310.20/291.51	WORST_CASE(Omega(n^1), ?)
310.20/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
310.20/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.20/291.52	
310.20/291.52	
310.20/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.20/291.52	
310.20/291.52	(0) CpxTRS
310.20/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
310.20/291.52	(2) CpxTRS
310.20/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
310.20/291.52	(4) typed CpxTrs
310.20/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
310.20/291.52	(6) typed CpxTrs
310.20/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 276 ms]
310.20/291.52	(8) BEST
310.20/291.52	    (9) proven lower bound
310.20/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
310.20/291.52	        (11) BOUNDS(n^1, INF)
310.20/291.52	    (12) typed CpxTrs
310.20/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 862 ms]
310.20/291.52	        (14) typed CpxTrs
310.20/291.52	
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(0)
310.20/291.52	Obligation:
310.20/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.20/291.52	
310.20/291.52	
310.20/291.52	The TRS R consists of the following rules:
310.20/291.52	
310.20/291.52	   le(0, y) -> true
310.20/291.52	   le(s(x), 0) -> false
310.20/291.52	   le(s(x), s(y)) -> le(x, y)
310.20/291.52	   pred(s(x)) -> x
310.20/291.52	   minus(x, 0) -> x
310.20/291.52	   minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	   gcd(0, y) -> y
310.20/291.52	   gcd(s(x), 0) -> s(x)
310.20/291.52	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	S is empty.
310.20/291.52	Rewrite Strategy: FULL
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
310.20/291.52	Renamed function symbols to avoid clashes with predefined symbol.
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(2)
310.20/291.52	Obligation:
310.20/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.20/291.52	
310.20/291.52	
310.20/291.52	The TRS R consists of the following rules:
310.20/291.52	
310.20/291.52	   le(0', y) -> true
310.20/291.52	   le(s(x), 0') -> false
310.20/291.52	   le(s(x), s(y)) -> le(x, y)
310.20/291.52	   pred(s(x)) -> x
310.20/291.52	   minus(x, 0') -> x
310.20/291.52	   minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	   gcd(0', y) -> y
310.20/291.52	   gcd(s(x), 0') -> s(x)
310.20/291.52	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	S is empty.
310.20/291.52	Rewrite Strategy: FULL
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
310.20/291.52	Infered types.
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(4)
310.20/291.52	Obligation:
310.20/291.52	TRS:
310.20/291.52	Rules:
310.20/291.52	le(0', y) -> true
310.20/291.52	le(s(x), 0') -> false
310.20/291.52	le(s(x), s(y)) -> le(x, y)
310.20/291.52	pred(s(x)) -> x
310.20/291.52	minus(x, 0') -> x
310.20/291.52	minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	gcd(0', y) -> y
310.20/291.52	gcd(s(x), 0') -> s(x)
310.20/291.52	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	Types:
310.20/291.52	le :: 0':s -> 0':s -> true:false
310.20/291.52	0' :: 0':s
310.20/291.52	true :: true:false
310.20/291.52	s :: 0':s -> 0':s
310.20/291.52	false :: true:false
310.20/291.52	pred :: 0':s -> 0':s
310.20/291.52	minus :: 0':s -> 0':s -> 0':s
310.20/291.52	gcd :: 0':s -> 0':s -> 0':s
310.20/291.52	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
310.20/291.52	hole_true:false1_0 :: true:false
310.20/291.52	hole_0':s2_0 :: 0':s
310.20/291.52	gen_0':s3_0 :: Nat -> 0':s
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(5) OrderProof (LOWER BOUND(ID))
310.20/291.52	Heuristically decided to analyse the following defined symbols:
310.20/291.52	le, minus, gcd
310.20/291.52	
310.20/291.52	They will be analysed ascendingly in the following order:
310.20/291.52	le < gcd
310.20/291.52	minus < gcd
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(6)
310.20/291.52	Obligation:
310.20/291.52	TRS:
310.20/291.52	Rules:
310.20/291.52	le(0', y) -> true
310.20/291.52	le(s(x), 0') -> false
310.20/291.52	le(s(x), s(y)) -> le(x, y)
310.20/291.52	pred(s(x)) -> x
310.20/291.52	minus(x, 0') -> x
310.20/291.52	minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	gcd(0', y) -> y
310.20/291.52	gcd(s(x), 0') -> s(x)
310.20/291.52	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	Types:
310.20/291.52	le :: 0':s -> 0':s -> true:false
310.20/291.52	0' :: 0':s
310.20/291.52	true :: true:false
310.20/291.52	s :: 0':s -> 0':s
310.20/291.52	false :: true:false
310.20/291.52	pred :: 0':s -> 0':s
310.20/291.52	minus :: 0':s -> 0':s -> 0':s
310.20/291.52	gcd :: 0':s -> 0':s -> 0':s
310.20/291.52	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
310.20/291.52	hole_true:false1_0 :: true:false
310.20/291.52	hole_0':s2_0 :: 0':s
310.20/291.52	gen_0':s3_0 :: Nat -> 0':s
310.20/291.52	
310.20/291.52	
310.20/291.52	Generator Equations:
310.20/291.52	gen_0':s3_0(0) <=> 0'
310.20/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.20/291.52	
310.20/291.52	
310.20/291.52	The following defined symbols remain to be analysed:
310.20/291.52	le, minus, gcd
310.20/291.52	
310.20/291.52	They will be analysed ascendingly in the following order:
310.20/291.52	le < gcd
310.20/291.52	minus < gcd
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
310.20/291.52	Proved the following rewrite lemma:
310.20/291.52	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.20/291.52	
310.20/291.52	Induction Base:
310.20/291.52	le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
310.20/291.52	true
310.20/291.52	
310.20/291.52	Induction Step:
310.20/291.52	le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
310.20/291.52	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
310.20/291.52	true
310.20/291.52	
310.20/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(8)
310.20/291.52	Complex Obligation (BEST)
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(9)
310.20/291.52	Obligation:
310.20/291.52	Proved the lower bound n^1 for the following obligation:
310.20/291.52	
310.20/291.52	TRS:
310.20/291.52	Rules:
310.20/291.52	le(0', y) -> true
310.20/291.52	le(s(x), 0') -> false
310.20/291.52	le(s(x), s(y)) -> le(x, y)
310.20/291.52	pred(s(x)) -> x
310.20/291.52	minus(x, 0') -> x
310.20/291.52	minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	gcd(0', y) -> y
310.20/291.52	gcd(s(x), 0') -> s(x)
310.20/291.52	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	Types:
310.20/291.52	le :: 0':s -> 0':s -> true:false
310.20/291.52	0' :: 0':s
310.20/291.52	true :: true:false
310.20/291.52	s :: 0':s -> 0':s
310.20/291.52	false :: true:false
310.20/291.52	pred :: 0':s -> 0':s
310.20/291.52	minus :: 0':s -> 0':s -> 0':s
310.20/291.52	gcd :: 0':s -> 0':s -> 0':s
310.20/291.52	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
310.20/291.52	hole_true:false1_0 :: true:false
310.20/291.52	hole_0':s2_0 :: 0':s
310.20/291.52	gen_0':s3_0 :: Nat -> 0':s
310.20/291.52	
310.20/291.52	
310.20/291.52	Generator Equations:
310.20/291.52	gen_0':s3_0(0) <=> 0'
310.20/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.20/291.52	
310.20/291.52	
310.20/291.52	The following defined symbols remain to be analysed:
310.20/291.52	le, minus, gcd
310.20/291.52	
310.20/291.52	They will be analysed ascendingly in the following order:
310.20/291.52	le < gcd
310.20/291.52	minus < gcd
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(10) LowerBoundPropagationProof (FINISHED)
310.20/291.52	Propagated lower bound.
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(11)
310.20/291.52	BOUNDS(n^1, INF)
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(12)
310.20/291.52	Obligation:
310.20/291.52	TRS:
310.20/291.52	Rules:
310.20/291.52	le(0', y) -> true
310.20/291.52	le(s(x), 0') -> false
310.20/291.52	le(s(x), s(y)) -> le(x, y)
310.20/291.52	pred(s(x)) -> x
310.20/291.52	minus(x, 0') -> x
310.20/291.52	minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	gcd(0', y) -> y
310.20/291.52	gcd(s(x), 0') -> s(x)
310.20/291.52	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	Types:
310.20/291.52	le :: 0':s -> 0':s -> true:false
310.20/291.52	0' :: 0':s
310.20/291.52	true :: true:false
310.20/291.52	s :: 0':s -> 0':s
310.20/291.52	false :: true:false
310.20/291.52	pred :: 0':s -> 0':s
310.20/291.52	minus :: 0':s -> 0':s -> 0':s
310.20/291.52	gcd :: 0':s -> 0':s -> 0':s
310.20/291.52	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
310.20/291.52	hole_true:false1_0 :: true:false
310.20/291.52	hole_0':s2_0 :: 0':s
310.20/291.52	gen_0':s3_0 :: Nat -> 0':s
310.20/291.52	
310.20/291.52	
310.20/291.52	Lemmas:
310.20/291.52	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.20/291.52	
310.20/291.52	
310.20/291.52	Generator Equations:
310.20/291.52	gen_0':s3_0(0) <=> 0'
310.20/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.20/291.52	
310.20/291.52	
310.20/291.52	The following defined symbols remain to be analysed:
310.20/291.52	minus, gcd
310.20/291.52	
310.20/291.52	They will be analysed ascendingly in the following order:
310.20/291.52	minus < gcd
310.20/291.52	
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
310.20/291.52	Proved the following rewrite lemma:
310.20/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0))) -> *4_0, rt in Omega(n258_0)
310.20/291.52	
310.20/291.52	Induction Base:
310.20/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))
310.20/291.52	
310.20/291.52	Induction Step:
310.20/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n258_0, 1)))) ->_R^Omega(1)
310.20/291.52	pred(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0)))) ->_IH
310.20/291.52	pred(*4_0)
310.20/291.52	
310.20/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
310.20/291.52	----------------------------------------
310.20/291.52	
310.20/291.52	(14)
310.20/291.52	Obligation:
310.20/291.52	TRS:
310.20/291.52	Rules:
310.20/291.52	le(0', y) -> true
310.20/291.52	le(s(x), 0') -> false
310.20/291.52	le(s(x), s(y)) -> le(x, y)
310.20/291.52	pred(s(x)) -> x
310.20/291.52	minus(x, 0') -> x
310.20/291.52	minus(x, s(y)) -> pred(minus(x, y))
310.20/291.52	gcd(0', y) -> y
310.20/291.52	gcd(s(x), 0') -> s(x)
310.20/291.52	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.20/291.52	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
310.20/291.52	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
310.20/291.52	
310.20/291.52	Types:
310.20/291.52	le :: 0':s -> 0':s -> true:false
310.20/291.52	0' :: 0':s
310.20/291.52	true :: true:false
310.20/291.52	s :: 0':s -> 0':s
310.20/291.52	false :: true:false
310.20/291.52	pred :: 0':s -> 0':s
310.20/291.52	minus :: 0':s -> 0':s -> 0':s
310.20/291.52	gcd :: 0':s -> 0':s -> 0':s
310.20/291.52	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
310.20/291.52	hole_true:false1_0 :: true:false
310.20/291.52	hole_0':s2_0 :: 0':s
310.20/291.52	gen_0':s3_0 :: Nat -> 0':s
310.20/291.52	
310.20/291.52	
310.20/291.52	Lemmas:
310.20/291.52	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
310.20/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0))) -> *4_0, rt in Omega(n258_0)
310.20/291.52	
310.20/291.52	
310.20/291.52	Generator Equations:
310.20/291.52	gen_0':s3_0(0) <=> 0'
310.20/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
310.20/291.52	
310.20/291.52	
310.20/291.52	The following defined symbols remain to be analysed:
310.20/291.52	gcd
310.28/291.55	EOF