Runtime_Complexity_Full_Rewriting 2019-04-01 06.11 pair #433308519

details

property	value
status	complete
benchmark	#4.20a.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n018.star.cs.uiowa.edu
space	Strategy_removed_AG01

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	6.82395 seconds
cpu usage	20.0761
user time	18.2045
system time	1.87159
max virtual memory	1.9341208E7
max residence set size	4327692.0

stage attributes

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

output

19.82/6.78	WORST_CASE(Omega(n^1), O(n^1))
19.82/6.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.82/6.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.82/6.78	
19.82/6.78	
19.82/6.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.82/6.78	
19.82/6.78	(0) CpxTRS
19.82/6.78	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
19.82/6.78	(2) CpxTRS
19.82/6.78	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
19.82/6.78	    (4) CpxTRS
19.82/6.78	    (5) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
19.82/6.78	    (6) BOUNDS(1, n^1)
19.82/6.78	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
19.82/6.78	(8) TRS for Loop Detection
19.82/6.78	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
19.82/6.78	    (10) BEST
19.82/6.78	        (11) proven lower bound
19.82/6.78	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
19.82/6.78	            (13) BOUNDS(n^1, INF)
19.82/6.78	        (14) TRS for Loop Detection
19.82/6.78	
19.82/6.78	
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(0)
19.82/6.78	Obligation:
19.82/6.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
19.82/6.78	
19.82/6.78	
19.82/6.78	The TRS R consists of the following rules:
19.82/6.78	
19.82/6.78	   f(f(x)) -> f(x)
19.82/6.78	   f(s(x)) -> f(x)
19.82/6.78	   g(s(0)) -> g(f(s(0)))
19.82/6.78	
19.82/6.78	S is empty.
19.82/6.78	Rewrite Strategy: FULL
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
19.82/6.78	The following defined symbols can occur below the 0th argument of g: f
19.82/6.78	
19.82/6.78	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
19.82/6.78	f(f(x)) -> f(x)
19.82/6.78	
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(2)
19.82/6.78	Obligation:
19.82/6.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.82/6.78	
19.82/6.78	
19.82/6.78	The TRS R consists of the following rules:
19.82/6.78	
19.82/6.78	   f(s(x)) -> f(x)
19.82/6.78	   g(s(0)) -> g(f(s(0)))
19.82/6.78	
19.82/6.78	S is empty.
19.82/6.78	Rewrite Strategy: FULL
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(3) RelTrsToTrsProof (UPPER BOUND(ID))
19.82/6.78	transformed relative TRS to TRS
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(4)
19.82/6.78	Obligation:
19.82/6.78	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
19.82/6.78	
19.82/6.78	
19.82/6.78	The TRS R consists of the following rules:
19.82/6.78	
19.82/6.78	   f(s(x)) -> f(x)
19.82/6.78	   g(s(0)) -> g(f(s(0)))
19.82/6.78	
19.82/6.78	S is empty.
19.82/6.78	Rewrite Strategy: FULL
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(5) CpxTrsMatchBoundsProof (FINISHED)
19.82/6.78	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. 
19.82/6.78	The certificate found is represented by the following graph.
19.82/6.78	
19.82/6.78	"[7, 8, 9, 10, 11]
19.82/6.78	{(7,8,[f_1|0, g_1|0, f_1|1]), (7,9,[g_1|1]), (8,8,[s_1|0, 0|0]), (9,10,[f_1|1]), (9,11,[f_1|2]), (10,11,[s_1|1]), (11,8,[0|1])}"
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(6)
19.82/6.78	BOUNDS(1, n^1)
19.82/6.78	
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
19.82/6.78	Transformed a relative TRS into a decreasing-loop problem.
19.82/6.78	----------------------------------------
19.82/6.78	
19.82/6.78	(8)
19.82/6.78	Obligation:

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