Runtime_Complexity_Full_Rewriting 2019-04-01 06.11 pair #433308223

details

property	value
status	complete
benchmark	PEANO_nokinds_C.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n031.star.cs.uiowa.edu
space	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	12.8005 seconds
cpu usage	32.5344
user time	30.6886
system time	1.84583
max virtual memory	1.914574E7
max residence set size	3916332.0

stage attributes

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

output

32.24/9.49	WORST_CASE(Omega(n^1), O(n^1))
32.46/9.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.46/9.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.46/9.50	
32.46/9.50	
32.46/9.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.46/9.50	
32.46/9.50	(0) CpxTRS
32.46/9.50	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms]
32.46/9.50	(2) CpxTRS
32.46/9.50	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
32.46/9.50	    (4) CpxTRS
32.46/9.50	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 85 ms]
32.46/9.50	    (6) BOUNDS(1, n^1)
32.46/9.50	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
32.46/9.50	(8) TRS for Loop Detection
32.46/9.50	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
32.46/9.50	    (10) BEST
32.46/9.50	        (11) proven lower bound
32.46/9.50	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
32.46/9.50	            (13) BOUNDS(n^1, INF)
32.46/9.50	        (14) TRS for Loop Detection
32.46/9.50	
32.46/9.50	
32.46/9.50	----------------------------------------
32.46/9.50	
32.46/9.50	(0)
32.46/9.50	Obligation:
32.46/9.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
32.46/9.50	
32.46/9.50	
32.46/9.50	The TRS R consists of the following rules:
32.46/9.50	
32.46/9.50	   active(U11(tt, N)) -> mark(N)
32.46/9.50	   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
32.46/9.50	   active(and(tt, X)) -> mark(X)
32.46/9.50	   active(isNat(0)) -> mark(tt)
32.46/9.50	   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
32.46/9.50	   active(isNat(s(V1))) -> mark(isNat(V1))
32.46/9.50	   active(plus(N, 0)) -> mark(U11(isNat(N), N))
32.46/9.50	   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
32.46/9.50	   active(U11(X1, X2)) -> U11(active(X1), X2)
32.46/9.50	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
32.46/9.50	   active(s(X)) -> s(active(X))
32.46/9.50	   active(plus(X1, X2)) -> plus(active(X1), X2)
32.46/9.50	   active(plus(X1, X2)) -> plus(X1, active(X2))
32.46/9.50	   active(and(X1, X2)) -> and(active(X1), X2)
32.46/9.50	   U11(mark(X1), X2) -> mark(U11(X1, X2))
32.46/9.50	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
32.46/9.50	   s(mark(X)) -> mark(s(X))
32.46/9.50	   plus(mark(X1), X2) -> mark(plus(X1, X2))
32.46/9.50	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
32.46/9.50	   and(mark(X1), X2) -> mark(and(X1, X2))
32.46/9.50	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
32.46/9.50	   proper(tt) -> ok(tt)
32.46/9.50	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
32.46/9.50	   proper(s(X)) -> s(proper(X))
32.46/9.50	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
32.46/9.50	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
32.46/9.50	   proper(isNat(X)) -> isNat(proper(X))
32.46/9.50	   proper(0) -> ok(0)
32.46/9.50	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
32.46/9.50	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
32.46/9.50	   s(ok(X)) -> ok(s(X))
32.46/9.50	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
32.46/9.50	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
32.46/9.50	   isNat(ok(X)) -> ok(isNat(X))
32.46/9.50	   top(mark(X)) -> top(proper(X))
32.46/9.50	   top(ok(X)) -> top(active(X))
32.46/9.50	
32.46/9.50	S is empty.
32.46/9.50	Rewrite Strategy: FULL
32.46/9.50	----------------------------------------
32.46/9.50	
32.46/9.50	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
32.46/9.50	The following defined symbols can occur below the 0th argument of top: proper, active
32.46/9.50	The following defined symbols can occur below the 0th argument of proper: proper, active
32.46/9.50	The following defined symbols can occur below the 0th argument of active: proper, active
32.46/9.50	
32.46/9.50	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
32.46/9.50	active(U11(tt, N)) -> mark(N)
32.46/9.50	active(U21(tt, M, N)) -> mark(s(plus(N, M)))
32.46/9.50	active(and(tt, X)) -> mark(X)
32.46/9.50	active(isNat(0)) -> mark(tt)
32.46/9.50	active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
32.46/9.50	active(isNat(s(V1))) -> mark(isNat(V1))
32.46/9.50	active(plus(N, 0)) -> mark(U11(isNat(N), N))
32.46/9.50	active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
32.46/9.50	active(U11(X1, X2)) -> U11(active(X1), X2)
32.46/9.50	active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
32.46/9.50	active(s(X)) -> s(active(X))
32.46/9.50	active(plus(X1, X2)) -> plus(active(X1), X2)
32.46/9.50	active(plus(X1, X2)) -> plus(X1, active(X2))
32.46/9.50	active(and(X1, X2)) -> and(active(X1), X2)
32.46/9.50	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
32.46/9.50	proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
32.46/9.50	proper(s(X)) -> s(proper(X))
32.46/9.50	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
32.46/9.50	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
32.46/9.50	proper(isNat(X)) -> isNat(proper(X))

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