/1.63	(6) TRS for Loop Detection
3.28/1.63	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.28/1.63	    (8) BEST
3.28/1.63	        (9) proven lower bound
3.28/1.63	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.28/1.63	            (11) BOUNDS(n^1, INF)
3.28/1.63	        (12) TRS for Loop Detection
3.28/1.63	
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(0)
3.28/1.63	Obligation:
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.28/1.63	
3.28/1.63	
3.28/1.63	The TRS R consists of the following rules:
3.28/1.63	
3.28/1.63	   f(x, y) -> g(x, y)
3.28/1.63	   g(h(x), y) -> h(f(x, y))
3.28/1.63	   g(h(x), y) -> h(g(x, y))
3.28/1.63	
3.28/1.63	S is empty.
3.28/1.63	Rewrite Strategy: FULL
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.28/1.63	transformed relative TRS to TRS
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(2)
3.28/1.63	Obligation:
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.28/1.63	
3.28/1.63	
3.28/1.63	The TRS R consists of the following rules:
3.28/1.63	
3.28/1.63	   f(x, y) -> g(x, y)
3.28/1.63	   g(h(x), y) -> h(f(x, y))
3.28/1.63	   g(h(x), y) -> h(g(x, y))
3.28/1.63	
3.28/1.63	S is empty.
3.28/1.63	Rewrite Strategy: FULL
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.28/1.63	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. 
3.28/1.63	
3.28/1.63	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.28/1.63	final states : [1, 2]
3.28/1.63	transitions: 
3.28/1.63	h0(0) -> 0
3.28/1.63	f0(0, 0) -> 1
3.28/1.63	g0(0, 0) -> 2
3.28/1.63	g1(0, 0) -> 1
3.28/1.63	f1(0, 0) -> 3
3.28/1.63	h1(3) -> 2
3.28/1.63	g1(0, 0) -> 4
3.28/1.63	h1(4) -> 2
3.28/1.63	g2(0, 0) -> 3
3.28/1.63	h1(3) -> 1
3.28/1.63	h1(3) -> 4
3.28/1.63	h1(4) -> 1
3.28/1.63	h1(4) -> 4
3.28/1.63	h1(3) -> 3
3.28/1.63	h1(4) -> 3
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(4)
3.28/1.63	BOUNDS(1, n^1)
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.28/1.63	Transformed a relative TRS into a decreasing-loop problem.
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(6)
3.28/1.63	Obligation:
3.28/1.63	Analyzing the following TRS for decreasing loops:
3.28/1.63	
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.28/1.63	
3.28/1.63	
3.28/1.63	The TRS R consists of the following rules:
3.28/1.63	
3.28/1.63	   f(x, y) -> g(x, y)
3.28/1.63	   g(h(x), y) -> h(f(x, y))
3.28/1.63	   g(h(x), y) -> h(g(x, y))
3.28/1.63	
3.28/1.63	S is empty.
3.28/1.63	Rewrite Strategy: FULL
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.28/1.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.28/1.63	
3.28/1.63	The rewrite sequence
3.28/1.63	
3.28/1.63	g(h(x), y) ->^+ h(g(x, y))
3.28/1.63	
3.28/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.28/1.63	
3.28/1.63	The pumping substitution is [x / h(x)].
3.28/1.63	
3.28/1.63	The result substitution is [ ].
3.28/1.63	
3.28/1.63	
3.28/1.63	
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(8)
3.28/1.63	Complex Obligation (BEST)
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(9)
3.28/1.63	Obligation:
3.28/1.63	Proved the lower bound n^1 for the following obligation:
3.28/1.63	
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.28/1.63	
3.28/1.63	
3.28/1.63	The TRS R consists of the following rules:
3.28/1.63	
3.28/1.63	   f(x, y) -> g(x, y)
3.28/1.63	   g(h(x), y) -> h(f(x, y))
3.28/1.63	   g(h(x), y) -> h(g(x, y))
3.28/1.63	
3.28/1.63	S is empty.
3.28/1.63	Rewrite Strategy: FULL
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(10) LowerBoundPropagationProof (FINISHED)
3.28/1.63	Propagated lower bound.
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(11)
3.28/1.63	BOUNDS(n^1, INF)
3.28/1.63	
3.28/1.63	----------------------------------------
3.28/1.63	
3.28/1.63	(12)
3.28/1.63	Obligation:
3.28/1.63	Analyzing the following TRS for decreasing loops:
3.28/1.63	
3.28/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.28/1.63	
3.28/1.63	
3.28/1.63	The TRS R consists of the following rules:
3.28/1.63	
3.28/1.63	   f(x, y) -> g(x, y)
3.28/1.63	   g(h(x), y) -> h(f(x, y))
3.28/1.63	   g(h(x), y) -> h(g(x, y))
3.28/1.63	
3.28/1.63	S is empty.
3.28/1.63	Rewrite Strategy: FULL
3.28/1.67	EOF