310.94/291.55	WORST_CASE(Omega(n^1), ?)
310.98/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
310.98/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.98/291.56	
310.98/291.56	
310.98/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.98/291.56	
310.98/291.56	(0) CpxTRS
310.98/291.56	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
310.98/291.56	(2) CpxTRS
310.98/291.56	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
310.98/291.56	(4) typed CpxTrs
310.98/291.56	(5) OrderProof [LOWER BOUND(ID), 0 ms]
310.98/291.56	(6) typed CpxTrs
310.98/291.56	(7) RewriteLemmaProof [LOWER BOUND(ID), 264 ms]
310.98/291.56	(8) BEST
310.98/291.56	    (9) proven lower bound
310.98/291.56	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
310.98/291.56	        (11) BOUNDS(n^1, INF)
310.98/291.56	    (12) typed CpxTrs
310.98/291.56	        (13) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
310.98/291.56	        (14) typed CpxTrs
310.98/291.56	        (15) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
310.98/291.56	        (16) typed CpxTrs
310.98/291.56	
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(0)
310.98/291.56	Obligation:
310.98/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.98/291.56	
310.98/291.56	
310.98/291.56	The TRS R consists of the following rules:
310.98/291.56	
310.98/291.56	   le(0, y) -> true
310.98/291.56	   le(s(x), 0) -> false
310.98/291.56	   le(s(x), s(y)) -> le(x, y)
310.98/291.56	   minus(x, 0) -> x
310.98/291.56	   minus(0, s(y)) -> 0
310.98/291.56	   minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	   plus(x, 0) -> x
310.98/291.56	   plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	   mod(s(x), 0) -> 0
310.98/291.56	   mod(x, s(y)) -> help(x, s(y), 0)
310.98/291.56	   help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	   if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	   if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	S is empty.
310.98/291.56	Rewrite Strategy: FULL
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(1) RenamingProof (BOTH BOUNDS(ID, ID))
310.98/291.56	Renamed function symbols to avoid clashes with predefined symbol.
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(2)
310.98/291.56	Obligation:
310.98/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.98/291.56	
310.98/291.56	
310.98/291.56	The TRS R consists of the following rules:
310.98/291.56	
310.98/291.56	   le(0', y) -> true
310.98/291.56	   le(s(x), 0') -> false
310.98/291.56	   le(s(x), s(y)) -> le(x, y)
310.98/291.56	   minus(x, 0') -> x
310.98/291.56	   minus(0', s(y)) -> 0'
310.98/291.56	   minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	   plus(x, 0') -> x
310.98/291.56	   plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	   mod(s(x), 0') -> 0'
310.98/291.56	   mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	   help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	   if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	   if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	S is empty.
310.98/291.56	Rewrite Strategy: FULL
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
310.98/291.56	Infered types.
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(4)
310.98/291.56	Obligation:
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(5) OrderProof (LOWER BOUND(ID))
310.98/291.56	Heuristically decided to analyse the following defined symbols:
310.98/291.56	le, minus, plus, help
310.98/291.56	
310.98/291.56	They will be analysed ascendingly in the following order:
310.98/291.56	le < help
310.98/291.56	minus < help
310.98/291.56	plus < help
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(6)
310.98/291.56	Obligation:
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	
310.98/291.56	Generator Equations:
310.98/291.56	gen_0':s3_0(0) <=> 0'
310.98/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.98/291.56	
310.98/291.56	
310.98/291.56	The following defined symbols remain to be analysed:
310.98/291.56	le, minus, plus, help
310.98/291.56	
310.98/291.56	They will be analysed ascendingly in the following order:
310.98/291.56	le < help
310.98/291.56	minus < help
310.98/291.56	plus < help
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(7) RewriteLemmaProof (LOWER BOUND(ID))
310.98/291.56	Proved the following rewrite lemma:
310.98/291.56	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.98/291.56	
310.98/291.56	Induction Base:
310.98/291.56	le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
310.98/291.56	true
310.98/291.56	
310.98/291.56	Induction Step:
310.98/291.56	le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
310.98/291.56	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
310.98/291.56	true
310.98/291.56	
310.98/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(8)
310.98/291.56	Complex Obligation (BEST)
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(9)
310.98/291.56	Obligation:
310.98/291.56	Proved the lower bound n^1 for the following obligation:
310.98/291.56	
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	
310.98/291.56	Generator Equations:
310.98/291.56	gen_0':s3_0(0) <=> 0'
310.98/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.98/291.56	
310.98/291.56	
310.98/291.56	The following defined symbols remain to be analysed:
310.98/291.56	le, minus, plus, help
310.98/291.56	
310.98/291.56	They will be analysed ascendingly in the following order:
310.98/291.56	le < help
310.98/291.56	minus < help
310.98/291.56	plus < help
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(10) LowerBoundPropagationProof (FINISHED)
310.98/291.56	Propagated lower bound.
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(11)
310.98/291.56	BOUNDS(n^1, INF)
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(12)
310.98/291.56	Obligation:
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	
310.98/291.56	Lemmas:
310.98/291.56	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.98/291.56	
310.98/291.56	
310.98/291.56	Generator Equations:
310.98/291.56	gen_0':s3_0(0) <=> 0'
310.98/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.98/291.56	
310.98/291.56	
310.98/291.56	The following defined symbols remain to be analysed:
310.98/291.56	minus, plus, help
310.98/291.56	
310.98/291.56	They will be analysed ascendingly in the following order:
310.98/291.56	minus < help
310.98/291.56	plus < help
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(13) RewriteLemmaProof (LOWER BOUND(ID))
310.98/291.56	Proved the following rewrite lemma:
310.98/291.56	minus(gen_0':s3_0(n270_0), gen_0':s3_0(n270_0)) -> gen_0':s3_0(0), rt in Omega(1 + n270_0)
310.98/291.56	
310.98/291.56	Induction Base:
310.98/291.56	minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
310.98/291.56	gen_0':s3_0(0)
310.98/291.56	
310.98/291.56	Induction Step:
310.98/291.56	minus(gen_0':s3_0(+(n270_0, 1)), gen_0':s3_0(+(n270_0, 1))) ->_R^Omega(1)
310.98/291.56	minus(gen_0':s3_0(n270_0), gen_0':s3_0(n270_0)) ->_IH
310.98/291.56	gen_0':s3_0(0)
310.98/291.56	
310.98/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(14)
310.98/291.56	Obligation:
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	
310.98/291.56	Lemmas:
310.98/291.56	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.98/291.56	minus(gen_0':s3_0(n270_0), gen_0':s3_0(n270_0)) -> gen_0':s3_0(0), rt in Omega(1 + n270_0)
310.98/291.56	
310.98/291.56	
310.98/291.56	Generator Equations:
310.98/291.56	gen_0':s3_0(0) <=> 0'
310.98/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.98/291.56	
310.98/291.56	
310.98/291.56	The following defined symbols remain to be analysed:
310.98/291.56	plus, help
310.98/291.56	
310.98/291.56	They will be analysed ascendingly in the following order:
310.98/291.56	plus < help
310.98/291.56	
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(15) RewriteLemmaProof (LOWER BOUND(ID))
310.98/291.56	Proved the following rewrite lemma:
310.98/291.56	plus(gen_0':s3_0(a), gen_0':s3_0(n657_0)) -> gen_0':s3_0(+(n657_0, a)), rt in Omega(1 + n657_0)
310.98/291.56	
310.98/291.56	Induction Base:
310.98/291.56	plus(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
310.98/291.56	gen_0':s3_0(a)
310.98/291.56	
310.98/291.56	Induction Step:
310.98/291.56	plus(gen_0':s3_0(a), gen_0':s3_0(+(n657_0, 1))) ->_R^Omega(1)
310.98/291.56	s(plus(gen_0':s3_0(a), gen_0':s3_0(n657_0))) ->_IH
310.98/291.56	s(gen_0':s3_0(+(a, c658_0)))
310.98/291.56	
310.98/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
310.98/291.56	----------------------------------------
310.98/291.56	
310.98/291.56	(16)
310.98/291.56	Obligation:
310.98/291.56	TRS:
310.98/291.56	Rules:
310.98/291.56	le(0', y) -> true
310.98/291.56	le(s(x), 0') -> false
310.98/291.56	le(s(x), s(y)) -> le(x, y)
310.98/291.56	minus(x, 0') -> x
310.98/291.56	minus(0', s(y)) -> 0'
310.98/291.56	minus(s(x), s(y)) -> minus(x, y)
310.98/291.56	plus(x, 0') -> x
310.98/291.56	plus(x, s(y)) -> s(plus(x, y))
310.98/291.56	mod(s(x), 0') -> 0'
310.98/291.56	mod(x, s(y)) -> help(x, s(y), 0')
310.98/291.56	help(x, s(y), c) -> if(le(c, x), x, s(y), c)
310.98/291.56	if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
310.98/291.56	if(false, x, s(y), c) -> minus(x, minus(c, s(y)))
310.98/291.56	
310.98/291.56	Types:
310.98/291.56	le :: 0':s -> 0':s -> true:false
310.98/291.56	0' :: 0':s
310.98/291.56	true :: true:false
310.98/291.56	s :: 0':s -> 0':s
310.98/291.56	false :: true:false
310.98/291.56	minus :: 0':s -> 0':s -> 0':s
310.98/291.56	plus :: 0':s -> 0':s -> 0':s
310.98/291.56	mod :: 0':s -> 0':s -> 0':s
310.98/291.56	help :: 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
310.98/291.56	hole_true:false1_0 :: true:false
310.98/291.56	hole_0':s2_0 :: 0':s
310.98/291.56	gen_0':s3_0 :: Nat -> 0':s
310.98/291.56	
310.98/291.56	
310.98/291.56	Lemmas:
310.98/291.56	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.98/291.56	minus(gen_0':s3_0(n270_0), gen_0':s3_0(n270_0)) -> gen_0':s3_0(0), rt in Omega(1 + n270_0)
310.98/291.56	plus(gen_0':s3_0(a), gen_0':s3_0(n657_0)) -> gen_0':s3_0(+(n657_0, a)), rt in Omega(1 + n657_0)
310.98/291.56	
310.98/291.56	
310.98/291.56	Generator Equations:
310.98/291.56	gen_0':s3_0(0) <=> 0'
310.98/291.56	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.98/291.56	
310.98/291.56	
310.98/291.56	The following defined symbols remain to be analysed:
310.98/291.56	help
310.98/291.60	EOF