Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.40 pair #433313605

details

property	value
status	complete
benchmark	2.41.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n119.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	1.63625 seconds
cpu usage	3.33481
user time	3.18663
system time	0.148175
max virtual memory	1.8277336E7
max residence set size	222268.0

stage attributes

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

output

3.01/1.59	WORST_CASE(Omega(n^1), O(n^1))
3.01/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.01/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.01/1.60	
3.01/1.60	
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	(0) CpxTRS
3.01/1.60	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.01/1.60	(2) CpxTRS
3.01/1.60	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.01/1.60	    (4) BOUNDS(1, n^1)
3.01/1.60	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.01/1.60	(6) TRS for Loop Detection
3.01/1.60	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.01/1.60	    (8) BEST
3.01/1.60	        (9) proven lower bound
3.01/1.60	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.01/1.60	            (11) BOUNDS(n^1, INF)
3.01/1.60	        (12) TRS for Loop Detection
3.01/1.60	
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(0)
3.01/1.60	Obligation:
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.01/1.60	transformed relative TRS to TRS
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(2)
3.01/1.60	Obligation:
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.01/1.60	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. 
3.01/1.60	
3.01/1.60	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.01/1.60	final states : [1, 2, 3]
3.01/1.60	transitions: 
3.01/1.60	nil0() -> 0
3.01/1.60	00() -> 0
3.01/1.60	g0(0, 0) -> 0
3.01/1.60	s0(0) -> 0
3.01/1.60	norm0(0) -> 1
3.01/1.60	f0(0, 0) -> 2
3.01/1.60	rem0(0, 0) -> 3
3.01/1.60	01() -> 1
3.01/1.60	norm1(0) -> 4
3.01/1.60	s1(4) -> 1
3.01/1.60	nil1() -> 5
3.01/1.60	g1(5, 0) -> 2
3.01/1.60	f1(0, 0) -> 6
3.01/1.60	g1(6, 0) -> 2
3.01/1.60	nil1() -> 3
3.01/1.60	g1(0, 0) -> 3
3.01/1.60	rem1(0, 0) -> 3
3.01/1.60	01() -> 4
3.01/1.60	s1(4) -> 4
3.01/1.60	g1(5, 0) -> 6
3.01/1.60	g1(6, 0) -> 6
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(4)
3.01/1.60	BOUNDS(1, n^1)
3.01/1.60	
3.01/1.60	----------------------------------------

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