Runtime_Complexity_Full_Rewriting 2019-04-01 06.11 pair #433308160

details

property	value
status	complete
benchmark	Ex1_Luc02b_C.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n157.star.cs.uiowa.edu
space	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	11.488 seconds
cpu usage	30.4497
user time	28.1845
system time	2.26512
max virtual memory	1.9538856E7
max residence set size	4230920.0

stage attributes

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

output

30.17/11.41	WORST_CASE(Omega(n^1), O(n^1))
30.17/11.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.17/11.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.17/11.42	
30.17/11.42	
30.17/11.42	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
30.17/11.42	
30.17/11.42	(0) CpxTRS
30.17/11.42	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 10 ms]
30.17/11.42	(2) CpxTRS
30.17/11.42	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
30.17/11.42	    (4) CpxTRS
30.17/11.42	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 82 ms]
30.17/11.42	    (6) BOUNDS(1, n^1)
30.17/11.42	(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
30.17/11.42	(8) CpxTRS
30.17/11.42	    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
30.17/11.42	    (10) typed CpxTrs
30.17/11.42	    (11) OrderProof [LOWER BOUND(ID), 0 ms]
30.17/11.42	    (12) typed CpxTrs
30.17/11.42	    (13) RewriteLemmaProof [LOWER BOUND(ID), 472 ms]
30.17/11.42	    (14) BEST
30.17/11.42	        (15) proven lower bound
30.17/11.42	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
30.17/11.42	            (17) BOUNDS(n^1, INF)
30.17/11.42	        (18) typed CpxTrs
30.17/11.42	            (19) RewriteLemmaProof [LOWER BOUND(ID), 93 ms]
30.17/11.42	            (20) typed CpxTrs
30.17/11.42	            (21) RewriteLemmaProof [LOWER BOUND(ID), 74 ms]
30.17/11.42	            (22) typed CpxTrs
30.17/11.42	            (23) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
30.17/11.42	            (24) typed CpxTrs
30.17/11.42	            (25) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
30.17/11.42	            (26) typed CpxTrs
30.17/11.42	
30.17/11.42	
30.17/11.42	----------------------------------------
30.17/11.42	
30.17/11.42	(0)
30.17/11.42	Obligation:
30.17/11.42	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
30.17/11.42	
30.17/11.42	
30.17/11.42	The TRS R consists of the following rules:
30.17/11.42	
30.17/11.42	   active(from(X)) -> mark(cons(X, from(s(X))))
30.17/11.42	   active(first(0, Z)) -> mark(nil)
30.17/11.42	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
30.17/11.42	   active(sel(0, cons(X, Z))) -> mark(X)
30.17/11.42	   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
30.17/11.42	   active(from(X)) -> from(active(X))
30.17/11.42	   active(cons(X1, X2)) -> cons(active(X1), X2)
30.17/11.42	   active(s(X)) -> s(active(X))
30.17/11.42	   active(first(X1, X2)) -> first(active(X1), X2)
30.17/11.42	   active(first(X1, X2)) -> first(X1, active(X2))
30.17/11.42	   active(sel(X1, X2)) -> sel(active(X1), X2)
30.17/11.42	   active(sel(X1, X2)) -> sel(X1, active(X2))
30.17/11.42	   from(mark(X)) -> mark(from(X))
30.17/11.42	   cons(mark(X1), X2) -> mark(cons(X1, X2))
30.17/11.42	   s(mark(X)) -> mark(s(X))
30.17/11.42	   first(mark(X1), X2) -> mark(first(X1, X2))
30.17/11.42	   first(X1, mark(X2)) -> mark(first(X1, X2))
30.17/11.42	   sel(mark(X1), X2) -> mark(sel(X1, X2))
30.17/11.42	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
30.17/11.42	   proper(from(X)) -> from(proper(X))
30.17/11.42	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
30.17/11.42	   proper(s(X)) -> s(proper(X))
30.17/11.42	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
30.17/11.42	   proper(0) -> ok(0)
30.17/11.42	   proper(nil) -> ok(nil)
30.17/11.42	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
30.17/11.42	   from(ok(X)) -> ok(from(X))
30.17/11.42	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
30.17/11.42	   s(ok(X)) -> ok(s(X))
30.17/11.42	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
30.17/11.42	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
30.17/11.42	   top(mark(X)) -> top(proper(X))
30.17/11.42	   top(ok(X)) -> top(active(X))
30.17/11.42	
30.17/11.42	S is empty.
30.17/11.42	Rewrite Strategy: FULL
30.17/11.42	----------------------------------------
30.17/11.42	
30.17/11.42	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
30.17/11.42	The following defined symbols can occur below the 0th argument of top: proper, active
30.17/11.42	The following defined symbols can occur below the 0th argument of proper: proper, active
30.17/11.42	The following defined symbols can occur below the 0th argument of active: proper, active
30.17/11.42	
30.17/11.42	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
30.17/11.42	active(from(X)) -> mark(cons(X, from(s(X))))
30.17/11.42	active(first(0, Z)) -> mark(nil)
30.17/11.42	active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
30.17/11.42	active(sel(0, cons(X, Z))) -> mark(X)
30.17/11.42	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
30.17/11.42	active(from(X)) -> from(active(X))
30.17/11.42	active(cons(X1, X2)) -> cons(active(X1), X2)
30.17/11.42	active(s(X)) -> s(active(X))
30.17/11.42	active(first(X1, X2)) -> first(active(X1), X2)
30.17/11.42	active(first(X1, X2)) -> first(X1, active(X2))
30.17/11.42	active(sel(X1, X2)) -> sel(active(X1), X2)

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