7.63/2.68	WORST_CASE(Omega(n^1), O(n^1))
7.63/2.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.63/2.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.63/2.68	
7.63/2.68	
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.63/2.68	
7.63/2.68	(0) CpxTRS
7.63/2.68	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
7.63/2.68	(2) CpxTRS
7.63/2.68	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
7.63/2.68	    (4) BOUNDS(1, n^1)
7.63/2.68	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
7.63/2.68	(6) TRS for Loop Detection
7.63/2.68	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
7.63/2.68	    (8) BEST
7.63/2.68	        (9) proven lower bound
7.63/2.68	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
7.63/2.68	            (11) BOUNDS(n^1, INF)
7.63/2.68	        (12) TRS for Loop Detection
7.63/2.68	
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(0)
7.63/2.68	Obligation:
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.63/2.68	
7.63/2.68	
7.63/2.68	The TRS R consists of the following rules:
7.63/2.68	
7.63/2.68	   h(f(x, y)) -> f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	S is empty.
7.63/2.68	Rewrite Strategy: FULL
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(1) RelTrsToTrsProof (UPPER BOUND(ID))
7.63/2.68	transformed relative TRS to TRS
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(2)
7.63/2.68	Obligation:
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
7.63/2.68	
7.63/2.68	
7.63/2.68	The TRS R consists of the following rules:
7.63/2.68	
7.63/2.68	   h(f(x, y)) -> f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	S is empty.
7.63/2.68	Rewrite Strategy: FULL
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(3) CpxTrsMatchBoundsTAProof (FINISHED)
7.63/2.68	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. 
7.63/2.68	
7.63/2.68	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
7.63/2.68	final states : [1]
7.63/2.68	transitions: 
7.63/2.68	f0(0, 0) -> 0
7.63/2.68	a0() -> 0
7.63/2.68	h0(0) -> 1
7.63/2.68	h1(0) -> 4
7.63/2.68	h1(4) -> 3
7.63/2.68	a1() -> 5
7.63/2.68	f1(3, 5) -> 2
7.63/2.68	f1(0, 2) -> 1
7.63/2.68	f1(0, 2) -> 4
7.63/2.68	h2(0) -> 8
7.63/2.68	h2(8) -> 7
7.63/2.68	a2() -> 9
7.63/2.68	f2(7, 9) -> 6
7.63/2.68	f2(2, 6) -> 3
7.63/2.68	f1(0, 2) -> 8
7.63/2.68	f2(2, 6) -> 7
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(4)
7.63/2.68	BOUNDS(1, n^1)
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
7.63/2.68	Transformed a relative TRS into a decreasing-loop problem.
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(6)
7.63/2.68	Obligation:
7.63/2.68	Analyzing the following TRS for decreasing loops:
7.63/2.68	
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.63/2.68	
7.63/2.68	
7.63/2.68	The TRS R consists of the following rules:
7.63/2.68	
7.63/2.68	   h(f(x, y)) -> f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	S is empty.
7.63/2.68	Rewrite Strategy: FULL
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(7) DecreasingLoopProof (LOWER BOUND(ID))
7.63/2.68	The following loop(s) give(s) rise to the lower bound Omega(n^1):
7.63/2.68	
7.63/2.68	The rewrite sequence
7.63/2.68	
7.63/2.68	h(f(x, y)) ->^+ f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
7.63/2.68	
7.63/2.68	The pumping substitution is [x / f(x, y)].
7.63/2.68	
7.63/2.68	The result substitution is [ ].
7.63/2.68	
7.63/2.68	
7.63/2.68	
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(8)
7.63/2.68	Complex Obligation (BEST)
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(9)
7.63/2.68	Obligation:
7.63/2.68	Proved the lower bound n^1 for the following obligation:
7.63/2.68	
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.63/2.68	
7.63/2.68	
7.63/2.68	The TRS R consists of the following rules:
7.63/2.68	
7.63/2.68	   h(f(x, y)) -> f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	S is empty.
7.63/2.68	Rewrite Strategy: FULL
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(10) LowerBoundPropagationProof (FINISHED)
7.63/2.68	Propagated lower bound.
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(11)
7.63/2.68	BOUNDS(n^1, INF)
7.63/2.68	
7.63/2.68	----------------------------------------
7.63/2.68	
7.63/2.68	(12)
7.63/2.68	Obligation:
7.63/2.68	Analyzing the following TRS for decreasing loops:
7.63/2.68	
7.63/2.68	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
7.63/2.68	
7.63/2.68	
7.63/2.68	The TRS R consists of the following rules:
7.63/2.68	
7.63/2.68	   h(f(x, y)) -> f(y, f(h(h(x)), a))
7.63/2.68	
7.63/2.68	S is empty.
7.63/2.68	Rewrite Strategy: FULL
7.67/2.72	EOF