302.72/291.48	WORST_CASE(Omega(n^1), ?)
302.72/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
302.72/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
302.72/291.49	
302.72/291.49	
302.72/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.72/291.49	
302.72/291.49	(0) CpxTRS
302.72/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
302.72/291.49	(2) CpxTRS
302.72/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
302.72/291.49	(4) typed CpxTrs
302.72/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
302.72/291.49	(6) typed CpxTrs
302.72/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
302.72/291.49	(8) BEST
302.72/291.49	    (9) proven lower bound
302.72/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
302.72/291.49	        (11) BOUNDS(n^1, INF)
302.72/291.49	    (12) typed CpxTrs
302.72/291.49	
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(0)
302.72/291.49	Obligation:
302.72/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.72/291.49	
302.72/291.49	
302.72/291.49	The TRS R consists of the following rules:
302.72/291.49	
302.72/291.49	   half(x) -> if(ge(x, s(s(0))), x)
302.72/291.49	   if(false, x) -> 0
302.72/291.49	   if(true, x) -> s(half(p(p(x))))
302.72/291.49	   p(0) -> 0
302.72/291.49	   p(s(x)) -> x
302.72/291.49	   ge(x, 0) -> true
302.72/291.49	   ge(0, s(x)) -> false
302.72/291.49	   ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	   log(0) -> 0
302.72/291.49	   log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	S is empty.
302.72/291.49	Rewrite Strategy: FULL
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
302.72/291.49	Renamed function symbols to avoid clashes with predefined symbol.
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(2)
302.72/291.49	Obligation:
302.72/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.72/291.49	
302.72/291.49	
302.72/291.49	The TRS R consists of the following rules:
302.72/291.49	
302.72/291.49	   half(x) -> if(ge(x, s(s(0'))), x)
302.72/291.49	   if(false, x) -> 0'
302.72/291.49	   if(true, x) -> s(half(p(p(x))))
302.72/291.49	   p(0') -> 0'
302.72/291.49	   p(s(x)) -> x
302.72/291.49	   ge(x, 0') -> true
302.72/291.49	   ge(0', s(x)) -> false
302.72/291.49	   ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	   log(0') -> 0'
302.72/291.49	   log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	S is empty.
302.72/291.49	Rewrite Strategy: FULL
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
302.72/291.49	Infered types.
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(4)
302.72/291.49	Obligation:
302.72/291.49	TRS:
302.72/291.49	Rules:
302.72/291.49	half(x) -> if(ge(x, s(s(0'))), x)
302.72/291.49	if(false, x) -> 0'
302.72/291.49	if(true, x) -> s(half(p(p(x))))
302.72/291.49	p(0') -> 0'
302.72/291.49	p(s(x)) -> x
302.72/291.49	ge(x, 0') -> true
302.72/291.49	ge(0', s(x)) -> false
302.72/291.49	ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	log(0') -> 0'
302.72/291.49	log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	Types:
302.72/291.49	half :: 0':s -> 0':s
302.72/291.49	if :: false:true -> 0':s -> 0':s
302.72/291.49	ge :: 0':s -> 0':s -> false:true
302.72/291.49	s :: 0':s -> 0':s
302.72/291.49	0' :: 0':s
302.72/291.49	false :: false:true
302.72/291.49	true :: false:true
302.72/291.49	p :: 0':s -> 0':s
302.72/291.49	log :: 0':s -> 0':s
302.72/291.49	hole_0':s1_0 :: 0':s
302.72/291.49	hole_false:true2_0 :: false:true
302.72/291.49	gen_0':s3_0 :: Nat -> 0':s
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(5) OrderProof (LOWER BOUND(ID))
302.72/291.49	Heuristically decided to analyse the following defined symbols:
302.72/291.49	half, ge, log
302.72/291.49	
302.72/291.49	They will be analysed ascendingly in the following order:
302.72/291.49	ge < half
302.72/291.49	half < log
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(6)
302.72/291.49	Obligation:
302.72/291.49	TRS:
302.72/291.49	Rules:
302.72/291.49	half(x) -> if(ge(x, s(s(0'))), x)
302.72/291.49	if(false, x) -> 0'
302.72/291.49	if(true, x) -> s(half(p(p(x))))
302.72/291.49	p(0') -> 0'
302.72/291.49	p(s(x)) -> x
302.72/291.49	ge(x, 0') -> true
302.72/291.49	ge(0', s(x)) -> false
302.72/291.49	ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	log(0') -> 0'
302.72/291.49	log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	Types:
302.72/291.49	half :: 0':s -> 0':s
302.72/291.49	if :: false:true -> 0':s -> 0':s
302.72/291.49	ge :: 0':s -> 0':s -> false:true
302.72/291.49	s :: 0':s -> 0':s
302.72/291.49	0' :: 0':s
302.72/291.49	false :: false:true
302.72/291.49	true :: false:true
302.72/291.49	p :: 0':s -> 0':s
302.72/291.49	log :: 0':s -> 0':s
302.72/291.49	hole_0':s1_0 :: 0':s
302.72/291.49	hole_false:true2_0 :: false:true
302.72/291.49	gen_0':s3_0 :: Nat -> 0':s
302.72/291.49	
302.72/291.49	
302.72/291.49	Generator Equations:
302.72/291.49	gen_0':s3_0(0) <=> 0'
302.72/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.72/291.49	
302.72/291.49	
302.72/291.49	The following defined symbols remain to be analysed:
302.72/291.49	ge, half, log
302.72/291.49	
302.72/291.49	They will be analysed ascendingly in the following order:
302.72/291.49	ge < half
302.72/291.49	half < log
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
302.72/291.49	Proved the following rewrite lemma:
302.72/291.49	ge(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
302.72/291.49	
302.72/291.49	Induction Base:
302.72/291.49	ge(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
302.72/291.49	true
302.72/291.49	
302.72/291.49	Induction Step:
302.72/291.49	ge(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
302.72/291.49	ge(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
302.72/291.49	true
302.72/291.49	
302.72/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(8)
302.72/291.49	Complex Obligation (BEST)
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(9)
302.72/291.49	Obligation:
302.72/291.49	Proved the lower bound n^1 for the following obligation:
302.72/291.49	
302.72/291.49	TRS:
302.72/291.49	Rules:
302.72/291.49	half(x) -> if(ge(x, s(s(0'))), x)
302.72/291.49	if(false, x) -> 0'
302.72/291.49	if(true, x) -> s(half(p(p(x))))
302.72/291.49	p(0') -> 0'
302.72/291.49	p(s(x)) -> x
302.72/291.49	ge(x, 0') -> true
302.72/291.49	ge(0', s(x)) -> false
302.72/291.49	ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	log(0') -> 0'
302.72/291.49	log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	Types:
302.72/291.49	half :: 0':s -> 0':s
302.72/291.49	if :: false:true -> 0':s -> 0':s
302.72/291.49	ge :: 0':s -> 0':s -> false:true
302.72/291.49	s :: 0':s -> 0':s
302.72/291.49	0' :: 0':s
302.72/291.49	false :: false:true
302.72/291.49	true :: false:true
302.72/291.49	p :: 0':s -> 0':s
302.72/291.49	log :: 0':s -> 0':s
302.72/291.49	hole_0':s1_0 :: 0':s
302.72/291.49	hole_false:true2_0 :: false:true
302.72/291.49	gen_0':s3_0 :: Nat -> 0':s
302.72/291.49	
302.72/291.49	
302.72/291.49	Generator Equations:
302.72/291.49	gen_0':s3_0(0) <=> 0'
302.72/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.72/291.49	
302.72/291.49	
302.72/291.49	The following defined symbols remain to be analysed:
302.72/291.49	ge, half, log
302.72/291.49	
302.72/291.49	They will be analysed ascendingly in the following order:
302.72/291.49	ge < half
302.72/291.49	half < log
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(10) LowerBoundPropagationProof (FINISHED)
302.72/291.49	Propagated lower bound.
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(11)
302.72/291.49	BOUNDS(n^1, INF)
302.72/291.49	
302.72/291.49	----------------------------------------
302.72/291.49	
302.72/291.49	(12)
302.72/291.49	Obligation:
302.72/291.49	TRS:
302.72/291.49	Rules:
302.72/291.49	half(x) -> if(ge(x, s(s(0'))), x)
302.72/291.49	if(false, x) -> 0'
302.72/291.49	if(true, x) -> s(half(p(p(x))))
302.72/291.49	p(0') -> 0'
302.72/291.49	p(s(x)) -> x
302.72/291.49	ge(x, 0') -> true
302.72/291.49	ge(0', s(x)) -> false
302.72/291.49	ge(s(x), s(y)) -> ge(x, y)
302.72/291.49	log(0') -> 0'
302.72/291.49	log(s(x)) -> s(log(half(s(x))))
302.72/291.49	
302.72/291.49	Types:
302.72/291.49	half :: 0':s -> 0':s
302.72/291.49	if :: false:true -> 0':s -> 0':s
302.72/291.49	ge :: 0':s -> 0':s -> false:true
302.72/291.49	s :: 0':s -> 0':s
302.72/291.49	0' :: 0':s
302.72/291.49	false :: false:true
302.72/291.49	true :: false:true
302.72/291.49	p :: 0':s -> 0':s
302.72/291.49	log :: 0':s -> 0':s
302.72/291.49	hole_0':s1_0 :: 0':s
302.72/291.49	hole_false:true2_0 :: false:true
302.72/291.49	gen_0':s3_0 :: Nat -> 0':s
302.72/291.49	
302.72/291.49	
302.72/291.49	Lemmas:
302.72/291.49	ge(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
302.72/291.49	
302.72/291.49	
302.72/291.49	Generator Equations:
302.72/291.49	gen_0':s3_0(0) <=> 0'
302.72/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
302.72/291.49	
302.72/291.49	
302.72/291.49	The following defined symbols remain to be analysed:
302.72/291.49	half, log
302.72/291.49	
302.72/291.49	They will be analysed ascendingly in the following order:
302.72/291.49	half < log
302.72/291.52	EOF