315.81/291.51	WORST_CASE(Omega(n^1), ?)
315.81/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
315.81/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
315.81/291.52	
315.81/291.52	
315.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.81/291.52	
315.81/291.52	(0) CpxTRS
315.81/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
315.81/291.52	(2) CpxTRS
315.81/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
315.81/291.52	(4) typed CpxTrs
315.81/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
315.81/291.52	(6) typed CpxTrs
315.81/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 218 ms]
315.81/291.52	(8) BEST
315.81/291.52	    (9) proven lower bound
315.81/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
315.81/291.52	        (11) BOUNDS(n^1, INF)
315.81/291.52	    (12) typed CpxTrs
315.81/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 57 ms]
315.81/291.52	        (14) typed CpxTrs
315.81/291.52	        (15) RewriteLemmaProof [LOWER BOUND(ID), 14 ms]
315.81/291.52	        (16) typed CpxTrs
315.81/291.52	
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(0)
315.81/291.52	Obligation:
315.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.81/291.52	
315.81/291.52	
315.81/291.52	The TRS R consists of the following rules:
315.81/291.52	
315.81/291.52	   even(0) -> true
315.81/291.52	   even(s(0)) -> false
315.81/291.52	   even(s(s(x))) -> even(x)
315.81/291.52	   half(0) -> 0
315.81/291.52	   half(s(s(x))) -> s(half(x))
315.81/291.52	   plus(0, y) -> y
315.81/291.52	   plus(s(x), y) -> s(plus(x, y))
315.81/291.52	   times(0, y) -> 0
315.81/291.52	   times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	   if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	S is empty.
315.81/291.52	Rewrite Strategy: FULL
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
315.81/291.52	Renamed function symbols to avoid clashes with predefined symbol.
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(2)
315.81/291.52	Obligation:
315.81/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.81/291.52	
315.81/291.52	
315.81/291.52	The TRS R consists of the following rules:
315.81/291.52	
315.81/291.52	   even(0') -> true
315.81/291.52	   even(s(0')) -> false
315.81/291.52	   even(s(s(x))) -> even(x)
315.81/291.52	   half(0') -> 0'
315.81/291.52	   half(s(s(x))) -> s(half(x))
315.81/291.52	   plus(0', y) -> y
315.81/291.52	   plus(s(x), y) -> s(plus(x, y))
315.81/291.52	   times(0', y) -> 0'
315.81/291.52	   times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	   if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	S is empty.
315.81/291.52	Rewrite Strategy: FULL
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
315.81/291.52	Infered types.
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(4)
315.81/291.52	Obligation:
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(5) OrderProof (LOWER BOUND(ID))
315.81/291.52	Heuristically decided to analyse the following defined symbols:
315.81/291.52	even, half, plus, times
315.81/291.52	
315.81/291.52	They will be analysed ascendingly in the following order:
315.81/291.52	even < times
315.81/291.52	half < times
315.81/291.52	plus < times
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(6)
315.81/291.52	Obligation:
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	
315.81/291.52	Generator Equations:
315.81/291.52	gen_0':s3_0(0) <=> 0'
315.81/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
315.81/291.52	
315.81/291.52	
315.81/291.52	The following defined symbols remain to be analysed:
315.81/291.52	even, half, plus, times
315.81/291.52	
315.81/291.52	They will be analysed ascendingly in the following order:
315.81/291.52	even < times
315.81/291.52	half < times
315.81/291.52	plus < times
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
315.81/291.52	Proved the following rewrite lemma:
315.81/291.52	even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
315.81/291.52	
315.81/291.52	Induction Base:
315.81/291.52	even(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
315.81/291.52	true
315.81/291.52	
315.81/291.52	Induction Step:
315.81/291.52	even(gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
315.81/291.52	even(gen_0':s3_0(*(2, n5_0))) ->_IH
315.81/291.52	true
315.81/291.52	
315.81/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(8)
315.81/291.52	Complex Obligation (BEST)
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(9)
315.81/291.52	Obligation:
315.81/291.52	Proved the lower bound n^1 for the following obligation:
315.81/291.52	
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	
315.81/291.52	Generator Equations:
315.81/291.52	gen_0':s3_0(0) <=> 0'
315.81/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
315.81/291.52	
315.81/291.52	
315.81/291.52	The following defined symbols remain to be analysed:
315.81/291.52	even, half, plus, times
315.81/291.52	
315.81/291.52	They will be analysed ascendingly in the following order:
315.81/291.52	even < times
315.81/291.52	half < times
315.81/291.52	plus < times
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(10) LowerBoundPropagationProof (FINISHED)
315.81/291.52	Propagated lower bound.
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(11)
315.81/291.52	BOUNDS(n^1, INF)
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(12)
315.81/291.52	Obligation:
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	
315.81/291.52	Lemmas:
315.81/291.52	even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
315.81/291.52	
315.81/291.52	
315.81/291.52	Generator Equations:
315.81/291.52	gen_0':s3_0(0) <=> 0'
315.81/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
315.81/291.52	
315.81/291.52	
315.81/291.52	The following defined symbols remain to be analysed:
315.81/291.52	half, plus, times
315.81/291.52	
315.81/291.52	They will be analysed ascendingly in the following order:
315.81/291.52	half < times
315.81/291.52	plus < times
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
315.81/291.52	Proved the following rewrite lemma:
315.81/291.52	half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)
315.81/291.52	
315.81/291.52	Induction Base:
315.81/291.52	half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
315.81/291.52	0'
315.81/291.52	
315.81/291.52	Induction Step:
315.81/291.52	half(gen_0':s3_0(*(2, +(n149_0, 1)))) ->_R^Omega(1)
315.81/291.52	s(half(gen_0':s3_0(*(2, n149_0)))) ->_IH
315.81/291.52	s(gen_0':s3_0(c150_0))
315.81/291.52	
315.81/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(14)
315.81/291.52	Obligation:
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	
315.81/291.52	Lemmas:
315.81/291.52	even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
315.81/291.52	half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)
315.81/291.52	
315.81/291.52	
315.81/291.52	Generator Equations:
315.81/291.52	gen_0':s3_0(0) <=> 0'
315.81/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
315.81/291.52	
315.81/291.52	
315.81/291.52	The following defined symbols remain to be analysed:
315.81/291.52	plus, times
315.81/291.52	
315.81/291.52	They will be analysed ascendingly in the following order:
315.81/291.52	plus < times
315.81/291.52	
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(15) RewriteLemmaProof (LOWER BOUND(ID))
315.81/291.52	Proved the following rewrite lemma:
315.81/291.52	plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n355_0, b)), rt in Omega(1 + n355_0)
315.81/291.52	
315.81/291.52	Induction Base:
315.81/291.52	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
315.81/291.52	gen_0':s3_0(b)
315.81/291.52	
315.81/291.52	Induction Step:
315.81/291.52	plus(gen_0':s3_0(+(n355_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
315.81/291.52	s(plus(gen_0':s3_0(n355_0), gen_0':s3_0(b))) ->_IH
315.81/291.52	s(gen_0':s3_0(+(b, c356_0)))
315.81/291.52	
315.81/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
315.81/291.52	----------------------------------------
315.81/291.52	
315.81/291.52	(16)
315.81/291.52	Obligation:
315.81/291.52	TRS:
315.81/291.52	Rules:
315.81/291.52	even(0') -> true
315.81/291.52	even(s(0')) -> false
315.81/291.52	even(s(s(x))) -> even(x)
315.81/291.52	half(0') -> 0'
315.81/291.52	half(s(s(x))) -> s(half(x))
315.81/291.52	plus(0', y) -> y
315.81/291.52	plus(s(x), y) -> s(plus(x, y))
315.81/291.52	times(0', y) -> 0'
315.81/291.52	times(s(x), y) -> if_times(even(s(x)), s(x), y)
315.81/291.52	if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
315.81/291.52	if_times(false, s(x), y) -> plus(y, times(x, y))
315.81/291.52	
315.81/291.52	Types:
315.81/291.52	even :: 0':s -> true:false
315.81/291.52	0' :: 0':s
315.81/291.52	true :: true:false
315.81/291.52	s :: 0':s -> 0':s
315.81/291.52	false :: true:false
315.81/291.52	half :: 0':s -> 0':s
315.81/291.52	plus :: 0':s -> 0':s -> 0':s
315.81/291.52	times :: 0':s -> 0':s -> 0':s
315.81/291.52	if_times :: true:false -> 0':s -> 0':s -> 0':s
315.81/291.52	hole_true:false1_0 :: true:false
315.81/291.52	hole_0':s2_0 :: 0':s
315.81/291.52	gen_0':s3_0 :: Nat -> 0':s
315.81/291.52	
315.81/291.52	
315.81/291.52	Lemmas:
315.81/291.52	even(gen_0':s3_0(*(2, n5_0))) -> true, rt in Omega(1 + n5_0)
315.81/291.52	half(gen_0':s3_0(*(2, n149_0))) -> gen_0':s3_0(n149_0), rt in Omega(1 + n149_0)
315.81/291.52	plus(gen_0':s3_0(n355_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n355_0, b)), rt in Omega(1 + n355_0)
315.81/291.52	
315.81/291.52	
315.81/291.52	Generator Equations:
315.81/291.52	gen_0':s3_0(0) <=> 0'
315.81/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
315.81/291.52	
315.81/291.52	
315.81/291.52	The following defined symbols remain to be analysed:
315.81/291.52	times
315.90/291.55	EOF