314.31/291.51	WORST_CASE(Omega(n^1), ?)
314.31/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
314.31/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.31/291.52	
314.31/291.52	
314.31/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.31/291.52	
314.31/291.52	(0) CpxTRS
314.31/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
314.31/291.52	(2) CpxTRS
314.31/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
314.31/291.52	(4) typed CpxTrs
314.31/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
314.31/291.52	(6) typed CpxTrs
314.31/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 308 ms]
314.31/291.52	(8) BEST
314.31/291.52	    (9) proven lower bound
314.31/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
314.31/291.52	        (11) BOUNDS(n^1, INF)
314.31/291.52	    (12) typed CpxTrs
314.31/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
314.31/291.52	        (14) typed CpxTrs
314.31/291.52	        (15) RewriteLemmaProof [LOWER BOUND(ID), 157 ms]
314.31/291.52	        (16) typed CpxTrs
314.31/291.52	
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(0)
314.31/291.52	Obligation:
314.31/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.31/291.52	
314.31/291.52	
314.31/291.52	The TRS R consists of the following rules:
314.31/291.52	
314.31/291.52	   min(x, 0) -> 0
314.31/291.52	   min(0, y) -> 0
314.31/291.52	   min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	   max(x, 0) -> x
314.31/291.52	   max(0, y) -> y
314.31/291.52	   max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	   minus(x, 0) -> x
314.31/291.52	   minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	   gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	   any(s(x)) -> s(s(any(x)))
314.31/291.52	   any(x) -> x
314.31/291.52	
314.31/291.52	S is empty.
314.31/291.52	Rewrite Strategy: FULL
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
314.31/291.52	Renamed function symbols to avoid clashes with predefined symbol.
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(2)
314.31/291.52	Obligation:
314.31/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.31/291.52	
314.31/291.52	
314.31/291.52	The TRS R consists of the following rules:
314.31/291.52	
314.31/291.52	   min(x, 0') -> 0'
314.31/291.52	   min(0', y) -> 0'
314.31/291.52	   min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	   max(x, 0') -> x
314.31/291.52	   max(0', y) -> y
314.31/291.52	   max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	   minus(x, 0') -> x
314.31/291.52	   minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	   gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	   any(s(x)) -> s(s(any(x)))
314.31/291.52	   any(x) -> x
314.31/291.52	
314.31/291.52	S is empty.
314.31/291.52	Rewrite Strategy: FULL
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
314.31/291.52	Infered types.
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(4)
314.31/291.52	Obligation:
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(5) OrderProof (LOWER BOUND(ID))
314.31/291.52	Heuristically decided to analyse the following defined symbols:
314.31/291.52	min, max, minus, any, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	min < gcd
314.31/291.52	max < gcd
314.31/291.52	any < minus
314.31/291.52	minus < gcd
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(6)
314.31/291.52	Obligation:
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	
314.31/291.52	Generator Equations:
314.31/291.52	gen_0':s3_0(0) <=> 0'
314.31/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.31/291.52	
314.31/291.52	
314.31/291.52	The following defined symbols remain to be analysed:
314.31/291.52	min, max, minus, any, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	min < gcd
314.31/291.52	max < gcd
314.31/291.52	any < minus
314.31/291.52	minus < gcd
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
314.31/291.52	Proved the following rewrite lemma:
314.31/291.52	min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
314.31/291.52	
314.31/291.52	Induction Base:
314.31/291.52	min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
314.31/291.52	0'
314.31/291.52	
314.31/291.52	Induction Step:
314.31/291.52	min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
314.31/291.52	s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH
314.31/291.52	s(gen_0':s3_0(c6_0))
314.31/291.52	
314.31/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(8)
314.31/291.52	Complex Obligation (BEST)
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(9)
314.31/291.52	Obligation:
314.31/291.52	Proved the lower bound n^1 for the following obligation:
314.31/291.52	
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	
314.31/291.52	Generator Equations:
314.31/291.52	gen_0':s3_0(0) <=> 0'
314.31/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.31/291.52	
314.31/291.52	
314.31/291.52	The following defined symbols remain to be analysed:
314.31/291.52	min, max, minus, any, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	min < gcd
314.31/291.52	max < gcd
314.31/291.52	any < minus
314.31/291.52	minus < gcd
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(10) LowerBoundPropagationProof (FINISHED)
314.31/291.52	Propagated lower bound.
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(11)
314.31/291.52	BOUNDS(n^1, INF)
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(12)
314.31/291.52	Obligation:
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	
314.31/291.52	Lemmas:
314.31/291.52	min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
314.31/291.52	
314.31/291.52	
314.31/291.52	Generator Equations:
314.31/291.52	gen_0':s3_0(0) <=> 0'
314.31/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.31/291.52	
314.31/291.52	
314.31/291.52	The following defined symbols remain to be analysed:
314.31/291.52	max, minus, any, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	max < gcd
314.31/291.52	any < minus
314.31/291.52	minus < gcd
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
314.31/291.52	Proved the following rewrite lemma:
314.31/291.52	max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)
314.31/291.52	
314.31/291.52	Induction Base:
314.31/291.52	max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
314.31/291.52	gen_0':s3_0(0)
314.31/291.52	
314.31/291.52	Induction Step:
314.31/291.52	max(gen_0':s3_0(+(n301_0, 1)), gen_0':s3_0(+(n301_0, 1))) ->_R^Omega(1)
314.31/291.52	s(max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0))) ->_IH
314.31/291.52	s(gen_0':s3_0(c302_0))
314.31/291.52	
314.31/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(14)
314.31/291.52	Obligation:
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	
314.31/291.52	Lemmas:
314.31/291.52	min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
314.31/291.52	max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)
314.31/291.52	
314.31/291.52	
314.31/291.52	Generator Equations:
314.31/291.52	gen_0':s3_0(0) <=> 0'
314.31/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.31/291.52	
314.31/291.52	
314.31/291.52	The following defined symbols remain to be analysed:
314.31/291.52	any, minus, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	any < minus
314.31/291.52	minus < gcd
314.31/291.52	
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(15) RewriteLemmaProof (LOWER BOUND(ID))
314.31/291.52	Proved the following rewrite lemma:
314.31/291.52	any(gen_0':s3_0(+(1, n677_0))) -> *4_0, rt in Omega(n677_0)
314.31/291.52	
314.31/291.52	Induction Base:
314.31/291.52	any(gen_0':s3_0(+(1, 0)))
314.31/291.52	
314.31/291.52	Induction Step:
314.31/291.52	any(gen_0':s3_0(+(1, +(n677_0, 1)))) ->_R^Omega(1)
314.31/291.52	s(s(any(gen_0':s3_0(+(1, n677_0))))) ->_IH
314.31/291.52	s(s(*4_0))
314.31/291.52	
314.31/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.31/291.52	----------------------------------------
314.31/291.52	
314.31/291.52	(16)
314.31/291.52	Obligation:
314.31/291.52	TRS:
314.31/291.52	Rules:
314.31/291.52	min(x, 0') -> 0'
314.31/291.52	min(0', y) -> 0'
314.31/291.52	min(s(x), s(y)) -> s(min(x, y))
314.31/291.52	max(x, 0') -> x
314.31/291.52	max(0', y) -> y
314.31/291.52	max(s(x), s(y)) -> s(max(x, y))
314.31/291.52	minus(x, 0') -> x
314.31/291.52	minus(s(x), s(y)) -> s(minus(x, any(y)))
314.31/291.52	gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
314.31/291.52	any(s(x)) -> s(s(any(x)))
314.31/291.52	any(x) -> x
314.31/291.52	
314.31/291.52	Types:
314.31/291.52	min :: 0':s -> 0':s -> 0':s
314.31/291.52	0' :: 0':s
314.31/291.52	s :: 0':s -> 0':s
314.31/291.52	max :: 0':s -> 0':s -> 0':s
314.31/291.52	minus :: 0':s -> 0':s -> 0':s
314.31/291.52	any :: 0':s -> 0':s
314.31/291.52	gcd :: 0':s -> 0':s -> gcd
314.31/291.52	hole_0':s1_0 :: 0':s
314.31/291.52	hole_gcd2_0 :: gcd
314.31/291.52	gen_0':s3_0 :: Nat -> 0':s
314.31/291.52	
314.31/291.52	
314.31/291.52	Lemmas:
314.31/291.52	min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
314.31/291.52	max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)
314.31/291.52	any(gen_0':s3_0(+(1, n677_0))) -> *4_0, rt in Omega(n677_0)
314.31/291.52	
314.31/291.52	
314.31/291.52	Generator Equations:
314.31/291.52	gen_0':s3_0(0) <=> 0'
314.31/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
314.31/291.52	
314.31/291.52	
314.31/291.52	The following defined symbols remain to be analysed:
314.31/291.52	minus, gcd
314.31/291.52	
314.31/291.52	They will be analysed ascendingly in the following order:
314.31/291.52	minus < gcd
314.42/291.55	EOF