id	sid	tid	token	lemma	pos
ejpam-4750	1	1	european	european	PROPN
ejpam-4750	1	2	journal	journal	PROPN
ejpam-4750	1	3	of	of	ADP
ejpam-4750	1	4	pure	pure	ADJ
ejpam-4750	1	5	and	and	CCONJ
ejpam-4750	1	6	applied	apply	VERB
ejpam-4750	1	7	mathematics	mathematic	NOUN
ejpam-4750	1	8	vol	vol	NOUN
ejpam-4750	1	9	.	.	PUNCT
ejpam-4750	2	1	16	16	NUM
ejpam-4750	2	2	,	,	PUNCT
ejpam-4750	2	3	no	no	INTJ
ejpam-4750	2	4	.	.	NOUN
ejpam-4750	2	5	2	2	NUM
ejpam-4750	2	6	,	,	PUNCT
ejpam-4750	2	7	2023	2023	NUM
ejpam-4750	2	8	,	,	PUNCT
ejpam-4750	2	9	1068	1068	NUM
ejpam-4750	2	10	-	-	SYM
ejpam-4750	2	11	1083	1083	NUM
ejpam-4750	2	12	issn	issn	PROPN
ejpam-4750	2	13	1307	1307	NUM
ejpam-4750	2	14	-	-	SYM
ejpam-4750	2	15	5543	5543	NUM
ejpam-4750	2	16	–	–	PUNCT
ejpam-4750	2	17	ejpam.com	ejpam.com	X
ejpam-4750	2	18	published	publish	VERB
ejpam-4750	2	19	by	by	ADP
ejpam-4750	2	20	new	new	PROPN
ejpam-4750	2	21	york	york	PROPN
ejpam-4750	2	22	business	business	PROPN
ejpam-4750	2	23	global	global	ADJ
ejpam-4750	2	24	forcing	force	VERB
ejpam-4750	2	25	2	2	NUM
ejpam-4750	2	26	-	-	PUNCT
ejpam-4750	2	27	metric	metric	ADJ
ejpam-4750	2	28	dimension	dimension	NOUN
ejpam-4750	2	29	in	in	ADP
ejpam-4750	2	30	the	the	DET
ejpam-4750	2	31	join	join	NOUN
ejpam-4750	2	32	and	and	CCONJ
ejpam-4750	2	33	corona	corona	NOUN
ejpam-4750	2	34	of	of	ADP
ejpam-4750	2	35	graphs	graph	NOUN
ejpam-4750	2	36	dennis	dennis	PROPN
ejpam-4750	2	37	b.	b.	PROPN
ejpam-4750	2	38	managbanag1,∗	managbanag1,∗	PROPN
ejpam-4750	2	39	,	,	PUNCT
ejpam-4750	2	40	helen	helen	PROPN
ejpam-4750	2	41	m.	m.	PROPN
ejpam-4750	2	42	rara2	rara2	PROPN
ejpam-4750	3	1	1	1	NUM
ejpam-4750	3	2	department	department	NOUN
ejpam-4750	3	3	of	of	ADP
ejpam-4750	3	4	mathematics	mathematic	NOUN
ejpam-4750	3	5	and	and	CCONJ
ejpam-4750	3	6	statistics	statistic	NOUN
ejpam-4750	3	7	,	,	PUNCT
ejpam-4750	3	8	college	college	NOUN
ejpam-4750	3	9	of	of	ADP
ejpam-4750	3	10	science	science	NOUN
ejpam-4750	3	11	and	and	CCONJ
ejpam-4750	3	12	mathematics	mathematic	NOUN
ejpam-4750	3	13	,	,	PUNCT
ejpam-4750	3	14	mindanao	mindanao	PROPN
ejpam-4750	3	15	state	state	PROPN
ejpam-4750	3	16	university	university	PROPN
ejpam-4750	3	17	-	-	PUNCT
ejpam-4750	3	18	iligan	iligan	PROPN
ejpam-4750	3	19	institute	institute	PROPN
ejpam-4750	3	20	of	of	ADP
ejpam-4750	3	21	technology	technology	PROPN
ejpam-4750	3	22	,	,	PUNCT
ejpam-4750	3	23	9200	9200	NUM
ejpam-4750	3	24	iligan	iligan	ADJ
ejpam-4750	3	25	city	city	NOUN
ejpam-4750	3	26	,	,	PUNCT
ejpam-4750	3	27	philippines	philippines	PROPN
ejpam-4750	3	28	2	2	NUM
ejpam-4750	3	29	department	department	NOUN
ejpam-4750	3	30	of	of	ADP
ejpam-4750	3	31	mathematics	mathematic	NOUN
ejpam-4750	3	32	and	and	CCONJ
ejpam-4750	3	33	statistics	statistic	NOUN
ejpam-4750	3	34	,	,	PUNCT
ejpam-4750	3	35	college	college	NOUN
ejpam-4750	3	36	of	of	ADP
ejpam-4750	3	37	science	science	NOUN
ejpam-4750	3	38	and	and	CCONJ
ejpam-4750	3	39	mathematics	mathematic	NOUN
ejpam-4750	3	40	,	,	PUNCT
ejpam-4750	3	41	center	center	NOUN
ejpam-4750	3	42	of	of	ADP
ejpam-4750	3	43	graph	graph	NOUN
ejpam-4750	3	44	theory	theory	NOUN
ejpam-4750	3	45	,	,	PUNCT
ejpam-4750	3	46	algebra	algebra	NOUN
ejpam-4750	3	47	,	,	PUNCT
ejpam-4750	3	48	and	and	CCONJ
ejpam-4750	3	49	analysis	analysis	NOUN
ejpam-4750	3	50	-	-	PUNCT
ejpam-4750	3	51	premier	premier	NOUN
ejpam-4750	3	52	research	research	NOUN
ejpam-4750	3	53	institute	institute	PROPN
ejpam-4750	3	54	of	of	ADP
ejpam-4750	3	55	science	science	NOUN
ejpam-4750	3	56	and	and	CCONJ
ejpam-4750	3	57	mathematics	mathematic	NOUN
ejpam-4750	3	58	,	,	PUNCT
ejpam-4750	3	59	mindanao	mindanao	PROPN
ejpam-4750	3	60	state	state	PROPN
ejpam-4750	3	61	university	university	PROPN
ejpam-4750	3	62	-	-	PUNCT
ejpam-4750	3	63	iligan	iligan	PROPN
ejpam-4750	3	64	institute	institute	PROPN
ejpam-4750	3	65	of	of	ADP
ejpam-4750	3	66	technology	technology	PROPN
ejpam-4750	3	67	,	,	PUNCT
ejpam-4750	3	68	9200	9200	NUM
ejpam-4750	3	69	iligan	iligan	ADJ
ejpam-4750	3	70	city	city	NOUN
ejpam-4750	3	71	,	,	PUNCT
ejpam-4750	3	72	philippines	philippine	NOUN
ejpam-4750	3	73	abstract	abstract	ADJ
ejpam-4750	3	74	.	.	PUNCT
ejpam-4750	4	1	this	this	DET
ejpam-4750	4	2	study	study	NOUN
ejpam-4750	4	3	deals	deal	VERB
ejpam-4750	4	4	with	with	ADP
ejpam-4750	4	5	the	the	DET
ejpam-4750	4	6	forcing	force	VERB
ejpam-4750	4	7	subsets	subset	NOUN
ejpam-4750	4	8	of	of	ADP
ejpam-4750	4	9	2	2	NUM
ejpam-4750	4	10	-	-	PUNCT
ejpam-4750	4	11	metric	metric	ADJ
ejpam-4750	4	12	basis	basis	NOUN
ejpam-4750	4	13	in	in	ADP
ejpam-4750	4	14	graphs	graph	NOUN
ejpam-4750	4	15	.	.	PUNCT
ejpam-4750	5	1	some	some	DET
ejpam-4750	5	2	main	main	ADJ
ejpam-4750	5	3	results	result	NOUN
ejpam-4750	5	4	generated	generate	VERB
ejpam-4750	5	5	in	in	ADP
ejpam-4750	5	6	this	this	DET
ejpam-4750	5	7	study	study	NOUN
ejpam-4750	5	8	include	include	VERB
ejpam-4750	5	9	the	the	DET
ejpam-4750	5	10	characterization	characterization	NOUN
ejpam-4750	5	11	of	of	ADP
ejpam-4750	5	12	a	a	DET
ejpam-4750	5	13	2	2	NUM
ejpam-4750	5	14	-	-	PUNCT
ejpam-4750	5	15	metric	metric	ADJ
ejpam-4750	5	16	basis	basis	NOUN
ejpam-4750	5	17	in	in	ADP
ejpam-4750	5	18	graphs	graph	NOUN
ejpam-4750	5	19	and	and	CCONJ
ejpam-4750	5	20	the	the	DET
ejpam-4750	5	21	characterization	characterization	NOUN
ejpam-4750	5	22	of	of	ADP
ejpam-4750	5	23	the	the	DET
ejpam-4750	5	24	forcing	force	VERB
ejpam-4750	5	25	subsets	subset	NOUN
ejpam-4750	5	26	of	of	ADP
ejpam-4750	5	27	these	these	DET
ejpam-4750	5	28	2	2	NUM
ejpam-4750	5	29	-	-	PUNCT
ejpam-4750	5	30	metric	metric	ADJ
ejpam-4750	5	31	bases	basis	NOUN
ejpam-4750	5	32	.	.	PUNCT
ejpam-4750	6	1	these	these	DET
ejpam-4750	6	2	characterizations	characterization	NOUN
ejpam-4750	6	3	are	be	AUX
ejpam-4750	6	4	used	use	VERB
ejpam-4750	6	5	to	to	PART
ejpam-4750	6	6	determine	determine	VERB
ejpam-4750	6	7	values	value	NOUN
ejpam-4750	6	8	for	for	ADP
ejpam-4750	6	9	the	the	DET
ejpam-4750	6	10	forcing	force	VERB
ejpam-4750	6	11	2	2	NUM
ejpam-4750	6	12	-	-	PUNCT
ejpam-4750	6	13	metric	metric	ADJ
ejpam-4750	6	14	dimension	dimension	NOUN
ejpam-4750	6	15	of	of	ADP
ejpam-4750	6	16	graphs	graph	NOUN
ejpam-4750	6	17	resulting	result	VERB
ejpam-4750	6	18	from	from	ADP
ejpam-4750	6	19	some	some	DET
ejpam-4750	6	20	binary	binary	ADJ
ejpam-4750	6	21	operations	operation	NOUN
ejpam-4750	6	22	such	such	ADJ
ejpam-4750	6	23	as	as	ADP
ejpam-4750	6	24	the	the	DET
ejpam-4750	6	25	join	join	NOUN
ejpam-4750	6	26	and	and	CCONJ
ejpam-4750	6	27	corona	corona	NOUN
ejpam-4750	6	28	of	of	ADP
ejpam-4750	6	29	graphs	graph	NOUN
ejpam-4750	6	30	.	.	PUNCT
ejpam-4750	7	1	2020	2020	NUM
ejpam-4750	7	2	mathematics	mathematic	NOUN
ejpam-4750	7	3	subject	subject	NOUN
ejpam-4750	7	4	classifications	classification	NOUN
ejpam-4750	7	5	:	:	PUNCT
ejpam-4750	7	6	05c69	05c69	X
ejpam-4750	7	7	key	key	ADJ
ejpam-4750	7	8	words	word	NOUN
ejpam-4750	7	9	and	and	CCONJ
ejpam-4750	7	10	phrases	phrase	NOUN
ejpam-4750	7	11	:	:	PUNCT
ejpam-4750	7	12	2	2	NUM
ejpam-4750	7	13	-	-	PUNCT
ejpam-4750	7	14	resolving	resolve	VERB
ejpam-4750	7	15	set	set	NOUN
ejpam-4750	7	16	,	,	PUNCT
ejpam-4750	7	17	2	2	NUM
ejpam-4750	7	18	-	-	PUNCT
ejpam-4750	7	19	metric	metric	ADJ
ejpam-4750	7	20	basis	basis	NOUN
ejpam-4750	7	21	,	,	PUNCT
ejpam-4750	7	22	2	2	NUM
ejpam-4750	7	23	-	-	PUNCT
ejpam-4750	7	24	metric	metric	ADJ
ejpam-4750	7	25	dimension	dimension	NOUN
ejpam-4750	7	26	,	,	PUNCT
ejpam-4750	7	27	forcing	force	VERB
ejpam-4750	7	28	subsets	subset	NOUN
ejpam-4750	7	29	,	,	PUNCT
ejpam-4750	7	30	forcing	force	VERB
ejpam-4750	7	31	number	number	NOUN
ejpam-4750	7	32	,	,	PUNCT
ejpam-4750	7	33	join	join	NOUN
ejpam-4750	7	34	,	,	PUNCT
ejpam-4750	7	35	corona	corona	PROPN
ejpam-4750	7	36	1	1	NUM
ejpam-4750	7	37	.	.	PUNCT
ejpam-4750	8	1	introduction	introduction	NOUN
ejpam-4750	8	2	metric	metric	ADJ
ejpam-4750	8	3	dimension	dimension	NOUN
ejpam-4750	8	4	and	and	CCONJ
ejpam-4750	8	5	resolving	resolving	NOUN
ejpam-4750	8	6	sets	set	NOUN
ejpam-4750	8	7	,	,	PUNCT
ejpam-4750	8	8	concepts	concept	NOUN
ejpam-4750	8	9	initially	initially	ADV
ejpam-4750	8	10	drafted	draft	VERB
ejpam-4750	8	11	for	for	ADP
ejpam-4750	8	12	the	the	DET
ejpam-4750	8	13	metric	metric	ADJ
ejpam-4750	8	14	spaces	space	NOUN
ejpam-4750	8	15	introduced	introduce	VERB
ejpam-4750	8	16	by	by	ADP
ejpam-4750	8	17	blumenthal	blumenthal	PROPN
ejpam-4750	8	18	[	[	X
ejpam-4750	8	19	3	3	X
ejpam-4750	8	20	]	]	PUNCT
ejpam-4750	8	21	in	in	ADP
ejpam-4750	8	22	1953	1953	NUM
ejpam-4750	8	23	.	.	PUNCT
ejpam-4750	9	1	since	since	SCONJ
ejpam-4750	9	2	then	then	ADV
ejpam-4750	9	3	,	,	PUNCT
ejpam-4750	9	4	the	the	DET
ejpam-4750	9	5	notion	notion	NOUN
ejpam-4750	9	6	of	of	ADP
ejpam-4750	9	7	metric	metric	ADJ
ejpam-4750	9	8	dimension	dimension	NOUN
ejpam-4750	9	9	has	have	AUX
ejpam-4750	9	10	been	be	AUX
ejpam-4750	9	11	broadened	broaden	VERB
ejpam-4750	9	12	to	to	PART
ejpam-4750	9	13	encompass	encompass	VERB
ejpam-4750	9	14	both	both	CCONJ
ejpam-4750	9	15	metric	metric	ADJ
ejpam-4750	9	16	and	and	CCONJ
ejpam-4750	9	17	geometric	geometric	ADJ
ejpam-4750	9	18	spaces	space	NOUN
ejpam-4750	9	19	[	[	X
ejpam-4750	9	20	2	2	NUM
ejpam-4750	9	21	,	,	PUNCT
ejpam-4750	9	22	7	7	NUM
ejpam-4750	9	23	]	]	PUNCT
ejpam-4750	9	24	.	.	PUNCT
ejpam-4750	10	1	nearly	nearly	ADV
ejpam-4750	10	2	20	20	NUM
ejpam-4750	10	3	years	year	NOUN
ejpam-4750	10	4	after	after	ADP
ejpam-4750	10	5	in	in	ADP
ejpam-4750	10	6	1976	1976	NUM
ejpam-4750	10	7	,	,	PUNCT
ejpam-4750	10	8	harary	harary	NOUN
ejpam-4750	10	9	,	,	PUNCT
ejpam-4750	10	10	melter	melter	NOUN
ejpam-4750	10	11	[	[	X
ejpam-4750	10	12	5	5	NUM
ejpam-4750	10	13	]	]	PUNCT
ejpam-4750	10	14	and	and	CCONJ
ejpam-4750	10	15	slater	slater	NOUN
ejpam-4750	10	16	[	[	X
ejpam-4750	10	17	11	11	NUM
ejpam-4750	10	18	,	,	PUNCT
ejpam-4750	10	19	12	12	NUM
ejpam-4750	10	20	]	]	PUNCT
ejpam-4750	10	21	each	each	DET
ejpam-4750	10	22	separatedly	separatedly	NOUN
ejpam-4750	10	23	discovered	discover	VERB
ejpam-4750	10	24	the	the	DET
ejpam-4750	10	25	idea	idea	NOUN
ejpam-4750	10	26	of	of	ADP
ejpam-4750	10	27	resolving	resolve	VERB
ejpam-4750	10	28	set	set	NOUN
ejpam-4750	10	29	.	.	PUNCT
ejpam-4750	11	1	in	in	ADP
ejpam-4750	11	2	2019	2019	NUM
ejpam-4750	11	3	,	,	PUNCT
ejpam-4750	11	4	bailey	bailey	NOUN
ejpam-4750	11	5	and	and	CCONJ
ejpam-4750	11	6	yero	yero	NOUN
ejpam-4750	11	7	[	[	X
ejpam-4750	11	8	1	1	NUM
ejpam-4750	11	9	]	]	PUNCT
ejpam-4750	11	10	demonstrated	demonstrate	VERB
ejpam-4750	11	11	the	the	DET
ejpam-4750	11	12	construction	construction	NOUN
ejpam-4750	11	13	of	of	ADP
ejpam-4750	11	14	error	error	NOUN
ejpam-4750	11	15	-	-	PUNCT
ejpam-4750	11	16	correcting	correct	VERB
ejpam-4750	11	17	codes	code	NOUN
ejpam-4750	11	18	out	out	ADP
ejpam-4750	11	19	of	of	ADP
ejpam-4750	11	20	graphs	graph	NOUN
ejpam-4750	11	21	using	use	VERB
ejpam-4750	11	22	k	k	ADJ
ejpam-4750	11	23	-	-	PUNCT
ejpam-4750	11	24	resolving	resolve	VERB
ejpam-4750	11	25	sets	set	NOUN
ejpam-4750	11	26	and	and	CCONJ
ejpam-4750	11	27	provided	provide	VERB
ejpam-4750	11	28	a	a	DET
ejpam-4750	11	29	decoding	decode	VERB
ejpam-4750	11	30	algorithm	algorithm	NOUN
ejpam-4750	11	31	that	that	PRON
ejpam-4750	11	32	used	use	VERB
ejpam-4750	11	33	covering	cover	VERB
ejpam-4750	11	34	designs	design	NOUN
ejpam-4750	11	35	.	.	PUNCT
ejpam-4750	12	1	a	a	DET
ejpam-4750	12	2	study	study	NOUN
ejpam-4750	12	3	on	on	ADP
ejpam-4750	12	4	the	the	DET
ejpam-4750	12	5	idea	idea	NOUN
ejpam-4750	12	6	of	of	ADP
ejpam-4750	12	7	the	the	DET
ejpam-4750	12	8	k	k	NOUN
ejpam-4750	12	9	-	-	PUNCT
ejpam-4750	12	10	resolving	resolving	ADJ
ejpam-4750	12	11	set	set	NOUN
ejpam-4750	12	12	,	,	PUNCT
ejpam-4750	12	13	also	also	ADV
ejpam-4750	12	14	known	know	VERB
ejpam-4750	12	15	as	as	ADP
ejpam-4750	12	16	“	"	PUNCT
ejpam-4750	12	17	on	on	ADP
ejpam-4750	12	18	2	2	NUM
ejpam-4750	12	19	-	-	PUNCT
ejpam-4750	12	20	resolving	resolve	VERB
ejpam-4750	12	21	sets	set	NOUN
ejpam-4750	12	22	in	in	ADP
ejpam-4750	12	23	the	the	DET
ejpam-4750	12	24	join	join	NOUN
ejpam-4750	12	25	and	and	CCONJ
ejpam-4750	12	26	corona	corona	NOUN
ejpam-4750	12	27	of	of	ADP
ejpam-4750	12	28	graphs	graph	NOUN
ejpam-4750	12	29	”	"	PUNCT
ejpam-4750	12	30	was	be	AUX
ejpam-4750	12	31	published	publish	VERB
ejpam-4750	12	32	by	by	ADP
ejpam-4750	12	33	j.	j.	PROPN
ejpam-4750	12	34	cabaro	cabaro	PROPN
ejpam-4750	12	35	and	and	CCONJ
ejpam-4750	12	36	h.	h.	PROPN
ejpam-4750	12	37	rara	rara	NOUN
ejpam-4750	13	1	[	[	X
ejpam-4750	13	2	4	4	X
ejpam-4750	13	3	]	]	PUNCT
ejpam-4750	13	4	in	in	ADP
ejpam-4750	13	5	2021	2021	NUM
ejpam-4750	13	6	.	.	PUNCT
ejpam-4750	14	1	the	the	DET
ejpam-4750	14	2	concept	concept	NOUN
ejpam-4750	14	3	of	of	ADP
ejpam-4750	14	4	forcing	force	VERB
ejpam-4750	14	5	numbers	number	NOUN
ejpam-4750	14	6	,	,	PUNCT
ejpam-4750	14	7	which	which	PRON
ejpam-4750	14	8	was	be	AUX
ejpam-4750	14	9	established	establish	VERB
ejpam-4750	14	10	in	in	ADP
ejpam-4750	14	11	1987	1987	NUM
ejpam-4750	14	12	as	as	ADP
ejpam-4750	14	13	a	a	DET
ejpam-4750	14	14	result	result	NOUN
ejpam-4750	14	15	of	of	ADP
ejpam-4750	14	16	klein	klein	PROPN
ejpam-4750	14	17	and	and	CCONJ
ejpam-4750	14	18	randic	randic	PROPN
ejpam-4750	14	19	’s	’s	PART
ejpam-4750	14	20	introduction	introduction	NOUN
ejpam-4750	14	21	of	of	ADP
ejpam-4750	14	22	the	the	DET
ejpam-4750	14	23	study	study	NOUN
ejpam-4750	14	24	of	of	ADP
ejpam-4750	14	25	molecular	molecular	ADJ
ejpam-4750	14	26	resonance	resonance	NOUN
ejpam-4750	14	27	structure	structure	NOUN
ejpam-4750	14	28	,	,	PUNCT
ejpam-4750	14	29	is	be	AUX
ejpam-4750	14	30	another	another	DET
ejpam-4750	14	31	intriguing	intriguing	ADJ
ejpam-4750	14	32	topic	topic	NOUN
ejpam-4750	14	33	that	that	PRON
ejpam-4750	14	34	has	have	AUX
ejpam-4750	14	35	drawn	draw	VERB
ejpam-4750	14	36	the	the	DET
ejpam-4750	14	37	interest	interest	NOUN
ejpam-4750	14	38	of	of	ADP
ejpam-4750	14	39	several	several	ADJ
ejpam-4750	14	40	researchers	researcher	NOUN
ejpam-4750	14	41	[	[	X
ejpam-4750	14	42	8	8	NUM
ejpam-4750	14	43	]	]	PUNCT
ejpam-4750	14	44	.	.	PUNCT
ejpam-4750	15	1	consequently	consequently	ADV
ejpam-4750	15	2	,	,	PUNCT
ejpam-4750	15	3	in	in	ADP
ejpam-4750	15	4	1991	1991	NUM
ejpam-4750	15	5	,	,	PUNCT
ejpam-4750	15	6	∗corresponding	∗corresponde	VERB
ejpam-4750	15	7	author	author	NOUN
ejpam-4750	15	8	.	.	PUNCT
ejpam-4750	16	1	doi	doi	NOUN
ejpam-4750	16	2	:	:	PUNCT
ejpam-4750	16	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4750	https://doi.org/10.29020/nybg.ejpam.v16i2.4750	NOUN
ejpam-4750	16	4	email	email	NOUN
ejpam-4750	16	5	addresses	address	NOUN
ejpam-4750	16	6	:	:	PUNCT
ejpam-4750	16	7	dennis.managbanag@g.msuiit.edu.ph	dennis.managbanag@g.msuiit.edu.ph	PROPN
ejpam-4750	16	8	(	(	PUNCT
ejpam-4750	16	9	d.	d.	PROPN
ejpam-4750	16	10	managbanag	managbanag	PROPN
ejpam-4750	16	11	)	)	PUNCT
ejpam-4750	16	12	,	,	PUNCT
ejpam-4750	16	13	helen.rara@g.msuiit.edu.ph	helen.rara@g.msuiit.edu.ph	PROPN
ejpam-4750	16	14	(	(	PUNCT
ejpam-4750	16	15	h.	h.	PROPN
ejpam-4750	16	16	rara	rara	PROPN
ejpam-4750	16	17	)	)	PUNCT
ejpam-4750	16	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4750	16	19	1068	1068	NUM
ejpam-4750	17	1	©	©	PROPN
ejpam-4750	17	2	2023	2023	NUM
ejpam-4750	17	3	ejpam	ejpam	NOUN
ejpam-4750	17	4	all	all	DET
ejpam-4750	17	5	rights	right	NOUN
ejpam-4750	17	6	reserved	reserve	VERB
ejpam-4750	17	7	.	.	PUNCT
ejpam-4750	18	1	d.	d.	PROPN
ejpam-4750	18	2	managbanag	managbanag	PROPN
ejpam-4750	18	3	,	,	PUNCT
ejpam-4750	18	4	h.	h.	PROPN
ejpam-4750	18	5	rara	rara	PROPN
ejpam-4750	18	6	/	/	SYM
ejpam-4750	18	7	eur	eur	PROPN
ejpam-4750	18	8	.	.	PUNCT
ejpam-4750	19	1	j.	j.	PROPN
ejpam-4750	19	2	pure	pure	PROPN
ejpam-4750	19	3	appl	appl	PROPN
ejpam-4750	19	4	.	.	PROPN
ejpam-4750	19	5	math	math	PROPN
ejpam-4750	19	6	,	,	PUNCT
ejpam-4750	19	7	16	16	NUM
ejpam-4750	19	8	(	(	PUNCT
ejpam-4750	19	9	2	2	NUM
ejpam-4750	19	10	)	)	PUNCT
ejpam-4750	19	11	(	(	PUNCT
ejpam-4750	19	12	2023	2023	NUM
ejpam-4750	19	13	)	)	PUNCT
ejpam-4750	19	14	,	,	PUNCT
ejpam-4750	19	15	1068	1068	NUM
ejpam-4750	19	16	-	-	SYM
ejpam-4750	19	17	1083	1083	NUM
ejpam-4750	19	18	1069	1069	NUM
ejpam-4750	19	19	harary	harary	PROPN
ejpam-4750	19	20	et	et	PROPN
ejpam-4750	19	21	.	.	PUNCT
ejpam-4750	20	1	al	al	PROPN
ejpam-4750	21	1	[	[	X
ejpam-4750	21	2	6	6	NUM
ejpam-4750	21	3	]	]	PUNCT
ejpam-4750	21	4	coined	coin	VERB
ejpam-4750	21	5	the	the	DET
ejpam-4750	21	6	term	term	NOUN
ejpam-4750	21	7	“	"	PUNCT
ejpam-4750	21	8	forcing	force	VERB
ejpam-4750	21	9	number	number	NOUN
ejpam-4750	21	10	”	"	PUNCT
ejpam-4750	21	11	and	and	CCONJ
ejpam-4750	21	12	presented	present	VERB
ejpam-4750	21	13	the	the	DET
ejpam-4750	21	14	idea	idea	NOUN
ejpam-4750	21	15	of	of	ADP
ejpam-4750	21	16	forcing	force	VERB
ejpam-4750	21	17	as	as	ADP
ejpam-4750	21	18	a	a	DET
ejpam-4750	21	19	perfect	perfect	ADJ
ejpam-4750	21	20	match	match	NOUN
ejpam-4750	21	21	.	.	PUNCT
ejpam-4750	22	1	in	in	ADP
ejpam-4750	22	2	1999	1999	NUM
ejpam-4750	22	3	,	,	PUNCT
ejpam-4750	22	4	chartrand	chartrand	PROPN
ejpam-4750	22	5	et	et	PROPN
ejpam-4750	22	6	.	.	PUNCT
ejpam-4750	23	1	al	al	PROPN
ejpam-4750	24	1	[	[	X
ejpam-4750	24	2	13	13	NUM
ejpam-4750	24	3	]	]	PUNCT
ejpam-4750	24	4	initiated	initiate	VERB
ejpam-4750	24	5	the	the	DET
ejpam-4750	24	6	investigation	investigation	NOUN
ejpam-4750	24	7	on	on	ADP
ejpam-4750	24	8	the	the	DET
ejpam-4750	24	9	relation	relation	NOUN
ejpam-4750	24	10	between	between	ADP
ejpam-4750	24	11	forcing	force	VERB
ejpam-4750	24	12	and	and	CCONJ
ejpam-4750	24	13	dimension	dimension	NOUN
ejpam-4750	24	14	of	of	ADP
ejpam-4750	24	15	a	a	DET
ejpam-4750	24	16	graph	graph	NOUN
ejpam-4750	24	17	.	.	PUNCT
ejpam-4750	25	1	the	the	DET
ejpam-4750	25	2	notions	notion	NOUN
ejpam-4750	25	3	of	of	ADP
ejpam-4750	25	4	a	a	DET
ejpam-4750	25	5	2	2	NUM
ejpam-4750	25	6	-	-	PUNCT
ejpam-4750	25	7	resolving	resolve	VERB
ejpam-4750	25	8	set	set	NOUN
ejpam-4750	25	9	and	and	CCONJ
ejpam-4750	25	10	the	the	DET
ejpam-4750	25	11	forcing	force	VERB
ejpam-4750	25	12	dimension	dimension	NOUN
ejpam-4750	25	13	of	of	ADP
ejpam-4750	25	14	a	a	DET
ejpam-4750	25	15	graph	graph	NOUN
ejpam-4750	25	16	serve	serve	VERB
ejpam-4750	25	17	as	as	ADP
ejpam-4750	25	18	the	the	DET
ejpam-4750	25	19	inspiration	inspiration	NOUN
ejpam-4750	25	20	for	for	ADP
ejpam-4750	25	21	this	this	DET
ejpam-4750	25	22	work	work	NOUN
ejpam-4750	25	23	.	.	PUNCT
ejpam-4750	26	1	we	we	PRON
ejpam-4750	26	2	believe	believe	VERB
ejpam-4750	26	3	that	that	SCONJ
ejpam-4750	26	4	this	this	DET
ejpam-4750	26	5	study	study	NOUN
ejpam-4750	26	6	will	will	AUX
ejpam-4750	26	7	be	be	AUX
ejpam-4750	26	8	tremendously	tremendously	ADV
ejpam-4750	26	9	beneficial	beneficial	ADJ
ejpam-4750	26	10	to	to	ADP
ejpam-4750	26	11	someone	someone	PRON
ejpam-4750	26	12	who	who	PRON
ejpam-4750	26	13	is	be	AUX
ejpam-4750	26	14	familiar	familiar	ADJ
ejpam-4750	26	15	with	with	ADP
ejpam-4750	26	16	the	the	DET
ejpam-4750	26	17	theory	theory	NOUN
ejpam-4750	26	18	of	of	ADP
ejpam-4750	26	19	the	the	DET
ejpam-4750	26	20	metric	metric	ADJ
ejpam-4750	26	21	dimension	dimension	NOUN
ejpam-4750	26	22	.	.	PUNCT
ejpam-4750	27	1	the	the	DET
ejpam-4750	27	2	findings	finding	NOUN
ejpam-4750	27	3	of	of	ADP
ejpam-4750	27	4	this	this	DET
ejpam-4750	27	5	work	work	NOUN
ejpam-4750	27	6	amplified	amplify	VERB
ejpam-4750	27	7	previously	previously	ADV
ejpam-4750	27	8	-	-	PUNCT
ejpam-4750	27	9	revealed	reveal	VERB
ejpam-4750	27	10	notions	notion	NOUN
ejpam-4750	27	11	to	to	PART
ejpam-4750	27	12	obtain	obtain	VERB
ejpam-4750	27	13	new	new	ADJ
ejpam-4750	27	14	applications	application	NOUN
ejpam-4750	27	15	in	in	ADP
ejpam-4750	27	16	graph	graph	NOUN
ejpam-4750	27	17	-	-	PUNCT
ejpam-4750	27	18	to	to	ADP
ejpam-4750	27	19	-	-	PUNCT
ejpam-4750	27	20	code	code	NOUN
ejpam-4750	27	21	theory	theory	NOUN
ejpam-4750	27	22	,	,	PUNCT
ejpam-4750	27	23	much	much	ADJ
ejpam-4750	27	24	like	like	ADP
ejpam-4750	27	25	the	the	DET
ejpam-4750	27	26	idea	idea	NOUN
ejpam-4750	27	27	of	of	ADP
ejpam-4750	27	28	a	a	DET
ejpam-4750	27	29	2	2	NUM
ejpam-4750	27	30	-	-	PUNCT
ejpam-4750	27	31	resolving	resolve	VERB
ejpam-4750	27	32	set	set	NOUN
ejpam-4750	27	33	,	,	PUNCT
ejpam-4750	27	34	by	by	ADP
ejpam-4750	27	35	developing	develop	VERB
ejpam-4750	27	36	a	a	DET
ejpam-4750	27	37	novel	novel	ADJ
ejpam-4750	27	38	method	method	NOUN
ejpam-4750	27	39	for	for	ADP
ejpam-4750	27	40	producing	produce	VERB
ejpam-4750	27	41	error	error	NOUN
ejpam-4750	27	42	-	-	PUNCT
ejpam-4750	27	43	correcting	correct	VERB
ejpam-4750	27	44	codes	code	NOUN
ejpam-4750	27	45	out	out	ADP
ejpam-4750	27	46	of	of	ADP
ejpam-4750	27	47	graphs	graph	NOUN
ejpam-4750	27	48	.	.	PUNCT
ejpam-4750	28	1	2	2	X
ejpam-4750	28	2	.	.	X
ejpam-4750	28	3	terminology	terminology	NOUN
ejpam-4750	28	4	and	and	CCONJ
ejpam-4750	28	5	notation	notation	NOUN
ejpam-4750	28	6	in	in	ADP
ejpam-4750	28	7	this	this	DET
ejpam-4750	28	8	study	study	NOUN
ejpam-4750	28	9	,	,	PUNCT
ejpam-4750	28	10	we	we	PRON
ejpam-4750	28	11	only	only	ADV
ejpam-4750	28	12	consider	consider	VERB
ejpam-4750	28	13	graphs	graph	NOUN
ejpam-4750	28	14	that	that	PRON
ejpam-4750	28	15	are	be	AUX
ejpam-4750	28	16	finite	finite	ADJ
ejpam-4750	28	17	,	,	PUNCT
ejpam-4750	28	18	simple	simple	ADJ
ejpam-4750	28	19	,	,	PUNCT
ejpam-4750	28	20	undirected	undirected	ADJ
ejpam-4750	28	21	and	and	CCONJ
ejpam-4750	28	22	connected	connected	ADJ
ejpam-4750	28	23	.	.	PUNCT
ejpam-4750	29	1	readers	reader	NOUN
ejpam-4750	29	2	are	be	AUX
ejpam-4750	29	3	referred	refer	VERB
ejpam-4750	29	4	to	to	ADP
ejpam-4750	29	5	[	[	X
ejpam-4750	29	6	1	1	NUM
ejpam-4750	29	7	,	,	PUNCT
ejpam-4750	29	8	4	4	NUM
ejpam-4750	29	9	,	,	PUNCT
ejpam-4750	29	10	9	9	NUM
ejpam-4750	29	11	,	,	PUNCT
ejpam-4750	29	12	10	10	NUM
ejpam-4750	29	13	]	]	PUNCT
ejpam-4750	29	14	for	for	ADP
ejpam-4750	29	15	elementary	elementary	ADJ
ejpam-4750	29	16	graph	graph	NOUN
ejpam-4750	29	17	theoretic	theoretic	ADJ
ejpam-4750	29	18	concepts	concept	NOUN
ejpam-4750	29	19	.	.	PUNCT
ejpam-4750	30	1	let	let	VERB
ejpam-4750	30	2	g	g	PRON
ejpam-4750	30	3	be	be	AUX
ejpam-4750	30	4	a	a	DET
ejpam-4750	30	5	connected	connected	ADJ
ejpam-4750	30	6	graph	graph	NOUN
ejpam-4750	30	7	of	of	ADP
ejpam-4750	30	8	order	order	NOUN
ejpam-4750	30	9	n.	n.	NOUN
ejpam-4750	30	10	for	for	ADP
ejpam-4750	30	11	an	an	DET
ejpam-4750	30	12	ordered	order	VERB
ejpam-4750	30	13	set	set	NOUN
ejpam-4750	30	14	of	of	ADP
ejpam-4750	30	15	vertices	vertex	NOUN
ejpam-4750	30	16	w	w	NOUN
ejpam-4750	30	17	=	=	SYM
ejpam-4750	30	18	{	{	PUNCT
ejpam-4750	30	19	w1	w1	NOUN
ejpam-4750	30	20	,	,	PUNCT
ejpam-4750	30	21	w2	w2	NOUN
ejpam-4750	30	22	,	,	PUNCT
ejpam-4750	30	23	...	...	PUNCT
ejpam-4750	30	24	,	,	PUNCT
ejpam-4750	30	25	wk	wk	ADP
ejpam-4750	30	26	}	}	PUNCT
ejpam-4750	30	27	⊆	⊆	NUM
ejpam-4750	30	28	v	v	NOUN
ejpam-4750	30	29	(	(	PUNCT
ejpam-4750	30	30	g	g	NOUN
ejpam-4750	30	31	)	)	PUNCT
ejpam-4750	30	32	and	and	CCONJ
ejpam-4750	30	33	a	a	DET
ejpam-4750	30	34	vertex	vertex	NOUN
ejpam-4750	30	35	v	v	NOUN
ejpam-4750	30	36	in	in	ADP
ejpam-4750	30	37	g	g	NOUN
ejpam-4750	30	38	,	,	PUNCT
ejpam-4750	30	39	we	we	PRON
ejpam-4750	30	40	refer	refer	VERB
ejpam-4750	30	41	to	to	ADP
ejpam-4750	30	42	the	the	DET
ejpam-4750	30	43	k	k	NOUN
ejpam-4750	30	44	-	-	NOUN
ejpam-4750	30	45	vector	vector	NOUN
ejpam-4750	30	46	(	(	PUNCT
ejpam-4750	30	47	ordered	order	VERB
ejpam-4750	30	48	k	k	NOUN
ejpam-4750	30	49	-	-	PUNCT
ejpam-4750	30	50	tuple	tuple	NOUN
ejpam-4750	30	51	)	)	PUNCT
ejpam-4750	30	52	rg(v	rg(v	PROPN
ejpam-4750	30	53	/	/	SYM
ejpam-4750	30	54	w	w	NOUN
ejpam-4750	30	55	)	)	PUNCT
ejpam-4750	31	1	=	=	SYM
ejpam-4750	31	2	(	(	PUNCT
ejpam-4750	31	3	dg(v	dg(v	X
ejpam-4750	31	4	,	,	PUNCT
ejpam-4750	31	5	w1	w1	NOUN
ejpam-4750	31	6	)	)	PUNCT
ejpam-4750	31	7	,	,	PUNCT
ejpam-4750	31	8	dg(v	dg(v	X
ejpam-4750	31	9	,	,	PUNCT
ejpam-4750	31	10	w2	w2	NOUN
ejpam-4750	31	11	)	)	PUNCT
ejpam-4750	31	12	,	,	PUNCT
ejpam-4750	31	13	...	...	PUNCT
ejpam-4750	31	14	,	,	PUNCT
ejpam-4750	31	15	dg(v	dg(v	X
ejpam-4750	31	16	,	,	PUNCT
ejpam-4750	31	17	wk	wk	NOUN
ejpam-4750	31	18	)	)	PUNCT
ejpam-4750	31	19	)	)	PUNCT
ejpam-4750	32	1	as	as	SCONJ
ejpam-4750	32	2	the	the	DET
ejpam-4750	32	3	(	(	PUNCT
ejpam-4750	32	4	metric	metric	ADJ
ejpam-4750	32	5	)	)	PUNCT
ejpam-4750	32	6	representation	representation	NOUN
ejpam-4750	32	7	of	of	ADP
ejpam-4750	32	8	v	v	NOUN
ejpam-4750	32	9	with	with	ADP
ejpam-4750	32	10	respect	respect	NOUN
ejpam-4750	32	11	to	to	ADP
ejpam-4750	32	12	w.	w.	PROPN
ejpam-4750	32	13	an	an	DET
ejpam-4750	32	14	ordered	order	VERB
ejpam-4750	32	15	set	set	NOUN
ejpam-4750	32	16	of	of	ADP
ejpam-4750	32	17	vertices	vertex	NOUN
ejpam-4750	32	18	w	w	NOUN
ejpam-4750	32	19	=	=	SYM
ejpam-4750	32	20	{	{	PUNCT
ejpam-4750	32	21	w1	w1	NOUN
ejpam-4750	32	22	,	,	PUNCT
ejpam-4750	32	23	w2	w2	NOUN
ejpam-4750	32	24	,	,	PUNCT
ejpam-4750	32	25	...	...	PUNCT
ejpam-4750	32	26	,	,	PUNCT
ejpam-4750	32	27	wk	wk	PROPN
ejpam-4750	32	28	}	}	PUNCT
ejpam-4750	32	29	is	be	AUX
ejpam-4750	32	30	a	a	DET
ejpam-4750	32	31	k	k	NOUN
ejpam-4750	32	32	-	-	PUNCT
ejpam-4750	32	33	resolving	resolving	NOUN
ejpam-4750	32	34	set	set	NOUN
ejpam-4750	32	35	for	for	ADP
ejpam-4750	32	36	g	g	PROPN
ejpam-4750	32	37	if	if	SCONJ
ejpam-4750	32	38	,	,	PUNCT
ejpam-4750	32	39	for	for	ADP
ejpam-4750	32	40	any	any	DET
ejpam-4750	32	41	distinct	distinct	ADJ
ejpam-4750	32	42	vertices	vertex	NOUN
ejpam-4750	32	43	u	u	NOUN
ejpam-4750	32	44	,	,	PUNCT
ejpam-4750	32	45	v	v	NOUN
ejpam-4750	32	46	∈	∈	PROPN
ejpam-4750	32	47	v	v	NOUN
ejpam-4750	32	48	(	(	PUNCT
ejpam-4750	32	49	g	g	NOUN
ejpam-4750	32	50	)	)	PUNCT
ejpam-4750	32	51	,	,	PUNCT
ejpam-4750	32	52	the	the	DET
ejpam-4750	32	53	(	(	PUNCT
ejpam-4750	32	54	metric	metric	ADJ
ejpam-4750	32	55	)	)	PUNCT
ejpam-4750	32	56	representations	representation	NOUN
ejpam-4750	32	57	rg(u	rg(u	NOUN
ejpam-4750	32	58	/	/	SYM
ejpam-4750	32	59	w	w	NOUN
ejpam-4750	32	60	)	)	PUNCT
ejpam-4750	32	61	and	and	CCONJ
ejpam-4750	32	62	rg(v	rg(v	PROPN
ejpam-4750	32	63	/	/	SYM
ejpam-4750	32	64	w	w	NOUN
ejpam-4750	32	65	)	)	PUNCT
ejpam-4750	32	66	of	of	ADP
ejpam-4750	32	67	u	u	NOUN
ejpam-4750	32	68	and	and	CCONJ
ejpam-4750	32	69	v	v	NOUN
ejpam-4750	32	70	,	,	PUNCT
ejpam-4750	32	71	respectively	respectively	ADV
ejpam-4750	32	72	differ	differ	VERB
ejpam-4750	32	73	in	in	ADP
ejpam-4750	32	74	at	at	ADP
ejpam-4750	32	75	least	least	ADJ
ejpam-4750	32	76	k	k	NOUN
ejpam-4750	32	77	positions	position	NOUN
ejpam-4750	32	78	.	.	PUNCT
ejpam-4750	33	1	if	if	SCONJ
ejpam-4750	33	2	k	k	PROPN
ejpam-4750	33	3	=	=	SYM
ejpam-4750	33	4	1	1	NUM
ejpam-4750	33	5	,	,	PUNCT
ejpam-4750	33	6	then	then	ADV
ejpam-4750	33	7	the	the	DET
ejpam-4750	33	8	k	k	NOUN
ejpam-4750	33	9	-	-	PUNCT
ejpam-4750	33	10	resolving	resolving	ADJ
ejpam-4750	33	11	set	set	NOUN
ejpam-4750	33	12	is	be	AUX
ejpam-4750	33	13	called	call	VERB
ejpam-4750	33	14	a	a	DET
ejpam-4750	33	15	resolving	resolving	NOUN
ejpam-4750	33	16	set	set	VERB
ejpam-4750	33	17	for	for	ADP
ejpam-4750	33	18	g.	g.	PROPN
ejpam-4750	33	19	if	if	SCONJ
ejpam-4750	33	20	k	k	PROPN
ejpam-4750	33	21	=	=	SYM
ejpam-4750	33	22	2	2	NUM
ejpam-4750	33	23	,	,	PUNCT
ejpam-4750	33	24	then	then	ADV
ejpam-4750	33	25	the	the	DET
ejpam-4750	33	26	k	k	NOUN
ejpam-4750	33	27	-	-	PUNCT
ejpam-4750	33	28	resolving	resolving	ADJ
ejpam-4750	33	29	set	set	NOUN
ejpam-4750	33	30	is	be	AUX
ejpam-4750	33	31	called	call	VERB
ejpam-4750	33	32	a	a	DET
ejpam-4750	33	33	2	2	NUM
ejpam-4750	33	34	-	-	PUNCT
ejpam-4750	33	35	resolving	resolving	NOUN
ejpam-4750	33	36	set	set	NOUN
ejpam-4750	33	37	for	for	ADP
ejpam-4750	33	38	g.	g.	PROPN
ejpam-4750	33	39	the	the	DET
ejpam-4750	33	40	least	least	ADJ
ejpam-4750	33	41	size	size	NOUN
ejpam-4750	33	42	of	of	ADP
ejpam-4750	33	43	a	a	DET
ejpam-4750	33	44	2	2	NUM
ejpam-4750	33	45	-	-	PUNCT
ejpam-4750	33	46	resolving	resolve	VERB
ejpam-4750	33	47	set	set	NOUN
ejpam-4750	33	48	is	be	AUX
ejpam-4750	33	49	called	call	VERB
ejpam-4750	33	50	a	a	DET
ejpam-4750	33	51	2	2	NUM
ejpam-4750	33	52	-	-	PUNCT
ejpam-4750	33	53	metric	metric	ADJ
ejpam-4750	33	54	dimension	dimension	NOUN
ejpam-4750	33	55	of	of	ADP
ejpam-4750	33	56	g	g	PROPN
ejpam-4750	33	57	and	and	CCONJ
ejpam-4750	33	58	we	we	PRON
ejpam-4750	33	59	denote	denote	VERB
ejpam-4750	33	60	it	it	PRON
ejpam-4750	33	61	by	by	ADP
ejpam-4750	33	62	dim2(g	dim2(g	NOUN
ejpam-4750	33	63	)	)	PUNCT
ejpam-4750	33	64	.	.	PUNCT
ejpam-4750	34	1	a	a	DET
ejpam-4750	34	2	resolving	resolving	NOUN
ejpam-4750	34	3	set	set	VERB
ejpam-4750	34	4	of	of	ADP
ejpam-4750	34	5	size	size	NOUN
ejpam-4750	34	6	dim2(g	dim2(g	NOUN
ejpam-4750	34	7	)	)	PUNCT
ejpam-4750	34	8	is	be	AUX
ejpam-4750	34	9	called	call	VERB
ejpam-4750	34	10	a	a	DET
ejpam-4750	34	11	2	2	NUM
ejpam-4750	34	12	-	-	PUNCT
ejpam-4750	34	13	metric	metric	ADJ
ejpam-4750	34	14	basis	basis	NOUN
ejpam-4750	34	15	for	for	ADP
ejpam-4750	34	16	g.	g.	PROPN
ejpam-4750	34	17	let	let	VERB
ejpam-4750	34	18	g	g	NOUN
ejpam-4750	34	19	be	be	AUX
ejpam-4750	34	20	any	any	DET
ejpam-4750	34	21	nontrivial	nontrivial	ADJ
ejpam-4750	34	22	connected	connect	VERB
ejpam-4750	34	23	graph	graph	NOUN
ejpam-4750	34	24	and	and	CCONJ
ejpam-4750	34	25	s	s	VERB
ejpam-4750	34	26	⊆	⊆	NUM
ejpam-4750	34	27	v	v	NOUN
ejpam-4750	34	28	(	(	PUNCT
ejpam-4750	34	29	g	g	NOUN
ejpam-4750	34	30	)	)	PUNCT
ejpam-4750	34	31	.	.	PUNCT
ejpam-4750	35	1	a	a	DET
ejpam-4750	35	2	set	set	NOUN
ejpam-4750	35	3	s	s	NOUN
ejpam-4750	35	4	⊆	⊆	NUM
ejpam-4750	35	5	v	v	NOUN
ejpam-4750	35	6	(	(	PUNCT
ejpam-4750	35	7	g	g	NOUN
ejpam-4750	35	8	)	)	PUNCT
ejpam-4750	35	9	is	be	AUX
ejpam-4750	35	10	2	2	NUM
ejpam-4750	35	11	-	-	PUNCT
ejpam-4750	35	12	locating	locate	VERB
ejpam-4750	35	13	set	set	NOUN
ejpam-4750	35	14	of	of	ADP
ejpam-4750	35	15	g	g	NOUN
ejpam-4750	35	16	if	if	SCONJ
ejpam-4750	35	17	it	it	PRON
ejpam-4750	35	18	satisfies	satisfy	VERB
ejpam-4750	35	19	the	the	DET
ejpam-4750	35	20	following	follow	VERB
ejpam-4750	35	21	conditions	condition	NOUN
ejpam-4750	35	22	:	:	PUNCT
ejpam-4750	35	23	(	(	PUNCT
ejpam-4750	35	24	i	i	NOUN
ejpam-4750	35	25	)	)	PUNCT
ejpam-4750	35	26	|[(ng(x	|[(ng(x	NUM
ejpam-4750	35	27	)	)	PUNCT
ejpam-4750	35	28	\	\	NOUN
ejpam-4750	35	29	ng(y	ng(y	NOUN
ejpam-4750	35	30	)	)	PUNCT
ejpam-4750	35	31	)	)	PUNCT
ejpam-4750	36	1	∩	∩	PROPN
ejpam-4750	36	2	s	s	X
ejpam-4750	36	3	]	]	X
ejpam-4750	36	4	∪	∪	X
ejpam-4750	36	5	[	[	X
ejpam-4750	36	6	(	(	PUNCT
ejpam-4750	36	7	ng(y	ng(y	NOUN
ejpam-4750	36	8	)	)	PUNCT
ejpam-4750	36	9	\ng(x))∩	\ng(x))∩	VERB
ejpam-4750	36	10	s]|	s]|	PROPN
ejpam-4750	36	11	≥	≥	PROPN
ejpam-4750	36	12	2	2	NUM
ejpam-4750	36	13	,	,	PUNCT
ejpam-4750	36	14	for	for	ADP
ejpam-4750	36	15	all	all	DET
ejpam-4750	36	16	x	x	NOUN
ejpam-4750	36	17	,	,	PUNCT
ejpam-4750	36	18	y	y	PROPN
ejpam-4750	36	19	∈	∈	PROPN
ejpam-4750	36	20	v	v	ADP
ejpam-4750	36	21	(	(	PUNCT
ejpam-4750	36	22	g	g	NOUN
ejpam-4750	36	23	)	)	PUNCT
ejpam-4750	36	24	\	\	PROPN
ejpam-4750	37	1	s	s	PART
ejpam-4750	37	2	with	with	ADP
ejpam-4750	37	3	x	x	SYM
ejpam-4750	37	4	̸=	̸=	PROPN
ejpam-4750	37	5	y	y	PROPN
ejpam-4750	37	6	and	and	CCONJ
ejpam-4750	37	7	(	(	PUNCT
ejpam-4750	37	8	ii	ii	NOUN
ejpam-4750	37	9	)	)	PUNCT
ejpam-4750	37	10	(	(	PUNCT
ejpam-4750	37	11	ng(v	ng(v	NOUN
ejpam-4750	37	12	)	)	PUNCT
ejpam-4750	37	13	\ng(w	\ng(w	NUM
ejpam-4750	37	14	)	)	PUNCT
ejpam-4750	37	15	)	)	PUNCT
ejpam-4750	38	1	∩s	∩s	PROPN
ejpam-4750	38	2	̸=	̸=	PROPN
ejpam-4750	38	3	∅	∅	NOUN
ejpam-4750	38	4	or	or	CCONJ
ejpam-4750	38	5	(	(	PUNCT
ejpam-4750	38	6	ng(w)\ng[v])∩s	ng(w)\ng[v])∩s	NUM
ejpam-4750	38	7	̸=	̸=	PROPN
ejpam-4750	38	8	∅	∅	NOUN
ejpam-4750	38	9	for	for	ADP
ejpam-4750	38	10	all	all	PRON
ejpam-4750	38	11	v	v	ADP
ejpam-4750	38	12	∈	∈	NOUN
ejpam-4750	38	13	s	s	NOUN
ejpam-4750	38	14	and	and	CCONJ
ejpam-4750	38	15	for	for	ADP
ejpam-4750	38	16	all	all	PRON
ejpam-4750	38	17	w	w	PROPN
ejpam-4750	38	18	∈	∈	PROPN
ejpam-4750	38	19	v	v	NOUN
ejpam-4750	38	20	(	(	PUNCT
ejpam-4750	38	21	g)\s	g)\s	NOUN
ejpam-4750	38	22	.	.	PUNCT
ejpam-4750	39	1	the	the	DET
ejpam-4750	39	2	2	2	NUM
ejpam-4750	39	3	-	-	PUNCT
ejpam-4750	39	4	locating	locate	VERB
ejpam-4750	39	5	number	number	NOUN
ejpam-4750	39	6	of	of	ADP
ejpam-4750	39	7	g	g	NOUN
ejpam-4750	39	8	,	,	PUNCT
ejpam-4750	39	9	denoted	denote	VERB
ejpam-4750	39	10	by	by	ADP
ejpam-4750	39	11	ln2(g	ln2(g	NOUN
ejpam-4750	39	12	)	)	PUNCT
ejpam-4750	39	13	,	,	PUNCT
ejpam-4750	39	14	is	be	AUX
ejpam-4750	39	15	the	the	DET
ejpam-4750	39	16	smallest	small	ADJ
ejpam-4750	39	17	cardinality	cardinality	NOUN
ejpam-4750	39	18	of	of	ADP
ejpam-4750	39	19	a	a	DET
ejpam-4750	39	20	2	2	NUM
ejpam-4750	39	21	-	-	PUNCT
ejpam-4750	39	22	locating	locate	VERB
ejpam-4750	39	23	set	set	NOUN
ejpam-4750	39	24	of	of	ADP
ejpam-4750	39	25	g.	g.	PROPN
ejpam-4750	39	26	a	a	DET
ejpam-4750	39	27	2	2	NUM
ejpam-4750	39	28	-	-	PUNCT
ejpam-4750	39	29	locating	locate	VERB
ejpam-4750	39	30	set	set	NOUN
ejpam-4750	39	31	of	of	ADP
ejpam-4750	39	32	g	g	NOUN
ejpam-4750	39	33	of	of	ADP
ejpam-4750	39	34	cardinality	cardinality	PROPN
ejpam-4750	39	35	ln2(g	ln2(g	PROPN
ejpam-4750	39	36	)	)	PUNCT
ejpam-4750	39	37	is	be	AUX
ejpam-4750	39	38	referred	refer	VERB
ejpam-4750	39	39	to	to	ADP
ejpam-4750	39	40	as	as	ADP
ejpam-4750	39	41	an	an	DET
ejpam-4750	39	42	ln2	ln2	NOUN
ejpam-4750	39	43	-	-	PUNCT
ejpam-4750	39	44	set	set	NOUN
ejpam-4750	39	45	of	of	ADP
ejpam-4750	39	46	g.	g.	PROPN
ejpam-4750	39	47	let	let	VERB
ejpam-4750	39	48	g	g	NOUN
ejpam-4750	39	49	be	be	AUX
ejpam-4750	39	50	any	any	DET
ejpam-4750	39	51	nontrivial	nontrivial	ADJ
ejpam-4750	39	52	connected	connect	VERB
ejpam-4750	39	53	graph	graph	NOUN
ejpam-4750	39	54	and	and	CCONJ
ejpam-4750	39	55	s	s	VERB
ejpam-4750	39	56	⊆	⊆	NUM
ejpam-4750	39	57	v	v	NOUN
ejpam-4750	39	58	(	(	PUNCT
ejpam-4750	39	59	g	g	NOUN
ejpam-4750	39	60	)	)	PUNCT
ejpam-4750	39	61	.	.	PUNCT
ejpam-4750	40	1	s	s	PART
ejpam-4750	40	2	is	be	AUX
ejpam-4750	40	3	a	a	DET
ejpam-4750	40	4	(	(	PUNCT
ejpam-4750	40	5	2,2)-locating	2,2)-locating	NUM
ejpam-4750	40	6	(	(	PUNCT
ejpam-4750	40	7	respectively	respectively	ADV
ejpam-4750	40	8	(	(	PUNCT
ejpam-4750	40	9	2,1)-locating	2,1)-locating	NUM
ejpam-4750	40	10	)	)	PUNCT
ejpam-4750	40	11	set	set	VERB
ejpam-4750	40	12	in	in	ADP
ejpam-4750	40	13	g	g	PROPN
ejpam-4750	40	14	if	if	SCONJ
ejpam-4750	40	15	s	s	VERB
ejpam-4750	40	16	is	be	AUX
ejpam-4750	40	17	a	a	DET
ejpam-4750	40	18	2	2	NUM
ejpam-4750	40	19	-	-	PUNCT
ejpam-4750	40	20	locating	locate	VERB
ejpam-4750	40	21	and	and	CCONJ
ejpam-4750	40	22	|ng(y	|ng(y	NUM
ejpam-4750	40	23	)	)	PUNCT
ejpam-4750	40	24	∩	∩	NOUN
ejpam-4750	40	25	s|	s|	VERB
ejpam-4750	40	26	≤	≤	NUM
ejpam-4750	40	27	|s|	|s|	PROPN
ejpam-4750	40	28	−	−	PROPN
ejpam-4750	40	29	2	2	NUM
ejpam-4750	40	30	(	(	PUNCT
ejpam-4750	40	31	|ng(y	|ng(y	NUM
ejpam-4750	40	32	)	)	PUNCT
ejpam-4750	40	33	∩	∩	NOUN
ejpam-4750	40	34	s|	s|	VERB
ejpam-4750	40	35	≤	≤	NUM
ejpam-4750	40	36	|s|	|s|	PROPN
ejpam-4750	40	37	−	−	PROPN
ejpam-4750	40	38	1	1	NUM
ejpam-4750	40	39	,	,	PUNCT
ejpam-4750	40	40	respectively	respectively	ADV
ejpam-4750	40	41	)	)	PUNCT
ejpam-4750	40	42	,	,	PUNCT
ejpam-4750	40	43	for	for	ADP
ejpam-4750	40	44	all	all	DET
ejpam-4750	40	45	y	y	PROPN
ejpam-4750	40	46	∈	∈	PROPN
ejpam-4750	40	47	v	v	NOUN
ejpam-4750	40	48	(	(	PUNCT
ejpam-4750	40	49	g	g	NOUN
ejpam-4750	40	50	)	)	PUNCT
ejpam-4750	40	51	.	.	PUNCT
ejpam-4750	41	1	the	the	DET
ejpam-4750	41	2	(	(	PUNCT
ejpam-4750	41	3	2,2)-locating	2,2)-locating	NUM
ejpam-4750	41	4	(	(	PUNCT
ejpam-4750	41	5	respectively	respectively	ADV
ejpam-4750	41	6	(	(	PUNCT
ejpam-4750	41	7	2,1)-locating	2,1)-locating	NUM
ejpam-4750	41	8	)	)	PUNCT
ejpam-4750	41	9	number	number	NOUN
ejpam-4750	41	10	of	of	ADP
ejpam-4750	41	11	g	g	NOUN
ejpam-4750	41	12	,	,	PUNCT
ejpam-4750	41	13	denoted	denote	VERB
ejpam-4750	41	14	by	by	ADP
ejpam-4750	41	15	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	41	16	)	)	PUNCT
ejpam-4750	41	17	(	(	PUNCT
ejpam-4750	41	18	respectively	respectively	ADV
ejpam-4750	41	19	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	41	20	)	)	PUNCT
ejpam-4750	41	21	)	)	PUNCT
ejpam-4750	41	22	,	,	PUNCT
ejpam-4750	41	23	is	be	AUX
ejpam-4750	41	24	the	the	DET
ejpam-4750	41	25	smallest	small	ADJ
ejpam-4750	41	26	cardinality	cardinality	NOUN
ejpam-4750	41	27	of	of	ADP
ejpam-4750	41	28	a	a	DET
ejpam-4750	41	29	(	(	PUNCT
ejpam-4750	41	30	2,2)-locating	2,2)-locating	NUM
ejpam-4750	41	31	(	(	PUNCT
ejpam-4750	41	32	respectively	respectively	ADV
ejpam-4750	41	33	(	(	PUNCT
ejpam-4750	41	34	2,1)-locating	2,1)-locating	NUM
ejpam-4750	41	35	)	)	PUNCT
ejpam-4750	41	36	set	set	VERB
ejpam-4750	41	37	in	in	ADP
ejpam-4750	41	38	g.	g.	PROPN
ejpam-4750	41	39	a	a	PRON
ejpam-4750	41	40	(	(	PUNCT
ejpam-4750	41	41	2,2)-locating	2,2)-locating	NUM
ejpam-4750	41	42	(	(	PUNCT
ejpam-4750	41	43	respectively	respectively	ADV
ejpam-4750	41	44	(	(	PUNCT
ejpam-4750	41	45	2,1)-locating	2,1)-locating	NUM
ejpam-4750	41	46	)	)	PUNCT
ejpam-4750	41	47	set	set	VERB
ejpam-4750	41	48	in	in	ADP
ejpam-4750	41	49	g	g	NOUN
ejpam-4750	41	50	of	of	ADP
ejpam-4750	41	51	cardinality	cardinality	NOUN
ejpam-4750	41	52	ln(2,2)(g	ln(2,2)(g	PROPN
ejpam-4750	41	53	)	)	PUNCT
ejpam-4750	41	54	(	(	PUNCT
ejpam-4750	41	55	respectively	respectively	ADV
ejpam-4750	41	56	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	41	57	)	)	PUNCT
ejpam-4750	41	58	)	)	PUNCT
ejpam-4750	41	59	is	be	AUX
ejpam-4750	41	60	referred	refer	VERB
ejpam-4750	41	61	to	to	ADP
ejpam-4750	41	62	as	as	ADP
ejpam-4750	41	63	an	an	DET
ejpam-4750	41	64	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	41	65	(	(	PUNCT
ejpam-4750	41	66	respectively	respectively	ADV
ejpam-4750	41	67	ln(2,1)-set	ln(2,1)-set	VERB
ejpam-4750	41	68	)	)	PUNCT
ejpam-4750	41	69	in	in	ADP
ejpam-4750	41	70	g.	g.	PROPN
ejpam-4750	41	71	let	let	VERB
ejpam-4750	41	72	w	w	NOUN
ejpam-4750	41	73	be	be	AUX
ejpam-4750	41	74	a	a	DET
ejpam-4750	41	75	2	2	NUM
ejpam-4750	41	76	-	-	PUNCT
ejpam-4750	41	77	metric	metric	ADJ
ejpam-4750	41	78	basis	basis	NOUN
ejpam-4750	41	79	of	of	ADP
ejpam-4750	41	80	a	a	DET
ejpam-4750	41	81	graph	graph	NOUN
ejpam-4750	41	82	g.	g.	NOUN
ejpam-4750	41	83	a	a	DET
ejpam-4750	41	84	subset	subset	NOUN
ejpam-4750	41	85	s	s	NOUN
ejpam-4750	41	86	of	of	ADP
ejpam-4750	41	87	w	w	NOUN
ejpam-4750	41	88	is	be	AUX
ejpam-4750	41	89	said	say	VERB
ejpam-4750	41	90	to	to	PART
ejpam-4750	41	91	be	be	AUX
ejpam-4750	41	92	a	a	DET
ejpam-4750	41	93	forcing	forcing	NOUN
ejpam-4750	41	94	subset	subset	NOUN
ejpam-4750	41	95	for	for	ADP
ejpam-4750	41	96	w	w	PROPN
ejpam-4750	41	97	if	if	SCONJ
ejpam-4750	41	98	w	w	PROPN
ejpam-4750	41	99	is	be	AUX
ejpam-4750	41	100	the	the	DET
ejpam-4750	41	101	unique	unique	ADJ
ejpam-4750	41	102	2	2	NUM
ejpam-4750	41	103	-	-	PUNCT
ejpam-4750	41	104	metric	metric	ADJ
ejpam-4750	41	105	basis	basis	NOUN
ejpam-4750	41	106	containing	contain	VERB
ejpam-4750	41	107	s.	s.	PROPN
ejpam-4750	41	108	the	the	DET
ejpam-4750	41	109	forcing	force	VERB
ejpam-4750	41	110	2	2	NUM
ejpam-4750	41	111	-	-	PUNCT
ejpam-4750	41	112	metric	metric	ADJ
ejpam-4750	41	113	dimension	dimension	NOUN
ejpam-4750	41	114	of	of	ADP
ejpam-4750	41	115	w	w	PROPN
ejpam-4750	41	116	is	be	AUX
ejpam-4750	41	117	given	give	VERB
ejpam-4750	41	118	by	by	ADP
ejpam-4750	41	119	fdim2(w	fdim2(w	NOUN
ejpam-4750	41	120	)	)	PUNCT
ejpam-4750	42	1	=	=	NOUN
ejpam-4750	42	2	min{|s|	min{|s|	NOUN
ejpam-4750	42	3	:	:	PUNCT
ejpam-4750	42	4	s	s	VERB
ejpam-4750	42	5	is	be	AUX
ejpam-4750	42	6	a	a	DET
ejpam-4750	42	7	forcing	forcing	NOUN
ejpam-4750	42	8	subset	subset	NOUN
ejpam-4750	42	9	for	for	ADP
ejpam-4750	42	10	w	w	NOUN
ejpam-4750	42	11	}	}	PUNCT
ejpam-4750	42	12	.	.	PUNCT
ejpam-4750	43	1	the	the	DET
ejpam-4750	43	2	forcing	force	VERB
ejpam-4750	43	3	2	2	NUM
ejpam-4750	43	4	-	-	PUNCT
ejpam-4750	43	5	metric	metric	ADJ
ejpam-4750	43	6	dimension	dimension	NOUN
ejpam-4750	43	7	of	of	ADP
ejpam-4750	43	8	g	g	PROPN
ejpam-4750	43	9	is	be	AUX
ejpam-4750	43	10	given	give	VERB
ejpam-4750	43	11	by	by	ADP
ejpam-4750	43	12	fdim2(g	fdim2(g	NOUN
ejpam-4750	43	13	)	)	PUNCT
ejpam-4750	43	14	=	=	SYM
ejpam-4750	43	15	min{fdim2(w	min{fdim2(w	PROPN
ejpam-4750	43	16	)	)	PUNCT
ejpam-4750	43	17	:	:	PUNCT
ejpam-4750	44	1	w	w	NOUN
ejpam-4750	44	2	is	be	AUX
ejpam-4750	44	3	a	a	DET
ejpam-4750	44	4	2	2	NUM
ejpam-4750	44	5	-	-	PUNCT
ejpam-4750	44	6	metric	metric	ADJ
ejpam-4750	44	7	basis	basis	NOUN
ejpam-4750	44	8	for	for	ADP
ejpam-4750	44	9	g	g	NOUN
ejpam-4750	44	10	}	}	PUNCT
ejpam-4750	44	11	.	.	PUNCT
ejpam-4750	45	1	let	let	VERB
ejpam-4750	45	2	w	w	NOUN
ejpam-4750	45	3	be	be	AUX
ejpam-4750	45	4	an	an	DET
ejpam-4750	45	5	ln2	ln2	NOUN
ejpam-4750	45	6	-	-	PUNCT
ejpam-4750	45	7	set	set	NOUN
ejpam-4750	45	8	of	of	ADP
ejpam-4750	45	9	a	a	DET
ejpam-4750	45	10	graph	graph	NOUN
ejpam-4750	45	11	g.	g.	NOUN
ejpam-4750	45	12	a	a	DET
ejpam-4750	45	13	subset	subset	NOUN
ejpam-4750	45	14	s	s	NOUN
ejpam-4750	45	15	of	of	ADP
ejpam-4750	45	16	w	w	NOUN
ejpam-4750	45	17	is	be	AUX
ejpam-4750	45	18	said	say	VERB
ejpam-4750	45	19	to	to	PART
ejpam-4750	45	20	be	be	AUX
ejpam-4750	45	21	a	a	DET
ejpam-4750	45	22	forcing	forcing	NOUN
ejpam-4750	45	23	subset	subset	NOUN
ejpam-4750	45	24	for	for	ADP
ejpam-4750	45	25	d.	d.	PROPN
ejpam-4750	45	26	managbanag	managbanag	PROPN
ejpam-4750	45	27	,	,	PUNCT
ejpam-4750	45	28	h.	h.	PROPN
ejpam-4750	45	29	rara	rara	PROPN
ejpam-4750	45	30	/	/	SYM
ejpam-4750	45	31	eur	eur	PROPN
ejpam-4750	45	32	.	.	PUNCT
ejpam-4750	46	1	j.	j.	PROPN
ejpam-4750	46	2	pure	pure	PROPN
ejpam-4750	46	3	appl	appl	PROPN
ejpam-4750	46	4	.	.	PROPN
ejpam-4750	46	5	math	math	PROPN
ejpam-4750	46	6	,	,	PUNCT
ejpam-4750	46	7	16	16	NUM
ejpam-4750	46	8	(	(	PUNCT
ejpam-4750	46	9	2	2	NUM
ejpam-4750	46	10	)	)	PUNCT
ejpam-4750	46	11	(	(	PUNCT
ejpam-4750	46	12	2023	2023	NUM
ejpam-4750	46	13	)	)	PUNCT
ejpam-4750	46	14	,	,	PUNCT
ejpam-4750	46	15	1068	1068	NUM
ejpam-4750	46	16	-	-	SYM
ejpam-4750	46	17	1083	1083	NUM
ejpam-4750	46	18	1070	1070	NUM
ejpam-4750	46	19	w	w	NOUN
ejpam-4750	47	1	if	if	SCONJ
ejpam-4750	47	2	w	w	NOUN
ejpam-4750	47	3	is	be	AUX
ejpam-4750	47	4	the	the	DET
ejpam-4750	47	5	unique	unique	ADJ
ejpam-4750	47	6	ln2	ln2	ADJ
ejpam-4750	47	7	-	-	PUNCT
ejpam-4750	47	8	set	set	NOUN
ejpam-4750	47	9	containing	contain	VERB
ejpam-4750	47	10	s.	s.	PROPN
ejpam-4750	47	11	the	the	DET
ejpam-4750	47	12	forcing	force	VERB
ejpam-4750	47	13	2	2	NUM
ejpam-4750	47	14	-	-	PUNCT
ejpam-4750	47	15	locating	locate	VERB
ejpam-4750	47	16	number	number	NOUN
ejpam-4750	47	17	of	of	ADP
ejpam-4750	47	18	w	w	NOUN
ejpam-4750	47	19	is	be	AUX
ejpam-4750	47	20	given	give	VERB
ejpam-4750	47	21	by	by	ADP
ejpam-4750	47	22	fln2(w	fln2(w	NOUN
ejpam-4750	47	23	)	)	PUNCT
ejpam-4750	47	24	=	=	NOUN
ejpam-4750	47	25	min{|s|	min{|s|	NOUN
ejpam-4750	47	26	:	:	PUNCT
ejpam-4750	47	27	s	s	VERB
ejpam-4750	47	28	is	be	AUX
ejpam-4750	47	29	a	a	DET
ejpam-4750	47	30	forcing	forcing	NOUN
ejpam-4750	47	31	subset	subset	NOUN
ejpam-4750	47	32	for	for	ADP
ejpam-4750	47	33	w	w	NOUN
ejpam-4750	47	34	}	}	PUNCT
ejpam-4750	47	35	.	.	PUNCT
ejpam-4750	48	1	the	the	DET
ejpam-4750	48	2	forcing	force	VERB
ejpam-4750	48	3	2	2	NUM
ejpam-4750	48	4	-	-	PUNCT
ejpam-4750	48	5	locating	locate	VERB
ejpam-4750	48	6	number	number	NOUN
ejpam-4750	48	7	of	of	ADP
ejpam-4750	48	8	g	g	PROPN
ejpam-4750	48	9	is	be	AUX
ejpam-4750	48	10	given	give	VERB
ejpam-4750	48	11	by	by	ADP
ejpam-4750	48	12	fln2(g	fln2(g	NOUN
ejpam-4750	48	13	)	)	PUNCT
ejpam-4750	48	14	=	=	SYM
ejpam-4750	48	15	min{fln2(w	min{fln2(w	PROPN
ejpam-4750	48	16	)	)	PUNCT
ejpam-4750	48	17	:	:	PUNCT
ejpam-4750	49	1	w	w	NOUN
ejpam-4750	49	2	is	be	AUX
ejpam-4750	49	3	a	a	DET
ejpam-4750	49	4	ln2	ln2	NOUN
ejpam-4750	49	5	-	-	PUNCT
ejpam-4750	49	6	set	set	NOUN
ejpam-4750	49	7	of	of	ADP
ejpam-4750	49	8	g	g	NOUN
ejpam-4750	49	9	}	}	PUNCT
ejpam-4750	49	10	.	.	PUNCT
ejpam-4750	50	1	let	let	VERB
ejpam-4750	50	2	w	w	NOUN
ejpam-4750	50	3	be	be	AUX
ejpam-4750	50	4	an	an	DET
ejpam-4750	50	5	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	50	6	of	of	ADP
ejpam-4750	50	7	a	a	DET
ejpam-4750	50	8	graph	graph	NOUN
ejpam-4750	50	9	g.	g.	NOUN
ejpam-4750	50	10	a	a	DET
ejpam-4750	50	11	subset	subset	NOUN
ejpam-4750	50	12	s	s	NOUN
ejpam-4750	50	13	of	of	ADP
ejpam-4750	50	14	w	w	NOUN
ejpam-4750	50	15	is	be	AUX
ejpam-4750	50	16	said	say	VERB
ejpam-4750	50	17	to	to	PART
ejpam-4750	50	18	be	be	AUX
ejpam-4750	50	19	a	a	DET
ejpam-4750	50	20	forcing	forcing	NOUN
ejpam-4750	50	21	subset	subset	NOUN
ejpam-4750	50	22	for	for	ADP
ejpam-4750	50	23	w	w	PROPN
ejpam-4750	50	24	if	if	SCONJ
ejpam-4750	50	25	w	w	PROPN
ejpam-4750	50	26	is	be	AUX
ejpam-4750	50	27	the	the	DET
ejpam-4750	50	28	unique	unique	ADJ
ejpam-4750	50	29	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	50	30	containing	contain	VERB
ejpam-4750	50	31	s.	s.	PROPN
ejpam-4750	50	32	the	the	DET
ejpam-4750	50	33	forcing	force	VERB
ejpam-4750	50	34	(	(	PUNCT
ejpam-4750	50	35	2	2	NUM
ejpam-4750	50	36	,	,	PUNCT
ejpam-4750	50	37	2)-locating	2)-locating	NUM
ejpam-4750	50	38	number	number	NOUN
ejpam-4750	50	39	of	of	ADP
ejpam-4750	50	40	w	w	NOUN
ejpam-4750	50	41	is	be	AUX
ejpam-4750	50	42	given	give	VERB
ejpam-4750	50	43	by	by	ADP
ejpam-4750	50	44	fln(2,2)(w	fln(2,2)(w	PROPN
ejpam-4750	50	45	)	)	PUNCT
ejpam-4750	50	46	=	=	NOUN
ejpam-4750	50	47	min{|s|	min{|s|	NOUN
ejpam-4750	50	48	:	:	PUNCT
ejpam-4750	50	49	s	s	VERB
ejpam-4750	50	50	is	be	AUX
ejpam-4750	50	51	a	a	DET
ejpam-4750	50	52	forcing	forcing	NOUN
ejpam-4750	50	53	subset	subset	NOUN
ejpam-4750	50	54	for	for	ADP
ejpam-4750	50	55	w	w	NOUN
ejpam-4750	50	56	}	}	PUNCT
ejpam-4750	50	57	.	.	PUNCT
ejpam-4750	51	1	the	the	DET
ejpam-4750	51	2	forcing	force	VERB
ejpam-4750	51	3	(	(	PUNCT
ejpam-4750	51	4	2	2	NUM
ejpam-4750	51	5	,	,	PUNCT
ejpam-4750	51	6	2)-locating	2)-locating	NUM
ejpam-4750	51	7	number	number	NOUN
ejpam-4750	51	8	of	of	ADP
ejpam-4750	51	9	g	g	PROPN
ejpam-4750	51	10	is	be	AUX
ejpam-4750	51	11	given	give	VERB
ejpam-4750	51	12	by	by	ADP
ejpam-4750	51	13	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	51	14	)	)	PUNCT
ejpam-4750	51	15	=	=	SYM
ejpam-4750	51	16	min{fln(2,2)(w	min{fln(2,2)(w	NOUN
ejpam-4750	51	17	)	)	PUNCT
ejpam-4750	51	18	:	:	PUNCT
ejpam-4750	52	1	w	w	NOUN
ejpam-4750	52	2	is	be	AUX
ejpam-4750	52	3	a	a	DET
ejpam-4750	52	4	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	52	5	of	of	ADP
ejpam-4750	52	6	g	g	NOUN
ejpam-4750	52	7	}	}	PUNCT
ejpam-4750	52	8	.	.	PUNCT
ejpam-4750	53	1	3	3	X
ejpam-4750	53	2	.	.	X
ejpam-4750	53	3	known	know	VERB
ejpam-4750	53	4	results	result	VERB
ejpam-4750	53	5	the	the	DET
ejpam-4750	53	6	following	follow	VERB
ejpam-4750	53	7	known	know	VERB
ejpam-4750	53	8	results	result	NOUN
ejpam-4750	53	9	are	be	AUX
ejpam-4750	53	10	taken	take	VERB
ejpam-4750	53	11	from	from	ADP
ejpam-4750	53	12	[	[	X
ejpam-4750	53	13	4	4	NUM
ejpam-4750	53	14	]	]	PUNCT
ejpam-4750	53	15	.	.	PUNCT
ejpam-4750	54	1	remark	remark	PROPN
ejpam-4750	54	2	1	1	NUM
ejpam-4750	54	3	.	.	PUNCT
ejpam-4750	55	1	for	for	ADP
ejpam-4750	55	2	any	any	DET
ejpam-4750	55	3	connected	connected	ADJ
ejpam-4750	55	4	nontrivial	nontrivial	NOUN
ejpam-4750	55	5	graph	graph	NOUN
ejpam-4750	55	6	g	g	NOUN
ejpam-4750	55	7	of	of	ADP
ejpam-4750	55	8	order	order	NOUN
ejpam-4750	55	9	n	n	PRON
ejpam-4750	55	10	≥	≥	NOUN
ejpam-4750	55	11	2	2	NUM
ejpam-4750	55	12	,	,	PUNCT
ejpam-4750	55	13	2	2	NUM
ejpam-4750	55	14	≤	≤	NUM
ejpam-4750	55	15	ln2(g	ln2(g	PROPN
ejpam-4750	55	16	)	)	PUNCT
ejpam-4750	55	17	≤	≤	PUNCT
ejpam-4750	55	18	n.	n.	NOUN
ejpam-4750	55	19	moreover	moreover	ADV
ejpam-4750	55	20	,	,	PUNCT
ejpam-4750	55	21	ln2(kn	ln2(kn	NUM
ejpam-4750	55	22	)	)	PUNCT
ejpam-4750	55	23	=	=	SYM
ejpam-4750	55	24	n	n	CCONJ
ejpam-4750	55	25	,	,	PUNCT
ejpam-4750	55	26	for	for	ADP
ejpam-4750	55	27	n	n	PRON
ejpam-4750	55	28	≥	≥	NUM
ejpam-4750	55	29	2	2	NUM
ejpam-4750	55	30	.	.	PUNCT
ejpam-4750	55	31	theorem	theorem	NOUN
ejpam-4750	55	32	1	1	NUM
ejpam-4750	55	33	.	.	PUNCT
ejpam-4750	55	34	let	let	VERB
ejpam-4750	55	35	g	g	PRON
ejpam-4750	55	36	be	be	AUX
ejpam-4750	55	37	a	a	DET
ejpam-4750	55	38	connected	connected	ADJ
ejpam-4750	55	39	nontrivial	nontrivial	ADJ
ejpam-4750	55	40	graph	graph	NOUN
ejpam-4750	55	41	.	.	PUNCT
ejpam-4750	56	1	then	then	ADV
ejpam-4750	56	2	ln2(g	ln2(g	PROPN
ejpam-4750	56	3	)	)	PUNCT
ejpam-4750	56	4	=	=	SYM
ejpam-4750	56	5	2	2	NUM
ejpam-4750	56	6	if	if	SCONJ
ejpam-4750	56	7	and	and	CCONJ
ejpam-4750	56	8	only	only	ADV
ejpam-4750	56	9	if	if	SCONJ
ejpam-4750	56	10	g	g	PROPN
ejpam-4750	56	11	∼=	∼=	NOUN
ejpam-4750	56	12	p2	p2	NOUN
ejpam-4750	56	13	or	or	CCONJ
ejpam-4750	56	14	g	g	NOUN
ejpam-4750	56	15	∼=	∼=	PROPN
ejpam-4750	56	16	p3	p3	NOUN
ejpam-4750	56	17	.	.	PUNCT
ejpam-4750	57	1	proposition	proposition	NOUN
ejpam-4750	57	2	1	1	NUM
ejpam-4750	57	3	.	.	NUM
ejpam-4750	57	4	dim2(g	dim2(g	NOUN
ejpam-4750	57	5	)	)	PUNCT
ejpam-4750	57	6	=	=	SYM
ejpam-4750	57	7	2	2	NUM
ejpam-4750	57	8	if	if	SCONJ
ejpam-4750	57	9	and	and	CCONJ
ejpam-4750	57	10	only	only	ADV
ejpam-4750	57	11	if	if	SCONJ
ejpam-4750	57	12	g	g	PROPN
ejpam-4750	57	13	∼=	∼=	PROPN
ejpam-4750	57	14	pn	pn	NOUN
ejpam-4750	57	15	,	,	PUNCT
ejpam-4750	57	16	n	n	PRON
ejpam-4750	57	17	≥	≥	NOUN
ejpam-4750	57	18	2	2	NUM
ejpam-4750	57	19	.	.	PUNCT
ejpam-4750	57	20	example	example	NOUN
ejpam-4750	58	1	1	1	NUM
ejpam-4750	58	2	.	.	PUNCT
ejpam-4750	59	1	let	let	VERB
ejpam-4750	59	2	n	n	PRON
ejpam-4750	59	3	be	be	AUX
ejpam-4750	59	4	a	a	DET
ejpam-4750	59	5	positive	positive	ADJ
ejpam-4750	59	6	integer	integer	NOUN
ejpam-4750	59	7	.	.	PUNCT
ejpam-4750	60	1	then	then	ADV
ejpam-4750	60	2	pn	pn	PROPN
ejpam-4750	60	3	and	and	CCONJ
ejpam-4750	60	4	cn	cn	PROPN
ejpam-4750	60	5	ln2(pn	ln2(pn	PROPN
ejpam-4750	60	6	)	)	PUNCT
ejpam-4750	60	7	=	=	PUNCT
ejpam-4750	61	1			PROPN
ejpam-4750	61	2	n	n	PRON
ejpam-4750	61	3	2	2	NUM
ejpam-4750	61	4	+	+	NUM
ejpam-4750	61	5	1	1	NUM
ejpam-4750	61	6	,	,	PUNCT
ejpam-4750	61	7	if	if	SCONJ
ejpam-4750	61	8	n	n	PRON
ejpam-4750	61	9	≥	≥	NOUN
ejpam-4750	61	10	2	2	NUM
ejpam-4750	61	11	and	and	CCONJ
ejpam-4750	61	12	n	n	PRON
ejpam-4750	61	13	is	be	AUX
ejpam-4750	61	14	even	even	ADV
ejpam-4750	61	15	,	,	PUNCT
ejpam-4750	61	16	n+	n+	ADP
ejpam-4750	61	17	1	1	NUM
ejpam-4750	61	18	2	2	NUM
ejpam-4750	61	19	,	,	PUNCT
ejpam-4750	61	20	if	if	SCONJ
ejpam-4750	61	21	n	n	PRON
ejpam-4750	61	22	≥	≥	NOUN
ejpam-4750	61	23	3	3	NUM
ejpam-4750	61	24	and	and	CCONJ
ejpam-4750	61	25	n	n	PROPN
ejpam-4750	61	26	is	be	AUX
ejpam-4750	61	27	odd	odd	ADJ
ejpam-4750	61	28	,	,	PUNCT
ejpam-4750	61	29	and	and	CCONJ
ejpam-4750	61	30	ln2(cn	ln2(cn	ADJ
ejpam-4750	61	31	)	)	PUNCT
ejpam-4750	61	32	=	=	PUNCT
ejpam-4750	62	1			PROPN
ejpam-4750	62	2	n	n	PRON
ejpam-4750	62	3	2	2	NUM
ejpam-4750	62	4	,	,	PUNCT
ejpam-4750	62	5	if	if	SCONJ
ejpam-4750	62	6	n	n	PRON
ejpam-4750	62	7	≥	≥	NOUN
ejpam-4750	62	8	6	6	NUM
ejpam-4750	62	9	and	and	CCONJ
ejpam-4750	62	10	n	n	NUM
ejpam-4750	62	11	is	be	AUX
ejpam-4750	62	12	even	even	ADV
ejpam-4750	62	13	,	,	PUNCT
ejpam-4750	62	14	n+	n+	ADP
ejpam-4750	62	15	1	1	NUM
ejpam-4750	62	16	2	2	NUM
ejpam-4750	62	17	,	,	PUNCT
ejpam-4750	62	18	if	if	SCONJ
ejpam-4750	62	19	n	n	PRON
ejpam-4750	62	20	≥	≥	NOUN
ejpam-4750	62	21	5	5	NUM
ejpam-4750	62	22	and	and	CCONJ
ejpam-4750	62	23	n	n	PRON
ejpam-4750	62	24	is	be	AUX
ejpam-4750	62	25	odd	odd	ADJ
ejpam-4750	62	26	.	.	PUNCT
ejpam-4750	62	27	example	example	NOUN
ejpam-4750	63	1	2	2	NUM
ejpam-4750	63	2	.	.	PUNCT
ejpam-4750	63	3	the	the	DET
ejpam-4750	63	4	formulas	formula	NOUN
ejpam-4750	63	5	below	below	ADP
ejpam-4750	63	6	give	give	VERB
ejpam-4750	63	7	the	the	DET
ejpam-4750	63	8	(	(	PUNCT
ejpam-4750	63	9	2,2)-locating	2,2)-locating	NUM
ejpam-4750	63	10	number	number	NOUN
ejpam-4750	63	11	of	of	ADP
ejpam-4750	63	12	the	the	DET
ejpam-4750	63	13	path	path	NOUN
ejpam-4750	63	14	pn	pn	NOUN
ejpam-4750	63	15	and	and	CCONJ
ejpam-4750	63	16	cycle	cycle	NOUN
ejpam-4750	63	17	cn	cn	PROPN
ejpam-4750	63	18	.	.	PUNCT
ejpam-4750	64	1	ln(2,2)(pn	ln(2,2)(pn	NOUN
ejpam-4750	64	2	)	)	PUNCT
ejpam-4750	65	1	=	=	PRON
ejpam-4750	65	2			NUM
ejpam-4750	65	3	4	4	NUM
ejpam-4750	65	4	,	,	PUNCT
ejpam-4750	65	5	if	if	SCONJ
ejpam-4750	65	6	n	n	NOUN
ejpam-4750	65	7	=	=	SYM
ejpam-4750	65	8	5	5	NUM
ejpam-4750	65	9	,	,	PUNCT
ejpam-4750	65	10	n	n	PRON
ejpam-4750	65	11	2	2	NUM
ejpam-4750	65	12	+	+	NUM
ejpam-4750	65	13	1	1	NUM
ejpam-4750	65	14	,	,	PUNCT
ejpam-4750	65	15	if	if	SCONJ
ejpam-4750	65	16	n	n	PRON
ejpam-4750	65	17	≥	≥	NOUN
ejpam-4750	65	18	6	6	NUM
ejpam-4750	65	19	and	and	CCONJ
ejpam-4750	65	20	n	n	NUM
ejpam-4750	65	21	is	be	AUX
ejpam-4750	65	22	even	even	ADV
ejpam-4750	65	23	,	,	PUNCT
ejpam-4750	65	24	n+	n+	ADP
ejpam-4750	65	25	1	1	NUM
ejpam-4750	65	26	2	2	NUM
ejpam-4750	65	27	,	,	PUNCT
ejpam-4750	65	28	if	if	SCONJ
ejpam-4750	65	29	n	n	PRON
ejpam-4750	65	30	≥	≥	VERB
ejpam-4750	65	31	7	7	NUM
ejpam-4750	65	32	and	and	CCONJ
ejpam-4750	65	33	n	n	PRON
ejpam-4750	65	34	is	be	AUX
ejpam-4750	65	35	odd	odd	ADJ
ejpam-4750	65	36	,	,	PUNCT
ejpam-4750	65	37	and	and	CCONJ
ejpam-4750	65	38	ln(2,2)(cn	ln(2,2)(cn	NOUN
ejpam-4750	65	39	)	)	PUNCT
ejpam-4750	65	40	=	=	PUNCT
ejpam-4750	66	1			PROPN
ejpam-4750	66	2	n	n	PRON
ejpam-4750	66	3	2	2	NUM
ejpam-4750	66	4	,	,	PUNCT
ejpam-4750	66	5	if	if	SCONJ
ejpam-4750	66	6	n	n	PRON
ejpam-4750	66	7	≥	≥	NOUN
ejpam-4750	66	8	8	8	NUM
ejpam-4750	66	9	and	and	CCONJ
ejpam-4750	66	10	n	n	PRON
ejpam-4750	66	11	is	be	AUX
ejpam-4750	66	12	even	even	ADV
ejpam-4750	66	13	,	,	PUNCT
ejpam-4750	66	14	n+	n+	ADP
ejpam-4750	66	15	1	1	NUM
ejpam-4750	66	16	2	2	NUM
ejpam-4750	66	17	,	,	PUNCT
ejpam-4750	66	18	if	if	SCONJ
ejpam-4750	66	19	n	n	PRON
ejpam-4750	66	20	≥	≥	VERB
ejpam-4750	66	21	7	7	NUM
ejpam-4750	66	22	and	and	CCONJ
ejpam-4750	66	23	n	n	PRON
ejpam-4750	66	24	is	be	AUX
ejpam-4750	66	25	odd	odd	ADJ
ejpam-4750	66	26	.	.	PUNCT
ejpam-4750	67	1	d.	d.	PROPN
ejpam-4750	67	2	managbanag	managbanag	PROPN
ejpam-4750	67	3	,	,	PUNCT
ejpam-4750	67	4	h.	h.	PROPN
ejpam-4750	67	5	rara	rara	PROPN
ejpam-4750	67	6	/	/	SYM
ejpam-4750	67	7	eur	eur	PROPN
ejpam-4750	67	8	.	.	PUNCT
ejpam-4750	68	1	j.	j.	PROPN
ejpam-4750	68	2	pure	pure	PROPN
ejpam-4750	68	3	appl	appl	PROPN
ejpam-4750	68	4	.	.	PROPN
ejpam-4750	68	5	math	math	PROPN
ejpam-4750	68	6	,	,	PUNCT
ejpam-4750	68	7	16	16	NUM
ejpam-4750	68	8	(	(	PUNCT
ejpam-4750	68	9	2	2	NUM
ejpam-4750	68	10	)	)	PUNCT
ejpam-4750	68	11	(	(	PUNCT
ejpam-4750	68	12	2023	2023	NUM
ejpam-4750	68	13	)	)	PUNCT
ejpam-4750	68	14	,	,	PUNCT
ejpam-4750	68	15	1068	1068	NUM
ejpam-4750	68	16	-	-	SYM
ejpam-4750	68	17	1083	1083	NUM
ejpam-4750	68	18	1071	1071	NUM
ejpam-4750	68	19	theorem	theorem	NOUN
ejpam-4750	68	20	2	2	NUM
ejpam-4750	68	21	.	.	PUNCT
ejpam-4750	69	1	let	let	VERB
ejpam-4750	69	2	g	g	PRON
ejpam-4750	69	3	be	be	AUX
ejpam-4750	69	4	a	a	DET
ejpam-4750	69	5	connected	connected	ADJ
ejpam-4750	69	6	graph	graph	NOUN
ejpam-4750	69	7	of	of	ADP
ejpam-4750	69	8	order	order	NOUN
ejpam-4750	69	9	greater	great	ADJ
ejpam-4750	69	10	than	than	ADP
ejpam-4750	69	11	3	3	NUM
ejpam-4750	69	12	and	and	CCONJ
ejpam-4750	69	13	let	let	VERB
ejpam-4750	69	14	k1	k1	NOUN
ejpam-4750	69	15	=	=	SYM
ejpam-4750	69	16	⟨v⟩.	⟨v⟩.	PROPN
ejpam-4750	69	17	then	then	ADV
ejpam-4750	69	18	s	s	VERB
ejpam-4750	69	19	⊆	⊆	NUM
ejpam-4750	69	20	v	v	NOUN
ejpam-4750	69	21	(	(	PUNCT
ejpam-4750	69	22	k1	k1	NOUN
ejpam-4750	69	23	+	+	CCONJ
ejpam-4750	69	24	g	g	NOUN
ejpam-4750	69	25	)	)	PUNCT
ejpam-4750	69	26	is	be	AUX
ejpam-4750	69	27	a	a	DET
ejpam-4750	69	28	2	2	NUM
ejpam-4750	69	29	-	-	PUNCT
ejpam-4750	69	30	resolving	resolve	VERB
ejpam-4750	69	31	set	set	NOUN
ejpam-4750	69	32	of	of	ADP
ejpam-4750	69	33	k1	k1	NOUN
ejpam-4750	69	34	+	+	CCONJ
ejpam-4750	69	35	g	g	NOUN
ejpam-4750	69	36	if	if	SCONJ
ejpam-4750	70	1	and	and	CCONJ
ejpam-4750	70	2	only	only	ADV
ejpam-4750	70	3	if	if	SCONJ
ejpam-4750	70	4	either	either	PRON
ejpam-4750	70	5	v	v	NOUN
ejpam-4750	70	6	/∈	/∈	PUNCT
ejpam-4750	70	7	s	s	PART
ejpam-4750	70	8	and	and	CCONJ
ejpam-4750	70	9	s	s	VERB
ejpam-4750	70	10	is	be	AUX
ejpam-4750	70	11	a	a	DET
ejpam-4750	70	12	(	(	PUNCT
ejpam-4750	70	13	2,2)-locating	2,2)-locating	NUM
ejpam-4750	70	14	set	set	VERB
ejpam-4750	70	15	in	in	ADP
ejpam-4750	70	16	g	g	PROPN
ejpam-4750	70	17	or	or	CCONJ
ejpam-4750	70	18	s	s	NOUN
ejpam-4750	70	19	=	=	PUNCT
ejpam-4750	70	20	{	{	PUNCT
ejpam-4750	70	21	v	v	NOUN
ejpam-4750	70	22	}	}	PUNCT
ejpam-4750	70	23	∪	∪	NOUN
ejpam-4750	70	24	t	t	PROPN
ejpam-4750	70	25	is	be	AUX
ejpam-4750	70	26	(	(	PUNCT
ejpam-4750	70	27	2,1)-locating	2,1)-locating	NUM
ejpam-4750	70	28	set	set	NOUN
ejpam-4750	70	29	in	in	ADP
ejpam-4750	70	30	g.	g.	PROPN
ejpam-4750	70	31	theorem	theorem	PROPN
ejpam-4750	70	32	3	3	X
ejpam-4750	70	33	.	.	PUNCT
ejpam-4750	71	1	let	let	VERB
ejpam-4750	71	2	g	g	NOUN
ejpam-4750	71	3	and	and	CCONJ
ejpam-4750	71	4	h	h	NOUN
ejpam-4750	71	5	be	be	AUX
ejpam-4750	71	6	nontrivial	nontrivial	ADJ
ejpam-4750	71	7	connected	connected	ADJ
ejpam-4750	71	8	graphs	graph	NOUN
ejpam-4750	71	9	.	.	PUNCT
ejpam-4750	72	1	a	a	DET
ejpam-4750	72	2	proper	proper	ADJ
ejpam-4750	72	3	subset	subset	NOUN
ejpam-4750	72	4	s	s	NOUN
ejpam-4750	72	5	of	of	ADP
ejpam-4750	72	6	v	v	NOUN
ejpam-4750	72	7	(	(	PUNCT
ejpam-4750	72	8	g+h	g+h	PROPN
ejpam-4750	72	9	)	)	PUNCT
ejpam-4750	72	10	is	be	AUX
ejpam-4750	72	11	a	a	DET
ejpam-4750	72	12	2	2	NUM
ejpam-4750	72	13	-	-	PUNCT
ejpam-4750	72	14	resolving	resolving	NOUN
ejpam-4750	72	15	set	set	NOUN
ejpam-4750	72	16	in	in	ADP
ejpam-4750	72	17	g	g	PROPN
ejpam-4750	73	1	+	+	NOUN
ejpam-4750	73	2	h	h	NOUN
ejpam-4750	73	3	if	if	SCONJ
ejpam-4750	73	4	and	and	CCONJ
ejpam-4750	73	5	only	only	ADV
ejpam-4750	73	6	if	if	SCONJ
ejpam-4750	73	7	sg	sg	PROPN
ejpam-4750	73	8	=	=	SYM
ejpam-4750	73	9	v	v	PROPN
ejpam-4750	73	10	(	(	PUNCT
ejpam-4750	73	11	g	g	NOUN
ejpam-4750	73	12	)	)	PUNCT
ejpam-4750	73	13	∩	∩	NOUN
ejpam-4750	73	14	s	s	NOUN
ejpam-4750	73	15	and	and	CCONJ
ejpam-4750	73	16	sh	sh	PROPN
ejpam-4750	73	17	=	=	SYM
ejpam-4750	73	18	v	v	PROPN
ejpam-4750	73	19	(	(	PUNCT
ejpam-4750	73	20	h	h	NOUN
ejpam-4750	73	21	)	)	PUNCT
ejpam-4750	73	22	∩	∩	NOUN
ejpam-4750	73	23	s	s	NOUN
ejpam-4750	73	24	are	be	AUX
ejpam-4750	73	25	2	2	NUM
ejpam-4750	73	26	-	-	PUNCT
ejpam-4750	73	27	locating	locate	VERB
ejpam-4750	73	28	sets	set	NOUN
ejpam-4750	73	29	in	in	ADP
ejpam-4750	73	30	g	g	PROPN
ejpam-4750	73	31	and	and	CCONJ
ejpam-4750	73	32	h	h	NOUN
ejpam-4750	73	33	,	,	PUNCT
ejpam-4750	73	34	respectively	respectively	ADV
ejpam-4750	73	35	,	,	PUNCT
ejpam-4750	73	36	where	where	SCONJ
ejpam-4750	73	37	sg	sg	NOUN
ejpam-4750	73	38	or	or	CCONJ
ejpam-4750	73	39	sh	sh	PROPN
ejpam-4750	73	40	is	be	AUX
ejpam-4750	73	41	(	(	PUNCT
ejpam-4750	73	42	2,2)-locating	2,2)-locating	NUM
ejpam-4750	73	43	set	set	NOUN
ejpam-4750	73	44	or	or	CCONJ
ejpam-4750	73	45	sg	sg	PROPN
ejpam-4750	74	1	and	and	CCONJ
ejpam-4750	74	2	sh	sh	PROPN
ejpam-4750	74	3	are	be	AUX
ejpam-4750	74	4	(	(	PUNCT
ejpam-4750	74	5	2,1)-locating	2,1)-locating	NUM
ejpam-4750	74	6	sets	set	NOUN
ejpam-4750	74	7	.	.	PUNCT
ejpam-4750	75	1	theorem	theorem	NOUN
ejpam-4750	75	2	4	4	NUM
ejpam-4750	75	3	.	.	PUNCT
ejpam-4750	76	1	let	let	VERB
ejpam-4750	76	2	g	g	NOUN
ejpam-4750	76	3	and	and	CCONJ
ejpam-4750	76	4	h	h	NOUN
ejpam-4750	76	5	be	be	AUX
ejpam-4750	76	6	nontrivial	nontrivial	ADJ
ejpam-4750	76	7	connected	connected	ADJ
ejpam-4750	76	8	graphs	graph	NOUN
ejpam-4750	76	9	.	.	PUNCT
ejpam-4750	77	1	a	a	DET
ejpam-4750	77	2	set	set	NOUN
ejpam-4750	77	3	s	s	NOUN
ejpam-4750	77	4	⊆	⊆	NUM
ejpam-4750	77	5	v	v	NOUN
ejpam-4750	77	6	(	(	PUNCT
ejpam-4750	77	7	g	g	PROPN
ejpam-4750	77	8	◦	◦	NOUN
ejpam-4750	77	9	h	h	NOUN
ejpam-4750	77	10	)	)	PUNCT
ejpam-4750	77	11	is	be	AUX
ejpam-4750	77	12	a	a	DET
ejpam-4750	77	13	2	2	NUM
ejpam-4750	77	14	-	-	PUNCT
ejpam-4750	77	15	resolving	resolve	VERB
ejpam-4750	77	16	set	set	NOUN
ejpam-4750	77	17	of	of	ADP
ejpam-4750	77	18	g	g	PROPN
ejpam-4750	77	19	◦	◦	NOUN
ejpam-4750	77	20	h	h	NOUN
ejpam-4750	77	21	if	if	SCONJ
ejpam-4750	78	1	and	and	CCONJ
ejpam-4750	78	2	only	only	ADV
ejpam-4750	78	3	if	if	SCONJ
ejpam-4750	78	4	s	s	VERB
ejpam-4750	78	5	=	=	X
ejpam-4750	78	6	a	a	DET
ejpam-4750	78	7	∪	∪	X
ejpam-4750	78	8	b	b	NOUN
ejpam-4750	78	9	,	,	PUNCT
ejpam-4750	78	10	where	where	SCONJ
ejpam-4750	78	11	a	a	DET
ejpam-4750	78	12	⊆	⊆	NUM
ejpam-4750	78	13	v	v	NOUN
ejpam-4750	78	14	(	(	PUNCT
ejpam-4750	78	15	g	g	NOUN
ejpam-4750	78	16	)	)	PUNCT
ejpam-4750	78	17	and	and	CCONJ
ejpam-4750	78	18	b	b	X
ejpam-4750	78	19	=	=	SYM
ejpam-4750	78	20	⋃	⋃	NOUN
ejpam-4750	78	21	{	{	PUNCT
ejpam-4750	78	22	sv	sv	NOUN
ejpam-4750	78	23	:	:	PUNCT
ejpam-4750	78	24	sv	sv	PROPN
ejpam-4750	78	25	is	be	AUX
ejpam-4750	78	26	a	a	DET
ejpam-4750	78	27	2	2	NUM
ejpam-4750	78	28	-	-	PUNCT
ejpam-4750	78	29	resolving	resolve	VERB
ejpam-4750	78	30	set	set	NOUN
ejpam-4750	78	31	of	of	ADP
ejpam-4750	78	32	hv	hv	PROPN
ejpam-4750	78	33	,	,	PUNCT
ejpam-4750	78	34	for	for	ADP
ejpam-4750	78	35	all	all	PRON
ejpam-4750	78	36	v	v	ADP
ejpam-4750	78	37	∈	∈	NUM
ejpam-4750	78	38	v	v	NOUN
ejpam-4750	78	39	(	(	PUNCT
ejpam-4750	78	40	g	g	NOUN
ejpam-4750	78	41	)	)	PUNCT
ejpam-4750	78	42	}	}	PUNCT
ejpam-4750	78	43	.	.	PUNCT
ejpam-4750	79	1	remark	remark	NOUN
ejpam-4750	79	2	2	2	NUM
ejpam-4750	79	3	.	.	PUNCT
ejpam-4750	80	1	let	let	VERB
ejpam-4750	80	2	g	g	NOUN
ejpam-4750	80	3	and	and	CCONJ
ejpam-4750	80	4	h	h	PROPN
ejpam-4750	80	5	be	be	VERB
ejpam-4750	80	6	non	non	ADJ
ejpam-4750	80	7	-	-	ADJ
ejpam-4750	80	8	trivial	trivial	ADJ
ejpam-4750	80	9	connected	connected	ADJ
ejpam-4750	80	10	graphs	graph	NOUN
ejpam-4750	80	11	,	,	PUNCT
ejpam-4750	80	12	c	c	PROPN
ejpam-4750	80	13	⊆	⊆	NUM
ejpam-4750	80	14	v	v	NOUN
ejpam-4750	80	15	(	(	PUNCT
ejpam-4750	80	16	g	g	PROPN
ejpam-4750	80	17	◦	◦	NOUN
ejpam-4750	80	18	h	h	NOUN
ejpam-4750	80	19	)	)	PUNCT
ejpam-4750	80	20	and	and	CCONJ
ejpam-4750	80	21	sv	sv	X
ejpam-4750	80	22	=	=	SYM
ejpam-4750	80	23	v	v	PROPN
ejpam-4750	80	24	(	(	PUNCT
ejpam-4750	80	25	hv	hv	NOUN
ejpam-4750	80	26	)	)	PUNCT
ejpam-4750	80	27	∩	∩	NOUN
ejpam-4750	80	28	c	c	X
ejpam-4750	80	29	where	where	SCONJ
ejpam-4750	80	30	v	v	X
ejpam-4750	80	31	∈	∈	NOUN
ejpam-4750	80	32	v	v	NOUN
ejpam-4750	80	33	(	(	PUNCT
ejpam-4750	80	34	g	g	NOUN
ejpam-4750	80	35	)	)	PUNCT
ejpam-4750	80	36	.	.	PUNCT
ejpam-4750	81	1	for	for	ADP
ejpam-4750	81	2	each	each	DET
ejpam-4750	81	3	x	x	SYM
ejpam-4750	81	4	∈	∈	PROPN
ejpam-4750	81	5	v	v	ADP
ejpam-4750	81	6	(	(	PUNCT
ejpam-4750	81	7	hv	hv	PROPN
ejpam-4750	81	8	)	)	PUNCT
ejpam-4750	81	9	\	\	PROPN
ejpam-4750	81	10	sv	sv	PROPN
ejpam-4750	81	11	and	and	CCONJ
ejpam-4750	81	12	z	z	PROPN
ejpam-4750	81	13	∈	∈	PROPN
ejpam-4750	81	14	sv	sv	PROPN
ejpam-4750	81	15	,	,	PUNCT
ejpam-4750	81	16	dg	dg	PROPN
ejpam-4750	81	17	◦	◦	NOUN
ejpam-4750	81	18	h(x	h(x	PROPN
ejpam-4750	81	19	,	,	PUNCT
ejpam-4750	81	20	z	z	NOUN
ejpam-4750	81	21	)	)	PUNCT
ejpam-4750	81	22	=	=	SYM
ejpam-4750	81	23	®	®	NOUN
ejpam-4750	81	24	1	1	NUM
ejpam-4750	81	25	,	,	PUNCT
ejpam-4750	81	26	if	if	SCONJ
ejpam-4750	81	27	z	z	PROPN
ejpam-4750	81	28	∈	∈	PROPN
ejpam-4750	81	29	nhv(x	nhv(x	PROPN
ejpam-4750	81	30	)	)	PUNCT
ejpam-4750	81	31	,	,	PUNCT
ejpam-4750	81	32	2	2	NUM
ejpam-4750	81	33	,	,	PUNCT
ejpam-4750	81	34	otherwise	otherwise	ADV
ejpam-4750	81	35	.	.	PUNCT
ejpam-4750	82	1	4	4	X
ejpam-4750	82	2	.	.	X
ejpam-4750	82	3	forcing	force	VERB
ejpam-4750	82	4	2	2	NUM
ejpam-4750	82	5	-	-	PUNCT
ejpam-4750	82	6	metric	metric	ADJ
ejpam-4750	82	7	dimension	dimension	NOUN
ejpam-4750	82	8	of	of	ADP
ejpam-4750	82	9	some	some	DET
ejpam-4750	82	10	special	special	ADJ
ejpam-4750	82	11	graphs	graph	NOUN
ejpam-4750	82	12	remark	remark	VERB
ejpam-4750	82	13	3	3	NUM
ejpam-4750	82	14	.	.	PUNCT
ejpam-4750	83	1	let	let	VERB
ejpam-4750	83	2	g	g	PRON
ejpam-4750	83	3	be	be	AUX
ejpam-4750	83	4	a	a	DET
ejpam-4750	83	5	nontrivial	nontrivial	ADJ
ejpam-4750	83	6	connected	connect	VERB
ejpam-4750	83	7	graph	graph	NOUN
ejpam-4750	83	8	.	.	PUNCT
ejpam-4750	84	1	then	then	ADV
ejpam-4750	84	2	0	0	NUM
ejpam-4750	84	3	≤	≤	NUM
ejpam-4750	84	4	fdim2(g	fdim2(g	NOUN
ejpam-4750	84	5	)	)	PUNCT
ejpam-4750	84	6	≤	≤	NOUN
ejpam-4750	84	7	dim2(g	dim2(g	NOUN
ejpam-4750	84	8	)	)	PUNCT
ejpam-4750	84	9	.	.	PUNCT
ejpam-4750	85	1	remark	remark	PROPN
ejpam-4750	85	2	4	4	NUM
ejpam-4750	85	3	.	.	PUNCT
ejpam-4750	86	1	let	let	VERB
ejpam-4750	86	2	g	g	PRON
ejpam-4750	86	3	be	be	AUX
ejpam-4750	86	4	a	a	DET
ejpam-4750	86	5	connected	connected	ADJ
ejpam-4750	86	6	graph	graph	NOUN
ejpam-4750	86	7	.	.	PUNCT
ejpam-4750	87	1	then	then	ADV
ejpam-4750	87	2	(	(	PUNCT
ejpam-4750	87	3	i	i	NOUN
ejpam-4750	87	4	)	)	PUNCT
ejpam-4750	87	5	fdim2(g	fdim2(g	PROPN
ejpam-4750	87	6	)	)	PUNCT
ejpam-4750	87	7	=	=	SYM
ejpam-4750	87	8	0	0	PUNCT
ejpam-4750	88	1	if	if	SCONJ
ejpam-4750	88	2	and	and	CCONJ
ejpam-4750	88	3	only	only	ADV
ejpam-4750	88	4	if	if	SCONJ
ejpam-4750	88	5	g	g	PROPN
ejpam-4750	88	6	has	have	VERB
ejpam-4750	88	7	a	a	DET
ejpam-4750	88	8	unique	unique	ADJ
ejpam-4750	88	9	2	2	NUM
ejpam-4750	88	10	-	-	PUNCT
ejpam-4750	88	11	metric	metric	ADJ
ejpam-4750	88	12	basis	basis	NOUN
ejpam-4750	88	13	,	,	PUNCT
ejpam-4750	88	14	and	and	CCONJ
ejpam-4750	88	15	(	(	PUNCT
ejpam-4750	88	16	ii	ii	NOUN
ejpam-4750	88	17	)	)	PUNCT
ejpam-4750	88	18	fdim2(g	fdim2(g	NOUN
ejpam-4750	88	19	)	)	PUNCT
ejpam-4750	88	20	=	=	SYM
ejpam-4750	88	21	1	1	NUM
ejpam-4750	88	22	if	if	SCONJ
ejpam-4750	88	23	and	and	CCONJ
ejpam-4750	88	24	only	only	ADV
ejpam-4750	88	25	if	if	SCONJ
ejpam-4750	88	26	g	g	PROPN
ejpam-4750	88	27	has	have	VERB
ejpam-4750	88	28	at	at	ADV
ejpam-4750	88	29	least	least	ADV
ejpam-4750	88	30	two	two	NUM
ejpam-4750	88	31	2	2	NUM
ejpam-4750	88	32	-	-	PUNCT
ejpam-4750	88	33	metric	metric	ADJ
ejpam-4750	88	34	bases	basis	NOUN
ejpam-4750	88	35	,	,	PUNCT
ejpam-4750	88	36	one	one	NUM
ejpam-4750	88	37	of	of	ADP
ejpam-4750	88	38	which	which	PRON
ejpam-4750	88	39	,	,	PUNCT
ejpam-4750	88	40	say	say	VERB
ejpam-4750	88	41	b	b	X
ejpam-4750	88	42	,	,	PUNCT
ejpam-4750	88	43	that	that	PRON
ejpam-4750	88	44	contains	contain	VERB
ejpam-4750	88	45	an	an	DET
ejpam-4750	88	46	element	element	NOUN
ejpam-4750	88	47	not	not	PART
ejpam-4750	88	48	in	in	ADP
ejpam-4750	88	49	any	any	DET
ejpam-4750	88	50	2	2	NUM
ejpam-4750	88	51	-	-	PUNCT
ejpam-4750	88	52	metric	metric	ADJ
ejpam-4750	88	53	basis	basis	NOUN
ejpam-4750	88	54	for	for	ADP
ejpam-4750	88	55	g.	g.	PROPN
ejpam-4750	88	56	theorem	theorem	PROPN
ejpam-4750	88	57	5	5	X
ejpam-4750	88	58	.	.	PUNCT
ejpam-4750	89	1	let	let	VERB
ejpam-4750	89	2	g	g	PRON
ejpam-4750	89	3	be	be	AUX
ejpam-4750	89	4	a	a	DET
ejpam-4750	89	5	connected	connected	ADJ
ejpam-4750	89	6	graph	graph	NOUN
ejpam-4750	89	7	.	.	PUNCT
ejpam-4750	90	1	then	then	ADV
ejpam-4750	90	2	fdim2(g	fdim2(g	NUM
ejpam-4750	90	3	)	)	PUNCT
ejpam-4750	91	1	=	=	PUNCT
ejpam-4750	91	2	dim2(g	dim2(g	NOUN
ejpam-4750	91	3	)	)	PUNCT
ejpam-4750	91	4	if	if	SCONJ
ejpam-4750	91	5	and	and	CCONJ
ejpam-4750	91	6	only	only	ADV
ejpam-4750	91	7	if	if	SCONJ
ejpam-4750	91	8	for	for	ADP
ejpam-4750	91	9	all	all	DET
ejpam-4750	91	10	2	2	NUM
ejpam-4750	91	11	-	-	PUNCT
ejpam-4750	91	12	metric	metric	ADJ
ejpam-4750	91	13	basis	basis	NOUN
ejpam-4750	91	14	d	d	NOUN
ejpam-4750	91	15	for	for	ADP
ejpam-4750	91	16	g	g	PROPN
ejpam-4750	91	17	and	and	CCONJ
ejpam-4750	91	18	for	for	ADP
ejpam-4750	91	19	each	each	DET
ejpam-4750	91	20	u	u	PROPN
ejpam-4750	91	21	∈	∈	PROPN
ejpam-4750	91	22	d	d	NOUN
ejpam-4750	91	23	,	,	PUNCT
ejpam-4750	91	24	there	there	PRON
ejpam-4750	91	25	exists	exist	VERB
ejpam-4750	91	26	vu	vu	PROPN
ejpam-4750	91	27	∈	∈	PROPN
ejpam-4750	91	28	v	v	ADP
ejpam-4750	91	29	(	(	PUNCT
ejpam-4750	91	30	g	g	NOUN
ejpam-4750	91	31	)	)	PUNCT
ejpam-4750	91	32	\d	\d	NOUN
ejpam-4750	91	33	such	such	ADJ
ejpam-4750	91	34	that	that	SCONJ
ejpam-4750	91	35	[	[	PUNCT
ejpam-4750	91	36	d	d	X
ejpam-4750	91	37	\	\	X
ejpam-4750	91	38	{	{	PUNCT
ejpam-4750	91	39	u	u	NOUN
ejpam-4750	91	40	}	}	PUNCT
ejpam-4750	91	41	]	]	PUNCT
ejpam-4750	91	42	∪	∪	X
ejpam-4750	91	43	{	{	PUNCT
ejpam-4750	91	44	vu	vu	INTJ
ejpam-4750	91	45	}	}	PUNCT
ejpam-4750	91	46	is	be	AUX
ejpam-4750	91	47	a	a	DET
ejpam-4750	91	48	2	2	NUM
ejpam-4750	91	49	-	-	PUNCT
ejpam-4750	91	50	metric	metric	ADJ
ejpam-4750	91	51	basis	basis	NOUN
ejpam-4750	91	52	for	for	ADP
ejpam-4750	91	53	g.	g.	NOUN
ejpam-4750	91	54	proof	proof	NOUN
ejpam-4750	91	55	:	:	PUNCT
ejpam-4750	91	56	suppose	suppose	VERB
ejpam-4750	91	57	that	that	SCONJ
ejpam-4750	91	58	fdim2(g	fdim2(g	NOUN
ejpam-4750	91	59	)	)	PUNCT
ejpam-4750	91	60	=	=	PUNCT
ejpam-4750	91	61	dim2(g	dim2(g	NOUN
ejpam-4750	91	62	)	)	PUNCT
ejpam-4750	91	63	.	.	PUNCT
ejpam-4750	92	1	let	let	VERB
ejpam-4750	92	2	d	d	PRON
ejpam-4750	92	3	be	be	AUX
ejpam-4750	92	4	a	a	DET
ejpam-4750	92	5	2	2	NUM
ejpam-4750	92	6	-	-	PUNCT
ejpam-4750	92	7	metric	metric	ADJ
ejpam-4750	92	8	basis	basis	NOUN
ejpam-4750	92	9	for	for	ADP
ejpam-4750	92	10	g	g	PROPN
ejpam-4750	92	11	such	such	ADJ
ejpam-4750	92	12	that	that	SCONJ
ejpam-4750	92	13	fdim2(g	fdim2(g	NOUN
ejpam-4750	92	14	)	)	PUNCT
ejpam-4750	92	15	=	=	SYM
ejpam-4750	92	16	|d|	|d|	PROPN
ejpam-4750	92	17	=	=	PUNCT
ejpam-4750	92	18	dim2(g	dim2(g	NOUN
ejpam-4750	92	19	)	)	PUNCT
ejpam-4750	92	20	,	,	PUNCT
ejpam-4750	92	21	that	that	ADV
ejpam-4750	92	22	is	is	ADV
ejpam-4750	92	23	,	,	PUNCT
ejpam-4750	92	24	d	d	PRON
ejpam-4750	92	25	is	be	AUX
ejpam-4750	92	26	the	the	DET
ejpam-4750	92	27	only	only	ADJ
ejpam-4750	92	28	forcing	forcing	NOUN
ejpam-4750	92	29	subset	subset	NOUN
ejpam-4750	92	30	for	for	ADP
ejpam-4750	92	31	itself	itself	PRON
ejpam-4750	92	32	.	.	PUNCT
ejpam-4750	93	1	let	let	VERB
ejpam-4750	93	2	u	u	PRON
ejpam-4750	93	3	∈	∈	PROPN
ejpam-4750	93	4	d.	d.	NOUN
ejpam-4750	93	5	since	since	SCONJ
ejpam-4750	93	6	d	d	PROPN
ejpam-4750	93	7	\	\	PROPN
ejpam-4750	93	8	{	{	PUNCT
ejpam-4750	93	9	u	u	NOUN
ejpam-4750	93	10	}	}	PUNCT
ejpam-4750	93	11	is	be	AUX
ejpam-4750	93	12	not	not	PART
ejpam-4750	93	13	a	a	DET
ejpam-4750	93	14	forcing	forcing	NOUN
ejpam-4750	93	15	subset	subset	NOUN
ejpam-4750	93	16	for	for	ADP
ejpam-4750	93	17	d	d	PROPN
ejpam-4750	93	18	,	,	PUNCT
ejpam-4750	93	19	there	there	PRON
ejpam-4750	93	20	exists	exist	VERB
ejpam-4750	93	21	a	a	DET
ejpam-4750	93	22	vu	vu	X
ejpam-4750	93	23	∈	∈	PROPN
ejpam-4750	93	24	v	v	NOUN
ejpam-4750	93	25	(	(	PUNCT
ejpam-4750	93	26	g	g	NOUN
ejpam-4750	93	27	)	)	PUNCT
ejpam-4750	93	28	\d	\d	NOUN
ejpam-4750	93	29	such	such	ADJ
ejpam-4750	93	30	that	that	SCONJ
ejpam-4750	93	31	[	[	PUNCT
ejpam-4750	93	32	d	d	X
ejpam-4750	93	33	\	\	X
ejpam-4750	93	34	{	{	PUNCT
ejpam-4750	93	35	u	u	NOUN
ejpam-4750	93	36	}	}	PUNCT
ejpam-4750	93	37	]	]	PUNCT
ejpam-4750	93	38	∪	∪	X
ejpam-4750	93	39	{	{	PUNCT
ejpam-4750	93	40	vu	vu	INTJ
ejpam-4750	93	41	}	}	PUNCT
ejpam-4750	93	42	is	be	AUX
ejpam-4750	93	43	a	a	DET
ejpam-4750	93	44	2	2	NUM
ejpam-4750	93	45	-	-	PUNCT
ejpam-4750	93	46	metric	metric	ADJ
ejpam-4750	93	47	basis	basis	NOUN
ejpam-4750	93	48	for	for	ADP
ejpam-4750	93	49	g.	g.	NOUN
ejpam-4750	93	50	conversely	conversely	ADV
ejpam-4750	93	51	,	,	PUNCT
ejpam-4750	93	52	suppose	suppose	VERB
ejpam-4750	93	53	that	that	SCONJ
ejpam-4750	93	54	every	every	DET
ejpam-4750	93	55	2	2	NUM
ejpam-4750	93	56	-	-	PUNCT
ejpam-4750	93	57	metric	metric	ADJ
ejpam-4750	93	58	basis	basis	NOUN
ejpam-4750	93	59	for	for	ADP
ejpam-4750	93	60	g	g	NOUN
ejpam-4750	93	61	satisfies	satisfie	NOUN
ejpam-4750	93	62	the	the	DET
ejpam-4750	93	63	given	give	VERB
ejpam-4750	93	64	condition	condition	NOUN
ejpam-4750	93	65	.	.	PUNCT
ejpam-4750	94	1	let	let	VERB
ejpam-4750	94	2	d	d	PRON
ejpam-4750	94	3	be	be	AUX
ejpam-4750	94	4	a	a	DET
ejpam-4750	94	5	2	2	NUM
ejpam-4750	94	6	-	-	PUNCT
ejpam-4750	94	7	metric	metric	ADJ
ejpam-4750	94	8	basis	basis	NOUN
ejpam-4750	94	9	for	for	ADP
ejpam-4750	94	10	g	g	PROPN
ejpam-4750	94	11	such	such	ADJ
ejpam-4750	94	12	that	that	SCONJ
ejpam-4750	94	13	fdim2(g	fdim2(g	NOUN
ejpam-4750	94	14	)	)	PUNCT
ejpam-4750	94	15	=	=	SYM
ejpam-4750	94	16	fdim2(d	fdim2(d	X
ejpam-4750	94	17	)	)	PUNCT
ejpam-4750	94	18	.	.	PUNCT
ejpam-4750	95	1	suppose	suppose	VERB
ejpam-4750	95	2	further	far	ADV
ejpam-4750	95	3	that	that	SCONJ
ejpam-4750	95	4	d	d	PROPN
ejpam-4750	95	5	has	have	VERB
ejpam-4750	95	6	a	a	DET
ejpam-4750	95	7	forcing	forcing	NOUN
ejpam-4750	95	8	subset	subset	NOUN
ejpam-4750	95	9	j	j	PROPN
ejpam-4750	95	10	with	with	ADP
ejpam-4750	95	11	|j	|j	PROPN
ejpam-4750	95	12	|	|	ADV
ejpam-4750	95	13	<	<	X
ejpam-4750	95	14	|d|	|d|	PROPN
ejpam-4750	95	15	,	,	PUNCT
ejpam-4750	95	16	that	that	ADV
ejpam-4750	95	17	is	be	AUX
ejpam-4750	95	18	,	,	PUNCT
ejpam-4750	95	19	d	d	PROPN
ejpam-4750	95	20	=	=	SYM
ejpam-4750	95	21	j	j	PROPN
ejpam-4750	95	22	∪k	∪k	NUM
ejpam-4750	95	23	where	where	SCONJ
ejpam-4750	95	24	k	k	PROPN
ejpam-4750	95	25	=	=	PRON
ejpam-4750	95	26	{	{	PUNCT
ejpam-4750	95	27	w	w	PROPN
ejpam-4750	95	28	∈	∈	PROPN
ejpam-4750	95	29	b	b	PROPN
ejpam-4750	95	30	:	:	PUNCT
ejpam-4750	95	31	w	w	PROPN
ejpam-4750	95	32	/∈	/∈	PROPN
ejpam-4750	95	33	j	j	NOUN
ejpam-4750	95	34	}	}	PUNCT
ejpam-4750	95	35	.	.	PUNCT
ejpam-4750	96	1	pick	pick	VERB
ejpam-4750	96	2	w	w	PROPN
ejpam-4750	96	3	∈	∈	PROPN
ejpam-4750	96	4	k.	k.	NOUN
ejpam-4750	96	5	by	by	ADP
ejpam-4750	96	6	assumption	assumption	NOUN
ejpam-4750	96	7	,	,	PUNCT
ejpam-4750	96	8	there	there	PRON
ejpam-4750	96	9	exists	exist	VERB
ejpam-4750	96	10	vw	vw	PROPN
ejpam-4750	96	11	∈	∈	PROPN
ejpam-4750	96	12	v	v	ADP
ejpam-4750	96	13	(	(	PUNCT
ejpam-4750	96	14	g	g	NOUN
ejpam-4750	96	15	)	)	PUNCT
ejpam-4750	96	16	\d	\d	NOUN
ejpam-4750	96	17	such	such	ADJ
ejpam-4750	96	18	that	that	SCONJ
ejpam-4750	96	19	[	[	PUNCT
ejpam-4750	96	20	d	d	X
ejpam-4750	96	21	\	\	X
ejpam-4750	96	22	{	{	PUNCT
ejpam-4750	96	23	w	w	NOUN
ejpam-4750	96	24	}	}	PUNCT
ejpam-4750	96	25	]	]	PUNCT
ejpam-4750	96	26	∪	∪	X
ejpam-4750	96	27	{	{	PUNCT
ejpam-4750	96	28	vw	vw	NOUN
ejpam-4750	96	29	}	}	PUNCT
ejpam-4750	96	30	=	=	SYM
ejpam-4750	96	31	t	t	PROPN
ejpam-4750	96	32	is	be	AUX
ejpam-4750	96	33	a	a	DET
ejpam-4750	96	34	2	2	NUM
ejpam-4750	96	35	-	-	PUNCT
ejpam-4750	96	36	metric	metric	ADJ
ejpam-4750	96	37	basis	basis	NOUN
ejpam-4750	96	38	for	for	ADP
ejpam-4750	96	39	g.	g.	PROPN
ejpam-4750	96	40	hence	hence	ADV
ejpam-4750	96	41	,	,	PUNCT
ejpam-4750	96	42	t	t	PROPN
ejpam-4750	96	43	=	=	SYM
ejpam-4750	96	44	j	j	PROPN
ejpam-4750	96	45	∪m	∪m	NUM
ejpam-4750	96	46	,	,	PUNCT
ejpam-4750	96	47	where	where	SCONJ
ejpam-4750	96	48	m	m	VERB
ejpam-4750	96	49	=	=	PRON
ejpam-4750	97	1	[	[	PUNCT
ejpam-4750	97	2	k	k	X
ejpam-4750	97	3	\	\	PROPN
ejpam-4750	97	4	{	{	PUNCT
ejpam-4750	97	5	w	w	NOUN
ejpam-4750	97	6	}	}	PUNCT
ejpam-4750	97	7	]	]	PUNCT
ejpam-4750	97	8	∪	∪	X
ejpam-4750	97	9	{	{	PUNCT
ejpam-4750	97	10	vw	vw	NOUN
ejpam-4750	97	11	}	}	PUNCT
ejpam-4750	97	12	,	,	PUNCT
ejpam-4750	97	13	is	be	AUX
ejpam-4750	97	14	a	a	DET
ejpam-4750	97	15	2	2	NUM
ejpam-4750	97	16	-	-	PUNCT
ejpam-4750	97	17	metric	metric	ADJ
ejpam-4750	97	18	basis	basis	NOUN
ejpam-4750	97	19	containing	contain	VERB
ejpam-4750	97	20	j	j	PROPN
ejpam-4750	97	21	,	,	PUNCT
ejpam-4750	97	22	a	a	DET
ejpam-4750	97	23	contradiction	contradiction	NOUN
ejpam-4750	97	24	.	.	PUNCT
ejpam-4750	98	1	hence	hence	ADV
ejpam-4750	98	2	,	,	PUNCT
ejpam-4750	98	3	d	d	PROPN
ejpam-4750	98	4	is	be	AUX
ejpam-4750	98	5	the	the	DET
ejpam-4750	98	6	only	only	ADJ
ejpam-4750	98	7	forcing	forcing	NOUN
ejpam-4750	98	8	subset	subset	NOUN
ejpam-4750	98	9	for	for	ADP
ejpam-4750	98	10	d.	d.	PROPN
ejpam-4750	98	11	therefore	therefore	ADV
ejpam-4750	98	12	,	,	PUNCT
ejpam-4750	98	13	fdim2(g	fdim2(g	ADV
ejpam-4750	98	14	)	)	PUNCT
ejpam-4750	98	15	=	=	PUNCT
ejpam-4750	98	16	dim2(g	dim2(g	NOUN
ejpam-4750	98	17	)	)	PUNCT
ejpam-4750	98	18	.	.	PUNCT
ejpam-4750	99	1	d.	d.	PROPN
ejpam-4750	99	2	managbanag	managbanag	PROPN
ejpam-4750	99	3	,	,	PUNCT
ejpam-4750	99	4	h.	h.	PROPN
ejpam-4750	99	5	rara	rara	PROPN
ejpam-4750	99	6	/	/	SYM
ejpam-4750	99	7	eur	eur	PROPN
ejpam-4750	99	8	.	.	PUNCT
ejpam-4750	100	1	j.	j.	PROPN
ejpam-4750	100	2	pure	pure	PROPN
ejpam-4750	100	3	appl	appl	PROPN
ejpam-4750	100	4	.	.	PROPN
ejpam-4750	100	5	math	math	PROPN
ejpam-4750	100	6	,	,	PUNCT
ejpam-4750	100	7	16	16	NUM
ejpam-4750	100	8	(	(	PUNCT
ejpam-4750	100	9	2	2	NUM
ejpam-4750	100	10	)	)	PUNCT
ejpam-4750	100	11	(	(	PUNCT
ejpam-4750	100	12	2023	2023	NUM
ejpam-4750	100	13	)	)	PUNCT
ejpam-4750	100	14	,	,	PUNCT
ejpam-4750	100	15	1068	1068	NUM
ejpam-4750	100	16	-	-	SYM
ejpam-4750	100	17	1083	1083	NUM
ejpam-4750	100	18	1072	1072	NUM
ejpam-4750	100	19	proposition	proposition	NOUN
ejpam-4750	100	20	2	2	NUM
ejpam-4750	100	21	.	.	X
ejpam-4750	101	1	for	for	ADP
ejpam-4750	101	2	any	any	DET
ejpam-4750	101	3	complete	complete	ADJ
ejpam-4750	101	4	graph	graph	NOUN
ejpam-4750	101	5	kn	kn	PROPN
ejpam-4750	101	6	with	with	ADP
ejpam-4750	101	7	n	n	PRON
ejpam-4750	101	8	≥	≥	NUM
ejpam-4750	101	9	1	1	NUM
ejpam-4750	101	10	vertices	vertex	NOUN
ejpam-4750	101	11	,	,	PUNCT
ejpam-4750	101	12	fdim2(kn	fdim2(kn	NUM
ejpam-4750	101	13	)	)	PUNCT
ejpam-4750	101	14	=	=	SYM
ejpam-4750	101	15	0	0	X
ejpam-4750	101	16	.	.	X
ejpam-4750	102	1	proof	proof	NOUN
ejpam-4750	102	2	:	:	PUNCT
ejpam-4750	102	3	by	by	ADP
ejpam-4750	102	4	definition	definition	NOUN
ejpam-4750	102	5	of	of	ADP
ejpam-4750	102	6	kn	kn	PROPN
ejpam-4750	102	7	,	,	PUNCT
ejpam-4750	102	8	v	v	PROPN
ejpam-4750	102	9	(	(	PUNCT
ejpam-4750	102	10	kn	kn	PROPN
ejpam-4750	102	11	)	)	PUNCT
ejpam-4750	102	12	is	be	AUX
ejpam-4750	102	13	the	the	DET
ejpam-4750	102	14	only	only	ADJ
ejpam-4750	102	15	minimum	minimum	ADJ
ejpam-4750	102	16	2	2	NUM
ejpam-4750	102	17	-	-	PUNCT
ejpam-4750	102	18	resolving	resolve	VERB
ejpam-4750	102	19	set	set	NOUN
ejpam-4750	102	20	for	for	ADP
ejpam-4750	102	21	kn	kn	PROPN
ejpam-4750	102	22	.	.	PUNCT
ejpam-4750	103	1	by	by	ADP
ejpam-4750	103	2	remark	remark	NOUN
ejpam-4750	103	3	4	4	NUM
ejpam-4750	103	4	(	(	PUNCT
ejpam-4750	103	5	i	i	NOUN
ejpam-4750	103	6	)	)	PUNCT
ejpam-4750	103	7	,	,	PUNCT
ejpam-4750	103	8	fdim2(kn	fdim2(kn	X
ejpam-4750	103	9	)	)	PUNCT
ejpam-4750	103	10	=	=	SYM
ejpam-4750	103	11	0	0	X
ejpam-4750	103	12	.	.	PUNCT
ejpam-4750	103	13	proposition	proposition	NOUN
ejpam-4750	103	14	3	3	NUM
ejpam-4750	103	15	.	.	X
ejpam-4750	104	1	for	for	ADP
ejpam-4750	104	2	any	any	DET
ejpam-4750	104	3	path	path	NOUN
ejpam-4750	104	4	pn	pn	NOUN
ejpam-4750	104	5	with	with	ADP
ejpam-4750	104	6	n	n	PRON
ejpam-4750	104	7	≥	≥	NUM
ejpam-4750	104	8	2	2	NUM
ejpam-4750	104	9	vertices	vertex	NOUN
ejpam-4750	104	10	,	,	PUNCT
ejpam-4750	104	11	fdim2(pn	fdim2(pn	NOUN
ejpam-4750	104	12	)	)	PUNCT
ejpam-4750	104	13	=	=	SYM
ejpam-4750	104	14	0	0	X
ejpam-4750	104	15	.	.	PUNCT
ejpam-4750	105	1	proof	proof	NOUN
ejpam-4750	105	2	:	:	PUNCT
ejpam-4750	105	3	suppose	suppose	VERB
ejpam-4750	105	4	that	that	SCONJ
ejpam-4750	105	5	pn	pn	PROPN
ejpam-4750	105	6	=	=	PUNCT
ejpam-4750	106	1	[	[	X
ejpam-4750	106	2	v1	v1	NOUN
ejpam-4750	106	3	,	,	PUNCT
ejpam-4750	106	4	v2	v2	NOUN
ejpam-4750	106	5	,	,	PUNCT
ejpam-4750	106	6	.	.	PUNCT
ejpam-4750	106	7	.	.	PUNCT
ejpam-4750	106	8	.	.	PUNCT
ejpam-4750	107	1	,	,	PUNCT
ejpam-4750	107	2	vn	vn	X
ejpam-4750	107	3	]	]	PUNCT
ejpam-4750	107	4	.	.	PUNCT
ejpam-4750	108	1	by	by	ADP
ejpam-4750	108	2	proposition	proposition	NOUN
ejpam-4750	108	3	1	1	NUM
ejpam-4750	108	4	,	,	PUNCT
ejpam-4750	108	5	dim2(pn	dim2(pn	PROPN
ejpam-4750	108	6	)	)	PUNCT
ejpam-4750	108	7	=	=	SYM
ejpam-4750	108	8	2	2	NUM
ejpam-4750	108	9	for	for	ADP
ejpam-4750	108	10	all	all	DET
ejpam-4750	108	11	n	n	PRON
ejpam-4750	108	12	≥	≥	NOUN
ejpam-4750	108	13	2	2	NUM
ejpam-4750	108	14	.	.	PUNCT
ejpam-4750	108	15	we	we	PRON
ejpam-4750	108	16	claim	claim	VERB
ejpam-4750	108	17	that	that	SCONJ
ejpam-4750	108	18	s	s	VERB
ejpam-4750	108	19	=	=	PUNCT
ejpam-4750	108	20	{	{	PUNCT
ejpam-4750	108	21	v1	v1	PROPN
ejpam-4750	108	22	,	,	PUNCT
ejpam-4750	108	23	vn	vn	PROPN
ejpam-4750	108	24	}	}	PUNCT
ejpam-4750	108	25	is	be	AUX
ejpam-4750	108	26	the	the	DET
ejpam-4750	108	27	unique	unique	ADJ
ejpam-4750	108	28	2	2	NUM
ejpam-4750	108	29	-	-	PUNCT
ejpam-4750	108	30	metric	metric	ADJ
ejpam-4750	108	31	basis	basis	NOUN
ejpam-4750	108	32	of	of	ADP
ejpam-4750	108	33	pn	pn	PROPN
ejpam-4750	108	34	.	.	PROPN
ejpam-4750	108	35	suppose	suppose	VERB
ejpam-4750	108	36	s	s	AUX
ejpam-4750	108	37	′	′	NOUN
ejpam-4750	108	38	=	=	SYM
ejpam-4750	108	39	{	{	PUNCT
ejpam-4750	108	40	vi	vi	PROPN
ejpam-4750	108	41	,	,	PUNCT
ejpam-4750	108	42	vj	vj	ADP
ejpam-4750	108	43	}	}	PUNCT
ejpam-4750	108	44	where	where	SCONJ
ejpam-4750	108	45	1	1	NUM
ejpam-4750	108	46	≤	≤	PUNCT
ejpam-4750	109	1	i	i	PRON
ejpam-4750	109	2	<	<	X
ejpam-4750	109	3	j	j	PROPN
ejpam-4750	109	4	≤	≤	PROPN
ejpam-4750	109	5	n	n	ADP
ejpam-4750	109	6	and	and	CCONJ
ejpam-4750	109	7	s′	s′	VERB
ejpam-4750	109	8	̸=	̸=	PROPN
ejpam-4750	109	9	s.	s.	PROPN
ejpam-4750	109	10	consider	consider	VERB
ejpam-4750	109	11	the	the	DET
ejpam-4750	109	12	following	follow	VERB
ejpam-4750	109	13	cases	case	NOUN
ejpam-4750	109	14	:	:	PUNCT
ejpam-4750	109	15	case	case	NOUN
ejpam-4750	109	16	1	1	NUM
ejpam-4750	109	17	.	.	PUNCT
ejpam-4750	109	18	suppose	suppose	VERB
ejpam-4750	109	19	i	i	PRON
ejpam-4750	109	20	=	=	NOUN
ejpam-4750	109	21	1	1	X
ejpam-4750	109	22	.	.	PUNCT
ejpam-4750	110	1	if	if	SCONJ
ejpam-4750	110	2	j	j	PROPN
ejpam-4750	110	3	=	=	SYM
ejpam-4750	110	4	2	2	NUM
ejpam-4750	110	5	,	,	PUNCT
ejpam-4750	110	6	then	then	ADV
ejpam-4750	110	7	rpn(v1	rpn(v1	VERB
ejpam-4750	110	8	/	/	SYM
ejpam-4750	110	9	s	s	PART
ejpam-4750	110	10	′	′	NUM
ejpam-4750	110	11	)	)	PUNCT
ejpam-4750	110	12	=	=	PUNCT
ejpam-4750	110	13	(	(	PUNCT
ejpam-4750	110	14	0	0	NUM
ejpam-4750	110	15	,	,	PUNCT
ejpam-4750	110	16	1	1	NUM
ejpam-4750	110	17	)	)	PUNCT
ejpam-4750	110	18	and	and	CCONJ
ejpam-4750	110	19	rpn(v3	rpn(v3	PROPN
ejpam-4750	110	20	/	/	SYM
ejpam-4750	110	21	s	s	PART
ejpam-4750	110	22	′	′	NOUN
ejpam-4750	110	23	)	)	PUNCT
ejpam-4750	110	24	=	=	PUNCT
ejpam-4750	110	25	(	(	PUNCT
ejpam-4750	110	26	2	2	NUM
ejpam-4750	110	27	,	,	PUNCT
ejpam-4750	110	28	1	1	NUM
ejpam-4750	110	29	)	)	PUNCT
ejpam-4750	110	30	.	.	PUNCT
ejpam-4750	111	1	if	if	SCONJ
ejpam-4750	111	2	2	2	NUM
ejpam-4750	111	3	<	<	X
ejpam-4750	111	4	j	j	X
ejpam-4750	111	5	<	<	X
ejpam-4750	111	6	n	n	CCONJ
ejpam-4750	111	7	,	,	PUNCT
ejpam-4750	111	8	then	then	ADV
ejpam-4750	111	9	rpn(vj−1	rpn(vj−1	PROPN
ejpam-4750	111	10	/	/	SYM
ejpam-4750	111	11	s	s	PART
ejpam-4750	111	12	′	′	NUM
ejpam-4750	111	13	)	)	PUNCT
ejpam-4750	112	1	=	=	PUNCT
ejpam-4750	113	1	(	(	PUNCT
ejpam-4750	113	2	j	j	PROPN
ejpam-4750	113	3	−	−	PROPN
ejpam-4750	113	4	2	2	NUM
ejpam-4750	113	5	,	,	PUNCT
ejpam-4750	113	6	1	1	NUM
ejpam-4750	113	7	)	)	PUNCT
ejpam-4750	113	8	and	and	CCONJ
ejpam-4750	113	9	rpn(vj+1	rpn(vj+1	PROPN
ejpam-4750	113	10	/	/	SYM
ejpam-4750	113	11	s	s	PART
ejpam-4750	113	12	′	′	NOUN
ejpam-4750	113	13	)	)	PUNCT
ejpam-4750	113	14	=	=	PUNCT
ejpam-4750	113	15	(	(	PUNCT
ejpam-4750	113	16	j	j	NOUN
ejpam-4750	113	17	,	,	PUNCT
ejpam-4750	113	18	1	1	NUM
ejpam-4750	113	19	)	)	PUNCT
ejpam-4750	113	20	.	.	PUNCT
ejpam-4750	114	1	case	case	NOUN
ejpam-4750	114	2	2	2	X
ejpam-4750	114	3	.	.	PUNCT
ejpam-4750	114	4	suppose	suppose	VERB
ejpam-4750	114	5	1	1	NUM
ejpam-4750	114	6	<	<	X
ejpam-4750	114	7	i	i	X
ejpam-4750	114	8	<	<	X
ejpam-4750	114	9	j	j	X
ejpam-4750	114	10	<	<	X
ejpam-4750	114	11	n.	n.	PROPN
ejpam-4750	114	12	if	if	SCONJ
ejpam-4750	114	13	j	j	PROPN
ejpam-4750	114	14	=	=	VERB
ejpam-4750	115	1	i	i	PRON
ejpam-4750	115	2	+	+	NOUN
ejpam-4750	115	3	1	1	NUM
ejpam-4750	115	4	,	,	PUNCT
ejpam-4750	115	5	then	then	ADV
ejpam-4750	115	6	rpn(vi	rpn(vi	PROPN
ejpam-4750	115	7	/	/	SYM
ejpam-4750	115	8	s	s	PART
ejpam-4750	115	9	′	′	NUM
ejpam-4750	115	10	)	)	PUNCT
ejpam-4750	115	11	=	=	PUNCT
ejpam-4750	115	12	(	(	PUNCT
ejpam-4750	115	13	0	0	NUM
ejpam-4750	115	14	,	,	PUNCT
ejpam-4750	115	15	1	1	NUM
ejpam-4750	115	16	)	)	PUNCT
ejpam-4750	115	17	,	,	PUNCT
ejpam-4750	115	18	rpn(vj+1	rpn(vj+1	PROPN
ejpam-4750	115	19	/	/	SYM
ejpam-4750	115	20	s	s	PART
ejpam-4750	115	21	′	′	NUM
ejpam-4750	115	22	)	)	PUNCT
ejpam-4750	115	23	=	=	PUNCT
ejpam-4750	115	24	(	(	PUNCT
ejpam-4750	115	25	2	2	NUM
ejpam-4750	115	26	,	,	PUNCT
ejpam-4750	115	27	1	1	NUM
ejpam-4750	115	28	)	)	PUNCT
ejpam-4750	115	29	,	,	PUNCT
ejpam-4750	115	30	rpn(vi−1	rpn(vi−1	NOUN
ejpam-4750	115	31	/	/	SYM
ejpam-4750	115	32	s	s	PART
ejpam-4750	115	33	′	′	NUM
ejpam-4750	115	34	)	)	PUNCT
ejpam-4750	116	1	=	=	PUNCT
ejpam-4750	116	2	(	(	PUNCT
ejpam-4750	116	3	1	1	NUM
ejpam-4750	116	4	,	,	PUNCT
ejpam-4750	116	5	2	2	NUM
ejpam-4750	116	6	)	)	PUNCT
ejpam-4750	116	7	,	,	PUNCT
ejpam-4750	116	8	and	and	CCONJ
ejpam-4750	116	9	rpn(vj	rpn(vj	X
ejpam-4750	116	10	/	/	SYM
ejpam-4750	116	11	s	s	PART
ejpam-4750	116	12	′	′	NOUN
ejpam-4750	116	13	)	)	PUNCT
ejpam-4750	116	14	=	=	PUNCT
ejpam-4750	116	15	(	(	PUNCT
ejpam-4750	116	16	1	1	NUM
ejpam-4750	116	17	,	,	PUNCT
ejpam-4750	116	18	0	0	NUM
ejpam-4750	116	19	)	)	PUNCT
ejpam-4750	116	20	.	.	PUNCT
ejpam-4750	117	1	if	if	SCONJ
ejpam-4750	117	2	j	j	PROPN
ejpam-4750	117	3	>	>	X
ejpam-4750	117	4	i	i	PRON
ejpam-4750	118	1	+	+	NOUN
ejpam-4750	118	2	1	1	NUM
ejpam-4750	118	3	,	,	PUNCT
ejpam-4750	118	4	then	then	ADV
ejpam-4750	118	5	rpn(vi−1	rpn(vi−1	PROPN
ejpam-4750	118	6	/	/	SYM
ejpam-4750	118	7	s	s	PART
ejpam-4750	118	8	′	′	NUM
ejpam-4750	118	9	)	)	PUNCT
ejpam-4750	119	1	=	=	PUNCT
ejpam-4750	119	2	(	(	PUNCT
ejpam-4750	119	3	1	1	NUM
ejpam-4750	119	4	,	,	PUNCT
ejpam-4750	119	5	j	j	PROPN
ejpam-4750	120	1	−	−	NOUN
ejpam-4750	120	2	i	i	PRON
ejpam-4750	120	3	+	+	NOUN
ejpam-4750	120	4	1	1	X
ejpam-4750	120	5	)	)	PUNCT
ejpam-4750	120	6	and	and	CCONJ
ejpam-4750	120	7	rpn(vi+1	rpn(vi+1	PROPN
ejpam-4750	120	8	/	/	SYM
ejpam-4750	120	9	s	s	PART
ejpam-4750	120	10	′	′	NOUN
ejpam-4750	120	11	)	)	PUNCT
ejpam-4750	120	12	=	=	PUNCT
ejpam-4750	120	13	(	(	PUNCT
ejpam-4750	120	14	1	1	NUM
ejpam-4750	120	15	,	,	PUNCT
ejpam-4750	120	16	j	j	PROPN
ejpam-4750	120	17	−	−	PROPN
ejpam-4750	120	18	i−	i−	PROPN
ejpam-4750	120	19	1	1	NUM
ejpam-4750	120	20	)	)	PUNCT
ejpam-4750	120	21	.	.	PUNCT
ejpam-4750	121	1	by	by	ADP
ejpam-4750	121	2	cases	case	NOUN
ejpam-4750	121	3	1	1	NUM
ejpam-4750	121	4	and	and	CCONJ
ejpam-4750	121	5	2	2	NUM
ejpam-4750	121	6	,	,	PUNCT
ejpam-4750	121	7	s′	s′	ADJ
ejpam-4750	121	8	is	be	AUX
ejpam-4750	121	9	not	not	PART
ejpam-4750	121	10	a	a	DET
ejpam-4750	121	11	2	2	NUM
ejpam-4750	121	12	-	-	PUNCT
ejpam-4750	121	13	resolving	resolving	NOUN
ejpam-4750	121	14	set	set	NOUN
ejpam-4750	121	15	for	for	ADP
ejpam-4750	121	16	pn	pn	PROPN
ejpam-4750	121	17	.	.	PUNCT
ejpam-4750	122	1	thus	thus	ADV
ejpam-4750	122	2	,	,	PUNCT
ejpam-4750	122	3	s	s	VERB
ejpam-4750	122	4	is	be	AUX
ejpam-4750	122	5	unique	unique	ADJ
ejpam-4750	122	6	.	.	PUNCT
ejpam-4750	123	1	hence	hence	ADV
ejpam-4750	123	2	,	,	PUNCT
ejpam-4750	123	3	by	by	ADP
ejpam-4750	123	4	remark	remark	NOUN
ejpam-4750	123	5	4	4	NUM
ejpam-4750	123	6	(	(	PUNCT
ejpam-4750	123	7	i	i	NOUN
ejpam-4750	123	8	)	)	PUNCT
ejpam-4750	123	9	,	,	PUNCT
ejpam-4750	123	10	fdim2(pn	fdim2(pn	NOUN
ejpam-4750	123	11	)	)	PUNCT
ejpam-4750	123	12	=	=	SYM
ejpam-4750	123	13	0	0	NUM
ejpam-4750	123	14	for	for	ADP
ejpam-4750	123	15	all	all	DET
ejpam-4750	123	16	n	n	PRON
ejpam-4750	123	17	≥	≥	NOUN
ejpam-4750	123	18	4	4	NUM
ejpam-4750	123	19	.	.	PUNCT
ejpam-4750	124	1	therefore	therefore	ADV
ejpam-4750	124	2	,	,	PUNCT
ejpam-4750	124	3	fdim2(pn	fdim2(pn	PRON
ejpam-4750	124	4	)	)	PUNCT
ejpam-4750	124	5	=	=	SYM
ejpam-4750	124	6	0	0	NUM
ejpam-4750	124	7	for	for	ADP
ejpam-4750	124	8	all	all	DET
ejpam-4750	124	9	n	n	PRON
ejpam-4750	124	10	≥	≥	NUM
ejpam-4750	124	11	2	2	NUM
ejpam-4750	124	12	vertices	vertex	NOUN
ejpam-4750	124	13	.	.	PUNCT
ejpam-4750	125	1	proposition	proposition	NOUN
ejpam-4750	125	2	4	4	NUM
ejpam-4750	125	3	.	.	X
ejpam-4750	126	1	for	for	ADP
ejpam-4750	126	2	any	any	DET
ejpam-4750	126	3	cycle	cycle	NOUN
ejpam-4750	126	4	cn	cn	NOUN
ejpam-4750	126	5	with	with	ADP
ejpam-4750	126	6	n	n	NUM
ejpam-4750	126	7	≥	≥	NUM
ejpam-4750	126	8	3	3	NUM
ejpam-4750	126	9	vertices	vertex	NOUN
ejpam-4750	126	10	,	,	PUNCT
ejpam-4750	126	11	fdim2(cn	fdim2(cn	NOUN
ejpam-4750	126	12	)	)	PUNCT
ejpam-4750	126	13	=	=	SYM
ejpam-4750	126	14	®	®	NOUN
ejpam-4750	126	15	0	0	NUM
ejpam-4750	126	16	,	,	PUNCT
ejpam-4750	126	17	if	if	SCONJ
ejpam-4750	126	18	n	n	NOUN
ejpam-4750	126	19	=	=	SYM
ejpam-4750	126	20	3	3	NUM
ejpam-4750	126	21	,	,	PUNCT
ejpam-4750	126	22	4	4	NUM
ejpam-4750	126	23	3	3	NUM
ejpam-4750	126	24	,	,	PUNCT
ejpam-4750	126	25	if	if	SCONJ
ejpam-4750	126	26	n	n	PRON
ejpam-4750	126	27	≥	≥	NOUN
ejpam-4750	126	28	5	5	NUM
ejpam-4750	126	29	.	.	PUNCT
ejpam-4750	127	1	proof	proof	NOUN
ejpam-4750	127	2	:	:	PUNCT
ejpam-4750	127	3	suppose	suppose	VERB
ejpam-4750	127	4	that	that	SCONJ
ejpam-4750	127	5	cn	cn	PROPN
ejpam-4750	127	6	=	=	PUNCT
ejpam-4750	128	1	[	[	X
ejpam-4750	128	2	u1	u1	NOUN
ejpam-4750	128	3	,	,	PUNCT
ejpam-4750	128	4	u2	u2	NOUN
ejpam-4750	128	5	,	,	PUNCT
ejpam-4750	128	6	.	.	PUNCT
ejpam-4750	128	7	.	.	PUNCT
ejpam-4750	128	8	.	.	PUNCT
ejpam-4750	129	1	,	,	PUNCT
ejpam-4750	129	2	un	un	PROPN
ejpam-4750	129	3	]	]	X
ejpam-4750	129	4	.	.	PUNCT
ejpam-4750	130	1	if	if	SCONJ
ejpam-4750	130	2	n	n	NUM
ejpam-4750	130	3	=	=	SYM
ejpam-4750	130	4	3	3	NUM
ejpam-4750	130	5	or	or	CCONJ
ejpam-4750	130	6	4	4	NUM
ejpam-4750	130	7	,	,	PUNCT
ejpam-4750	130	8	then	then	ADV
ejpam-4750	130	9	v	v	INTJ
ejpam-4750	130	10	(	(	PUNCT
ejpam-4750	130	11	cn	cn	PROPN
ejpam-4750	130	12	)	)	PUNCT
ejpam-4750	130	13	is	be	AUX
ejpam-4750	130	14	the	the	DET
ejpam-4750	130	15	only	only	ADJ
ejpam-4750	130	16	2	2	NUM
ejpam-4750	130	17	-	-	PUNCT
ejpam-4750	130	18	resolving	resolve	VERB
ejpam-4750	130	19	set	set	NOUN
ejpam-4750	130	20	for	for	ADP
ejpam-4750	130	21	cn	cn	PROPN
ejpam-4750	130	22	.	.	PUNCT
ejpam-4750	131	1	by	by	ADP
ejpam-4750	131	2	remark	remark	NOUN
ejpam-4750	131	3	4	4	NUM
ejpam-4750	131	4	(	(	PUNCT
ejpam-4750	131	5	i	i	NOUN
ejpam-4750	131	6	)	)	PUNCT
ejpam-4750	131	7	,	,	PUNCT
ejpam-4750	131	8	fdim2(cn	fdim2(cn	NOUN
ejpam-4750	131	9	)	)	PUNCT
ejpam-4750	131	10	=	=	SYM
ejpam-4750	132	1	0	0	X
ejpam-4750	132	2	.	.	PUNCT
ejpam-4750	133	1	let	let	VERB
ejpam-4750	133	2	n	n	PRON
ejpam-4750	133	3	≥	≥	NOUN
ejpam-4750	133	4	5	5	NUM
ejpam-4750	133	5	.	.	PUNCT
ejpam-4750	134	1	by	by	ADP
ejpam-4750	134	2	proposition	proposition	NOUN
ejpam-4750	134	3	1	1	NUM
ejpam-4750	134	4	,	,	PUNCT
ejpam-4750	134	5	dim2(cn	dim2(cn	PROPN
ejpam-4750	134	6	)	)	PUNCT
ejpam-4750	134	7	>	>	X
ejpam-4750	135	1	2	2	X
ejpam-4750	135	2	.	.	PUNCT
ejpam-4750	135	3	since	since	SCONJ
ejpam-4750	135	4	{	{	PUNCT
ejpam-4750	135	5	u1	u1	PROPN
ejpam-4750	135	6	,	,	PUNCT
ejpam-4750	135	7	u2	u2	NOUN
ejpam-4750	135	8	,	,	PUNCT
ejpam-4750	135	9	u3	u3	PROPN
ejpam-4750	135	10	}	}	PUNCT
ejpam-4750	135	11	is	be	AUX
ejpam-4750	135	12	a	a	DET
ejpam-4750	135	13	2	2	NUM
ejpam-4750	135	14	-	-	PUNCT
ejpam-4750	135	15	resolving	resolving	NOUN
ejpam-4750	135	16	set	set	NOUN
ejpam-4750	135	17	for	for	ADP
ejpam-4750	135	18	cn	cn	PROPN
ejpam-4750	135	19	,	,	PUNCT
ejpam-4750	135	20	dim2(cn	dim2(cn	PROPN
ejpam-4750	135	21	)	)	PUNCT
ejpam-4750	135	22	=	=	SYM
ejpam-4750	136	1	3	3	X
ejpam-4750	136	2	.	.	X
ejpam-4750	136	3	let	let	VERB
ejpam-4750	136	4	s	s	PRON
ejpam-4750	136	5	=	=	PUNCT
ejpam-4750	136	6	{	{	PUNCT
ejpam-4750	136	7	ua	ua	PROPN
ejpam-4750	136	8	,	,	PUNCT
ejpam-4750	136	9	ub	ub	PROPN
ejpam-4750	136	10	,	,	PUNCT
ejpam-4750	136	11	uc	uc	AUX
ejpam-4750	136	12	}	}	PUNCT
ejpam-4750	136	13	be	be	AUX
ejpam-4750	136	14	a	a	DET
ejpam-4750	136	15	2	2	NUM
ejpam-4750	136	16	-	-	PUNCT
ejpam-4750	136	17	metric	metric	ADJ
ejpam-4750	136	18	basis	basis	NOUN
ejpam-4750	136	19	for	for	ADP
ejpam-4750	136	20	cn	cn	PROPN
ejpam-4750	136	21	.	.	PUNCT
ejpam-4750	136	22	hence	hence	ADV
ejpam-4750	136	23	,	,	PUNCT
ejpam-4750	136	24	for	for	ADP
ejpam-4750	136	25	all	all	PRON
ejpam-4750	136	26	2	2	NUM
ejpam-4750	136	27	-	-	PUNCT
ejpam-4750	136	28	metric	metric	ADJ
ejpam-4750	136	29	basis	basis	NOUN
ejpam-4750	136	30	s	s	NOUN
ejpam-4750	136	31	for	for	ADP
ejpam-4750	136	32	cn	cn	PROPN
ejpam-4750	136	33	and	and	CCONJ
ejpam-4750	136	34	for	for	ADP
ejpam-4750	136	35	each	each	DET
ejpam-4750	136	36	uk	uk	PROPN
ejpam-4750	136	37	∈	∈	PROPN
ejpam-4750	136	38	s	s	PART
ejpam-4750	136	39	,	,	PUNCT
ejpam-4750	136	40	there	there	PRON
ejpam-4750	136	41	exists	exist	VERB
ejpam-4750	136	42	ul	ul	INTJ
ejpam-4750	136	43	∈	∈	PROPN
ejpam-4750	136	44	v	v	NOUN
ejpam-4750	136	45	(	(	PUNCT
ejpam-4750	136	46	cn	cn	PROPN
ejpam-4750	136	47	)	)	PUNCT
ejpam-4750	136	48	\s	\	VERB
ejpam-4750	136	49	such	such	ADJ
ejpam-4750	136	50	that	that	SCONJ
ejpam-4750	136	51	(	(	PUNCT
ejpam-4750	136	52	s	s	NOUN
ejpam-4750	136	53	\	\	X
ejpam-4750	136	54	{	{	PUNCT
ejpam-4750	136	55	uk})∪{ul	uk})∪{ul	NOUN
ejpam-4750	136	56	}	}	PUNCT
ejpam-4750	136	57	is	be	AUX
ejpam-4750	136	58	a	a	DET
ejpam-4750	136	59	2	2	NUM
ejpam-4750	136	60	-	-	PUNCT
ejpam-4750	136	61	metric	metric	ADJ
ejpam-4750	136	62	basis	basis	NOUN
ejpam-4750	136	63	for	for	ADP
ejpam-4750	136	64	cn	cn	PROPN
ejpam-4750	136	65	.	.	PUNCT
ejpam-4750	136	66	by	by	ADP
ejpam-4750	136	67	theorem	theorem	NOUN
ejpam-4750	136	68	5	5	NUM
ejpam-4750	136	69	,	,	PUNCT
ejpam-4750	136	70	fdim2(cn	fdim2(cn	NOUN
ejpam-4750	136	71	)	)	PUNCT
ejpam-4750	136	72	=	=	SYM
ejpam-4750	136	73	dim2(cn	dim2(cn	PROPN
ejpam-4750	136	74	)	)	PUNCT
ejpam-4750	136	75	=	=	PUNCT
ejpam-4750	137	1	3	3	NUM
ejpam-4750	137	2	.	.	NOUN
ejpam-4750	137	3	5	5	NUM
ejpam-4750	137	4	.	.	X
ejpam-4750	138	1	forcing	force	VERB
ejpam-4750	138	2	2	2	NUM
ejpam-4750	138	3	-	-	PUNCT
ejpam-4750	138	4	locating	locate	VERB
ejpam-4750	138	5	and	and	CCONJ
ejpam-4750	138	6	(	(	PUNCT
ejpam-4750	138	7	2,2)-locating	2,2)-locating	NUM
ejpam-4750	138	8	numbers	number	NOUN
ejpam-4750	138	9	of	of	ADP
ejpam-4750	138	10	some	some	DET
ejpam-4750	138	11	special	special	ADJ
ejpam-4750	138	12	graphs	graph	NOUN
ejpam-4750	138	13	remark	remark	VERB
ejpam-4750	138	14	5	5	NUM
ejpam-4750	138	15	.	.	PUNCT
ejpam-4750	139	1	let	let	VERB
ejpam-4750	139	2	g	g	PRON
ejpam-4750	139	3	be	be	AUX
ejpam-4750	139	4	a	a	DET
ejpam-4750	139	5	connected	connected	ADJ
ejpam-4750	139	6	graph	graph	NOUN
ejpam-4750	139	7	.	.	PUNCT
ejpam-4750	140	1	then	then	ADV
ejpam-4750	140	2	(	(	PUNCT
ejpam-4750	140	3	i	i	NOUN
ejpam-4750	140	4	)	)	PUNCT
ejpam-4750	140	5	fln2(g	fln2(g	NOUN
ejpam-4750	140	6	)	)	PUNCT
ejpam-4750	140	7	=	=	SYM
ejpam-4750	140	8	0	0	PUNCT
ejpam-4750	141	1	if	if	SCONJ
ejpam-4750	141	2	and	and	CCONJ
ejpam-4750	141	3	only	only	ADV
ejpam-4750	141	4	if	if	SCONJ
ejpam-4750	141	5	g	g	PROPN
ejpam-4750	141	6	has	have	VERB
ejpam-4750	141	7	a	a	DET
ejpam-4750	141	8	unique	unique	ADJ
ejpam-4750	141	9	ln2	ln2	NOUN
ejpam-4750	141	10	-	-	PUNCT
ejpam-4750	141	11	set	set	NOUN
ejpam-4750	141	12	,	,	PUNCT
ejpam-4750	141	13	and	and	CCONJ
ejpam-4750	141	14	(	(	PUNCT
ejpam-4750	141	15	ii	ii	NOUN
ejpam-4750	141	16	)	)	PUNCT
ejpam-4750	141	17	fln2(g	fln2(g	NOUN
ejpam-4750	141	18	)	)	PUNCT
ejpam-4750	141	19	=	=	SYM
ejpam-4750	141	20	1	1	NUM
ejpam-4750	141	21	if	if	SCONJ
ejpam-4750	141	22	and	and	CCONJ
ejpam-4750	141	23	only	only	ADV
ejpam-4750	141	24	if	if	SCONJ
ejpam-4750	141	25	g	g	PROPN
ejpam-4750	141	26	has	have	VERB
ejpam-4750	141	27	at	at	ADV
ejpam-4750	141	28	least	least	ADJ
ejpam-4750	141	29	two	two	NUM
ejpam-4750	141	30	ln2	ln2	ADJ
ejpam-4750	141	31	-	-	PUNCT
ejpam-4750	141	32	sets	set	NOUN
ejpam-4750	141	33	,	,	PUNCT
ejpam-4750	141	34	one	one	NUM
ejpam-4750	141	35	of	of	ADP
ejpam-4750	141	36	which	which	PRON
ejpam-4750	141	37	,	,	PUNCT
ejpam-4750	141	38	say	say	VERB
ejpam-4750	141	39	b	b	X
ejpam-4750	141	40	,	,	PUNCT
ejpam-4750	141	41	that	that	PRON
ejpam-4750	141	42	contains	contain	VERB
ejpam-4750	141	43	an	an	DET
ejpam-4750	141	44	element	element	NOUN
ejpam-4750	141	45	not	not	PART
ejpam-4750	141	46	in	in	ADP
ejpam-4750	141	47	any	any	DET
ejpam-4750	141	48	ln2	ln2	ADJ
ejpam-4750	141	49	-	-	PUNCT
ejpam-4750	141	50	sets	set	NOUN
ejpam-4750	141	51	of	of	ADP
ejpam-4750	141	52	g.	g.	PROPN
ejpam-4750	141	53	d.	d.	PROPN
ejpam-4750	141	54	managbanag	managbanag	PROPN
ejpam-4750	141	55	,	,	PUNCT
ejpam-4750	141	56	h.	h.	PROPN
ejpam-4750	141	57	rara	rara	PROPN
ejpam-4750	141	58	/	/	SYM
ejpam-4750	141	59	eur	eur	PROPN
ejpam-4750	141	60	.	.	PUNCT
ejpam-4750	142	1	j.	j.	PROPN
ejpam-4750	142	2	pure	pure	PROPN
ejpam-4750	142	3	appl	appl	PROPN
ejpam-4750	142	4	.	.	PROPN
ejpam-4750	142	5	math	math	PROPN
ejpam-4750	142	6	,	,	PUNCT
ejpam-4750	142	7	16	16	NUM
ejpam-4750	142	8	(	(	PUNCT
ejpam-4750	142	9	2	2	NUM
ejpam-4750	142	10	)	)	PUNCT
ejpam-4750	142	11	(	(	PUNCT
ejpam-4750	142	12	2023	2023	NUM
ejpam-4750	142	13	)	)	PUNCT
ejpam-4750	142	14	,	,	PUNCT
ejpam-4750	142	15	1068	1068	NUM
ejpam-4750	142	16	-	-	SYM
ejpam-4750	142	17	1083	1083	NUM
ejpam-4750	142	18	1073	1073	NUM
ejpam-4750	142	19	theorem	theorem	NOUN
ejpam-4750	142	20	6	6	NUM
ejpam-4750	142	21	.	.	PUNCT
ejpam-4750	143	1	let	let	VERB
ejpam-4750	143	2	g	g	PRON
ejpam-4750	143	3	be	be	AUX
ejpam-4750	143	4	a	a	DET
ejpam-4750	143	5	connected	connected	ADJ
ejpam-4750	143	6	graph	graph	NOUN
ejpam-4750	143	7	.	.	PUNCT
ejpam-4750	144	1	then	then	ADV
ejpam-4750	144	2	fln2(g	fln2(g	NOUN
ejpam-4750	144	3	)	)	PUNCT
ejpam-4750	144	4	=	=	PUNCT
ejpam-4750	144	5	ln2(g	ln2(g	VERB
ejpam-4750	144	6	)	)	PUNCT
ejpam-4750	144	7	if	if	SCONJ
ejpam-4750	144	8	and	and	CCONJ
ejpam-4750	144	9	only	only	ADV
ejpam-4750	144	10	if	if	SCONJ
ejpam-4750	144	11	for	for	ADP
ejpam-4750	144	12	all	all	DET
ejpam-4750	144	13	ln2	ln2	NOUN
ejpam-4750	144	14	-	-	PUNCT
ejpam-4750	144	15	set	set	NOUN
ejpam-4750	144	16	s	s	NOUN
ejpam-4750	144	17	of	of	ADP
ejpam-4750	144	18	g	g	NOUN
ejpam-4750	144	19	and	and	CCONJ
ejpam-4750	144	20	for	for	ADP
ejpam-4750	144	21	each	each	DET
ejpam-4750	144	22	u	u	PROPN
ejpam-4750	144	23	∈	∈	PROPN
ejpam-4750	144	24	s	s	PART
ejpam-4750	144	25	,	,	PUNCT
ejpam-4750	144	26	there	there	PRON
ejpam-4750	144	27	exists	exist	VERB
ejpam-4750	144	28	vu	vu	PROPN
ejpam-4750	144	29	∈	∈	PROPN
ejpam-4750	144	30	v	v	ADP
ejpam-4750	144	31	(	(	PUNCT
ejpam-4750	144	32	g	g	NOUN
ejpam-4750	144	33	)	)	PUNCT
ejpam-4750	144	34	\	\	PUNCT
ejpam-4750	144	35	s	s	VERB
ejpam-4750	144	36	such	such	ADJ
ejpam-4750	144	37	that	that	SCONJ
ejpam-4750	144	38	[	[	PUNCT
ejpam-4750	144	39	s	s	X
ejpam-4750	144	40	\	\	X
ejpam-4750	144	41	{	{	PUNCT
ejpam-4750	144	42	u	u	NOUN
ejpam-4750	144	43	}	}	PUNCT
ejpam-4750	144	44	]	]	PUNCT
ejpam-4750	144	45	∪	∪	X
ejpam-4750	144	46	{	{	PUNCT
ejpam-4750	144	47	vu	vu	INTJ
ejpam-4750	144	48	}	}	PUNCT
ejpam-4750	144	49	is	be	AUX
ejpam-4750	144	50	a	a	DET
ejpam-4750	144	51	2	2	NUM
ejpam-4750	144	52	-	-	PUNCT
ejpam-4750	144	53	locating	locate	VERB
ejpam-4750	144	54	set	set	NOUN
ejpam-4750	144	55	of	of	ADP
ejpam-4750	144	56	g.	g.	PROPN
ejpam-4750	144	57	proof	proof	PROPN
ejpam-4750	144	58	:	:	PUNCT
ejpam-4750	144	59	suppose	suppose	VERB
ejpam-4750	144	60	that	that	SCONJ
ejpam-4750	144	61	fln2(g	fln2(g	NOUN
ejpam-4750	144	62	)	)	PUNCT
ejpam-4750	144	63	=	=	SYM
ejpam-4750	144	64	ln2(g	ln2(g	PROPN
ejpam-4750	144	65	)	)	PUNCT
ejpam-4750	144	66	.	.	PUNCT
ejpam-4750	145	1	let	let	VERB
ejpam-4750	145	2	s	s	PRON
ejpam-4750	145	3	be	be	AUX
ejpam-4750	145	4	an	an	DET
ejpam-4750	145	5	ln2	ln2	NOUN
ejpam-4750	145	6	-	-	PUNCT
ejpam-4750	145	7	set	set	NOUN
ejpam-4750	145	8	of	of	ADP
ejpam-4750	145	9	g	g	NOUN
ejpam-4750	145	10	such	such	ADJ
ejpam-4750	145	11	that	that	SCONJ
ejpam-4750	145	12	fln2(g	fln2(g	ADJ
ejpam-4750	145	13	)	)	PUNCT
ejpam-4750	145	14	=	=	SYM
ejpam-4750	145	15	|s|	|s|	PROPN
ejpam-4750	145	16	=	=	PUNCT
ejpam-4750	145	17	ln2(g	ln2(g	PROPN
ejpam-4750	145	18	)	)	PUNCT
ejpam-4750	145	19	,	,	PUNCT
ejpam-4750	145	20	that	that	ADV
ejpam-4750	145	21	is	is	ADV
ejpam-4750	145	22	,	,	PUNCT
ejpam-4750	145	23	s	s	VERB
ejpam-4750	145	24	is	be	AUX
ejpam-4750	145	25	the	the	DET
ejpam-4750	145	26	only	only	ADJ
ejpam-4750	145	27	forcing	forcing	NOUN
ejpam-4750	145	28	subset	subset	NOUN
ejpam-4750	145	29	for	for	ADP
ejpam-4750	145	30	itself	itself	PRON
ejpam-4750	145	31	.	.	PUNCT
ejpam-4750	146	1	let	let	VERB
ejpam-4750	146	2	u	u	PRON
ejpam-4750	146	3	∈	∈	PROPN
ejpam-4750	146	4	s.	s.	PROPN
ejpam-4750	146	5	since	since	SCONJ
ejpam-4750	146	6	s	s	PRON
ejpam-4750	146	7	\{u	\{u	X
ejpam-4750	146	8	}	}	PUNCT
ejpam-4750	146	9	is	be	AUX
ejpam-4750	146	10	not	not	PART
ejpam-4750	146	11	a	a	DET
ejpam-4750	146	12	forcing	forcing	NOUN
ejpam-4750	146	13	subset	subset	NOUN
ejpam-4750	146	14	for	for	ADP
ejpam-4750	146	15	s	s	PROPN
ejpam-4750	146	16	,	,	PUNCT
ejpam-4750	146	17	there	there	PRON
ejpam-4750	146	18	exists	exist	VERB
ejpam-4750	146	19	a	a	DET
ejpam-4750	146	20	vu	vu	X
ejpam-4750	146	21	∈	∈	PROPN
ejpam-4750	146	22	v	v	NOUN
ejpam-4750	146	23	(	(	PUNCT
ejpam-4750	146	24	g)\s	g)\s	VERB
ejpam-4750	146	25	such	such	ADJ
ejpam-4750	146	26	that	that	SCONJ
ejpam-4750	146	27	[	[	PUNCT
ejpam-4750	146	28	s	s	AUX
ejpam-4750	146	29	\{u	\{u	ADV
ejpam-4750	146	30	}	}	PUNCT
ejpam-4750	146	31	]	]	SYM
ejpam-4750	146	32	∪{vu	∪{vu	NUM
ejpam-4750	146	33	}	}	PUNCT
ejpam-4750	146	34	is	be	AUX
ejpam-4750	146	35	an	an	DET
ejpam-4750	146	36	ln2	ln2	NOUN
ejpam-4750	146	37	-	-	PUNCT
ejpam-4750	146	38	set	set	NOUN
ejpam-4750	146	39	of	of	ADP
ejpam-4750	146	40	g.	g.	NOUN
ejpam-4750	146	41	conversely	conversely	ADV
ejpam-4750	146	42	,	,	PUNCT
ejpam-4750	146	43	suppose	suppose	VERB
ejpam-4750	146	44	that	that	SCONJ
ejpam-4750	146	45	every	every	DET
ejpam-4750	146	46	ln2	ln2	NOUN
ejpam-4750	146	47	-	-	PUNCT
ejpam-4750	146	48	set	set	NOUN
ejpam-4750	146	49	of	of	ADP
ejpam-4750	146	50	g	g	NOUN
ejpam-4750	146	51	satisfies	satisfy	VERB
ejpam-4750	146	52	the	the	DET
ejpam-4750	146	53	given	give	VERB
ejpam-4750	146	54	condition	condition	NOUN
ejpam-4750	146	55	.	.	PUNCT
ejpam-4750	147	1	let	let	VERB
ejpam-4750	147	2	s	s	PRON
ejpam-4750	147	3	be	be	AUX
ejpam-4750	147	4	a	a	DET
ejpam-4750	147	5	2	2	NUM
ejpam-4750	147	6	-	-	PUNCT
ejpam-4750	147	7	locating	locate	VERB
ejpam-4750	147	8	set	set	NOUN
ejpam-4750	147	9	of	of	ADP
ejpam-4750	147	10	g	g	NOUN
ejpam-4750	147	11	such	such	ADJ
ejpam-4750	147	12	that	that	SCONJ
ejpam-4750	147	13	fln2(g	fln2(g	ADJ
ejpam-4750	147	14	)	)	PUNCT
ejpam-4750	147	15	=	=	SYM
ejpam-4750	147	16	fln2(s	fln2(s	NOUN
ejpam-4750	147	17	)	)	PUNCT
ejpam-4750	147	18	.	.	PUNCT
ejpam-4750	148	1	suppose	suppose	VERB
ejpam-4750	148	2	further	far	ADV
ejpam-4750	148	3	that	that	SCONJ
ejpam-4750	148	4	s	s	VERB
ejpam-4750	148	5	has	have	VERB
ejpam-4750	148	6	a	a	DET
ejpam-4750	148	7	forcing	force	VERB
ejpam-4750	148	8	subset	subset	NOUN
ejpam-4750	148	9	h	h	NOUN
ejpam-4750	148	10	with	with	ADP
ejpam-4750	148	11	|h|	|h|	PROPN
ejpam-4750	148	12	<	<	X
ejpam-4750	148	13	|s|	|s|	NOUN
ejpam-4750	148	14	,	,	PUNCT
ejpam-4750	148	15	that	that	ADV
ejpam-4750	148	16	is	be	AUX
ejpam-4750	148	17	,	,	PUNCT
ejpam-4750	148	18	s	s	PART
ejpam-4750	148	19	=	=	NOUN
ejpam-4750	148	20	h	h	NOUN
ejpam-4750	148	21	∪	∪	NOUN
ejpam-4750	148	22	i	i	PRON
ejpam-4750	148	23	where	where	SCONJ
ejpam-4750	148	24	i	i	PRON
ejpam-4750	148	25	=	=	PUNCT
ejpam-4750	148	26	{	{	PUNCT
ejpam-4750	148	27	w	w	PROPN
ejpam-4750	148	28	∈	∈	PROPN
ejpam-4750	148	29	s	s	PART
ejpam-4750	148	30	:	:	PUNCT
ejpam-4750	148	31	w	w	PROPN
ejpam-4750	148	32	/∈	/∈	PROPN
ejpam-4750	148	33	h	h	NOUN
ejpam-4750	148	34	}	}	PUNCT
ejpam-4750	148	35	.	.	PUNCT
ejpam-4750	149	1	pick	pick	VERB
ejpam-4750	149	2	w	w	PROPN
ejpam-4750	149	3	∈	∈	PROPN
ejpam-4750	149	4	i.	i.	NOUN
ejpam-4750	149	5	by	by	ADP
ejpam-4750	149	6	assumption	assumption	NOUN
ejpam-4750	149	7	,	,	PUNCT
ejpam-4750	149	8	there	there	PRON
ejpam-4750	149	9	exists	exist	VERB
ejpam-4750	149	10	vw	vw	PROPN
ejpam-4750	149	11	∈	∈	PROPN
ejpam-4750	149	12	v	v	ADP
ejpam-4750	149	13	(	(	PUNCT
ejpam-4750	149	14	g	g	NOUN
ejpam-4750	149	15	)	)	PUNCT
ejpam-4750	149	16	\s	\	VERB
ejpam-4750	149	17	such	such	ADJ
ejpam-4750	149	18	that	that	SCONJ
ejpam-4750	149	19	[	[	PUNCT
ejpam-4750	149	20	s	s	X
ejpam-4750	149	21	\	\	X
ejpam-4750	149	22	{	{	PUNCT
ejpam-4750	149	23	w	w	NOUN
ejpam-4750	149	24	}	}	PUNCT
ejpam-4750	149	25	]	]	PUNCT
ejpam-4750	149	26	∪{vw	∪{vw	NOUN
ejpam-4750	149	27	}	}	PUNCT
ejpam-4750	149	28	=	=	SYM
ejpam-4750	149	29	j	j	PROPN
ejpam-4750	149	30	is	be	AUX
ejpam-4750	149	31	an	an	DET
ejpam-4750	149	32	ln2	ln2	NOUN
ejpam-4750	149	33	-	-	PUNCT
ejpam-4750	149	34	set	set	NOUN
ejpam-4750	149	35	of	of	ADP
ejpam-4750	149	36	g.	g.	PROPN
ejpam-4750	149	37	hence	hence	ADV
ejpam-4750	149	38	,	,	PUNCT
ejpam-4750	149	39	j	j	PROPN
ejpam-4750	149	40	=	=	PUNCT
ejpam-4750	149	41	h∪t	h∪t	PROPN
ejpam-4750	149	42	,	,	PUNCT
ejpam-4750	149	43	where	where	SCONJ
ejpam-4750	149	44	t	t	NOUN
ejpam-4750	149	45	=	=	PUNCT
ejpam-4750	149	46	[	[	PUNCT
ejpam-4750	149	47	i	i	PRON
ejpam-4750	149	48	\{w	\{w	NOUN
ejpam-4750	149	49	}	}	PUNCT
ejpam-4750	149	50	]	]	PUNCT
ejpam-4750	149	51	∪{vw	∪{vw	NOUN
ejpam-4750	149	52	}	}	PUNCT
ejpam-4750	149	53	,	,	PUNCT
ejpam-4750	149	54	is	be	AUX
ejpam-4750	149	55	an	an	DET
ejpam-4750	149	56	ln2	ln2	NOUN
ejpam-4750	149	57	-	-	PUNCT
ejpam-4750	149	58	set	set	NOUN
ejpam-4750	149	59	containing	contain	VERB
ejpam-4750	149	60	h	h	NOUN
ejpam-4750	149	61	,	,	PUNCT
ejpam-4750	149	62	a	a	DET
ejpam-4750	149	63	contradiction	contradiction	NOUN
ejpam-4750	149	64	.	.	PUNCT
ejpam-4750	150	1	hence	hence	ADV
ejpam-4750	150	2	,	,	PUNCT
ejpam-4750	150	3	s	s	VERB
ejpam-4750	150	4	is	be	AUX
ejpam-4750	150	5	the	the	DET
ejpam-4750	150	6	only	only	ADJ
ejpam-4750	150	7	forcing	forcing	NOUN
ejpam-4750	150	8	subset	subset	NOUN
ejpam-4750	150	9	for	for	ADP
ejpam-4750	150	10	s.	s.	PROPN
ejpam-4750	150	11	therefore	therefore	ADV
ejpam-4750	150	12	,	,	PUNCT
ejpam-4750	150	13	fln2(g	fln2(g	NOUN
ejpam-4750	150	14	)	)	PUNCT
ejpam-4750	150	15	=	=	SYM
ejpam-4750	150	16	ln2(g	ln2(g	PROPN
ejpam-4750	150	17	)	)	PUNCT
ejpam-4750	150	18	.	.	PUNCT
ejpam-4750	151	1	proposition	proposition	NOUN
ejpam-4750	151	2	5	5	NUM
ejpam-4750	151	3	.	.	PUNCT
ejpam-4750	152	1	for	for	ADP
ejpam-4750	152	2	any	any	DET
ejpam-4750	152	3	complete	complete	ADJ
ejpam-4750	152	4	graph	graph	NOUN
ejpam-4750	152	5	kn	kn	PROPN
ejpam-4750	152	6	with	with	ADP
ejpam-4750	152	7	n	n	CCONJ
ejpam-4750	152	8	>	>	SYM
ejpam-4750	152	9	1	1	NUM
ejpam-4750	152	10	vertices	vertex	NOUN
ejpam-4750	152	11	,	,	PUNCT
ejpam-4750	152	12	fln2(kn	fln2(kn	NOUN
ejpam-4750	152	13	)	)	PUNCT
ejpam-4750	152	14	=	=	SYM
ejpam-4750	152	15	0	0	X
ejpam-4750	152	16	.	.	X
ejpam-4750	153	1	proof	proof	NOUN
ejpam-4750	153	2	:	:	PUNCT
ejpam-4750	153	3	by	by	ADP
ejpam-4750	153	4	remark	remark	NOUN
ejpam-4750	153	5	1	1	NUM
ejpam-4750	153	6	,	,	PUNCT
ejpam-4750	153	7	v	v	PROPN
ejpam-4750	153	8	(	(	PUNCT
ejpam-4750	153	9	kn	kn	PROPN
ejpam-4750	153	10	)	)	PUNCT
ejpam-4750	153	11	is	be	AUX
ejpam-4750	153	12	the	the	DET
ejpam-4750	153	13	only	only	ADJ
ejpam-4750	153	14	ln2	ln2	NOUN
ejpam-4750	153	15	-	-	PUNCT
ejpam-4750	153	16	set	set	NOUN
ejpam-4750	153	17	of	of	ADP
ejpam-4750	153	18	kn	kn	PROPN
ejpam-4750	153	19	.	.	PUNCT
ejpam-4750	154	1	by	by	ADP
ejpam-4750	154	2	remark	remark	NOUN
ejpam-4750	154	3	5	5	NUM
ejpam-4750	154	4	(	(	PUNCT
ejpam-4750	154	5	i	i	NOUN
ejpam-4750	154	6	)	)	PUNCT
ejpam-4750	154	7	,	,	PUNCT
ejpam-4750	154	8	fln2(kn	fln2(kn	PROPN
ejpam-4750	154	9	)	)	PUNCT
ejpam-4750	154	10	=	=	SYM
ejpam-4750	155	1	0	0	X
ejpam-4750	155	2	.	.	NOUN
ejpam-4750	155	3	remark	remark	PROPN
ejpam-4750	155	4	6	6	NUM
ejpam-4750	155	5	.	.	PUNCT
ejpam-4750	156	1	let	let	VERB
ejpam-4750	156	2	s	s	PRON
ejpam-4750	156	3	be	be	AUX
ejpam-4750	156	4	a	a	DET
ejpam-4750	156	5	2	2	NUM
ejpam-4750	156	6	-	-	PUNCT
ejpam-4750	156	7	locating	locate	VERB
ejpam-4750	156	8	set	set	NOUN
ejpam-4750	156	9	of	of	ADP
ejpam-4750	156	10	pn	pn	NOUN
ejpam-4750	156	11	=	=	PUNCT
ejpam-4750	157	1	[	[	X
ejpam-4750	157	2	u1	u1	NOUN
ejpam-4750	157	3	,	,	PUNCT
ejpam-4750	157	4	u2	u2	NOUN
ejpam-4750	157	5	,	,	PUNCT
ejpam-4750	157	6	.	.	PUNCT
ejpam-4750	157	7	.	.	PUNCT
ejpam-4750	157	8	.	.	PUNCT
ejpam-4750	158	1	,	,	PUNCT
ejpam-4750	158	2	un−1	un−1	PROPN
ejpam-4750	158	3	,	,	PUNCT
ejpam-4750	158	4	un	un	NOUN
ejpam-4750	158	5	]	]	X
ejpam-4750	158	6	where	where	SCONJ
ejpam-4750	158	7	n	n	PRON
ejpam-4750	158	8	≥	≥	NOUN
ejpam-4750	158	9	2	2	NUM
ejpam-4750	158	10	.	.	PUNCT
ejpam-4750	159	1	then	then	ADV
ejpam-4750	159	2	(	(	PUNCT
ejpam-4750	159	3	i	i	NOUN
ejpam-4750	159	4	)	)	PUNCT
ejpam-4750	159	5	{	{	PUNCT
ejpam-4750	159	6	u1	u1	NOUN
ejpam-4750	159	7	,	,	PUNCT
ejpam-4750	159	8	u2	u2	NOUN
ejpam-4750	159	9	}	}	PUNCT
ejpam-4750	159	10	∩	∩	NOUN
ejpam-4750	159	11	s	s	PART
ejpam-4750	159	12	̸=	̸=	PROPN
ejpam-4750	159	13	∅.	∅.	X
ejpam-4750	159	14	(	(	PUNCT
ejpam-4750	159	15	ii	ii	NOUN
ejpam-4750	159	16	)	)	PUNCT
ejpam-4750	159	17	{	{	PUNCT
ejpam-4750	159	18	un−1	un−1	PROPN
ejpam-4750	159	19	,	,	PUNCT
ejpam-4750	159	20	un	un	ADJ
ejpam-4750	159	21	}	}	PUNCT
ejpam-4750	159	22	∩	∩	NOUN
ejpam-4750	159	23	s	s	PART
ejpam-4750	159	24	̸=	̸=	PROPN
ejpam-4750	159	25	∅.	∅.	X
ejpam-4750	159	26	(	(	PUNCT
ejpam-4750	159	27	iii	iii	NOUN
ejpam-4750	159	28	)	)	PUNCT
ejpam-4750	159	29	{	{	PUNCT
ejpam-4750	159	30	ui	ui	PROPN
ejpam-4750	159	31	,	,	PUNCT
ejpam-4750	159	32	ui+1	ui+1	PROPN
ejpam-4750	159	33	,	,	PUNCT
ejpam-4750	159	34	ui+2	ui+2	ADJ
ejpam-4750	159	35	}	}	PUNCT
ejpam-4750	159	36	∩	∩	NOUN
ejpam-4750	159	37	s	s	PART
ejpam-4750	159	38	̸=	̸=	PROPN
ejpam-4750	159	39	∅	∅	NOUN
ejpam-4750	159	40	where	where	SCONJ
ejpam-4750	159	41	1	1	NUM
ejpam-4750	159	42	≤	≤	PUNCT
ejpam-4750	159	43	i	i	PRON
ejpam-4750	159	44	<	<	X
ejpam-4750	159	45	n.	n.	NOUN
ejpam-4750	159	46	proposition	proposition	NOUN
ejpam-4750	159	47	6	6	NUM
ejpam-4750	159	48	.	.	PUNCT
ejpam-4750	160	1	for	for	ADP
ejpam-4750	160	2	any	any	DET
ejpam-4750	160	3	path	path	NOUN
ejpam-4750	160	4	pn	pn	NOUN
ejpam-4750	160	5	with	with	ADP
ejpam-4750	160	6	n	n	PRON
ejpam-4750	160	7	≥	≥	NUM
ejpam-4750	160	8	2	2	NUM
ejpam-4750	160	9	vertices	vertex	NOUN
ejpam-4750	160	10	,	,	PUNCT
ejpam-4750	160	11	fln2(pn	fln2(pn	NOUN
ejpam-4750	160	12	)	)	PUNCT
ejpam-4750	160	13	=	=	PUNCT
ejpam-4750	160	14			NOUN
ejpam-4750	160	15	0	0	NUM
ejpam-4750	160	16	,	,	PUNCT
ejpam-4750	160	17	if	if	SCONJ
ejpam-4750	160	18	n	n	NOUN
ejpam-4750	160	19	=	=	SYM
ejpam-4750	160	20	2	2	NUM
ejpam-4750	160	21	,	,	PUNCT
ejpam-4750	160	22	3	3	NUM
ejpam-4750	160	23	and	and	CCONJ
ejpam-4750	160	24	n	n	PRON
ejpam-4750	160	25	≥	≥	NOUN
ejpam-4750	160	26	7	7	NUM
ejpam-4750	160	27	is	be	AUX
ejpam-4750	160	28	odd	odd	ADJ
ejpam-4750	160	29	,	,	PUNCT
ejpam-4750	160	30	1	1	NUM
ejpam-4750	160	31	,	,	PUNCT
ejpam-4750	160	32	if	if	SCONJ
ejpam-4750	160	33	n	n	NOUN
ejpam-4750	160	34	=	=	SYM
ejpam-4750	160	35	5	5	NUM
ejpam-4750	160	36	,	,	PUNCT
ejpam-4750	160	37	2	2	NUM
ejpam-4750	160	38	,	,	PUNCT
ejpam-4750	160	39	if	if	SCONJ
ejpam-4750	160	40	n	n	PRON
ejpam-4750	160	41	≥	≥	NOUN
ejpam-4750	160	42	6	6	NUM
ejpam-4750	160	43	is	be	AUX
ejpam-4750	160	44	even	even	ADV
ejpam-4750	160	45	,	,	PUNCT
ejpam-4750	160	46	3	3	X
ejpam-4750	160	47	,	,	PUNCT
ejpam-4750	160	48	if	if	SCONJ
ejpam-4750	160	49	n	n	NOUN
ejpam-4750	160	50	=	=	SYM
ejpam-4750	160	51	4	4	X
ejpam-4750	160	52	.	.	X
ejpam-4750	161	1	proof	proof	NOUN
ejpam-4750	161	2	:	:	PUNCT
ejpam-4750	161	3	suppose	suppose	VERB
ejpam-4750	161	4	that	that	SCONJ
ejpam-4750	161	5	pn	pn	PROPN
ejpam-4750	161	6	=	=	PUNCT
ejpam-4750	162	1	[	[	X
ejpam-4750	162	2	u1	u1	NOUN
ejpam-4750	162	3	,	,	PUNCT
ejpam-4750	162	4	u2	u2	NOUN
ejpam-4750	162	5	,	,	PUNCT
ejpam-4750	162	6	.	.	PUNCT
ejpam-4750	162	7	.	.	PUNCT
ejpam-4750	162	8	.	.	PUNCT
ejpam-4750	163	1	,	,	PUNCT
ejpam-4750	163	2	un	un	PROPN
ejpam-4750	163	3	]	]	X
ejpam-4750	163	4	.	.	PUNCT
ejpam-4750	164	1	by	by	ADP
ejpam-4750	164	2	theorem	theorem	NOUN
ejpam-4750	164	3	1	1	NUM
ejpam-4750	164	4	,	,	PUNCT
ejpam-4750	164	5	if	if	SCONJ
ejpam-4750	164	6	n	n	NOUN
ejpam-4750	164	7	=	=	SYM
ejpam-4750	164	8	2	2	NUM
ejpam-4750	164	9	,	,	PUNCT
ejpam-4750	164	10	then	then	ADV
ejpam-4750	164	11	v	v	NOUN
ejpam-4750	164	12	(	(	PUNCT
ejpam-4750	164	13	p2	p2	PROPN
ejpam-4750	164	14	)	)	PUNCT
ejpam-4750	164	15	is	be	AUX
ejpam-4750	164	16	the	the	DET
ejpam-4750	164	17	only	only	ADJ
ejpam-4750	164	18	ln2	ln2	NOUN
ejpam-4750	164	19	-	-	PUNCT
ejpam-4750	164	20	set	set	NOUN
ejpam-4750	164	21	of	of	ADP
ejpam-4750	164	22	p2	p2	PROPN
ejpam-4750	164	23	and	and	CCONJ
ejpam-4750	164	24	if	if	SCONJ
ejpam-4750	164	25	n	n	X
ejpam-4750	164	26	=	=	SYM
ejpam-4750	164	27	3	3	NUM
ejpam-4750	164	28	,	,	PUNCT
ejpam-4750	164	29	then	then	ADV
ejpam-4750	164	30	m	m	VERB
ejpam-4750	164	31	=	=	PUNCT
ejpam-4750	164	32	{	{	PUNCT
ejpam-4750	164	33	u1	u1	NOUN
ejpam-4750	164	34	,	,	PUNCT
ejpam-4750	164	35	u3	u3	PROPN
ejpam-4750	164	36	}	}	PUNCT
ejpam-4750	164	37	is	be	AUX
ejpam-4750	164	38	the	the	DET
ejpam-4750	164	39	only	only	ADJ
ejpam-4750	164	40	ln2	ln2	NOUN
ejpam-4750	164	41	-	-	PUNCT
ejpam-4750	164	42	set	set	NOUN
ejpam-4750	164	43	of	of	ADP
ejpam-4750	164	44	p3	p3	PROPN
ejpam-4750	164	45	.	.	PUNCT
ejpam-4750	165	1	thus	thus	ADV
ejpam-4750	165	2	,	,	PUNCT
ejpam-4750	165	3	by	by	ADP
ejpam-4750	165	4	remark	remark	NOUN
ejpam-4750	165	5	5	5	NUM
ejpam-4750	165	6	(	(	PUNCT
ejpam-4750	165	7	i	i	NOUN
ejpam-4750	165	8	)	)	PUNCT
ejpam-4750	165	9	,	,	PUNCT
ejpam-4750	165	10	fln2(p2	fln2(p2	PROPN
ejpam-4750	165	11	)	)	PUNCT
ejpam-4750	165	12	=	=	SYM
ejpam-4750	165	13	fln2(p3	fln2(p3	PROPN
ejpam-4750	165	14	)	)	PUNCT
ejpam-4750	166	1	=	=	SYM
ejpam-4750	166	2	0	0	X
ejpam-4750	166	3	.	.	PUNCT
ejpam-4750	166	4	suppose	suppose	VERB
ejpam-4750	166	5	that	that	SCONJ
ejpam-4750	166	6	n	n	NOUN
ejpam-4750	166	7	=	=	SYM
ejpam-4750	166	8	5	5	X
ejpam-4750	166	9	.	.	PUNCT
ejpam-4750	166	10	by	by	ADP
ejpam-4750	166	11	example	example	NOUN
ejpam-4750	166	12	1	1	NUM
ejpam-4750	166	13	,	,	PUNCT
ejpam-4750	166	14	ln2(p5	ln2(p5	ADV
ejpam-4750	166	15	)	)	PUNCT
ejpam-4750	166	16	=	=	SYM
ejpam-4750	167	1	3	3	X
ejpam-4750	167	2	.	.	PUNCT
ejpam-4750	167	3	then	then	ADV
ejpam-4750	167	4	by	by	ADP
ejpam-4750	167	5	remark	remark	NOUN
ejpam-4750	167	6	6	6	NUM
ejpam-4750	167	7	,	,	PUNCT
ejpam-4750	167	8	the	the	DET
ejpam-4750	167	9	ln2	ln2	NOUN
ejpam-4750	167	10	-	-	PUNCT
ejpam-4750	167	11	sets	set	NOUN
ejpam-4750	167	12	of	of	ADP
ejpam-4750	167	13	p5	p5	NOUN
ejpam-4750	167	14	are	be	AUX
ejpam-4750	167	15	n1	n1	ADJ
ejpam-4750	167	16	=	=	SYM
ejpam-4750	167	17	{	{	PUNCT
ejpam-4750	167	18	u1	u1	NOUN
ejpam-4750	167	19	,	,	PUNCT
ejpam-4750	167	20	u3	u3	PROPN
ejpam-4750	167	21	,	,	PUNCT
ejpam-4750	167	22	u5	u5	PROPN
ejpam-4750	167	23	}	}	PUNCT
ejpam-4750	167	24	and	and	CCONJ
ejpam-4750	167	25	n2	n2	ADJ
ejpam-4750	167	26	=	=	PUNCT
ejpam-4750	167	27	{	{	PUNCT
ejpam-4750	167	28	u2	u2	PROPN
ejpam-4750	167	29	,	,	PUNCT
ejpam-4750	167	30	u3	u3	NOUN
ejpam-4750	167	31	,	,	PUNCT
ejpam-4750	167	32	u4	u4	PROPN
ejpam-4750	167	33	}	}	PUNCT
ejpam-4750	167	34	with	with	ADP
ejpam-4750	167	35	u1	u1	PROPN
ejpam-4750	167	36	∈	∈	PROPN
ejpam-4750	167	37	n1	n1	PROPN
ejpam-4750	167	38	and	and	CCONJ
ejpam-4750	167	39	u1	u1	PROPN
ejpam-4750	167	40	/∈	/∈	SYM
ejpam-4750	167	41	n2	n2	PROPN
ejpam-4750	167	42	.	.	PUNCT
ejpam-4750	168	1	thus	thus	ADV
ejpam-4750	168	2	,	,	PUNCT
ejpam-4750	168	3	by	by	ADP
ejpam-4750	168	4	remark	remark	NOUN
ejpam-4750	168	5	5	5	NUM
ejpam-4750	168	6	(	(	PUNCT
ejpam-4750	168	7	ii	ii	NOUN
ejpam-4750	168	8	)	)	PUNCT
ejpam-4750	168	9	,	,	PUNCT
ejpam-4750	168	10	fln2(n1	fln2(n1	NOUN
ejpam-4750	168	11	)	)	PUNCT
ejpam-4750	168	12	=	=	SYM
ejpam-4750	168	13	1	1	NUM
ejpam-4750	168	14	=	=	SYM
ejpam-4750	168	15	fln2(p5	fln2(p5	PROPN
ejpam-4750	168	16	)	)	PUNCT
ejpam-4750	168	17	.	.	PUNCT
ejpam-4750	169	1	if	if	SCONJ
ejpam-4750	169	2	n	n	NOUN
ejpam-4750	169	3	=	=	SYM
ejpam-4750	169	4	4	4	NUM
ejpam-4750	169	5	,	,	PUNCT
ejpam-4750	169	6	then	then	ADV
ejpam-4750	169	7	by	by	ADP
ejpam-4750	169	8	remark	remark	NOUN
ejpam-4750	169	9	6	6	NUM
ejpam-4750	169	10	,	,	PUNCT
ejpam-4750	169	11	r1	r1	NOUN
ejpam-4750	169	12	=	=	SYM
ejpam-4750	169	13	{	{	PUNCT
ejpam-4750	169	14	u1	u1	NOUN
ejpam-4750	169	15	,	,	PUNCT
ejpam-4750	169	16	u2	u2	PROPN
ejpam-4750	169	17	,	,	PUNCT
ejpam-4750	169	18	u4	u4	PROPN
ejpam-4750	169	19	}	}	PUNCT
ejpam-4750	169	20	,	,	PUNCT
ejpam-4750	169	21	r2	r2	PROPN
ejpam-4750	169	22	=	=	SYM
ejpam-4750	169	23	{	{	PUNCT
ejpam-4750	169	24	u1	u1	NOUN
ejpam-4750	169	25	,	,	PUNCT
ejpam-4750	169	26	u3	u3	PROPN
ejpam-4750	169	27	,	,	PUNCT
ejpam-4750	169	28	u4	u4	PROPN
ejpam-4750	169	29	}	}	PUNCT
ejpam-4750	169	30	,	,	PUNCT
ejpam-4750	169	31	r3	r3	PROPN
ejpam-4750	169	32	=	=	SYM
ejpam-4750	169	33	{	{	PUNCT
ejpam-4750	169	34	u1	u1	NOUN
ejpam-4750	169	35	,	,	PUNCT
ejpam-4750	169	36	u2	u2	NOUN
ejpam-4750	169	37	,	,	PUNCT
ejpam-4750	169	38	u3	u3	NOUN
ejpam-4750	169	39	}	}	PUNCT
ejpam-4750	169	40	and	and	CCONJ
ejpam-4750	169	41	r4	r4	VERB
ejpam-4750	169	42	=	=	PUNCT
ejpam-4750	169	43	{	{	PUNCT
ejpam-4750	169	44	u2	u2	PROPN
ejpam-4750	169	45	,	,	PUNCT
ejpam-4750	169	46	u3	u3	PROPN
ejpam-4750	169	47	,	,	PUNCT
ejpam-4750	169	48	u4	u4	PROPN
ejpam-4750	169	49	}	}	PUNCT
ejpam-4750	169	50	are	be	AUX
ejpam-4750	169	51	the	the	DET
ejpam-4750	169	52	ln2	ln2	ADJ
ejpam-4750	169	53	-	-	PUNCT
ejpam-4750	169	54	sets	set	NOUN
ejpam-4750	169	55	of	of	ADP
ejpam-4750	169	56	p4	p4	NOUN
ejpam-4750	169	57	.	.	PUNCT
ejpam-4750	170	1	clearly	clearly	ADV
ejpam-4750	170	2	,	,	PUNCT
ejpam-4750	170	3	none	none	NOUN
ejpam-4750	170	4	of	of	ADP
ejpam-4750	170	5	the	the	DET
ejpam-4750	170	6	singletons	singleton	NOUN
ejpam-4750	170	7	and	and	CCONJ
ejpam-4750	170	8	doubletons	doubleton	NOUN
ejpam-4750	170	9	is	be	AUX
ejpam-4750	170	10	a	a	DET
ejpam-4750	170	11	forcing	forcing	NOUN
ejpam-4750	170	12	subset	subset	NOUN
ejpam-4750	170	13	for	for	ADP
ejpam-4750	170	14	an	an	DET
ejpam-4750	170	15	ln2	ln2	NOUN
ejpam-4750	170	16	-	-	PUNCT
ejpam-4750	170	17	set	set	NOUN
ejpam-4750	170	18	.	.	PUNCT
ejpam-4750	171	1	thus	thus	ADV
ejpam-4750	171	2	,	,	PUNCT
ejpam-4750	171	3	fln2(p4	fln2(p4	PROPN
ejpam-4750	171	4	)	)	PUNCT
ejpam-4750	171	5	=	=	SYM
ejpam-4750	172	1	3	3	X
ejpam-4750	172	2	.	.	PUNCT
ejpam-4750	172	3	suppose	suppose	VERB
ejpam-4750	172	4	that	that	SCONJ
ejpam-4750	172	5	n	n	PROPN
ejpam-4750	172	6	=	=	SYM
ejpam-4750	172	7	6	6	NUM
ejpam-4750	172	8	.	.	PUNCT
ejpam-4750	172	9	by	by	ADP
ejpam-4750	172	10	remark	remark	NOUN
ejpam-4750	172	11	6	6	NUM
ejpam-4750	172	12	,	,	PUNCT
ejpam-4750	172	13	the	the	DET
ejpam-4750	172	14	ln2	ln2	NOUN
ejpam-4750	172	15	-	-	PUNCT
ejpam-4750	172	16	sets	set	NOUN
ejpam-4750	172	17	of	of	ADP
ejpam-4750	172	18	p6	p6	PROPN
ejpam-4750	172	19	are	be	AUX
ejpam-4750	172	20	s1	s1	NOUN
ejpam-4750	172	21	=	=	SYM
ejpam-4750	172	22	{	{	PUNCT
ejpam-4750	172	23	u1	u1	NOUN
ejpam-4750	172	24	,	,	PUNCT
ejpam-4750	172	25	u2	u2	PROPN
ejpam-4750	172	26	,	,	PUNCT
ejpam-4750	172	27	u4	u4	PROPN
ejpam-4750	172	28	,	,	PUNCT
ejpam-4750	172	29	u6	u6	PROPN
ejpam-4750	172	30	}	}	PUNCT
ejpam-4750	172	31	,	,	PUNCT
ejpam-4750	172	32	d.	d.	PROPN
ejpam-4750	172	33	managbanag	managbanag	PROPN
ejpam-4750	172	34	,	,	PUNCT
ejpam-4750	172	35	h.	h.	PROPN
ejpam-4750	172	36	rara	rara	PROPN
ejpam-4750	172	37	/	/	SYM
ejpam-4750	172	38	eur	eur	PROPN
ejpam-4750	172	39	.	.	PUNCT
ejpam-4750	173	1	j.	j.	PROPN
ejpam-4750	173	2	pure	pure	PROPN
ejpam-4750	173	3	appl	appl	PROPN
ejpam-4750	173	4	.	.	PROPN
ejpam-4750	173	5	math	math	PROPN
ejpam-4750	173	6	,	,	PUNCT
ejpam-4750	173	7	16	16	NUM
ejpam-4750	173	8	(	(	PUNCT
ejpam-4750	173	9	2	2	NUM
ejpam-4750	173	10	)	)	PUNCT
ejpam-4750	173	11	(	(	PUNCT
ejpam-4750	173	12	2023	2023	NUM
ejpam-4750	173	13	)	)	PUNCT
ejpam-4750	173	14	,	,	PUNCT
ejpam-4750	173	15	1068	1068	NUM
ejpam-4750	173	16	-	-	SYM
ejpam-4750	173	17	1083	1083	NUM
ejpam-4750	173	18	1074	1074	NUM
ejpam-4750	173	19	s2	s2	NOUN
ejpam-4750	173	20	=	=	SYM
ejpam-4750	173	21	{	{	PUNCT
ejpam-4750	173	22	u1	u1	NOUN
ejpam-4750	173	23	,	,	PUNCT
ejpam-4750	173	24	u3	u3	PROPN
ejpam-4750	173	25	,	,	PUNCT
ejpam-4750	173	26	u4	u4	PROPN
ejpam-4750	173	27	,	,	PUNCT
ejpam-4750	173	28	u5	u5	PROPN
ejpam-4750	173	29	}	}	PUNCT
ejpam-4750	173	30	,	,	PUNCT
ejpam-4750	173	31	s3	s3	PROPN
ejpam-4750	173	32	=	=	SYM
ejpam-4750	173	33	{	{	PUNCT
ejpam-4750	173	34	u1	u1	NOUN
ejpam-4750	173	35	,	,	PUNCT
ejpam-4750	173	36	u3	u3	PROPN
ejpam-4750	173	37	,	,	PUNCT
ejpam-4750	173	38	u4	u4	PROPN
ejpam-4750	173	39	,	,	PUNCT
ejpam-4750	173	40	u6	u6	PROPN
ejpam-4750	173	41	}	}	PUNCT
ejpam-4750	173	42	,	,	PUNCT
ejpam-4750	173	43	s4	s4	PROPN
ejpam-4750	173	44	=	=	SYM
ejpam-4750	173	45	{	{	PUNCT
ejpam-4750	173	46	u1	u1	NOUN
ejpam-4750	173	47	,	,	PUNCT
ejpam-4750	173	48	u3	u3	PROPN
ejpam-4750	173	49	,	,	PUNCT
ejpam-4750	173	50	u5	u5	PROPN
ejpam-4750	173	51	,	,	PUNCT
ejpam-4750	173	52	u6	u6	NOUN
ejpam-4750	173	53	}	}	PUNCT
ejpam-4750	173	54	,	,	PUNCT
ejpam-4750	173	55	s5	s5	X
ejpam-4750	173	56	=	=	PUNCT
ejpam-4750	173	57	{	{	PUNCT
ejpam-4750	173	58	u2	u2	PROPN
ejpam-4750	173	59	,	,	PUNCT
ejpam-4750	173	60	u3	u3	PROPN
ejpam-4750	173	61	,	,	PUNCT
ejpam-4750	173	62	u4	u4	PROPN
ejpam-4750	173	63	,	,	PUNCT
ejpam-4750	173	64	u5	u5	PROPN
ejpam-4750	173	65	}	}	PUNCT
ejpam-4750	173	66	and	and	CCONJ
ejpam-4750	173	67	s6	s6	PROPN
ejpam-4750	173	68	=	=	SYM
ejpam-4750	173	69	{	{	PUNCT
ejpam-4750	173	70	u2	u2	PROPN
ejpam-4750	173	71	,	,	PUNCT
ejpam-4750	173	72	u3	u3	PROPN
ejpam-4750	173	73	,	,	PUNCT
ejpam-4750	173	74	u4	u4	PROPN
ejpam-4750	173	75	,	,	PUNCT
ejpam-4750	173	76	u6	u6	PROPN
ejpam-4750	173	77	}	}	PUNCT
ejpam-4750	173	78	.	.	PUNCT
ejpam-4750	174	1	it	it	PRON
ejpam-4750	174	2	can	can	AUX
ejpam-4750	174	3	be	be	AUX
ejpam-4750	174	4	verified	verify	VERB
ejpam-4750	174	5	that	that	SCONJ
ejpam-4750	174	6	{	{	PUNCT
ejpam-4750	174	7	u1	u1	NOUN
ejpam-4750	174	8	,	,	PUNCT
ejpam-4750	174	9	u2	u2	PROPN
ejpam-4750	174	10	}	}	PUNCT
ejpam-4750	174	11	is	be	AUX
ejpam-4750	174	12	the	the	DET
ejpam-4750	174	13	forcing	force	VERB
ejpam-4750	174	14	subset	subset	NOUN
ejpam-4750	174	15	of	of	ADP
ejpam-4750	174	16	s1	s1	PROPN
ejpam-4750	174	17	and	and	CCONJ
ejpam-4750	174	18	the	the	DET
ejpam-4750	174	19	minimum	minimum	ADJ
ejpam-4750	174	20	forcing	forcing	NOUN
ejpam-4750	174	21	subset	subset	NOUN
ejpam-4750	174	22	of	of	ADP
ejpam-4750	174	23	p6	p6	PROPN
ejpam-4750	174	24	.	.	PUNCT
ejpam-4750	175	1	thus	thus	ADV
ejpam-4750	175	2	,	,	PUNCT
ejpam-4750	175	3	fln2(p6	fln2(p6	NOUN
ejpam-4750	175	4	)	)	PUNCT
ejpam-4750	175	5	=	=	SYM
ejpam-4750	176	1	2	2	X
ejpam-4750	176	2	.	.	PUNCT
ejpam-4750	177	1	next	next	ADV
ejpam-4750	177	2	,	,	PUNCT
ejpam-4750	177	3	suppose	suppose	VERB
ejpam-4750	177	4	that	that	SCONJ
ejpam-4750	177	5	n	n	PROPN
ejpam-4750	177	6	≥	≥	X
ejpam-4750	177	7	7	7	NUM
ejpam-4750	177	8	and	and	CCONJ
ejpam-4750	177	9	n	n	PRON
ejpam-4750	177	10	is	be	AUX
ejpam-4750	177	11	odd	odd	ADJ
ejpam-4750	177	12	.	.	PUNCT
ejpam-4750	178	1	by	by	ADP
ejpam-4750	178	2	remark	remark	NOUN
ejpam-4750	178	3	6	6	NUM
ejpam-4750	178	4	,	,	PUNCT
ejpam-4750	178	5	t	t	NOUN
ejpam-4750	178	6	=	=	SYM
ejpam-4750	178	7	{	{	PUNCT
ejpam-4750	178	8	u1	u1	NOUN
ejpam-4750	178	9	,	,	PUNCT
ejpam-4750	178	10	u3	u3	PROPN
ejpam-4750	178	11	,	,	PUNCT
ejpam-4750	178	12	u5	u5	PROPN
ejpam-4750	178	13	,	,	PUNCT
ejpam-4750	178	14	.	.	PUNCT
ejpam-4750	178	15	.	.	PUNCT
ejpam-4750	179	1	.	.	PUNCT
ejpam-4750	180	1	,	,	PUNCT
ejpam-4750	180	2	un−2	un−2	VERB
ejpam-4750	180	3	,	,	PUNCT
ejpam-4750	180	4	un	un	ADJ
ejpam-4750	180	5	}	}	PUNCT
ejpam-4750	180	6	is	be	AUX
ejpam-4750	180	7	the	the	DET
ejpam-4750	180	8	only	only	ADJ
ejpam-4750	180	9	ln2	ln2	NOUN
ejpam-4750	180	10	-	-	PUNCT
ejpam-4750	180	11	set	set	NOUN
ejpam-4750	180	12	of	of	ADP
ejpam-4750	180	13	pn	pn	PROPN
ejpam-4750	180	14	.	.	PUNCT
ejpam-4750	181	1	thus	thus	ADV
ejpam-4750	181	2	,	,	PUNCT
ejpam-4750	181	3	fln2(t	fln2(t	NOUN
ejpam-4750	181	4	)	)	PUNCT
ejpam-4750	181	5	=	=	SYM
ejpam-4750	182	1	fln2(pn	fln2(pn	NOUN
ejpam-4750	182	2	)	)	PUNCT
ejpam-4750	182	3	=	=	SYM
ejpam-4750	182	4	0	0	NUM
ejpam-4750	182	5	,	,	PUNCT
ejpam-4750	182	6	by	by	ADP
ejpam-4750	182	7	remark	remark	NOUN
ejpam-4750	182	8	5	5	NUM
ejpam-4750	182	9	(	(	PUNCT
ejpam-4750	182	10	i	i	NOUN
ejpam-4750	182	11	)	)	PUNCT
ejpam-4750	182	12	.	.	PUNCT
ejpam-4750	183	1	now	now	ADV
ejpam-4750	183	2	,	,	PUNCT
ejpam-4750	183	3	suppose	suppose	VERB
ejpam-4750	183	4	that	that	SCONJ
ejpam-4750	183	5	n	n	PROPN
ejpam-4750	183	6	≥	≥	NUM
ejpam-4750	183	7	8	8	NUM
ejpam-4750	183	8	and	and	CCONJ
ejpam-4750	183	9	n	n	PRON
ejpam-4750	183	10	is	be	AUX
ejpam-4750	183	11	even	even	ADV
ejpam-4750	183	12	.	.	PUNCT
ejpam-4750	184	1	then	then	ADV
ejpam-4750	184	2	the	the	DET
ejpam-4750	184	3	ln2	ln2	ADJ
ejpam-4750	184	4	-	-	PUNCT
ejpam-4750	184	5	sets	set	NOUN
ejpam-4750	184	6	of	of	ADP
ejpam-4750	184	7	pn	pn	PROPN
ejpam-4750	184	8	are	be	AUX
ejpam-4750	184	9	m1	m1	NOUN
ejpam-4750	184	10	=	=	SYM
ejpam-4750	184	11	{	{	PUNCT
ejpam-4750	184	12	u1	u1	NOUN
ejpam-4750	184	13	,	,	PUNCT
ejpam-4750	184	14	u2	u2	PROPN
ejpam-4750	184	15	,	,	PUNCT
ejpam-4750	184	16	u4	u4	PROPN
ejpam-4750	184	17	,	,	PUNCT
ejpam-4750	184	18	.	.	PUNCT
ejpam-4750	184	19	.	.	PUNCT
ejpam-4750	185	1	.	.	PUNCT
ejpam-4750	186	1	,	,	PUNCT
ejpam-4750	186	2	un−2	un−2	VERB
ejpam-4750	186	3	,	,	PUNCT
ejpam-4750	186	4	un	un	ADJ
ejpam-4750	186	5	}	}	PUNCT
ejpam-4750	186	6	,	,	PUNCT
ejpam-4750	186	7	m2	m2	PROPN
ejpam-4750	186	8	=	=	PROPN
ejpam-4750	186	9	{	{	PUNCT
ejpam-4750	186	10	u1	u1	NOUN
ejpam-4750	186	11	,	,	PUNCT
ejpam-4750	186	12	u3	u3	PROPN
ejpam-4750	186	13	,	,	PUNCT
ejpam-4750	186	14	u4	u4	PROPN
ejpam-4750	186	15	,	,	PUNCT
ejpam-4750	186	16	.	.	PUNCT
ejpam-4750	186	17	.	.	PUNCT
ejpam-4750	187	1	.	.	PUNCT
ejpam-4750	188	1	,	,	PUNCT
ejpam-4750	188	2	un−2	un−2	VERB
ejpam-4750	188	3	,	,	PUNCT
ejpam-4750	188	4	un	un	ADJ
ejpam-4750	188	5	}	}	PUNCT
ejpam-4750	188	6	,	,	PUNCT
ejpam-4750	188	7	m3	m3	PROPN
ejpam-4750	188	8	=	=	SYM
ejpam-4750	188	9	{	{	PUNCT
ejpam-4750	188	10	u1	u1	NOUN
ejpam-4750	188	11	,	,	PUNCT
ejpam-4750	188	12	u3	u3	PROPN
ejpam-4750	188	13	,	,	PUNCT
ejpam-4750	188	14	u5	u5	PROPN
ejpam-4750	188	15	,	,	PUNCT
ejpam-4750	188	16	.	.	PUNCT
ejpam-4750	188	17	.	.	PUNCT
ejpam-4750	189	1	.	.	PUNCT
ejpam-4750	190	1	,	,	PUNCT
ejpam-4750	190	2	un−2	un−2	VERB
ejpam-4750	190	3	,	,	PUNCT
ejpam-4750	190	4	un−1	un−1	PROPN
ejpam-4750	190	5	}	}	PUNCT
ejpam-4750	190	6	,	,	PUNCT
ejpam-4750	190	7	m4	m4	PROPN
ejpam-4750	190	8	=	=	SYM
ejpam-4750	190	9	{	{	PUNCT
ejpam-4750	190	10	u1	u1	NOUN
ejpam-4750	190	11	,	,	PUNCT
ejpam-4750	190	12	u3	u3	PROPN
ejpam-4750	190	13	,	,	PUNCT
ejpam-4750	190	14	u5	u5	PROPN
ejpam-4750	190	15	,	,	PUNCT
ejpam-4750	190	16	.	.	PUNCT
ejpam-4750	190	17	.	.	PUNCT
ejpam-4750	191	1	.	.	PUNCT
ejpam-4750	192	1	,	,	PUNCT
ejpam-4750	192	2	un−2	un−2	VERB
ejpam-4750	192	3	,	,	PUNCT
ejpam-4750	192	4	un	un	ADJ
ejpam-4750	192	5	}	}	PUNCT
ejpam-4750	192	6	,	,	PUNCT
ejpam-4750	192	7	m5	m5	PROPN
ejpam-4750	192	8	=	=	SYM
ejpam-4750	192	9	{	{	PUNCT
ejpam-4750	192	10	u1	u1	NOUN
ejpam-4750	192	11	,	,	PUNCT
ejpam-4750	192	12	u3	u3	PROPN
ejpam-4750	192	13	,	,	PUNCT
ejpam-4750	192	14	u5	u5	PROPN
ejpam-4750	192	15	,	,	PUNCT
ejpam-4750	192	16	.	.	PUNCT
ejpam-4750	192	17	.	.	PUNCT
ejpam-4750	193	1	.	.	PUNCT
ejpam-4750	194	1	,	,	PUNCT
ejpam-4750	194	2	un−1	un−1	PROPN
ejpam-4750	194	3	,	,	PUNCT
ejpam-4750	194	4	un	un	ADJ
ejpam-4750	194	5	}	}	PUNCT
ejpam-4750	194	6	and	and	CCONJ
ejpam-4750	194	7	m6	m6	ADJ
ejpam-4750	194	8	=	=	SYM
ejpam-4750	194	9	{	{	PUNCT
ejpam-4750	194	10	u2	u2	PROPN
ejpam-4750	194	11	,	,	PUNCT
ejpam-4750	194	12	u3	u3	PROPN
ejpam-4750	194	13	,	,	PUNCT
ejpam-4750	194	14	u4	u4	PROPN
ejpam-4750	194	15	,	,	PUNCT
ejpam-4750	194	16	.	.	PUNCT
ejpam-4750	194	17	.	.	PUNCT
ejpam-4750	195	1	.	.	PUNCT
ejpam-4750	196	1	,	,	PUNCT
ejpam-4750	196	2	un−2	un−2	VERB
ejpam-4750	196	3	,	,	PUNCT
ejpam-4750	196	4	un	un	ADJ
ejpam-4750	196	5	}	}	PUNCT
ejpam-4750	196	6	by	by	ADP
ejpam-4750	196	7	remark	remark	NOUN
ejpam-4750	196	8	6	6	NUM
ejpam-4750	196	9	.	.	PUNCT
ejpam-4750	197	1	since	since	SCONJ
ejpam-4750	197	2	{	{	PUNCT
ejpam-4750	197	3	u2	u2	NOUN
ejpam-4750	197	4	,	,	PUNCT
ejpam-4750	197	5	u3	u3	NOUN
ejpam-4750	197	6	}	}	PUNCT
ejpam-4750	197	7	⊆	⊆	NUM
ejpam-4750	197	8	m6	m6	ADJ
ejpam-4750	197	9	and	and	CCONJ
ejpam-4750	197	10	not	not	PART
ejpam-4750	197	11	contained	contain	VERB
ejpam-4750	197	12	in	in	ADP
ejpam-4750	197	13	any	any	DET
ejpam-4750	197	14	other	other	ADJ
ejpam-4750	197	15	ln2	ln2	ADJ
ejpam-4750	197	16	-	-	PUNCT
ejpam-4750	197	17	sets	set	NOUN
ejpam-4750	197	18	of	of	ADP
ejpam-4750	197	19	pn	pn	NOUN
ejpam-4750	197	20	,	,	PUNCT
ejpam-4750	197	21	{	{	PUNCT
ejpam-4750	197	22	u2	u2	NOUN
ejpam-4750	197	23	,	,	PUNCT
ejpam-4750	197	24	u3	u3	PROPN
ejpam-4750	197	25	}	}	PUNCT
ejpam-4750	197	26	is	be	AUX
ejpam-4750	197	27	the	the	DET
ejpam-4750	197	28	forcing	force	VERB
ejpam-4750	197	29	subset	subset	NOUN
ejpam-4750	197	30	of	of	ADP
ejpam-4750	197	31	m6	m6	PROPN
ejpam-4750	197	32	and	and	CCONJ
ejpam-4750	197	33	the	the	DET
ejpam-4750	197	34	minimum	minimum	ADJ
ejpam-4750	197	35	forcing	forcing	NOUN
ejpam-4750	197	36	subset	subset	NOUN
ejpam-4750	197	37	of	of	ADP
ejpam-4750	197	38	pn	pn	PROPN
ejpam-4750	197	39	.	.	PROPN
ejpam-4750	197	40	hence	hence	ADV
ejpam-4750	197	41	,	,	PUNCT
ejpam-4750	197	42	fln2(pn	fln2(pn	PROPN
ejpam-4750	197	43	)	)	PUNCT
ejpam-4750	197	44	=	=	SYM
ejpam-4750	197	45	2	2	X
ejpam-4750	197	46	.	.	NOUN
ejpam-4750	197	47	remark	remark	NOUN
ejpam-4750	197	48	7	7	NUM
ejpam-4750	197	49	.	.	PUNCT
ejpam-4750	198	1	let	let	VERB
ejpam-4750	198	2	w	w	NOUN
ejpam-4750	198	3	be	be	AUX
ejpam-4750	198	4	a	a	DET
ejpam-4750	198	5	2	2	NUM
ejpam-4750	198	6	-	-	PUNCT
ejpam-4750	198	7	locating	locate	VERB
ejpam-4750	198	8	set	set	NOUN
ejpam-4750	198	9	of	of	ADP
ejpam-4750	198	10	cn	cn	PROPN
ejpam-4750	198	11	=	=	PUNCT
ejpam-4750	199	1	[	[	X
ejpam-4750	199	2	v1	v1	NOUN
ejpam-4750	199	3	,	,	PUNCT
ejpam-4750	199	4	v2	v2	NOUN
ejpam-4750	199	5	,	,	PUNCT
ejpam-4750	199	6	.	.	PUNCT
ejpam-4750	199	7	.	.	PUNCT
ejpam-4750	199	8	.	.	PUNCT
ejpam-4750	200	1	,	,	PUNCT
ejpam-4750	200	2	vn	vn	X
ejpam-4750	200	3	,	,	PUNCT
ejpam-4750	200	4	v1	v1	PROPN
ejpam-4750	200	5	]	]	PUNCT
ejpam-4750	200	6	where	where	SCONJ
ejpam-4750	200	7	n	n	PRON
ejpam-4750	200	8	≥	≥	NOUN
ejpam-4750	200	9	3	3	NUM
ejpam-4750	200	10	.	.	PUNCT
ejpam-4750	201	1	then	then	ADV
ejpam-4750	201	2	(	(	PUNCT
ejpam-4750	201	3	i	i	NOUN
ejpam-4750	201	4	)	)	PUNCT
ejpam-4750	201	5	s	s	AUX
ejpam-4750	201	6	∩w	∩w	VERB
ejpam-4750	201	7	̸=	̸=	NOUN
ejpam-4750	201	8	∅	∅	NOUN
ejpam-4750	201	9	for	for	ADP
ejpam-4750	201	10	all	all	PRON
ejpam-4750	201	11	s	s	PART
ejpam-4750	201	12	⊆	⊆	NUM
ejpam-4750	201	13	v	v	NOUN
ejpam-4750	201	14	(	(	PUNCT
ejpam-4750	201	15	cn	cn	PROPN
ejpam-4750	201	16	)	)	PUNCT
ejpam-4750	201	17	with	with	ADP
ejpam-4750	201	18	⟨s⟩	⟨s⟩	PROPN
ejpam-4750	201	19	=	=	SYM
ejpam-4750	201	20	p3	p3	PROPN
ejpam-4750	201	21	.	.	PUNCT
ejpam-4750	202	1	(	(	PUNCT
ejpam-4750	202	2	ii	ii	NOUN
ejpam-4750	202	3	)	)	PUNCT
ejpam-4750	202	4	if	if	SCONJ
ejpam-4750	202	5	vj	vj	INTJ
ejpam-4750	202	6	,	,	PUNCT
ejpam-4750	202	7	vj+1	vj+1	PROPN
ejpam-4750	202	8	∈	∈	PROPN
ejpam-4750	202	9	w	w	NOUN
ejpam-4750	202	10	,	,	PUNCT
ejpam-4750	202	11	then	then	ADV
ejpam-4750	202	12	vj+3	vj+3	NOUN
ejpam-4750	202	13	,	,	PUNCT
ejpam-4750	202	14	vj+5	vj+5	NOUN
ejpam-4750	202	15	,	,	PUNCT
ejpam-4750	202	16	.	.	PUNCT
ejpam-4750	202	17	.	.	PUNCT
ejpam-4750	202	18	.	.	PUNCT
ejpam-4750	203	1	,	,	PUNCT
ejpam-4750	203	2	vj−2	vj−2	NOUN
ejpam-4750	203	3	∈	∈	PROPN
ejpam-4750	203	4	w	w	NOUN
ejpam-4750	203	5	where	where	SCONJ
ejpam-4750	203	6	1	1	NUM
ejpam-4750	203	7	≤	≤	NUM
ejpam-4750	203	8	j	j	PROPN
ejpam-4750	203	9	≤	≤	PROPN
ejpam-4750	204	1	n	n	ADP
ejpam-4750	205	1	and	and	CCONJ
ejpam-4750	205	2	n	n	PROPN
ejpam-4750	205	3	+	+	CCONJ
ejpam-4750	205	4	k	k	PROPN
ejpam-4750	205	5	≡	≡	PROPN
ejpam-4750	205	6	k	k	PROPN
ejpam-4750	205	7	(	(	PUNCT
ejpam-4750	205	8	mod	mod	PROPN
ejpam-4750	205	9	n	n	CCONJ
ejpam-4750	205	10	)	)	PUNCT
ejpam-4750	205	11	for	for	ADP
ejpam-4750	205	12	any	any	DET
ejpam-4750	205	13	positive	positive	ADJ
ejpam-4750	205	14	integer	integer	NOUN
ejpam-4750	205	15	k.	k.	PROPN
ejpam-4750	205	16	proposition	proposition	PROPN
ejpam-4750	205	17	7	7	NUM
ejpam-4750	205	18	.	.	X
ejpam-4750	206	1	for	for	ADP
ejpam-4750	206	2	any	any	DET
ejpam-4750	206	3	cycle	cycle	NOUN
ejpam-4750	206	4	cn	cn	NOUN
ejpam-4750	206	5	with	with	ADP
ejpam-4750	206	6	n	n	NUM
ejpam-4750	206	7	≥	≥	NUM
ejpam-4750	206	8	3	3	NUM
ejpam-4750	206	9	vertices	vertex	NOUN
ejpam-4750	206	10	,	,	PUNCT
ejpam-4750	206	11	fln2(cn	fln2(cn	PROPN
ejpam-4750	206	12	)	)	PUNCT
ejpam-4750	206	13	=	=	PUNCT
ejpam-4750	206	14			NOUN
ejpam-4750	206	15	0	0	NUM
ejpam-4750	206	16	,	,	PUNCT
ejpam-4750	206	17	if	if	SCONJ
ejpam-4750	206	18	n	n	NOUN
ejpam-4750	206	19	=	=	SYM
ejpam-4750	206	20	3	3	NUM
ejpam-4750	206	21	,	,	PUNCT
ejpam-4750	206	22	4	4	NUM
ejpam-4750	206	23	,	,	PUNCT
ejpam-4750	206	24	1	1	NUM
ejpam-4750	206	25	,	,	PUNCT
ejpam-4750	206	26	if	if	SCONJ
ejpam-4750	206	27	n	n	PRON
ejpam-4750	206	28	≥	≥	NOUN
ejpam-4750	206	29	6	6	NUM
ejpam-4750	206	30	is	be	AUX
ejpam-4750	206	31	even	even	ADV
ejpam-4750	206	32	,	,	PUNCT
ejpam-4750	206	33	2	2	NUM
ejpam-4750	206	34	,	,	PUNCT
ejpam-4750	206	35	if	if	SCONJ
ejpam-4750	206	36	n	n	PRON
ejpam-4750	206	37	≥	≥	NOUN
ejpam-4750	206	38	7	7	NUM
ejpam-4750	206	39	is	be	AUX
ejpam-4750	206	40	odd	odd	ADJ
ejpam-4750	206	41	,	,	PUNCT
ejpam-4750	206	42	3	3	NUM
ejpam-4750	206	43	,	,	PUNCT
ejpam-4750	206	44	if	if	SCONJ
ejpam-4750	206	45	n	n	NOUN
ejpam-4750	206	46	=	=	SYM
ejpam-4750	206	47	5	5	X
ejpam-4750	206	48	.	.	PUNCT
ejpam-4750	207	1	proof	proof	NOUN
ejpam-4750	207	2	:	:	PUNCT
ejpam-4750	207	3	suppose	suppose	VERB
ejpam-4750	207	4	that	that	SCONJ
ejpam-4750	207	5	cn	cn	PROPN
ejpam-4750	207	6	=	=	PUNCT
ejpam-4750	208	1	[	[	X
ejpam-4750	208	2	u1	u1	NOUN
ejpam-4750	208	3	,	,	PUNCT
ejpam-4750	208	4	u2	u2	NOUN
ejpam-4750	208	5	,	,	PUNCT
ejpam-4750	208	6	.	.	PUNCT
ejpam-4750	208	7	.	.	PUNCT
ejpam-4750	208	8	.	.	PUNCT
ejpam-4750	209	1	,	,	PUNCT
ejpam-4750	209	2	un	un	PROPN
ejpam-4750	209	3	,	,	PUNCT
ejpam-4750	209	4	u1	u1	NOUN
ejpam-4750	209	5	]	]	PUNCT
ejpam-4750	209	6	.	.	PUNCT
ejpam-4750	210	1	note	note	VERB
ejpam-4750	210	2	that	that	SCONJ
ejpam-4750	210	3	c3	c3	PROPN
ejpam-4750	210	4	=	=	PUNCT
ejpam-4750	210	5	k3	k3	X
ejpam-4750	210	6	and	and	CCONJ
ejpam-4750	210	7	by	by	ADP
ejpam-4750	210	8	proposition	proposition	NOUN
ejpam-4750	210	9	5	5	NUM
ejpam-4750	210	10	,	,	PUNCT
ejpam-4750	210	11	fln2(c3	fln2(c3	PROPN
ejpam-4750	210	12	)	)	PUNCT
ejpam-4750	210	13	=	=	NOUN
ejpam-4750	211	1	0	0	X
ejpam-4750	211	2	.	.	PUNCT
ejpam-4750	212	1	if	if	SCONJ
ejpam-4750	212	2	n	n	NOUN
ejpam-4750	212	3	=	=	SYM
ejpam-4750	212	4	4	4	NUM
ejpam-4750	212	5	,	,	PUNCT
ejpam-4750	212	6	then	then	ADV
ejpam-4750	212	7	v	v	X
ejpam-4750	212	8	(	(	PUNCT
ejpam-4750	212	9	c4	c4	NOUN
ejpam-4750	212	10	)	)	PUNCT
ejpam-4750	212	11	is	be	AUX
ejpam-4750	212	12	the	the	DET
ejpam-4750	212	13	only	only	ADJ
ejpam-4750	212	14	2	2	NUM
ejpam-4750	212	15	-	-	PUNCT
ejpam-4750	212	16	locating	locate	VERB
ejpam-4750	212	17	set	set	NOUN
ejpam-4750	212	18	of	of	ADP
ejpam-4750	212	19	c4	c4	NOUN
ejpam-4750	212	20	.	.	PUNCT
ejpam-4750	213	1	thus	thus	ADV
ejpam-4750	213	2	,	,	PUNCT
ejpam-4750	213	3	by	by	ADP
ejpam-4750	213	4	remark	remark	NOUN
ejpam-4750	213	5	5	5	NUM
ejpam-4750	213	6	(	(	PUNCT
ejpam-4750	213	7	i	i	NOUN
ejpam-4750	213	8	)	)	PUNCT
ejpam-4750	213	9	,	,	PUNCT
ejpam-4750	213	10	fln2(c4	fln2(c4	PROPN
ejpam-4750	213	11	)	)	PUNCT
ejpam-4750	213	12	=	=	SYM
ejpam-4750	213	13	0	0	X
ejpam-4750	213	14	.	.	PUNCT
ejpam-4750	213	15	suppose	suppose	VERB
ejpam-4750	213	16	that	that	SCONJ
ejpam-4750	213	17	n	n	NOUN
ejpam-4750	213	18	=	=	SYM
ejpam-4750	213	19	5	5	X
ejpam-4750	213	20	.	.	PUNCT
ejpam-4750	213	21	by	by	ADP
ejpam-4750	213	22	example	example	NOUN
ejpam-4750	213	23	1	1	NUM
ejpam-4750	213	24	,	,	PUNCT
ejpam-4750	213	25	ln2(c5	ln2(c5	PROPN
ejpam-4750	213	26	)	)	PUNCT
ejpam-4750	213	27	=	=	SYM
ejpam-4750	214	1	3	3	X
ejpam-4750	214	2	.	.	X
ejpam-4750	214	3	then	then	ADV
ejpam-4750	214	4	bi	bi	PROPN
ejpam-4750	214	5	,	,	PUNCT
ejpam-4750	214	6	j	j	PROPN
ejpam-4750	214	7	=	=	SYM
ejpam-4750	214	8	v	v	PROPN
ejpam-4750	214	9	(	(	PUNCT
ejpam-4750	214	10	c5	c5	PROPN
ejpam-4750	214	11	)	)	PUNCT
ejpam-4750	214	12	\	\	PROPN
ejpam-4750	214	13	{	{	PUNCT
ejpam-4750	214	14	vi	vi	PROPN
ejpam-4750	214	15	,	,	PUNCT
ejpam-4750	214	16	vj	vj	ADP
ejpam-4750	214	17	}	}	PUNCT
ejpam-4750	214	18	for	for	ADP
ejpam-4750	214	19	all	all	DET
ejpam-4750	214	20	i	i	PROPN
ejpam-4750	214	21	,	,	PUNCT
ejpam-4750	214	22	j	j	PROPN
ejpam-4750	214	23	∈	∈	PROPN
ejpam-4750	214	24	{	{	PUNCT
ejpam-4750	214	25	1	1	NUM
ejpam-4750	214	26	,	,	PUNCT
ejpam-4750	214	27	2	2	NUM
ejpam-4750	214	28	,	,	PUNCT
ejpam-4750	214	29	.	.	PUNCT
ejpam-4750	214	30	.	.	PUNCT
ejpam-4750	214	31	.	.	PUNCT
ejpam-4750	215	1	,	,	PUNCT
ejpam-4750	215	2	5	5	X
ejpam-4750	215	3	}	}	PUNCT
ejpam-4750	215	4	and	and	CCONJ
ejpam-4750	215	5	i	i	PRON
ejpam-4750	215	6	̸=	̸=	PROPN
ejpam-4750	215	7	j	j	PROPN
ejpam-4750	215	8	are	be	AUX
ejpam-4750	215	9	the	the	DET
ejpam-4750	215	10	ln2	ln2	ADJ
ejpam-4750	215	11	-	-	PUNCT
ejpam-4750	215	12	sets	set	NOUN
ejpam-4750	215	13	of	of	ADP
ejpam-4750	215	14	c5	c5	PROPN
ejpam-4750	215	15	.	.	PUNCT
ejpam-4750	216	1	thus	thus	ADV
ejpam-4750	216	2	,	,	PUNCT
ejpam-4750	216	3	for	for	ADP
ejpam-4750	216	4	every	every	DET
ejpam-4750	216	5	vk	vk	PROPN
ejpam-4750	216	6	∈	∈	PROPN
ejpam-4750	216	7	bi	bi	PROPN
ejpam-4750	216	8	,	,	PUNCT
ejpam-4750	216	9	j	j	PROPN
ejpam-4750	216	10	where	where	SCONJ
ejpam-4750	216	11	k	k	PROPN
ejpam-4750	216	12	̸=	̸=	PROPN
ejpam-4750	216	13	i	i	PROPN
ejpam-4750	216	14	,	,	PUNCT
ejpam-4750	216	15	j	j	PROPN
ejpam-4750	216	16	there	there	PRON
ejpam-4750	216	17	exists	exist	VERB
ejpam-4750	216	18	vi	vi	PROPN
ejpam-4750	216	19	∈	∈	PROPN
ejpam-4750	216	20	bj	bj	NOUN
ejpam-4750	216	21	,	,	PUNCT
ejpam-4750	216	22	k	k	PROPN
ejpam-4750	216	23	such	such	ADJ
ejpam-4750	216	24	that	that	SCONJ
ejpam-4750	216	25	[	[	X
ejpam-4750	216	26	bi	bi	NOUN
ejpam-4750	216	27	,	,	PUNCT
ejpam-4750	216	28	j	j	PROPN
ejpam-4750	216	29	\	\	PROPN
ejpam-4750	216	30	{	{	PUNCT
ejpam-4750	216	31	vk	vk	PROPN
ejpam-4750	216	32	}	}	PUNCT
ejpam-4750	216	33	]	]	PUNCT
ejpam-4750	216	34	∪	∪	X
ejpam-4750	216	35	{	{	PUNCT
ejpam-4750	216	36	vi	vi	NOUN
ejpam-4750	216	37	}	}	PUNCT
ejpam-4750	216	38	=	=	PUNCT
ejpam-4750	216	39	bj	bj	NOUN
ejpam-4750	216	40	,	,	PUNCT
ejpam-4750	216	41	k	k	PROPN
ejpam-4750	216	42	is	be	AUX
ejpam-4750	216	43	an	an	DET
ejpam-4750	216	44	ln2	ln2	NOUN
ejpam-4750	216	45	-	-	PUNCT
ejpam-4750	216	46	set	set	NOUN
ejpam-4750	216	47	of	of	ADP
ejpam-4750	216	48	c5	c5	PROPN
ejpam-4750	216	49	.	.	PUNCT
ejpam-4750	217	1	hence	hence	ADV
ejpam-4750	217	2	,	,	PUNCT
ejpam-4750	217	3	by	by	ADP
ejpam-4750	217	4	theorem	theorem	NOUN
ejpam-4750	217	5	6	6	NUM
ejpam-4750	217	6	,	,	PUNCT
ejpam-4750	217	7	fln2(c5	fln2(c5	PROPN
ejpam-4750	217	8	)	)	PUNCT
ejpam-4750	217	9	=	=	PUNCT
ejpam-4750	218	1	3	3	X
ejpam-4750	218	2	.	.	PUNCT
ejpam-4750	219	1	next	next	ADV
ejpam-4750	219	2	,	,	PUNCT
ejpam-4750	219	3	suppose	suppose	VERB
ejpam-4750	219	4	n	n	PRON
ejpam-4750	219	5	≥	≥	NUM
ejpam-4750	219	6	6	6	NUM
ejpam-4750	219	7	and	and	CCONJ
ejpam-4750	219	8	n	n	NUM
ejpam-4750	219	9	is	be	AUX
ejpam-4750	219	10	even	even	ADV
ejpam-4750	219	11	.	.	PUNCT
ejpam-4750	220	1	then	then	ADV
ejpam-4750	220	2	by	by	ADP
ejpam-4750	220	3	example	example	NOUN
ejpam-4750	220	4	1	1	NUM
ejpam-4750	220	5	,	,	PUNCT
ejpam-4750	220	6	ln2(cn	ln2(cn	ADJ
ejpam-4750	220	7	)	)	PUNCT
ejpam-4750	220	8	=	=	SYM
ejpam-4750	221	1	n	n	DET
ejpam-4750	221	2	2	2	NUM
ejpam-4750	221	3	.	.	PUNCT
ejpam-4750	222	1	thus	thus	ADV
ejpam-4750	222	2	,	,	PUNCT
ejpam-4750	222	3	cn	cn	PROPN
ejpam-4750	222	4	has	have	VERB
ejpam-4750	222	5	ln2	ln2	ADJ
ejpam-4750	222	6	-	-	PUNCT
ejpam-4750	222	7	sets	set	NOUN
ejpam-4750	222	8	d1	d1	NOUN
ejpam-4750	222	9	=	=	SYM
ejpam-4750	222	10	{	{	PUNCT
ejpam-4750	222	11	u1	u1	NOUN
ejpam-4750	222	12	,	,	PUNCT
ejpam-4750	222	13	u3	u3	PROPN
ejpam-4750	222	14	,	,	PUNCT
ejpam-4750	222	15	u5	u5	PROPN
ejpam-4750	222	16	,	,	PUNCT
ejpam-4750	222	17	.	.	PUNCT
ejpam-4750	222	18	.	.	PUNCT
ejpam-4750	223	1	.	.	PUNCT
ejpam-4750	224	1	,	,	PUNCT
ejpam-4750	224	2	un−1	un−1	ADJ
ejpam-4750	224	3	}	}	PUNCT
ejpam-4750	224	4	and	and	CCONJ
ejpam-4750	224	5	d2	d2	PROPN
ejpam-4750	224	6	=	=	SYM
ejpam-4750	224	7	{	{	PUNCT
ejpam-4750	224	8	u2	u2	PROPN
ejpam-4750	224	9	,	,	PUNCT
ejpam-4750	224	10	u4	u4	PROPN
ejpam-4750	224	11	,	,	PUNCT
ejpam-4750	224	12	u6	u6	NOUN
ejpam-4750	224	13	,	,	PUNCT
ejpam-4750	224	14	.	.	PUNCT
ejpam-4750	224	15	.	.	PUNCT
ejpam-4750	225	1	.	.	PUNCT
ejpam-4750	226	1	,	,	PUNCT
ejpam-4750	226	2	un	un	PROPN
ejpam-4750	226	3	}	}	PUNCT
ejpam-4750	226	4	.	.	PUNCT
ejpam-4750	227	1	it	it	PRON
ejpam-4750	227	2	can	can	AUX
ejpam-4750	227	3	be	be	AUX
ejpam-4750	227	4	seen	see	VERB
ejpam-4750	227	5	that	that	SCONJ
ejpam-4750	227	6	d1	d1	PROPN
ejpam-4750	227	7	d.	d.	PROPN
ejpam-4750	227	8	managbanag	managbanag	PROPN
ejpam-4750	227	9	,	,	PUNCT
ejpam-4750	227	10	h.	h.	PROPN
ejpam-4750	227	11	rara	rara	PROPN
ejpam-4750	227	12	/	/	SYM
ejpam-4750	227	13	eur	eur	PROPN
ejpam-4750	227	14	.	.	PUNCT
ejpam-4750	228	1	j.	j.	PROPN
ejpam-4750	228	2	pure	pure	PROPN
ejpam-4750	228	3	appl	appl	PROPN
ejpam-4750	228	4	.	.	PROPN
ejpam-4750	228	5	math	math	PROPN
ejpam-4750	228	6	,	,	PUNCT
ejpam-4750	228	7	16	16	NUM
ejpam-4750	228	8	(	(	PUNCT
ejpam-4750	228	9	2	2	NUM
ejpam-4750	228	10	)	)	PUNCT
ejpam-4750	228	11	(	(	PUNCT
ejpam-4750	228	12	2023	2023	NUM
ejpam-4750	228	13	)	)	PUNCT
ejpam-4750	228	14	,	,	PUNCT
ejpam-4750	228	15	1068	1068	NUM
ejpam-4750	228	16	-	-	SYM
ejpam-4750	228	17	1083	1083	NUM
ejpam-4750	228	18	1075	1075	NUM
ejpam-4750	228	19	is	be	AUX
ejpam-4750	228	20	the	the	DET
ejpam-4750	228	21	only	only	ADJ
ejpam-4750	228	22	ln2	ln2	NOUN
ejpam-4750	228	23	-	-	PUNCT
ejpam-4750	228	24	set	set	NOUN
ejpam-4750	228	25	containing	contain	VERB
ejpam-4750	228	26	the	the	DET
ejpam-4750	228	27	vertex	vertex	NOUN
ejpam-4750	228	28	u1	u1	NOUN
ejpam-4750	228	29	.	.	PUNCT
ejpam-4750	229	1	thus	thus	ADV
ejpam-4750	229	2	,	,	PUNCT
ejpam-4750	229	3	by	by	ADP
ejpam-4750	229	4	remark	remark	NOUN
ejpam-4750	229	5	5	5	NUM
ejpam-4750	229	6	(	(	PUNCT
ejpam-4750	229	7	ii	ii	NOUN
ejpam-4750	229	8	)	)	PUNCT
ejpam-4750	229	9	,	,	PUNCT
ejpam-4750	229	10	fln2(cn	fln2(cn	PROPN
ejpam-4750	229	11	)	)	PUNCT
ejpam-4750	229	12	=	=	SYM
ejpam-4750	230	1	1	1	X
ejpam-4750	230	2	.	.	PUNCT
ejpam-4750	230	3	now	now	ADV
ejpam-4750	230	4	,	,	PUNCT
ejpam-4750	230	5	suppose	suppose	VERB
ejpam-4750	230	6	that	that	SCONJ
ejpam-4750	230	7	n	n	PROPN
ejpam-4750	230	8	≥	≥	X
ejpam-4750	230	9	7	7	NUM
ejpam-4750	230	10	and	and	CCONJ
ejpam-4750	230	11	n	n	PRON
ejpam-4750	230	12	is	be	AUX
ejpam-4750	230	13	odd	odd	ADJ
ejpam-4750	230	14	.	.	PUNCT
ejpam-4750	231	1	by	by	ADP
ejpam-4750	231	2	example	example	NOUN
ejpam-4750	231	3	1	1	NUM
ejpam-4750	231	4	,	,	PUNCT
ejpam-4750	231	5	ln2(cn	ln2(cn	ADJ
ejpam-4750	231	6	)	)	PUNCT
ejpam-4750	231	7	=	=	PUNCT
ejpam-4750	232	1	n+	n+	PUNCT
ejpam-4750	232	2	1	1	NUM
ejpam-4750	232	3	2	2	NUM
ejpam-4750	232	4	.	.	PUNCT
ejpam-4750	233	1	hence	hence	ADV
ejpam-4750	233	2	,	,	PUNCT
ejpam-4750	233	3	by	by	ADP
ejpam-4750	233	4	remark	remark	NOUN
ejpam-4750	233	5	7	7	NUM
ejpam-4750	233	6	,	,	PUNCT
ejpam-4750	233	7	the	the	DET
ejpam-4750	233	8	ln2	ln2	NOUN
ejpam-4750	233	9	-	-	PUNCT
ejpam-4750	233	10	sets	set	NOUN
ejpam-4750	233	11	of	of	ADP
ejpam-4750	233	12	cn	cn	PROPN
ejpam-4750	233	13	is	be	AUX
ejpam-4750	233	14	of	of	ADP
ejpam-4750	233	15	the	the	DET
ejpam-4750	233	16	form	form	NOUN
ejpam-4750	233	17	si	si	X
ejpam-4750	233	18	=	=	ADJ
ejpam-4750	233	19	{	{	PUNCT
ejpam-4750	233	20	ui	ui	PROPN
ejpam-4750	233	21	,	,	PUNCT
ejpam-4750	233	22	ui+1	ui+1	PROPN
ejpam-4750	233	23	,	,	PUNCT
ejpam-4750	233	24	ui+3	ui+3	NOUN
ejpam-4750	233	25	,	,	PUNCT
ejpam-4750	233	26	ui+5	ui+5	NOUN
ejpam-4750	233	27	,	,	PUNCT
ejpam-4750	233	28	.	.	PUNCT
ejpam-4750	233	29	.	.	PUNCT
ejpam-4750	234	1	.	.	PUNCT
ejpam-4750	235	1	,	,	PUNCT
ejpam-4750	235	2	ui−4	ui−4	NOUN
ejpam-4750	235	3	,	,	PUNCT
ejpam-4750	235	4	ui−2	ui−2	PROPN
ejpam-4750	235	5	}	}	PUNCT
ejpam-4750	235	6	where	where	SCONJ
ejpam-4750	235	7	1	1	NUM
ejpam-4750	235	8	≤	≤	NUM
ejpam-4750	235	9	i	i	NOUN
ejpam-4750	235	10	≤	≤	ADJ
ejpam-4750	235	11	n	n	CCONJ
ejpam-4750	235	12	and	and	CCONJ
ejpam-4750	235	13	n+k	n+k	PROPN
ejpam-4750	235	14	≡	≡	PROPN
ejpam-4750	235	15	k	k	PROPN
ejpam-4750	235	16	(	(	PUNCT
ejpam-4750	235	17	mod	mod	PROPN
ejpam-4750	235	18	n	n	CCONJ
ejpam-4750	235	19	)	)	PUNCT
ejpam-4750	235	20	for	for	ADP
ejpam-4750	235	21	any	any	DET
ejpam-4750	235	22	positive	positive	ADJ
ejpam-4750	235	23	integer	integer	NOUN
ejpam-4750	236	1	k.	k.	PROPN
ejpam-4750	236	2	observe	observe	VERB
ejpam-4750	236	3	that	that	SCONJ
ejpam-4750	236	4	no	no	DET
ejpam-4750	236	5	single	single	ADJ
ejpam-4750	236	6	vertex	vertex	NOUN
ejpam-4750	236	7	is	be	AUX
ejpam-4750	236	8	contained	contain	VERB
ejpam-4750	236	9	in	in	ADP
ejpam-4750	236	10	a	a	DET
ejpam-4750	236	11	unique	unique	ADJ
ejpam-4750	236	12	ln2	ln2	NOUN
ejpam-4750	236	13	-	-	PUNCT
ejpam-4750	236	14	set	set	NOUN
ejpam-4750	236	15	of	of	ADP
ejpam-4750	236	16	cn	cn	PROPN
ejpam-4750	236	17	.	.	PUNCT
ejpam-4750	236	18	thus	thus	ADV
ejpam-4750	236	19	,	,	PUNCT
ejpam-4750	236	20	fln2(cn	fln2(cn	PROPN
ejpam-4750	236	21	)	)	PUNCT
ejpam-4750	236	22	>	>	X
ejpam-4750	237	1	1	1	X
ejpam-4750	237	2	.	.	PUNCT
ejpam-4750	237	3	it	it	PRON
ejpam-4750	237	4	can	can	AUX
ejpam-4750	237	5	be	be	AUX
ejpam-4750	237	6	verified	verify	VERB
ejpam-4750	237	7	that	that	SCONJ
ejpam-4750	237	8	{	{	PUNCT
ejpam-4750	237	9	ui	ui	PROPN
ejpam-4750	237	10	,	,	PUNCT
ejpam-4750	237	11	ui+1	ui+1	PROPN
ejpam-4750	237	12	}	}	PUNCT
ejpam-4750	237	13	is	be	AUX
ejpam-4750	237	14	uniquely	uniquely	ADV
ejpam-4750	237	15	contained	contain	VERB
ejpam-4750	237	16	in	in	ADP
ejpam-4750	237	17	si	si	PROPN
ejpam-4750	237	18	.	.	PROPN
ejpam-4750	237	19	hence	hence	ADV
ejpam-4750	237	20	,	,	PUNCT
ejpam-4750	237	21	fln2(si	fln2(si	PROPN
ejpam-4750	237	22	)	)	PUNCT
ejpam-4750	237	23	=	=	SYM
ejpam-4750	237	24	2	2	NUM
ejpam-4750	237	25	=	=	SYM
ejpam-4750	237	26	fln2(cn	fln2(cn	PROPN
ejpam-4750	237	27	)	)	PUNCT
ejpam-4750	237	28	.	.	PUNCT
ejpam-4750	238	1	remark	remark	PROPN
ejpam-4750	238	2	8	8	NUM
ejpam-4750	238	3	.	.	PUNCT
ejpam-4750	239	1	let	let	VERB
ejpam-4750	239	2	g	g	PRON
ejpam-4750	239	3	be	be	AUX
ejpam-4750	239	4	a	a	DET
ejpam-4750	239	5	connected	connected	ADJ
ejpam-4750	239	6	graph	graph	NOUN
ejpam-4750	239	7	.	.	PUNCT
ejpam-4750	240	1	then	then	ADV
ejpam-4750	240	2	(	(	PUNCT
ejpam-4750	240	3	i	i	NOUN
ejpam-4750	240	4	)	)	PUNCT
ejpam-4750	240	5	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	240	6	)	)	PUNCT
ejpam-4750	240	7	=	=	SYM
ejpam-4750	240	8	0	0	PUNCT
ejpam-4750	241	1	if	if	SCONJ
ejpam-4750	241	2	and	and	CCONJ
ejpam-4750	241	3	only	only	ADV
ejpam-4750	241	4	if	if	SCONJ
ejpam-4750	241	5	g	g	PROPN
ejpam-4750	241	6	has	have	VERB
ejpam-4750	241	7	a	a	DET
ejpam-4750	241	8	unique	unique	ADJ
ejpam-4750	241	9	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	241	10	,	,	PUNCT
ejpam-4750	241	11	and	and	CCONJ
ejpam-4750	241	12	(	(	PUNCT
ejpam-4750	241	13	ii	ii	NOUN
ejpam-4750	241	14	)	)	PUNCT
ejpam-4750	241	15	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	241	16	)	)	PUNCT
ejpam-4750	241	17	=	=	SYM
ejpam-4750	241	18	1	1	NUM
ejpam-4750	241	19	if	if	SCONJ
ejpam-4750	241	20	and	and	CCONJ
ejpam-4750	241	21	only	only	ADV
ejpam-4750	241	22	if	if	SCONJ
ejpam-4750	241	23	g	g	PROPN
ejpam-4750	241	24	has	have	VERB
ejpam-4750	241	25	at	at	ADV
ejpam-4750	241	26	least	least	ADV
ejpam-4750	241	27	two	two	NUM
ejpam-4750	241	28	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	241	29	,	,	PUNCT
ejpam-4750	241	30	one	one	NUM
ejpam-4750	241	31	of	of	ADP
ejpam-4750	241	32	which	which	PRON
ejpam-4750	241	33	,	,	PUNCT
ejpam-4750	241	34	say	say	VERB
ejpam-4750	241	35	b	b	X
ejpam-4750	241	36	,	,	PUNCT
ejpam-4750	241	37	that	that	PRON
ejpam-4750	241	38	contains	contain	VERB
ejpam-4750	241	39	an	an	DET
ejpam-4750	241	40	element	element	NOUN
ejpam-4750	241	41	not	not	PART
ejpam-4750	241	42	in	in	ADP
ejpam-4750	241	43	any	any	DET
ejpam-4750	241	44	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	241	45	of	of	ADP
ejpam-4750	241	46	g.	g.	PROPN
ejpam-4750	241	47	theorem	theorem	VERB
ejpam-4750	241	48	7	7	NUM
ejpam-4750	241	49	.	.	PUNCT
ejpam-4750	242	1	let	let	VERB
ejpam-4750	242	2	g	g	PRON
ejpam-4750	242	3	be	be	AUX
ejpam-4750	242	4	a	a	DET
ejpam-4750	242	5	connected	connected	ADJ
ejpam-4750	242	6	graph	graph	NOUN
ejpam-4750	242	7	.	.	PUNCT
ejpam-4750	243	1	then	then	ADV
ejpam-4750	243	2	fln(2,2)(g	fln(2,2)(g	PROPN
ejpam-4750	243	3	)	)	PUNCT
ejpam-4750	243	4	=	=	SYM
ejpam-4750	244	1	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	244	2	)	)	PUNCT
ejpam-4750	245	1	if	if	SCONJ
ejpam-4750	245	2	and	and	CCONJ
ejpam-4750	245	3	only	only	ADV
ejpam-4750	245	4	if	if	SCONJ
ejpam-4750	245	5	for	for	ADP
ejpam-4750	245	6	all	all	DET
ejpam-4750	245	7	(	(	PUNCT
ejpam-4750	245	8	2	2	NUM
ejpam-4750	245	9	,	,	PUNCT
ejpam-4750	245	10	2)-locating	2)-locating	NUM
ejpam-4750	245	11	set	set	NOUN
ejpam-4750	245	12	s	s	NOUN
ejpam-4750	245	13	of	of	ADP
ejpam-4750	245	14	g	g	NOUN
ejpam-4750	245	15	and	and	CCONJ
ejpam-4750	245	16	for	for	ADP
ejpam-4750	245	17	each	each	DET
ejpam-4750	245	18	u	u	PROPN
ejpam-4750	245	19	∈	∈	PROPN
ejpam-4750	245	20	s	s	PART
ejpam-4750	245	21	,	,	PUNCT
ejpam-4750	245	22	there	there	PRON
ejpam-4750	245	23	exists	exist	VERB
ejpam-4750	245	24	vu	vu	PROPN
ejpam-4750	245	25	∈	∈	PROPN
ejpam-4750	245	26	v	v	ADP
ejpam-4750	245	27	(	(	PUNCT
ejpam-4750	245	28	g	g	NOUN
ejpam-4750	245	29	)	)	PUNCT
ejpam-4750	245	30	\	\	PUNCT
ejpam-4750	246	1	s	s	VERB
ejpam-4750	246	2	such	such	ADJ
ejpam-4750	246	3	that	that	SCONJ
ejpam-4750	246	4	[	[	PUNCT
ejpam-4750	246	5	s	s	X
ejpam-4750	246	6	\	\	X
ejpam-4750	246	7	{	{	PUNCT
ejpam-4750	246	8	u	u	NOUN
ejpam-4750	246	9	}	}	PUNCT
ejpam-4750	246	10	]	]	PUNCT
ejpam-4750	246	11	∪	∪	X
ejpam-4750	246	12	{	{	PUNCT
ejpam-4750	246	13	vu	vu	INTJ
ejpam-4750	246	14	}	}	PUNCT
ejpam-4750	246	15	is	be	AUX
ejpam-4750	246	16	an	an	DET
ejpam-4750	246	17	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	246	18	of	of	ADP
ejpam-4750	246	19	g.	g.	PROPN
ejpam-4750	246	20	proof	proof	PROPN
ejpam-4750	246	21	:	:	PUNCT
ejpam-4750	246	22	suppose	suppose	VERB
ejpam-4750	246	23	that	that	SCONJ
ejpam-4750	246	24	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	246	25	)	)	PUNCT
ejpam-4750	247	1	=	=	SYM
ejpam-4750	247	2	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	247	3	)	)	PUNCT
ejpam-4750	247	4	.	.	PUNCT
ejpam-4750	248	1	let	let	VERB
ejpam-4750	248	2	s	s	PRON
ejpam-4750	248	3	be	be	AUX
ejpam-4750	248	4	an	an	DET
ejpam-4750	248	5	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	248	6	of	of	ADP
ejpam-4750	248	7	g	g	NOUN
ejpam-4750	248	8	such	such	ADJ
ejpam-4750	248	9	that	that	DET
ejpam-4750	248	10	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	248	11	)	)	PUNCT
ejpam-4750	248	12	=	=	PUNCT
ejpam-4750	248	13	|s|	|s|	PROPN
ejpam-4750	248	14	=	=	SYM
ejpam-4750	248	15	ln(2,2)(g	ln(2,2)(g	PROPN
ejpam-4750	248	16	)	)	PUNCT
ejpam-4750	248	17	that	that	PRON
ejpam-4750	248	18	is	be	AUX
ejpam-4750	248	19	,	,	PUNCT
ejpam-4750	248	20	s	s	VERB
ejpam-4750	248	21	is	be	AUX
ejpam-4750	248	22	the	the	DET
ejpam-4750	248	23	only	only	ADJ
ejpam-4750	248	24	forcing	forcing	NOUN
ejpam-4750	248	25	subset	subset	NOUN
ejpam-4750	248	26	for	for	ADP
ejpam-4750	248	27	itself	itself	PRON
ejpam-4750	248	28	.	.	PUNCT
ejpam-4750	249	1	let	let	VERB
ejpam-4750	249	2	u	u	PRON
ejpam-4750	249	3	∈	∈	PROPN
ejpam-4750	249	4	s.	s.	PROPN
ejpam-4750	249	5	since	since	SCONJ
ejpam-4750	249	6	s	s	PRON
ejpam-4750	249	7	\{u	\{u	X
ejpam-4750	249	8	}	}	PUNCT
ejpam-4750	249	9	is	be	AUX
ejpam-4750	249	10	not	not	PART
ejpam-4750	249	11	a	a	DET
ejpam-4750	249	12	forcing	forcing	NOUN
ejpam-4750	249	13	subset	subset	NOUN
ejpam-4750	249	14	for	for	ADP
ejpam-4750	249	15	s	s	PROPN
ejpam-4750	249	16	,	,	PUNCT
ejpam-4750	249	17	there	there	PRON
ejpam-4750	249	18	exists	exist	VERB
ejpam-4750	249	19	a	a	DET
ejpam-4750	249	20	vu	vu	X
ejpam-4750	249	21	∈	∈	PROPN
ejpam-4750	249	22	v	v	NOUN
ejpam-4750	249	23	(	(	PUNCT
ejpam-4750	249	24	g)\s	g)\s	VERB
ejpam-4750	249	25	such	such	ADJ
ejpam-4750	249	26	that	that	SCONJ
ejpam-4750	249	27	[	[	PUNCT
ejpam-4750	249	28	s	s	AUX
ejpam-4750	249	29	\{u	\{u	ADV
ejpam-4750	249	30	}	}	PUNCT
ejpam-4750	249	31	]	]	SYM
ejpam-4750	249	32	∪{vu	∪{vu	NUM
ejpam-4750	249	33	}	}	PUNCT
ejpam-4750	249	34	is	be	AUX
ejpam-4750	249	35	an	an	DET
ejpam-4750	249	36	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	249	37	of	of	ADP
ejpam-4750	249	38	g.	g.	NOUN
ejpam-4750	249	39	conversely	conversely	ADV
ejpam-4750	249	40	,	,	PUNCT
ejpam-4750	249	41	suppose	suppose	VERB
ejpam-4750	249	42	that	that	SCONJ
ejpam-4750	249	43	every	every	DET
ejpam-4750	249	44	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	249	45	of	of	ADP
ejpam-4750	249	46	g	g	PROPN
ejpam-4750	249	47	satisfies	satisfy	VERB
ejpam-4750	249	48	the	the	DET
ejpam-4750	249	49	given	give	VERB
ejpam-4750	249	50	condition	condition	NOUN
ejpam-4750	249	51	.	.	PUNCT
ejpam-4750	250	1	let	let	VERB
ejpam-4750	250	2	s	s	PRON
ejpam-4750	250	3	be	be	AUX
ejpam-4750	250	4	an	an	DET
ejpam-4750	250	5	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	250	6	of	of	ADP
ejpam-4750	250	7	g	g	NOUN
ejpam-4750	250	8	such	such	ADJ
ejpam-4750	250	9	that	that	DET
ejpam-4750	250	10	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	250	11	)	)	PUNCT
ejpam-4750	250	12	=	=	SYM
ejpam-4750	250	13	fln(2,2)(s	fln(2,2)(s	PROPN
ejpam-4750	250	14	)	)	PUNCT
ejpam-4750	250	15	.	.	PUNCT
ejpam-4750	251	1	suppose	suppose	VERB
ejpam-4750	251	2	further	far	ADV
ejpam-4750	251	3	that	that	SCONJ
ejpam-4750	251	4	s	s	VERB
ejpam-4750	251	5	has	have	VERB
ejpam-4750	251	6	a	a	DET
ejpam-4750	251	7	forcing	force	VERB
ejpam-4750	251	8	subset	subset	NOUN
ejpam-4750	251	9	d	d	NOUN
ejpam-4750	251	10	with	with	ADP
ejpam-4750	251	11	|d|	|d|	PROPN
ejpam-4750	251	12	<	<	X
ejpam-4750	251	13	|s|	|s|	PROPN
ejpam-4750	251	14	,	,	PUNCT
ejpam-4750	251	15	that	that	ADV
ejpam-4750	251	16	is	be	AUX
ejpam-4750	251	17	,	,	PUNCT
ejpam-4750	251	18	s	s	PART
ejpam-4750	251	19	=	=	X
ejpam-4750	251	20	d	d	X
ejpam-4750	251	21	∪	∪	ADP
ejpam-4750	251	22	i	i	PRON
ejpam-4750	251	23	where	where	SCONJ
ejpam-4750	251	24	i	i	PRON
ejpam-4750	251	25	=	=	PUNCT
ejpam-4750	251	26	{	{	PUNCT
ejpam-4750	251	27	w	w	PROPN
ejpam-4750	251	28	∈	∈	PROPN
ejpam-4750	251	29	s	s	PART
ejpam-4750	251	30	:	:	PUNCT
ejpam-4750	251	31	w	w	X
ejpam-4750	251	32	/∈	/∈	PUNCT
ejpam-4750	252	1	d	d	NOUN
ejpam-4750	252	2	}	}	PUNCT
ejpam-4750	252	3	.	.	PUNCT
ejpam-4750	253	1	pick	pick	VERB
ejpam-4750	253	2	w	w	PROPN
ejpam-4750	253	3	∈	∈	PROPN
ejpam-4750	253	4	i.	i.	NOUN
ejpam-4750	253	5	by	by	ADP
ejpam-4750	253	6	assumption	assumption	NOUN
ejpam-4750	253	7	,	,	PUNCT
ejpam-4750	253	8	there	there	PRON
ejpam-4750	253	9	exists	exist	VERB
ejpam-4750	253	10	vw	vw	PROPN
ejpam-4750	253	11	∈	∈	PROPN
ejpam-4750	253	12	v	v	ADP
ejpam-4750	253	13	(	(	PUNCT
ejpam-4750	253	14	g	g	NOUN
ejpam-4750	253	15	)	)	PUNCT
ejpam-4750	253	16	\	\	PUNCT
ejpam-4750	254	1	s	s	VERB
ejpam-4750	254	2	such	such	ADJ
ejpam-4750	254	3	that	that	SCONJ
ejpam-4750	254	4	[	[	PUNCT
ejpam-4750	254	5	s	s	X
ejpam-4750	254	6	\	\	X
ejpam-4750	254	7	{	{	PUNCT
ejpam-4750	254	8	w	w	NOUN
ejpam-4750	254	9	}	}	PUNCT
ejpam-4750	254	10	]	]	PUNCT
ejpam-4750	254	11	∪	∪	X
ejpam-4750	254	12	{	{	PUNCT
ejpam-4750	254	13	vw	vw	NOUN
ejpam-4750	254	14	}	}	PUNCT
ejpam-4750	254	15	=	=	SYM
ejpam-4750	254	16	j	j	PROPN
ejpam-4750	254	17	is	be	AUX
ejpam-4750	254	18	an	an	DET
ejpam-4750	254	19	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	254	20	of	of	ADP
ejpam-4750	254	21	g.	g.	PROPN
ejpam-4750	254	22	hence	hence	ADV
ejpam-4750	254	23	,	,	PUNCT
ejpam-4750	254	24	j	j	PROPN
ejpam-4750	254	25	=	=	SYM
ejpam-4750	254	26	d	d	X
ejpam-4750	254	27	∪	∪	ADP
ejpam-4750	254	28	t	t	PROPN
ejpam-4750	254	29	,	,	PUNCT
ejpam-4750	254	30	where	where	SCONJ
ejpam-4750	254	31	t	t	NOUN
ejpam-4750	254	32	=	=	PUNCT
ejpam-4750	255	1	[	[	PUNCT
ejpam-4750	255	2	i	i	PRON
ejpam-4750	255	3	\	\	PROPN
ejpam-4750	255	4	{	{	PUNCT
ejpam-4750	255	5	w	w	NOUN
ejpam-4750	255	6	}	}	PUNCT
ejpam-4750	255	7	]	]	PUNCT
ejpam-4750	255	8	∪	∪	X
ejpam-4750	255	9	{	{	PUNCT
ejpam-4750	255	10	vw	vw	NOUN
ejpam-4750	255	11	}	}	PUNCT
ejpam-4750	255	12	,	,	PUNCT
ejpam-4750	255	13	is	be	AUX
ejpam-4750	255	14	an	an	DET
ejpam-4750	255	15	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	255	16	containing	contain	VERB
ejpam-4750	255	17	d	d	PROPN
ejpam-4750	255	18	,	,	PUNCT
ejpam-4750	255	19	a	a	DET
ejpam-4750	255	20	contradiction	contradiction	NOUN
ejpam-4750	255	21	.	.	PUNCT
ejpam-4750	256	1	hence	hence	ADV
ejpam-4750	256	2	,	,	PUNCT
ejpam-4750	256	3	s	s	VERB
ejpam-4750	256	4	is	be	AUX
ejpam-4750	256	5	the	the	DET
ejpam-4750	256	6	only	only	ADJ
ejpam-4750	256	7	forcing	forcing	NOUN
ejpam-4750	256	8	subset	subset	NOUN
ejpam-4750	256	9	for	for	ADP
ejpam-4750	256	10	s.	s.	PROPN
ejpam-4750	256	11	therefore	therefore	ADV
ejpam-4750	256	12	,	,	PUNCT
ejpam-4750	256	13	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	256	14	)	)	PUNCT
ejpam-4750	256	15	=	=	SYM
ejpam-4750	256	16	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	256	17	)	)	PUNCT
ejpam-4750	256	18	.	.	PUNCT
ejpam-4750	257	1	remark	remark	NOUN
ejpam-4750	257	2	9	9	NUM
ejpam-4750	257	3	.	.	PUNCT
ejpam-4750	258	1	a	a	DET
ejpam-4750	258	2	(	(	PUNCT
ejpam-4750	258	3	2,2)-locating	2,2)-locating	NUM
ejpam-4750	258	4	set	set	VERB
ejpam-4750	258	5	in	in	ADP
ejpam-4750	258	6	g	g	NOUN
ejpam-4750	258	7	does	do	AUX
ejpam-4750	258	8	not	not	PART
ejpam-4750	258	9	exist	exist	VERB
ejpam-4750	258	10	for	for	ADP
ejpam-4750	258	11	some	some	DET
ejpam-4750	258	12	graph	graph	NOUN
ejpam-4750	258	13	g.	g.	NOUN
ejpam-4750	258	14	in	in	ADP
ejpam-4750	258	15	particular	particular	ADJ
ejpam-4750	258	16	,	,	PUNCT
ejpam-4750	258	17	if	if	SCONJ
ejpam-4750	258	18	γ(g	γ(g	PROPN
ejpam-4750	258	19	)	)	PUNCT
ejpam-4750	259	1	=	=	SYM
ejpam-4750	259	2	1	1	NUM
ejpam-4750	259	3	,	,	PUNCT
ejpam-4750	259	4	then	then	ADV
ejpam-4750	259	5	g	g	PROPN
ejpam-4750	259	6	has	have	VERB
ejpam-4750	259	7	no	no	DET
ejpam-4750	259	8	(	(	PUNCT
ejpam-4750	259	9	2,2)-locating	2,2)-locating	NUM
ejpam-4750	259	10	set	set	NOUN
ejpam-4750	259	11	.	.	PUNCT
ejpam-4750	260	1	proposition	proposition	NOUN
ejpam-4750	260	2	8	8	NUM
ejpam-4750	260	3	.	.	PUNCT
ejpam-4750	261	1	for	for	ADP
ejpam-4750	261	2	any	any	DET
ejpam-4750	261	3	path	path	NOUN
ejpam-4750	261	4	pn	pn	NOUN
ejpam-4750	261	5	with	with	ADP
ejpam-4750	261	6	n	n	PRON
ejpam-4750	261	7	≥	≥	NUM
ejpam-4750	261	8	4	4	NUM
ejpam-4750	261	9	vertices	vertex	NOUN
ejpam-4750	261	10	,	,	PUNCT
ejpam-4750	261	11	fln(2,2)(pn	fln(2,2)(pn	NOUN
ejpam-4750	261	12	)	)	PUNCT
ejpam-4750	261	13	=	=	PUNCT
ejpam-4750	261	14			NOUN
ejpam-4750	261	15	0	0	NUM
ejpam-4750	261	16	,	,	PUNCT
ejpam-4750	261	17	if	if	SCONJ
ejpam-4750	261	18	n	n	NOUN
ejpam-4750	261	19	=	=	SYM
ejpam-4750	261	20	4	4	NUM
ejpam-4750	261	21	and	and	CCONJ
ejpam-4750	261	22	n	n	PRON
ejpam-4750	261	23	≥	≥	NOUN
ejpam-4750	261	24	7	7	NUM
ejpam-4750	261	25	is	be	AUX
ejpam-4750	261	26	odd	odd	ADJ
ejpam-4750	261	27	,	,	PUNCT
ejpam-4750	261	28	2	2	NUM
ejpam-4750	261	29	,	,	PUNCT
ejpam-4750	261	30	if	if	SCONJ
ejpam-4750	261	31	n	n	PRON
ejpam-4750	261	32	≥	≥	NOUN
ejpam-4750	261	33	8	8	NUM
ejpam-4750	261	34	is	be	AUX
ejpam-4750	261	35	even	even	ADV
ejpam-4750	261	36	,	,	PUNCT
ejpam-4750	261	37	3	3	X
ejpam-4750	261	38	,	,	PUNCT
ejpam-4750	261	39	if	if	SCONJ
ejpam-4750	261	40	n	n	NOUN
ejpam-4750	261	41	=	=	SYM
ejpam-4750	261	42	6	6	NUM
ejpam-4750	261	43	,	,	PUNCT
ejpam-4750	261	44	4	4	NUM
ejpam-4750	261	45	,	,	PUNCT
ejpam-4750	261	46	if	if	SCONJ
ejpam-4750	261	47	n	n	NOUN
ejpam-4750	261	48	=	=	SYM
ejpam-4750	261	49	5	5	X
ejpam-4750	261	50	.	.	PUNCT
ejpam-4750	262	1	proof	proof	NOUN
ejpam-4750	262	2	:	:	PUNCT
ejpam-4750	262	3	suppose	suppose	VERB
ejpam-4750	262	4	that	that	SCONJ
ejpam-4750	262	5	pn	pn	PROPN
ejpam-4750	262	6	=	=	PUNCT
ejpam-4750	263	1	[	[	X
ejpam-4750	263	2	v1	v1	NOUN
ejpam-4750	263	3	,	,	PUNCT
ejpam-4750	263	4	v2	v2	NOUN
ejpam-4750	263	5	,	,	PUNCT
ejpam-4750	263	6	.	.	PUNCT
ejpam-4750	263	7	.	.	PUNCT
ejpam-4750	263	8	.	.	PUNCT
ejpam-4750	264	1	,	,	PUNCT
ejpam-4750	264	2	vn	vn	X
ejpam-4750	264	3	]	]	PUNCT
ejpam-4750	264	4	.	.	PUNCT
ejpam-4750	265	1	if	if	SCONJ
ejpam-4750	265	2	n	n	NOUN
ejpam-4750	265	3	=	=	SYM
ejpam-4750	265	4	4	4	NUM
ejpam-4750	265	5	,	,	PUNCT
ejpam-4750	265	6	then	then	ADV
ejpam-4750	265	7	v	v	X
ejpam-4750	265	8	(	(	PUNCT
ejpam-4750	265	9	p4	p4	ADJ
ejpam-4750	265	10	)	)	PUNCT
ejpam-4750	265	11	is	be	AUX
ejpam-4750	265	12	the	the	DET
ejpam-4750	265	13	only	only	ADJ
ejpam-4750	265	14	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	265	15	of	of	ADP
ejpam-4750	265	16	p4	p4	NOUN
ejpam-4750	265	17	.	.	PUNCT
ejpam-4750	266	1	thus	thus	ADV
ejpam-4750	266	2	,	,	PUNCT
ejpam-4750	266	3	by	by	ADP
ejpam-4750	266	4	remark	remark	NOUN
ejpam-4750	266	5	8	8	NUM
ejpam-4750	266	6	(	(	PUNCT
ejpam-4750	266	7	i	i	NOUN
ejpam-4750	266	8	)	)	PUNCT
ejpam-4750	266	9	,	,	PUNCT
ejpam-4750	266	10	fln(2,2)(p4	fln(2,2)(p4	X
ejpam-4750	266	11	)	)	PUNCT
ejpam-4750	266	12	=	=	SYM
ejpam-4750	266	13	0	0	X
ejpam-4750	266	14	.	.	PUNCT
ejpam-4750	266	15	suppose	suppose	VERB
ejpam-4750	266	16	that	that	SCONJ
ejpam-4750	266	17	n	n	NOUN
ejpam-4750	266	18	=	=	SYM
ejpam-4750	266	19	5	5	X
ejpam-4750	266	20	.	.	PUNCT
ejpam-4750	266	21	by	by	ADP
ejpam-4750	266	22	example	example	NOUN
ejpam-4750	266	23	2	2	NUM
ejpam-4750	266	24	,	,	PUNCT
ejpam-4750	266	25	ln(2,2)(p5	ln(2,2)(p5	ADJ
ejpam-4750	266	26	)	)	PUNCT
ejpam-4750	266	27	=	=	SYM
ejpam-4750	266	28	4	4	X
ejpam-4750	266	29	.	.	X
ejpam-4750	266	30	then	then	ADV
ejpam-4750	266	31	q1	q1	VERB
ejpam-4750	266	32	=	=	SYM
ejpam-4750	266	33	{	{	PUNCT
ejpam-4750	266	34	v1	v1	PROPN
ejpam-4750	266	35	,	,	PUNCT
ejpam-4750	266	36	v2	v2	PROPN
ejpam-4750	266	37	,	,	PUNCT
ejpam-4750	266	38	v3	v3	PROPN
ejpam-4750	266	39	,	,	PUNCT
ejpam-4750	266	40	v4	v4	PROPN
ejpam-4750	266	41	}	}	PUNCT
ejpam-4750	266	42	,	,	PUNCT
ejpam-4750	266	43	q2	q2	NOUN
ejpam-4750	266	44	=	=	SYM
ejpam-4750	266	45	{	{	PUNCT
ejpam-4750	266	46	v1	v1	PROPN
ejpam-4750	266	47	,	,	PUNCT
ejpam-4750	266	48	v2	v2	PROPN
ejpam-4750	266	49	,	,	PUNCT
ejpam-4750	266	50	v3	v3	PROPN
ejpam-4750	266	51	,	,	PUNCT
ejpam-4750	266	52	v5	v5	PROPN
ejpam-4750	266	53	}	}	PUNCT
ejpam-4750	266	54	,	,	PUNCT
ejpam-4750	266	55	q3	q3	NOUN
ejpam-4750	266	56	=	=	SYM
ejpam-4750	266	57	{	{	PUNCT
ejpam-4750	266	58	v1	v1	PROPN
ejpam-4750	266	59	,	,	PUNCT
ejpam-4750	266	60	v2	v2	PROPN
ejpam-4750	266	61	,	,	PUNCT
ejpam-4750	266	62	v4	v4	NOUN
ejpam-4750	266	63	,	,	PUNCT
ejpam-4750	266	64	v5	v5	PROPN
ejpam-4750	266	65	}	}	PUNCT
ejpam-4750	266	66	,	,	PUNCT
ejpam-4750	266	67	d.	d.	PROPN
ejpam-4750	266	68	managbanag	managbanag	PROPN
ejpam-4750	266	69	,	,	PUNCT
ejpam-4750	266	70	h.	h.	PROPN
ejpam-4750	266	71	rara	rara	PROPN
ejpam-4750	266	72	/	/	SYM
ejpam-4750	266	73	eur	eur	PROPN
ejpam-4750	266	74	.	.	PUNCT
ejpam-4750	267	1	j.	j.	PROPN
ejpam-4750	267	2	pure	pure	PROPN
ejpam-4750	267	3	appl	appl	PROPN
ejpam-4750	267	4	.	.	PROPN
ejpam-4750	267	5	math	math	PROPN
ejpam-4750	267	6	,	,	PUNCT
ejpam-4750	267	7	16	16	NUM
ejpam-4750	267	8	(	(	PUNCT
ejpam-4750	267	9	2	2	NUM
ejpam-4750	267	10	)	)	PUNCT
ejpam-4750	267	11	(	(	PUNCT
ejpam-4750	267	12	2023	2023	NUM
ejpam-4750	267	13	)	)	PUNCT
ejpam-4750	267	14	,	,	PUNCT
ejpam-4750	267	15	1068	1068	NUM
ejpam-4750	267	16	-	-	SYM
ejpam-4750	267	17	1083	1083	NUM
ejpam-4750	267	18	1076	1076	NUM
ejpam-4750	267	19	q4	q4	PROPN
ejpam-4750	267	20	=	=	PUNCT
ejpam-4750	267	21	{	{	PUNCT
ejpam-4750	267	22	v1	v1	PROPN
ejpam-4750	267	23	,	,	PUNCT
ejpam-4750	267	24	v3	v3	PROPN
ejpam-4750	267	25	,	,	PUNCT
ejpam-4750	267	26	v4	v4	PROPN
ejpam-4750	267	27	,	,	PUNCT
ejpam-4750	267	28	v5	v5	PROPN
ejpam-4750	267	29	}	}	PUNCT
ejpam-4750	267	30	and	and	CCONJ
ejpam-4750	267	31	q5	q5	PROPN
ejpam-4750	267	32	=	=	SYM
ejpam-4750	267	33	{	{	PUNCT
ejpam-4750	267	34	v2	v2	PROPN
ejpam-4750	267	35	,	,	PUNCT
ejpam-4750	267	36	v3	v3	PROPN
ejpam-4750	267	37	,	,	PUNCT
ejpam-4750	267	38	v4	v4	PROPN
ejpam-4750	267	39	,	,	PUNCT
ejpam-4750	267	40	v5	v5	PROPN
ejpam-4750	267	41	}	}	PUNCT
ejpam-4750	267	42	are	be	AUX
ejpam-4750	267	43	the	the	DET
ejpam-4750	267	44	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	267	45	of	of	ADP
ejpam-4750	267	46	p5	p5	NOUN
ejpam-4750	267	47	.	.	PUNCT
ejpam-4750	268	1	clearly	clearly	ADV
ejpam-4750	268	2	,	,	PUNCT
ejpam-4750	268	3	for	for	ADP
ejpam-4750	268	4	every	every	DET
ejpam-4750	268	5	vi	vi	PROPN
ejpam-4750	268	6	∈	∈	PROPN
ejpam-4750	268	7	qj	qj	NOUN
ejpam-4750	268	8	there	there	PRON
ejpam-4750	268	9	exists	exist	VERB
ejpam-4750	268	10	vk	vk	ADP
ejpam-4750	268	11	∈	∈	PROPN
ejpam-4750	268	12	v	v	ADP
ejpam-4750	268	13	(	(	PUNCT
ejpam-4750	268	14	p5	p5	ADJ
ejpam-4750	268	15	)	)	PUNCT
ejpam-4750	268	16	\	\	PROPN
ejpam-4750	269	1	qj	qj	PROPN
ejpam-4750	269	2	such	such	ADJ
ejpam-4750	269	3	that	that	SCONJ
ejpam-4750	269	4	[	[	X
ejpam-4750	269	5	qj	qj	X
ejpam-4750	269	6	\	\	PROPN
ejpam-4750	269	7	{	{	PUNCT
ejpam-4750	269	8	vi	vi	NOUN
ejpam-4750	269	9	}	}	PUNCT
ejpam-4750	269	10	]	]	PUNCT
ejpam-4750	269	11	∪	∪	X
ejpam-4750	269	12	{	{	PUNCT
ejpam-4750	269	13	vk	vk	NOUN
ejpam-4750	269	14	}	}	PUNCT
ejpam-4750	269	15	where	where	SCONJ
ejpam-4750	269	16	i	i	PRON
ejpam-4750	269	17	,	,	PUNCT
ejpam-4750	269	18	j	j	PROPN
ejpam-4750	269	19	,	,	PUNCT
ejpam-4750	269	20	k	k	PROPN
ejpam-4750	269	21	∈	∈	PROPN
ejpam-4750	269	22	{	{	PUNCT
ejpam-4750	269	23	1	1	NUM
ejpam-4750	269	24	,	,	PUNCT
ejpam-4750	269	25	2	2	NUM
ejpam-4750	269	26	,	,	PUNCT
ejpam-4750	269	27	3	3	NUM
ejpam-4750	269	28	,	,	PUNCT
ejpam-4750	269	29	4	4	NUM
ejpam-4750	269	30	,	,	PUNCT
ejpam-4750	269	31	5	5	NUM
ejpam-4750	269	32	}	}	PUNCT
ejpam-4750	269	33	is	be	AUX
ejpam-4750	269	34	an	an	DET
ejpam-4750	269	35	ln(2	ln(2	PROPN
ejpam-4750	269	36	,	,	PUNCT
ejpam-4750	269	37	2)-set	2)-set	NOUN
ejpam-4750	269	38	of	of	ADP
ejpam-4750	269	39	p5	p5	PROPN
ejpam-4750	269	40	.	.	PUNCT
ejpam-4750	270	1	thus	thus	ADV
ejpam-4750	270	2	,	,	PUNCT
ejpam-4750	270	3	by	by	ADP
ejpam-4750	270	4	theorem	theorem	VERB
ejpam-4750	270	5	7	7	NUM
ejpam-4750	270	6	,	,	PUNCT
ejpam-4750	270	7	fln(2,2)(p5	fln(2,2)(p5	NOUN
ejpam-4750	270	8	)	)	PUNCT
ejpam-4750	270	9	=	=	SYM
ejpam-4750	270	10	4	4	X
ejpam-4750	270	11	.	.	PUNCT
ejpam-4750	270	12	suppose	suppose	VERB
ejpam-4750	270	13	that	that	SCONJ
ejpam-4750	270	14	n	n	PROPN
ejpam-4750	270	15	=	=	SYM
ejpam-4750	270	16	6	6	NUM
ejpam-4750	270	17	.	.	PUNCT
ejpam-4750	270	18	by	by	ADP
ejpam-4750	270	19	example	example	NOUN
ejpam-4750	270	20	2	2	NUM
ejpam-4750	270	21	,	,	PUNCT
ejpam-4750	270	22	ln(2,2)(p6	ln(2,2)(p6	PROPN
ejpam-4750	270	23	)	)	PUNCT
ejpam-4750	270	24	=	=	PUNCT
ejpam-4750	271	1	4	4	X
ejpam-4750	271	2	.	.	PUNCT
ejpam-4750	271	3	then	then	ADV
ejpam-4750	271	4	the	the	DET
ejpam-4750	271	5	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	271	6	of	of	ADP
ejpam-4750	271	7	p6	p6	PROPN
ejpam-4750	271	8	are	be	AUX
ejpam-4750	271	9	r1	r1	NOUN
ejpam-4750	271	10	=	=	SYM
ejpam-4750	271	11	{	{	PUNCT
ejpam-4750	271	12	v1	v1	PROPN
ejpam-4750	271	13	,	,	PUNCT
ejpam-4750	271	14	v2	v2	PROPN
ejpam-4750	271	15	,	,	PUNCT
ejpam-4750	271	16	v4	v4	NOUN
ejpam-4750	271	17	,	,	PUNCT
ejpam-4750	271	18	v5	v5	PROPN
ejpam-4750	271	19	}	}	PUNCT
ejpam-4750	271	20	,	,	PUNCT
ejpam-4750	271	21	r2	r2	PROPN
ejpam-4750	271	22	=	=	SYM
ejpam-4750	271	23	{	{	PUNCT
ejpam-4750	271	24	v1	v1	PROPN
ejpam-4750	271	25	,	,	PUNCT
ejpam-4750	271	26	v2	v2	PROPN
ejpam-4750	271	27	,	,	PUNCT
ejpam-4750	271	28	v4	v4	NOUN
ejpam-4750	271	29	,	,	PUNCT
ejpam-4750	271	30	v6	v6	PROPN
ejpam-4750	271	31	}	}	PUNCT
ejpam-4750	271	32	,	,	PUNCT
ejpam-4750	271	33	r3	r3	PROPN
ejpam-4750	271	34	=	=	SYM
ejpam-4750	271	35	{	{	PUNCT
ejpam-4750	271	36	v1	v1	PROPN
ejpam-4750	271	37	,	,	PUNCT
ejpam-4750	271	38	v3	v3	PROPN
ejpam-4750	271	39	,	,	PUNCT
ejpam-4750	271	40	v4	v4	PROPN
ejpam-4750	271	41	,	,	PUNCT
ejpam-4750	271	42	v5	v5	PROPN
ejpam-4750	271	43	}	}	PUNCT
ejpam-4750	271	44	,	,	PUNCT
ejpam-4750	271	45	r4	r4	NOUN
ejpam-4750	271	46	=	=	SYM
ejpam-4750	271	47	{	{	PUNCT
ejpam-4750	271	48	v1	v1	PROPN
ejpam-4750	271	49	,	,	PUNCT
ejpam-4750	271	50	v3	v3	PROPN
ejpam-4750	271	51	,	,	PUNCT
ejpam-4750	271	52	v4	v4	PROPN
ejpam-4750	271	53	,	,	PUNCT
ejpam-4750	271	54	v6	v6	PROPN
ejpam-4750	271	55	}	}	PUNCT
ejpam-4750	271	56	,	,	PUNCT
ejpam-4750	271	57	r5	r5	PROPN
ejpam-4750	271	58	=	=	SYM
ejpam-4750	271	59	{	{	PUNCT
ejpam-4750	271	60	v1	v1	PROPN
ejpam-4750	271	61	,	,	PUNCT
ejpam-4750	271	62	v3	v3	PROPN
ejpam-4750	271	63	,	,	PUNCT
ejpam-4750	271	64	v5	v5	PROPN
ejpam-4750	271	65	,	,	PUNCT
ejpam-4750	271	66	v6	v6	NOUN
ejpam-4750	271	67	}	}	PUNCT
ejpam-4750	271	68	,	,	PUNCT
ejpam-4750	271	69	r6	r6	NOUN
ejpam-4750	271	70	=	=	SYM
ejpam-4750	271	71	{	{	PUNCT
ejpam-4750	271	72	v2	v2	PROPN
ejpam-4750	271	73	,	,	PUNCT
ejpam-4750	271	74	v3	v3	PROPN
ejpam-4750	271	75	,	,	PUNCT
ejpam-4750	271	76	v4	v4	PROPN
ejpam-4750	271	77	,	,	PUNCT
ejpam-4750	271	78	v5	v5	NOUN
ejpam-4750	271	79	}	}	PUNCT
ejpam-4750	271	80	and	and	CCONJ
ejpam-4750	271	81	r7	r7	NOUN
ejpam-4750	271	82	=	=	SYM
ejpam-4750	271	83	{	{	PUNCT
ejpam-4750	271	84	v2	v2	PROPN
ejpam-4750	271	85	,	,	PUNCT
ejpam-4750	271	86	v3	v3	PROPN
ejpam-4750	271	87	,	,	PUNCT
ejpam-4750	271	88	v4	v4	PROPN
ejpam-4750	271	89	,	,	PUNCT
ejpam-4750	271	90	v6	v6	NOUN
ejpam-4750	271	91	}	}	PUNCT
ejpam-4750	271	92	.	.	PUNCT
ejpam-4750	272	1	note	note	VERB
ejpam-4750	272	2	that	that	SCONJ
ejpam-4750	272	3	{	{	PUNCT
ejpam-4750	272	4	v3	v3	PROPN
ejpam-4750	272	5	,	,	PUNCT
ejpam-4750	272	6	v5	v5	PROPN
ejpam-4750	272	7	,	,	PUNCT
ejpam-4750	272	8	v6	v6	NOUN
ejpam-4750	272	9	}	}	PUNCT
ejpam-4750	272	10	is	be	AUX
ejpam-4750	272	11	the	the	DET
ejpam-4750	272	12	forcing	force	VERB
ejpam-4750	272	13	subset	subset	NOUN
ejpam-4750	272	14	of	of	ADP
ejpam-4750	272	15	r5	r5	PROPN
ejpam-4750	272	16	and	and	CCONJ
ejpam-4750	272	17	the	the	DET
ejpam-4750	272	18	minimum	minimum	ADJ
ejpam-4750	272	19	forcing	forcing	NOUN
ejpam-4750	272	20	subset	subset	NOUN
ejpam-4750	272	21	of	of	ADP
ejpam-4750	272	22	p6	p6	PROPN
ejpam-4750	272	23	.	.	PUNCT
ejpam-4750	273	1	thus	thus	ADV
ejpam-4750	273	2	,	,	PUNCT
ejpam-4750	273	3	fln(2,2)(p6	fln(2,2)(p6	NOUN
ejpam-4750	273	4	)	)	PUNCT
ejpam-4750	273	5	=	=	SYM
ejpam-4750	274	1	3	3	X
ejpam-4750	274	2	.	.	PUNCT
ejpam-4750	274	3	now	now	ADV
ejpam-4750	274	4	,	,	PUNCT
ejpam-4750	274	5	suppose	suppose	VERB
ejpam-4750	274	6	that	that	SCONJ
ejpam-4750	274	7	n	n	PROPN
ejpam-4750	274	8	≥	≥	X
ejpam-4750	274	9	7	7	NUM
ejpam-4750	274	10	and	and	CCONJ
ejpam-4750	274	11	n	n	PRON
ejpam-4750	274	12	is	be	AUX
ejpam-4750	274	13	odd	odd	ADJ
ejpam-4750	274	14	.	.	PUNCT
ejpam-4750	275	1	by	by	ADP
ejpam-4750	275	2	example	example	NOUN
ejpam-4750	275	3	2	2	NUM
ejpam-4750	275	4	,	,	PUNCT
ejpam-4750	275	5	ln(2,2)(pn	ln(2,2)(pn	NOUN
ejpam-4750	275	6	)	)	PUNCT
ejpam-4750	275	7	=	=	PUNCT
ejpam-4750	276	1	n+	n+	PUNCT
ejpam-4750	276	2	1	1	NUM
ejpam-4750	276	3	2	2	NUM
ejpam-4750	276	4	.	.	PUNCT
ejpam-4750	277	1	then	then	ADV
ejpam-4750	277	2	s	s	VERB
ejpam-4750	277	3	=	=	SYM
ejpam-4750	277	4	{	{	PUNCT
ejpam-4750	277	5	v1	v1	PROPN
ejpam-4750	277	6	,	,	PUNCT
ejpam-4750	277	7	v3	v3	PROPN
ejpam-4750	277	8	,	,	PUNCT
ejpam-4750	277	9	v5	v5	PROPN
ejpam-4750	277	10	,	,	PUNCT
ejpam-4750	277	11	.	.	PUNCT
ejpam-4750	277	12	.	.	PUNCT
ejpam-4750	277	13	.	.	PUNCT
ejpam-4750	278	1	,	,	PUNCT
ejpam-4750	278	2	vn−2	vn−2	PROPN
ejpam-4750	278	3	,	,	PUNCT
ejpam-4750	278	4	vn	vn	PROPN
ejpam-4750	278	5	}	}	PUNCT
ejpam-4750	278	6	is	be	AUX
ejpam-4750	278	7	the	the	DET
ejpam-4750	278	8	only	only	ADJ
ejpam-4750	278	9	ln(2	ln(2	PROPN
ejpam-4750	278	10	,	,	PUNCT
ejpam-4750	278	11	2)-set	2)-set	NUM
ejpam-4750	278	12	of	of	ADP
ejpam-4750	278	13	pn	pn	PROPN
ejpam-4750	278	14	.	.	PUNCT
ejpam-4750	279	1	thus	thus	ADV
ejpam-4750	279	2	,	,	PUNCT
ejpam-4750	279	3	by	by	ADP
ejpam-4750	279	4	remark	remark	NOUN
ejpam-4750	279	5	8	8	NUM
ejpam-4750	279	6	(	(	PUNCT
ejpam-4750	279	7	i	i	NOUN
ejpam-4750	279	8	)	)	PUNCT
ejpam-4750	279	9	,	,	PUNCT
ejpam-4750	279	10	fln(2,2)(s	fln(2,2)(s	PROPN
ejpam-4750	279	11	)	)	PUNCT
ejpam-4750	279	12	=	=	SYM
ejpam-4750	279	13	0	0	NUM
ejpam-4750	279	14	=	=	SYM
ejpam-4750	279	15	fln(2,2)(pn	fln(2,2)(pn	NOUN
ejpam-4750	279	16	)	)	PUNCT
ejpam-4750	279	17	.	.	PUNCT
ejpam-4750	280	1	next	next	ADV
ejpam-4750	280	2	,	,	PUNCT
ejpam-4750	280	3	suppose	suppose	VERB
ejpam-4750	280	4	that	that	SCONJ
ejpam-4750	280	5	n	n	PROPN
ejpam-4750	280	6	≥	≥	NUM
ejpam-4750	280	7	8	8	NUM
ejpam-4750	280	8	and	and	CCONJ
ejpam-4750	280	9	n	n	PRON
ejpam-4750	280	10	is	be	AUX
ejpam-4750	280	11	even	even	ADV
ejpam-4750	280	12	.	.	PUNCT
ejpam-4750	281	1	then	then	ADV
ejpam-4750	281	2	t1	t1	NOUN
ejpam-4750	281	3	=	=	PUNCT
ejpam-4750	281	4	{	{	PUNCT
ejpam-4750	281	5	v1	v1	PROPN
ejpam-4750	281	6	,	,	PUNCT
ejpam-4750	281	7	v2	v2	PROPN
ejpam-4750	281	8	,	,	PUNCT
ejpam-4750	281	9	v4	v4	NOUN
ejpam-4750	281	10	,	,	PUNCT
ejpam-4750	281	11	.	.	PUNCT
ejpam-4750	281	12	.	.	PUNCT
ejpam-4750	281	13	.	.	PUNCT
ejpam-4750	282	1	,	,	PUNCT
ejpam-4750	282	2	vn−2	vn−2	PROPN
ejpam-4750	282	3	,	,	PUNCT
ejpam-4750	282	4	vn	vn	NOUN
ejpam-4750	282	5	}	}	PUNCT
ejpam-4750	282	6	,	,	PUNCT
ejpam-4750	282	7	t2	t2	NOUN
ejpam-4750	282	8	=	=	SYM
ejpam-4750	282	9	{	{	PUNCT
ejpam-4750	282	10	v1	v1	PROPN
ejpam-4750	282	11	,	,	PUNCT
ejpam-4750	282	12	v3	v3	PROPN
ejpam-4750	282	13	,	,	PUNCT
ejpam-4750	282	14	v4	v4	PROPN
ejpam-4750	282	15	,	,	PUNCT
ejpam-4750	282	16	.	.	PUNCT
ejpam-4750	282	17	.	.	PUNCT
ejpam-4750	282	18	.	.	PUNCT
ejpam-4750	283	1	,	,	PUNCT
ejpam-4750	283	2	vn−2	vn−2	PROPN
ejpam-4750	283	3	,	,	PUNCT
ejpam-4750	283	4	vn	vn	NOUN
ejpam-4750	283	5	}	}	PUNCT
ejpam-4750	283	6	,	,	PUNCT
ejpam-4750	283	7	t3	t3	PROPN
ejpam-4750	283	8	=	=	PUNCT
ejpam-4750	283	9	{	{	PUNCT
ejpam-4750	283	10	v1	v1	PROPN
ejpam-4750	283	11	,	,	PUNCT
ejpam-4750	283	12	v3	v3	PROPN
ejpam-4750	283	13	,	,	PUNCT
ejpam-4750	283	14	v5	v5	PROPN
ejpam-4750	283	15	,	,	PUNCT
ejpam-4750	283	16	.	.	PUNCT
ejpam-4750	283	17	.	.	PUNCT
ejpam-4750	284	1	.	.	PUNCT
ejpam-4750	285	1	,	,	PUNCT
ejpam-4750	285	2	vn−3	vn−3	PROPN
ejpam-4750	285	3	,	,	PUNCT
ejpam-4750	285	4	vn−2	vn−2	PROPN
ejpam-4750	285	5	,	,	PUNCT
ejpam-4750	285	6	vn−1	vn−1	ADJ
ejpam-4750	285	7	}	}	PUNCT
ejpam-4750	285	8	,	,	PUNCT
ejpam-4750	285	9	t4	t4	PROPN
ejpam-4750	285	10	=	=	PROPN
ejpam-4750	285	11	{	{	PUNCT
ejpam-4750	285	12	v1	v1	PROPN
ejpam-4750	285	13	,	,	PUNCT
ejpam-4750	285	14	v3	v3	PROPN
ejpam-4750	285	15	,	,	PUNCT
ejpam-4750	285	16	v5	v5	PROPN
ejpam-4750	285	17	,	,	PUNCT
ejpam-4750	285	18	.	.	PUNCT
ejpam-4750	285	19	.	.	PUNCT
ejpam-4750	286	1	.	.	PUNCT
ejpam-4750	287	1	,	,	PUNCT
ejpam-4750	287	2	vn−3	vn−3	PROPN
ejpam-4750	287	3	,	,	PUNCT
ejpam-4750	287	4	vn−2	vn−2	PROPN
ejpam-4750	287	5	,	,	PUNCT
ejpam-4750	287	6	vn	vn	NOUN
ejpam-4750	287	7	}	}	PUNCT
ejpam-4750	287	8	,	,	PUNCT
ejpam-4750	287	9	t5	t5	PROPN
ejpam-4750	287	10	=	=	SYM
ejpam-4750	287	11	{	{	PUNCT
ejpam-4750	287	12	v1	v1	PROPN
ejpam-4750	287	13	,	,	PUNCT
ejpam-4750	287	14	v3	v3	PROPN
ejpam-4750	287	15	,	,	PUNCT
ejpam-4750	287	16	v5	v5	PROPN
ejpam-4750	287	17	,	,	PUNCT
ejpam-4750	287	18	.	.	PUNCT
ejpam-4750	287	19	.	.	PUNCT
ejpam-4750	288	1	.	.	PUNCT
ejpam-4750	289	1	,	,	PUNCT
ejpam-4750	289	2	vn−1	vn−1	PROPN
ejpam-4750	289	3	,	,	PUNCT
ejpam-4750	289	4	vn	vn	NOUN
ejpam-4750	289	5	}	}	PUNCT
ejpam-4750	289	6	and	and	CCONJ
ejpam-4750	289	7	t6	t6	PROPN
ejpam-4750	289	8	=	=	PUNCT
ejpam-4750	289	9	{	{	PUNCT
ejpam-4750	289	10	v2	v2	PROPN
ejpam-4750	289	11	,	,	PUNCT
ejpam-4750	289	12	v3	v3	PROPN
ejpam-4750	289	13	,	,	PUNCT
ejpam-4750	289	14	v4	v4	PROPN
ejpam-4750	289	15	,	,	PUNCT
ejpam-4750	289	16	.	.	PUNCT
ejpam-4750	289	17	.	.	PUNCT
ejpam-4750	290	1	.	.	PUNCT
ejpam-4750	291	1	,	,	PUNCT
ejpam-4750	291	2	vn−2	vn−2	PROPN
ejpam-4750	291	3	,	,	PUNCT
ejpam-4750	291	4	vn	vn	PROPN
ejpam-4750	291	5	}	}	PUNCT
ejpam-4750	291	6	are	be	AUX
ejpam-4750	291	7	the	the	DET
ejpam-4750	291	8	ln(2	ln(2	PROPN
ejpam-4750	291	9	,	,	PUNCT
ejpam-4750	291	10	2)-sets	2)-sets	NUM
ejpam-4750	291	11	of	of	ADP
ejpam-4750	291	12	pn	pn	PROPN
ejpam-4750	291	13	.	.	PUNCT
ejpam-4750	292	1	hence	hence	ADV
ejpam-4750	292	2	,	,	PUNCT
ejpam-4750	292	3	no	no	DET
ejpam-4750	292	4	vertex	vertex	NOUN
ejpam-4750	292	5	of	of	ADP
ejpam-4750	292	6	pn	pn	PROPN
ejpam-4750	292	7	is	be	AUX
ejpam-4750	292	8	contained	contain	VERB
ejpam-4750	292	9	in	in	ADP
ejpam-4750	292	10	a	a	DET
ejpam-4750	292	11	unique	unique	ADJ
ejpam-4750	292	12	ln(2	ln(2	PROPN
ejpam-4750	292	13	,	,	PUNCT
ejpam-4750	292	14	2)-set	2)-set	NUM
ejpam-4750	292	15	.	.	PUNCT
ejpam-4750	293	1	thus	thus	ADV
ejpam-4750	293	2	,	,	PUNCT
ejpam-4750	293	3	fln(2,2)(pn	fln(2,2)(pn	NOUN
ejpam-4750	293	4	)	)	PUNCT
ejpam-4750	293	5	≥	≥	NOUN
ejpam-4750	294	1	2	2	NUM
ejpam-4750	294	2	.	.	PUNCT
ejpam-4750	294	3	it	it	PRON
ejpam-4750	294	4	can	can	AUX
ejpam-4750	294	5	be	be	AUX
ejpam-4750	294	6	seen	see	VERB
ejpam-4750	294	7	that	that	SCONJ
ejpam-4750	294	8	{	{	PUNCT
ejpam-4750	294	9	v1	v1	NOUN
ejpam-4750	294	10	,	,	PUNCT
ejpam-4750	294	11	v2	v2	PROPN
ejpam-4750	294	12	}	}	PUNCT
ejpam-4750	294	13	,	,	PUNCT
ejpam-4750	294	14	{	{	PUNCT
ejpam-4750	294	15	vn−2	vn−2	PROPN
ejpam-4750	294	16	,	,	PUNCT
ejpam-4750	294	17	vn−1	vn−1	ADJ
ejpam-4750	294	18	}	}	PUNCT
ejpam-4750	294	19	,	,	PUNCT
ejpam-4750	294	20	{	{	PUNCT
ejpam-4750	294	21	vn−1	vn−1	ADJ
ejpam-4750	294	22	,	,	PUNCT
ejpam-4750	294	23	vn	vn	NOUN
ejpam-4750	294	24	}	}	PUNCT
ejpam-4750	294	25	and	and	CCONJ
ejpam-4750	294	26	{	{	PUNCT
ejpam-4750	294	27	v2	v2	PROPN
ejpam-4750	294	28	,	,	PUNCT
ejpam-4750	294	29	v3	v3	PROPN
ejpam-4750	294	30	}	}	PUNCT
ejpam-4750	294	31	are	be	AUX
ejpam-4750	294	32	uniquely	uniquely	ADV
ejpam-4750	294	33	contained	contain	VERB
ejpam-4750	294	34	in	in	ADP
ejpam-4750	294	35	t1	t1	PROPN
ejpam-4750	294	36	,	,	PUNCT
ejpam-4750	294	37	t3	t3	PROPN
ejpam-4750	294	38	,	,	PUNCT
ejpam-4750	294	39	t4	t4	PROPN
ejpam-4750	294	40	and	and	CCONJ
ejpam-4750	294	41	t5	t5	PROPN
ejpam-4750	294	42	,	,	PUNCT
ejpam-4750	294	43	respectively	respectively	ADV
ejpam-4750	294	44	.	.	PUNCT
ejpam-4750	295	1	therefore	therefore	ADV
ejpam-4750	295	2	,	,	PUNCT
ejpam-4750	295	3	fln(2,2)(t1	fln(2,2)(t1	PROPN
ejpam-4750	295	4	)	)	PUNCT
ejpam-4750	295	5	=	=	SYM
ejpam-4750	295	6	fln(2,2)(t3	fln(2,2)(t3	X
ejpam-4750	295	7	)	)	PUNCT
ejpam-4750	295	8	=	=	SYM
ejpam-4750	295	9	fln(2,2)(t4	fln(2,2)(t4	X
ejpam-4750	295	10	)	)	PUNCT
ejpam-4750	295	11	=	=	SYM
ejpam-4750	295	12	fln(2,2)(t5	fln(2,2)(t5	NOUN
ejpam-4750	295	13	)	)	PUNCT
ejpam-4750	295	14	=	=	SYM
ejpam-4750	295	15	2	2	NUM
ejpam-4750	295	16	=	=	SYM
ejpam-4750	295	17	fln(2,2)(pn	fln(2,2)(pn	NOUN
ejpam-4750	295	18	)	)	PUNCT
ejpam-4750	295	19	.	.	PUNCT
ejpam-4750	296	1	proposition	proposition	NOUN
ejpam-4750	296	2	9	9	NUM
ejpam-4750	296	3	.	.	PUNCT
ejpam-4750	297	1	for	for	ADP
ejpam-4750	297	2	any	any	DET
ejpam-4750	297	3	cycle	cycle	NOUN
ejpam-4750	297	4	cn	cn	NOUN
ejpam-4750	297	5	with	with	ADP
ejpam-4750	297	6	n	n	NUM
ejpam-4750	297	7	≥	≥	NUM
ejpam-4750	297	8	4	4	NUM
ejpam-4750	297	9	vertices	vertex	NOUN
ejpam-4750	297	10	,	,	PUNCT
ejpam-4750	297	11	fln(2,2)(cn	fln(2,2)(cn	NOUN
ejpam-4750	297	12	)	)	PUNCT
ejpam-4750	297	13	=	=	PUNCT
ejpam-4750	297	14			NOUN
ejpam-4750	297	15	0	0	NUM
ejpam-4750	297	16	,	,	PUNCT
ejpam-4750	297	17	if	if	SCONJ
ejpam-4750	297	18	n	n	NOUN
ejpam-4750	297	19	=	=	SYM
ejpam-4750	297	20	4	4	NUM
ejpam-4750	297	21	,	,	PUNCT
ejpam-4750	297	22	1	1	NUM
ejpam-4750	297	23	,	,	PUNCT
ejpam-4750	297	24	if	if	SCONJ
ejpam-4750	297	25	n	n	PRON
ejpam-4750	297	26	≥	≥	NOUN
ejpam-4750	297	27	8	8	NUM
ejpam-4750	297	28	is	be	AUX
ejpam-4750	297	29	even	even	ADV
ejpam-4750	297	30	,	,	PUNCT
ejpam-4750	297	31	2	2	NUM
ejpam-4750	297	32	,	,	PUNCT
ejpam-4750	297	33	if	if	SCONJ
ejpam-4750	297	34	n	n	PRON
ejpam-4750	297	35	≥	≥	NOUN
ejpam-4750	297	36	7	7	NUM
ejpam-4750	297	37	is	be	AUX
ejpam-4750	297	38	odd	odd	ADJ
ejpam-4750	297	39	,	,	PUNCT
ejpam-4750	297	40	4	4	NUM
ejpam-4750	297	41	,	,	PUNCT
ejpam-4750	297	42	if	if	SCONJ
ejpam-4750	297	43	n	n	NOUN
ejpam-4750	297	44	=	=	SYM
ejpam-4750	297	45	5	5	NUM
ejpam-4750	297	46	and	and	CCONJ
ejpam-4750	297	47	6	6	NUM
ejpam-4750	297	48	.	.	X
ejpam-4750	298	1	proof	proof	NOUN
ejpam-4750	298	2	:	:	PUNCT
ejpam-4750	298	3	suppose	suppose	VERB
ejpam-4750	298	4	that	that	SCONJ
ejpam-4750	298	5	cn	cn	PROPN
ejpam-4750	298	6	=	=	PUNCT
ejpam-4750	298	7	[	[	X
ejpam-4750	298	8	v1	v1	NOUN
ejpam-4750	298	9	,	,	PUNCT
ejpam-4750	298	10	v2	v2	NOUN
ejpam-4750	298	11	,	,	PUNCT
ejpam-4750	298	12	.	.	PUNCT
ejpam-4750	298	13	.	.	PUNCT
ejpam-4750	298	14	.	.	PUNCT
ejpam-4750	299	1	,	,	PUNCT
ejpam-4750	299	2	vn	vn	X
ejpam-4750	299	3	,	,	PUNCT
ejpam-4750	299	4	v1	v1	PROPN
ejpam-4750	299	5	]	]	PUNCT
ejpam-4750	299	6	.	.	PUNCT
ejpam-4750	300	1	if	if	SCONJ
ejpam-4750	300	2	n	n	NOUN
ejpam-4750	300	3	=	=	SYM
ejpam-4750	300	4	4	4	NUM
ejpam-4750	300	5	,	,	PUNCT
ejpam-4750	300	6	then	then	ADV
ejpam-4750	300	7	v	v	X
ejpam-4750	300	8	(	(	PUNCT
ejpam-4750	300	9	c4	c4	NOUN
ejpam-4750	300	10	)	)	PUNCT
ejpam-4750	300	11	is	be	AUX
ejpam-4750	300	12	the	the	DET
ejpam-4750	300	13	only	only	ADJ
ejpam-4750	300	14	ln(2	ln(2	PROPN
ejpam-4750	300	15	,	,	PUNCT
ejpam-4750	300	16	2)-set	2)-set	NOUN
ejpam-4750	300	17	of	of	ADP
ejpam-4750	300	18	c4	c4	NOUN
ejpam-4750	300	19	.	.	PUNCT
ejpam-4750	301	1	thus	thus	ADV
ejpam-4750	301	2	,	,	PUNCT
ejpam-4750	301	3	by	by	ADP
ejpam-4750	301	4	remark	remark	NOUN
ejpam-4750	301	5	8	8	NUM
ejpam-4750	301	6	(	(	PUNCT
ejpam-4750	301	7	i	i	NOUN
ejpam-4750	301	8	)	)	PUNCT
ejpam-4750	301	9	,	,	PUNCT
ejpam-4750	301	10	fln(2,2)(c4	fln(2,2)(c4	PROPN
ejpam-4750	301	11	)	)	PUNCT
ejpam-4750	301	12	=	=	SYM
ejpam-4750	301	13	0	0	X
ejpam-4750	301	14	.	.	PUNCT
ejpam-4750	302	1	if	if	SCONJ
ejpam-4750	302	2	n	n	NOUN
ejpam-4750	302	3	=	=	SYM
ejpam-4750	302	4	5	5	NUM
ejpam-4750	302	5	,	,	PUNCT
ejpam-4750	302	6	then	then	ADV
ejpam-4750	302	7	the	the	DET
ejpam-4750	302	8	ln(2	ln(2	PROPN
ejpam-4750	302	9	,	,	PUNCT
ejpam-4750	302	10	2)-sets	2)-sets	NUM
ejpam-4750	302	11	are	be	AUX
ejpam-4750	302	12	b1	b1	NOUN
ejpam-4750	302	13	=	=	SYM
ejpam-4750	302	14	{	{	PUNCT
ejpam-4750	302	15	v1	v1	PROPN
ejpam-4750	302	16	,	,	PUNCT
ejpam-4750	302	17	v2	v2	PROPN
ejpam-4750	302	18	,	,	PUNCT
ejpam-4750	302	19	v3	v3	PROPN
ejpam-4750	302	20	,	,	PUNCT
ejpam-4750	302	21	v4	v4	NOUN
ejpam-4750	302	22	}	}	PUNCT
ejpam-4750	302	23	,	,	PUNCT
ejpam-4750	302	24	b2	b2	NOUN
ejpam-4750	302	25	=	=	SYM
ejpam-4750	302	26	{	{	PUNCT
ejpam-4750	302	27	v1	v1	PROPN
ejpam-4750	302	28	,	,	PUNCT
ejpam-4750	302	29	v2	v2	PROPN
ejpam-4750	302	30	,	,	PUNCT
ejpam-4750	302	31	v3	v3	PROPN
ejpam-4750	302	32	,	,	PUNCT
ejpam-4750	302	33	v5	v5	PROPN
ejpam-4750	302	34	}	}	PUNCT
ejpam-4750	302	35	,	,	PUNCT
ejpam-4750	302	36	b3	b3	PROPN
ejpam-4750	302	37	=	=	SYM
ejpam-4750	302	38	{	{	PUNCT
ejpam-4750	302	39	v1	v1	PROPN
ejpam-4750	302	40	,	,	PUNCT
ejpam-4750	302	41	v2	v2	PROPN
ejpam-4750	302	42	,	,	PUNCT
ejpam-4750	302	43	v4	v4	NOUN
ejpam-4750	302	44	,	,	PUNCT
ejpam-4750	302	45	v5	v5	PROPN
ejpam-4750	302	46	}	}	PUNCT
ejpam-4750	302	47	,	,	PUNCT
ejpam-4750	302	48	b4	b4	NOUN
ejpam-4750	302	49	=	=	SYM
ejpam-4750	302	50	{	{	PUNCT
ejpam-4750	302	51	v1	v1	PROPN
ejpam-4750	302	52	,	,	PUNCT
ejpam-4750	302	53	v3	v3	PROPN
ejpam-4750	302	54	,	,	PUNCT
ejpam-4750	302	55	v4	v4	PROPN
ejpam-4750	302	56	,	,	PUNCT
ejpam-4750	302	57	v5	v5	NOUN
ejpam-4750	302	58	}	}	PUNCT
ejpam-4750	302	59	and	and	CCONJ
ejpam-4750	302	60	b5	b5	PROPN
ejpam-4750	302	61	=	=	PUNCT
ejpam-4750	302	62	{	{	PUNCT
ejpam-4750	302	63	v2	v2	PROPN
ejpam-4750	302	64	,	,	PUNCT
ejpam-4750	302	65	v3	v3	PROPN
ejpam-4750	302	66	,	,	PUNCT
ejpam-4750	302	67	v4	v4	PROPN
ejpam-4750	302	68	,	,	PUNCT
ejpam-4750	302	69	v5	v5	PROPN
ejpam-4750	302	70	}	}	PUNCT
ejpam-4750	302	71	.	.	PUNCT
ejpam-4750	303	1	clearly	clearly	ADV
ejpam-4750	303	2	,	,	PUNCT
ejpam-4750	303	3	for	for	ADP
ejpam-4750	303	4	each	each	DET
ejpam-4750	303	5	vi	vi	PROPN
ejpam-4750	303	6	∈	∈	NOUN
ejpam-4750	303	7	bj	bj	NOUN
ejpam-4750	303	8	there	there	ADV
ejpam-4750	303	9	exists	exist	VERB
ejpam-4750	303	10	vk	vk	ADP
ejpam-4750	303	11	∈	∈	PROPN
ejpam-4750	303	12	v	v	PROPN
ejpam-4750	303	13	(	(	PUNCT
ejpam-4750	303	14	c5	c5	PROPN
ejpam-4750	303	15	)	)	PUNCT
ejpam-4750	303	16	\	\	NOUN
ejpam-4750	303	17	bj	bj	ADP
ejpam-4750	303	18	such	such	ADJ
ejpam-4750	303	19	that	that	SCONJ
ejpam-4750	303	20	[	[	X
ejpam-4750	303	21	bj	bj	ADP
ejpam-4750	303	22	\	\	PROPN
ejpam-4750	303	23	{	{	PUNCT
ejpam-4750	303	24	vi	vi	NOUN
ejpam-4750	303	25	}	}	PUNCT
ejpam-4750	303	26	]	]	PUNCT
ejpam-4750	303	27	∪	∪	X
ejpam-4750	303	28	{	{	PUNCT
ejpam-4750	303	29	vk	vk	NOUN
ejpam-4750	303	30	}	}	PUNCT
ejpam-4750	303	31	where	where	SCONJ
ejpam-4750	303	32	i	i	PRON
ejpam-4750	303	33	,	,	PUNCT
ejpam-4750	303	34	j	j	PROPN
ejpam-4750	303	35	,	,	PUNCT
ejpam-4750	303	36	k	k	PROPN
ejpam-4750	303	37	∈	∈	PROPN
ejpam-4750	303	38	{	{	PUNCT
ejpam-4750	303	39	1	1	NUM
ejpam-4750	303	40	,	,	PUNCT
ejpam-4750	303	41	2	2	NUM
ejpam-4750	303	42	,	,	PUNCT
ejpam-4750	303	43	3	3	NUM
ejpam-4750	303	44	,	,	PUNCT
ejpam-4750	303	45	4	4	NUM
ejpam-4750	303	46	,	,	PUNCT
ejpam-4750	303	47	5	5	NUM
ejpam-4750	303	48	}	}	PUNCT
ejpam-4750	303	49	is	be	AUX
ejpam-4750	303	50	an	an	DET
ejpam-4750	303	51	ln(2	ln(2	PROPN
ejpam-4750	303	52	,	,	PUNCT
ejpam-4750	303	53	2)-set	2)-set	NUM
ejpam-4750	303	54	of	of	ADP
ejpam-4750	303	55	c5	c5	PROPN
ejpam-4750	303	56	.	.	PUNCT
ejpam-4750	304	1	thus	thus	ADV
ejpam-4750	304	2	,	,	PUNCT
ejpam-4750	304	3	by	by	ADP
ejpam-4750	304	4	theorem	theorem	ADJ
ejpam-4750	304	5	7	7	NUM
ejpam-4750	304	6	,	,	PUNCT
ejpam-4750	304	7	fln(2,2)(c5	fln(2,2)(c5	NOUN
ejpam-4750	304	8	)	)	PUNCT
ejpam-4750	304	9	=	=	SYM
ejpam-4750	304	10	4	4	X
ejpam-4750	304	11	.	.	PUNCT
ejpam-4750	304	12	suppose	suppose	VERB
ejpam-4750	304	13	that	that	SCONJ
ejpam-4750	304	14	n	n	PROPN
ejpam-4750	304	15	=	=	SYM
ejpam-4750	304	16	6	6	NUM
ejpam-4750	304	17	.	.	PUNCT
ejpam-4750	304	18	then	then	ADV
ejpam-4750	304	19	ei	ei	PROPN
ejpam-4750	304	20	,	,	PUNCT
ejpam-4750	304	21	j	j	PROPN
ejpam-4750	304	22	=	=	SYM
ejpam-4750	304	23	v	v	PROPN
ejpam-4750	304	24	(	(	PUNCT
ejpam-4750	304	25	c6	c6	PROPN
ejpam-4750	304	26	)	)	PUNCT
ejpam-4750	304	27	\	\	PROPN
ejpam-4750	304	28	{	{	PUNCT
ejpam-4750	304	29	vi	vi	PROPN
ejpam-4750	304	30	,	,	PUNCT
ejpam-4750	304	31	vj	vj	ADP
ejpam-4750	304	32	}	}	PUNCT
ejpam-4750	304	33	for	for	ADP
ejpam-4750	304	34	all	all	DET
ejpam-4750	304	35	i	i	PROPN
ejpam-4750	304	36	,	,	PUNCT
ejpam-4750	304	37	j	j	PROPN
ejpam-4750	304	38	∈	∈	PROPN
ejpam-4750	304	39	{	{	PUNCT
ejpam-4750	304	40	1	1	NUM
ejpam-4750	304	41	,	,	PUNCT
ejpam-4750	304	42	2	2	NUM
ejpam-4750	304	43	,	,	PUNCT
ejpam-4750	304	44	.	.	PUNCT
ejpam-4750	304	45	.	.	PUNCT
ejpam-4750	305	1	.	.	PUNCT
ejpam-4750	306	1	,	,	PUNCT
ejpam-4750	306	2	6	6	X
ejpam-4750	306	3	}	}	PUNCT
ejpam-4750	306	4	are	be	AUX
ejpam-4750	306	5	the	the	DET
ejpam-4750	306	6	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	306	7	of	of	ADP
ejpam-4750	306	8	c6	c6	PROPN
ejpam-4750	306	9	.	.	PUNCT
ejpam-4750	307	1	thus	thus	ADV
ejpam-4750	307	2	,	,	PUNCT
ejpam-4750	307	3	for	for	ADP
ejpam-4750	307	4	every	every	DET
ejpam-4750	307	5	vk	vk	NOUN
ejpam-4750	307	6	∈	∈	PROPN
ejpam-4750	307	7	ei	ei	PROPN
ejpam-4750	307	8	,	,	PUNCT
ejpam-4750	307	9	j	j	PROPN
ejpam-4750	307	10	where	where	SCONJ
ejpam-4750	307	11	k	k	PROPN
ejpam-4750	307	12	̸=	̸=	PROPN
ejpam-4750	307	13	i	i	PROPN
ejpam-4750	307	14	,	,	PUNCT
ejpam-4750	307	15	j	j	PROPN
ejpam-4750	307	16	there	there	PRON
ejpam-4750	307	17	exists	exist	VERB
ejpam-4750	307	18	vi	vi	PROPN
ejpam-4750	307	19	∈	∈	PROPN
ejpam-4750	307	20	v	v	NOUN
ejpam-4750	307	21	(	(	PUNCT
ejpam-4750	307	22	c6	c6	PROPN
ejpam-4750	307	23	)	)	PUNCT
ejpam-4750	307	24	\ei	\ei	PROPN
ejpam-4750	307	25	,	,	PUNCT
ejpam-4750	307	26	j	j	PROPN
ejpam-4750	307	27	such	such	ADJ
ejpam-4750	307	28	that	that	SCONJ
ejpam-4750	307	29	[	[	X
ejpam-4750	307	30	ei	ei	X
ejpam-4750	307	31	,	,	PUNCT
ejpam-4750	307	32	j	j	PROPN
ejpam-4750	307	33	\	\	PROPN
ejpam-4750	307	34	{	{	PUNCT
ejpam-4750	307	35	vk	vk	PROPN
ejpam-4750	307	36	}	}	PUNCT
ejpam-4750	307	37	]	]	PUNCT
ejpam-4750	307	38	∪	∪	X
ejpam-4750	307	39	{	{	PUNCT
ejpam-4750	307	40	vi	vi	NOUN
ejpam-4750	307	41	}	}	PUNCT
ejpam-4750	307	42	=	=	SYM
ejpam-4750	307	43	ej	ej	PROPN
ejpam-4750	307	44	,	,	PUNCT
ejpam-4750	307	45	k	k	PROPN
ejpam-4750	307	46	is	be	AUX
ejpam-4750	307	47	an	an	DET
ejpam-4750	307	48	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	307	49	of	of	ADP
ejpam-4750	307	50	c6	c6	PROPN
ejpam-4750	307	51	.	.	PUNCT
ejpam-4750	308	1	hence	hence	ADV
ejpam-4750	308	2	,	,	PUNCT
ejpam-4750	308	3	d.	d.	PROPN
ejpam-4750	308	4	managbanag	managbanag	PROPN
ejpam-4750	308	5	,	,	PUNCT
ejpam-4750	308	6	h.	h.	PROPN
ejpam-4750	308	7	rara	rara	PROPN
ejpam-4750	308	8	/	/	SYM
ejpam-4750	308	9	eur	eur	PROPN
ejpam-4750	308	10	.	.	PUNCT
ejpam-4750	309	1	j.	j.	PROPN
ejpam-4750	309	2	pure	pure	PROPN
ejpam-4750	309	3	appl	appl	PROPN
ejpam-4750	309	4	.	.	PROPN
ejpam-4750	309	5	math	math	PROPN
ejpam-4750	309	6	,	,	PUNCT
ejpam-4750	309	7	16	16	NUM
ejpam-4750	309	8	(	(	PUNCT
ejpam-4750	309	9	2	2	NUM
ejpam-4750	309	10	)	)	PUNCT
ejpam-4750	309	11	(	(	PUNCT
ejpam-4750	309	12	2023	2023	NUM
ejpam-4750	309	13	)	)	PUNCT
ejpam-4750	309	14	,	,	PUNCT
ejpam-4750	309	15	1068	1068	NUM
ejpam-4750	309	16	-	-	SYM
ejpam-4750	309	17	1083	1083	NUM
ejpam-4750	309	18	1077	1077	NUM
ejpam-4750	309	19	by	by	ADP
ejpam-4750	309	20	theorem	theorem	ADJ
ejpam-4750	309	21	7	7	NUM
ejpam-4750	309	22	,	,	PUNCT
ejpam-4750	309	23	fln(2,2)(c6	fln(2,2)(c6	NOUN
ejpam-4750	309	24	)	)	PUNCT
ejpam-4750	309	25	=	=	SYM
ejpam-4750	310	1	4	4	X
ejpam-4750	310	2	.	.	PUNCT
ejpam-4750	311	1	next	next	ADV
ejpam-4750	311	2	,	,	PUNCT
ejpam-4750	311	3	suppose	suppose	VERB
ejpam-4750	311	4	that	that	SCONJ
ejpam-4750	311	5	n	n	PROPN
ejpam-4750	311	6	≥	≥	X
ejpam-4750	311	7	7	7	NUM
ejpam-4750	311	8	and	and	CCONJ
ejpam-4750	311	9	n	n	PRON
ejpam-4750	311	10	is	be	AUX
ejpam-4750	311	11	odd	odd	ADJ
ejpam-4750	311	12	.	.	PUNCT
ejpam-4750	312	1	by	by	ADP
ejpam-4750	312	2	examples	example	NOUN
ejpam-4750	312	3	1	1	NUM
ejpam-4750	312	4	and	and	CCONJ
ejpam-4750	312	5	2	2	NUM
ejpam-4750	312	6	,	,	PUNCT
ejpam-4750	312	7	ln2(cn	ln2(cn	ADJ
ejpam-4750	312	8	)	)	PUNCT
ejpam-4750	312	9	=	=	PUNCT
ejpam-4750	312	10	n+	n+	PUNCT
ejpam-4750	312	11	1	1	NUM
ejpam-4750	312	12	2	2	NUM
ejpam-4750	312	13	=	=	SYM
ejpam-4750	312	14	ln(2,2)(cn	ln(2,2)(cn	NOUN
ejpam-4750	312	15	)	)	PUNCT
ejpam-4750	312	16	.	.	PUNCT
ejpam-4750	313	1	by	by	ADP
ejpam-4750	313	2	similar	similar	ADJ
ejpam-4750	313	3	argument	argument	NOUN
ejpam-4750	313	4	as	as	ADP
ejpam-4750	313	5	in	in	ADP
ejpam-4750	313	6	the	the	DET
ejpam-4750	313	7	proof	proof	NOUN
ejpam-4750	313	8	of	of	ADP
ejpam-4750	313	9	proposition	proposition	NOUN
ejpam-4750	313	10	7	7	NUM
ejpam-4750	313	11	,	,	PUNCT
ejpam-4750	313	12	fln(2,2)(cn	fln(2,2)(cn	NOUN
ejpam-4750	313	13	)	)	PUNCT
ejpam-4750	313	14	=	=	SYM
ejpam-4750	313	15	2	2	X
ejpam-4750	313	16	.	.	PUNCT
ejpam-4750	313	17	now	now	ADV
ejpam-4750	313	18	,	,	PUNCT
ejpam-4750	313	19	suppose	suppose	VERB
ejpam-4750	313	20	that	that	SCONJ
ejpam-4750	313	21	n	n	PROPN
ejpam-4750	313	22	≥	≥	NUM
ejpam-4750	313	23	8	8	NUM
ejpam-4750	313	24	and	and	CCONJ
ejpam-4750	313	25	n	n	PRON
ejpam-4750	313	26	is	be	AUX
ejpam-4750	313	27	even	even	ADV
ejpam-4750	313	28	.	.	PUNCT
ejpam-4750	314	1	by	by	ADP
ejpam-4750	314	2	example	example	NOUN
ejpam-4750	314	3	2	2	NUM
ejpam-4750	314	4	,	,	PUNCT
ejpam-4750	314	5	ln(2,2)(cn	ln(2,2)(cn	NOUN
ejpam-4750	314	6	)	)	PUNCT
ejpam-4750	314	7	=	=	SYM
ejpam-4750	315	1	n	n	DET
ejpam-4750	315	2	2	2	NUM
ejpam-4750	315	3	.	.	PUNCT
ejpam-4750	316	1	then	then	ADV
ejpam-4750	316	2	f1	f1	PROPN
ejpam-4750	316	3	=	=	SYM
ejpam-4750	316	4	{	{	PUNCT
ejpam-4750	316	5	v1	v1	PROPN
ejpam-4750	316	6	,	,	PUNCT
ejpam-4750	316	7	v3	v3	PROPN
ejpam-4750	316	8	,	,	PUNCT
ejpam-4750	316	9	v5	v5	PROPN
ejpam-4750	316	10	,	,	PUNCT
ejpam-4750	316	11	.	.	PUNCT
ejpam-4750	316	12	.	.	PUNCT
ejpam-4750	316	13	.	.	PUNCT
ejpam-4750	317	1	,	,	PUNCT
ejpam-4750	317	2	vn−3	vn−3	PROPN
ejpam-4750	317	3	,	,	PUNCT
ejpam-4750	317	4	vn−1	vn−1	ADJ
ejpam-4750	317	5	}	}	PUNCT
ejpam-4750	317	6	and	and	CCONJ
ejpam-4750	317	7	f2	f2	PROPN
ejpam-4750	317	8	=	=	SYM
ejpam-4750	317	9	{	{	PUNCT
ejpam-4750	317	10	v2	v2	PROPN
ejpam-4750	317	11	,	,	PUNCT
ejpam-4750	317	12	v4	v4	PROPN
ejpam-4750	317	13	,	,	PUNCT
ejpam-4750	317	14	v6	v6	NOUN
ejpam-4750	317	15	,	,	PUNCT
ejpam-4750	317	16	.	.	PUNCT
ejpam-4750	317	17	.	.	PUNCT
ejpam-4750	318	1	.	.	PUNCT
ejpam-4750	319	1	,	,	PUNCT
ejpam-4750	319	2	vn−2	vn−2	PROPN
ejpam-4750	319	3	,	,	PUNCT
ejpam-4750	319	4	vn	vn	PROPN
ejpam-4750	319	5	}	}	PUNCT
ejpam-4750	319	6	are	be	AUX
ejpam-4750	319	7	the	the	DET
ejpam-4750	319	8	only	only	ADJ
ejpam-4750	319	9	ln(2	ln(2	PROPN
ejpam-4750	319	10	,	,	PUNCT
ejpam-4750	319	11	2)-sets	2)-sets	NUM
ejpam-4750	319	12	of	of	ADP
ejpam-4750	319	13	cn	cn	PROPN
ejpam-4750	319	14	with	with	ADP
ejpam-4750	319	15	v1	v1	PROPN
ejpam-4750	319	16	∈	∈	PROPN
ejpam-4750	319	17	f1	f1	NOUN
ejpam-4750	319	18	and	and	CCONJ
ejpam-4750	319	19	v1	v1	NOUN
ejpam-4750	319	20	/∈	/∈	PUNCT
ejpam-4750	320	1	f2	f2	PROPN
ejpam-4750	320	2	.	.	PUNCT
ejpam-4750	321	1	thus	thus	ADV
ejpam-4750	321	2	,	,	PUNCT
ejpam-4750	321	3	by	by	ADP
ejpam-4750	321	4	remark	remark	NOUN
ejpam-4750	321	5	8(ii	8(ii	PROPN
ejpam-4750	321	6	)	)	PUNCT
ejpam-4750	321	7	,	,	PUNCT
ejpam-4750	321	8	fln(2,2)(f1	fln(2,2)(f1	PROPN
ejpam-4750	321	9	)	)	PUNCT
ejpam-4750	321	10	=	=	SYM
ejpam-4750	321	11	1	1	NUM
ejpam-4750	321	12	=	=	SYM
ejpam-4750	321	13	fln(2,2)(cn	fln(2,2)(cn	NOUN
ejpam-4750	321	14	)	)	PUNCT
ejpam-4750	321	15	.	.	PUNCT
ejpam-4750	322	1	6	6	X
ejpam-4750	322	2	.	.	X
ejpam-4750	322	3	forcing	force	VERB
ejpam-4750	322	4	2	2	NUM
ejpam-4750	322	5	-	-	PUNCT
ejpam-4750	322	6	metric	metric	ADJ
ejpam-4750	322	7	dimension	dimension	NOUN
ejpam-4750	322	8	in	in	ADP
ejpam-4750	322	9	the	the	DET
ejpam-4750	322	10	join	join	NOUN
ejpam-4750	322	11	of	of	ADP
ejpam-4750	322	12	graphs	graph	NOUN
ejpam-4750	322	13	the	the	DET
ejpam-4750	322	14	join	join	NOUN
ejpam-4750	322	15	of	of	ADP
ejpam-4750	322	16	two	two	NUM
ejpam-4750	322	17	graphs	graph	NOUN
ejpam-4750	322	18	g	g	NOUN
ejpam-4750	322	19	and	and	CCONJ
ejpam-4750	322	20	h	h	NOUN
ejpam-4750	322	21	,	,	PUNCT
ejpam-4750	322	22	denoted	denote	VERB
ejpam-4750	322	23	by	by	ADP
ejpam-4750	322	24	g	g	PROPN
ejpam-4750	322	25	+	+	PROPN
ejpam-4750	322	26	h	h	NOUN
ejpam-4750	322	27	,	,	PUNCT
ejpam-4750	322	28	is	be	AUX
ejpam-4750	322	29	the	the	DET
ejpam-4750	322	30	graph	graph	NOUN
ejpam-4750	322	31	with	with	ADP
ejpam-4750	322	32	vertex	vertex	NOUN
ejpam-4750	322	33	-	-	PUNCT
ejpam-4750	322	34	set	set	VERB
ejpam-4750	322	35	v	v	NOUN
ejpam-4750	322	36	(	(	PUNCT
ejpam-4750	322	37	g	g	PROPN
ejpam-4750	322	38	+	+	NOUN
ejpam-4750	322	39	h	h	NOUN
ejpam-4750	322	40	)	)	PUNCT
ejpam-4750	322	41	=	=	NOUN
ejpam-4750	322	42	v	v	X
ejpam-4750	322	43	(	(	PUNCT
ejpam-4750	322	44	g)∪̇v	g)∪̇v	X
ejpam-4750	322	45	(	(	PUNCT
ejpam-4750	322	46	h	h	NOUN
ejpam-4750	322	47	)	)	PUNCT
ejpam-4750	322	48	and	and	CCONJ
ejpam-4750	322	49	edge	edge	NOUN
ejpam-4750	322	50	-	-	PUNCT
ejpam-4750	322	51	set	set	VERB
ejpam-4750	322	52	e(g	e(g	NOUN
ejpam-4750	322	53	+	+	CCONJ
ejpam-4750	322	54	h	h	NOUN
ejpam-4750	322	55	)	)	PUNCT
ejpam-4750	322	56	=	=	SYM
ejpam-4750	322	57	e(g)∪̇e(h	e(g)∪̇e(h	PROPN
ejpam-4750	322	58	)	)	PUNCT
ejpam-4750	322	59	∪	∪	NOUN
ejpam-4750	322	60	{	{	PUNCT
ejpam-4750	322	61	uv	uv	NOUN
ejpam-4750	322	62	:	:	PUNCT
ejpam-4750	322	63	u	u	PROPN
ejpam-4750	322	64	∈	∈	PROPN
ejpam-4750	322	65	v	v	ADP
ejpam-4750	322	66	(	(	PUNCT
ejpam-4750	322	67	g	g	NOUN
ejpam-4750	322	68	)	)	PUNCT
ejpam-4750	322	69	,	,	PUNCT
ejpam-4750	322	70	v	v	X
ejpam-4750	322	71	∈	∈	PROPN
ejpam-4750	322	72	v	v	NOUN
ejpam-4750	322	73	(	(	PUNCT
ejpam-4750	322	74	h	h	NOUN
ejpam-4750	322	75	)	)	PUNCT
ejpam-4750	322	76	}	}	PUNCT
ejpam-4750	322	77	.	.	PUNCT
ejpam-4750	323	1	in	in	ADP
ejpam-4750	323	2	view	view	NOUN
ejpam-4750	323	3	of	of	ADP
ejpam-4750	323	4	theorem	theorem	NOUN
ejpam-4750	323	5	2	2	NUM
ejpam-4750	323	6	,	,	PUNCT
ejpam-4750	323	7	we	we	PRON
ejpam-4750	323	8	have	have	VERB
ejpam-4750	323	9	the	the	DET
ejpam-4750	323	10	following	follow	VERB
ejpam-4750	323	11	theorem	theorem	VERB
ejpam-4750	323	12	.	.	PUNCT
ejpam-4750	323	13	theorem	theorem	NOUN
ejpam-4750	323	14	8	8	NUM
ejpam-4750	323	15	.	.	PUNCT
ejpam-4750	324	1	let	let	VERB
ejpam-4750	324	2	g	g	PRON
ejpam-4750	324	3	be	be	AUX
ejpam-4750	324	4	a	a	DET
ejpam-4750	324	5	connected	connected	ADJ
ejpam-4750	324	6	graph	graph	NOUN
ejpam-4750	324	7	with	with	ADP
ejpam-4750	324	8	|v	|v	PROPN
ejpam-4750	324	9	(	(	PUNCT
ejpam-4750	324	10	g)|	g)|	X
ejpam-4750	324	11	≥	≥	NOUN
ejpam-4750	324	12	3	3	NUM
ejpam-4750	324	13	and	and	CCONJ
ejpam-4750	324	14	let	let	VERB
ejpam-4750	324	15	k1	k1	NOUN
ejpam-4750	324	16	=	=	SYM
ejpam-4750	324	17	⟨v⟩.	⟨v⟩.	PROPN
ejpam-4750	324	18	then	then	ADV
ejpam-4750	324	19	a	a	DET
ejpam-4750	324	20	proper	proper	ADJ
ejpam-4750	324	21	subset	subset	NOUN
ejpam-4750	324	22	s	s	NOUN
ejpam-4750	324	23	of	of	ADP
ejpam-4750	324	24	v	v	NOUN
ejpam-4750	324	25	(	(	PUNCT
ejpam-4750	324	26	k1	k1	NOUN
ejpam-4750	324	27	+	+	CCONJ
ejpam-4750	324	28	g	g	NOUN
ejpam-4750	324	29	)	)	PUNCT
ejpam-4750	324	30	is	be	AUX
ejpam-4750	324	31	a	a	DET
ejpam-4750	324	32	2	2	NUM
ejpam-4750	324	33	-	-	PUNCT
ejpam-4750	324	34	metric	metric	ADJ
ejpam-4750	324	35	basis	basis	NOUN
ejpam-4750	324	36	of	of	ADP
ejpam-4750	324	37	k1	k1	NOUN
ejpam-4750	324	38	+	+	CCONJ
ejpam-4750	324	39	g	g	NOUN
ejpam-4750	324	40	if	if	SCONJ
ejpam-4750	325	1	and	and	CCONJ
ejpam-4750	325	2	only	only	ADV
ejpam-4750	325	3	if	if	SCONJ
ejpam-4750	325	4	one	one	NUM
ejpam-4750	325	5	of	of	ADP
ejpam-4750	325	6	the	the	DET
ejpam-4750	325	7	following	follow	VERB
ejpam-4750	325	8	holds	hold	VERB
ejpam-4750	325	9	:	:	PUNCT
ejpam-4750	325	10	(	(	PUNCT
ejpam-4750	325	11	i	i	NOUN
ejpam-4750	325	12	)	)	PUNCT
ejpam-4750	325	13	s	s	VERB
ejpam-4750	325	14	is	be	AUX
ejpam-4750	325	15	an	an	DET
ejpam-4750	325	16	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	325	17	of	of	ADP
ejpam-4750	325	18	g	g	NOUN
ejpam-4750	325	19	,	,	PUNCT
ejpam-4750	325	20	(	(	PUNCT
ejpam-4750	325	21	ii	ii	NOUN
ejpam-4750	325	22	)	)	PUNCT
ejpam-4750	325	23	s	s	PART
ejpam-4750	326	1	=	=	VERB
ejpam-4750	326	2	{	{	PUNCT
ejpam-4750	326	3	v	v	NOUN
ejpam-4750	326	4	}	}	PUNCT
ejpam-4750	326	5	∪	∪	NOUN
ejpam-4750	326	6	t	t	PROPN
ejpam-4750	326	7	,	,	PUNCT
ejpam-4750	326	8	where	where	SCONJ
ejpam-4750	326	9	t	t	PROPN
ejpam-4750	326	10	is	be	AUX
ejpam-4750	326	11	an	an	DET
ejpam-4750	326	12	ln(2,1)-set	ln(2,1)-set	NOUN
ejpam-4750	326	13	of	of	ADP
ejpam-4750	326	14	g.	g.	PROPN
ejpam-4750	326	15	theorem	theorem	VERB
ejpam-4750	326	16	9	9	NUM
ejpam-4750	326	17	.	.	PUNCT
ejpam-4750	327	1	let	let	VERB
ejpam-4750	327	2	k1	k1	NOUN
ejpam-4750	327	3	=	=	PROPN
ejpam-4750	327	4	⟨v⟩	⟨v⟩	PROPN
ejpam-4750	327	5	and	and	CCONJ
ejpam-4750	327	6	g	g	ADP
ejpam-4750	327	7	a	a	DET
ejpam-4750	327	8	connected	connected	ADJ
ejpam-4750	327	9	graph	graph	NOUN
ejpam-4750	327	10	with	with	ADP
ejpam-4750	327	11	|v	|v	PROPN
ejpam-4750	327	12	(	(	PUNCT
ejpam-4750	327	13	g)|	g)|	X
ejpam-4750	327	14	≥	≥	NUM
ejpam-4750	327	15	3	3	NUM
ejpam-4750	327	16	and	and	CCONJ
ejpam-4750	327	17	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	327	18	)	)	PUNCT
ejpam-4750	327	19	=	=	PUNCT
ejpam-4750	328	1	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	328	2	)	)	PUNCT
ejpam-4750	328	3	.	.	PUNCT
ejpam-4750	329	1	then	then	ADV
ejpam-4750	329	2	fdim2(k1	fdim2(k1	PROPN
ejpam-4750	330	1	+	+	NOUN
ejpam-4750	330	2	g	g	NOUN
ejpam-4750	330	3	)	)	PUNCT
ejpam-4750	330	4	=	=	SYM
ejpam-4750	330	5	®	®	NOUN
ejpam-4750	330	6	0	0	NUM
ejpam-4750	330	7	,	,	PUNCT
ejpam-4750	330	8	if	if	SCONJ
ejpam-4750	330	9	g	g	PROPN
ejpam-4750	330	10	has	have	VERB
ejpam-4750	330	11	a	a	DET
ejpam-4750	330	12	unique	unique	ADJ
ejpam-4750	330	13	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	330	14	,	,	PUNCT
ejpam-4750	330	15	f	f	PROPN
ejpam-4750	330	16	ln(2,2)(g	ln(2,2)(g	PROPN
ejpam-4750	330	17	)	)	PUNCT
ejpam-4750	330	18	,	,	PUNCT
ejpam-4750	330	19	if	if	SCONJ
ejpam-4750	330	20	g	g	PROPN
ejpam-4750	330	21	has	have	VERB
ejpam-4750	330	22	no	no	DET
ejpam-4750	330	23	unique	unique	ADJ
ejpam-4750	330	24	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	330	25	.	.	PUNCT
ejpam-4750	331	1	proof	proof	NOUN
ejpam-4750	331	2	:	:	PUNCT
ejpam-4750	331	3	suppose	suppose	VERB
ejpam-4750	331	4	that	that	SCONJ
ejpam-4750	331	5	g	g	PROPN
ejpam-4750	331	6	has	have	VERB
ejpam-4750	331	7	a	a	DET
ejpam-4750	331	8	unique	unique	ADJ
ejpam-4750	331	9	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	331	10	,	,	PUNCT
ejpam-4750	331	11	say	say	VERB
ejpam-4750	331	12	t	t	NOUN
ejpam-4750	331	13	.	.	PUNCT
ejpam-4750	332	1	since	since	SCONJ
ejpam-4750	332	2	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	332	3	)	)	PUNCT
ejpam-4750	332	4	=	=	PUNCT
ejpam-4750	332	5	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	332	6	)	)	PUNCT
ejpam-4750	332	7	,	,	PUNCT
ejpam-4750	332	8	by	by	ADP
ejpam-4750	332	9	theorem	theorem	NOUN
ejpam-4750	332	10	8	8	NUM
ejpam-4750	332	11	,	,	PUNCT
ejpam-4750	332	12	t	t	PROPN
ejpam-4750	332	13	is	be	AUX
ejpam-4750	332	14	a	a	DET
ejpam-4750	332	15	unique	unique	ADJ
ejpam-4750	332	16	2	2	NUM
ejpam-4750	332	17	-	-	PUNCT
ejpam-4750	332	18	metric	metric	ADJ
ejpam-4750	332	19	basis	basis	NOUN
ejpam-4750	332	20	for	for	ADP
ejpam-4750	332	21	k1+g	k1+g	NOUN
ejpam-4750	332	22	.	.	PUNCT
ejpam-4750	333	1	by	by	ADP
ejpam-4750	333	2	remark	remark	NOUN
ejpam-4750	333	3	4	4	NUM
ejpam-4750	333	4	(	(	PUNCT
ejpam-4750	333	5	i	i	NOUN
ejpam-4750	333	6	)	)	PUNCT
ejpam-4750	333	7	,	,	PUNCT
ejpam-4750	333	8	fdim2(k1+g	fdim2(k1+g	NOUN
ejpam-4750	333	9	)	)	PUNCT
ejpam-4750	333	10	=	=	SYM
ejpam-4750	333	11	0	0	X
ejpam-4750	333	12	.	.	PUNCT
ejpam-4750	334	1	now	now	ADV
ejpam-4750	334	2	,	,	PUNCT
ejpam-4750	334	3	suppose	suppose	VERB
ejpam-4750	334	4	that	that	SCONJ
ejpam-4750	334	5	g	g	PROPN
ejpam-4750	334	6	has	have	VERB
ejpam-4750	334	7	at	at	ADV
ejpam-4750	334	8	least	least	ADV
ejpam-4750	334	9	two	two	NUM
ejpam-4750	334	10	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	334	11	.	.	PUNCT
ejpam-4750	335	1	let	let	VERB
ejpam-4750	335	2	a	a	DET
ejpam-4750	335	3	be	be	AUX
ejpam-4750	335	4	an	an	DET
ejpam-4750	335	5	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	335	6	of	of	ADP
ejpam-4750	335	7	g	g	NOUN
ejpam-4750	335	8	and	and	CCONJ
ejpam-4750	335	9	let	let	VERB
ejpam-4750	335	10	f	f	PRON
ejpam-4750	335	11	be	be	AUX
ejpam-4750	335	12	a	a	DET
ejpam-4750	335	13	forcing	forcing	NOUN
ejpam-4750	335	14	subset	subset	NOUN
ejpam-4750	335	15	for	for	ADP
ejpam-4750	335	16	a	a	DET
ejpam-4750	335	17	such	such	ADJ
ejpam-4750	335	18	that	that	DET
ejpam-4750	335	19	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	335	20	)	)	PUNCT
ejpam-4750	335	21	=	=	SYM
ejpam-4750	335	22	fln(2,2)(a	fln(2,2)(a	PROPN
ejpam-4750	335	23	)	)	PUNCT
ejpam-4750	335	24	=	=	SYM
ejpam-4750	335	25	|f	|f	PROPN
ejpam-4750	336	1	|	|	NOUN
ejpam-4750	336	2	.	.	PUNCT
ejpam-4750	337	1	by	by	ADP
ejpam-4750	337	2	theorem	theorem	NOUN
ejpam-4750	337	3	8	8	NUM
ejpam-4750	337	4	,	,	PUNCT
ejpam-4750	337	5	a	a	PRON
ejpam-4750	337	6	is	be	AUX
ejpam-4750	337	7	a	a	DET
ejpam-4750	337	8	2	2	NUM
ejpam-4750	337	9	-	-	PUNCT
ejpam-4750	337	10	metric	metric	ADJ
ejpam-4750	337	11	basis	basis	NOUN
ejpam-4750	337	12	of	of	ADP
ejpam-4750	337	13	k1	k1	PROPN
ejpam-4750	337	14	+	+	PROPN
ejpam-4750	337	15	g.	g.	PROPN
ejpam-4750	337	16	thus	thus	ADV
ejpam-4750	337	17	,	,	PUNCT
ejpam-4750	337	18	fdim2(k1	fdim2(k1	PROPN
ejpam-4750	338	1	+	+	NOUN
ejpam-4750	338	2	g	g	NOUN
ejpam-4750	338	3	)	)	PUNCT
ejpam-4750	338	4	≤	≤	NOUN
ejpam-4750	338	5	fln(2,2)(a	fln(2,2)(a	NUM
ejpam-4750	338	6	)	)	PUNCT
ejpam-4750	338	7	=	=	SYM
ejpam-4750	338	8	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	338	9	)	)	PUNCT
ejpam-4750	338	10	.	.	PUNCT
ejpam-4750	339	1	let	let	VERB
ejpam-4750	339	2	s0	s0	PROPN
ejpam-4750	339	3	be	be	AUX
ejpam-4750	339	4	a	a	DET
ejpam-4750	339	5	2	2	NUM
ejpam-4750	339	6	-	-	PUNCT
ejpam-4750	339	7	metric	metric	ADJ
ejpam-4750	339	8	basis	basis	NOUN
ejpam-4750	339	9	for	for	ADP
ejpam-4750	339	10	k1	k1	NOUN
ejpam-4750	339	11	+	+	CCONJ
ejpam-4750	339	12	g	g	NOUN
ejpam-4750	339	13	such	such	ADJ
ejpam-4750	339	14	that	that	DET
ejpam-4750	339	15	fdim2(k1	fdim2(k1	PROPN
ejpam-4750	340	1	+	+	CCONJ
ejpam-4750	340	2	g	g	NOUN
ejpam-4750	340	3	)	)	PUNCT
ejpam-4750	340	4	=	=	SYM
ejpam-4750	340	5	fdim2(s0	fdim2(s0	PROPN
ejpam-4750	340	6	)	)	PUNCT
ejpam-4750	340	7	.	.	PUNCT
ejpam-4750	341	1	by	by	ADP
ejpam-4750	341	2	theorem	theorem	NOUN
ejpam-4750	341	3	8	8	NUM
ejpam-4750	341	4	,	,	PUNCT
ejpam-4750	341	5	s0	s0	PROPN
ejpam-4750	341	6	is	be	AUX
ejpam-4750	341	7	an	an	DET
ejpam-4750	341	8	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	341	9	of	of	ADP
ejpam-4750	341	10	g.	g.	PROPN
ejpam-4750	341	11	let	let	VERB
ejpam-4750	341	12	f0	f0	PROPN
ejpam-4750	341	13	be	be	AUX
ejpam-4750	341	14	a	a	DET
ejpam-4750	341	15	forcing	forcing	NOUN
ejpam-4750	341	16	subset	subset	NOUN
ejpam-4750	341	17	for	for	ADP
ejpam-4750	341	18	s0	s0	PROPN
ejpam-4750	341	19	with	with	ADP
ejpam-4750	341	20	|f0|	|f0|	NOUN
ejpam-4750	341	21	=	=	SYM
ejpam-4750	341	22	fdim2(s0	fdim2(s0	PROPN
ejpam-4750	341	23	)	)	PUNCT
ejpam-4750	341	24	.	.	PUNCT
ejpam-4750	342	1	hence	hence	ADV
ejpam-4750	342	2	,	,	PUNCT
ejpam-4750	342	3	fdim2(k1	fdim2(k1	PROPN
ejpam-4750	342	4	+	+	NOUN
ejpam-4750	342	5	g	g	NOUN
ejpam-4750	342	6	)	)	PUNCT
ejpam-4750	342	7	=	=	SYM
ejpam-4750	342	8	fdim2(s0	fdim2(s0	PROPN
ejpam-4750	342	9	)	)	PUNCT
ejpam-4750	342	10	=	=	PUNCT
ejpam-4750	342	11	|f0|	|f0|	NOUN
ejpam-4750	342	12	≥	≥	NOUN
ejpam-4750	342	13	fln(2,2)(s0	fln(2,2)(s0	SYM
ejpam-4750	342	14	)	)	PUNCT
ejpam-4750	342	15	≥	≥	NOUN
ejpam-4750	342	16	fln(2,2)(g	fln(2,2)(g	PROPN
ejpam-4750	342	17	)	)	PUNCT
ejpam-4750	342	18	.	.	PUNCT
ejpam-4750	343	1	therefore	therefore	ADV
ejpam-4750	343	2	,	,	PUNCT
ejpam-4750	343	3	fdim2(k1	fdim2(k1	PROPN
ejpam-4750	344	1	+	+	NOUN
ejpam-4750	344	2	g	g	NOUN
ejpam-4750	344	3	)	)	PUNCT
ejpam-4750	344	4	=	=	SYM
ejpam-4750	344	5	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	344	6	)	)	PUNCT
ejpam-4750	344	7	.	.	PUNCT
ejpam-4750	345	1	d.	d.	PROPN
ejpam-4750	345	2	managbanag	managbanag	PROPN
ejpam-4750	345	3	,	,	PUNCT
ejpam-4750	345	4	h.	h.	PROPN
ejpam-4750	345	5	rara	rara	PROPN
ejpam-4750	345	6	/	/	SYM
ejpam-4750	345	7	eur	eur	PROPN
ejpam-4750	345	8	.	.	PUNCT
ejpam-4750	346	1	j.	j.	PROPN
ejpam-4750	346	2	pure	pure	PROPN
ejpam-4750	346	3	appl	appl	PROPN
ejpam-4750	346	4	.	.	PROPN
ejpam-4750	346	5	math	math	PROPN
ejpam-4750	346	6	,	,	PUNCT
ejpam-4750	346	7	16	16	NUM
ejpam-4750	346	8	(	(	PUNCT
ejpam-4750	346	9	2	2	NUM
ejpam-4750	346	10	)	)	PUNCT
ejpam-4750	346	11	(	(	PUNCT
ejpam-4750	346	12	2023	2023	NUM
ejpam-4750	346	13	)	)	PUNCT
ejpam-4750	346	14	,	,	PUNCT
ejpam-4750	346	15	1068	1068	NUM
ejpam-4750	346	16	-	-	SYM
ejpam-4750	346	17	1083	1083	NUM
ejpam-4750	346	18	1078	1078	NUM
ejpam-4750	346	19	example	example	NOUN
ejpam-4750	346	20	3	3	NUM
ejpam-4750	346	21	.	.	PUNCT
ejpam-4750	347	1	(	(	PUNCT
ejpam-4750	347	2	1	1	NUM
ejpam-4750	347	3	.	.	PUNCT
ejpam-4750	347	4	)	)	PUNCT
ejpam-4750	348	1	for	for	ADP
ejpam-4750	348	2	the	the	DET
ejpam-4750	348	3	fan	fan	NOUN
ejpam-4750	348	4	fn	fn	PROPN
ejpam-4750	348	5	=	=	PROPN
ejpam-4750	348	6	k1	k1	PROPN
ejpam-4750	348	7	+	+	CCONJ
ejpam-4750	348	8	pn	pn	NOUN
ejpam-4750	348	9	,	,	PUNCT
ejpam-4750	348	10	where	where	SCONJ
ejpam-4750	348	11	n	n	PRON
ejpam-4750	348	12	≥	≥	X
ejpam-4750	348	13	2	2	NUM
ejpam-4750	348	14	,	,	PUNCT
ejpam-4750	348	15	fdim2(fn	fdim2(fn	NOUN
ejpam-4750	348	16	)	)	PUNCT
ejpam-4750	348	17	=	=	PUNCT
ejpam-4750	348	18			NOUN
ejpam-4750	348	19	0	0	NUM
ejpam-4750	348	20	,	,	PUNCT
ejpam-4750	348	21	if	if	SCONJ
ejpam-4750	348	22	n	n	NOUN
ejpam-4750	348	23	=	=	SYM
ejpam-4750	348	24	2	2	NUM
ejpam-4750	348	25	,	,	PUNCT
ejpam-4750	348	26	3	3	NUM
ejpam-4750	348	27	and	and	CCONJ
ejpam-4750	348	28	n	n	PRON
ejpam-4750	348	29	≥	≥	NOUN
ejpam-4750	348	30	7	7	NUM
ejpam-4750	348	31	is	be	AUX
ejpam-4750	348	32	odd	odd	ADJ
ejpam-4750	348	33	,	,	PUNCT
ejpam-4750	348	34	1	1	NUM
ejpam-4750	348	35	,	,	PUNCT
ejpam-4750	348	36	if	if	SCONJ
ejpam-4750	348	37	n	n	NOUN
ejpam-4750	348	38	=	=	SYM
ejpam-4750	348	39	5	5	NUM
ejpam-4750	348	40	,	,	PUNCT
ejpam-4750	348	41	2	2	NUM
ejpam-4750	348	42	,	,	PUNCT
ejpam-4750	348	43	if	if	SCONJ
ejpam-4750	348	44	n	n	PRON
ejpam-4750	348	45	≥	≥	NOUN
ejpam-4750	348	46	8	8	NUM
ejpam-4750	348	47	is	be	AUX
ejpam-4750	348	48	even	even	ADV
ejpam-4750	348	49	,	,	PUNCT
ejpam-4750	348	50	3	3	X
ejpam-4750	348	51	,	,	PUNCT
ejpam-4750	348	52	if	if	SCONJ
ejpam-4750	348	53	n	n	NOUN
ejpam-4750	348	54	=	=	SYM
ejpam-4750	348	55	4	4	NUM
ejpam-4750	348	56	,	,	PUNCT
ejpam-4750	348	57	6	6	NUM
ejpam-4750	348	58	.	.	PUNCT
ejpam-4750	349	1	(	(	PUNCT
ejpam-4750	349	2	2	2	NUM
ejpam-4750	349	3	.	.	PUNCT
ejpam-4750	349	4	)	)	PUNCT
ejpam-4750	350	1	for	for	ADP
ejpam-4750	350	2	the	the	DET
ejpam-4750	350	3	wheel	wheel	NOUN
ejpam-4750	350	4	wn	wn	PROPN
ejpam-4750	350	5	=	=	PROPN
ejpam-4750	350	6	k1	k1	PROPN
ejpam-4750	350	7	+	+	CCONJ
ejpam-4750	350	8	cn	cn	PROPN
ejpam-4750	350	9	,	,	PUNCT
ejpam-4750	350	10	where	where	SCONJ
ejpam-4750	350	11	n	n	PRON
ejpam-4750	350	12	≥	≥	NOUN
ejpam-4750	350	13	3	3	NUM
ejpam-4750	350	14	,	,	PUNCT
ejpam-4750	350	15	fdim2(wn	fdim2(wn	NOUN
ejpam-4750	350	16	)	)	PUNCT
ejpam-4750	350	17	=	=	PUNCT
ejpam-4750	350	18			NOUN
ejpam-4750	350	19	0	0	NUM
ejpam-4750	350	20	,	,	PUNCT
ejpam-4750	350	21	if	if	SCONJ
ejpam-4750	350	22	n	n	NOUN
ejpam-4750	350	23	=	=	SYM
ejpam-4750	350	24	3	3	NUM
ejpam-4750	350	25	,	,	PUNCT
ejpam-4750	350	26	4	4	NUM
ejpam-4750	350	27	,	,	PUNCT
ejpam-4750	350	28	1	1	NUM
ejpam-4750	350	29	,	,	PUNCT
ejpam-4750	350	30	if	if	SCONJ
ejpam-4750	350	31	n	n	PRON
ejpam-4750	350	32	≥	≥	NOUN
ejpam-4750	350	33	8	8	NUM
ejpam-4750	350	34	is	be	AUX
ejpam-4750	350	35	even	even	ADV
ejpam-4750	350	36	,	,	PUNCT
ejpam-4750	350	37	2	2	NUM
ejpam-4750	350	38	,	,	PUNCT
ejpam-4750	350	39	if	if	SCONJ
ejpam-4750	350	40	n	n	PRON
ejpam-4750	350	41	≥	≥	NOUN
ejpam-4750	350	42	7	7	NUM
ejpam-4750	350	43	is	be	AUX
ejpam-4750	350	44	odd	odd	ADJ
ejpam-4750	350	45	,	,	PUNCT
ejpam-4750	350	46	3	3	NUM
ejpam-4750	350	47	,	,	PUNCT
ejpam-4750	350	48	if	if	SCONJ
ejpam-4750	350	49	n	n	NOUN
ejpam-4750	350	50	=	=	SYM
ejpam-4750	350	51	5	5	NUM
ejpam-4750	350	52	,	,	PUNCT
ejpam-4750	350	53	6	6	NUM
ejpam-4750	350	54	.	.	PUNCT
ejpam-4750	351	1	(	(	PUNCT
ejpam-4750	351	2	3	3	NUM
ejpam-4750	351	3	.	.	PUNCT
ejpam-4750	351	4	)	)	PUNCT
ejpam-4750	352	1	for	for	ADP
ejpam-4750	352	2	the	the	DET
ejpam-4750	352	3	star	star	NOUN
ejpam-4750	352	4	sn	sn	PROPN
ejpam-4750	352	5	=	=	PUNCT
ejpam-4750	352	6	k1	k1	PROPN
ejpam-4750	353	1	+	+	NOUN
ejpam-4750	353	2	kn	kn	NOUN
ejpam-4750	353	3	of	of	ADP
ejpam-4750	353	4	order	order	NOUN
ejpam-4750	353	5	n+	n+	PUNCT
ejpam-4750	353	6	1	1	NUM
ejpam-4750	353	7	,	,	PUNCT
ejpam-4750	353	8	fdim2(sn	fdim2(sn	PROPN
ejpam-4750	353	9	)	)	PUNCT
ejpam-4750	353	10	=	=	SYM
ejpam-4750	354	1	0	0	X
ejpam-4750	354	2	.	.	PUNCT
ejpam-4750	355	1	as	as	ADP
ejpam-4750	355	2	a	a	DET
ejpam-4750	355	3	consequence	consequence	NOUN
ejpam-4750	355	4	of	of	ADP
ejpam-4750	355	5	theorem	theorem	NOUN
ejpam-4750	355	6	3	3	NUM
ejpam-4750	355	7	,	,	PUNCT
ejpam-4750	355	8	we	we	PRON
ejpam-4750	355	9	have	have	VERB
ejpam-4750	355	10	the	the	DET
ejpam-4750	355	11	following	follow	VERB
ejpam-4750	355	12	results	result	NOUN
ejpam-4750	355	13	.	.	PUNCT
ejpam-4750	356	1	theorem	theorem	ADJ
ejpam-4750	356	2	10	10	NUM
ejpam-4750	356	3	.	.	PUNCT
ejpam-4750	357	1	let	let	VERB
ejpam-4750	357	2	g	g	NOUN
ejpam-4750	357	3	and	and	CCONJ
ejpam-4750	357	4	h	h	NOUN
ejpam-4750	357	5	be	be	AUX
ejpam-4750	357	6	nontrivial	nontrivial	ADJ
ejpam-4750	357	7	connected	connect	VERB
ejpam-4750	357	8	graphs	graph	NOUN
ejpam-4750	357	9	such	such	ADJ
ejpam-4750	357	10	that	that	DET
ejpam-4750	357	11	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	357	12	)	)	PUNCT
ejpam-4750	357	13	=	=	PUNCT
ejpam-4750	358	1	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	358	2	)	)	PUNCT
ejpam-4750	358	3	and	and	CCONJ
ejpam-4750	358	4	ln(2,2)(h	ln(2,2)(h	PROPN
ejpam-4750	358	5	)	)	PUNCT
ejpam-4750	358	6	=	=	PUNCT
ejpam-4750	358	7	ln(2,1)(h	ln(2,1)(h	NOUN
ejpam-4750	358	8	)	)	PUNCT
ejpam-4750	358	9	.	.	PUNCT
ejpam-4750	359	1	a	a	DET
ejpam-4750	359	2	proper	proper	ADJ
ejpam-4750	359	3	subset	subset	NOUN
ejpam-4750	359	4	s	s	NOUN
ejpam-4750	359	5	of	of	ADP
ejpam-4750	359	6	v	v	NOUN
ejpam-4750	359	7	(	(	PUNCT
ejpam-4750	359	8	g	g	PROPN
ejpam-4750	359	9	+	+	NOUN
ejpam-4750	359	10	h	h	NOUN
ejpam-4750	359	11	)	)	PUNCT
ejpam-4750	359	12	is	be	AUX
ejpam-4750	359	13	a	a	DET
ejpam-4750	359	14	2	2	NUM
ejpam-4750	359	15	-	-	PUNCT
ejpam-4750	359	16	metric	metric	ADJ
ejpam-4750	359	17	basis	basis	NOUN
ejpam-4750	359	18	for	for	ADP
ejpam-4750	359	19	g	g	PROPN
ejpam-4750	360	1	+	+	NOUN
ejpam-4750	360	2	h	h	NOUN
ejpam-4750	360	3	if	if	SCONJ
ejpam-4750	360	4	and	and	CCONJ
ejpam-4750	360	5	only	only	ADV
ejpam-4750	360	6	if	if	SCONJ
ejpam-4750	360	7	s	s	VERB
ejpam-4750	360	8	=	=	PUNCT
ejpam-4750	360	9	sg	sg	X
ejpam-4750	360	10	∪	∪	NOUN
ejpam-4750	360	11	sh	sh	PROPN
ejpam-4750	360	12	where	where	SCONJ
ejpam-4750	360	13	sg	sg	PROPN
ejpam-4750	360	14	=	=	SYM
ejpam-4750	360	15	v	v	PROPN
ejpam-4750	360	16	(	(	PUNCT
ejpam-4750	360	17	g	g	NOUN
ejpam-4750	360	18	)	)	PUNCT
ejpam-4750	360	19	∩	∩	NOUN
ejpam-4750	360	20	s	s	NOUN
ejpam-4750	360	21	and	and	CCONJ
ejpam-4750	360	22	sh	sh	PROPN
ejpam-4750	360	23	=	=	SYM
ejpam-4750	360	24	v	v	PROPN
ejpam-4750	360	25	(	(	PUNCT
ejpam-4750	360	26	h)∩s	h)∩s	PROPN
ejpam-4750	360	27	are	be	AUX
ejpam-4750	360	28	ln2	ln2	ADJ
ejpam-4750	360	29	-	-	PUNCT
ejpam-4750	360	30	sets	set	NOUN
ejpam-4750	360	31	of	of	ADP
ejpam-4750	360	32	g	g	PROPN
ejpam-4750	360	33	and	and	CCONJ
ejpam-4750	360	34	h	h	NOUN
ejpam-4750	360	35	,	,	PUNCT
ejpam-4750	360	36	respectively	respectively	ADV
ejpam-4750	360	37	such	such	ADJ
ejpam-4750	360	38	that	that	SCONJ
ejpam-4750	360	39	sg	sg	NOUN
ejpam-4750	360	40	or	or	CCONJ
ejpam-4750	360	41	sh	sh	PROPN
ejpam-4750	360	42	is	be	AUX
ejpam-4750	360	43	an	an	DET
ejpam-4750	360	44	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	360	45	.	.	PUNCT
ejpam-4750	361	1	in	in	ADP
ejpam-4750	361	2	particular	particular	ADJ
ejpam-4750	361	3	,	,	PUNCT
ejpam-4750	361	4	dim2(g+h	dim2(g+h	PROPN
ejpam-4750	361	5	)	)	PUNCT
ejpam-4750	361	6	=	=	SYM
ejpam-4750	361	7	min{ln(2,2)(g	min{ln(2,2)(g	NOUN
ejpam-4750	361	8	)	)	PUNCT
ejpam-4750	361	9	+	+	CCONJ
ejpam-4750	361	10	ln2(h	ln2(h	PROPN
ejpam-4750	361	11	)	)	PUNCT
ejpam-4750	361	12	,	,	PUNCT
ejpam-4750	361	13	ln2(g	ln2(g	PROPN
ejpam-4750	361	14	)	)	PUNCT
ejpam-4750	361	15	+	+	CCONJ
ejpam-4750	361	16	ln(2,2)(h	ln(2,2)(h	ADJ
ejpam-4750	361	17	)	)	PUNCT
ejpam-4750	361	18	}	}	PUNCT
ejpam-4750	361	19	.	.	PUNCT
ejpam-4750	362	1	theorem	theorem	VERB
ejpam-4750	362	2	11	11	NUM
ejpam-4750	362	3	.	.	PUNCT
ejpam-4750	363	1	let	let	VERB
ejpam-4750	363	2	g	g	NOUN
ejpam-4750	363	3	and	and	CCONJ
ejpam-4750	363	4	h	h	NOUN
ejpam-4750	363	5	be	be	AUX
ejpam-4750	363	6	nontrivial	nontrivial	ADJ
ejpam-4750	363	7	graphs	graph	NOUN
ejpam-4750	363	8	such	such	ADJ
ejpam-4750	363	9	that	that	SCONJ
ejpam-4750	363	10	(	(	PUNCT
ejpam-4750	363	11	2	2	NUM
ejpam-4750	363	12	,	,	PUNCT
ejpam-4750	363	13	2)-locating	2)-locating	NUM
ejpam-4750	363	14	set	set	NOUN
ejpam-4750	363	15	of	of	ADP
ejpam-4750	363	16	g	g	PROPN
ejpam-4750	363	17	and	and	CCONJ
ejpam-4750	363	18	h	h	NOUN
ejpam-4750	363	19	do	do	AUX
ejpam-4750	363	20	not	not	PART
ejpam-4750	363	21	exist	exist	VERB
ejpam-4750	363	22	.	.	PUNCT
ejpam-4750	364	1	then	then	ADV
ejpam-4750	364	2	s	s	VERB
ejpam-4750	364	3	⊆	⊆	NUM
ejpam-4750	364	4	v	v	NOUN
ejpam-4750	364	5	(	(	PUNCT
ejpam-4750	364	6	g	g	PROPN
ejpam-4750	364	7	+	+	NOUN
ejpam-4750	364	8	h	h	NOUN
ejpam-4750	364	9	)	)	PUNCT
ejpam-4750	364	10	is	be	AUX
ejpam-4750	364	11	a	a	DET
ejpam-4750	364	12	2	2	NUM
ejpam-4750	364	13	-	-	PUNCT
ejpam-4750	364	14	metric	metric	ADJ
ejpam-4750	364	15	basis	basis	NOUN
ejpam-4750	364	16	for	for	ADP
ejpam-4750	364	17	g	g	PROPN
ejpam-4750	365	1	+	+	NOUN
ejpam-4750	365	2	h	h	NOUN
ejpam-4750	365	3	if	if	SCONJ
ejpam-4750	365	4	and	and	CCONJ
ejpam-4750	365	5	only	only	ADV
ejpam-4750	365	6	if	if	SCONJ
ejpam-4750	365	7	s	s	VERB
ejpam-4750	365	8	=	=	PUNCT
ejpam-4750	365	9	sg	sg	X
ejpam-4750	365	10	∪	∪	NOUN
ejpam-4750	365	11	sh	sh	PROPN
ejpam-4750	365	12	where	where	SCONJ
ejpam-4750	365	13	sg	sg	PROPN
ejpam-4750	365	14	and	and	CCONJ
ejpam-4750	365	15	sh	sh	PROPN
ejpam-4750	365	16	are	be	AUX
ejpam-4750	365	17	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	365	18	of	of	ADP
ejpam-4750	365	19	g	g	NOUN
ejpam-4750	365	20	and	and	CCONJ
ejpam-4750	365	21	h	h	NOUN
ejpam-4750	365	22	,	,	PUNCT
ejpam-4750	365	23	respectively	respectively	ADV
ejpam-4750	365	24	.	.	PUNCT
ejpam-4750	366	1	in	in	ADP
ejpam-4750	366	2	particular	particular	ADJ
ejpam-4750	366	3	,	,	PUNCT
ejpam-4750	366	4	dim2(g+h	dim2(g+h	PROPN
ejpam-4750	366	5	)	)	PUNCT
ejpam-4750	366	6	=	=	PUNCT
ejpam-4750	366	7	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	366	8	)	)	PUNCT
ejpam-4750	366	9	+	+	PUNCT
ejpam-4750	366	10	ln(2,1)(h	ln(2,1)(h	NUM
ejpam-4750	366	11	)	)	PUNCT
ejpam-4750	366	12	.	.	PUNCT
ejpam-4750	367	1	theorem	theorem	NOUN
ejpam-4750	367	2	12	12	NUM
ejpam-4750	367	3	.	.	PUNCT
ejpam-4750	368	1	let	let	VERB
ejpam-4750	368	2	g	g	NOUN
ejpam-4750	368	3	and	and	CCONJ
ejpam-4750	368	4	h	h	NOUN
ejpam-4750	368	5	be	be	AUX
ejpam-4750	368	6	nontrivial	nontrivial	ADJ
ejpam-4750	368	7	graphs	graph	NOUN
ejpam-4750	368	8	such	such	ADJ
ejpam-4750	368	9	that	that	DET
ejpam-4750	368	10	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	368	11	)	)	PUNCT
ejpam-4750	368	12	=	=	PUNCT
ejpam-4750	369	1	ln(2,1)(g	ln(2,1)(g	NOUN
ejpam-4750	369	2	)	)	PUNCT
ejpam-4750	369	3	and	and	CCONJ
ejpam-4750	369	4	ln(2,2)(h	ln(2,2)(h	PROPN
ejpam-4750	369	5	)	)	PUNCT
ejpam-4750	369	6	=	=	PUNCT
ejpam-4750	369	7	ln(2,1)(h	ln(2,1)(h	PRON
ejpam-4750	369	8	)	)	PUNCT
ejpam-4750	369	9	.	.	PUNCT
ejpam-4750	370	1	then	then	ADV
ejpam-4750	370	2	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	370	3	)	)	PUNCT
ejpam-4750	370	4	=	=	SYM
ejpam-4750	370	5	min{fln(2,2)(g	min{fln(2,2)(g	NOUN
ejpam-4750	370	6	)	)	PUNCT
ejpam-4750	370	7	+	+	NUM
ejpam-4750	370	8	fln2(h	fln2(h	NOUN
ejpam-4750	370	9	)	)	PUNCT
ejpam-4750	370	10	,	,	PUNCT
ejpam-4750	371	1	f	f	PROPN
ejpam-4750	371	2	ln2(g	ln2(g	PROPN
ejpam-4750	371	3	)	)	PUNCT
ejpam-4750	371	4	+	+	CCONJ
ejpam-4750	371	5	fln(2,2)(h	fln(2,2)(h	PROPN
ejpam-4750	371	6	)	)	PUNCT
ejpam-4750	371	7	}	}	PUNCT
ejpam-4750	371	8	.	.	PUNCT
ejpam-4750	372	1	proof	proof	NOUN
ejpam-4750	372	2	:	:	PUNCT
ejpam-4750	372	3	suppose	suppose	VERB
ejpam-4750	372	4	that	that	SCONJ
ejpam-4750	372	5	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	372	6	)	)	PUNCT
ejpam-4750	373	1	+	+	CCONJ
ejpam-4750	373	2	ln2(h	ln2(h	PROPN
ejpam-4750	373	3	)	)	PUNCT
ejpam-4750	373	4	<	<	X
ejpam-4750	373	5	ln2(g	ln2(g	X
ejpam-4750	373	6	)	)	PUNCT
ejpam-4750	373	7	+	+	CCONJ
ejpam-4750	373	8	ln(2,2)(h	ln(2,2)(h	ADJ
ejpam-4750	373	9	)	)	PUNCT
ejpam-4750	373	10	.	.	PUNCT
ejpam-4750	374	1	by	by	ADP
ejpam-4750	374	2	theorem	theorem	NOUN
ejpam-4750	374	3	10	10	NUM
ejpam-4750	374	4	,	,	PUNCT
ejpam-4750	374	5	dim2(g	dim2(g	X
ejpam-4750	374	6	+	+	CCONJ
ejpam-4750	374	7	h	h	NOUN
ejpam-4750	374	8	)	)	PUNCT
ejpam-4750	374	9	=	=	SYM
ejpam-4750	374	10	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	374	11	)	)	PUNCT
ejpam-4750	374	12	+	+	CCONJ
ejpam-4750	374	13	ln2(h	ln2(h	PROPN
ejpam-4750	374	14	)	)	PUNCT
ejpam-4750	374	15	.	.	PUNCT
ejpam-4750	375	1	let	let	VERB
ejpam-4750	375	2	sg	sg	PART
ejpam-4750	375	3	be	be	AUX
ejpam-4750	375	4	an	an	DET
ejpam-4750	375	5	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	375	6	of	of	ADP
ejpam-4750	375	7	g	g	NOUN
ejpam-4750	375	8	with	with	ADP
ejpam-4750	375	9	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	375	10	)	)	PUNCT
ejpam-4750	375	11	=	=	SYM
ejpam-4750	375	12	fln(2,2)(sg	fln(2,2)(sg	ADJ
ejpam-4750	375	13	)	)	PUNCT
ejpam-4750	375	14	and	and	CCONJ
ejpam-4750	375	15	fg	fg	PROPN
ejpam-4750	375	16	be	be	AUX
ejpam-4750	375	17	a	a	DET
ejpam-4750	375	18	forcing	force	VERB
ejpam-4750	375	19	subset	subset	NOUN
ejpam-4750	375	20	of	of	ADP
ejpam-4750	375	21	sg	sg	PROPN
ejpam-4750	375	22	with	with	ADP
ejpam-4750	375	23	fln(2,2)(sg	fln(2,2)(sg	NOUN
ejpam-4750	375	24	)	)	PUNCT
ejpam-4750	376	1	=	=	SYM
ejpam-4750	376	2	|fg|	|fg|	PROPN
ejpam-4750	376	3	.	.	PUNCT
ejpam-4750	377	1	let	let	VERB
ejpam-4750	377	2	sh	sh	PRON
ejpam-4750	377	3	be	be	AUX
ejpam-4750	377	4	an	an	DET
ejpam-4750	377	5	ln2	ln2	NOUN
ejpam-4750	377	6	-	-	PUNCT
ejpam-4750	377	7	set	set	NOUN
ejpam-4750	377	8	of	of	ADP
ejpam-4750	377	9	h	h	NOUN
ejpam-4750	377	10	such	such	ADJ
ejpam-4750	377	11	that	that	DET
ejpam-4750	377	12	fln2(h	fln2(h	NUM
ejpam-4750	377	13	)	)	PUNCT
ejpam-4750	377	14	=	=	SYM
ejpam-4750	378	1	fln2(sh	fln2(sh	PROPN
ejpam-4750	378	2	)	)	PUNCT
ejpam-4750	378	3	and	and	CCONJ
ejpam-4750	378	4	fh	fh	PROPN
ejpam-4750	378	5	be	be	AUX
ejpam-4750	378	6	a	a	DET
ejpam-4750	378	7	forcing	force	VERB
ejpam-4750	378	8	subset	subset	NOUN
ejpam-4750	378	9	of	of	ADP
ejpam-4750	378	10	sh	sh	PROPN
ejpam-4750	378	11	with	with	ADP
ejpam-4750	378	12	fln2(sh	fln2(sh	PROPN
ejpam-4750	378	13	)	)	PUNCT
ejpam-4750	378	14	=	=	PUNCT
ejpam-4750	379	1	|fh	|fh	NUM
ejpam-4750	379	2	|	|	ADV
ejpam-4750	379	3	.	.	PUNCT
ejpam-4750	380	1	by	by	ADP
ejpam-4750	380	2	theorem	theorem	NOUN
ejpam-4750	380	3	10	10	NUM
ejpam-4750	380	4	,	,	PUNCT
ejpam-4750	380	5	s	s	PART
ejpam-4750	380	6	=	=	PUNCT
ejpam-4750	380	7	sg	sg	PROPN
ejpam-4750	380	8	∪	∪	NOUN
ejpam-4750	380	9	sh	sh	PROPN
ejpam-4750	380	10	is	be	AUX
ejpam-4750	380	11	a	a	DET
ejpam-4750	380	12	2	2	NUM
ejpam-4750	380	13	-	-	PUNCT
ejpam-4750	380	14	metric	metric	ADJ
ejpam-4750	380	15	basis	basis	NOUN
ejpam-4750	380	16	of	of	ADP
ejpam-4750	380	17	g+h	g+h	PROPN
ejpam-4750	380	18	.	.	PUNCT
ejpam-4750	381	1	we	we	PRON
ejpam-4750	381	2	claim	claim	VERB
ejpam-4750	381	3	that	that	SCONJ
ejpam-4750	381	4	fg	fg	PROPN
ejpam-4750	381	5	∪	∪	PROPN
ejpam-4750	381	6	fh	fh	PROPN
ejpam-4750	381	7	is	be	AUX
ejpam-4750	381	8	a	a	DET
ejpam-4750	381	9	forcing	force	VERB
ejpam-4750	381	10	subset	subset	NOUN
ejpam-4750	381	11	of	of	ADP
ejpam-4750	381	12	s.	s.	PROPN
ejpam-4750	381	13	suppose	suppose	VERB
ejpam-4750	381	14	there	there	PRON
ejpam-4750	381	15	exists	exist	VERB
ejpam-4750	381	16	a	a	DET
ejpam-4750	381	17	2	2	NUM
ejpam-4750	381	18	-	-	PUNCT
ejpam-4750	381	19	metric	metric	ADJ
ejpam-4750	381	20	basis	basis	NOUN
ejpam-4750	381	21	s′	s′	VERB
ejpam-4750	381	22	̸=	̸=	PROPN
ejpam-4750	381	23	s	s	NOUN
ejpam-4750	381	24	of	of	ADP
ejpam-4750	381	25	g+h	g+h	PROPN
ejpam-4750	381	26	with	with	ADP
ejpam-4750	381	27	fg∪fh	fg∪fh	PROPN
ejpam-4750	381	28	⊆	⊆	NUM
ejpam-4750	381	29	s′.	s′.	PROPN
ejpam-4750	381	30	by	by	ADP
ejpam-4750	381	31	theorem	theorem	NOUN
ejpam-4750	381	32	10	10	NUM
ejpam-4750	381	33	,	,	PUNCT
ejpam-4750	381	34	s′	s′	PUNCT
ejpam-4750	381	35	=	=	PUNCT
ejpam-4750	381	36	s′	s′	NOUN
ejpam-4750	381	37	g∪s′	g∪s′	NOUN
ejpam-4750	381	38	h	h	PROPN
ejpam-4750	381	39	d.	d.	PROPN
ejpam-4750	381	40	managbanag	managbanag	PROPN
ejpam-4750	381	41	,	,	PUNCT
ejpam-4750	381	42	h.	h.	PROPN
ejpam-4750	381	43	rara	rara	PROPN
ejpam-4750	381	44	/	/	SYM
ejpam-4750	381	45	eur	eur	PROPN
ejpam-4750	381	46	.	.	PUNCT
ejpam-4750	382	1	j.	j.	PROPN
ejpam-4750	382	2	pure	pure	PROPN
ejpam-4750	382	3	appl	appl	PROPN
ejpam-4750	382	4	.	.	PROPN
ejpam-4750	382	5	math	math	PROPN
ejpam-4750	382	6	,	,	PUNCT
ejpam-4750	382	7	16	16	NUM
ejpam-4750	382	8	(	(	PUNCT
ejpam-4750	382	9	2	2	NUM
ejpam-4750	382	10	)	)	PUNCT
ejpam-4750	382	11	(	(	PUNCT
ejpam-4750	382	12	2023	2023	NUM
ejpam-4750	382	13	)	)	PUNCT
ejpam-4750	382	14	,	,	PUNCT
ejpam-4750	382	15	1068	1068	NUM
ejpam-4750	382	16	-	-	SYM
ejpam-4750	382	17	1083	1083	NUM
ejpam-4750	382	18	1079	1079	NUM
ejpam-4750	382	19	where	where	SCONJ
ejpam-4750	382	20	s′	s′	VERB
ejpam-4750	382	21	g	g	NOUN
ejpam-4750	382	22	is	be	AUX
ejpam-4750	382	23	an	an	DET
ejpam-4750	382	24	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	382	25	of	of	ADP
ejpam-4750	382	26	g	g	PROPN
ejpam-4750	382	27	and	and	CCONJ
ejpam-4750	382	28	s′	s′	ADJ
ejpam-4750	382	29	h	h	NOUN
ejpam-4750	382	30	is	be	AUX
ejpam-4750	382	31	an	an	DET
ejpam-4750	382	32	ln2	ln2	NOUN
ejpam-4750	382	33	-	-	PUNCT
ejpam-4750	382	34	set	set	NOUN
ejpam-4750	382	35	of	of	ADP
ejpam-4750	382	36	h.	h.	NOUN
ejpam-4750	382	37	since	since	SCONJ
ejpam-4750	382	38	fg∪fh	fg∪fh	PROPN
ejpam-4750	382	39	⊆	⊆	NUM
ejpam-4750	382	40	s′	s′	NOUN
ejpam-4750	382	41	,	,	PUNCT
ejpam-4750	382	42	fg∩fh	fg∩fh	PROPN
ejpam-4750	382	43	=	=	SYM
ejpam-4750	382	44	∅	∅	NOUN
ejpam-4750	382	45	and	and	CCONJ
ejpam-4750	382	46	s′	s′	ADJ
ejpam-4750	382	47	g	g	PROPN
ejpam-4750	382	48	∩	∩	NOUN
ejpam-4750	382	49	s′	s′	ADJ
ejpam-4750	382	50	h	h	NOUN
ejpam-4750	382	51	=	=	NOUN
ejpam-4750	382	52	∅	∅	NOUN
ejpam-4750	382	53	,	,	PUNCT
ejpam-4750	382	54	hence	hence	ADV
ejpam-4750	382	55	,	,	PUNCT
ejpam-4750	382	56	fg	fg	PROPN
ejpam-4750	382	57	is	be	AUX
ejpam-4750	382	58	not	not	PART
ejpam-4750	382	59	a	a	DET
ejpam-4750	382	60	forcing	force	VERB
ejpam-4750	382	61	subset	subset	NOUN
ejpam-4750	382	62	of	of	ADP
ejpam-4750	382	63	sg	sg	PROPN
ejpam-4750	382	64	or	or	CCONJ
ejpam-4750	382	65	fh	fh	PROPN
ejpam-4750	382	66	is	be	AUX
ejpam-4750	382	67	not	not	PART
ejpam-4750	382	68	a	a	DET
ejpam-4750	382	69	forcing	force	VERB
ejpam-4750	382	70	subset	subset	NOUN
ejpam-4750	382	71	of	of	ADP
ejpam-4750	382	72	sh	sh	PROPN
ejpam-4750	382	73	,	,	PUNCT
ejpam-4750	382	74	a	a	DET
ejpam-4750	382	75	contradiction	contradiction	NOUN
ejpam-4750	382	76	.	.	PUNCT
ejpam-4750	383	1	thus	thus	ADV
ejpam-4750	383	2	,	,	PUNCT
ejpam-4750	383	3	fg	fg	PROPN
ejpam-4750	383	4	∪	∪	PROPN
ejpam-4750	383	5	fh	fh	PROPN
ejpam-4750	383	6	is	be	AUX
ejpam-4750	383	7	a	a	DET
ejpam-4750	383	8	forcing	force	VERB
ejpam-4750	383	9	subset	subset	NOUN
ejpam-4750	383	10	of	of	ADP
ejpam-4750	383	11	s.	s.	PROPN
ejpam-4750	383	12	this	this	PRON
ejpam-4750	383	13	implies	imply	VERB
ejpam-4750	383	14	that	that	SCONJ
ejpam-4750	383	15	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	383	16	)	)	PUNCT
ejpam-4750	383	17	≤	≤	NUM
ejpam-4750	383	18	fdim2(s	fdim2(s	NOUN
ejpam-4750	383	19	)	)	PUNCT
ejpam-4750	383	20	≤	≤	NOUN
ejpam-4750	384	1	|fg	|fg	NUM
ejpam-4750	384	2	∪	∪	ADP
ejpam-4750	384	3	fh	fh	PROPN
ejpam-4750	384	4	|	|	NOUN
ejpam-4750	384	5	=	=	PUNCT
ejpam-4750	384	6	|fg|+	|fg|+	NOUN
ejpam-4750	384	7	|fh	|fh	NUM
ejpam-4750	384	8	|	|	ADV
ejpam-4750	384	9	=	=	SYM
ejpam-4750	384	10	fln(2,2)(sg	fln(2,2)(sg	ADJ
ejpam-4750	384	11	)	)	PUNCT
ejpam-4750	384	12	+	+	CCONJ
ejpam-4750	384	13	fln2(sh	fln2(sh	PROPN
ejpam-4750	384	14	)	)	PUNCT
ejpam-4750	384	15	=	=	SYM
ejpam-4750	384	16	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	384	17	)	)	PUNCT
ejpam-4750	384	18	+	+	NUM
ejpam-4750	384	19	fln2(h	fln2(h	NOUN
ejpam-4750	384	20	)	)	PUNCT
ejpam-4750	384	21	.	.	PUNCT
ejpam-4750	385	1	now	now	ADV
ejpam-4750	385	2	,	,	PUNCT
ejpam-4750	385	3	let	let	VERB
ejpam-4750	385	4	w	w	NOUN
ejpam-4750	385	5	be	be	AUX
ejpam-4750	385	6	a	a	DET
ejpam-4750	385	7	2	2	NUM
ejpam-4750	385	8	-	-	PUNCT
ejpam-4750	385	9	metric	metric	ADJ
ejpam-4750	385	10	basis	basis	NOUN
ejpam-4750	385	11	for	for	ADP
ejpam-4750	385	12	g+h	g+h	PROPN
ejpam-4750	385	13	with	with	ADP
ejpam-4750	385	14	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	385	15	)	)	PUNCT
ejpam-4750	385	16	=	=	SYM
ejpam-4750	385	17	fdim2(w	fdim2(w	NOUN
ejpam-4750	385	18	)	)	PUNCT
ejpam-4750	385	19	and	and	CCONJ
ejpam-4750	385	20	let	let	VERB
ejpam-4750	385	21	f	f	PRON
ejpam-4750	385	22	be	be	AUX
ejpam-4750	385	23	a	a	DET
ejpam-4750	385	24	forcing	force	VERB
ejpam-4750	385	25	subset	subset	NOUN
ejpam-4750	385	26	of	of	ADP
ejpam-4750	385	27	w	w	ADP
ejpam-4750	385	28	such	such	ADJ
ejpam-4750	385	29	that	that	PRON
ejpam-4750	385	30	fdim2(w	fdim2(w	PUNCT
ejpam-4750	385	31	)	)	PUNCT
ejpam-4750	386	1	=	=	PRON
ejpam-4750	386	2	|f	|f	PROPN
ejpam-4750	387	1	|	|	NOUN
ejpam-4750	387	2	.	.	PUNCT
ejpam-4750	388	1	then	then	ADV
ejpam-4750	388	2	by	by	ADP
ejpam-4750	388	3	theorem	theorem	NOUN
ejpam-4750	388	4	10	10	NUM
ejpam-4750	388	5	,	,	PUNCT
ejpam-4750	388	6	w	w	PROPN
ejpam-4750	388	7	=	=	PROPN
ejpam-4750	388	8	wg∪wh	wg∪wh	PROPN
ejpam-4750	388	9	where	where	SCONJ
ejpam-4750	388	10	wg	wg	PROPN
ejpam-4750	388	11	is	be	AUX
ejpam-4750	388	12	an	an	DET
ejpam-4750	388	13	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	388	14	of	of	ADP
ejpam-4750	388	15	g	g	PROPN
ejpam-4750	388	16	and	and	CCONJ
ejpam-4750	388	17	wh	wh	PROPN
ejpam-4750	388	18	is	be	AUX
ejpam-4750	388	19	an	an	DET
ejpam-4750	388	20	ln2	ln2	NOUN
ejpam-4750	388	21	-	-	PUNCT
ejpam-4750	388	22	set	set	NOUN
ejpam-4750	388	23	of	of	ADP
ejpam-4750	388	24	h	h	NOUN
ejpam-4750	388	25	or	or	CCONJ
ejpam-4750	388	26	wg	wg	PROPN
ejpam-4750	388	27	is	be	AUX
ejpam-4750	388	28	an	an	DET
ejpam-4750	388	29	ln2	ln2	NOUN
ejpam-4750	388	30	-	-	PUNCT
ejpam-4750	388	31	set	set	NOUN
ejpam-4750	388	32	of	of	ADP
ejpam-4750	388	33	g	g	PROPN
ejpam-4750	388	34	and	and	CCONJ
ejpam-4750	388	35	wh	wh	PROPN
ejpam-4750	388	36	is	be	AUX
ejpam-4750	388	37	an	an	DET
ejpam-4750	388	38	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	388	39	of	of	ADP
ejpam-4750	388	40	h.	h.	PROPN
ejpam-4750	388	41	since	since	SCONJ
ejpam-4750	388	42	f	f	PROPN
ejpam-4750	389	1	⊆	⊆	NUM
ejpam-4750	389	2	w	w	PROPN
ejpam-4750	389	3	=	=	PUNCT
ejpam-4750	389	4	wg	wg	PROPN
ejpam-4750	389	5	∪wh	∪wh	NOUN
ejpam-4750	389	6	,	,	PUNCT
ejpam-4750	389	7	consider	consider	VERB
ejpam-4750	389	8	the	the	DET
ejpam-4750	389	9	following	follow	VERB
ejpam-4750	389	10	cases	case	NOUN
ejpam-4750	389	11	:	:	PUNCT
ejpam-4750	389	12	case	case	NOUN
ejpam-4750	389	13	1	1	NUM
ejpam-4750	389	14	.	.	PUNCT
ejpam-4750	390	1	f	f	PROPN
ejpam-4750	391	1	⊆	⊆	NUM
ejpam-4750	391	2	wg	wg	INTJ
ejpam-4750	391	3	then	then	ADV
ejpam-4750	391	4	wh	wh	PROPN
ejpam-4750	391	5	is	be	AUX
ejpam-4750	391	6	a	a	DET
ejpam-4750	391	7	unique	unique	ADJ
ejpam-4750	391	8	ln2	ln2	NOUN
ejpam-4750	391	9	-	-	PUNCT
ejpam-4750	391	10	set	set	NOUN
ejpam-4750	391	11	of	of	ADP
ejpam-4750	391	12	h	h	NOUN
ejpam-4750	391	13	and	and	CCONJ
ejpam-4750	391	14	f	f	PROPN
ejpam-4750	391	15	is	be	AUX
ejpam-4750	391	16	a	a	DET
ejpam-4750	391	17	forcing	force	VERB
ejpam-4750	391	18	subset	subset	NOUN
ejpam-4750	391	19	of	of	ADP
ejpam-4750	391	20	wg	wg	PROPN
ejpam-4750	391	21	.	.	PUNCT
ejpam-4750	392	1	for	for	ADP
ejpam-4750	392	2	if	if	SCONJ
ejpam-4750	392	3	there	there	PRON
ejpam-4750	392	4	exists	exist	VERB
ejpam-4750	392	5	ln(2,2)-set	ln(2,2)-set	VERB
ejpam-4750	392	6	w	w	NOUN
ejpam-4750	392	7	′	′	NOUN
ejpam-4750	392	8	g	g	PROPN
ejpam-4750	392	9	̸=	̸=	PROPN
ejpam-4750	392	10	wg	wg	NOUN
ejpam-4750	392	11	of	of	ADP
ejpam-4750	392	12	g	g	PROPN
ejpam-4750	392	13	and	and	CCONJ
ejpam-4750	392	14	f	f	PROPN
ejpam-4750	392	15	⊆	⊆	NUM
ejpam-4750	392	16	w	w	NOUN
ejpam-4750	392	17	′	′	NUM
ejpam-4750	392	18	g	g	NOUN
ejpam-4750	392	19	,	,	PUNCT
ejpam-4750	392	20	then	then	ADV
ejpam-4750	392	21	f	f	PROPN
ejpam-4750	393	1	⊆	⊆	NUM
ejpam-4750	393	2	w	w	NOUN
ejpam-4750	393	3	′	′	NUM
ejpam-4750	394	1	=	=	PUNCT
ejpam-4750	394	2	w	w	NOUN
ejpam-4750	394	3	′	′	NUM
ejpam-4750	394	4	g	g	NOUN
ejpam-4750	394	5	∪	∪	ADJ
ejpam-4750	394	6	wh	wh	NOUN
ejpam-4750	394	7	.	.	PUNCT
ejpam-4750	395	1	by	by	ADP
ejpam-4750	395	2	theorem	theorem	NOUN
ejpam-4750	395	3	10	10	NUM
ejpam-4750	395	4	,	,	PUNCT
ejpam-4750	395	5	w	w	NOUN
ejpam-4750	395	6	′	′	NOUN
ejpam-4750	395	7	̸=	̸=	PROPN
ejpam-4750	395	8	w	w	NOUN
ejpam-4750	395	9	is	be	AUX
ejpam-4750	395	10	a	a	DET
ejpam-4750	395	11	2	2	NUM
ejpam-4750	395	12	-	-	PUNCT
ejpam-4750	395	13	metric	metric	ADJ
ejpam-4750	395	14	basis	basis	NOUN
ejpam-4750	395	15	for	for	ADP
ejpam-4750	395	16	g+h	g+h	PROPN
ejpam-4750	395	17	,	,	PUNCT
ejpam-4750	395	18	a	a	DET
ejpam-4750	395	19	contradiction	contradiction	NOUN
ejpam-4750	395	20	.	.	PUNCT
ejpam-4750	396	1	case	case	NOUN
ejpam-4750	396	2	2	2	NUM
ejpam-4750	396	3	.	.	PUNCT
ejpam-4750	396	4	f	f	NOUN
ejpam-4750	397	1	⊆	⊆	NUM
ejpam-4750	397	2	wh	wh	NOUN
ejpam-4750	397	3	then	then	ADV
ejpam-4750	397	4	wg	wg	PROPN
ejpam-4750	397	5	is	be	AUX
ejpam-4750	397	6	a	a	DET
ejpam-4750	397	7	unique	unique	ADJ
ejpam-4750	397	8	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	397	9	of	of	ADP
ejpam-4750	397	10	g.	g.	NOUN
ejpam-4750	397	11	by	by	ADP
ejpam-4750	397	12	similar	similar	ADJ
ejpam-4750	397	13	argument	argument	NOUN
ejpam-4750	397	14	as	as	ADP
ejpam-4750	397	15	in	in	ADP
ejpam-4750	397	16	case	case	NOUN
ejpam-4750	397	17	1	1	NUM
ejpam-4750	397	18	,	,	PUNCT
ejpam-4750	397	19	f	f	PROPN
ejpam-4750	397	20	is	be	AUX
ejpam-4750	397	21	a	a	DET
ejpam-4750	397	22	forcing	force	VERB
ejpam-4750	397	23	subset	subset	NOUN
ejpam-4750	397	24	of	of	ADP
ejpam-4750	397	25	wh	wh	PROPN
ejpam-4750	397	26	.	.	PUNCT
ejpam-4750	398	1	case	case	NOUN
ejpam-4750	398	2	3	3	NUM
ejpam-4750	398	3	.	.	PUNCT
ejpam-4750	398	4	f	f	X
ejpam-4750	399	1	=	=	PUNCT
ejpam-4750	399	2	a	a	PRON
ejpam-4750	399	3	∪b	∪b	PUNCT
ejpam-4750	399	4	where	where	SCONJ
ejpam-4750	399	5	a	a	DET
ejpam-4750	399	6	⊆	⊆	NUM
ejpam-4750	399	7	wg	wg	NOUN
ejpam-4750	399	8	and	and	CCONJ
ejpam-4750	399	9	b	b	NOUN
ejpam-4750	399	10	⊆	⊆	NUM
ejpam-4750	399	11	wh	wh	NOUN
ejpam-4750	399	12	.	.	PUNCT
ejpam-4750	400	1	then	then	ADV
ejpam-4750	400	2	a	a	PRON
ejpam-4750	400	3	is	be	AUX
ejpam-4750	400	4	a	a	DET
ejpam-4750	400	5	forcing	force	VERB
ejpam-4750	400	6	subset	subset	NOUN
ejpam-4750	400	7	of	of	ADP
ejpam-4750	400	8	wg	wg	PRON
ejpam-4750	400	9	since	since	SCONJ
ejpam-4750	400	10	if	if	SCONJ
ejpam-4750	400	11	there	there	PRON
ejpam-4750	400	12	exists	exist	VERB
ejpam-4750	400	13	an	an	DET
ejpam-4750	400	14	ln(2,2)-set	ln(2,2)-set	NOUN
ejpam-4750	400	15	w	w	NOUN
ejpam-4750	400	16	′	′	NOUN
ejpam-4750	400	17	g	g	PROPN
ejpam-4750	400	18	̸=	̸=	PROPN
ejpam-4750	400	19	wg	wg	NOUN
ejpam-4750	400	20	of	of	ADP
ejpam-4750	400	21	g	g	PROPN
ejpam-4750	400	22	and	and	CCONJ
ejpam-4750	400	23	a	a	DET
ejpam-4750	400	24	⊆	⊆	NUM
ejpam-4750	400	25	w	w	NOUN
ejpam-4750	400	26	′	′	NUM
ejpam-4750	400	27	g	g	NOUN
ejpam-4750	400	28	,	,	PUNCT
ejpam-4750	400	29	then	then	ADV
ejpam-4750	400	30	w	w	ADP
ejpam-4750	400	31	′	′	NUM
ejpam-4750	401	1	=	=	PUNCT
ejpam-4750	401	2	w	w	NOUN
ejpam-4750	401	3	′	′	NUM
ejpam-4750	401	4	g	g	NOUN
ejpam-4750	401	5	∪	∪	ADJ
ejpam-4750	401	6	wh	wh	NOUN
ejpam-4750	401	7	is	be	AUX
ejpam-4750	401	8	a	a	DET
ejpam-4750	401	9	2	2	NUM
ejpam-4750	401	10	-	-	PUNCT
ejpam-4750	401	11	metric	metric	ADJ
ejpam-4750	401	12	basis	basis	NOUN
ejpam-4750	401	13	of	of	ADP
ejpam-4750	401	14	g	g	PROPN
ejpam-4750	401	15	+	+	CCONJ
ejpam-4750	401	16	h	h	NOUN
ejpam-4750	401	17	with	with	ADP
ejpam-4750	401	18	f	f	PROPN
ejpam-4750	401	19	⊆	⊆	NUM
ejpam-4750	401	20	w	w	NOUN
ejpam-4750	401	21	′	′	NOUN
ejpam-4750	401	22	̸=	̸=	PROPN
ejpam-4750	401	23	w	w	PROPN
ejpam-4750	401	24	,	,	PUNCT
ejpam-4750	401	25	a	a	DET
ejpam-4750	401	26	contradiction	contradiction	NOUN
ejpam-4750	401	27	.	.	PUNCT
ejpam-4750	402	1	similary	similary	ADJ
ejpam-4750	402	2	,	,	PUNCT
ejpam-4750	402	3	b	b	NOUN
ejpam-4750	402	4	is	be	AUX
ejpam-4750	402	5	a	a	DET
ejpam-4750	402	6	forcing	force	VERB
ejpam-4750	402	7	subset	subset	NOUN
ejpam-4750	402	8	of	of	ADP
ejpam-4750	402	9	wh	wh	PROPN
ejpam-4750	402	10	.	.	PUNCT
ejpam-4750	403	1	therefore	therefore	ADV
ejpam-4750	403	2	in	in	ADP
ejpam-4750	403	3	any	any	DET
ejpam-4750	403	4	case	case	NOUN
ejpam-4750	403	5	f	f	NOUN
ejpam-4750	403	6	=	=	NOUN
ejpam-4750	403	7	fwg	fwg	PROPN
ejpam-4750	403	8	∪	∪	ADP
ejpam-4750	403	9	fwh	fwh	PROPN
ejpam-4750	403	10	where	where	SCONJ
ejpam-4750	403	11	fwg	fwg	NOUN
ejpam-4750	403	12	and	and	CCONJ
ejpam-4750	403	13	fwh	fwh	NOUN
ejpam-4750	403	14	are	be	AUX
ejpam-4750	403	15	forcing	force	VERB
ejpam-4750	403	16	subsets	subset	NOUN
ejpam-4750	403	17	of	of	ADP
ejpam-4750	403	18	wg	wg	PROPN
ejpam-4750	403	19	and	and	CCONJ
ejpam-4750	403	20	wh	wh	VERB
ejpam-4750	403	21	,	,	PUNCT
ejpam-4750	403	22	respectively	respectively	ADV
ejpam-4750	403	23	where	where	SCONJ
ejpam-4750	403	24	it	it	PRON
ejpam-4750	403	25	may	may	AUX
ejpam-4750	403	26	happen	happen	VERB
ejpam-4750	403	27	that	that	SCONJ
ejpam-4750	403	28	fwg	fwg	NOUN
ejpam-4750	403	29	or	or	CCONJ
ejpam-4750	403	30	fwh	fwh	NOUN
ejpam-4750	403	31	is	be	AUX
ejpam-4750	403	32	empty	empty	ADJ
ejpam-4750	403	33	.	.	PUNCT
ejpam-4750	404	1	hence	hence	ADV
ejpam-4750	404	2	,	,	PUNCT
ejpam-4750	404	3	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	404	4	)	)	PUNCT
ejpam-4750	404	5	=	=	SYM
ejpam-4750	405	1	fdim2(w	fdim2(w	NOUN
ejpam-4750	405	2	)	)	PUNCT
ejpam-4750	406	1	=	=	SYM
ejpam-4750	406	2	|f	|f	PROPN
ejpam-4750	407	1	|	|	NOUN
ejpam-4750	407	2	=	=	PUNCT
ejpam-4750	407	3	|fwg	|fwg	X
ejpam-4750	407	4	∪	∪	VERB
ejpam-4750	407	5	fwh	fwh	NOUN
ejpam-4750	407	6	|	|	NOUN
ejpam-4750	407	7	=	=	PUNCT
ejpam-4750	407	8	|fwg	|fwg	X
ejpam-4750	407	9	|+	|+	X
ejpam-4750	407	10	|fwh	|fwh	NOUN
ejpam-4750	407	11	|	|	PRON
ejpam-4750	407	12	≥	≥	NOUN
ejpam-4750	407	13	fln(2,2)(wg	fln(2,2)(wg	NOUN
ejpam-4750	407	14	)	)	PUNCT
ejpam-4750	407	15	+	+	SYM
ejpam-4750	407	16	fln2(wh	fln2(wh	PROPN
ejpam-4750	407	17	)	)	PUNCT
ejpam-4750	407	18	≥	≥	NOUN
ejpam-4750	407	19	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	407	20	)	)	PUNCT
ejpam-4750	407	21	+	+	NUM
ejpam-4750	407	22	fln2(h	fln2(h	NOUN
ejpam-4750	407	23	)	)	PUNCT
ejpam-4750	407	24	.	.	PUNCT
ejpam-4750	408	1	consequently	consequently	ADV
ejpam-4750	408	2	,	,	PUNCT
ejpam-4750	408	3	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	408	4	)	)	PUNCT
ejpam-4750	408	5	=	=	SYM
ejpam-4750	408	6	fln(2,2)(g	fln(2,2)(g	NOUN
ejpam-4750	408	7	)	)	PUNCT
ejpam-4750	408	8	+	+	NUM
ejpam-4750	408	9	fln2(h	fln2(h	NOUN
ejpam-4750	408	10	)	)	PUNCT
ejpam-4750	408	11	.	.	PUNCT
ejpam-4750	409	1	similarly	similarly	ADV
ejpam-4750	409	2	,	,	PUNCT
ejpam-4750	409	3	if	if	SCONJ
ejpam-4750	409	4	ln2(g	ln2(g	NOUN
ejpam-4750	409	5	)	)	PUNCT
ejpam-4750	409	6	+	+	PUNCT
ejpam-4750	409	7	ln(2,2)(h	ln(2,2)(h	X
ejpam-4750	409	8	)	)	PUNCT
ejpam-4750	409	9	<	<	X
ejpam-4750	409	10	ln(2,2)(g	ln(2,2)(g	NOUN
ejpam-4750	409	11	)	)	PUNCT
ejpam-4750	410	1	+	+	CCONJ
ejpam-4750	410	2	ln2(h	ln2(h	PROPN
ejpam-4750	410	3	)	)	PUNCT
ejpam-4750	410	4	,	,	PUNCT
ejpam-4750	410	5	then	then	ADV
ejpam-4750	410	6	,	,	PUNCT
ejpam-4750	410	7	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	410	8	)	)	PUNCT
ejpam-4750	410	9	=	=	PUNCT
ejpam-4750	410	10	fln2(g	fln2(g	NOUN
ejpam-4750	410	11	)	)	PUNCT
ejpam-4750	411	1	+	+	CCONJ
ejpam-4750	411	2	fln(2,2)(h	fln(2,2)(h	PROPN
ejpam-4750	411	3	)	)	PUNCT
ejpam-4750	411	4	.	.	PUNCT
ejpam-4750	412	1	example	example	NOUN
ejpam-4750	413	1	4	4	X
ejpam-4750	413	2	.	.	PUNCT
ejpam-4750	413	3	consider	consider	VERB
ejpam-4750	413	4	the	the	DET
ejpam-4750	413	5	join	join	NOUN
ejpam-4750	413	6	of	of	ADP
ejpam-4750	413	7	two	two	NUM
ejpam-4750	413	8	graphs	graph	NOUN
ejpam-4750	413	9	c4	c4	NOUN
ejpam-4750	413	10	and	and	CCONJ
ejpam-4750	413	11	p7	p7	ADJ
ejpam-4750	413	12	.	.	PUNCT
ejpam-4750	414	1	note	note	VERB
ejpam-4750	414	2	that	that	SCONJ
ejpam-4750	414	3	ln(2,2)(c4	ln(2,2)(c4	NOUN
ejpam-4750	414	4	)	)	PUNCT
ejpam-4750	414	5	=	=	SYM
ejpam-4750	414	6	4	4	NUM
ejpam-4750	414	7	=	=	SYM
ejpam-4750	414	8	ln(2,1)(c4	ln(2,1)(c4	NOUN
ejpam-4750	414	9	)	)	PUNCT
ejpam-4750	414	10	and	and	CCONJ
ejpam-4750	414	11	ln(2,2)(p7	ln(2,2)(p7	PROPN
ejpam-4750	414	12	)	)	PUNCT
ejpam-4750	414	13	=	=	SYM
ejpam-4750	414	14	4	4	NUM
ejpam-4750	414	15	=	=	SYM
ejpam-4750	414	16	ln(2,1)(p7	ln(2,1)(p7	PROPN
ejpam-4750	414	17	)	)	PUNCT
ejpam-4750	414	18	.	.	PUNCT
ejpam-4750	415	1	since	since	SCONJ
ejpam-4750	415	2	c4	c4	NOUN
ejpam-4750	415	3	and	and	CCONJ
ejpam-4750	415	4	p7	p7	VERB
ejpam-4750	415	5	both	both	PRON
ejpam-4750	415	6	having	have	VERB
ejpam-4750	415	7	unique	unique	ADJ
ejpam-4750	415	8	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	415	9	,	,	PUNCT
ejpam-4750	415	10	fdim2(c4	fdim2(c4	PROPN
ejpam-4750	415	11	+	+	CCONJ
ejpam-4750	415	12	p7	p7	ADJ
ejpam-4750	415	13	)	)	PUNCT
ejpam-4750	415	14	=	=	SYM
ejpam-4750	415	15	fln(2,2)(c4	fln(2,2)(c4	X
ejpam-4750	415	16	)	)	PUNCT
ejpam-4750	416	1	=	=	SYM
ejpam-4750	416	2	fln(2,2)(p7	fln(2,2)(p7	X
ejpam-4750	416	3	)	)	PUNCT
ejpam-4750	416	4	=	=	SYM
ejpam-4750	417	1	0	0	X
ejpam-4750	417	2	.	.	X
ejpam-4750	417	3	d.	d.	PROPN
ejpam-4750	417	4	managbanag	managbanag	PROPN
ejpam-4750	417	5	,	,	PUNCT
ejpam-4750	417	6	h.	h.	PROPN
ejpam-4750	417	7	rara	rara	PROPN
ejpam-4750	417	8	/	/	SYM
ejpam-4750	417	9	eur	eur	PROPN
ejpam-4750	417	10	.	.	PUNCT
ejpam-4750	418	1	j.	j.	PROPN
ejpam-4750	418	2	pure	pure	PROPN
ejpam-4750	418	3	appl	appl	PROPN
ejpam-4750	418	4	.	.	PROPN
ejpam-4750	418	5	math	math	PROPN
ejpam-4750	418	6	,	,	PUNCT
ejpam-4750	418	7	16	16	NUM
ejpam-4750	418	8	(	(	PUNCT
ejpam-4750	418	9	2	2	NUM
ejpam-4750	418	10	)	)	PUNCT
ejpam-4750	418	11	(	(	PUNCT
ejpam-4750	418	12	2023	2023	NUM
ejpam-4750	418	13	)	)	PUNCT
ejpam-4750	418	14	,	,	PUNCT
ejpam-4750	418	15	1068	1068	NUM
ejpam-4750	418	16	-	-	SYM
ejpam-4750	418	17	1083	1083	NUM
ejpam-4750	418	18	1080	1080	NUM
ejpam-4750	418	19	example	example	NOUN
ejpam-4750	418	20	5	5	NUM
ejpam-4750	418	21	.	.	X
ejpam-4750	418	22	consider	consider	VERB
ejpam-4750	418	23	the	the	DET
ejpam-4750	418	24	join	join	NOUN
ejpam-4750	418	25	of	of	ADP
ejpam-4750	418	26	two	two	NUM
ejpam-4750	418	27	graphs	graph	NOUN
ejpam-4750	418	28	c7	c7	PROPN
ejpam-4750	418	29	and	and	CCONJ
ejpam-4750	418	30	p6	p6	PROPN
ejpam-4750	418	31	.	.	PUNCT
ejpam-4750	419	1	then	then	ADV
ejpam-4750	419	2	ln(2,2)(c7	ln(2,2)(c7	NOUN
ejpam-4750	419	3	)	)	PUNCT
ejpam-4750	419	4	=	=	SYM
ejpam-4750	419	5	4	4	NUM
ejpam-4750	419	6	=	=	SYM
ejpam-4750	419	7	ln(2,1)(c7	ln(2,1)(c7	NOUN
ejpam-4750	419	8	)	)	PUNCT
ejpam-4750	419	9	and	and	CCONJ
ejpam-4750	419	10	ln(2,2)(p6	ln(2,2)(p6	PROPN
ejpam-4750	419	11	)	)	PUNCT
ejpam-4750	419	12	=	=	SYM
ejpam-4750	419	13	3	3	NUM
ejpam-4750	419	14	=	=	SYM
ejpam-4750	419	15	ln(2,1)(p6	ln(2,1)(p6	NOUN
ejpam-4750	419	16	)	)	PUNCT
ejpam-4750	419	17	.	.	PUNCT
ejpam-4750	420	1	since	since	SCONJ
ejpam-4750	420	2	c7	c7	PROPN
ejpam-4750	420	3	and	and	CCONJ
ejpam-4750	420	4	p6	p6	PROPN
ejpam-4750	420	5	have	have	VERB
ejpam-4750	420	6	no	no	DET
ejpam-4750	420	7	unique	unique	ADJ
ejpam-4750	420	8	ln(2,2)-sets	ln(2,2)-set	NOUN
ejpam-4750	420	9	,	,	PUNCT
ejpam-4750	420	10	fdim2(c7	fdim2(c7	NOUN
ejpam-4750	420	11	+	+	CCONJ
ejpam-4750	420	12	p6	p6	ADJ
ejpam-4750	420	13	)	)	PUNCT
ejpam-4750	420	14	=	=	SYM
ejpam-4750	420	15	min{fln(2,2)(c7	min{fln(2,2)(c7	NOUN
ejpam-4750	420	16	)	)	PUNCT
ejpam-4750	421	1	+	+	CCONJ
ejpam-4750	421	2	fln2(p6	fln2(p6	PROPN
ejpam-4750	421	3	)	)	PUNCT
ejpam-4750	421	4	,	,	PUNCT
ejpam-4750	421	5	f	f	PROPN
ejpam-4750	421	6	ln2(c7	ln2(c7	PROPN
ejpam-4750	421	7	)	)	PUNCT
ejpam-4750	422	1	+	+	CCONJ
ejpam-4750	422	2	fln(2,2)(p6	fln(2,2)(p6	NOUN
ejpam-4750	422	3	)	)	PUNCT
ejpam-4750	422	4	}	}	PUNCT
ejpam-4750	422	5	=	=	SYM
ejpam-4750	423	1	min{2	min{2	ADJ
ejpam-4750	423	2	+	+	CCONJ
ejpam-4750	423	3	2	2	NUM
ejpam-4750	423	4	,	,	PUNCT
ejpam-4750	423	5	2	2	NUM
ejpam-4750	423	6	+	+	CCONJ
ejpam-4750	423	7	3	3	NUM
ejpam-4750	423	8	}	}	PUNCT
ejpam-4750	423	9	=	=	SYM
ejpam-4750	423	10	min{4	min{4	NOUN
ejpam-4750	423	11	,	,	PUNCT
ejpam-4750	423	12	5	5	NUM
ejpam-4750	423	13	}	}	PUNCT
ejpam-4750	423	14	=	=	SYM
ejpam-4750	423	15	4	4	X
ejpam-4750	423	16	.	.	PUNCT
ejpam-4750	423	17	theorem	theorem	VERB
ejpam-4750	423	18	13	13	NUM
ejpam-4750	423	19	.	.	PUNCT
ejpam-4750	424	1	let	let	VERB
ejpam-4750	424	2	g	g	NOUN
ejpam-4750	424	3	and	and	CCONJ
ejpam-4750	424	4	h	h	NOUN
ejpam-4750	424	5	be	be	AUX
ejpam-4750	424	6	nontrivial	nontrivial	ADJ
ejpam-4750	424	7	connected	connect	VERB
ejpam-4750	424	8	graphs	graph	NOUN
ejpam-4750	424	9	where	where	SCONJ
ejpam-4750	424	10	both	both	DET
ejpam-4750	424	11	g	g	PROPN
ejpam-4750	424	12	and	and	CCONJ
ejpam-4750	424	13	h	h	NOUN
ejpam-4750	424	14	have	have	VERB
ejpam-4750	424	15	no	no	DET
ejpam-4750	424	16	(	(	PUNCT
ejpam-4750	424	17	2	2	NUM
ejpam-4750	424	18	,	,	PUNCT
ejpam-4750	424	19	2)-locating	2)-locating	NUM
ejpam-4750	424	20	sets	set	NOUN
ejpam-4750	424	21	.	.	PUNCT
ejpam-4750	425	1	then	then	ADV
ejpam-4750	425	2	fdim2(g+h	fdim2(g+h	PRON
ejpam-4750	425	3	)	)	PUNCT
ejpam-4750	425	4	=	=	SYM
ejpam-4750	426	1			NOUN
ejpam-4750	426	2	0	0	NUM
ejpam-4750	426	3	,	,	PUNCT
ejpam-4750	426	4	if	if	SCONJ
ejpam-4750	426	5	both	both	DET
ejpam-4750	426	6	g	g	PROPN
ejpam-4750	426	7	and	and	CCONJ
ejpam-4750	426	8	h	h	NOUN
ejpam-4750	426	9	have	have	VERB
ejpam-4750	426	10	unique	unique	ADJ
ejpam-4750	426	11	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	426	12	,	,	PUNCT
ejpam-4750	426	13	f	f	PROPN
ejpam-4750	426	14	ln(2,1)(g	ln(2,1)(g	PROPN
ejpam-4750	426	15	)	)	PUNCT
ejpam-4750	426	16	+	+	CCONJ
ejpam-4750	426	17	fln(2,1)(h	fln(2,1)(h	PROPN
ejpam-4750	426	18	)	)	PUNCT
ejpam-4750	426	19	,	,	PUNCT
ejpam-4750	426	20	otherwise	otherwise	ADV
ejpam-4750	426	21	.	.	PUNCT
ejpam-4750	427	1	proof	proof	NOUN
ejpam-4750	427	2	:	:	PUNCT
ejpam-4750	427	3	suppose	suppose	VERB
ejpam-4750	427	4	that	that	SCONJ
ejpam-4750	427	5	both	both	PRON
ejpam-4750	427	6	g	g	PROPN
ejpam-4750	427	7	and	and	CCONJ
ejpam-4750	427	8	h	h	NOUN
ejpam-4750	427	9	have	have	VERB
ejpam-4750	427	10	unique	unique	ADJ
ejpam-4750	427	11	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	427	12	.	.	PUNCT
ejpam-4750	428	1	let	let	VERB
ejpam-4750	428	2	sg	sg	INTJ
ejpam-4750	429	1	and	and	CCONJ
ejpam-4750	429	2	sh	sh	INTJ
ejpam-4750	429	3	be	be	AUX
ejpam-4750	429	4	the	the	DET
ejpam-4750	429	5	unique	unique	ADJ
ejpam-4750	429	6	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	429	7	of	of	ADP
ejpam-4750	429	8	g	g	NOUN
ejpam-4750	429	9	and	and	CCONJ
ejpam-4750	429	10	h	h	NOUN
ejpam-4750	429	11	,	,	PUNCT
ejpam-4750	429	12	respectively	respectively	ADV
ejpam-4750	429	13	.	.	PUNCT
ejpam-4750	430	1	by	by	ADP
ejpam-4750	430	2	theorem	theorem	NOUN
ejpam-4750	430	3	11	11	NUM
ejpam-4750	430	4	,	,	PUNCT
ejpam-4750	430	5	s	s	PART
ejpam-4750	430	6	=	=	PUNCT
ejpam-4750	430	7	sg	sg	PROPN
ejpam-4750	430	8	∪	∪	NOUN
ejpam-4750	430	9	sh	sh	PROPN
ejpam-4750	430	10	is	be	AUX
ejpam-4750	430	11	the	the	DET
ejpam-4750	430	12	unique	unique	ADJ
ejpam-4750	430	13	2	2	NUM
ejpam-4750	430	14	-	-	PUNCT
ejpam-4750	430	15	metric	metric	ADJ
ejpam-4750	430	16	basis	basis	NOUN
ejpam-4750	430	17	for	for	ADP
ejpam-4750	430	18	g	g	PROPN
ejpam-4750	430	19	+	+	CCONJ
ejpam-4750	430	20	h.	h.	PROPN
ejpam-4750	430	21	by	by	ADP
ejpam-4750	430	22	remark	remark	NOUN
ejpam-4750	430	23	4	4	NUM
ejpam-4750	430	24	(	(	PUNCT
ejpam-4750	430	25	i	i	NOUN
ejpam-4750	430	26	)	)	PUNCT
ejpam-4750	430	27	,	,	PUNCT
ejpam-4750	430	28	fdim2(g	fdim2(g	PROPN
ejpam-4750	431	1	+	+	CCONJ
ejpam-4750	431	2	h	h	X
ejpam-4750	431	3	)	)	PUNCT
ejpam-4750	431	4	=	=	NOUN
ejpam-4750	431	5	0	0	X
ejpam-4750	431	6	.	.	PUNCT
ejpam-4750	432	1	next	next	ADV
ejpam-4750	432	2	,	,	PUNCT
ejpam-4750	432	3	suppose	suppose	VERB
ejpam-4750	432	4	that	that	SCONJ
ejpam-4750	432	5	g	g	PROPN
ejpam-4750	432	6	has	have	VERB
ejpam-4750	432	7	no	no	DET
ejpam-4750	432	8	unique	unique	ADJ
ejpam-4750	432	9	ln(2,1)-set	ln(2,1)-set	NOUN
ejpam-4750	432	10	.	.	PUNCT
ejpam-4750	433	1	let	let	VERB
ejpam-4750	433	2	sg	sg	INTJ
ejpam-4750	434	1	and	and	CCONJ
ejpam-4750	434	2	sh	sh	INTJ
ejpam-4750	434	3	be	be	VERB
ejpam-4750	434	4	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	434	5	of	of	ADP
ejpam-4750	434	6	g	g	NOUN
ejpam-4750	434	7	and	and	CCONJ
ejpam-4750	434	8	h	h	NOUN
ejpam-4750	434	9	,	,	PUNCT
ejpam-4750	434	10	respectively	respectively	ADV
ejpam-4750	434	11	.	.	PUNCT
ejpam-4750	435	1	thus	thus	ADV
ejpam-4750	435	2	,	,	PUNCT
ejpam-4750	435	3	by	by	ADP
ejpam-4750	435	4	theorem	theorem	NOUN
ejpam-4750	435	5	11	11	NUM
ejpam-4750	435	6	,	,	PUNCT
ejpam-4750	435	7	s	s	PART
ejpam-4750	435	8	=	=	PUNCT
ejpam-4750	435	9	sg	sg	PROPN
ejpam-4750	435	10	∪	∪	NOUN
ejpam-4750	435	11	sh	sh	PROPN
ejpam-4750	435	12	is	be	AUX
ejpam-4750	435	13	a	a	DET
ejpam-4750	435	14	2	2	NUM
ejpam-4750	435	15	-	-	PUNCT
ejpam-4750	435	16	metric	metric	ADJ
ejpam-4750	435	17	basis	basis	NOUN
ejpam-4750	435	18	for	for	ADP
ejpam-4750	435	19	g	g	PROPN
ejpam-4750	435	20	+	+	CCONJ
ejpam-4750	435	21	h.	h.	PROPN
ejpam-4750	435	22	let	let	VERB
ejpam-4750	435	23	fg	fg	PRON
ejpam-4750	435	24	and	and	CCONJ
ejpam-4750	435	25	fh	fh	PROPN
ejpam-4750	435	26	be	be	AUX
ejpam-4750	435	27	forcing	force	VERB
ejpam-4750	435	28	subsets	subset	NOUN
ejpam-4750	435	29	of	of	ADP
ejpam-4750	435	30	sg	sg	PROPN
ejpam-4750	435	31	and	and	CCONJ
ejpam-4750	435	32	sh	sh	INTJ
ejpam-4750	435	33	,	,	PUNCT
ejpam-4750	435	34	respectively	respectively	ADV
ejpam-4750	435	35	such	such	ADJ
ejpam-4750	435	36	that	that	PRON
ejpam-4750	435	37	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	435	38	)	)	PUNCT
ejpam-4750	435	39	=	=	SYM
ejpam-4750	435	40	fln(2,1)(sg	fln(2,1)(sg	NOUN
ejpam-4750	435	41	)	)	PUNCT
ejpam-4750	435	42	=	=	SYM
ejpam-4750	435	43	|fg|	|fg|	PROPN
ejpam-4750	435	44	and	and	CCONJ
ejpam-4750	435	45	fln(2,1)(h	fln(2,1)(h	NUM
ejpam-4750	435	46	)	)	PUNCT
ejpam-4750	435	47	=	=	PUNCT
ejpam-4750	435	48	fln(2,1)(sh	fln(2,1)(sh	X
ejpam-4750	435	49	)	)	PUNCT
ejpam-4750	435	50	=	=	PUNCT
ejpam-4750	436	1	|fh	|fh	NUM
ejpam-4750	436	2	|	|	ADV
ejpam-4750	436	3	.	.	PUNCT
ejpam-4750	437	1	we	we	PRON
ejpam-4750	437	2	claim	claim	VERB
ejpam-4750	437	3	that	that	SCONJ
ejpam-4750	437	4	fg	fg	PROPN
ejpam-4750	437	5	∪	∪	PROPN
ejpam-4750	437	6	fh	fh	PROPN
ejpam-4750	437	7	is	be	AUX
ejpam-4750	437	8	a	a	DET
ejpam-4750	437	9	forcing	force	VERB
ejpam-4750	437	10	subset	subset	NOUN
ejpam-4750	437	11	of	of	ADP
ejpam-4750	437	12	s	s	NOUN
ejpam-4750	437	13	=	=	PUNCT
ejpam-4750	437	14	sg	sg	X
ejpam-4750	437	15	∪	∪	VERB
ejpam-4750	437	16	sh	sh	PROPN
ejpam-4750	437	17	.	.	PUNCT
ejpam-4750	438	1	clearly	clearly	ADV
ejpam-4750	438	2	,	,	PUNCT
ejpam-4750	438	3	fg	fg	PROPN
ejpam-4750	438	4	∪	∪	ADP
ejpam-4750	438	5	fh	fh	PROPN
ejpam-4750	438	6	⊆	⊆	NUM
ejpam-4750	438	7	sg	sg	NOUN
ejpam-4750	438	8	∪	∪	ADJ
ejpam-4750	438	9	sh	sh	PROPN
ejpam-4750	438	10	=	=	SYM
ejpam-4750	438	11	s.	s.	PROPN
ejpam-4750	438	12	suppose	suppose	VERB
ejpam-4750	438	13	there	there	PRON
ejpam-4750	438	14	exists	exist	VERB
ejpam-4750	438	15	2	2	NUM
ejpam-4750	438	16	-	-	PUNCT
ejpam-4750	438	17	metric	metric	ADJ
ejpam-4750	438	18	basis	basis	NOUN
ejpam-4750	438	19	s′	s′	VERB
ejpam-4750	438	20	̸=	̸=	PROPN
ejpam-4750	438	21	s	s	PART
ejpam-4750	438	22	with	with	ADP
ejpam-4750	438	23	fg	fg	PROPN
ejpam-4750	438	24	∪	∪	PROPN
ejpam-4750	438	25	fh	fh	PROPN
ejpam-4750	438	26	⊆	⊆	NUM
ejpam-4750	438	27	s′	s′	PROPN
ejpam-4750	438	28	for	for	ADP
ejpam-4750	438	29	g	g	PROPN
ejpam-4750	438	30	+	+	CCONJ
ejpam-4750	438	31	h.	h.	PROPN
ejpam-4750	438	32	by	by	ADP
ejpam-4750	438	33	theorem	theorem	NOUN
ejpam-4750	438	34	11	11	NUM
ejpam-4750	438	35	,	,	PUNCT
ejpam-4750	438	36	s′	s′	PUNCT
ejpam-4750	438	37	=	=	PUNCT
ejpam-4750	438	38	s′	s′	ADJ
ejpam-4750	438	39	g	g	NOUN
ejpam-4750	438	40	∪	∪	ADJ
ejpam-4750	438	41	s′	s′	ADJ
ejpam-4750	438	42	h	h	NOUN
ejpam-4750	438	43	where	where	SCONJ
ejpam-4750	438	44	s′	s′	ADJ
ejpam-4750	438	45	g	g	NOUN
ejpam-4750	438	46	and	and	CCONJ
ejpam-4750	438	47	s′	s′	ADJ
ejpam-4750	438	48	h	h	NOUN
ejpam-4750	438	49	are	be	AUX
ejpam-4750	438	50	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	438	51	of	of	ADP
ejpam-4750	438	52	g	g	NOUN
ejpam-4750	438	53	and	and	CCONJ
ejpam-4750	438	54	h	h	NOUN
ejpam-4750	438	55	,	,	PUNCT
ejpam-4750	438	56	respectively	respectively	ADV
ejpam-4750	438	57	.	.	PUNCT
ejpam-4750	439	1	since	since	SCONJ
ejpam-4750	439	2	s′	s′	ADJ
ejpam-4750	439	3	̸=	̸=	PROPN
ejpam-4750	439	4	s	s	PART
ejpam-4750	439	5	,	,	PUNCT
ejpam-4750	439	6	sg	sg	ADP
ejpam-4750	439	7	̸=	̸=	PROPN
ejpam-4750	439	8	s′	s′	VERB
ejpam-4750	439	9	g	g	PROPN
ejpam-4750	439	10	or	or	CCONJ
ejpam-4750	439	11	sh	sh	PROPN
ejpam-4750	439	12	̸=	̸=	PROPN
ejpam-4750	439	13	s′	s′	NUM
ejpam-4750	439	14	h	h	NOUN
ejpam-4750	439	15	.	.	PUNCT
ejpam-4750	440	1	also	also	ADV
ejpam-4750	440	2	,	,	PUNCT
ejpam-4750	440	3	since	since	SCONJ
ejpam-4750	440	4	fg	fg	PROPN
ejpam-4750	440	5	∪	∪	ADP
ejpam-4750	440	6	fh	fh	PROPN
ejpam-4750	440	7	⊆	⊆	NUM
ejpam-4750	440	8	s′	s′	NOUN
ejpam-4750	440	9	,	,	PUNCT
ejpam-4750	440	10	fh	fh	PROPN
ejpam-4750	440	11	⊆	⊆	NUM
ejpam-4750	440	12	s′	s′	PROPN
ejpam-4750	440	13	h	h	NOUN
ejpam-4750	440	14	and	and	CCONJ
ejpam-4750	440	15	fg	fg	PROPN
ejpam-4750	440	16	⊆	⊆	NUM
ejpam-4750	440	17	s′	s′	ADJ
ejpam-4750	440	18	g.	g.	NOUN
ejpam-4750	441	1	this	this	PRON
ejpam-4750	441	2	is	be	AUX
ejpam-4750	441	3	a	a	DET
ejpam-4750	441	4	contradiction	contradiction	NOUN
ejpam-4750	441	5	since	since	SCONJ
ejpam-4750	441	6	fg	fg	PROPN
ejpam-4750	441	7	and	and	CCONJ
ejpam-4750	441	8	fh	fh	PROPN
ejpam-4750	441	9	are	be	AUX
ejpam-4750	441	10	forcing	force	VERB
ejpam-4750	441	11	subsets	subset	NOUN
ejpam-4750	441	12	of	of	ADP
ejpam-4750	441	13	sg	sg	PROPN
ejpam-4750	441	14	and	and	CCONJ
ejpam-4750	441	15	sh	sh	INTJ
ejpam-4750	441	16	,	,	PUNCT
ejpam-4750	441	17	respectively	respectively	ADV
ejpam-4750	441	18	.	.	PUNCT
ejpam-4750	442	1	hence	hence	ADV
ejpam-4750	442	2	,	,	PUNCT
ejpam-4750	442	3	fg	fg	PROPN
ejpam-4750	442	4	∪	∪	PROPN
ejpam-4750	442	5	fh	fh	PROPN
ejpam-4750	442	6	is	be	AUX
ejpam-4750	442	7	a	a	DET
ejpam-4750	442	8	forcing	force	VERB
ejpam-4750	442	9	subset	subset	NOUN
ejpam-4750	442	10	of	of	ADP
ejpam-4750	442	11	s.	s.	PROPN
ejpam-4750	442	12	thus	thus	ADV
ejpam-4750	442	13	,	,	PUNCT
ejpam-4750	442	14	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	442	15	)	)	PUNCT
ejpam-4750	442	16	≤	≤	NUM
ejpam-4750	442	17	fdim2(s	fdim2(s	NOUN
ejpam-4750	442	18	)	)	PUNCT
ejpam-4750	442	19	≤	≤	NOUN
ejpam-4750	443	1	|fg	|fg	NUM
ejpam-4750	443	2	∪	∪	ADP
ejpam-4750	443	3	fh	fh	PROPN
ejpam-4750	443	4	|	|	NOUN
ejpam-4750	443	5	=	=	PUNCT
ejpam-4750	443	6	|fg|+	|fg|+	NOUN
ejpam-4750	443	7	|fh	|fh	NUM
ejpam-4750	443	8	|	|	ADV
ejpam-4750	443	9	=	=	PUNCT
ejpam-4750	443	10	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	443	11	)	)	PUNCT
ejpam-4750	443	12	+	+	CCONJ
ejpam-4750	443	13	fln(2,1)(h	fln(2,1)(h	PROPN
ejpam-4750	443	14	)	)	PUNCT
ejpam-4750	443	15	.	.	PUNCT
ejpam-4750	444	1	next	next	ADV
ejpam-4750	444	2	,	,	PUNCT
ejpam-4750	444	3	suppose	suppose	VERB
ejpam-4750	444	4	that	that	SCONJ
ejpam-4750	444	5	s0	s0	PROPN
ejpam-4750	444	6	is	be	AUX
ejpam-4750	444	7	a	a	DET
ejpam-4750	444	8	2	2	NUM
ejpam-4750	444	9	-	-	PUNCT
ejpam-4750	444	10	metric	metric	ADJ
ejpam-4750	444	11	basis	basis	NOUN
ejpam-4750	444	12	for	for	ADP
ejpam-4750	444	13	g	g	PROPN
ejpam-4750	444	14	+	+	NOUN
ejpam-4750	444	15	h	h	NOUN
ejpam-4750	444	16	such	such	ADJ
ejpam-4750	444	17	that	that	SCONJ
ejpam-4750	444	18	fdim2(g	fdim2(g	NOUN
ejpam-4750	444	19	+	+	CCONJ
ejpam-4750	444	20	h	h	X
ejpam-4750	444	21	)	)	PUNCT
ejpam-4750	444	22	=	=	SYM
ejpam-4750	444	23	fdim2(s0	fdim2(s0	PROPN
ejpam-4750	444	24	)	)	PUNCT
ejpam-4750	444	25	.	.	PUNCT
ejpam-4750	445	1	by	by	ADP
ejpam-4750	445	2	theorem	theorem	NOUN
ejpam-4750	445	3	11	11	NUM
ejpam-4750	445	4	,	,	PUNCT
ejpam-4750	445	5	s0	s0	NOUN
ejpam-4750	445	6	=	=	SYM
ejpam-4750	445	7	s0	s0	PROPN
ejpam-4750	445	8	g	g	PROPN
ejpam-4750	445	9	∪	∪	PROPN
ejpam-4750	445	10	s0	s0	PROPN
ejpam-4750	445	11	h	h	PROPN
ejpam-4750	445	12	where	where	SCONJ
ejpam-4750	445	13	s0	s0	PROPN
ejpam-4750	445	14	g	g	PROPN
ejpam-4750	445	15	and	and	CCONJ
ejpam-4750	445	16	s0	s0	PROPN
ejpam-4750	445	17	h	h	PROPN
ejpam-4750	445	18	are	be	AUX
ejpam-4750	445	19	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	445	20	of	of	ADP
ejpam-4750	445	21	g	g	NOUN
ejpam-4750	445	22	and	and	CCONJ
ejpam-4750	445	23	h	h	NOUN
ejpam-4750	445	24	,	,	PUNCT
ejpam-4750	445	25	respectively	respectively	ADV
ejpam-4750	445	26	.	.	PUNCT
ejpam-4750	446	1	let	let	VERB
ejpam-4750	446	2	f0	f0	PROPN
ejpam-4750	446	3	=	=	SYM
ejpam-4750	446	4	f	f	PROPN
ejpam-4750	446	5	0	0	NUM
ejpam-4750	446	6	g∪f	g∪f	NOUN
ejpam-4750	446	7	0	0	NUM
ejpam-4750	446	8	h	h	NOUN
ejpam-4750	446	9	,	,	PUNCT
ejpam-4750	446	10	where	where	SCONJ
ejpam-4750	446	11	f	f	PROPN
ejpam-4750	446	12	0	0	NUM
ejpam-4750	446	13	g	g	PROPN
ejpam-4750	446	14	⊆	⊆	NUM
ejpam-4750	446	15	s0	s0	PROPN
ejpam-4750	446	16	g	g	PROPN
ejpam-4750	446	17	and	and	CCONJ
ejpam-4750	446	18	f	f	PROPN
ejpam-4750	446	19	0	0	PROPN
ejpam-4750	446	20	h	h	NOUN
ejpam-4750	446	21	⊆	⊆	NUM
ejpam-4750	446	22	s0	s0	PROPN
ejpam-4750	446	23	h	h	PROPN
ejpam-4750	446	24	,	,	PUNCT
ejpam-4750	446	25	be	be	AUX
ejpam-4750	446	26	a	a	DET
ejpam-4750	446	27	forcing	force	VERB
ejpam-4750	446	28	subset	subset	NOUN
ejpam-4750	446	29	of	of	ADP
ejpam-4750	446	30	s0	s0	PROPN
ejpam-4750	446	31	such	such	ADJ
ejpam-4750	446	32	that	that	SCONJ
ejpam-4750	446	33	fdim2(s0	fdim2(s0	NOUN
ejpam-4750	446	34	)	)	PUNCT
ejpam-4750	447	1	=	=	NOUN
ejpam-4750	447	2	|f0|	|f0|	NOUN
ejpam-4750	447	3	.	.	PUNCT
ejpam-4750	448	1	suppose	suppose	VERB
ejpam-4750	448	2	f	f	PROPN
ejpam-4750	448	3	0	0	NUM
ejpam-4750	448	4	g	g	PROPN
ejpam-4750	448	5	⊆	⊆	NUM
ejpam-4750	448	6	d0	d0	NOUN
ejpam-4750	448	7	g	g	NOUN
ejpam-4750	448	8	for	for	ADP
ejpam-4750	448	9	some	some	DET
ejpam-4750	448	10	ln(2,1)-set	ln(2,1)-set	VERB
ejpam-4750	448	11	d0	d0	NOUN
ejpam-4750	448	12	g	g	NOUN
ejpam-4750	448	13	of	of	ADP
ejpam-4750	448	14	g	g	NOUN
ejpam-4750	448	15	with	with	ADP
ejpam-4750	448	16	d0	d0	NOUN
ejpam-4750	448	17	g	g	PROPN
ejpam-4750	448	18	̸=	̸=	PROPN
ejpam-4750	448	19	s0	s0	PROPN
ejpam-4750	448	20	g.	g.	PROPN
ejpam-4750	448	21	then	then	ADV
ejpam-4750	448	22	s′	s′	NUM
ejpam-4750	448	23	0	0	NUM
ejpam-4750	448	24	=	=	SYM
ejpam-4750	448	25	d0	d0	PROPN
ejpam-4750	448	26	g	g	PROPN
ejpam-4750	448	27	∪	∪	ADJ
ejpam-4750	448	28	s0	s0	PROPN
ejpam-4750	448	29	h	h	PROPN
ejpam-4750	448	30	is	be	AUX
ejpam-4750	448	31	a	a	DET
ejpam-4750	448	32	2	2	NUM
ejpam-4750	448	33	-	-	PUNCT
ejpam-4750	448	34	metric	metric	ADJ
ejpam-4750	448	35	basis	basis	NOUN
ejpam-4750	448	36	for	for	ADP
ejpam-4750	448	37	g+h	g+h	PROPN
ejpam-4750	448	38	,	,	PUNCT
ejpam-4750	448	39	s′	s′	X
ejpam-4750	448	40	0	0	NUM
ejpam-4750	448	41	̸=	̸=	PROPN
ejpam-4750	448	42	s0	s0	NOUN
ejpam-4750	448	43	,	,	PUNCT
ejpam-4750	448	44	and	and	CCONJ
ejpam-4750	448	45	f0	f0	VERB
ejpam-4750	448	46	⊆	⊆	NUM
ejpam-4750	448	47	s′	s′	NUM
ejpam-4750	448	48	0	0	NUM
ejpam-4750	448	49	.	.	PUNCT
ejpam-4750	449	1	this	this	PRON
ejpam-4750	449	2	contradicts	contradict	VERB
ejpam-4750	449	3	the	the	DET
ejpam-4750	449	4	assumption	assumption	NOUN
ejpam-4750	449	5	that	that	SCONJ
ejpam-4750	449	6	f0	f0	PROPN
ejpam-4750	449	7	is	be	AUX
ejpam-4750	449	8	a	a	DET
ejpam-4750	449	9	forcing	force	VERB
ejpam-4750	449	10	subset	subset	NOUN
ejpam-4750	449	11	of	of	ADP
ejpam-4750	449	12	s0	s0	PROPN
ejpam-4750	449	13	.	.	PUNCT
ejpam-4750	450	1	thus	thus	ADV
ejpam-4750	450	2	,	,	PUNCT
ejpam-4750	450	3	f	f	PROPN
ejpam-4750	450	4	0	0	NUM
ejpam-4750	450	5	g	g	NOUN
ejpam-4750	450	6	is	be	AUX
ejpam-4750	450	7	a	a	DET
ejpam-4750	450	8	forcing	force	VERB
ejpam-4750	450	9	subset	subset	NOUN
ejpam-4750	450	10	of	of	ADP
ejpam-4750	450	11	s0	s0	PROPN
ejpam-4750	450	12	g.	g.	PROPN
ejpam-4750	450	13	similarly	similarly	ADV
ejpam-4750	450	14	,	,	PUNCT
ejpam-4750	450	15	f	f	PROPN
ejpam-4750	450	16	0	0	NUM
ejpam-4750	450	17	h	h	NOUN
ejpam-4750	450	18	is	be	AUX
ejpam-4750	450	19	a	a	DET
ejpam-4750	450	20	forcing	force	VERB
ejpam-4750	450	21	subset	subset	NOUN
ejpam-4750	450	22	of	of	ADP
ejpam-4750	450	23	s0	s0	PROPN
ejpam-4750	450	24	h	h	PROPN
ejpam-4750	450	25	.	.	PUNCT
ejpam-4750	451	1	hence	hence	ADV
ejpam-4750	451	2	,	,	PUNCT
ejpam-4750	451	3	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	451	4	)	)	PUNCT
ejpam-4750	451	5	=	=	SYM
ejpam-4750	451	6	fdim2(s0	fdim2(s0	PROPN
ejpam-4750	451	7	)	)	PUNCT
ejpam-4750	451	8	=	=	NOUN
ejpam-4750	451	9	|f0|	|f0|	NOUN
ejpam-4750	451	10	=	=	SYM
ejpam-4750	451	11	|f	|f	PROPN
ejpam-4750	451	12	0	0	NUM
ejpam-4750	452	1	g	g	PROPN
ejpam-4750	452	2	∪	∪	PROPN
ejpam-4750	452	3	f	f	PROPN
ejpam-4750	452	4	0	0	NUM
ejpam-4750	452	5	h	h	NOUN
ejpam-4750	453	1	|	|	ADV
ejpam-4750	453	2	=	=	SYM
ejpam-4750	453	3	|f	|f	PROPN
ejpam-4750	453	4	0	0	PUNCT
ejpam-4750	454	1	g|+	g|+	PROPN
ejpam-4750	454	2	|f	|f	PROPN
ejpam-4750	454	3	0	0	NUM
ejpam-4750	455	1	h	h	NOUN
ejpam-4750	456	1	|	|	ADV
ejpam-4750	456	2	≥	≥	NOUN
ejpam-4750	456	3	fln(2,1)(s	fln(2,1)(s	PROPN
ejpam-4750	456	4	0	0	NUM
ejpam-4750	456	5	g	g	NOUN
ejpam-4750	456	6	)	)	PUNCT
ejpam-4750	457	1	+	+	CCONJ
ejpam-4750	457	2	fln(2,1)(s	fln(2,1)(s	PROPN
ejpam-4750	457	3	0	0	NUM
ejpam-4750	457	4	h	h	NOUN
ejpam-4750	457	5	)	)	PUNCT
ejpam-4750	457	6	≥	≥	NOUN
ejpam-4750	457	7	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	457	8	)	)	PUNCT
ejpam-4750	457	9	+	+	CCONJ
ejpam-4750	457	10	fln(2,1)(h	fln(2,1)(h	PROPN
ejpam-4750	457	11	)	)	PUNCT
ejpam-4750	457	12	.	.	PUNCT
ejpam-4750	458	1	therefore	therefore	ADV
ejpam-4750	458	2	,	,	PUNCT
ejpam-4750	458	3	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	458	4	)	)	PUNCT
ejpam-4750	458	5	=	=	SYM
ejpam-4750	458	6	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	458	7	)	)	PUNCT
ejpam-4750	458	8	+	+	CCONJ
ejpam-4750	458	9	fln(2,1)(h	fln(2,1)(h	PROPN
ejpam-4750	458	10	)	)	PUNCT
ejpam-4750	458	11	.	.	PUNCT
ejpam-4750	459	1	d.	d.	PROPN
ejpam-4750	459	2	managbanag	managbanag	PROPN
ejpam-4750	459	3	,	,	PUNCT
ejpam-4750	459	4	h.	h.	PROPN
ejpam-4750	459	5	rara	rara	PROPN
ejpam-4750	459	6	/	/	SYM
ejpam-4750	459	7	eur	eur	PROPN
ejpam-4750	459	8	.	.	PUNCT
ejpam-4750	460	1	j.	j.	PROPN
ejpam-4750	460	2	pure	pure	PROPN
ejpam-4750	460	3	appl	appl	PROPN
ejpam-4750	460	4	.	.	PROPN
ejpam-4750	460	5	math	math	PROPN
ejpam-4750	460	6	,	,	PUNCT
ejpam-4750	460	7	16	16	NUM
ejpam-4750	460	8	(	(	PUNCT
ejpam-4750	460	9	2	2	NUM
ejpam-4750	460	10	)	)	PUNCT
ejpam-4750	460	11	(	(	PUNCT
ejpam-4750	460	12	2023	2023	NUM
ejpam-4750	460	13	)	)	PUNCT
ejpam-4750	460	14	,	,	PUNCT
ejpam-4750	460	15	1068	1068	NUM
ejpam-4750	460	16	-	-	SYM
ejpam-4750	460	17	1083	1083	NUM
ejpam-4750	460	18	1081	1081	NUM
ejpam-4750	460	19	example	example	NOUN
ejpam-4750	460	20	6	6	NUM
ejpam-4750	460	21	.	.	PUNCT
ejpam-4750	460	22	consider	consider	VERB
ejpam-4750	460	23	the	the	DET
ejpam-4750	460	24	join	join	NOUN
ejpam-4750	460	25	of	of	ADP
ejpam-4750	460	26	two	two	NUM
ejpam-4750	460	27	graphs	graph	NOUN
ejpam-4750	460	28	g	g	NOUN
ejpam-4750	460	29	and	and	CCONJ
ejpam-4750	460	30	h	h	NOUN
ejpam-4750	460	31	,	,	PUNCT
ejpam-4750	460	32	where	where	SCONJ
ejpam-4750	460	33	(	(	PUNCT
ejpam-4750	460	34	2,2)-locating	2,2)-locating	NUM
ejpam-4750	460	35	sets	set	NOUN
ejpam-4750	460	36	do	do	AUX
ejpam-4750	460	37	not	not	PART
ejpam-4750	460	38	exist	exist	VERB
ejpam-4750	460	39	for	for	ADP
ejpam-4750	460	40	both	both	DET
ejpam-4750	460	41	graphs	graph	NOUN
ejpam-4750	460	42	.	.	PUNCT
ejpam-4750	461	1	a	a	DET
ejpam-4750	461	2	b	b	X
ejpam-4750	461	3	c	c	NOUN
ejpam-4750	461	4	d	d	X
ejpam-4750	461	5	f	f	PROPN
ejpam-4750	461	6	g	g	NOUN
ejpam-4750	462	1	h	h	NOUN
ejpam-4750	463	1	i	i	PRON
ejpam-4750	463	2	j	j	PROPN
ejpam-4750	464	1	k	k	PROPN
ejpam-4750	464	2	g	g	NOUN
ejpam-4750	464	3	:	:	PUNCT
ejpam-4750	464	4	h	h	NOUN
ejpam-4750	464	5	:	:	PUNCT
ejpam-4750	464	6	e	e	X
ejpam-4750	464	7	the	the	DET
ejpam-4750	464	8	join	join	NOUN
ejpam-4750	464	9	g	g	PROPN
ejpam-4750	464	10	+	+	CCONJ
ejpam-4750	464	11	h	h	NOUN
ejpam-4750	464	12	note	note	NOUN
ejpam-4750	464	13	that	that	SCONJ
ejpam-4750	464	14	g	g	PROPN
ejpam-4750	464	15	has	have	VERB
ejpam-4750	464	16	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	464	17	w1	w1	NOUN
ejpam-4750	464	18	=	=	SYM
ejpam-4750	464	19	{	{	PUNCT
ejpam-4750	464	20	a	a	X
ejpam-4750	464	21	,	,	PUNCT
ejpam-4750	464	22	c	c	NOUN
ejpam-4750	464	23	,	,	PUNCT
ejpam-4750	464	24	d	d	NOUN
ejpam-4750	464	25	,	,	PUNCT
ejpam-4750	464	26	uf	uf	NOUN
ejpam-4750	464	27	}	}	PUNCT
ejpam-4750	464	28	,	,	PUNCT
ejpam-4750	464	29	w2	w2	NOUN
ejpam-4750	464	30	=	=	SYM
ejpam-4750	464	31	{	{	PUNCT
ejpam-4750	464	32	b	b	PROPN
ejpam-4750	464	33	,	,	PUNCT
ejpam-4750	464	34	c	c	NOUN
ejpam-4750	464	35	,	,	PUNCT
ejpam-4750	464	36	d	d	NOUN
ejpam-4750	464	37	,	,	PUNCT
ejpam-4750	464	38	e	e	NOUN
ejpam-4750	464	39	}	}	PUNCT
ejpam-4750	464	40	,	,	PUNCT
ejpam-4750	464	41	w3	w3	PROPN
ejpam-4750	464	42	=	=	SYM
ejpam-4750	464	43	{	{	PUNCT
ejpam-4750	464	44	b	b	PROPN
ejpam-4750	464	45	,	,	PUNCT
ejpam-4750	464	46	c	c	NOUN
ejpam-4750	464	47	,	,	PUNCT
ejpam-4750	464	48	d	d	NOUN
ejpam-4750	464	49	,	,	PUNCT
ejpam-4750	464	50	f	f	NOUN
ejpam-4750	464	51	}	}	PUNCT
ejpam-4750	464	52	,	,	PUNCT
ejpam-4750	464	53	w4	w4	NOUN
ejpam-4750	464	54	=	=	SYM
ejpam-4750	464	55	{	{	PUNCT
ejpam-4750	464	56	b	b	PROPN
ejpam-4750	464	57	,	,	PUNCT
ejpam-4750	464	58	d	d	NOUN
ejpam-4750	464	59	,	,	PUNCT
ejpam-4750	464	60	e	e	NOUN
ejpam-4750	464	61	,	,	PUNCT
ejpam-4750	464	62	f	f	NOUN
ejpam-4750	464	63	}	}	PUNCT
ejpam-4750	464	64	and	and	CCONJ
ejpam-4750	464	65	w5	w5	PROPN
ejpam-4750	464	66	=	=	PUNCT
ejpam-4750	464	67	{	{	PUNCT
ejpam-4750	464	68	c	c	NOUN
ejpam-4750	464	69	,	,	PUNCT
ejpam-4750	464	70	d	d	NOUN
ejpam-4750	464	71	,	,	PUNCT
ejpam-4750	464	72	e	e	NOUN
ejpam-4750	464	73	,	,	PUNCT
ejpam-4750	464	74	f	f	NOUN
ejpam-4750	464	75	}	}	PUNCT
ejpam-4750	464	76	.	.	PUNCT
ejpam-4750	465	1	on	on	ADP
ejpam-4750	465	2	the	the	DET
ejpam-4750	465	3	other	other	ADJ
ejpam-4750	465	4	hand	hand	NOUN
ejpam-4750	465	5	,	,	PUNCT
ejpam-4750	465	6	h	h	NOUN
ejpam-4750	465	7	have	have	VERB
ejpam-4750	465	8	ln(2,1)-sets	ln(2,1)-set	NOUN
ejpam-4750	465	9	z1	z1	NOUN
ejpam-4750	465	10	=	=	SYM
ejpam-4750	465	11	{	{	PUNCT
ejpam-4750	465	12	g	g	PROPN
ejpam-4750	465	13	,	,	PUNCT
ejpam-4750	465	14	h	h	NOUN
ejpam-4750	465	15	,	,	PUNCT
ejpam-4750	465	16	j	j	PROPN
ejpam-4750	465	17	,	,	PUNCT
ejpam-4750	465	18	k	k	NOUN
ejpam-4750	465	19	}	}	PUNCT
ejpam-4750	465	20	,	,	PUNCT
ejpam-4750	465	21	z2	z2	PROPN
ejpam-4750	465	22	=	=	SYM
ejpam-4750	465	23	{	{	PUNCT
ejpam-4750	465	24	g	g	PROPN
ejpam-4750	465	25	,	,	PUNCT
ejpam-4750	465	26	h	h	NOUN
ejpam-4750	465	27	,	,	PUNCT
ejpam-4750	465	28	i	i	PRON
ejpam-4750	465	29	,	,	PUNCT
ejpam-4750	465	30	k	k	NOUN
ejpam-4750	465	31	}	}	PUNCT
ejpam-4750	465	32	and	and	CCONJ
ejpam-4750	465	33	z3	z3	PROPN
ejpam-4750	465	34	=	=	PUNCT
ejpam-4750	465	35	{	{	PUNCT
ejpam-4750	465	36	g	g	PROPN
ejpam-4750	465	37	,	,	PUNCT
ejpam-4750	465	38	i	i	PROPN
ejpam-4750	465	39	,	,	PUNCT
ejpam-4750	465	40	j	j	PROPN
ejpam-4750	465	41	,	,	PUNCT
ejpam-4750	465	42	k	k	NOUN
ejpam-4750	465	43	}	}	PUNCT
ejpam-4750	465	44	.	.	PUNCT
ejpam-4750	466	1	clearly	clearly	ADV
ejpam-4750	466	2	,	,	PUNCT
ejpam-4750	466	3	a	a	DET
ejpam-4750	466	4	∈	∈	PROPN
ejpam-4750	466	5	w1	w1	NOUN
ejpam-4750	466	6	and	and	CCONJ
ejpam-4750	466	7	a	a	DET
ejpam-4750	466	8	/∈	/∈	NOUN
ejpam-4750	466	9	w2	w2	NOUN
ejpam-4750	466	10	,	,	PUNCT
ejpam-4750	466	11	w3,w4,w5	w3,w4,w5	PROPN
ejpam-4750	466	12	.	.	PUNCT
ejpam-4750	467	1	also	also	ADV
ejpam-4750	467	2	,	,	PUNCT
ejpam-4750	467	3	{	{	PUNCT
ejpam-4750	467	4	h	h	NOUN
ejpam-4750	467	5	,	,	PUNCT
ejpam-4750	467	6	j	j	PROPN
ejpam-4750	467	7	}	}	PUNCT
ejpam-4750	467	8	⊆	⊆	NUM
ejpam-4750	467	9	z1	z1	PROPN
ejpam-4750	467	10	and	and	CCONJ
ejpam-4750	467	11	{	{	PUNCT
ejpam-4750	467	12	h	h	NOUN
ejpam-4750	467	13	,	,	PUNCT
ejpam-4750	467	14	j	j	PROPN
ejpam-4750	467	15	}	}	PUNCT
ejpam-4750	467	16	⊈	⊈	PROPN
ejpam-4750	467	17	z2	z2	PROPN
ejpam-4750	467	18	,	,	PUNCT
ejpam-4750	467	19	z3	z3	PROPN
ejpam-4750	467	20	.	.	PUNCT
ejpam-4750	468	1	thus	thus	ADV
ejpam-4750	468	2	,	,	PUNCT
ejpam-4750	468	3	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	468	4	)	)	PUNCT
ejpam-4750	468	5	=	=	SYM
ejpam-4750	468	6	1	1	NUM
ejpam-4750	468	7	and	and	CCONJ
ejpam-4750	468	8	fln(2,1)(h	fln(2,1)(h	NUM
ejpam-4750	468	9	)	)	PUNCT
ejpam-4750	468	10	=	=	SYM
ejpam-4750	468	11	2	2	X
ejpam-4750	468	12	.	.	X
ejpam-4750	468	13	hence	hence	ADV
ejpam-4750	468	14	,	,	PUNCT
ejpam-4750	468	15	fdim2(g+h	fdim2(g+h	NOUN
ejpam-4750	468	16	)	)	PUNCT
ejpam-4750	468	17	=	=	SYM
ejpam-4750	468	18	fln(2,1)(g	fln(2,1)(g	PROPN
ejpam-4750	468	19	)	)	PUNCT
ejpam-4750	468	20	+	+	CCONJ
ejpam-4750	468	21	fln(2,1)(h	fln(2,1)(h	PROPN
ejpam-4750	468	22	)	)	PUNCT
ejpam-4750	468	23	=	=	SYM
ejpam-4750	468	24	1	1	NUM
ejpam-4750	468	25	+	+	NUM
ejpam-4750	468	26	2	2	NUM
ejpam-4750	468	27	=	=	SYM
ejpam-4750	468	28	3	3	NUM
ejpam-4750	468	29	.	.	NOUN
ejpam-4750	468	30	7	7	NUM
ejpam-4750	468	31	.	.	X
ejpam-4750	469	1	forcing	force	VERB
ejpam-4750	469	2	2	2	NUM
ejpam-4750	469	3	-	-	PUNCT
ejpam-4750	469	4	metric	metric	ADJ
ejpam-4750	469	5	dimension	dimension	NOUN
ejpam-4750	469	6	in	in	ADP
ejpam-4750	469	7	the	the	DET
ejpam-4750	469	8	corona	corona	NOUN
ejpam-4750	469	9	of	of	ADP
ejpam-4750	469	10	graphs	graph	NOUN
ejpam-4750	469	11	the	the	DET
ejpam-4750	469	12	corona	corona	NOUN
ejpam-4750	469	13	of	of	ADP
ejpam-4750	469	14	two	two	NUM
ejpam-4750	469	15	graphs	graph	NOUN
ejpam-4750	469	16	g	g	NOUN
ejpam-4750	469	17	and	and	CCONJ
ejpam-4750	469	18	h	h	NOUN
ejpam-4750	469	19	,	,	PUNCT
ejpam-4750	469	20	denoted	denote	VERB
ejpam-4750	469	21	by	by	ADP
ejpam-4750	469	22	g	g	PROPN
ejpam-4750	469	23	◦	◦	NOUN
ejpam-4750	469	24	h	h	NOUN
ejpam-4750	469	25	,	,	PUNCT
ejpam-4750	469	26	is	be	AUX
ejpam-4750	469	27	the	the	DET
ejpam-4750	469	28	graph	graph	NOUN
ejpam-4750	469	29	obtained	obtain	VERB
ejpam-4750	469	30	by	by	ADP
ejpam-4750	469	31	taking	take	VERB
ejpam-4750	469	32	one	one	NUM
ejpam-4750	469	33	copy	copy	NOUN
ejpam-4750	469	34	of	of	ADP
ejpam-4750	469	35	g	g	NOUN
ejpam-4750	469	36	of	of	ADP
ejpam-4750	469	37	order	order	NOUN
ejpam-4750	469	38	n	n	NOUN
ejpam-4750	469	39	and	and	CCONJ
ejpam-4750	469	40	n	n	PRON
ejpam-4750	469	41	copies	copy	NOUN
ejpam-4750	469	42	of	of	ADP
ejpam-4750	469	43	h	h	NOUN
ejpam-4750	469	44	,	,	PUNCT
ejpam-4750	469	45	and	and	CCONJ
ejpam-4750	469	46	then	then	ADV
ejpam-4750	469	47	joining	join	VERB
ejpam-4750	469	48	every	every	DET
ejpam-4750	469	49	vertex	vertex	NOUN
ejpam-4750	469	50	of	of	ADP
ejpam-4750	469	51	the	the	DET
ejpam-4750	469	52	ith	ith	PROPN
ejpam-4750	469	53	copy	copy	NOUN
ejpam-4750	469	54	of	of	ADP
ejpam-4750	469	55	h	h	NOUN
ejpam-4750	469	56	to	to	ADP
ejpam-4750	469	57	the	the	DET
ejpam-4750	469	58	ith	ith	PROPN
ejpam-4750	469	59	vertex	vertex	NOUN
ejpam-4750	469	60	of	of	ADP
ejpam-4750	469	61	g.	g.	PROPN
ejpam-4750	469	62	for	for	ADP
ejpam-4750	469	63	v	v	NOUN
ejpam-4750	469	64	∈	∈	PROPN
ejpam-4750	469	65	v	v	NOUN
ejpam-4750	469	66	(	(	PUNCT
ejpam-4750	469	67	g	g	NOUN
ejpam-4750	469	68	)	)	PUNCT
ejpam-4750	469	69	,	,	PUNCT
ejpam-4750	469	70	denote	denote	VERB
ejpam-4750	469	71	by	by	ADP
ejpam-4750	469	72	hv	hv	PROPN
ejpam-4750	469	73	the	the	DET
ejpam-4750	469	74	copy	copy	NOUN
ejpam-4750	469	75	of	of	ADP
ejpam-4750	469	76	h	h	NOUN
ejpam-4750	469	77	whose	whose	DET
ejpam-4750	469	78	vertices	vertex	NOUN
ejpam-4750	469	79	are	be	AUX
ejpam-4750	469	80	attached	attach	VERB
ejpam-4750	469	81	one	one	NUM
ejpam-4750	469	82	by	by	ADP
ejpam-4750	469	83	one	one	NUM
ejpam-4750	469	84	to	to	ADP
ejpam-4750	469	85	the	the	DET
ejpam-4750	469	86	vertex	vertex	NOUN
ejpam-4750	469	87	v.	v.	ADP
ejpam-4750	469	88	subsequently	subsequently	ADV
ejpam-4750	469	89	,	,	PUNCT
ejpam-4750	469	90	denote	denote	VERB
ejpam-4750	469	91	by	by	ADP
ejpam-4750	469	92	v+hv	v+hv	NOUN
ejpam-4750	469	93	the	the	DET
ejpam-4750	469	94	subgraph	subgraph	NOUN
ejpam-4750	469	95	of	of	ADP
ejpam-4750	469	96	the	the	DET
ejpam-4750	469	97	corona	corona	NOUN
ejpam-4750	469	98	g	g	PROPN
ejpam-4750	469	99	◦	◦	NOUN
ejpam-4750	469	100	h	h	NOUN
ejpam-4750	469	101	corresponding	correspond	VERB
ejpam-4750	469	102	to	to	ADP
ejpam-4750	469	103	the	the	DET
ejpam-4750	469	104	join	join	NOUN
ejpam-4750	469	105	⟨{v}⟩+hv	⟨{v}⟩+hv	PROPN
ejpam-4750	469	106	,	,	PUNCT
ejpam-4750	469	107	v	v	PROPN
ejpam-4750	469	108	∈	∈	PROPN
ejpam-4750	469	109	v	v	NOUN
ejpam-4750	469	110	(	(	PUNCT
ejpam-4750	469	111	g	g	NOUN
ejpam-4750	469	112	)	)	PUNCT
ejpam-4750	469	113	.	.	PUNCT
ejpam-4750	470	1	let	let	VERB
ejpam-4750	470	2	s	s	PRON
ejpam-4750	470	3	=	=	PUNCT
ejpam-4750	470	4	⋃	⋃	NOUN
ejpam-4750	470	5	v∈v	v∈v	NOUN
ejpam-4750	470	6	(	(	PUNCT
ejpam-4750	470	7	g	g	NOUN
ejpam-4750	470	8	)	)	PUNCT
ejpam-4750	470	9	sv	sv	VERB
ejpam-4750	471	1	⊆	⊆	NUM
ejpam-4750	471	2	v	v	NOUN
ejpam-4750	471	3	(	(	PUNCT
ejpam-4750	471	4	g	g	PROPN
ejpam-4750	471	5	◦	◦	NOUN
ejpam-4750	471	6	h	h	NOUN
ejpam-4750	471	7	)	)	PUNCT
ejpam-4750	471	8	be	be	VERB
ejpam-4750	471	9	a	a	DET
ejpam-4750	471	10	2	2	NUM
ejpam-4750	471	11	-	-	PUNCT
ejpam-4750	471	12	resolving	resolve	VERB
ejpam-4750	471	13	set	set	NOUN
ejpam-4750	471	14	of	of	ADP
ejpam-4750	471	15	g	g	PROPN
ejpam-4750	471	16	◦	◦	NOUN
ejpam-4750	471	17	h	h	NOUN
ejpam-4750	471	18	and	and	CCONJ
ejpam-4750	471	19	p	p	X
ejpam-4750	471	20	,	,	PUNCT
ejpam-4750	471	21	q	q	PROPN
ejpam-4750	471	22	∈	∈	PROPN
ejpam-4750	471	23	v	v	ADP
ejpam-4750	471	24	(	(	PUNCT
ejpam-4750	471	25	hv	hv	PROPN
ejpam-4750	471	26	)	)	PUNCT
ejpam-4750	471	27	where	where	SCONJ
ejpam-4750	471	28	p	p	PROPN
ejpam-4750	471	29	̸=	̸=	PROPN
ejpam-4750	471	30	q	q	NOUN
ejpam-4750	471	31	for	for	ADP
ejpam-4750	471	32	v	v	NOUN
ejpam-4750	471	33	∈	∈	PROPN
ejpam-4750	471	34	v	v	NOUN
ejpam-4750	471	35	(	(	PUNCT
ejpam-4750	471	36	g	g	NOUN
ejpam-4750	471	37	)	)	PUNCT
ejpam-4750	471	38	.	.	PUNCT
ejpam-4750	472	1	then	then	ADV
ejpam-4750	472	2	rg	rg	PROPN
ejpam-4750	472	3	◦	◦	NOUN
ejpam-4750	472	4	h(p	h(p	NOUN
ejpam-4750	472	5	/	/	SYM
ejpam-4750	472	6	s	s	NOUN
ejpam-4750	472	7	)	)	PUNCT
ejpam-4750	472	8	and	and	CCONJ
ejpam-4750	472	9	rg	rg	NOUN
ejpam-4750	472	10	◦	◦	NOUN
ejpam-4750	472	11	h(q	h(q	ADV
ejpam-4750	472	12	/	/	SYM
ejpam-4750	472	13	s	s	X
ejpam-4750	472	14	)	)	PUNCT
ejpam-4750	472	15	differ	differ	VERB
ejpam-4750	472	16	in	in	ADP
ejpam-4750	472	17	at	at	ADV
ejpam-4750	472	18	least	least	ADJ
ejpam-4750	472	19	2	2	NUM
ejpam-4750	472	20	-	-	PUNCT
ejpam-4750	472	21	positions	position	NOUN
ejpam-4750	472	22	.	.	PUNCT
ejpam-4750	473	1	by	by	ADP
ejpam-4750	473	2	remark	remark	NOUN
ejpam-4750	473	3	2	2	NUM
ejpam-4750	473	4	,	,	PUNCT
ejpam-4750	473	5	rhv(p	rhv(p	PROPN
ejpam-4750	473	6	/	/	SYM
ejpam-4750	473	7	sv	sv	NOUN
ejpam-4750	473	8	)	)	PUNCT
ejpam-4750	473	9	and	and	CCONJ
ejpam-4750	473	10	rhv(q	rhv(q	PROPN
ejpam-4750	473	11	/	/	SYM
ejpam-4750	473	12	sv	sv	NOUN
ejpam-4750	473	13	)	)	PUNCT
ejpam-4750	473	14	must	must	AUX
ejpam-4750	473	15	differ	differ	VERB
ejpam-4750	473	16	in	in	ADP
ejpam-4750	473	17	at	at	ADV
ejpam-4750	473	18	least	least	ADV
ejpam-4750	473	19	two	two	NUM
ejpam-4750	473	20	distinct	distinct	ADJ
ejpam-4750	473	21	positions	position	NOUN
ejpam-4750	473	22	.	.	PUNCT
ejpam-4750	474	1	by	by	ADP
ejpam-4750	474	2	definition	definition	NOUN
ejpam-4750	474	3	of	of	ADP
ejpam-4750	474	4	g	g	PROPN
ejpam-4750	474	5	◦	◦	NOUN
ejpam-4750	474	6	h	h	NOUN
ejpam-4750	474	7	,	,	PUNCT
ejpam-4750	474	8	if	if	SCONJ
ejpam-4750	474	9	p	p	X
ejpam-4750	474	10	,	,	PUNCT
ejpam-4750	474	11	q	q	PROPN
ejpam-4750	474	12	∈	∈	PROPN
ejpam-4750	474	13	v	v	ADP
ejpam-4750	474	14	(	(	PUNCT
ejpam-4750	474	15	hv	hv	PROPN
ejpam-4750	474	16	)	)	PUNCT
ejpam-4750	474	17	\	\	PROPN
ejpam-4750	475	1	sv	sv	PROPN
ejpam-4750	475	2	,	,	PUNCT
ejpam-4750	475	3	then	then	ADV
ejpam-4750	475	4	there	there	PRON
ejpam-4750	475	5	exist	exist	VERB
ejpam-4750	475	6	at	at	ADV
ejpam-4750	475	7	least	least	ADV
ejpam-4750	475	8	two	two	NUM
ejpam-4750	475	9	distinct	distinct	ADJ
ejpam-4750	475	10	vertices	vertex	NOUN
ejpam-4750	475	11	r	r	NOUN
ejpam-4750	475	12	,	,	PUNCT
ejpam-4750	475	13	s	s	NOUN
ejpam-4750	475	14	∈	∈	PROPN
ejpam-4750	475	15	v	v	ADP
ejpam-4750	475	16	(	(	PUNCT
ejpam-4750	475	17	hv	hv	NOUN
ejpam-4750	475	18	)	)	PUNCT
ejpam-4750	475	19	∩	∩	NOUN
ejpam-4750	475	20	sv	sv	INTJ
ejpam-4750	476	1	such	such	ADJ
ejpam-4750	476	2	that	that	SCONJ
ejpam-4750	476	3	either	either	DET
ejpam-4750	476	4	r	r	NOUN
ejpam-4750	476	5	,	,	PUNCT
ejpam-4750	476	6	s	s	PART
ejpam-4750	476	7	∈	∈	PROPN
ejpam-4750	476	8	nhv(p	nhv(p	PROPN
ejpam-4750	476	9	)	)	PUNCT
ejpam-4750	476	10	\	\	PROPN
ejpam-4750	476	11	nhv(q	nhv(q	PROPN
ejpam-4750	476	12	)	)	PUNCT
ejpam-4750	476	13	or	or	CCONJ
ejpam-4750	476	14	r	r	NOUN
ejpam-4750	476	15	,	,	PUNCT
ejpam-4750	476	16	s	s	NOUN
ejpam-4750	476	17	∈	∈	PROPN
ejpam-4750	476	18	nhv(q	nhv(q	PROPN
ejpam-4750	476	19	)	)	PUNCT
ejpam-4750	476	20	\	\	PROPN
ejpam-4750	476	21	nhv(p	nhv(p	PROPN
ejpam-4750	476	22	)	)	PUNCT
ejpam-4750	476	23	or	or	CCONJ
ejpam-4750	476	24	r	r	NOUN
ejpam-4750	476	25	∈	∈	PROPN
ejpam-4750	476	26	nhv(p	nhv(p	PROPN
ejpam-4750	476	27	)	)	PUNCT
ejpam-4750	476	28	\	\	PROPN
ejpam-4750	477	1	nhv(q	nhv(q	PROPN
ejpam-4750	477	2	)	)	PUNCT
ejpam-4750	477	3	and	and	CCONJ
ejpam-4750	478	1	s	s	NOUN
ejpam-4750	478	2	∈	∈	PROPN
ejpam-4750	478	3	nhv(q	nhv(q	PROPN
ejpam-4750	478	4	)	)	PUNCT
ejpam-4750	478	5	\	\	PROPN
ejpam-4750	478	6	nhv(p	nhv(p	PROPN
ejpam-4750	478	7	)	)	PUNCT
ejpam-4750	478	8	.	.	PUNCT
ejpam-4750	479	1	similarly	similarly	ADV
ejpam-4750	479	2	,	,	PUNCT
ejpam-4750	479	3	if	if	SCONJ
ejpam-4750	479	4	p	p	PROPN
ejpam-4750	479	5	∈	∈	PROPN
ejpam-4750	479	6	v	v	ADP
ejpam-4750	479	7	(	(	PUNCT
ejpam-4750	479	8	hv	hv	PROPN
ejpam-4750	479	9	)	)	PUNCT
ejpam-4750	479	10	\	\	PROPN
ejpam-4750	479	11	sv	sv	PROPN
ejpam-4750	479	12	and	and	CCONJ
ejpam-4750	479	13	q	q	PROPN
ejpam-4750	479	14	∈	∈	PROPN
ejpam-4750	479	15	sv	sv	ADP
ejpam-4750	479	16	,	,	PUNCT
ejpam-4750	479	17	there	there	PRON
ejpam-4750	479	18	exists	exist	VERB
ejpam-4750	479	19	a	a	DET
ejpam-4750	479	20	vertex	vertex	NOUN
ejpam-4750	479	21	w	w	ADP
ejpam-4750	479	22	∈	∈	PROPN
ejpam-4750	479	23	v	v	ADP
ejpam-4750	479	24	(	(	PUNCT
ejpam-4750	479	25	hv	hv	NOUN
ejpam-4750	479	26	)	)	PUNCT
ejpam-4750	479	27	∩	∩	NOUN
ejpam-4750	479	28	sv	sv	INTJ
ejpam-4750	479	29	such	such	ADJ
ejpam-4750	480	1	that	that	DET
ejpam-4750	480	2	w	w	PROPN
ejpam-4750	480	3	∈	∈	PROPN
ejpam-4750	480	4	nhv(p	nhv(p	PROPN
ejpam-4750	480	5	)	)	PUNCT
ejpam-4750	480	6	\	\	PROPN
ejpam-4750	480	7	nhv(q	nhv(q	PROPN
ejpam-4750	480	8	)	)	PUNCT
ejpam-4750	480	9	or	or	CCONJ
ejpam-4750	480	10	w	w	NOUN
ejpam-4750	480	11	∈	∈	PROPN
ejpam-4750	480	12	nhv(q	nhv(q	PROPN
ejpam-4750	480	13	)	)	PUNCT
ejpam-4750	480	14	\	\	PROPN
ejpam-4750	480	15	nhv(p	nhv(p	PROPN
ejpam-4750	480	16	)	)	PUNCT
ejpam-4750	480	17	.	.	PUNCT
ejpam-4750	481	1	hence	hence	ADV
ejpam-4750	481	2	,	,	PUNCT
ejpam-4750	481	3	sv	sv	PROPN
ejpam-4750	481	4	is	be	AUX
ejpam-4750	481	5	a	a	DET
ejpam-4750	481	6	2	2	NUM
ejpam-4750	481	7	-	-	PUNCT
ejpam-4750	481	8	locating	locate	VERB
ejpam-4750	481	9	set	set	NOUN
ejpam-4750	481	10	of	of	ADP
ejpam-4750	481	11	hv	hv	PROPN
ejpam-4750	481	12	.	.	PUNCT
ejpam-4750	482	1	thus	thus	ADV
ejpam-4750	482	2	,	,	PUNCT
ejpam-4750	482	3	theorem	theorem	VERB
ejpam-4750	482	4	4	4	NUM
ejpam-4750	482	5	is	be	AUX
ejpam-4750	482	6	modified	modify	VERB
ejpam-4750	482	7	in	in	ADP
ejpam-4750	482	8	the	the	DET
ejpam-4750	482	9	next	next	ADJ
ejpam-4750	482	10	result	result	NOUN
ejpam-4750	482	11	.	.	PUNCT
ejpam-4750	483	1	theorem	theorem	VERB
ejpam-4750	483	2	14	14	NUM
ejpam-4750	483	3	.	.	PUNCT
ejpam-4750	484	1	let	let	VERB
ejpam-4750	484	2	g	g	NOUN
ejpam-4750	484	3	and	and	CCONJ
ejpam-4750	484	4	h	h	NOUN
ejpam-4750	484	5	be	be	AUX
ejpam-4750	484	6	nontrivial	nontrivial	ADJ
ejpam-4750	484	7	connected	connected	ADJ
ejpam-4750	484	8	graphs	graph	NOUN
ejpam-4750	484	9	.	.	PUNCT
ejpam-4750	485	1	then	then	ADV
ejpam-4750	485	2	s	s	VERB
ejpam-4750	485	3	⊆	⊆	NUM
ejpam-4750	485	4	v	v	NOUN
ejpam-4750	485	5	(	(	PUNCT
ejpam-4750	485	6	g	g	PROPN
ejpam-4750	485	7	◦	◦	NOUN
ejpam-4750	485	8	h	h	NOUN
ejpam-4750	485	9	)	)	PUNCT
ejpam-4750	485	10	is	be	AUX
ejpam-4750	485	11	a	a	DET
ejpam-4750	485	12	2	2	NUM
ejpam-4750	485	13	-	-	PUNCT
ejpam-4750	485	14	metric	metric	ADJ
ejpam-4750	485	15	basis	basis	NOUN
ejpam-4750	485	16	for	for	ADP
ejpam-4750	485	17	g	g	PROPN
ejpam-4750	485	18	◦	◦	NOUN
ejpam-4750	485	19	h	h	NOUN
ejpam-4750	485	20	if	if	SCONJ
ejpam-4750	486	1	and	and	CCONJ
ejpam-4750	486	2	only	only	ADV
ejpam-4750	486	3	if	if	SCONJ
ejpam-4750	486	4	s	s	X
ejpam-4750	486	5	=	=	PUNCT
ejpam-4750	486	6	⋃	⋃	NOUN
ejpam-4750	486	7	v∈v	v∈v	NOUN
ejpam-4750	486	8	(	(	PUNCT
ejpam-4750	486	9	g	g	NOUN
ejpam-4750	486	10	)	)	PUNCT
ejpam-4750	486	11	sv	sv	PROPN
ejpam-4750	486	12	where	where	SCONJ
ejpam-4750	486	13	sv	sv	PROPN
ejpam-4750	486	14	is	be	AUX
ejpam-4750	486	15	an	an	DET
ejpam-4750	486	16	ln2	ln2	NOUN
ejpam-4750	486	17	-	-	PUNCT
ejpam-4750	486	18	set	set	NOUN
ejpam-4750	486	19	of	of	ADP
ejpam-4750	486	20	h	h	NOUN
ejpam-4750	486	21	v	v	NOUN
ejpam-4750	486	22	for	for	ADP
ejpam-4750	486	23	all	all	DET
ejpam-4750	486	24	v	v	ADP
ejpam-4750	486	25	∈	∈	NUM
ejpam-4750	486	26	v	v	NOUN
ejpam-4750	486	27	(	(	PUNCT
ejpam-4750	486	28	g	g	NOUN
ejpam-4750	486	29	)	)	PUNCT
ejpam-4750	486	30	.	.	PUNCT
ejpam-4750	487	1	in	in	ADP
ejpam-4750	487	2	particular	particular	ADJ
ejpam-4750	487	3	,	,	PUNCT
ejpam-4750	487	4	dim2(g	dim2(g	NOUN
ejpam-4750	487	5	◦	◦	NOUN
ejpam-4750	487	6	h	h	NOUN
ejpam-4750	487	7	)	)	PUNCT
ejpam-4750	487	8	=	=	SYM
ejpam-4750	487	9	|v	|v	PROPN
ejpam-4750	487	10	(	(	PUNCT
ejpam-4750	487	11	g)|ln2(h	g)|ln2(h	NOUN
ejpam-4750	487	12	)	)	PUNCT
ejpam-4750	487	13	.	.	PUNCT
ejpam-4750	488	1	d.	d.	PROPN
ejpam-4750	488	2	managbanag	managbanag	PROPN
ejpam-4750	488	3	,	,	PUNCT
ejpam-4750	488	4	h.	h.	PROPN
ejpam-4750	488	5	rara	rara	PROPN
ejpam-4750	488	6	/	/	SYM
ejpam-4750	488	7	eur	eur	PROPN
ejpam-4750	488	8	.	.	PUNCT
ejpam-4750	489	1	j.	j.	PROPN
ejpam-4750	489	2	pure	pure	PROPN
ejpam-4750	489	3	appl	appl	PROPN
ejpam-4750	489	4	.	.	PROPN
ejpam-4750	489	5	math	math	PROPN
ejpam-4750	489	6	,	,	PUNCT
ejpam-4750	489	7	16	16	NUM
ejpam-4750	489	8	(	(	PUNCT
ejpam-4750	489	9	2	2	NUM
ejpam-4750	489	10	)	)	PUNCT
ejpam-4750	489	11	(	(	PUNCT
ejpam-4750	489	12	2023	2023	NUM
ejpam-4750	489	13	)	)	PUNCT
ejpam-4750	489	14	,	,	PUNCT
ejpam-4750	489	15	1068	1068	NUM
ejpam-4750	489	16	-	-	SYM
ejpam-4750	489	17	1083	1083	NUM
ejpam-4750	489	18	1082	1082	NUM
ejpam-4750	489	19	theorem	theorem	VERB
ejpam-4750	489	20	15	15	NUM
ejpam-4750	489	21	.	.	PUNCT
ejpam-4750	490	1	let	let	VERB
ejpam-4750	490	2	g	g	NOUN
ejpam-4750	490	3	and	and	CCONJ
ejpam-4750	490	4	h	h	NOUN
ejpam-4750	490	5	be	be	AUX
ejpam-4750	490	6	nontrivial	nontrivial	ADJ
ejpam-4750	490	7	connected	connected	ADJ
ejpam-4750	490	8	graphs	graph	NOUN
ejpam-4750	490	9	.	.	PUNCT
ejpam-4750	491	1	then	then	ADV
ejpam-4750	491	2	fdim2(g	fdim2(g	CCONJ
ejpam-4750	491	3	◦	◦	NOUN
ejpam-4750	491	4	h	h	NOUN
ejpam-4750	491	5	)	)	PUNCT
ejpam-4750	491	6	=	=	SYM
ejpam-4750	491	7	®	®	NOUN
ejpam-4750	491	8	0	0	NUM
ejpam-4750	491	9	,	,	PUNCT
ejpam-4750	491	10	if	if	SCONJ
ejpam-4750	491	11	h	h	NOUN
ejpam-4750	491	12	has	have	VERB
ejpam-4750	491	13	a	a	DET
ejpam-4750	491	14	unique	unique	ADJ
ejpam-4750	491	15	ln2	ln2	NOUN
ejpam-4750	491	16	-	-	PUNCT
ejpam-4750	491	17	set	set	NOUN
ejpam-4750	491	18	,	,	PUNCT
ejpam-4750	491	19	|v	|v	PROPN
ejpam-4750	491	20	(	(	PUNCT
ejpam-4750	491	21	g)|fln2(h	g)|fln2(h	PROPN
ejpam-4750	491	22	)	)	PUNCT
ejpam-4750	491	23	,	,	PUNCT
ejpam-4750	491	24	if	if	SCONJ
ejpam-4750	491	25	h	h	NOUN
ejpam-4750	491	26	has	have	VERB
ejpam-4750	491	27	no	no	DET
ejpam-4750	491	28	unique	unique	ADJ
ejpam-4750	491	29	ln2	ln2	NOUN
ejpam-4750	491	30	-	-	PUNCT
ejpam-4750	491	31	set	set	NOUN
ejpam-4750	491	32	.	.	PUNCT
ejpam-4750	492	1	proof	proof	NOUN
ejpam-4750	492	2	:	:	PUNCT
ejpam-4750	492	3	suppose	suppose	VERB
ejpam-4750	492	4	h	h	NOUN
ejpam-4750	492	5	has	have	VERB
ejpam-4750	492	6	a	a	DET
ejpam-4750	492	7	unique	unique	ADJ
ejpam-4750	492	8	ln2	ln2	NOUN
ejpam-4750	492	9	-	-	PUNCT
ejpam-4750	492	10	set	set	NOUN
ejpam-4750	492	11	.	.	PUNCT
ejpam-4750	493	1	for	for	ADP
ejpam-4750	493	2	each	each	DET
ejpam-4750	493	3	v	v	NUM
ejpam-4750	493	4	∈	∈	PROPN
ejpam-4750	493	5	v	v	NOUN
ejpam-4750	493	6	(	(	PUNCT
ejpam-4750	493	7	g	g	NOUN
ejpam-4750	493	8	)	)	PUNCT
ejpam-4750	493	9	,	,	PUNCT
ejpam-4750	493	10	let	let	VERB
ejpam-4750	493	11	mv	mv	PROPN
ejpam-4750	493	12	⊆	⊆	NUM
ejpam-4750	493	13	v	v	X
ejpam-4750	493	14	(	(	PUNCT
ejpam-4750	493	15	hv	hv	NOUN
ejpam-4750	493	16	)	)	PUNCT
ejpam-4750	493	17	be	be	VERB
ejpam-4750	493	18	the	the	DET
ejpam-4750	493	19	unique	unique	ADJ
ejpam-4750	493	20	ln2	ln2	NOUN
ejpam-4750	493	21	-	-	PUNCT
ejpam-4750	493	22	set	set	NOUN
ejpam-4750	493	23	of	of	ADP
ejpam-4750	493	24	h	h	NOUN
ejpam-4750	493	25	v.	v.	ADP
ejpam-4750	493	26	by	by	ADP
ejpam-4750	493	27	theorem	theorem	NOUN
ejpam-4750	493	28	15	15	NUM
ejpam-4750	493	29	,	,	PUNCT
ejpam-4750	493	30	s	s	PART
ejpam-4750	493	31	=	=	PUNCT
ejpam-4750	493	32	⋃	⋃	NOUN
ejpam-4750	493	33	v∈v	v∈v	NOUN
ejpam-4750	493	34	(	(	PUNCT
ejpam-4750	493	35	g	g	NOUN
ejpam-4750	493	36	)	)	PUNCT
ejpam-4750	493	37	mv	mv	PROPN
ejpam-4750	493	38	is	be	AUX
ejpam-4750	493	39	the	the	DET
ejpam-4750	493	40	unique	unique	ADJ
ejpam-4750	493	41	2	2	NUM
ejpam-4750	493	42	-	-	PUNCT
ejpam-4750	493	43	metric	metric	ADJ
ejpam-4750	493	44	basis	basis	NOUN
ejpam-4750	493	45	for	for	ADP
ejpam-4750	493	46	g	g	PROPN
ejpam-4750	493	47	◦	◦	NOUN
ejpam-4750	493	48	h.	h.	NOUN
ejpam-4750	493	49	thus	thus	ADV
ejpam-4750	493	50	,	,	PUNCT
ejpam-4750	493	51	by	by	ADP
ejpam-4750	493	52	remark	remark	NOUN
ejpam-4750	493	53	4	4	NUM
ejpam-4750	493	54	(	(	PUNCT
ejpam-4750	493	55	i	i	NOUN
ejpam-4750	493	56	)	)	PUNCT
ejpam-4750	493	57	,	,	PUNCT
ejpam-4750	493	58	fdim2(g	fdim2(g	NOUN
ejpam-4750	493	59	◦	◦	NOUN
ejpam-4750	493	60	h	h	NOUN
ejpam-4750	493	61	)	)	PUNCT
ejpam-4750	493	62	=	=	SYM
ejpam-4750	493	63	0	0	X
ejpam-4750	493	64	.	.	PUNCT
ejpam-4750	493	65	suppose	suppose	VERB
ejpam-4750	493	66	h	h	NOUN
ejpam-4750	493	67	does	do	AUX
ejpam-4750	493	68	not	not	PART
ejpam-4750	493	69	have	have	VERB
ejpam-4750	493	70	unique	unique	ADJ
ejpam-4750	493	71	ln2	ln2	ADJ
ejpam-4750	493	72	-	-	PUNCT
ejpam-4750	493	73	set	set	NOUN
ejpam-4750	493	74	.	.	PUNCT
ejpam-4750	494	1	for	for	ADP
ejpam-4750	494	2	each	each	DET
ejpam-4750	494	3	v	v	NUM
ejpam-4750	494	4	∈	∈	PROPN
ejpam-4750	494	5	v	v	NOUN
ejpam-4750	494	6	(	(	PUNCT
ejpam-4750	494	7	g	g	NOUN
ejpam-4750	494	8	)	)	PUNCT
ejpam-4750	494	9	,	,	PUNCT
ejpam-4750	494	10	let	let	VERB
ejpam-4750	494	11	qv	qv	PRON
ejpam-4750	494	12	⊆	⊆	NUM
ejpam-4750	494	13	v	v	ADP
ejpam-4750	494	14	(	(	PUNCT
ejpam-4750	494	15	hv	hv	NOUN
ejpam-4750	494	16	)	)	PUNCT
ejpam-4750	494	17	be	be	VERB
ejpam-4750	494	18	an	an	DET
ejpam-4750	494	19	ln2	ln2	NOUN
ejpam-4750	494	20	-	-	PUNCT
ejpam-4750	494	21	set	set	NOUN
ejpam-4750	494	22	of	of	ADP
ejpam-4750	494	23	h	h	NOUN
ejpam-4750	494	24	v	v	ADP
ejpam-4750	494	25	such	such	ADJ
ejpam-4750	495	1	that	that	DET
ejpam-4750	495	2	fln2(h	fln2(h	NUM
ejpam-4750	495	3	v	v	NOUN
ejpam-4750	495	4	)	)	PUNCT
ejpam-4750	495	5	=	=	SYM
ejpam-4750	495	6	fln2(qv	fln2(qv	PROPN
ejpam-4750	495	7	)	)	PUNCT
ejpam-4750	495	8	.	.	PUNCT
ejpam-4750	496	1	let	let	VERB
ejpam-4750	496	2	m(qv	m(qv	NOUN
ejpam-4750	496	3	)	)	PUNCT
ejpam-4750	496	4	⊆	⊆	NUM
ejpam-4750	496	5	qv	qv	INTJ
ejpam-4750	496	6	be	be	AUX
ejpam-4750	496	7	a	a	DET
ejpam-4750	496	8	forcing	forcing	NOUN
ejpam-4750	496	9	subset	subset	NOUN
ejpam-4750	496	10	for	for	ADP
ejpam-4750	496	11	qv	qv	INTJ
ejpam-4750	496	12	with	with	ADP
ejpam-4750	496	13	fln2(qv	fln2(qv	PROPN
ejpam-4750	496	14	)	)	PUNCT
ejpam-4750	497	1	=	=	SYM
ejpam-4750	497	2	|m(qv)|	|m(qv)|	NOUN
ejpam-4750	497	3	.	.	PUNCT
ejpam-4750	498	1	then	then	ADV
ejpam-4750	498	2	by	by	ADP
ejpam-4750	498	3	theorem	theorem	NOUN
ejpam-4750	498	4	15	15	NUM
ejpam-4750	498	5	,	,	PUNCT
ejpam-4750	498	6	sq	sq	NOUN
ejpam-4750	498	7	=	=	SYM
ejpam-4750	498	8	⋃	⋃	NOUN
ejpam-4750	498	9	v∈v	v∈v	NOUN
ejpam-4750	498	10	(	(	PUNCT
ejpam-4750	498	11	g	g	NOUN
ejpam-4750	498	12	)	)	PUNCT
ejpam-4750	498	13	qv	qv	VERB
ejpam-4750	498	14	is	be	AUX
ejpam-4750	498	15	a	a	DET
ejpam-4750	498	16	2	2	NUM
ejpam-4750	498	17	-	-	PUNCT
ejpam-4750	498	18	metric	metric	ADJ
ejpam-4750	498	19	basis	basis	NOUN
ejpam-4750	498	20	of	of	ADP
ejpam-4750	498	21	g	g	PROPN
ejpam-4750	498	22	◦	◦	PROPN
ejpam-4750	498	23	h.	h.	PROPN
ejpam-4750	498	24	let	let	VERB
ejpam-4750	498	25	c	c	NOUN
ejpam-4750	498	26	=	=	PUNCT
ejpam-4750	498	27	⋃	⋃	NOUN
ejpam-4750	498	28	v∈v	v∈v	NOUN
ejpam-4750	498	29	(	(	PUNCT
ejpam-4750	498	30	g	g	NOUN
ejpam-4750	498	31	)	)	PUNCT
ejpam-4750	498	32	m(qv	m(qv	NOUN
ejpam-4750	498	33	)	)	PUNCT
ejpam-4750	498	34	.	.	PUNCT
ejpam-4750	499	1	then	then	ADV
ejpam-4750	499	2	c	c	PROPN
ejpam-4750	499	3	is	be	AUX
ejpam-4750	499	4	a	a	DET
ejpam-4750	499	5	forcing	forcing	NOUN
ejpam-4750	499	6	subset	subset	NOUN
ejpam-4750	499	7	for	for	ADP
ejpam-4750	499	8	sq	sq	PROPN
ejpam-4750	499	9	.	.	PROPN
ejpam-4750	499	10	thus	thus	ADV
ejpam-4750	499	11	,	,	PUNCT
ejpam-4750	499	12	fdim2(g	fdim2(g	ADV
ejpam-4750	499	13	◦	◦	NOUN
ejpam-4750	499	14	h	h	NOUN
ejpam-4750	499	15	)	)	PUNCT
ejpam-4750	499	16	≤	≤	NUM
ejpam-4750	499	17	fdim2(sq	fdim2(sq	NOUN
ejpam-4750	499	18	)	)	PUNCT
ejpam-4750	499	19	≤	≤	NOUN
ejpam-4750	499	20	|c|	|c|	PROPN
ejpam-4750	499	21	=	=	SYM
ejpam-4750	499	22	|v	|v	PROPN
ejpam-4750	499	23	(	(	PUNCT
ejpam-4750	499	24	g)||m(qv)|	g)||m(qv)|	NOUN
ejpam-4750	499	25	=	=	SYM
ejpam-4750	499	26	|v	|v	X
ejpam-4750	499	27	(	(	PUNCT
ejpam-4750	499	28	g)|fln2(qv	g)|fln2(qv	ADV
ejpam-4750	499	29	)	)	PUNCT
ejpam-4750	499	30	=	=	SYM
ejpam-4750	500	1	|v	|v	PROPN
ejpam-4750	500	2	(	(	PUNCT
ejpam-4750	500	3	g)|fln2(h	g)|fln2(h	PROPN
ejpam-4750	500	4	)	)	PUNCT
ejpam-4750	500	5	.	.	PUNCT
ejpam-4750	501	1	next	next	ADV
ejpam-4750	501	2	,	,	PUNCT
ejpam-4750	501	3	let	let	VERB
ejpam-4750	501	4	s′	s′	NOUN
ejpam-4750	501	5	be	be	AUX
ejpam-4750	501	6	a	a	DET
ejpam-4750	501	7	2	2	NUM
ejpam-4750	501	8	-	-	PUNCT
ejpam-4750	501	9	metric	metric	ADJ
ejpam-4750	501	10	basis	basis	NOUN
ejpam-4750	501	11	for	for	ADP
ejpam-4750	501	12	g	g	PROPN
ejpam-4750	501	13	◦	◦	NOUN
ejpam-4750	501	14	h	h	NOUN
ejpam-4750	501	15	such	such	ADJ
ejpam-4750	502	1	that	that	SCONJ
ejpam-4750	502	2	fdim2(g	fdim2(g	NOUN
ejpam-4750	502	3	◦	◦	NOUN
ejpam-4750	502	4	h	h	NOUN
ejpam-4750	502	5	)	)	PUNCT
ejpam-4750	502	6	=	=	SYM
ejpam-4750	502	7	fdim2(s	fdim2(s	NOUN
ejpam-4750	502	8	′	′	NUM
ejpam-4750	502	9	)	)	PUNCT
ejpam-4750	502	10	.	.	PUNCT
ejpam-4750	503	1	then	then	ADV
ejpam-4750	503	2	by	by	ADP
ejpam-4750	503	3	theorem	theorem	NOUN
ejpam-4750	503	4	15	15	NUM
ejpam-4750	503	5	,	,	PUNCT
ejpam-4750	503	6	s′	s′	ADJ
ejpam-4750	503	7	=	=	PUNCT
ejpam-4750	503	8	⋃	⋃	NOUN
ejpam-4750	503	9	v∈v	v∈v	NOUN
ejpam-4750	503	10	(	(	PUNCT
ejpam-4750	503	11	g	g	NOUN
ejpam-4750	503	12	)	)	PUNCT
ejpam-4750	503	13	rv	rv	NOUN
ejpam-4750	503	14	where	where	SCONJ
ejpam-4750	503	15	rv	rv	PROPN
ejpam-4750	503	16	is	be	AUX
ejpam-4750	503	17	an	an	DET
ejpam-4750	503	18	ln2	ln2	NOUN
ejpam-4750	503	19	-	-	PUNCT
ejpam-4750	503	20	set	set	NOUN
ejpam-4750	503	21	of	of	ADP
ejpam-4750	503	22	hv	hv	PROPN
ejpam-4750	503	23	for	for	ADP
ejpam-4750	503	24	each	each	DET
ejpam-4750	503	25	v	v	NUM
ejpam-4750	503	26	∈	∈	PROPN
ejpam-4750	503	27	v	v	NOUN
ejpam-4750	503	28	(	(	PUNCT
ejpam-4750	503	29	g	g	NOUN
ejpam-4750	503	30	)	)	PUNCT
ejpam-4750	503	31	.	.	PUNCT
ejpam-4750	504	1	let	let	VERB
ejpam-4750	504	2	c	c	NOUN
ejpam-4750	504	3	′	′	VERB
ejpam-4750	504	4	be	be	AUX
ejpam-4750	504	5	a	a	DET
ejpam-4750	504	6	forcing	force	VERB
ejpam-4750	504	7	subset	subset	NOUN
ejpam-4750	504	8	of	of	ADP
ejpam-4750	504	9	s′	s′	NUM
ejpam-4750	504	10	such	such	ADJ
ejpam-4750	504	11	that	that	DET
ejpam-4750	504	12	fdim2(s	fdim2(s	NOUN
ejpam-4750	504	13	′	′	NUM
ejpam-4750	504	14	)	)	PUNCT
ejpam-4750	505	1	=	=	PRON
ejpam-4750	505	2	|c	|c	VERB
ejpam-4750	505	3	′|	′|	PROPN
ejpam-4750	505	4	.	.	PUNCT
ejpam-4750	506	1	we	we	PRON
ejpam-4750	506	2	claim	claim	VERB
ejpam-4750	506	3	that	that	SCONJ
ejpam-4750	506	4	c	c	NOUN
ejpam-4750	506	5	′	′	NUM
ejpam-4750	507	1	∩rv	∩rv	NOUN
ejpam-4750	508	1	=	=	SYM
ejpam-4750	508	2	cv	cv	PROPN
ejpam-4750	508	3	is	be	AUX
ejpam-4750	508	4	a	a	DET
ejpam-4750	508	5	forcing	forcing	NOUN
ejpam-4750	508	6	subset	subset	NOUN
ejpam-4750	508	7	for	for	ADP
ejpam-4750	508	8	rv	rv	PROPN
ejpam-4750	508	9	for	for	ADP
ejpam-4750	508	10	all	all	DET
ejpam-4750	508	11	v	v	ADP
ejpam-4750	508	12	∈	∈	NUM
ejpam-4750	508	13	v	v	NOUN
ejpam-4750	508	14	(	(	PUNCT
ejpam-4750	508	15	g	g	NOUN
ejpam-4750	508	16	)	)	PUNCT
ejpam-4750	508	17	.	.	PUNCT
ejpam-4750	509	1	suppose	suppose	VERB
ejpam-4750	509	2	there	there	PRON
ejpam-4750	509	3	exists	exist	VERB
ejpam-4750	509	4	w	w	PROPN
ejpam-4750	509	5	∈	∈	PROPN
ejpam-4750	509	6	v	v	ADP
ejpam-4750	509	7	(	(	PUNCT
ejpam-4750	509	8	g	g	NOUN
ejpam-4750	509	9	)	)	PUNCT
ejpam-4750	509	10	such	such	ADJ
ejpam-4750	509	11	that	that	SCONJ
ejpam-4750	509	12	c	c	NOUN
ejpam-4750	509	13	′	′	NUM
ejpam-4750	509	14	∩	∩	ADJ
ejpam-4750	509	15	rw	rw	NOUN
ejpam-4750	509	16	=	=	NOUN
ejpam-4750	509	17	cw	cw	NOUN
ejpam-4750	509	18	is	be	AUX
ejpam-4750	509	19	not	not	PART
ejpam-4750	509	20	a	a	DET
ejpam-4750	509	21	forcing	forcing	NOUN
ejpam-4750	509	22	subset	subset	NOUN
ejpam-4750	509	23	for	for	ADP
ejpam-4750	509	24	rw	rw	NOUN
ejpam-4750	509	25	.	.	PUNCT
ejpam-4750	510	1	let	let	VERB
ejpam-4750	510	2	r′	r′	PROPN
ejpam-4750	510	3	w	w	AUX
ejpam-4750	510	4	be	be	AUX
ejpam-4750	510	5	an	an	DET
ejpam-4750	510	6	ln2	ln2	NOUN
ejpam-4750	510	7	-	-	PUNCT
ejpam-4750	510	8	set	set	NOUN
ejpam-4750	510	9	of	of	ADP
ejpam-4750	510	10	hw	hw	PRON
ejpam-4750	510	11	with	with	ADP
ejpam-4750	510	12	cw	cw	PROPN
ejpam-4750	510	13	⊆	⊆	NUM
ejpam-4750	510	14	r′	r′	PROPN
ejpam-4750	510	15	w	w	NOUN
ejpam-4750	510	16	and	and	CCONJ
ejpam-4750	510	17	r′	r′	PROPN
ejpam-4750	510	18	w	w	PROPN
ejpam-4750	510	19	̸=	̸=	PROPN
ejpam-4750	510	20	rw	rw	NOUN
ejpam-4750	510	21	.	.	PUNCT
ejpam-4750	511	1	then	then	ADV
ejpam-4750	511	2	s′′	s′′	PROPN
ejpam-4750	511	3	=	=	PROPN
ejpam-4750	511	4	ñ	ñ	PROPN
ejpam-4750	511	5	⋃	⋃	NOUN
ejpam-4750	511	6	v∈v	v∈v	NOUN
ejpam-4750	511	7	(	(	PUNCT
ejpam-4750	511	8	g)\{w	g)\{w	NOUN
ejpam-4750	511	9	}	}	PUNCT
ejpam-4750	511	10	rv	rv	NOUN
ejpam-4750	511	11	é	é	X
ejpam-4750	511	12	∪r′	∪r′	VERB
ejpam-4750	511	13	w	w	PROPN
ejpam-4750	511	14	is	be	AUX
ejpam-4750	511	15	a	a	DET
ejpam-4750	511	16	2	2	NUM
ejpam-4750	511	17	-	-	PUNCT
ejpam-4750	511	18	metric	metric	ADJ
ejpam-4750	511	19	basis	basis	NOUN
ejpam-4750	511	20	for	for	ADP
ejpam-4750	511	21	g	g	PROPN
ejpam-4750	511	22	◦	◦	NOUN
ejpam-4750	511	23	h	h	NOUN
ejpam-4750	511	24	with	with	ADP
ejpam-4750	511	25	s′	s′	ADJ
ejpam-4750	511	26	̸=	̸=	PROPN
ejpam-4750	511	27	s′′	s′′	PROPN
ejpam-4750	511	28	and	and	CCONJ
ejpam-4750	511	29	c	c	NOUN
ejpam-4750	511	30	′	′	NOUN
ejpam-4750	511	31	⊆	⊆	NUM
ejpam-4750	511	32	s′′	s′′	PROPN
ejpam-4750	511	33	,	,	PUNCT
ejpam-4750	511	34	a	a	DET
ejpam-4750	511	35	contradiction	contradiction	NOUN
ejpam-4750	511	36	.	.	PUNCT
ejpam-4750	512	1	thus	thus	ADV
ejpam-4750	512	2	,	,	PUNCT
ejpam-4750	512	3	cv	cv	PROPN
ejpam-4750	512	4	is	be	AUX
ejpam-4750	512	5	a	a	DET
ejpam-4750	512	6	forcing	forcing	NOUN
ejpam-4750	512	7	subset	subset	NOUN
ejpam-4750	512	8	for	for	ADP
ejpam-4750	512	9	rv	rv	PROPN
ejpam-4750	512	10	for	for	ADP
ejpam-4750	512	11	each	each	DET
ejpam-4750	512	12	v	v	NUM
ejpam-4750	512	13	∈	∈	PROPN
ejpam-4750	512	14	v	v	NOUN
ejpam-4750	512	15	(	(	PUNCT
ejpam-4750	512	16	g	g	NOUN
ejpam-4750	512	17	)	)	PUNCT
ejpam-4750	512	18	.	.	PUNCT
ejpam-4750	513	1	let	let	VERB
ejpam-4750	513	2	c	c	NOUN
ejpam-4750	513	3	′	′	VERB
ejpam-4750	514	1	=	=	PUNCT
ejpam-4750	515	1	⋃	⋃	VERB
ejpam-4750	515	2	v∈v	v∈v	NOUN
ejpam-4750	515	3	(	(	PUNCT
ejpam-4750	515	4	g	g	NOUN
ejpam-4750	515	5	)	)	PUNCT
ejpam-4750	515	6	cv	cv	PROPN
ejpam-4750	515	7	.	.	PROPN
ejpam-4750	515	8	then	then	ADV
ejpam-4750	515	9	fdim2(g	fdim2(g	CCONJ
ejpam-4750	515	10	◦	◦	NOUN
ejpam-4750	515	11	h	h	NOUN
ejpam-4750	515	12	)	)	PUNCT
ejpam-4750	515	13	=	=	NOUN
ejpam-4750	515	14	|c	|c	VERB
ejpam-4750	515	15	′|	′|	NUM
ejpam-4750	515	16	=	=	SYM
ejpam-4750	515	17	∑	∑	PUNCT
ejpam-4750	515	18	v∈v	v∈v	NOUN
ejpam-4750	515	19	(	(	PUNCT
ejpam-4750	515	20	g	g	NOUN
ejpam-4750	515	21	)	)	PUNCT
ejpam-4750	515	22	|cv|	|cv|	PROPN
ejpam-4750	515	23	≥	≥	NUM
ejpam-4750	515	24	∑	∑	PUNCT
ejpam-4750	515	25	v∈v	v∈v	PROPN
ejpam-4750	515	26	(	(	PUNCT
ejpam-4750	515	27	g	g	NOUN
ejpam-4750	515	28	)	)	PUNCT
ejpam-4750	515	29	fln2(h	fln2(h	X
ejpam-4750	515	30	v	v	NOUN
ejpam-4750	515	31	)	)	PUNCT
ejpam-4750	515	32	=	=	SYM
ejpam-4750	515	33	|v	|v	PROPN
ejpam-4750	515	34	(	(	PUNCT
ejpam-4750	515	35	g)|fln2(h	g)|fln2(h	PROPN
ejpam-4750	515	36	)	)	PUNCT
ejpam-4750	515	37	.	.	PUNCT
ejpam-4750	516	1	therefore	therefore	ADV
ejpam-4750	516	2	,	,	PUNCT
ejpam-4750	516	3	fdim2(g	fdim2(g	ADV
ejpam-4750	516	4	◦	◦	NOUN
ejpam-4750	516	5	h	h	NOUN
ejpam-4750	516	6	)	)	PUNCT
ejpam-4750	517	1	=	=	SYM
ejpam-4750	517	2	|v	|v	PROPN
ejpam-4750	517	3	(	(	PUNCT
ejpam-4750	517	4	g)|fln2(h	g)|fln2(h	PROPN
ejpam-4750	517	5	)	)	PUNCT
ejpam-4750	517	6	.	.	PUNCT
ejpam-4750	518	1	example	example	NOUN
ejpam-4750	519	1	7	7	X
ejpam-4750	519	2	.	.	X
ejpam-4750	519	3	consider	consider	VERB
ejpam-4750	519	4	the	the	DET
ejpam-4750	519	5	corona	corona	NOUN
ejpam-4750	519	6	of	of	ADP
ejpam-4750	519	7	two	two	NUM
ejpam-4750	519	8	graphs	graph	NOUN
ejpam-4750	519	9	p3	p3	NOUN
ejpam-4750	519	10	and	and	CCONJ
ejpam-4750	519	11	c4	c4	NOUN
ejpam-4750	519	12	.	.	PUNCT
ejpam-4750	520	1	since	since	SCONJ
ejpam-4750	520	2	c4	c4	NOUN
ejpam-4750	520	3	has	have	VERB
ejpam-4750	520	4	a	a	DET
ejpam-4750	520	5	unique	unique	ADJ
ejpam-4750	520	6	ln2	ln2	NOUN
ejpam-4750	520	7	-	-	PUNCT
ejpam-4750	520	8	set	set	NOUN
ejpam-4750	520	9	,	,	PUNCT
ejpam-4750	520	10	fdim2(p3	fdim2(p3	NOUN
ejpam-4750	520	11	◦	◦	NOUN
ejpam-4750	520	12	c4	c4	NOUN
ejpam-4750	520	13	)	)	PUNCT
ejpam-4750	520	14	=	=	SYM
ejpam-4750	520	15	|v	|v	X
ejpam-4750	520	16	(	(	PUNCT
ejpam-4750	520	17	p3)|fln2(c4	p3)|fln2(c4	NOUN
ejpam-4750	520	18	)	)	PUNCT
ejpam-4750	520	19	=	=	SYM
ejpam-4750	520	20	3	3	NUM
ejpam-4750	520	21	·	·	SYM
ejpam-4750	520	22	0	0	NUM
ejpam-4750	521	1	=	=	SYM
ejpam-4750	521	2	0	0	PROPN
ejpam-4750	521	3	.	.	NOUN
ejpam-4750	521	4	example	example	NOUN
ejpam-4750	521	5	8	8	NUM
ejpam-4750	521	6	.	.	PUNCT
ejpam-4750	522	1	consider	consider	VERB
ejpam-4750	522	2	the	the	DET
ejpam-4750	522	3	corona	corona	NOUN
ejpam-4750	522	4	of	of	ADP
ejpam-4750	522	5	two	two	NUM
ejpam-4750	522	6	graphs	graph	NOUN
ejpam-4750	522	7	k3	k3	VERB
ejpam-4750	522	8	and	and	CCONJ
ejpam-4750	522	9	c5	c5	PROPN
ejpam-4750	522	10	.	.	PUNCT
ejpam-4750	523	1	since	since	SCONJ
ejpam-4750	523	2	c5	c5	PROPN
ejpam-4750	523	3	has	have	VERB
ejpam-4750	523	4	no	no	DET
ejpam-4750	523	5	unique	unique	ADJ
ejpam-4750	523	6	ln2	ln2	NOUN
ejpam-4750	523	7	-	-	PUNCT
ejpam-4750	523	8	set	set	NOUN
ejpam-4750	523	9	,	,	PUNCT
ejpam-4750	523	10	fdim2(k3	fdim2(k3	PROPN
ejpam-4750	523	11	◦	◦	PROPN
ejpam-4750	523	12	c5	c5	PROPN
ejpam-4750	523	13	)	)	PUNCT
ejpam-4750	523	14	=	=	SYM
ejpam-4750	523	15	|v	|v	PROPN
ejpam-4750	523	16	(	(	PUNCT
ejpam-4750	523	17	k3)|fln2(c5	k3)|fln2(c5	PROPN
ejpam-4750	523	18	)	)	PUNCT
ejpam-4750	523	19	=	=	SYM
ejpam-4750	523	20	3	3	X
ejpam-4750	523	21	·	·	SYM
ejpam-4750	523	22	3	3	NUM
ejpam-4750	523	23	=	=	SYM
ejpam-4750	523	24	9	9	NUM
ejpam-4750	523	25	.	.	PUNCT
ejpam-4750	523	26	references	reference	NOUN
ejpam-4750	523	27	1083	1083	NUM
ejpam-4750	523	28	acknowledgements	acknowledgement	NOUN
ejpam-4750	523	29	this	this	DET
ejpam-4750	523	30	research	research	NOUN
ejpam-4750	523	31	is	be	AUX
ejpam-4750	523	32	funded	fund	VERB
ejpam-4750	523	33	by	by	ADP
ejpam-4750	523	34	the	the	DET
ejpam-4750	523	35	department	department	PROPN
ejpam-4750	523	36	of	of	ADP
ejpam-4750	523	37	science	science	NOUN
ejpam-4750	523	38	and	and	CCONJ
ejpam-4750	523	39	technology	technology	NOUN
ejpam-4750	523	40	accelerated	accelerate	VERB
ejpam-4750	523	41	science	science	NOUN
ejpam-4750	523	42	and	and	CCONJ
ejpam-4750	523	43	technology	technology	NOUN
ejpam-4750	523	44	human	human	ADJ
ejpam-4750	523	45	resource	resource	NOUN
ejpam-4750	523	46	development	development	NOUN
ejpam-4750	523	47	program	program	NOUN
ejpam-4750	523	48	(	(	PUNCT
ejpam-4750	523	49	dost	dost	NOUN
ejpam-4750	523	50	-	-	PUNCT
ejpam-4750	523	51	asthrdp	asthrdp	NOUN
ejpam-4750	523	52	)	)	PUNCT
ejpam-4750	523	53	,	,	PUNCT
ejpam-4750	523	54	philippines	philippine	NOUN
ejpam-4750	523	55	.	.	PUNCT
ejpam-4750	524	1	references	reference	NOUN
ejpam-4750	524	2	[	[	X
ejpam-4750	524	3	1	1	NUM
ejpam-4750	524	4	]	]	X
ejpam-4750	524	5	r.f	r.f	PROPN
ejpam-4750	524	6	.	.	PROPN
ejpam-4750	524	7	bailey	bailey	PROPN
ejpam-4750	524	8	and	and	CCONJ
ejpam-4750	524	9	i.	i.	PROPN
ejpam-4750	524	10	yero	yero	PROPN
ejpam-4750	524	11	.	.	PUNCT
ejpam-4750	525	1	error	error	NOUN
ejpam-4750	525	2	-	-	PUNCT
ejpam-4750	525	3	correcting	correct	VERB
ejpam-4750	525	4	codes	code	NOUN
ejpam-4750	525	5	from	from	ADP
ejpam-4750	525	6	kresolving	kresolve	VERB
ejpam-4750	525	7	sets	set	NOUN
ejpam-4750	525	8	.	.	PUNCT
ejpam-4750	526	1	discussiones	discussione	NOUN
ejpam-4750	526	2	mathematicae	mathematicae	VERB
ejpam-4750	526	3	,	,	PUNCT
ejpam-4750	526	4	graph	graph	NOUN
ejpam-4750	526	5	theory	theory	NOUN
ejpam-4750	526	6	,	,	PUNCT
ejpam-4750	526	7	39:341–355	39:341–355	PROPN
ejpam-4750	526	8	,	,	PUNCT
ejpam-4750	526	9	2019	2019	NUM
ejpam-4750	526	10	.	.	PUNCT
ejpam-4750	527	1	[	[	X
ejpam-4750	527	2	2	2	X
ejpam-4750	527	3	]	]	PUNCT
ejpam-4750	527	4	s.	s.	PROPN
ejpam-4750	527	5	bau	bau	PROPN
ejpam-4750	527	6	and	and	CCONJ
ejpam-4750	527	7	a.f	a.f	PROPN
ejpam-4750	527	8	.	.	PUNCT
ejpam-4750	528	1	beardon	beardon	PROPN
ejpam-4750	528	2	.	.	PUNCT
ejpam-4750	529	1	the	the	DET
ejpam-4750	529	2	metric	metric	ADJ
ejpam-4750	529	3	dimension	dimension	NOUN
ejpam-4750	529	4	of	of	ADP
ejpam-4750	529	5	metric	metric	ADJ
ejpam-4750	529	6	spaces	space	NOUN
ejpam-4750	529	7	.	.	PUNCT
ejpam-4750	530	1	comput	comput	NOUN
ejpam-4750	530	2	.	.	PUNCT
ejpam-4750	531	1	methods	method	NOUN
ejpam-4750	531	2	funct	funct	VERB
ejpam-4750	531	3	.	.	PUNCT
ejpam-4750	532	1	theory	theory	NOUN
ejpam-4750	532	2	,	,	PUNCT
ejpam-4750	532	3	13:295–305	13:295–305	NUM
ejpam-4750	532	4	,	,	PUNCT
ejpam-4750	532	5	2013	2013	NUM
ejpam-4750	532	6	.	.	PUNCT
ejpam-4750	533	1	[	[	X
ejpam-4750	533	2	3	3	X
ejpam-4750	533	3	]	]	X
ejpam-4750	533	4	l.m	l.m	PROPN
ejpam-4750	533	5	.	.	PROPN
ejpam-4750	533	6	blumenthal	blumenthal	PROPN
ejpam-4750	533	7	.	.	PUNCT
ejpam-4750	534	1	theory	theory	NOUN
ejpam-4750	534	2	and	and	CCONJ
ejpam-4750	534	3	applications	application	NOUN
ejpam-4750	534	4	of	of	ADP
ejpam-4750	534	5	distance	distance	NOUN
ejpam-4750	534	6	geometry	geometry	NOUN
ejpam-4750	534	7	.	.	PUNCT
ejpam-4750	535	1	clarendon	clarendon	PROPN
ejpam-4750	535	2	press	press	PROPN
ejpam-4750	535	3	,	,	PUNCT
ejpam-4750	535	4	oxford	oxford	PROPN
ejpam-4750	535	5	,	,	PUNCT
ejpam-4750	535	6	1953	1953	NUM
ejpam-4750	535	7	.	.	PUNCT
ejpam-4750	536	1	[	[	X
ejpam-4750	536	2	4	4	X
ejpam-4750	536	3	]	]	X
ejpam-4750	536	4	j.	j.	PROPN
ejpam-4750	536	5	cabaro	cabaro	PROPN
ejpam-4750	536	6	and	and	CCONJ
ejpam-4750	536	7	h.	h.	PROPN
ejpam-4750	536	8	rara	rara	PROPN
ejpam-4750	536	9	.	.	PUNCT
ejpam-4750	537	1	on	on	ADP
ejpam-4750	537	2	2resolving	2resolving	NUM
ejpam-4750	537	3	sets	set	NOUN
ejpam-4750	537	4	in	in	ADP
ejpam-4750	537	5	join	join	NOUN
ejpam-4750	537	6	and	and	CCONJ
ejpam-4750	537	7	corona	corona	NOUN
ejpam-4750	537	8	of	of	ADP
ejpam-4750	537	9	graphs	graph	NOUN
ejpam-4750	537	10	.	.	PUNCT
ejpam-4750	538	1	european	european	ADJ
ejpam-4750	538	2	journal	journal	PROPN
ejpam-4750	538	3	of	of	ADP
ejpam-4750	538	4	pure	pure	ADJ
ejpam-4750	538	5	and	and	CCONJ
ejpam-4750	538	6	applied	applied	ADJ
ejpam-4750	538	7	mathematics	mathematic	NOUN
ejpam-4750	538	8	,	,	PUNCT
ejpam-4750	538	9	14(3):773–782	14(3):773–782	PROPN
ejpam-4750	538	10	,	,	PUNCT
ejpam-4750	538	11	2021	2021	NUM
ejpam-4750	538	12	.	.	PUNCT
ejpam-4750	539	1	[	[	X
ejpam-4750	539	2	5	5	X
ejpam-4750	539	3	]	]	PUNCT
ejpam-4750	539	4	f.	f.	PROPN
ejpam-4750	539	5	harary	harary	PROPN
ejpam-4750	539	6	and	and	CCONJ
ejpam-4750	539	7	r.a	r.a	PROPN
ejpam-4750	539	8	.	.	PROPN
ejpam-4750	539	9	melter	melter	NOUN
ejpam-4750	539	10	.	.	PUNCT
ejpam-4750	540	1	on	on	ADP
ejpam-4750	540	2	the	the	DET
ejpam-4750	540	3	metric	metric	ADJ
ejpam-4750	540	4	dimension	dimension	NOUN
ejpam-4750	540	5	of	of	ADP
ejpam-4750	540	6	a	a	DET
ejpam-4750	540	7	graph	graph	NOUN
ejpam-4750	540	8	.	.	PUNCT
ejpam-4750	540	9	ars	ars	PROPN
ejpam-4750	540	10	combin	combin	PROPN
ejpam-4750	540	11	.	.	PROPN
ejpam-4750	540	12	,	,	PUNCT
ejpam-4750	540	13	2:191–195	2:191–195	NUM
ejpam-4750	540	14	,	,	PUNCT
ejpam-4750	540	15	1976	1976	NUM
ejpam-4750	540	16	.	.	PUNCT
ejpam-4750	541	1	[	[	X
ejpam-4750	541	2	6	6	NUM
ejpam-4750	541	3	]	]	X
ejpam-4750	541	4	t.p	t.p	PROPN
ejpam-4750	541	5	.	.	PROPN
ejpam-4750	541	6	zivkovic	zivkovic	PROPN
ejpam-4750	541	7	,	,	PUNCT
ejpam-4750	541	8	f.	f.	PROPN
ejpam-4750	541	9	harary	harary	PROPN
ejpam-4750	541	10	and	and	CCONJ
ejpam-4750	541	11	d.j	d.j	PROPN
ejpam-4750	541	12	.	.	PROPN
ejpam-4750	541	13	klein	klein	PROPN
ejpam-4750	541	14	.	.	PUNCT
ejpam-4750	542	1	graphical	graphical	ADJ
ejpam-4750	542	2	properties	property	NOUN
ejpam-4750	542	3	of	of	ADP
ejpam-4750	542	4	polyhexes	polyhexe	NOUN
ejpam-4750	542	5	:	:	PUNCT
ejpam-4750	542	6	perfect	perfect	ADJ
ejpam-4750	542	7	matching	matching	NOUN
ejpam-4750	542	8	vector	vector	NOUN
ejpam-4750	542	9	and	and	CCONJ
ejpam-4750	542	10	forcing	forcing	NOUN
ejpam-4750	542	11	.	.	PUNCT
ejpam-4750	543	1	j.	j.	PROPN
ejpam-4750	543	2	math	math	PROPN
ejpam-4750	543	3	.	.	PUNCT
ejpam-4750	544	1	chem	chem	PROPN
ejpam-4750	544	2	.	.	PUNCT
ejpam-4750	544	3	,	,	PUNCT
ejpam-4750	545	1	6:295–306	6:295–306	NUM
ejpam-4750	545	2	,	,	PUNCT
ejpam-4750	545	3	1991	1991	NUM
ejpam-4750	545	4	.	.	PUNCT
ejpam-4750	546	1	[	[	X
ejpam-4750	546	2	7	7	X
ejpam-4750	546	3	]	]	X
ejpam-4750	546	4	m.	m.	NOUN
ejpam-4750	546	5	heydarpour	heydarpour	PROPN
ejpam-4750	546	6	and	and	CCONJ
ejpam-4750	546	7	s.	s.	PROPN
ejpam-4750	546	8	maghsoudi	maghsoudi	PROPN
ejpam-4750	546	9	.	.	PUNCT
ejpam-4750	547	1	the	the	DET
ejpam-4750	547	2	metric	metric	ADJ
ejpam-4750	547	3	dimension	dimension	NOUN
ejpam-4750	547	4	of	of	ADP
ejpam-4750	547	5	geometric	geometric	ADJ
ejpam-4750	547	6	spaces	space	NOUN
ejpam-4750	547	7	.	.	PUNCT
ejpam-4750	548	1	topology	topology	NOUN
ejpam-4750	548	2	appl	appl	PROPN
ejpam-4750	548	3	.	.	PROPN
ejpam-4750	548	4	,	,	PUNCT
ejpam-4750	548	5	178:230–235	178:230–235	NUM
ejpam-4750	548	6	,	,	PUNCT
ejpam-4750	548	7	2014	2014	NUM
ejpam-4750	548	8	.	.	PUNCT
ejpam-4750	549	1	[	[	X
ejpam-4750	549	2	8	8	NUM
ejpam-4750	549	3	]	]	X
ejpam-4750	549	4	d.j	d.j	PROPN
ejpam-4750	549	5	.	.	PROPN
ejpam-4750	549	6	klein	klein	PROPN
ejpam-4750	549	7	and	and	CCONJ
ejpam-4750	549	8	m.	m.	PROPN
ejpam-4750	549	9	randic	randic	PROPN
ejpam-4750	549	10	.	.	PUNCT
ejpam-4750	549	11	innate	innate	ADJ
ejpam-4750	549	12	degree	degree	NOUN
ejpam-4750	549	13	of	of	ADP
ejpam-4750	549	14	freedom	freedom	NOUN
ejpam-4750	549	15	of	of	ADP
ejpam-4750	549	16	a	a	DET
ejpam-4750	549	17	graph	graph	NOUN
ejpam-4750	549	18	.	.	PUNCT
ejpam-4750	550	1	j.	j.	PROPN
ejpam-4750	550	2	comput	comput	PROPN
ejpam-4750	550	3	.	.	PUNCT
ejpam-4750	551	1	chem	chem	PROPN
ejpam-4750	551	2	.	.	PUNCT
ejpam-4750	551	3	,	,	PUNCT
ejpam-4750	551	4	8:516–521	8:516–521	NOUN
ejpam-4750	551	5	,	,	PUNCT
ejpam-4750	551	6	1987	1987	NUM
ejpam-4750	551	7	.	.	PUNCT
ejpam-4750	552	1	[	[	X
ejpam-4750	552	2	9	9	NUM
ejpam-4750	552	3	]	]	SYM
ejpam-4750	552	4	a.	a.	NOUN
ejpam-4750	552	5	estrada	estrada	PROPN
ejpam-4750	552	6	-	-	PUNCT
ejpam-4750	552	7	moreno	moreno	PROPN
ejpam-4750	552	8	,	,	PUNCT
ejpam-4750	552	9	j.	j.	PROPN
ejpam-4750	552	10	rodrıguez	rodrıguez	PROPN
ejpam-4750	552	11	-	-	PUNCT
ejpam-4750	552	12	velazquez	velazquez	PROPN
ejpam-4750	552	13	and	and	CCONJ
ejpam-4750	552	14	i.	i.	PROPN
ejpam-4750	552	15	yero	yero	PROPN
ejpam-4750	552	16	.	.	PUNCT
ejpam-4750	553	1	the	the	DET
ejpam-4750	553	2	kmetric	kmetric	ADJ
ejpam-4750	553	3	dimension	dimension	NOUN
ejpam-4750	553	4	of	of	ADP
ejpam-4750	553	5	agraph	agraph	NOUN
ejpam-4750	553	6	.	.	PUNCT
ejpam-4750	554	1	applied	apply	VERB
ejpam-4750	554	2	mathematics	mathematic	NOUN
ejpam-4750	554	3	and	and	CCONJ
ejpam-4750	554	4	information	information	NOUN
ejpam-4750	554	5	sciences	science	NOUN
ejpam-4750	554	6	,	,	PUNCT
ejpam-4750	554	7	9:2829–2840	9:2829–2840	NUM
ejpam-4750	554	8	,	,	PUNCT
ejpam-4750	554	9	2015	2015	NUM
ejpam-4750	554	10	.	.	PUNCT
ejpam-4750	555	1	[	[	X
ejpam-4750	555	2	10	10	NUM
ejpam-4750	555	3	]	]	X
ejpam-4750	555	4	v.	v.	ADP
ejpam-4750	555	5	saenpholphat	saenpholphat	PROPN
ejpam-4750	555	6	and	and	CCONJ
ejpam-4750	555	7	p.	p.	PROPN
ejpam-4750	555	8	zhang	zhang	PROPN
ejpam-4750	555	9	.	.	PUNCT
ejpam-4750	556	1	on	on	ADP
ejpam-4750	556	2	connected	connected	ADJ
ejpam-4750	556	3	resolvability	resolvability	NOUN
ejpam-4750	556	4	of	of	ADP
ejpam-4750	556	5	graphs	graph	NOUN
ejpam-4750	556	6	.	.	PUNCT
ejpam-4750	557	1	australian	australian	ADJ
ejpam-4750	557	2	journal	journal	NOUN
ejpam-4750	557	3	of	of	ADP
ejpam-4750	557	4	combinatorics	combinatoric	NOUN
ejpam-4750	557	5	,	,	PUNCT
ejpam-4750	557	6	28:25–37	28:25–37	NUM
ejpam-4750	557	7	,	,	PUNCT
ejpam-4750	557	8	2003	2003	NUM
ejpam-4750	557	9	.	.	PUNCT
ejpam-4750	558	1	[	[	X
ejpam-4750	558	2	11	11	NUM
ejpam-4750	558	3	]	]	X
ejpam-4750	558	4	p.j	p.j	PROPN
ejpam-4750	558	5	.	.	PROPN
ejpam-4750	558	6	slater	slater	PROPN
ejpam-4750	558	7	.	.	PUNCT
ejpam-4750	559	1	leaves	leave	NOUN
ejpam-4750	559	2	of	of	ADP
ejpam-4750	559	3	trees	tree	NOUN
ejpam-4750	559	4	.	.	PUNCT
ejpam-4750	560	1	congress	congress	PROPN
ejpam-4750	560	2	.	.	PUNCT
ejpam-4750	561	1	numer	numer	PROPN
ejpam-4750	561	2	.	.	PROPN
ejpam-4750	561	3	,	,	PUNCT
ejpam-4750	562	1	14:549–559	14:549–559	NUM
ejpam-4750	562	2	,	,	PUNCT
ejpam-4750	562	3	1975	1975	NUM
ejpam-4750	562	4	.	.	PUNCT
ejpam-4750	563	1	[	[	X
ejpam-4750	563	2	12	12	NUM
ejpam-4750	563	3	]	]	X
ejpam-4750	563	4	p.j	p.j	PROPN
ejpam-4750	563	5	.	.	PROPN
ejpam-4750	563	6	slater	slater	PROPN
ejpam-4750	563	7	.	.	PUNCT
ejpam-4750	564	1	dominating	dominating	NOUN
ejpam-4750	564	2	and	and	CCONJ
ejpam-4750	564	3	reference	reference	NOUN
ejpam-4750	564	4	sets	set	NOUN
ejpam-4750	564	5	in	in	ADP
ejpam-4750	564	6	graphs	graph	NOUN
ejpam-4750	564	7	.	.	PUNCT
ejpam-4750	565	1	j.	j.	PROPN
ejpam-4750	565	2	math	math	PROPN
ejpam-4750	565	3	.	.	PUNCT
ejpam-4750	566	1	phys	phy	NOUN
ejpam-4750	566	2	.	.	PUNCT
ejpam-4750	567	1	sci	sci	PROPN
ejpam-4750	567	2	.	.	PROPN
ejpam-4750	567	3	,	,	PUNCT
ejpam-4750	567	4	22:445–455	22:445–455	PROPN
ejpam-4750	567	5	,	,	PUNCT
ejpam-4750	567	6	1988	1988	NUM
ejpam-4750	567	7	.	.	PUNCT
ejpam-4750	568	1	[	[	X
ejpam-4750	568	2	13	13	NUM
ejpam-4750	568	3	]	]	PUNCT
ejpam-4750	568	4	g.	g.	PROPN
ejpam-4750	568	5	chartrand	chartrand	PROPN
ejpam-4750	568	6	,	,	PUNCT
ejpam-4750	568	7	p.	p.	PROPN
ejpam-4750	568	8	zhang	zhang	PROPN
ejpam-4750	568	9	and	and	CCONJ
ejpam-4750	568	10	kalamazoo	kalamazoo	PROPN
ejpam-4750	568	11	.	.	PUNCT
ejpam-4750	569	1	the	the	DET
ejpam-4750	569	2	forcing	force	VERB
ejpam-4750	569	3	dimension	dimension	NOUN
ejpam-4750	569	4	of	of	ADP
ejpam-4750	569	5	a	a	DET
ejpam-4750	569	6	graph	graph	NOUN
ejpam-4750	569	7	.	.	PUNCT
ejpam-4750	570	1	mathematica	mathematica	PROPN
ejpam-4750	570	2	bohemica	bohemica	PROPN
ejpam-4750	570	3	,	,	PUNCT
ejpam-4750	570	4	126(4):711–720	126(4):711–720	PROPN
ejpam-4750	570	5	,	,	PUNCT
ejpam-4750	570	6	2001	2001	NUM
ejpam-4750	570	7	.	.	PUNCT