Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.40 pair #433313440

details

property	value
status	complete
benchmark	Ex4_4_Luc96b_L.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n118.star.cs.uiowa.edu
space	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	3.10698 seconds
cpu usage	8.75649
user time	8.1759
system time	0.580589
max virtual memory	1.8279384E7
max residence set size	1516356.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), O(n^1))

output

6.67/2.55	WORST_CASE(Omega(n^1), O(n^1))
8.59/3.06	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.59/3.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.59/3.06	
8.59/3.06	
8.59/3.06	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.06	
8.59/3.06	(0) CpxTRS
8.59/3.06	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
8.59/3.06	(2) CpxTRS
8.59/3.06	    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
8.59/3.06	    (4) BOUNDS(1, n^1)
8.59/3.06	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
8.59/3.06	(6) TRS for Loop Detection
8.59/3.06	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
8.59/3.06	    (8) BEST
8.59/3.06	        (9) proven lower bound
8.59/3.06	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
8.59/3.06	            (11) BOUNDS(n^1, INF)
8.59/3.06	        (12) TRS for Loop Detection
8.59/3.06	
8.59/3.06	
8.59/3.06	----------------------------------------
8.59/3.06	
8.59/3.06	(0)
8.59/3.06	Obligation:
8.59/3.06	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.06	
8.59/3.06	
8.59/3.06	The TRS R consists of the following rules:
8.59/3.06	
8.59/3.06	   f(g(X)) -> f(X)
8.59/3.06	
8.59/3.06	S is empty.
8.59/3.06	Rewrite Strategy: INNERMOST
8.59/3.06	----------------------------------------
8.59/3.06	
8.59/3.06	(1) RelTrsToTrsProof (UPPER BOUND(ID))
8.59/3.07	transformed relative TRS to TRS
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(2)
8.59/3.07	Obligation:
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(3) CpxTrsMatchBoundsProof (FINISHED)
8.59/3.07	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
8.59/3.07	The certificate found is represented by the following graph.
8.59/3.07	
8.59/3.07	"[1, 2]
8.59/3.07	{(1,2,[f_1|0, f_1|1]), (2,2,[g_1|0])}"
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(4)
8.59/3.07	BOUNDS(1, n^1)
8.59/3.07	
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
8.59/3.07	Transformed a relative TRS into a decreasing-loop problem.
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(6)
8.59/3.07	Obligation:
8.59/3.07	Analyzing the following TRS for decreasing loops:
8.59/3.07	
8.59/3.07	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
8.59/3.07	
8.59/3.07	
8.59/3.07	The TRS R consists of the following rules:
8.59/3.07	
8.59/3.07	   f(g(X)) -> f(X)
8.59/3.07	
8.59/3.07	S is empty.
8.59/3.07	Rewrite Strategy: INNERMOST
8.59/3.07	----------------------------------------
8.59/3.07	
8.59/3.07	(7) DecreasingLoopProof (LOWER BOUND(ID))
8.59/3.07	The following loop(s) give(s) rise to the lower bound Omega(n^1):
8.59/3.07	
8.59/3.07	The rewrite sequence
8.59/3.07	
8.59/3.07	f(g(X)) ->^+ f(X)
8.59/3.07	
8.59/3.07	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
8.59/3.07	
8.59/3.07	The pumping substitution is [X / g(X)].
8.59/3.07	
8.59/3.07	The result substitution is [ ].
8.59/3.07	
8.59/3.07

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