311.12/291.48	WORST_CASE(Omega(n^1), ?)
311.12/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
311.12/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.12/291.49	
311.12/291.49	
311.12/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.12/291.49	
311.12/291.49	(0) CpxTRS
311.12/291.49	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
311.12/291.49	(2) CpxTRS
311.12/291.49	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
311.12/291.49	(4) typed CpxTrs
311.12/291.49	(5) OrderProof [LOWER BOUND(ID), 0 ms]
311.12/291.49	(6) typed CpxTrs
311.12/291.49	(7) RewriteLemmaProof [LOWER BOUND(ID), 287 ms]
311.12/291.49	(8) BEST
311.12/291.49	    (9) proven lower bound
311.12/291.49	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
311.12/291.49	        (11) BOUNDS(n^1, INF)
311.12/291.49	    (12) typed CpxTrs
311.12/291.49	        (13) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
311.12/291.49	        (14) typed CpxTrs
311.12/291.49	        (15) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
311.12/291.49	        (16) typed CpxTrs
311.12/291.49	
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(0)
311.12/291.49	Obligation:
311.12/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.12/291.49	
311.12/291.49	
311.12/291.49	The TRS R consists of the following rules:
311.12/291.49	
311.12/291.49	   gt(0, y) -> false
311.12/291.49	   gt(s(x), 0) -> true
311.12/291.49	   gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	   plus(0, y) -> y
311.12/291.49	   plus(s(x), y) -> s(plus(x, y))
311.12/291.49	   double(0) -> 0
311.12/291.49	   double(s(x)) -> s(s(double(x)))
311.12/291.49	   average(x, y) -> aver(plus(x, y), 0)
311.12/291.49	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	   if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	   if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	S is empty.
311.12/291.49	Rewrite Strategy: FULL
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(1) RenamingProof (BOTH BOUNDS(ID, ID))
311.12/291.49	Renamed function symbols to avoid clashes with predefined symbol.
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(2)
311.12/291.49	Obligation:
311.12/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
311.12/291.49	
311.12/291.49	
311.12/291.49	The TRS R consists of the following rules:
311.12/291.49	
311.12/291.49	   gt(0', y) -> false
311.12/291.49	   gt(s(x), 0') -> true
311.12/291.49	   gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	   plus(0', y) -> y
311.12/291.49	   plus(s(x), y) -> s(plus(x, y))
311.12/291.49	   double(0') -> 0'
311.12/291.49	   double(s(x)) -> s(s(double(x)))
311.12/291.49	   average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	   if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	   if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	S is empty.
311.12/291.49	Rewrite Strategy: FULL
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
311.12/291.49	Infered types.
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(4)
311.12/291.49	Obligation:
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(5) OrderProof (LOWER BOUND(ID))
311.12/291.49	Heuristically decided to analyse the following defined symbols:
311.12/291.49	gt, plus, double, aver
311.12/291.49	
311.12/291.49	They will be analysed ascendingly in the following order:
311.12/291.49	gt < aver
311.12/291.49	double < aver
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(6)
311.12/291.49	Obligation:
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	
311.12/291.49	Generator Equations:
311.12/291.49	gen_0':s3_0(0) <=> 0'
311.12/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.12/291.49	
311.12/291.49	
311.12/291.49	The following defined symbols remain to be analysed:
311.12/291.49	gt, plus, double, aver
311.12/291.49	
311.12/291.49	They will be analysed ascendingly in the following order:
311.12/291.49	gt < aver
311.12/291.49	double < aver
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(7) RewriteLemmaProof (LOWER BOUND(ID))
311.12/291.49	Proved the following rewrite lemma:
311.12/291.49	gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
311.12/291.49	
311.12/291.49	Induction Base:
311.12/291.49	gt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
311.12/291.49	false
311.12/291.49	
311.12/291.49	Induction Step:
311.12/291.49	gt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
311.12/291.49	gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
311.12/291.49	false
311.12/291.49	
311.12/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(8)
311.12/291.49	Complex Obligation (BEST)
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(9)
311.12/291.49	Obligation:
311.12/291.49	Proved the lower bound n^1 for the following obligation:
311.12/291.49	
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	
311.12/291.49	Generator Equations:
311.12/291.49	gen_0':s3_0(0) <=> 0'
311.12/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.12/291.49	
311.12/291.49	
311.12/291.49	The following defined symbols remain to be analysed:
311.12/291.49	gt, plus, double, aver
311.12/291.49	
311.12/291.49	They will be analysed ascendingly in the following order:
311.12/291.49	gt < aver
311.12/291.49	double < aver
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(10) LowerBoundPropagationProof (FINISHED)
311.12/291.49	Propagated lower bound.
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(11)
311.12/291.49	BOUNDS(n^1, INF)
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(12)
311.12/291.49	Obligation:
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	
311.12/291.49	Lemmas:
311.12/291.49	gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
311.12/291.49	
311.12/291.49	
311.12/291.49	Generator Equations:
311.12/291.49	gen_0':s3_0(0) <=> 0'
311.12/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.12/291.49	
311.12/291.49	
311.12/291.49	The following defined symbols remain to be analysed:
311.12/291.49	plus, double, aver
311.12/291.49	
311.12/291.49	They will be analysed ascendingly in the following order:
311.12/291.49	double < aver
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(13) RewriteLemmaProof (LOWER BOUND(ID))
311.12/291.49	Proved the following rewrite lemma:
311.12/291.49	plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_0)
311.12/291.49	
311.12/291.49	Induction Base:
311.12/291.49	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
311.12/291.49	gen_0':s3_0(b)
311.12/291.49	
311.12/291.49	Induction Step:
311.12/291.49	plus(gen_0':s3_0(+(n258_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
311.12/291.49	s(plus(gen_0':s3_0(n258_0), gen_0':s3_0(b))) ->_IH
311.12/291.49	s(gen_0':s3_0(+(b, c259_0)))
311.12/291.49	
311.12/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(14)
311.12/291.49	Obligation:
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	
311.12/291.49	Lemmas:
311.12/291.49	gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
311.12/291.49	plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_0)
311.12/291.49	
311.12/291.49	
311.12/291.49	Generator Equations:
311.12/291.49	gen_0':s3_0(0) <=> 0'
311.12/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.12/291.49	
311.12/291.49	
311.12/291.49	The following defined symbols remain to be analysed:
311.12/291.49	double, aver
311.12/291.49	
311.12/291.49	They will be analysed ascendingly in the following order:
311.12/291.49	double < aver
311.12/291.49	
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(15) RewriteLemmaProof (LOWER BOUND(ID))
311.12/291.49	Proved the following rewrite lemma:
311.12/291.49	double(gen_0':s3_0(n801_0)) -> gen_0':s3_0(*(2, n801_0)), rt in Omega(1 + n801_0)
311.12/291.49	
311.12/291.49	Induction Base:
311.12/291.49	double(gen_0':s3_0(0)) ->_R^Omega(1)
311.12/291.49	0'
311.12/291.49	
311.12/291.49	Induction Step:
311.12/291.49	double(gen_0':s3_0(+(n801_0, 1))) ->_R^Omega(1)
311.12/291.49	s(s(double(gen_0':s3_0(n801_0)))) ->_IH
311.12/291.49	s(s(gen_0':s3_0(*(2, c802_0))))
311.12/291.49	
311.12/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
311.12/291.49	----------------------------------------
311.12/291.49	
311.12/291.49	(16)
311.12/291.49	Obligation:
311.12/291.49	TRS:
311.12/291.49	Rules:
311.12/291.49	gt(0', y) -> false
311.12/291.49	gt(s(x), 0') -> true
311.12/291.49	gt(s(x), s(y)) -> gt(x, y)
311.12/291.49	plus(0', y) -> y
311.12/291.49	plus(s(x), y) -> s(plus(x, y))
311.12/291.49	double(0') -> 0'
311.12/291.49	double(s(x)) -> s(s(double(x)))
311.12/291.49	average(x, y) -> aver(plus(x, y), 0')
311.12/291.49	aver(sum, z) -> if(gt(sum, double(z)), sum, z)
311.12/291.49	if(true, sum, z) -> aver(sum, s(z))
311.12/291.49	if(false, sum, z) -> z
311.12/291.49	
311.12/291.49	Types:
311.12/291.49	gt :: 0':s -> 0':s -> false:true
311.12/291.49	0' :: 0':s
311.12/291.49	false :: false:true
311.12/291.49	s :: 0':s -> 0':s
311.12/291.49	true :: false:true
311.12/291.49	plus :: 0':s -> 0':s -> 0':s
311.12/291.49	double :: 0':s -> 0':s
311.12/291.49	average :: 0':s -> 0':s -> 0':s
311.12/291.49	aver :: 0':s -> 0':s -> 0':s
311.12/291.49	if :: false:true -> 0':s -> 0':s -> 0':s
311.12/291.49	hole_false:true1_0 :: false:true
311.12/291.49	hole_0':s2_0 :: 0':s
311.12/291.49	gen_0':s3_0 :: Nat -> 0':s
311.12/291.49	
311.12/291.49	
311.12/291.49	Lemmas:
311.12/291.49	gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
311.12/291.49	plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_0)
311.12/291.49	double(gen_0':s3_0(n801_0)) -> gen_0':s3_0(*(2, n801_0)), rt in Omega(1 + n801_0)
311.12/291.49	
311.12/291.49	
311.12/291.49	Generator Equations:
311.12/291.49	gen_0':s3_0(0) <=> 0'
311.12/291.49	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
311.12/291.49	
311.12/291.49	
311.12/291.49	The following defined symbols remain to be analysed:
311.12/291.49	aver
311.21/291.53	EOF