304.13/291.47	WORST_CASE(Omega(n^1), ?)
304.13/291.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
304.13/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.13/291.48	
304.13/291.48	
304.13/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.13/291.48	
304.13/291.48	(0) CpxTRS
304.13/291.48	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
304.13/291.48	(2) CpxTRS
304.13/291.48	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
304.13/291.48	(4) typed CpxTrs
304.13/291.48	(5) OrderProof [LOWER BOUND(ID), 0 ms]
304.13/291.48	(6) typed CpxTrs
304.13/291.48	(7) RewriteLemmaProof [LOWER BOUND(ID), 272 ms]
304.13/291.48	(8) BEST
304.13/291.48	    (9) proven lower bound
304.13/291.48	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
304.13/291.48	        (11) BOUNDS(n^1, INF)
304.13/291.48	    (12) typed CpxTrs
304.13/291.48	
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(0)
304.13/291.48	Obligation:
304.13/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.13/291.48	
304.13/291.48	
304.13/291.48	The TRS R consists of the following rules:
304.13/291.48	
304.13/291.48	   le(0, y) -> true
304.13/291.48	   le(s(x), 0) -> false
304.13/291.48	   le(s(x), s(y)) -> le(x, y)
304.13/291.48	   minus(0, y) -> 0
304.13/291.48	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	   if_minus(true, s(x), y) -> 0
304.13/291.48	   if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	   quot(0, s(y)) -> 0
304.13/291.48	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	   log(s(0)) -> 0
304.13/291.48	   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))
304.13/291.48	
304.13/291.48	S is empty.
304.13/291.48	Rewrite Strategy: FULL
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(1) RenamingProof (BOTH BOUNDS(ID, ID))
304.13/291.48	Renamed function symbols to avoid clashes with predefined symbol.
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(2)
304.13/291.48	Obligation:
304.13/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.13/291.48	
304.13/291.48	
304.13/291.48	The TRS R consists of the following rules:
304.13/291.48	
304.13/291.48	   le(0', y) -> true
304.13/291.48	   le(s(x), 0') -> false
304.13/291.48	   le(s(x), s(y)) -> le(x, y)
304.13/291.48	   minus(0', y) -> 0'
304.13/291.48	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	   if_minus(true, s(x), y) -> 0'
304.13/291.48	   if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	   quot(0', s(y)) -> 0'
304.13/291.48	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	   log(s(0')) -> 0'
304.13/291.48	   log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
304.13/291.48	
304.13/291.48	S is empty.
304.13/291.48	Rewrite Strategy: FULL
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
304.13/291.48	Infered types.
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(4)
304.13/291.48	Obligation:
304.13/291.48	TRS:
304.13/291.48	Rules:
304.13/291.48	le(0', y) -> true
304.13/291.48	le(s(x), 0') -> false
304.13/291.48	le(s(x), s(y)) -> le(x, y)
304.13/291.48	minus(0', y) -> 0'
304.13/291.48	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	if_minus(true, s(x), y) -> 0'
304.13/291.48	if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	quot(0', s(y)) -> 0'
304.13/291.48	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	log(s(0')) -> 0'
304.13/291.48	log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
304.13/291.48	
304.13/291.48	Types:
304.13/291.48	le :: 0':s -> 0':s -> true:false
304.13/291.48	0' :: 0':s
304.13/291.48	true :: true:false
304.13/291.48	s :: 0':s -> 0':s
304.13/291.48	false :: true:false
304.13/291.48	minus :: 0':s -> 0':s -> 0':s
304.13/291.48	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.13/291.48	quot :: 0':s -> 0':s -> 0':s
304.13/291.48	log :: 0':s -> 0':s
304.13/291.48	hole_true:false1_0 :: true:false
304.13/291.48	hole_0':s2_0 :: 0':s
304.13/291.48	gen_0':s3_0 :: Nat -> 0':s
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(5) OrderProof (LOWER BOUND(ID))
304.13/291.48	Heuristically decided to analyse the following defined symbols:
304.13/291.48	le, minus, quot, log
304.13/291.48	
304.13/291.48	They will be analysed ascendingly in the following order:
304.13/291.48	le < minus
304.13/291.48	minus < quot
304.13/291.48	quot < log
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(6)
304.13/291.48	Obligation:
304.13/291.48	TRS:
304.13/291.48	Rules:
304.13/291.48	le(0', y) -> true
304.13/291.48	le(s(x), 0') -> false
304.13/291.48	le(s(x), s(y)) -> le(x, y)
304.13/291.48	minus(0', y) -> 0'
304.13/291.48	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	if_minus(true, s(x), y) -> 0'
304.13/291.48	if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	quot(0', s(y)) -> 0'
304.13/291.48	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	log(s(0')) -> 0'
304.13/291.48	log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
304.13/291.48	
304.13/291.48	Types:
304.13/291.48	le :: 0':s -> 0':s -> true:false
304.13/291.48	0' :: 0':s
304.13/291.48	true :: true:false
304.13/291.48	s :: 0':s -> 0':s
304.13/291.48	false :: true:false
304.13/291.48	minus :: 0':s -> 0':s -> 0':s
304.13/291.48	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.13/291.48	quot :: 0':s -> 0':s -> 0':s
304.13/291.48	log :: 0':s -> 0':s
304.13/291.48	hole_true:false1_0 :: true:false
304.13/291.48	hole_0':s2_0 :: 0':s
304.13/291.48	gen_0':s3_0 :: Nat -> 0':s
304.13/291.48	
304.13/291.48	
304.13/291.48	Generator Equations:
304.13/291.48	gen_0':s3_0(0) <=> 0'
304.13/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.13/291.48	
304.13/291.48	
304.13/291.48	The following defined symbols remain to be analysed:
304.13/291.48	le, minus, quot, log
304.13/291.48	
304.13/291.48	They will be analysed ascendingly in the following order:
304.13/291.48	le < minus
304.13/291.48	minus < quot
304.13/291.48	quot < log
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(7) RewriteLemmaProof (LOWER BOUND(ID))
304.13/291.48	Proved the following rewrite lemma:
304.13/291.48	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
304.13/291.48	
304.13/291.48	Induction Base:
304.13/291.48	le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
304.13/291.48	true
304.13/291.48	
304.13/291.48	Induction Step:
304.13/291.48	le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
304.13/291.48	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
304.13/291.48	true
304.13/291.48	
304.13/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(8)
304.13/291.48	Complex Obligation (BEST)
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(9)
304.13/291.48	Obligation:
304.13/291.48	Proved the lower bound n^1 for the following obligation:
304.13/291.48	
304.13/291.48	TRS:
304.13/291.48	Rules:
304.13/291.48	le(0', y) -> true
304.13/291.48	le(s(x), 0') -> false
304.13/291.48	le(s(x), s(y)) -> le(x, y)
304.13/291.48	minus(0', y) -> 0'
304.13/291.48	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	if_minus(true, s(x), y) -> 0'
304.13/291.48	if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	quot(0', s(y)) -> 0'
304.13/291.48	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	log(s(0')) -> 0'
304.13/291.48	log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
304.13/291.48	
304.13/291.48	Types:
304.13/291.48	le :: 0':s -> 0':s -> true:false
304.13/291.48	0' :: 0':s
304.13/291.48	true :: true:false
304.13/291.48	s :: 0':s -> 0':s
304.13/291.48	false :: true:false
304.13/291.48	minus :: 0':s -> 0':s -> 0':s
304.13/291.48	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.13/291.48	quot :: 0':s -> 0':s -> 0':s
304.13/291.48	log :: 0':s -> 0':s
304.13/291.48	hole_true:false1_0 :: true:false
304.13/291.48	hole_0':s2_0 :: 0':s
304.13/291.48	gen_0':s3_0 :: Nat -> 0':s
304.13/291.48	
304.13/291.48	
304.13/291.48	Generator Equations:
304.13/291.48	gen_0':s3_0(0) <=> 0'
304.13/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.13/291.48	
304.13/291.48	
304.13/291.48	The following defined symbols remain to be analysed:
304.13/291.48	le, minus, quot, log
304.13/291.48	
304.13/291.48	They will be analysed ascendingly in the following order:
304.13/291.48	le < minus
304.13/291.48	minus < quot
304.13/291.48	quot < log
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(10) LowerBoundPropagationProof (FINISHED)
304.13/291.48	Propagated lower bound.
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(11)
304.13/291.48	BOUNDS(n^1, INF)
304.13/291.48	
304.13/291.48	----------------------------------------
304.13/291.48	
304.13/291.48	(12)
304.13/291.48	Obligation:
304.13/291.48	TRS:
304.13/291.48	Rules:
304.13/291.48	le(0', y) -> true
304.13/291.48	le(s(x), 0') -> false
304.13/291.48	le(s(x), s(y)) -> le(x, y)
304.13/291.48	minus(0', y) -> 0'
304.13/291.48	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.13/291.48	if_minus(true, s(x), y) -> 0'
304.13/291.48	if_minus(false, s(x), y) -> s(minus(x, y))
304.13/291.48	quot(0', s(y)) -> 0'
304.13/291.48	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
304.13/291.48	log(s(0')) -> 0'
304.13/291.48	log(s(s(x))) -> s(log(s(quot(x, s(s(0'))))))
304.13/291.48	
304.13/291.48	Types:
304.13/291.48	le :: 0':s -> 0':s -> true:false
304.13/291.48	0' :: 0':s
304.13/291.48	true :: true:false
304.13/291.48	s :: 0':s -> 0':s
304.13/291.48	false :: true:false
304.13/291.48	minus :: 0':s -> 0':s -> 0':s
304.13/291.48	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.13/291.48	quot :: 0':s -> 0':s -> 0':s
304.13/291.48	log :: 0':s -> 0':s
304.13/291.48	hole_true:false1_0 :: true:false
304.13/291.48	hole_0':s2_0 :: 0':s
304.13/291.48	gen_0':s3_0 :: Nat -> 0':s
304.13/291.48	
304.13/291.48	
304.13/291.48	Lemmas:
304.13/291.48	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
304.13/291.48	
304.13/291.48	
304.13/291.48	Generator Equations:
304.13/291.48	gen_0':s3_0(0) <=> 0'
304.13/291.48	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.13/291.48	
304.13/291.48	
304.13/291.48	The following defined symbols remain to be analysed:
304.13/291.48	minus, quot, log
304.13/291.48	
304.13/291.48	They will be analysed ascendingly in the following order:
304.13/291.48	minus < quot
304.13/291.48	quot < log
304.13/291.52	EOF