321.23/291.47	WORST_CASE(Omega(n^1), ?)
321.23/291.48	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
321.23/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
321.23/291.48	
321.23/291.48	
321.23/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
321.23/291.48	
321.23/291.48	(0) CpxTRS
321.23/291.48	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
321.23/291.48	(2) CpxTRS
321.23/291.48	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
321.23/291.48	(4) typed CpxTrs
321.23/291.48	(5) OrderProof [LOWER BOUND(ID), 0 ms]
321.23/291.48	(6) typed CpxTrs
321.23/291.48	(7) RewriteLemmaProof [LOWER BOUND(ID), 330 ms]
321.23/291.48	(8) BEST
321.23/291.48	    (9) proven lower bound
321.23/291.48	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
321.23/291.48	        (11) BOUNDS(n^1, INF)
321.23/291.48	    (12) typed CpxTrs
321.23/291.48	        (13) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
321.23/291.48	        (14) typed CpxTrs
321.23/291.48	
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(0)
321.23/291.48	Obligation:
321.23/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
321.23/291.48	
321.23/291.48	
321.23/291.48	The TRS R consists of the following rules:
321.23/291.48	
321.23/291.48	   half(0) -> 0
321.23/291.48	   half(s(0)) -> 0
321.23/291.48	   half(s(s(x))) -> s(half(x))
321.23/291.48	   lastbit(0) -> 0
321.23/291.48	   lastbit(s(0)) -> s(0)
321.23/291.48	   lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	   conv(0) -> cons(nil, 0)
321.23/291.48	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	S is empty.
321.23/291.48	Rewrite Strategy: FULL
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(1) RenamingProof (BOTH BOUNDS(ID, ID))
321.23/291.48	Renamed function symbols to avoid clashes with predefined symbol.
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(2)
321.23/291.48	Obligation:
321.23/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
321.23/291.48	
321.23/291.48	
321.23/291.48	The TRS R consists of the following rules:
321.23/291.48	
321.23/291.48	   half(0') -> 0'
321.23/291.48	   half(s(0')) -> 0'
321.23/291.48	   half(s(s(x))) -> s(half(x))
321.23/291.48	   lastbit(0') -> 0'
321.23/291.48	   lastbit(s(0')) -> s(0')
321.23/291.48	   lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	   conv(0') -> cons(nil, 0')
321.23/291.48	   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	S is empty.
321.23/291.48	Rewrite Strategy: FULL
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
321.23/291.48	Infered types.
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(4)
321.23/291.48	Obligation:
321.23/291.48	TRS:
321.23/291.48	Rules:
321.23/291.48	half(0') -> 0'
321.23/291.48	half(s(0')) -> 0'
321.23/291.48	half(s(s(x))) -> s(half(x))
321.23/291.48	lastbit(0') -> 0'
321.23/291.48	lastbit(s(0')) -> s(0')
321.23/291.48	lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	conv(0') -> cons(nil, 0')
321.23/291.48	conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	Types:
321.23/291.48	half :: 0':s -> 0':s
321.23/291.48	0' :: 0':s
321.23/291.48	s :: 0':s -> 0':s
321.23/291.48	lastbit :: 0':s -> 0':s
321.23/291.48	conv :: 0':s -> nil:cons
321.23/291.48	cons :: nil:cons -> 0':s -> nil:cons
321.23/291.48	nil :: nil:cons
321.23/291.48	hole_0':s1_0 :: 0':s
321.23/291.48	hole_nil:cons2_0 :: nil:cons
321.23/291.48	gen_0':s3_0 :: Nat -> 0':s
321.23/291.48	gen_nil:cons4_0 :: Nat -> nil:cons
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(5) OrderProof (LOWER BOUND(ID))
321.23/291.48	Heuristically decided to analyse the following defined symbols:
321.23/291.48	half, lastbit, conv
321.23/291.48	
321.23/291.48	They will be analysed ascendingly in the following order:
321.23/291.48	half < conv
321.23/291.48	lastbit < conv
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(6)
321.23/291.48	Obligation:
321.23/291.48	TRS:
321.23/291.48	Rules:
321.23/291.48	half(0') -> 0'
321.23/291.48	half(s(0')) -> 0'
321.23/291.48	half(s(s(x))) -> s(half(x))
321.23/291.48	lastbit(0') -> 0'
321.23/291.48	lastbit(s(0')) -> s(0')
321.23/291.48	lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	conv(0') -> cons(nil, 0')
321.23/291.48	conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	Types:
321.23/291.48	half :: 0':s -> 0':s
321.23/291.48	0' :: 0':s
321.23/291.48	s :: 0':s -> 0':s
321.23/291.48	lastbit :: 0':s -> 0':s
321.23/291.48	conv :: 0':s -> nil:cons
321.23/291.48	cons :: nil:cons -> 0':s -> nil:cons
321.23/291.48	nil :: nil:cons
321.23/291.48	hole_0':s1_0 :: 0':s
321.23/291.48	hole_nil:cons2_0 :: nil:cons
321.23/291.48	gen_0':s3_0 :: Nat -> 0':s
321.23/291.48	gen_nil:cons4_0 :: Nat -> nil:cons
321.23/291.48	
321.23/291.48	
321.23/291.48	Generator Equations:
321.23/291.48	gen_0':s3_0(0) <=> 0'
321.23/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.23/291.48	gen_nil:cons4_0(0) <=> nil
321.23/291.48	gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')
321.23/291.48	
321.23/291.48	
321.23/291.48	The following defined symbols remain to be analysed:
321.23/291.48	half, lastbit, conv
321.23/291.48	
321.23/291.48	They will be analysed ascendingly in the following order:
321.23/291.48	half < conv
321.23/291.48	lastbit < conv
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(7) RewriteLemmaProof (LOWER BOUND(ID))
321.23/291.48	Proved the following rewrite lemma:
321.23/291.48	half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)
321.23/291.48	
321.23/291.48	Induction Base:
321.23/291.48	half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
321.23/291.48	0'
321.23/291.48	
321.23/291.48	Induction Step:
321.23/291.48	half(gen_0':s3_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
321.23/291.48	s(half(gen_0':s3_0(*(2, n6_0)))) ->_IH
321.23/291.48	s(gen_0':s3_0(c7_0))
321.23/291.48	
321.23/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(8)
321.23/291.48	Complex Obligation (BEST)
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(9)
321.23/291.48	Obligation:
321.23/291.48	Proved the lower bound n^1 for the following obligation:
321.23/291.48	
321.23/291.48	TRS:
321.23/291.48	Rules:
321.23/291.48	half(0') -> 0'
321.23/291.48	half(s(0')) -> 0'
321.23/291.48	half(s(s(x))) -> s(half(x))
321.23/291.48	lastbit(0') -> 0'
321.23/291.48	lastbit(s(0')) -> s(0')
321.23/291.48	lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	conv(0') -> cons(nil, 0')
321.23/291.48	conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	Types:
321.23/291.48	half :: 0':s -> 0':s
321.23/291.48	0' :: 0':s
321.23/291.48	s :: 0':s -> 0':s
321.23/291.48	lastbit :: 0':s -> 0':s
321.23/291.48	conv :: 0':s -> nil:cons
321.23/291.48	cons :: nil:cons -> 0':s -> nil:cons
321.23/291.48	nil :: nil:cons
321.23/291.48	hole_0':s1_0 :: 0':s
321.23/291.48	hole_nil:cons2_0 :: nil:cons
321.23/291.48	gen_0':s3_0 :: Nat -> 0':s
321.23/291.48	gen_nil:cons4_0 :: Nat -> nil:cons
321.23/291.48	
321.23/291.48	
321.23/291.48	Generator Equations:
321.23/291.48	gen_0':s3_0(0) <=> 0'
321.23/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.23/291.48	gen_nil:cons4_0(0) <=> nil
321.23/291.48	gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')
321.23/291.48	
321.23/291.48	
321.23/291.48	The following defined symbols remain to be analysed:
321.23/291.48	half, lastbit, conv
321.23/291.48	
321.23/291.48	They will be analysed ascendingly in the following order:
321.23/291.48	half < conv
321.23/291.48	lastbit < conv
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(10) LowerBoundPropagationProof (FINISHED)
321.23/291.48	Propagated lower bound.
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(11)
321.23/291.48	BOUNDS(n^1, INF)
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(12)
321.23/291.48	Obligation:
321.23/291.48	TRS:
321.23/291.48	Rules:
321.23/291.48	half(0') -> 0'
321.23/291.48	half(s(0')) -> 0'
321.23/291.48	half(s(s(x))) -> s(half(x))
321.23/291.48	lastbit(0') -> 0'
321.23/291.48	lastbit(s(0')) -> s(0')
321.23/291.48	lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	conv(0') -> cons(nil, 0')
321.23/291.48	conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	Types:
321.23/291.48	half :: 0':s -> 0':s
321.23/291.48	0' :: 0':s
321.23/291.48	s :: 0':s -> 0':s
321.23/291.48	lastbit :: 0':s -> 0':s
321.23/291.48	conv :: 0':s -> nil:cons
321.23/291.48	cons :: nil:cons -> 0':s -> nil:cons
321.23/291.48	nil :: nil:cons
321.23/291.48	hole_0':s1_0 :: 0':s
321.23/291.48	hole_nil:cons2_0 :: nil:cons
321.23/291.48	gen_0':s3_0 :: Nat -> 0':s
321.23/291.48	gen_nil:cons4_0 :: Nat -> nil:cons
321.23/291.48	
321.23/291.48	
321.23/291.48	Lemmas:
321.23/291.48	half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)
321.23/291.48	
321.23/291.48	
321.23/291.48	Generator Equations:
321.23/291.48	gen_0':s3_0(0) <=> 0'
321.23/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.23/291.48	gen_nil:cons4_0(0) <=> nil
321.23/291.48	gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')
321.23/291.48	
321.23/291.48	
321.23/291.48	The following defined symbols remain to be analysed:
321.23/291.48	lastbit, conv
321.23/291.48	
321.23/291.48	They will be analysed ascendingly in the following order:
321.23/291.48	lastbit < conv
321.23/291.48	
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(13) RewriteLemmaProof (LOWER BOUND(ID))
321.23/291.48	Proved the following rewrite lemma:
321.23/291.48	lastbit(gen_0':s3_0(*(2, n300_0))) -> gen_0':s3_0(0), rt in Omega(1 + n300_0)
321.23/291.48	
321.23/291.48	Induction Base:
321.23/291.48	lastbit(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
321.23/291.48	0'
321.23/291.48	
321.23/291.48	Induction Step:
321.23/291.48	lastbit(gen_0':s3_0(*(2, +(n300_0, 1)))) ->_R^Omega(1)
321.23/291.48	lastbit(gen_0':s3_0(*(2, n300_0))) ->_IH
321.23/291.48	gen_0':s3_0(0)
321.23/291.48	
321.23/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
321.23/291.48	----------------------------------------
321.23/291.48	
321.23/291.48	(14)
321.23/291.48	Obligation:
321.23/291.48	TRS:
321.23/291.48	Rules:
321.23/291.48	half(0') -> 0'
321.23/291.48	half(s(0')) -> 0'
321.23/291.48	half(s(s(x))) -> s(half(x))
321.23/291.48	lastbit(0') -> 0'
321.23/291.48	lastbit(s(0')) -> s(0')
321.23/291.48	lastbit(s(s(x))) -> lastbit(x)
321.23/291.48	conv(0') -> cons(nil, 0')
321.23/291.48	conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))
321.23/291.48	
321.23/291.48	Types:
321.23/291.48	half :: 0':s -> 0':s
321.23/291.48	0' :: 0':s
321.23/291.48	s :: 0':s -> 0':s
321.23/291.48	lastbit :: 0':s -> 0':s
321.23/291.48	conv :: 0':s -> nil:cons
321.23/291.48	cons :: nil:cons -> 0':s -> nil:cons
321.23/291.48	nil :: nil:cons
321.23/291.48	hole_0':s1_0 :: 0':s
321.23/291.48	hole_nil:cons2_0 :: nil:cons
321.23/291.48	gen_0':s3_0 :: Nat -> 0':s
321.23/291.48	gen_nil:cons4_0 :: Nat -> nil:cons
321.23/291.48	
321.23/291.48	
321.23/291.48	Lemmas:
321.23/291.48	half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)
321.23/291.48	lastbit(gen_0':s3_0(*(2, n300_0))) -> gen_0':s3_0(0), rt in Omega(1 + n300_0)
321.23/291.48	
321.23/291.48	
321.23/291.48	Generator Equations:
321.23/291.48	gen_0':s3_0(0) <=> 0'
321.23/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.23/291.48	gen_nil:cons4_0(0) <=> nil
321.23/291.48	gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')
321.23/291.48	
321.23/291.48	
321.23/291.48	The following defined symbols remain to be analysed:
321.23/291.48	conv
321.23/291.51	EOF