304.04/292.21	WORST_CASE(Omega(n^1), ?)
304.04/292.22	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
304.04/292.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.04/292.22	
304.04/292.22	
304.04/292.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.04/292.22	
304.04/292.22	(0) CpxTRS
304.04/292.22	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
304.04/292.22	(2) CpxTRS
304.04/292.22	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
304.04/292.22	(4) typed CpxTrs
304.04/292.22	(5) OrderProof [LOWER BOUND(ID), 0 ms]
304.04/292.22	(6) typed CpxTrs
304.04/292.22	(7) RewriteLemmaProof [LOWER BOUND(ID), 253 ms]
304.04/292.22	(8) BEST
304.04/292.22	    (9) proven lower bound
304.04/292.22	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
304.04/292.22	        (11) BOUNDS(n^1, INF)
304.04/292.22	    (12) typed CpxTrs
304.04/292.22	
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(0)
304.04/292.22	Obligation:
304.04/292.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.04/292.22	
304.04/292.22	
304.04/292.22	The TRS R consists of the following rules:
304.04/292.22	
304.04/292.22	   le(0, y) -> true
304.04/292.22	   le(s(x), 0) -> false
304.04/292.22	   le(s(x), s(y)) -> le(x, y)
304.04/292.22	   minus(0, y) -> 0
304.04/292.22	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	   if_minus(true, s(x), y) -> 0
304.04/292.22	   if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	   gcd(0, y) -> y
304.04/292.22	   gcd(s(x), 0) -> s(x)
304.04/292.22	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	S is empty.
304.04/292.22	Rewrite Strategy: FULL
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(1) RenamingProof (BOTH BOUNDS(ID, ID))
304.04/292.22	Renamed function symbols to avoid clashes with predefined symbol.
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(2)
304.04/292.22	Obligation:
304.04/292.22	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.04/292.22	
304.04/292.22	
304.04/292.22	The TRS R consists of the following rules:
304.04/292.22	
304.04/292.22	   le(0', y) -> true
304.04/292.22	   le(s(x), 0') -> false
304.04/292.22	   le(s(x), s(y)) -> le(x, y)
304.04/292.22	   minus(0', y) -> 0'
304.04/292.22	   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	   if_minus(true, s(x), y) -> 0'
304.04/292.22	   if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	   gcd(0', y) -> y
304.04/292.22	   gcd(s(x), 0') -> s(x)
304.04/292.22	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	S is empty.
304.04/292.22	Rewrite Strategy: FULL
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
304.04/292.22	Infered types.
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(4)
304.04/292.22	Obligation:
304.04/292.22	TRS:
304.04/292.22	Rules:
304.04/292.22	le(0', y) -> true
304.04/292.22	le(s(x), 0') -> false
304.04/292.22	le(s(x), s(y)) -> le(x, y)
304.04/292.22	minus(0', y) -> 0'
304.04/292.22	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	if_minus(true, s(x), y) -> 0'
304.04/292.22	if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	gcd(0', y) -> y
304.04/292.22	gcd(s(x), 0') -> s(x)
304.04/292.22	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	Types:
304.04/292.22	le :: 0':s -> 0':s -> true:false
304.04/292.22	0' :: 0':s
304.04/292.22	true :: true:false
304.04/292.22	s :: 0':s -> 0':s
304.04/292.22	false :: true:false
304.04/292.22	minus :: 0':s -> 0':s -> 0':s
304.04/292.22	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	gcd :: 0':s -> 0':s -> 0':s
304.04/292.22	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	hole_true:false1_0 :: true:false
304.04/292.22	hole_0':s2_0 :: 0':s
304.04/292.22	gen_0':s3_0 :: Nat -> 0':s
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(5) OrderProof (LOWER BOUND(ID))
304.04/292.22	Heuristically decided to analyse the following defined symbols:
304.04/292.22	le, minus, gcd
304.04/292.22	
304.04/292.22	They will be analysed ascendingly in the following order:
304.04/292.22	le < minus
304.04/292.22	le < gcd
304.04/292.22	minus < gcd
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(6)
304.04/292.22	Obligation:
304.04/292.22	TRS:
304.04/292.22	Rules:
304.04/292.22	le(0', y) -> true
304.04/292.22	le(s(x), 0') -> false
304.04/292.22	le(s(x), s(y)) -> le(x, y)
304.04/292.22	minus(0', y) -> 0'
304.04/292.22	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	if_minus(true, s(x), y) -> 0'
304.04/292.22	if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	gcd(0', y) -> y
304.04/292.22	gcd(s(x), 0') -> s(x)
304.04/292.22	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	Types:
304.04/292.22	le :: 0':s -> 0':s -> true:false
304.04/292.22	0' :: 0':s
304.04/292.22	true :: true:false
304.04/292.22	s :: 0':s -> 0':s
304.04/292.22	false :: true:false
304.04/292.22	minus :: 0':s -> 0':s -> 0':s
304.04/292.22	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	gcd :: 0':s -> 0':s -> 0':s
304.04/292.22	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	hole_true:false1_0 :: true:false
304.04/292.22	hole_0':s2_0 :: 0':s
304.04/292.22	gen_0':s3_0 :: Nat -> 0':s
304.04/292.22	
304.04/292.22	
304.04/292.22	Generator Equations:
304.04/292.22	gen_0':s3_0(0) <=> 0'
304.04/292.22	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.04/292.22	
304.04/292.22	
304.04/292.22	The following defined symbols remain to be analysed:
304.04/292.22	le, minus, gcd
304.04/292.22	
304.04/292.22	They will be analysed ascendingly in the following order:
304.04/292.22	le < minus
304.04/292.22	le < gcd
304.04/292.22	minus < gcd
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(7) RewriteLemmaProof (LOWER BOUND(ID))
304.04/292.22	Proved the following rewrite lemma:
304.04/292.22	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
304.04/292.22	
304.04/292.22	Induction Base:
304.04/292.22	le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
304.04/292.22	true
304.04/292.22	
304.04/292.22	Induction Step:
304.04/292.22	le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
304.04/292.22	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
304.04/292.22	true
304.04/292.22	
304.04/292.22	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(8)
304.04/292.22	Complex Obligation (BEST)
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(9)
304.04/292.22	Obligation:
304.04/292.22	Proved the lower bound n^1 for the following obligation:
304.04/292.22	
304.04/292.22	TRS:
304.04/292.22	Rules:
304.04/292.22	le(0', y) -> true
304.04/292.22	le(s(x), 0') -> false
304.04/292.22	le(s(x), s(y)) -> le(x, y)
304.04/292.22	minus(0', y) -> 0'
304.04/292.22	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	if_minus(true, s(x), y) -> 0'
304.04/292.22	if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	gcd(0', y) -> y
304.04/292.22	gcd(s(x), 0') -> s(x)
304.04/292.22	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	Types:
304.04/292.22	le :: 0':s -> 0':s -> true:false
304.04/292.22	0' :: 0':s
304.04/292.22	true :: true:false
304.04/292.22	s :: 0':s -> 0':s
304.04/292.22	false :: true:false
304.04/292.22	minus :: 0':s -> 0':s -> 0':s
304.04/292.22	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	gcd :: 0':s -> 0':s -> 0':s
304.04/292.22	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	hole_true:false1_0 :: true:false
304.04/292.22	hole_0':s2_0 :: 0':s
304.04/292.22	gen_0':s3_0 :: Nat -> 0':s
304.04/292.22	
304.04/292.22	
304.04/292.22	Generator Equations:
304.04/292.22	gen_0':s3_0(0) <=> 0'
304.04/292.22	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.04/292.22	
304.04/292.22	
304.04/292.22	The following defined symbols remain to be analysed:
304.04/292.22	le, minus, gcd
304.04/292.22	
304.04/292.22	They will be analysed ascendingly in the following order:
304.04/292.22	le < minus
304.04/292.22	le < gcd
304.04/292.22	minus < gcd
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(10) LowerBoundPropagationProof (FINISHED)
304.04/292.22	Propagated lower bound.
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(11)
304.04/292.22	BOUNDS(n^1, INF)
304.04/292.22	
304.04/292.22	----------------------------------------
304.04/292.22	
304.04/292.22	(12)
304.04/292.22	Obligation:
304.04/292.22	TRS:
304.04/292.22	Rules:
304.04/292.22	le(0', y) -> true
304.04/292.22	le(s(x), 0') -> false
304.04/292.22	le(s(x), s(y)) -> le(x, y)
304.04/292.22	minus(0', y) -> 0'
304.04/292.22	minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
304.04/292.22	if_minus(true, s(x), y) -> 0'
304.04/292.22	if_minus(false, s(x), y) -> s(minus(x, y))
304.04/292.22	gcd(0', y) -> y
304.04/292.22	gcd(s(x), 0') -> s(x)
304.04/292.22	gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.04/292.22	if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.04/292.22	if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.04/292.22	
304.04/292.22	Types:
304.04/292.22	le :: 0':s -> 0':s -> true:false
304.04/292.22	0' :: 0':s
304.04/292.22	true :: true:false
304.04/292.22	s :: 0':s -> 0':s
304.04/292.22	false :: true:false
304.04/292.22	minus :: 0':s -> 0':s -> 0':s
304.04/292.22	if_minus :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	gcd :: 0':s -> 0':s -> 0':s
304.04/292.22	if_gcd :: true:false -> 0':s -> 0':s -> 0':s
304.04/292.22	hole_true:false1_0 :: true:false
304.04/292.22	hole_0':s2_0 :: 0':s
304.04/292.22	gen_0':s3_0 :: Nat -> 0':s
304.04/292.22	
304.04/292.22	
304.04/292.22	Lemmas:
304.04/292.22	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
304.04/292.22	
304.04/292.22	
304.04/292.22	Generator Equations:
304.04/292.22	gen_0':s3_0(0) <=> 0'
304.04/292.22	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
304.04/292.22	
304.04/292.22	
304.04/292.22	The following defined symbols remain to be analysed:
304.04/292.22	minus, gcd
304.04/292.22	
304.04/292.22	They will be analysed ascendingly in the following order:
304.04/292.22	minus < gcd
304.19/292.28	EOF