Runtime_Complexity_Full_Rewriting 2019-04-01 06.11 pair #433307984

details

property	value
status	complete
benchmark	Ex4_7_37_Bor03_C.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n091.star.cs.uiowa.edu
space	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	11.8489 seconds
cpu usage	38.4582
user time	36.2923
system time	2.16593
max virtual memory	3.7773928E7
max residence set size	4230204.0

stage attributes

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

output

38.35/11.79	WORST_CASE(Omega(n^1), O(n^1))
38.35/11.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.35/11.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.35/11.80	
38.35/11.80	
38.35/11.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
38.35/11.80	
38.35/11.80	(0) CpxTRS
38.35/11.80	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms]
38.35/11.80	(2) CpxTRS
38.35/11.80	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
38.35/11.80	    (4) CpxTRS
38.35/11.80	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 102 ms]
38.35/11.80	    (6) BOUNDS(1, n^1)
38.35/11.80	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
38.35/11.80	(8) TRS for Loop Detection
38.35/11.80	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
38.35/11.80	    (10) BEST
38.35/11.80	        (11) proven lower bound
38.35/11.80	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
38.35/11.80	            (13) BOUNDS(n^1, INF)
38.35/11.80	        (14) TRS for Loop Detection
38.35/11.80	
38.35/11.80	
38.35/11.80	----------------------------------------
38.35/11.80	
38.35/11.80	(0)
38.35/11.80	Obligation:
38.35/11.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
38.35/11.80	
38.35/11.80	
38.35/11.80	The TRS R consists of the following rules:
38.35/11.80	
38.35/11.80	   active(from(X)) -> mark(cons(X, from(s(X))))
38.35/11.80	   active(sel(0, cons(X, XS))) -> mark(X)
38.35/11.80	   active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
38.35/11.80	   active(minus(X, 0)) -> mark(0)
38.35/11.80	   active(minus(s(X), s(Y))) -> mark(minus(X, Y))
38.35/11.80	   active(quot(0, s(Y))) -> mark(0)
38.35/11.80	   active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
38.35/11.80	   active(zWquot(XS, nil)) -> mark(nil)
38.35/11.80	   active(zWquot(nil, XS)) -> mark(nil)
38.35/11.80	   active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
38.35/11.80	   active(from(X)) -> from(active(X))
38.35/11.80	   active(cons(X1, X2)) -> cons(active(X1), X2)
38.35/11.80	   active(s(X)) -> s(active(X))
38.35/11.80	   active(sel(X1, X2)) -> sel(active(X1), X2)
38.35/11.80	   active(sel(X1, X2)) -> sel(X1, active(X2))
38.35/11.80	   active(minus(X1, X2)) -> minus(active(X1), X2)
38.35/11.80	   active(minus(X1, X2)) -> minus(X1, active(X2))
38.35/11.80	   active(quot(X1, X2)) -> quot(active(X1), X2)
38.35/11.80	   active(quot(X1, X2)) -> quot(X1, active(X2))
38.35/11.80	   active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
38.35/11.80	   active(zWquot(X1, X2)) -> zWquot(X1, active(X2))
38.35/11.80	   from(mark(X)) -> mark(from(X))
38.35/11.80	   cons(mark(X1), X2) -> mark(cons(X1, X2))
38.35/11.80	   s(mark(X)) -> mark(s(X))
38.35/11.80	   sel(mark(X1), X2) -> mark(sel(X1, X2))
38.35/11.80	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.35/11.80	   minus(mark(X1), X2) -> mark(minus(X1, X2))
38.35/11.80	   minus(X1, mark(X2)) -> mark(minus(X1, X2))
38.35/11.80	   quot(mark(X1), X2) -> mark(quot(X1, X2))
38.35/11.80	   quot(X1, mark(X2)) -> mark(quot(X1, X2))
38.35/11.80	   zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
38.35/11.80	   zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
38.35/11.80	   proper(from(X)) -> from(proper(X))
38.35/11.80	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.35/11.80	   proper(s(X)) -> s(proper(X))
38.35/11.80	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.35/11.80	   proper(0) -> ok(0)
38.35/11.80	   proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
38.35/11.80	   proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
38.35/11.80	   proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
38.35/11.80	   proper(nil) -> ok(nil)
38.35/11.80	   from(ok(X)) -> ok(from(X))
38.35/11.80	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.35/11.80	   s(ok(X)) -> ok(s(X))
38.35/11.80	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.35/11.80	   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
38.35/11.80	   quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
38.35/11.80	   zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
38.35/11.80	   top(mark(X)) -> top(proper(X))
38.35/11.80	   top(ok(X)) -> top(active(X))
38.35/11.80	
38.35/11.80	S is empty.
38.35/11.80	Rewrite Strategy: FULL
38.35/11.80	----------------------------------------
38.35/11.80	
38.35/11.80	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
38.35/11.80	The following defined symbols can occur below the 0th argument of top: proper, active
38.35/11.80	The following defined symbols can occur below the 0th argument of proper: proper, active
38.35/11.80	The following defined symbols can occur below the 0th argument of active: proper, active
38.35/11.80	
38.35/11.80	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
38.35/11.80	active(from(X)) -> mark(cons(X, from(s(X))))
38.35/11.80	active(sel(0, cons(X, XS))) -> mark(X)
38.35/11.80	active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
38.35/11.80	active(minus(X, 0)) -> mark(0)
38.35/11.80	active(minus(s(X), s(Y))) -> mark(minus(X, Y))
38.35/11.80	active(quot(0, s(Y))) -> mark(0)

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