887.61/291.54	WORST_CASE(Omega(n^1), ?)
887.61/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
887.61/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
887.61/291.56	
887.61/291.56	
887.61/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
887.61/291.56	
887.61/291.56	(0) CpxTRS
887.61/291.56	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
887.61/291.56	(2) CpxTRS
887.61/291.56	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
887.61/291.56	(4) typed CpxTrs
887.61/291.56	(5) OrderProof [LOWER BOUND(ID), 0 ms]
887.61/291.56	(6) typed CpxTrs
887.61/291.56	(7) RewriteLemmaProof [LOWER BOUND(ID), 333 ms]
887.61/291.56	(8) BEST
887.61/291.56	    (9) proven lower bound
887.61/291.56	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
887.61/291.56	        (11) BOUNDS(n^1, INF)
887.61/291.56	    (12) typed CpxTrs
887.61/291.56	
887.61/291.56	
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(0)
887.61/291.56	Obligation:
887.61/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
887.61/291.56	
887.61/291.56	
887.61/291.56	The TRS R consists of the following rules:
887.61/291.56	
887.61/291.56	   g(x, 0) -> 0
887.61/291.56	   g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	   g(h, s(0)) -> 0
887.61/291.56	   g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	   double(x) -> g(d, x)
887.61/291.56	   half(x) -> g(h, x)
887.61/291.56	   f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	   f(s(0), y) -> y
887.61/291.56	   id(x) -> f(x, s(0))
887.61/291.56	
887.61/291.56	S is empty.
887.61/291.56	Rewrite Strategy: INNERMOST
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(1) RenamingProof (BOTH BOUNDS(ID, ID))
887.61/291.56	Renamed function symbols to avoid clashes with predefined symbol.
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(2)
887.61/291.56	Obligation:
887.61/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
887.61/291.56	
887.61/291.56	
887.61/291.56	The TRS R consists of the following rules:
887.61/291.56	
887.61/291.56	   g(x, 0') -> 0'
887.61/291.56	   g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	   g(h, s(0')) -> 0'
887.61/291.56	   g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	   double(x) -> g(d, x)
887.61/291.56	   half(x) -> g(h, x)
887.61/291.56	   f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	   f(s(0'), y) -> y
887.61/291.56	   id(x) -> f(x, s(0'))
887.61/291.56	
887.61/291.56	S is empty.
887.61/291.56	Rewrite Strategy: INNERMOST
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
887.61/291.56	Infered types.
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(4)
887.61/291.56	Obligation:
887.61/291.56	Innermost TRS:
887.61/291.56	Rules:
887.61/291.56	g(x, 0') -> 0'
887.61/291.56	g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	g(h, s(0')) -> 0'
887.61/291.56	g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	double(x) -> g(d, x)
887.61/291.56	half(x) -> g(h, x)
887.61/291.56	f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	f(s(0'), y) -> y
887.61/291.56	id(x) -> f(x, s(0'))
887.61/291.56	
887.61/291.56	Types:
887.61/291.56	g :: d:h -> 0':s -> 0':s
887.61/291.56	0' :: 0':s
887.61/291.56	d :: d:h
887.61/291.56	s :: 0':s -> 0':s
887.61/291.56	h :: d:h
887.61/291.56	double :: 0':s -> 0':s
887.61/291.56	half :: 0':s -> 0':s
887.61/291.56	f :: 0':s -> 0':s -> 0':s
887.61/291.56	id :: 0':s -> 0':s
887.61/291.56	hole_0':s1_0 :: 0':s
887.61/291.56	hole_d:h2_0 :: d:h
887.61/291.56	gen_0':s3_0 :: Nat -> 0':s
887.61/291.56	
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(5) OrderProof (LOWER BOUND(ID))
887.61/291.56	Heuristically decided to analyse the following defined symbols:
887.61/291.56	g, f
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(6)
887.61/291.56	Obligation:
887.61/291.56	Innermost TRS:
887.61/291.56	Rules:
887.61/291.56	g(x, 0') -> 0'
887.61/291.56	g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	g(h, s(0')) -> 0'
887.61/291.56	g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	double(x) -> g(d, x)
887.61/291.56	half(x) -> g(h, x)
887.61/291.56	f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	f(s(0'), y) -> y
887.61/291.56	id(x) -> f(x, s(0'))
887.61/291.56	
887.61/291.56	Types:
887.61/291.56	g :: d:h -> 0':s -> 0':s
887.61/291.56	0' :: 0':s
887.61/291.56	d :: d:h
887.61/291.56	s :: 0':s -> 0':s
887.61/291.56	h :: d:h
887.61/291.56	double :: 0':s -> 0':s
887.61/291.56	half :: 0':s -> 0':s
887.61/291.56	f :: 0':s -> 0':s -> 0':s
887.61/291.56	id :: 0':s -> 0':s
887.61/291.56	hole_0':s1_0 :: 0':s
887.61/291.56	hole_d:h2_0 :: d:h
887.61/291.56	gen_0':s3_0 :: Nat -> 0':s
887.61/291.56	
887.61/291.56	
887.61/291.56	Generator Equations:
887.61/291.56	gen_0':s3_0(0) <=> 0'
887.61/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
887.61/291.56	
887.61/291.56	
887.61/291.56	The following defined symbols remain to be analysed:
887.61/291.56	g, f
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(7) RewriteLemmaProof (LOWER BOUND(ID))
887.61/291.56	Proved the following rewrite lemma:
887.61/291.56	g(h, gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
887.61/291.56	
887.61/291.56	Induction Base:
887.61/291.56	g(h, gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
887.61/291.56	0'
887.61/291.56	
887.61/291.56	Induction Step:
887.61/291.56	g(h, gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
887.61/291.56	s(g(h, gen_0':s3_0(*(2, n5_0)))) ->_IH
887.61/291.56	s(gen_0':s3_0(c6_0))
887.61/291.56	
887.61/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(8)
887.61/291.56	Complex Obligation (BEST)
887.61/291.56	
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(9)
887.61/291.56	Obligation:
887.61/291.56	Proved the lower bound n^1 for the following obligation:
887.61/291.56	
887.61/291.56	Innermost TRS:
887.61/291.56	Rules:
887.61/291.56	g(x, 0') -> 0'
887.61/291.56	g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	g(h, s(0')) -> 0'
887.61/291.56	g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	double(x) -> g(d, x)
887.61/291.56	half(x) -> g(h, x)
887.61/291.56	f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	f(s(0'), y) -> y
887.61/291.56	id(x) -> f(x, s(0'))
887.61/291.56	
887.61/291.56	Types:
887.61/291.56	g :: d:h -> 0':s -> 0':s
887.61/291.56	0' :: 0':s
887.61/291.56	d :: d:h
887.61/291.56	s :: 0':s -> 0':s
887.61/291.56	h :: d:h
887.61/291.56	double :: 0':s -> 0':s
887.61/291.56	half :: 0':s -> 0':s
887.61/291.56	f :: 0':s -> 0':s -> 0':s
887.61/291.56	id :: 0':s -> 0':s
887.61/291.56	hole_0':s1_0 :: 0':s
887.61/291.56	hole_d:h2_0 :: d:h
887.61/291.56	gen_0':s3_0 :: Nat -> 0':s
887.61/291.56	
887.61/291.56	
887.61/291.56	Generator Equations:
887.61/291.56	gen_0':s3_0(0) <=> 0'
887.61/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
887.61/291.56	
887.61/291.56	
887.61/291.56	The following defined symbols remain to be analysed:
887.61/291.56	g, f
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(10) LowerBoundPropagationProof (FINISHED)
887.61/291.56	Propagated lower bound.
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(11)
887.61/291.56	BOUNDS(n^1, INF)
887.61/291.56	
887.61/291.56	----------------------------------------
887.61/291.56	
887.61/291.56	(12)
887.61/291.56	Obligation:
887.61/291.56	Innermost TRS:
887.61/291.56	Rules:
887.61/291.56	g(x, 0') -> 0'
887.61/291.56	g(d, s(x)) -> s(s(g(d, x)))
887.61/291.56	g(h, s(0')) -> 0'
887.61/291.56	g(h, s(s(x))) -> s(g(h, x))
887.61/291.56	double(x) -> g(d, x)
887.61/291.56	half(x) -> g(h, x)
887.61/291.56	f(s(x), y) -> f(half(s(x)), double(y))
887.61/291.56	f(s(0'), y) -> y
887.61/291.56	id(x) -> f(x, s(0'))
887.61/291.56	
887.61/291.56	Types:
887.61/291.56	g :: d:h -> 0':s -> 0':s
887.61/291.56	0' :: 0':s
887.61/291.56	d :: d:h
887.61/291.56	s :: 0':s -> 0':s
887.61/291.56	h :: d:h
887.61/291.56	double :: 0':s -> 0':s
887.61/291.56	half :: 0':s -> 0':s
887.61/291.56	f :: 0':s -> 0':s -> 0':s
887.61/291.56	id :: 0':s -> 0':s
887.61/291.56	hole_0':s1_0 :: 0':s
887.61/291.56	hole_d:h2_0 :: d:h
887.61/291.56	gen_0':s3_0 :: Nat -> 0':s
887.61/291.56	
887.61/291.56	
887.61/291.56	Lemmas:
887.61/291.56	g(h, gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
887.61/291.56	
887.61/291.56	
887.61/291.56	Generator Equations:
887.61/291.56	gen_0':s3_0(0) <=> 0'
887.61/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
887.61/291.56	
887.61/291.56	
887.61/291.56	The following defined symbols remain to be analysed:
887.61/291.56	f
887.91/291.61	EOF