3.32/1.58	WORST_CASE(Omega(n^1), O(n^1))
3.32/1.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.32/1.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.32/1.58	
3.32/1.58	
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.32/1.58	
3.32/1.58	(0) CpxTRS
3.32/1.58	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.32/1.58	(2) CpxTRS
3.32/1.58	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.32/1.58	    (4) BOUNDS(1, n^1)
3.32/1.58	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.32/1.58	(6) TRS for Loop Detection
3.32/1.58	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.32/1.58	    (8) BEST
3.32/1.58	        (9) proven lower bound
3.32/1.58	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.32/1.58	            (11) BOUNDS(n^1, INF)
3.32/1.58	        (12) TRS for Loop Detection
3.32/1.58	
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(0)
3.32/1.58	Obligation:
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.32/1.58	
3.32/1.58	
3.32/1.58	The TRS R consists of the following rules:
3.32/1.58	
3.32/1.58	   h(f(x, y)) -> f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	S is empty.
3.32/1.58	Rewrite Strategy: INNERMOST
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.32/1.58	transformed relative TRS to TRS
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(2)
3.32/1.58	Obligation:
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.32/1.58	
3.32/1.58	
3.32/1.58	The TRS R consists of the following rules:
3.32/1.58	
3.32/1.58	   h(f(x, y)) -> f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	S is empty.
3.32/1.58	Rewrite Strategy: INNERMOST
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.32/1.58	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. 
3.32/1.58	
3.32/1.58	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.32/1.58	final states : [1]
3.32/1.58	transitions: 
3.32/1.58	f0(0, 0) -> 0
3.32/1.58	a0() -> 0
3.32/1.58	h0(0) -> 1
3.32/1.58	h1(0) -> 4
3.32/1.58	h1(4) -> 3
3.32/1.58	a1() -> 5
3.32/1.58	f1(3, 5) -> 2
3.32/1.58	f1(0, 2) -> 1
3.32/1.58	f1(0, 2) -> 4
3.32/1.58	h2(0) -> 8
3.32/1.58	h2(8) -> 7
3.32/1.58	a2() -> 9
3.32/1.58	f2(7, 9) -> 6
3.32/1.58	f2(2, 6) -> 3
3.32/1.58	f1(0, 2) -> 8
3.32/1.58	f2(2, 6) -> 7
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(4)
3.32/1.58	BOUNDS(1, n^1)
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.32/1.58	Transformed a relative TRS into a decreasing-loop problem.
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(6)
3.32/1.58	Obligation:
3.32/1.58	Analyzing the following TRS for decreasing loops:
3.32/1.58	
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.32/1.58	
3.32/1.58	
3.32/1.58	The TRS R consists of the following rules:
3.32/1.58	
3.32/1.58	   h(f(x, y)) -> f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	S is empty.
3.32/1.58	Rewrite Strategy: INNERMOST
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.32/1.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.32/1.58	
3.32/1.58	The rewrite sequence
3.32/1.58	
3.32/1.58	h(f(x, y)) ->^+ f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.32/1.58	
3.32/1.58	The pumping substitution is [x / f(x, y)].
3.32/1.58	
3.32/1.58	The result substitution is [ ].
3.32/1.58	
3.32/1.58	
3.32/1.58	
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(8)
3.32/1.58	Complex Obligation (BEST)
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(9)
3.32/1.58	Obligation:
3.32/1.58	Proved the lower bound n^1 for the following obligation:
3.32/1.58	
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.32/1.58	
3.32/1.58	
3.32/1.58	The TRS R consists of the following rules:
3.32/1.58	
3.32/1.58	   h(f(x, y)) -> f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	S is empty.
3.32/1.58	Rewrite Strategy: INNERMOST
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(10) LowerBoundPropagationProof (FINISHED)
3.32/1.58	Propagated lower bound.
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(11)
3.32/1.58	BOUNDS(n^1, INF)
3.32/1.58	
3.32/1.58	----------------------------------------
3.32/1.58	
3.32/1.58	(12)
3.32/1.58	Obligation:
3.32/1.58	Analyzing the following TRS for decreasing loops:
3.32/1.58	
3.32/1.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.32/1.58	
3.32/1.58	
3.32/1.58	The TRS R consists of the following rules:
3.32/1.58	
3.32/1.58	   h(f(x, y)) -> f(y, f(h(h(x)), a))
3.32/1.58	
3.32/1.58	S is empty.
3.32/1.58	Rewrite Strategy: INNERMOST
3.32/1.61	EOF