2.97/1.54	WORST_CASE(Omega(n^1), O(n^1))
2.97/1.56	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.97/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.97/1.56	
2.97/1.56	
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.97/1.56	
2.97/1.56	(0) CpxTRS
2.97/1.56	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
2.97/1.56	(2) CpxTRS
2.97/1.56	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
2.97/1.56	    (4) BOUNDS(1, n^1)
2.97/1.56	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.97/1.56	(6) TRS for Loop Detection
2.97/1.56	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.97/1.56	    (8) BEST
2.97/1.56	        (9) proven lower bound
2.97/1.56	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
2.97/1.56	            (11) BOUNDS(n^1, INF)
2.97/1.56	        (12) TRS for Loop Detection
2.97/1.56	
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(0)
2.97/1.56	Obligation:
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.97/1.56	
2.97/1.56	
2.97/1.56	The TRS R consists of the following rules:
2.97/1.56	
2.97/1.56	   f(x, y) -> g(x, y)
2.97/1.56	   g(h(x), y) -> h(f(x, y))
2.97/1.56	   g(h(x), y) -> h(g(x, y))
2.97/1.56	
2.97/1.56	S is empty.
2.97/1.56	Rewrite Strategy: INNERMOST
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(1) RelTrsToTrsProof (UPPER BOUND(ID))
2.97/1.56	transformed relative TRS to TRS
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(2)
2.97/1.56	Obligation:
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
2.97/1.56	
2.97/1.56	
2.97/1.56	The TRS R consists of the following rules:
2.97/1.56	
2.97/1.56	   f(x, y) -> g(x, y)
2.97/1.56	   g(h(x), y) -> h(f(x, y))
2.97/1.56	   g(h(x), y) -> h(g(x, y))
2.97/1.56	
2.97/1.56	S is empty.
2.97/1.56	Rewrite Strategy: INNERMOST
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(3) CpxTrsMatchBoundsTAProof (FINISHED)
2.97/1.56	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. 
2.97/1.56	
2.97/1.56	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
2.97/1.56	final states : [1, 2]
2.97/1.56	transitions: 
2.97/1.56	h0(0) -> 0
2.97/1.56	f0(0, 0) -> 1
2.97/1.56	g0(0, 0) -> 2
2.97/1.56	g1(0, 0) -> 1
2.97/1.56	f1(0, 0) -> 3
2.97/1.56	h1(3) -> 2
2.97/1.56	g1(0, 0) -> 4
2.97/1.56	h1(4) -> 2
2.97/1.56	g2(0, 0) -> 3
2.97/1.56	h1(3) -> 1
2.97/1.56	h1(3) -> 4
2.97/1.56	h1(4) -> 1
2.97/1.56	h1(4) -> 4
2.97/1.56	h1(3) -> 3
2.97/1.56	h1(4) -> 3
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(4)
2.97/1.56	BOUNDS(1, n^1)
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.97/1.56	Transformed a relative TRS into a decreasing-loop problem.
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(6)
2.97/1.56	Obligation:
2.97/1.56	Analyzing the following TRS for decreasing loops:
2.97/1.56	
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.97/1.56	
2.97/1.56	
2.97/1.56	The TRS R consists of the following rules:
2.97/1.56	
2.97/1.56	   f(x, y) -> g(x, y)
2.97/1.56	   g(h(x), y) -> h(f(x, y))
2.97/1.56	   g(h(x), y) -> h(g(x, y))
2.97/1.56	
2.97/1.56	S is empty.
2.97/1.56	Rewrite Strategy: INNERMOST
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(7) DecreasingLoopProof (LOWER BOUND(ID))
2.97/1.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.97/1.56	
2.97/1.56	The rewrite sequence
2.97/1.56	
2.97/1.56	g(h(x), y) ->^+ h(g(x, y))
2.97/1.56	
2.97/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
2.97/1.56	
2.97/1.56	The pumping substitution is [x / h(x)].
2.97/1.56	
2.97/1.56	The result substitution is [ ].
2.97/1.56	
2.97/1.56	
2.97/1.56	
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(8)
2.97/1.56	Complex Obligation (BEST)
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(9)
2.97/1.56	Obligation:
2.97/1.56	Proved the lower bound n^1 for the following obligation:
2.97/1.56	
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.97/1.56	
2.97/1.56	
2.97/1.56	The TRS R consists of the following rules:
2.97/1.56	
2.97/1.56	   f(x, y) -> g(x, y)
2.97/1.56	   g(h(x), y) -> h(f(x, y))
2.97/1.56	   g(h(x), y) -> h(g(x, y))
2.97/1.56	
2.97/1.56	S is empty.
2.97/1.56	Rewrite Strategy: INNERMOST
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(10) LowerBoundPropagationProof (FINISHED)
2.97/1.56	Propagated lower bound.
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(11)
2.97/1.56	BOUNDS(n^1, INF)
2.97/1.56	
2.97/1.56	----------------------------------------
2.97/1.56	
2.97/1.56	(12)
2.97/1.56	Obligation:
2.97/1.56	Analyzing the following TRS for decreasing loops:
2.97/1.56	
2.97/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
2.97/1.56	
2.97/1.56	
2.97/1.56	The TRS R consists of the following rules:
2.97/1.56	
2.97/1.56	   f(x, y) -> g(x, y)
2.97/1.56	   g(h(x), y) -> h(f(x, y))
2.97/1.56	   g(h(x), y) -> h(g(x, y))
2.97/1.56	
2.97/1.56	S is empty.
2.97/1.56	Rewrite Strategy: INNERMOST
3.14/1.57	EOF