3.26/1.56	WORST_CASE(Omega(n^1), O(n^1))
3.26/1.56	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.26/1.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.56	
3.26/1.56	
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.26/1.56	
3.26/1.56	(0) CpxTRS
3.26/1.56	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.26/1.56	(2) CpxTRS
3.26/1.56	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.26/1.56	    (4) BOUNDS(1, n^1)
3.26/1.56	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.26/1.56	(6) TRS for Loop Detection
3.26/1.56	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.26/1.56	    (8) BEST
3.26/1.56	        (9) proven lower bound
3.26/1.56	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.26/1.56	            (11) BOUNDS(n^1, INF)
3.26/1.56	        (12) TRS for Loop Detection
3.26/1.56	
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(0)
3.26/1.56	Obligation:
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.26/1.56	
3.26/1.56	
3.26/1.56	The TRS R consists of the following rules:
3.26/1.56	
3.26/1.56	   g(f(x, y), z) -> f(x, g(y, z))
3.26/1.56	   g(h(x, y), z) -> g(x, f(y, z))
3.26/1.56	   g(x, h(y, z)) -> h(g(x, y), z)
3.26/1.56	
3.26/1.56	S is empty.
3.26/1.56	Rewrite Strategy: INNERMOST
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.26/1.56	transformed relative TRS to TRS
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(2)
3.26/1.56	Obligation:
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.26/1.56	
3.26/1.56	
3.26/1.56	The TRS R consists of the following rules:
3.26/1.56	
3.26/1.56	   g(f(x, y), z) -> f(x, g(y, z))
3.26/1.56	   g(h(x, y), z) -> g(x, f(y, z))
3.26/1.56	   g(x, h(y, z)) -> h(g(x, y), z)
3.26/1.56	
3.26/1.56	S is empty.
3.26/1.56	Rewrite Strategy: INNERMOST
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.26/1.56	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 1. 
3.26/1.56	
3.26/1.56	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.26/1.56	final states : [1]
3.26/1.56	transitions: 
3.26/1.56	f0(0, 0) -> 0
3.26/1.56	h0(0, 0) -> 0
3.26/1.56	g0(0, 0) -> 1
3.26/1.56	g1(0, 0) -> 2
3.26/1.56	f1(0, 2) -> 1
3.26/1.56	f1(0, 0) -> 3
3.26/1.56	g1(0, 3) -> 1
3.26/1.56	g1(0, 0) -> 4
3.26/1.56	h1(4, 0) -> 1
3.26/1.56	g1(0, 3) -> 2
3.26/1.56	f1(0, 2) -> 2
3.26/1.56	f1(0, 2) -> 4
3.26/1.56	f1(0, 3) -> 3
3.26/1.56	g1(0, 3) -> 4
3.26/1.56	h1(4, 0) -> 2
3.26/1.56	h1(4, 0) -> 4
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(4)
3.26/1.56	BOUNDS(1, n^1)
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.26/1.56	Transformed a relative TRS into a decreasing-loop problem.
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(6)
3.26/1.56	Obligation:
3.26/1.56	Analyzing the following TRS for decreasing loops:
3.26/1.56	
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.26/1.56	
3.26/1.56	
3.26/1.56	The TRS R consists of the following rules:
3.26/1.56	
3.26/1.56	   g(f(x, y), z) -> f(x, g(y, z))
3.26/1.56	   g(h(x, y), z) -> g(x, f(y, z))
3.26/1.56	   g(x, h(y, z)) -> h(g(x, y), z)
3.26/1.56	
3.26/1.56	S is empty.
3.26/1.56	Rewrite Strategy: INNERMOST
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.26/1.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.26/1.56	
3.26/1.56	The rewrite sequence
3.26/1.56	
3.26/1.56	g(h(x, y), z) ->^+ g(x, f(y, z))
3.26/1.56	
3.26/1.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.26/1.56	
3.26/1.56	The pumping substitution is [x / h(x, y)].
3.26/1.56	
3.26/1.56	The result substitution is [z / f(y, z)].
3.26/1.56	
3.26/1.56	
3.26/1.56	
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(8)
3.26/1.56	Complex Obligation (BEST)
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(9)
3.26/1.56	Obligation:
3.26/1.56	Proved the lower bound n^1 for the following obligation:
3.26/1.56	
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.26/1.56	
3.26/1.56	
3.26/1.56	The TRS R consists of the following rules:
3.26/1.56	
3.26/1.56	   g(f(x, y), z) -> f(x, g(y, z))
3.26/1.56	   g(h(x, y), z) -> g(x, f(y, z))
3.26/1.56	   g(x, h(y, z)) -> h(g(x, y), z)
3.26/1.56	
3.26/1.56	S is empty.
3.26/1.56	Rewrite Strategy: INNERMOST
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(10) LowerBoundPropagationProof (FINISHED)
3.26/1.56	Propagated lower bound.
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(11)
3.26/1.56	BOUNDS(n^1, INF)
3.26/1.56	
3.26/1.56	----------------------------------------
3.26/1.56	
3.26/1.56	(12)
3.26/1.56	Obligation:
3.26/1.56	Analyzing the following TRS for decreasing loops:
3.26/1.56	
3.26/1.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.26/1.56	
3.26/1.56	
3.26/1.56	The TRS R consists of the following rules:
3.26/1.56	
3.26/1.56	   g(f(x, y), z) -> f(x, g(y, z))
3.26/1.56	   g(h(x, y), z) -> g(x, f(y, z))
3.26/1.56	   g(x, h(y, z)) -> h(g(x, y), z)
3.26/1.56	
3.26/1.56	S is empty.
3.26/1.56	Rewrite Strategy: INNERMOST
3.47/1.59	EOF