id	sid	tid	token	lemma	pos
ejpam-4653	1	1	european	european	PROPN
ejpam-4653	1	2	journal	journal	PROPN
ejpam-4653	1	3	of	of	ADP
ejpam-4653	1	4	pure	pure	ADJ
ejpam-4653	1	5	and	and	CCONJ
ejpam-4653	1	6	applied	apply	VERB
ejpam-4653	1	7	mathematics	mathematic	NOUN
ejpam-4653	1	8	vol	vol	NOUN
ejpam-4653	1	9	.	.	PUNCT
ejpam-4653	2	1	16	16	NUM
ejpam-4653	2	2	,	,	PUNCT
ejpam-4653	2	3	no	no	INTJ
ejpam-4653	2	4	.	.	NOUN
ejpam-4653	2	5	2	2	NUM
ejpam-4653	2	6	,	,	PUNCT
ejpam-4653	2	7	2023	2023	NUM
ejpam-4653	2	8	,	,	PUNCT
ejpam-4653	2	9	847	847	NUM
ejpam-4653	2	10	-	-	SYM
ejpam-4653	2	11	863	863	NUM
ejpam-4653	2	12	issn	issn	PROPN
ejpam-4653	2	13	1307	1307	NUM
ejpam-4653	2	14	-	-	SYM
ejpam-4653	2	15	5543	5543	NUM
ejpam-4653	2	16	–	–	PUNCT
ejpam-4653	3	1	ejpam.com	ejpam.com	X
ejpam-4653	3	2	published	publish	VERB
ejpam-4653	3	3	by	by	ADP
ejpam-4653	3	4	new	new	PROPN
ejpam-4653	3	5	york	york	PROPN
ejpam-4653	3	6	business	business	PROPN
ejpam-4653	3	7	global	global	ADJ
ejpam-4653	3	8	on	on	ADP
ejpam-4653	3	9	double	double	ADJ
ejpam-4653	3	10	roman	roman	ADJ
ejpam-4653	3	11	dominating	dominating	NOUN
ejpam-4653	3	12	functions	function	NOUN
ejpam-4653	3	13	in	in	ADP
ejpam-4653	3	14	graphs	graph	NOUN
ejpam-4653	3	15	jerry	jerry	NOUN
ejpam-4653	3	16	boy	boy	NOUN
ejpam-4653	3	17	g.	g.	PROPN
ejpam-4653	3	18	cariaga1,∗	cariaga1,∗	PROPN
ejpam-4653	3	19	,	,	PUNCT
ejpam-4653	3	20	ferdinand	ferdinand	PROPN
ejpam-4653	3	21	p.	p.	PROPN
ejpam-4653	4	1	jamil1,2	jamil1,2	PROPN
ejpam-4653	4	2	1	1	NUM
ejpam-4653	4	3	department	department	NOUN
ejpam-4653	4	4	of	of	ADP
ejpam-4653	4	5	mathematics	mathematic	NOUN
ejpam-4653	4	6	and	and	CCONJ
ejpam-4653	4	7	statistics	statistic	NOUN
ejpam-4653	4	8	,	,	PUNCT
ejpam-4653	4	9	college	college	NOUN
ejpam-4653	4	10	of	of	ADP
ejpam-4653	4	11	science	science	NOUN
ejpam-4653	4	12	and	and	CCONJ
ejpam-4653	4	13	mathematics	mathematic	NOUN
ejpam-4653	4	14	,	,	PUNCT
ejpam-4653	4	15	2	2	NUM
ejpam-4653	4	16	center	center	NOUN
ejpam-4653	4	17	for	for	ADP
ejpam-4653	4	18	graph	graph	NOUN
ejpam-4653	4	19	theory	theory	NOUN
ejpam-4653	4	20	,	,	PUNCT
ejpam-4653	4	21	premier	premier	PROPN
ejpam-4653	4	22	research	research	PROPN
ejpam-4653	4	23	institute	institute	PROPN
ejpam-4653	4	24	of	of	ADP
ejpam-4653	4	25	science	science	NOUN
ejpam-4653	4	26	and	and	CCONJ
ejpam-4653	4	27	mathematics	mathematic	NOUN
ejpam-4653	4	28	,	,	PUNCT
ejpam-4653	4	29	msu	msu	PROPN
ejpam-4653	4	30	-	-	PUNCT
ejpam-4653	4	31	iligan	iligan	PROPN
ejpam-4653	4	32	institute	institute	PROPN
ejpam-4653	4	33	of	of	ADP
ejpam-4653	4	34	technology	technology	PROPN
ejpam-4653	4	35	,	,	PUNCT
ejpam-4653	4	36	9200	9200	NUM
ejpam-4653	4	37	iligan	iligan	ADJ
ejpam-4653	4	38	city	city	NOUN
ejpam-4653	4	39	,	,	PUNCT
ejpam-4653	4	40	philippines	philippine	NOUN
ejpam-4653	4	41	abstract	abstract	ADJ
ejpam-4653	4	42	.	.	PUNCT
ejpam-4653	5	1	let	let	VERB
ejpam-4653	5	2	g	g	PRON
ejpam-4653	5	3	be	be	AUX
ejpam-4653	5	4	a	a	DET
ejpam-4653	5	5	connected	connected	ADJ
ejpam-4653	5	6	graph	graph	NOUN
ejpam-4653	5	7	.	.	PUNCT
ejpam-4653	6	1	a	a	DET
ejpam-4653	6	2	function	function	NOUN
ejpam-4653	6	3	f	f	NOUN
ejpam-4653	6	4	:	:	PUNCT
ejpam-4653	6	5	v	v	X
ejpam-4653	6	6	(	(	PUNCT
ejpam-4653	6	7	g	g	NOUN
ejpam-4653	6	8	)	)	PUNCT
ejpam-4653	6	9	→	→	SYM
ejpam-4653	6	10	{	{	PUNCT
ejpam-4653	6	11	0	0	NUM
ejpam-4653	6	12	,	,	PUNCT
ejpam-4653	6	13	1	1	NUM
ejpam-4653	6	14	,	,	PUNCT
ejpam-4653	6	15	2	2	NUM
ejpam-4653	6	16	,	,	PUNCT
ejpam-4653	6	17	3	3	NUM
ejpam-4653	6	18	}	}	PUNCT
ejpam-4653	6	19	is	be	AUX
ejpam-4653	6	20	a	a	DET
ejpam-4653	6	21	double	double	ADJ
ejpam-4653	6	22	roman	roman	ADJ
ejpam-4653	6	23	dominating	dominating	NOUN
ejpam-4653	6	24	function	function	NOUN
ejpam-4653	6	25	of	of	ADP
ejpam-4653	6	26	g	g	PROPN
ejpam-4653	6	27	if	if	SCONJ
ejpam-4653	6	28	for	for	ADP
ejpam-4653	6	29	each	each	PRON
ejpam-4653	6	30	v	v	NUM
ejpam-4653	6	31	∈	∈	PROPN
ejpam-4653	6	32	v	v	NOUN
ejpam-4653	6	33	(	(	PUNCT
ejpam-4653	6	34	g	g	NOUN
ejpam-4653	6	35	)	)	PUNCT
ejpam-4653	6	36	with	with	ADP
ejpam-4653	6	37	f(v	f(v	NOUN
ejpam-4653	6	38	)	)	PUNCT
ejpam-4653	6	39	=	=	SYM
ejpam-4653	6	40	0	0	NUM
ejpam-4653	6	41	,	,	PUNCT
ejpam-4653	6	42	v	v	NOUN
ejpam-4653	6	43	has	have	VERB
ejpam-4653	6	44	two	two	NUM
ejpam-4653	6	45	adjacent	adjacent	ADJ
ejpam-4653	6	46	vertices	vertex	NOUN
ejpam-4653	6	47	u	u	NOUN
ejpam-4653	6	48	and	and	CCONJ
ejpam-4653	6	49	w	w	NOUN
ejpam-4653	6	50	for	for	ADP
ejpam-4653	6	51	which	which	PRON
ejpam-4653	6	52	f(u	f(u	PROPN
ejpam-4653	6	53	)	)	PUNCT
ejpam-4653	6	54	=	=	PUNCT
ejpam-4653	7	1	f(w	f(w	PROPN
ejpam-4653	7	2	)	)	PUNCT
ejpam-4653	7	3	=	=	SYM
ejpam-4653	7	4	2	2	NUM
ejpam-4653	7	5	or	or	CCONJ
ejpam-4653	7	6	v	v	NOUN
ejpam-4653	7	7	has	have	VERB
ejpam-4653	7	8	an	an	DET
ejpam-4653	7	9	adjacent	adjacent	ADJ
ejpam-4653	7	10	vertex	vertex	NOUN
ejpam-4653	7	11	u	u	NOUN
ejpam-4653	7	12	for	for	ADP
ejpam-4653	7	13	which	which	PRON
ejpam-4653	7	14	f(u	f(u	PROPN
ejpam-4653	7	15	)	)	PUNCT
ejpam-4653	7	16	=	=	SYM
ejpam-4653	7	17	3	3	NUM
ejpam-4653	7	18	,	,	PUNCT
ejpam-4653	7	19	and	and	CCONJ
ejpam-4653	7	20	for	for	ADP
ejpam-4653	7	21	each	each	PRON
ejpam-4653	7	22	v	v	NUM
ejpam-4653	7	23	∈	∈	PROPN
ejpam-4653	7	24	v	v	NOUN
ejpam-4653	7	25	(	(	PUNCT
ejpam-4653	7	26	g	g	NOUN
ejpam-4653	7	27	)	)	PUNCT
ejpam-4653	7	28	with	with	ADP
ejpam-4653	7	29	f(v	f(v	NOUN
ejpam-4653	7	30	)	)	PUNCT
ejpam-4653	7	31	=	=	SYM
ejpam-4653	7	32	1	1	NUM
ejpam-4653	7	33	,	,	PUNCT
ejpam-4653	7	34	v	v	NOUN
ejpam-4653	7	35	is	be	AUX
ejpam-4653	7	36	adjacent	adjacent	ADJ
ejpam-4653	7	37	to	to	ADP
ejpam-4653	7	38	a	a	DET
ejpam-4653	7	39	vertex	vertex	NOUN
ejpam-4653	7	40	u	u	NOUN
ejpam-4653	7	41	for	for	ADP
ejpam-4653	7	42	which	which	PRON
ejpam-4653	7	43	either	either	CCONJ
ejpam-4653	7	44	f(u	f(u	PROPN
ejpam-4653	7	45	)	)	PUNCT
ejpam-4653	8	1	=	=	SYM
ejpam-4653	8	2	2	2	NUM
ejpam-4653	8	3	or	or	CCONJ
ejpam-4653	8	4	f(u	f(u	PROPN
ejpam-4653	8	5	)	)	PUNCT
ejpam-4653	8	6	=	=	SYM
ejpam-4653	9	1	3	3	X
ejpam-4653	9	2	.	.	PUNCT
ejpam-4653	9	3	the	the	DET
ejpam-4653	9	4	minimum	minimum	ADJ
ejpam-4653	9	5	weight	weight	NOUN
ejpam-4653	9	6	ωg(f	ωg(f	PRON
ejpam-4653	9	7	)	)	PUNCT
ejpam-4653	9	8	=	=	SYM
ejpam-4653	9	9	∑	∑	PUNCT
ejpam-4653	9	10	v∈v	v∈v	PROPN
ejpam-4653	9	11	(	(	PUNCT
ejpam-4653	9	12	g	g	NOUN
ejpam-4653	9	13	)	)	PUNCT
ejpam-4653	9	14	f(v	f(v	NOUN
ejpam-4653	9	15	)	)	PUNCT
ejpam-4653	9	16	of	of	ADP
ejpam-4653	9	17	a	a	DET
ejpam-4653	9	18	double	double	ADJ
ejpam-4653	9	19	roman	roman	ADJ
ejpam-4653	9	20	dominating	dominating	NOUN
ejpam-4653	9	21	function	function	NOUN
ejpam-4653	9	22	f	f	PROPN
ejpam-4653	9	23	of	of	ADP
ejpam-4653	9	24	g	g	PROPN
ejpam-4653	9	25	is	be	AUX
ejpam-4653	9	26	the	the	DET
ejpam-4653	9	27	double	double	ADJ
ejpam-4653	9	28	roman	roman	ADJ
ejpam-4653	9	29	domination	domination	NOUN
ejpam-4653	9	30	number	number	NOUN
ejpam-4653	9	31	of	of	ADP
ejpam-4653	9	32	g.	g.	PROPN
ejpam-4653	9	33	in	in	ADP
ejpam-4653	9	34	this	this	DET
ejpam-4653	9	35	paper	paper	NOUN
ejpam-4653	9	36	,	,	PUNCT
ejpam-4653	9	37	we	we	PRON
ejpam-4653	9	38	continue	continue	VERB
ejpam-4653	9	39	the	the	DET
ejpam-4653	9	40	study	study	NOUN
ejpam-4653	9	41	of	of	ADP
ejpam-4653	9	42	double	double	ADJ
ejpam-4653	9	43	roman	roman	ADJ
ejpam-4653	9	44	domination	domination	NOUN
ejpam-4653	9	45	introduced	introduce	VERB
ejpam-4653	9	46	and	and	CCONJ
ejpam-4653	9	47	studied	study	VERB
ejpam-4653	9	48	by	by	ADP
ejpam-4653	9	49	r.a	r.a	PROPN
ejpam-4653	9	50	.	.	PROPN
ejpam-4653	9	51	beeler	beeler	PROPN
ejpam-4653	9	52	et	et	PROPN
ejpam-4653	9	53	al	al	PROPN
ejpam-4653	9	54	.	.	PUNCT
ejpam-4653	10	1	in	in	ADP
ejpam-4653	10	2	[	[	X
ejpam-4653	10	3	2	2	NUM
ejpam-4653	10	4	]	]	PUNCT
ejpam-4653	10	5	.	.	PUNCT
ejpam-4653	11	1	first	first	ADV
ejpam-4653	11	2	,	,	PUNCT
ejpam-4653	11	3	we	we	PRON
ejpam-4653	11	4	characterize	characterize	VERB
ejpam-4653	11	5	some	some	DET
ejpam-4653	11	6	double	double	ADJ
ejpam-4653	11	7	roman	roman	ADJ
ejpam-4653	11	8	domination	domination	NOUN
ejpam-4653	11	9	numbers	number	NOUN
ejpam-4653	11	10	with	with	ADP
ejpam-4653	11	11	small	small	ADJ
ejpam-4653	11	12	values	value	NOUN
ejpam-4653	11	13	in	in	ADP
ejpam-4653	11	14	terms	term	NOUN
ejpam-4653	11	15	of	of	ADP
ejpam-4653	11	16	the	the	DET
ejpam-4653	11	17	domination	domination	NOUN
ejpam-4653	11	18	numbers	number	NOUN
ejpam-4653	11	19	and	and	CCONJ
ejpam-4653	11	20	2	2	NUM
ejpam-4653	11	21	-	-	PUNCT
ejpam-4653	11	22	domination	domination	NOUN
ejpam-4653	11	23	numbers	number	NOUN
ejpam-4653	11	24	.	.	PUNCT
ejpam-4653	12	1	then	then	ADV
ejpam-4653	12	2	we	we	PRON
ejpam-4653	12	3	determine	determine	VERB
ejpam-4653	12	4	the	the	DET
ejpam-4653	12	5	double	double	ADJ
ejpam-4653	12	6	roman	roman	ADJ
ejpam-4653	12	7	domination	domination	NOUN
ejpam-4653	12	8	numbers	number	NOUN
ejpam-4653	12	9	of	of	ADP
ejpam-4653	12	10	the	the	DET
ejpam-4653	12	11	join	join	NOUN
ejpam-4653	12	12	,	,	PUNCT
ejpam-4653	12	13	corona	corona	PROPN
ejpam-4653	12	14	,	,	PUNCT
ejpam-4653	12	15	complementary	complementary	ADJ
ejpam-4653	12	16	prism	prism	NOUN
ejpam-4653	12	17	and	and	CCONJ
ejpam-4653	12	18	lexicographic	lexicographic	ADJ
ejpam-4653	12	19	product	product	NOUN
ejpam-4653	12	20	of	of	ADP
ejpam-4653	12	21	graphs	graph	NOUN
ejpam-4653	12	22	.	.	PUNCT
ejpam-4653	13	1	2020	2020	NUM
ejpam-4653	13	2	mathematics	mathematic	NOUN
ejpam-4653	13	3	subject	subject	NOUN
ejpam-4653	13	4	classifications	classification	NOUN
ejpam-4653	13	5	:	:	PUNCT
ejpam-4653	13	6	05c69	05c69	X
ejpam-4653	13	7	key	key	ADJ
ejpam-4653	13	8	words	word	NOUN
ejpam-4653	13	9	and	and	CCONJ
ejpam-4653	13	10	phrases	phrase	NOUN
ejpam-4653	13	11	:	:	PUNCT
ejpam-4653	13	12	domination	domination	NOUN
ejpam-4653	13	13	number	number	NOUN
ejpam-4653	13	14	,	,	PUNCT
ejpam-4653	13	15	2	2	NUM
ejpam-4653	13	16	-	-	PUNCT
ejpam-4653	13	17	domination	domination	NOUN
ejpam-4653	13	18	number	number	NOUN
ejpam-4653	13	19	,	,	PUNCT
ejpam-4653	13	20	double	double	ADJ
ejpam-4653	13	21	roman	roman	ADJ
ejpam-4653	13	22	dominating	dominating	NOUN
ejpam-4653	13	23	function	function	NOUN
ejpam-4653	13	24	,	,	PUNCT
ejpam-4653	13	25	double	double	ADJ
ejpam-4653	13	26	roman	roman	ADJ
ejpam-4653	13	27	domination	domination	NOUN
ejpam-4653	13	28	number	number	NOUN
ejpam-4653	13	29	1	1	NUM
ejpam-4653	13	30	.	.	PUNCT
ejpam-4653	14	1	introduction	introduction	NOUN
ejpam-4653	14	2	throughout	throughout	ADP
ejpam-4653	14	3	this	this	DET
ejpam-4653	14	4	paper	paper	NOUN
ejpam-4653	14	5	,	,	PUNCT
ejpam-4653	14	6	all	all	DET
ejpam-4653	14	7	graphs	graph	NOUN
ejpam-4653	14	8	considered	consider	VERB
ejpam-4653	14	9	are	be	AUX
ejpam-4653	14	10	finite	finite	ADJ
ejpam-4653	14	11	,	,	PUNCT
ejpam-4653	14	12	simple	simple	ADJ
ejpam-4653	14	13	and	and	CCONJ
ejpam-4653	14	14	undirected	undirected	ADJ
ejpam-4653	14	15	.	.	PUNCT
ejpam-4653	15	1	let	let	VERB
ejpam-4653	15	2	g	g	PROPN
ejpam-4653	15	3	=	=	SYM
ejpam-4653	15	4	(	(	PUNCT
ejpam-4653	15	5	v	v	NOUN
ejpam-4653	15	6	(	(	PUNCT
ejpam-4653	15	7	g	g	NOUN
ejpam-4653	15	8	)	)	PUNCT
ejpam-4653	15	9	,	,	PUNCT
ejpam-4653	15	10	e(g	e(g	PROPN
ejpam-4653	15	11	)	)	PUNCT
ejpam-4653	15	12	)	)	PUNCT
ejpam-4653	16	1	be	be	AUX
ejpam-4653	16	2	a	a	DET
ejpam-4653	16	3	graph	graph	NOUN
ejpam-4653	16	4	with	with	ADP
ejpam-4653	16	5	v	v	NOUN
ejpam-4653	16	6	(	(	PUNCT
ejpam-4653	16	7	g	g	NOUN
ejpam-4653	16	8	)	)	PUNCT
ejpam-4653	16	9	and	and	CCONJ
ejpam-4653	16	10	e(g	e(g	PROPN
ejpam-4653	16	11	)	)	PUNCT
ejpam-4653	16	12	being	be	AUX
ejpam-4653	16	13	the	the	DET
ejpam-4653	16	14	vertex	vertex	NOUN
ejpam-4653	16	15	set	set	NOUN
ejpam-4653	16	16	and	and	CCONJ
ejpam-4653	16	17	edge	edge	NOUN
ejpam-4653	16	18	set	set	NOUN
ejpam-4653	16	19	of	of	ADP
ejpam-4653	16	20	g	g	NOUN
ejpam-4653	16	21	,	,	PUNCT
ejpam-4653	16	22	respectively	respectively	ADV
ejpam-4653	16	23	.	.	PUNCT
ejpam-4653	17	1	for	for	ADP
ejpam-4653	17	2	s	s	PROPN
ejpam-4653	17	3	⊆	⊆	NUM
ejpam-4653	17	4	v	v	NOUN
ejpam-4653	17	5	(	(	PUNCT
ejpam-4653	17	6	g	g	NOUN
ejpam-4653	17	7	)	)	PUNCT
ejpam-4653	17	8	,	,	PUNCT
ejpam-4653	17	9	the	the	DET
ejpam-4653	17	10	symbol	symbol	NOUN
ejpam-4653	17	11	|s|	|s|	NOUN
ejpam-4653	17	12	refers	refer	VERB
ejpam-4653	17	13	to	to	ADP
ejpam-4653	17	14	the	the	DET
ejpam-4653	17	15	cardinality	cardinality	NOUN
ejpam-4653	17	16	of	of	ADP
ejpam-4653	17	17	s.	s.	PROPN
ejpam-4653	17	18	in	in	ADP
ejpam-4653	17	19	particular	particular	ADJ
ejpam-4653	17	20	,	,	PUNCT
ejpam-4653	17	21	|v	|v	PROPN
ejpam-4653	17	22	(	(	PUNCT
ejpam-4653	17	23	g)|	g)|	PROPN
ejpam-4653	17	24	is	be	AUX
ejpam-4653	17	25	the	the	DET
ejpam-4653	17	26	order	order	NOUN
ejpam-4653	17	27	of	of	ADP
ejpam-4653	17	28	g.	g.	PROPN
ejpam-4653	17	29	for	for	ADP
ejpam-4653	17	30	other	other	ADJ
ejpam-4653	17	31	basic	basic	ADJ
ejpam-4653	17	32	concepts	concept	NOUN
ejpam-4653	17	33	not	not	PART
ejpam-4653	17	34	presented	present	VERB
ejpam-4653	17	35	but	but	CCONJ
ejpam-4653	17	36	are	be	AUX
ejpam-4653	17	37	used	use	VERB
ejpam-4653	17	38	here	here	ADV
ejpam-4653	17	39	are	be	AUX
ejpam-4653	17	40	adopted	adopt	VERB
ejpam-4653	17	41	from	from	ADP
ejpam-4653	17	42	(	(	PUNCT
ejpam-4653	17	43	[	[	X
ejpam-4653	17	44	4	4	NUM
ejpam-4653	17	45	,	,	PUNCT
ejpam-4653	17	46	11	11	NUM
ejpam-4653	17	47	]	]	NUM
ejpam-4653	17	48	)	)	PUNCT
ejpam-4653	17	49	.	.	PUNCT
ejpam-4653	18	1	for	for	ADP
ejpam-4653	18	2	a	a	DET
ejpam-4653	18	3	vertex	vertex	NOUN
ejpam-4653	18	4	v	v	NOUN
ejpam-4653	18	5	of	of	ADP
ejpam-4653	18	6	a	a	DET
ejpam-4653	18	7	graph	graph	NOUN
ejpam-4653	18	8	g	g	NOUN
ejpam-4653	18	9	,	,	PUNCT
ejpam-4653	18	10	the	the	DET
ejpam-4653	18	11	open	open	ADJ
ejpam-4653	18	12	neighborhood	neighborhood	NOUN
ejpam-4653	18	13	of	of	ADP
ejpam-4653	18	14	v	v	NOUN
ejpam-4653	18	15	refers	refer	VERB
ejpam-4653	18	16	to	to	ADP
ejpam-4653	18	17	the	the	DET
ejpam-4653	18	18	set	set	NOUN
ejpam-4653	18	19	ng(v	ng(v	PUNCT
ejpam-4653	18	20	)	)	PUNCT
ejpam-4653	18	21	=	=	SYM
ejpam-4653	19	1	{	{	PUNCT
ejpam-4653	19	2	u	u	NOUN
ejpam-4653	19	3	∈	∈	PROPN
ejpam-4653	19	4	v	v	NOUN
ejpam-4653	19	5	(	(	PUNCT
ejpam-4653	19	6	g	g	NOUN
ejpam-4653	19	7	)	)	PUNCT
ejpam-4653	19	8	:	:	PUNCT
ejpam-4653	19	9	uv	uv	PROPN
ejpam-4653	19	10	∈	∈	PROPN
ejpam-4653	19	11	e(g	e(g	PROPN
ejpam-4653	19	12	)	)	PUNCT
ejpam-4653	19	13	}	}	PUNCT
ejpam-4653	19	14	while	while	SCONJ
ejpam-4653	19	15	its	its	PRON
ejpam-4653	19	16	closed	closed	ADJ
ejpam-4653	19	17	neighborhood	neighborhood	NOUN
ejpam-4653	19	18	is	be	AUX
ejpam-4653	19	19	the	the	DET
ejpam-4653	19	20	set	set	NOUN
ejpam-4653	19	21	ng[v	ng[v	NOUN
ejpam-4653	19	22	]	]	X
ejpam-4653	19	23	=	=	SYM
ejpam-4653	19	24	{	{	PUNCT
ejpam-4653	19	25	v	v	NOUN
ejpam-4653	19	26	}	}	PUNCT
ejpam-4653	19	27	∪ng(v	∪ng(v	ADJ
ejpam-4653	19	28	)	)	PUNCT
ejpam-4653	19	29	.	.	PUNCT
ejpam-4653	20	1	vertex	vertex	NOUN
ejpam-4653	20	2	v	v	NOUN
ejpam-4653	20	3	is	be	AUX
ejpam-4653	20	4	an	an	DET
ejpam-4653	20	5	isolated	isolated	ADJ
ejpam-4653	20	6	vertex	vertex	NOUN
ejpam-4653	20	7	if	if	SCONJ
ejpam-4653	20	8	ng(v	ng(v	NOUN
ejpam-4653	20	9	)	)	PUNCT
ejpam-4653	20	10	=	=	PUNCT
ejpam-4653	20	11	∅.	∅.	NOUN
ejpam-4653	20	12	for	for	ADP
ejpam-4653	20	13	s	s	PROPN
ejpam-4653	20	14	⊆	⊆	NUM
ejpam-4653	20	15	v	v	NOUN
ejpam-4653	20	16	(	(	PUNCT
ejpam-4653	20	17	g	g	NOUN
ejpam-4653	20	18	)	)	PUNCT
ejpam-4653	20	19	,	,	PUNCT
ejpam-4653	20	20	the	the	DET
ejpam-4653	20	21	open	open	ADJ
ejpam-4653	20	22	neighborhood	neighborhood	NOUN
ejpam-4653	20	23	and	and	CCONJ
ejpam-4653	20	24	closed	close	VERB
ejpam-4653	20	25	neighborhood	neighborhood	NOUN
ejpam-4653	20	26	of	of	ADP
ejpam-4653	20	27	s	s	NOUN
ejpam-4653	20	28	are	be	AUX
ejpam-4653	20	29	the	the	DET
ejpam-4653	20	30	sets	set	NOUN
ejpam-4653	20	31	ng(s	ng(s	PRON
ejpam-4653	20	32	)	)	PUNCT
ejpam-4653	20	33	=	=	SYM
ejpam-4653	20	34	∪v∈sng(v	∪v∈sng(v	PROPN
ejpam-4653	20	35	)	)	PUNCT
ejpam-4653	20	36	and	and	CCONJ
ejpam-4653	20	37	ng[s	ng[s	PROPN
ejpam-4653	20	38	]	]	PUNCT
ejpam-4653	20	39	=	=	SYM
ejpam-4653	20	40	∪v∈sng[u	∪v∈sng[u	X
ejpam-4653	20	41	]	]	PUNCT
ejpam-4653	20	42	,	,	PUNCT
ejpam-4653	20	43	respectively	respectively	ADV
ejpam-4653	20	44	.	.	PUNCT
ejpam-4653	21	1	∗corresponding	∗corresponde	VERB
ejpam-4653	21	2	author	author	NOUN
ejpam-4653	21	3	.	.	PUNCT
ejpam-4653	22	1	doi	doi	NOUN
ejpam-4653	22	2	:	:	PUNCT
ejpam-4653	22	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4653	https://doi.org/10.29020/nybg.ejpam.v16i2.4653	PROPN
ejpam-4653	22	4	email	email	NOUN
ejpam-4653	22	5	addresses	address	NOUN
ejpam-4653	22	6	:	:	PUNCT
ejpam-4653	22	7	jerryboy.cariaga@g.msuiit.edu.ph	jerryboy.cariaga@g.msuiit.edu.ph	PROPN
ejpam-4653	22	8	(	(	PUNCT
ejpam-4653	22	9	j.	j.	PROPN
ejpam-4653	22	10	b.	b.	PROPN
ejpam-4653	22	11	g.	g.	PROPN
ejpam-4653	22	12	cariaga	cariaga	PROPN
ejpam-4653	22	13	)	)	PUNCT
ejpam-4653	22	14	,	,	PUNCT
ejpam-4653	22	15	ferdinand.jamil@g.msuiit.edu.ph	ferdinand.jamil@g.msuiit.edu.ph	PROPN
ejpam-4653	22	16	(	(	PUNCT
ejpam-4653	22	17	f.	f.	PROPN
ejpam-4653	22	18	jamil	jamil	PROPN
ejpam-4653	22	19	)	)	PUNCT
ejpam-4653	22	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4653	22	21	847	847	NUM
ejpam-4653	23	1	©	©	PROPN
ejpam-4653	23	2	2023	2023	NUM
ejpam-4653	23	3	ejpam	ejpam	NOUN
ejpam-4653	23	4	all	all	DET
ejpam-4653	23	5	rights	right	NOUN
ejpam-4653	23	6	reserved	reserve	VERB
ejpam-4653	23	7	.	.	PUNCT
ejpam-4653	24	1	j.	j.	PROPN
ejpam-4653	24	2	b.g	b.g	PROPN
ejpam-4653	24	3	.	.	PROPN
ejpam-4653	24	4	cariaga	cariaga	PROPN
ejpam-4653	24	5	,	,	PUNCT
ejpam-4653	24	6	f.	f.	PROPN
ejpam-4653	24	7	jamil	jamil	PROPN
ejpam-4653	24	8	/	/	SYM
ejpam-4653	24	9	eur	eur	PROPN
ejpam-4653	24	10	.	.	PUNCT
ejpam-4653	25	1	j.	j.	PROPN
ejpam-4653	25	2	pure	pure	PROPN
ejpam-4653	25	3	appl	appl	PROPN
ejpam-4653	25	4	.	.	PROPN
ejpam-4653	25	5	math	math	PROPN
ejpam-4653	25	6	,	,	PUNCT
ejpam-4653	25	7	16	16	NUM
ejpam-4653	25	8	(	(	PUNCT
ejpam-4653	25	9	2	2	NUM
ejpam-4653	25	10	)	)	PUNCT
ejpam-4653	25	11	(	(	PUNCT
ejpam-4653	25	12	2023	2023	NUM
ejpam-4653	25	13	)	)	PUNCT
ejpam-4653	25	14	,	,	PUNCT
ejpam-4653	25	15	847	847	NUM
ejpam-4653	25	16	-	-	SYM
ejpam-4653	25	17	863	863	NUM
ejpam-4653	25	18	848	848	NUM
ejpam-4653	25	19	a	a	DET
ejpam-4653	25	20	set	set	NOUN
ejpam-4653	25	21	s	s	NOUN
ejpam-4653	25	22	⊆	⊆	NUM
ejpam-4653	25	23	v	v	NOUN
ejpam-4653	25	24	(	(	PUNCT
ejpam-4653	25	25	g	g	NOUN
ejpam-4653	25	26	)	)	PUNCT
ejpam-4653	25	27	is	be	AUX
ejpam-4653	25	28	said	say	VERB
ejpam-4653	25	29	to	to	PART
ejpam-4653	25	30	be	be	AUX
ejpam-4653	25	31	a	a	DET
ejpam-4653	25	32	dominating	dominating	NOUN
ejpam-4653	25	33	set	set	NOUN
ejpam-4653	25	34	of	of	ADP
ejpam-4653	25	35	g	g	PROPN
ejpam-4653	25	36	if	if	SCONJ
ejpam-4653	25	37	ng[s	ng[	NOUN
ejpam-4653	25	38	]	]	PUNCT
ejpam-4653	25	39	=	=	SYM
ejpam-4653	25	40	v	v	NOUN
ejpam-4653	25	41	(	(	PUNCT
ejpam-4653	25	42	g	g	NOUN
ejpam-4653	25	43	)	)	PUNCT
ejpam-4653	25	44	.	.	PUNCT
ejpam-4653	26	1	the	the	DET
ejpam-4653	26	2	minimum	minimum	ADJ
ejpam-4653	26	3	cardinality	cardinality	NOUN
ejpam-4653	26	4	of	of	ADP
ejpam-4653	26	5	a	a	DET
ejpam-4653	26	6	dominating	dominating	NOUN
ejpam-4653	26	7	set	set	NOUN
ejpam-4653	26	8	is	be	AUX
ejpam-4653	26	9	called	call	VERB
ejpam-4653	26	10	the	the	DET
ejpam-4653	26	11	domination	domination	NOUN
ejpam-4653	26	12	number	number	NOUN
ejpam-4653	26	13	of	of	ADP
ejpam-4653	26	14	g	g	NOUN
ejpam-4653	26	15	,	,	PUNCT
ejpam-4653	26	16	and	and	CCONJ
ejpam-4653	26	17	is	be	AUX
ejpam-4653	26	18	denoted	denote	VERB
ejpam-4653	26	19	by	by	ADP
ejpam-4653	26	20	γ(g	γ(g	PROPN
ejpam-4653	26	21	)	)	PUNCT
ejpam-4653	26	22	.	.	PUNCT
ejpam-4653	27	1	any	any	DET
ejpam-4653	27	2	dominating	dominating	NOUN
ejpam-4653	27	3	set	set	NOUN
ejpam-4653	27	4	of	of	ADP
ejpam-4653	27	5	cardinality	cardinality	PROPN
ejpam-4653	27	6	γ(g	γ(g	PROPN
ejpam-4653	27	7	)	)	PUNCT
ejpam-4653	27	8	is	be	AUX
ejpam-4653	27	9	referred	refer	VERB
ejpam-4653	27	10	to	to	ADP
ejpam-4653	27	11	as	as	ADP
ejpam-4653	27	12	a	a	DET
ejpam-4653	27	13	γ	γ	NOUN
ejpam-4653	27	14	-	-	PUNCT
ejpam-4653	27	15	set	set	NOUN
ejpam-4653	27	16	of	of	ADP
ejpam-4653	27	17	g.	g.	NOUN
ejpam-4653	27	18	we	we	PRON
ejpam-4653	27	19	refer	refer	VERB
ejpam-4653	27	20	to	to	ADP
ejpam-4653	27	21	[	[	X
ejpam-4653	27	22	1	1	NUM
ejpam-4653	27	23	,	,	PUNCT
ejpam-4653	27	24	3	3	NUM
ejpam-4653	27	25	,	,	PUNCT
ejpam-4653	27	26	5	5	NUM
ejpam-4653	27	27	,	,	PUNCT
ejpam-4653	27	28	7	7	NUM
ejpam-4653	27	29	,	,	PUNCT
ejpam-4653	27	30	8	8	NUM
ejpam-4653	27	31	,	,	PUNCT
ejpam-4653	27	32	12	12	NUM
ejpam-4653	27	33	,	,	PUNCT
ejpam-4653	27	34	14	14	NUM
ejpam-4653	27	35	]	]	PUNCT
ejpam-4653	27	36	for	for	ADP
ejpam-4653	27	37	the	the	DET
ejpam-4653	27	38	introduction	introduction	NOUN
ejpam-4653	27	39	,	,	PUNCT
ejpam-4653	27	40	fundamental	fundamental	ADJ
ejpam-4653	27	41	concepts	concept	NOUN
ejpam-4653	27	42	and	and	CCONJ
ejpam-4653	27	43	some	some	DET
ejpam-4653	27	44	studies	study	NOUN
ejpam-4653	27	45	on	on	ADP
ejpam-4653	27	46	domination	domination	NOUN
ejpam-4653	27	47	in	in	ADP
ejpam-4653	27	48	graphs	graph	NOUN
ejpam-4653	27	49	.	.	PUNCT
ejpam-4653	28	1	a	a	DET
ejpam-4653	28	2	set	set	NOUN
ejpam-4653	28	3	s	s	NOUN
ejpam-4653	28	4	⊆	⊆	NUM
ejpam-4653	28	5	v	v	NOUN
ejpam-4653	28	6	(	(	PUNCT
ejpam-4653	28	7	g	g	NOUN
ejpam-4653	28	8	)	)	PUNCT
ejpam-4653	28	9	is	be	AUX
ejpam-4653	28	10	called	call	VERB
ejpam-4653	28	11	a	a	DET
ejpam-4653	28	12	2	2	NUM
ejpam-4653	28	13	-	-	PUNCT
ejpam-4653	28	14	dominating	dominating	NOUN
ejpam-4653	28	15	set	set	NOUN
ejpam-4653	28	16	if	if	SCONJ
ejpam-4653	28	17	each	each	PRON
ejpam-4653	28	18	v	v	ADP
ejpam-4653	28	19	∈	∈	NOUN
ejpam-4653	28	20	v	v	NOUN
ejpam-4653	28	21	(	(	PUNCT
ejpam-4653	28	22	g)\s	g)\s	NOUN
ejpam-4653	28	23	,	,	PUNCT
ejpam-4653	28	24	|s∩ng(v)|	|s∩ng(v)|	ADP
ejpam-4653	28	25	≥	≥	NUM
ejpam-4653	28	26	2	2	NUM
ejpam-4653	28	27	.	.	PUNCT
ejpam-4653	29	1	the	the	DET
ejpam-4653	29	2	2	2	NUM
ejpam-4653	29	3	-	-	PUNCT
ejpam-4653	29	4	domination	domination	NOUN
ejpam-4653	29	5	number	number	NOUN
ejpam-4653	29	6	of	of	ADP
ejpam-4653	29	7	g	g	NOUN
ejpam-4653	29	8	,	,	PUNCT
ejpam-4653	29	9	denoted	denote	VERB
ejpam-4653	29	10	γ2(g	γ2(g	NOUN
ejpam-4653	29	11	)	)	PUNCT
ejpam-4653	29	12	,	,	PUNCT
ejpam-4653	29	13	is	be	AUX
ejpam-4653	29	14	the	the	DET
ejpam-4653	29	15	minimum	minimum	ADJ
ejpam-4653	29	16	cardinality	cardinality	NOUN
ejpam-4653	29	17	of	of	ADP
ejpam-4653	29	18	a	a	DET
ejpam-4653	29	19	2	2	NUM
ejpam-4653	29	20	-	-	PUNCT
ejpam-4653	29	21	dominating	dominating	NOUN
ejpam-4653	29	22	set	set	NOUN
ejpam-4653	29	23	of	of	ADP
ejpam-4653	29	24	g.	g.	PROPN
ejpam-4653	29	25	references	reference	NOUN
ejpam-4653	29	26	[	[	X
ejpam-4653	29	27	7	7	NUM
ejpam-4653	29	28	,	,	PUNCT
ejpam-4653	29	29	10	10	NUM
ejpam-4653	29	30	]	]	PUNCT
ejpam-4653	29	31	provide	provide	VERB
ejpam-4653	29	32	a	a	DET
ejpam-4653	29	33	good	good	ADJ
ejpam-4653	29	34	study	study	NOUN
ejpam-4653	29	35	on	on	ADP
ejpam-4653	29	36	2	2	NUM
ejpam-4653	29	37	-	-	PUNCT
ejpam-4653	29	38	domination	domination	NOUN
ejpam-4653	29	39	.	.	PUNCT
ejpam-4653	30	1	a	a	DET
ejpam-4653	30	2	roman	roman	ADJ
ejpam-4653	30	3	dominating	dominating	NOUN
ejpam-4653	30	4	function	function	NOUN
ejpam-4653	30	5	on	on	ADP
ejpam-4653	30	6	g	g	PROPN
ejpam-4653	30	7	is	be	AUX
ejpam-4653	30	8	a	a	DET
ejpam-4653	30	9	function	function	NOUN
ejpam-4653	30	10	f	f	NOUN
ejpam-4653	30	11	:	:	PUNCT
ejpam-4653	30	12	v	v	X
ejpam-4653	30	13	(	(	PUNCT
ejpam-4653	30	14	g	g	NOUN
ejpam-4653	30	15	)	)	PUNCT
ejpam-4653	30	16	→	→	SYM
ejpam-4653	30	17	{	{	PUNCT
ejpam-4653	30	18	0	0	NUM
ejpam-4653	30	19	,	,	PUNCT
ejpam-4653	30	20	1	1	NUM
ejpam-4653	30	21	,	,	PUNCT
ejpam-4653	30	22	2	2	NUM
ejpam-4653	30	23	}	}	PUNCT
ejpam-4653	30	24	satisfying	satisfy	VERB
ejpam-4653	30	25	the	the	DET
ejpam-4653	30	26	condition	condition	NOUN
ejpam-4653	30	27	that	that	SCONJ
ejpam-4653	30	28	for	for	ADP
ejpam-4653	30	29	each	each	DET
ejpam-4653	30	30	u	u	PROPN
ejpam-4653	30	31	∈	∈	PROPN
ejpam-4653	30	32	v	v	NOUN
ejpam-4653	30	33	(	(	PUNCT
ejpam-4653	30	34	g	g	NOUN
ejpam-4653	30	35	)	)	PUNCT
ejpam-4653	30	36	for	for	ADP
ejpam-4653	30	37	which	which	PRON
ejpam-4653	30	38	f(u	f(u	PROPN
ejpam-4653	30	39	)	)	PUNCT
ejpam-4653	30	40	=	=	SYM
ejpam-4653	30	41	0	0	NUM
ejpam-4653	30	42	,	,	PUNCT
ejpam-4653	30	43	there	there	PRON
ejpam-4653	30	44	exists	exist	VERB
ejpam-4653	30	45	v	v	ADP
ejpam-4653	30	46	∈	∈	PROPN
ejpam-4653	30	47	v	v	NOUN
ejpam-4653	30	48	(	(	PUNCT
ejpam-4653	30	49	g	g	NOUN
ejpam-4653	30	50	)	)	PUNCT
ejpam-4653	30	51	such	such	ADJ
ejpam-4653	30	52	that	that	SCONJ
ejpam-4653	30	53	f(v	f(v	NOUN
ejpam-4653	30	54	)	)	PUNCT
ejpam-4653	30	55	=	=	SYM
ejpam-4653	30	56	2	2	NUM
ejpam-4653	30	57	and	and	CCONJ
ejpam-4653	30	58	uv	uv	NOUN
ejpam-4653	30	59	∈	∈	PROPN
ejpam-4653	30	60	e(g	e(g	PROPN
ejpam-4653	30	61	)	)	PUNCT
ejpam-4653	30	62	.	.	PUNCT
ejpam-4653	31	1	the	the	DET
ejpam-4653	31	2	weight	weight	NOUN
ejpam-4653	31	3	of	of	ADP
ejpam-4653	31	4	f	f	PROPN
ejpam-4653	31	5	is	be	AUX
ejpam-4653	31	6	the	the	DET
ejpam-4653	31	7	value	value	NOUN
ejpam-4653	31	8	ωg(f	ωg(f	PRON
ejpam-4653	31	9	)	)	PUNCT
ejpam-4653	31	10	=	=	SYM
ejpam-4653	31	11	∑	∑	PUNCT
ejpam-4653	31	12	v∈v	v∈v	PROPN
ejpam-4653	31	13	(	(	PUNCT
ejpam-4653	31	14	g	g	NOUN
ejpam-4653	31	15	)	)	PUNCT
ejpam-4653	31	16	f(v	f(v	NOUN
ejpam-4653	31	17	)	)	PUNCT
ejpam-4653	31	18	.	.	PUNCT
ejpam-4653	32	1	the	the	DET
ejpam-4653	32	2	roman	roman	ADJ
ejpam-4653	32	3	domination	domination	NOUN
ejpam-4653	32	4	number	number	NOUN
ejpam-4653	32	5	of	of	ADP
ejpam-4653	32	6	g	g	NOUN
ejpam-4653	32	7	,	,	PUNCT
ejpam-4653	32	8	denoted	denote	VERB
ejpam-4653	32	9	by	by	ADP
ejpam-4653	32	10	γr(g	γr(g	PROPN
ejpam-4653	32	11	)	)	PUNCT
ejpam-4653	32	12	,	,	PUNCT
ejpam-4653	32	13	is	be	AUX
ejpam-4653	32	14	the	the	DET
ejpam-4653	32	15	minimum	minimum	ADJ
ejpam-4653	32	16	weight	weight	NOUN
ejpam-4653	32	17	of	of	ADP
ejpam-4653	32	18	a	a	DET
ejpam-4653	32	19	roman	roman	ADJ
ejpam-4653	32	20	dominating	dominating	NOUN
ejpam-4653	32	21	function	function	NOUN
ejpam-4653	32	22	of	of	ADP
ejpam-4653	32	23	g.	g.	PROPN
ejpam-4653	32	24	the	the	DET
ejpam-4653	32	25	history	history	NOUN
ejpam-4653	32	26	,	,	PUNCT
ejpam-4653	32	27	introduction	introduction	NOUN
ejpam-4653	32	28	and	and	CCONJ
ejpam-4653	32	29	some	some	PRON
ejpam-4653	32	30	of	of	ADP
ejpam-4653	32	31	the	the	DET
ejpam-4653	32	32	recent	recent	ADJ
ejpam-4653	32	33	studies	study	NOUN
ejpam-4653	32	34	in	in	ADP
ejpam-4653	32	35	roman	roman	ADJ
ejpam-4653	32	36	domination	domination	NOUN
ejpam-4653	32	37	have	have	AUX
ejpam-4653	32	38	been	be	AUX
ejpam-4653	32	39	provided	provide	VERB
ejpam-4653	32	40	in	in	ADP
ejpam-4653	32	41	[	[	X
ejpam-4653	32	42	6	6	NUM
ejpam-4653	32	43	,	,	PUNCT
ejpam-4653	32	44	13	13	NUM
ejpam-4653	32	45	,	,	PUNCT
ejpam-4653	32	46	15–17	15–17	NUM
ejpam-4653	32	47	]	]	PUNCT
ejpam-4653	32	48	.	.	PUNCT
ejpam-4653	33	1	a	a	DET
ejpam-4653	33	2	function	function	NOUN
ejpam-4653	33	3	f	f	NOUN
ejpam-4653	33	4	:	:	PUNCT
ejpam-4653	33	5	v	v	X
ejpam-4653	33	6	(	(	PUNCT
ejpam-4653	33	7	g	g	NOUN
ejpam-4653	33	8	)	)	PUNCT
ejpam-4653	33	9	→	→	SYM
ejpam-4653	33	10	{	{	PUNCT
ejpam-4653	33	11	0	0	NUM
ejpam-4653	33	12	,	,	PUNCT
ejpam-4653	33	13	1	1	NUM
ejpam-4653	33	14	,	,	PUNCT
ejpam-4653	33	15	2	2	NUM
ejpam-4653	33	16	,	,	PUNCT
ejpam-4653	33	17	3	3	NUM
ejpam-4653	33	18	}	}	PUNCT
ejpam-4653	33	19	is	be	AUX
ejpam-4653	33	20	a	a	DET
ejpam-4653	33	21	double	double	ADJ
ejpam-4653	33	22	roman	roman	ADJ
ejpam-4653	33	23	dominating	dominating	NOUN
ejpam-4653	33	24	function	function	NOUN
ejpam-4653	33	25	of	of	ADP
ejpam-4653	33	26	g	g	NOUN
ejpam-4653	33	27	,	,	PUNCT
ejpam-4653	33	28	written	write	VERB
ejpam-4653	33	29	f	f	PROPN
ejpam-4653	33	30	∈	∈	PROPN
ejpam-4653	33	31	drd(g	drd(g	PROPN
ejpam-4653	33	32	)	)	PUNCT
ejpam-4653	33	33	,	,	PUNCT
ejpam-4653	33	34	if	if	SCONJ
ejpam-4653	33	35	each	each	PRON
ejpam-4653	33	36	of	of	ADP
ejpam-4653	33	37	the	the	DET
ejpam-4653	33	38	following	follow	VERB
ejpam-4653	33	39	holds	hold	VERB
ejpam-4653	33	40	:	:	PUNCT
ejpam-4653	33	41	(	(	PUNCT
ejpam-4653	33	42	1	1	X
ejpam-4653	33	43	)	)	PUNCT
ejpam-4653	33	44	for	for	ADP
ejpam-4653	33	45	each	each	DET
ejpam-4653	33	46	v	v	NUM
ejpam-4653	33	47	∈	∈	PROPN
ejpam-4653	33	48	v	v	NOUN
ejpam-4653	33	49	(	(	PUNCT
ejpam-4653	33	50	g	g	NOUN
ejpam-4653	33	51	)	)	PUNCT
ejpam-4653	33	52	with	with	ADP
ejpam-4653	33	53	f(v	f(v	NOUN
ejpam-4653	33	54	)	)	PUNCT
ejpam-4653	34	1	=	=	SYM
ejpam-4653	34	2	0	0	PUNCT
ejpam-4653	34	3	at	at	ADV
ejpam-4653	34	4	least	least	ADJ
ejpam-4653	34	5	one	one	NUM
ejpam-4653	34	6	of	of	ADP
ejpam-4653	34	7	the	the	DET
ejpam-4653	34	8	following	following	NOUN
ejpam-4653	34	9	holds	hold	VERB
ejpam-4653	34	10	:	:	PUNCT
ejpam-4653	34	11	(	(	PUNCT
ejpam-4653	34	12	a	a	X
ejpam-4653	34	13	)	)	PUNCT
ejpam-4653	34	14	v	v	NOUN
ejpam-4653	34	15	has	have	VERB
ejpam-4653	34	16	two	two	NUM
ejpam-4653	34	17	adjacent	adjacent	ADJ
ejpam-4653	34	18	vertices	vertex	NOUN
ejpam-4653	34	19	u	u	NOUN
ejpam-4653	34	20	and	and	CCONJ
ejpam-4653	34	21	w	w	NOUN
ejpam-4653	34	22	for	for	ADP
ejpam-4653	34	23	which	which	PRON
ejpam-4653	34	24	f(u	f(u	PROPN
ejpam-4653	34	25	)	)	PUNCT
ejpam-4653	34	26	=	=	PUNCT
ejpam-4653	35	1	f(w	f(w	PROPN
ejpam-4653	35	2	)	)	PUNCT
ejpam-4653	35	3	=	=	SYM
ejpam-4653	35	4	2	2	NUM
ejpam-4653	35	5	;	;	PUNCT
ejpam-4653	35	6	or	or	CCONJ
ejpam-4653	35	7	(	(	PUNCT
ejpam-4653	35	8	b	b	X
ejpam-4653	35	9	)	)	PUNCT
ejpam-4653	35	10	v	v	NOUN
ejpam-4653	35	11	has	have	VERB
ejpam-4653	35	12	an	an	DET
ejpam-4653	35	13	adjacent	adjacent	ADJ
ejpam-4653	35	14	vertex	vertex	NOUN
ejpam-4653	35	15	u	u	NOUN
ejpam-4653	35	16	for	for	ADP
ejpam-4653	35	17	which	which	PRON
ejpam-4653	35	18	f(u	f(u	PROPN
ejpam-4653	35	19	)	)	PUNCT
ejpam-4653	35	20	=	=	SYM
ejpam-4653	35	21	3	3	NUM
ejpam-4653	35	22	,	,	PUNCT
ejpam-4653	35	23	and	and	CCONJ
ejpam-4653	35	24	(	(	PUNCT
ejpam-4653	35	25	2	2	X
ejpam-4653	35	26	)	)	PUNCT
ejpam-4653	35	27	for	for	ADP
ejpam-4653	35	28	each	each	DET
ejpam-4653	35	29	v	v	NUM
ejpam-4653	35	30	∈	∈	PROPN
ejpam-4653	35	31	v	v	NOUN
ejpam-4653	35	32	(	(	PUNCT
ejpam-4653	35	33	g	g	NOUN
ejpam-4653	35	34	)	)	PUNCT
ejpam-4653	35	35	with	with	ADP
ejpam-4653	35	36	f(v	f(v	NOUN
ejpam-4653	35	37	)	)	PUNCT
ejpam-4653	35	38	=	=	SYM
ejpam-4653	35	39	1	1	NUM
ejpam-4653	35	40	,	,	PUNCT
ejpam-4653	35	41	v	v	NOUN
ejpam-4653	35	42	is	be	AUX
ejpam-4653	35	43	adjacent	adjacent	ADJ
ejpam-4653	35	44	to	to	ADP
ejpam-4653	35	45	a	a	DET
ejpam-4653	35	46	vertex	vertex	NOUN
ejpam-4653	35	47	u	u	NOUN
ejpam-4653	35	48	for	for	ADP
ejpam-4653	35	49	which	which	PRON
ejpam-4653	35	50	either	either	CCONJ
ejpam-4653	35	51	f(u	f(u	PROPN
ejpam-4653	35	52	)	)	PUNCT
ejpam-4653	35	53	=	=	SYM
ejpam-4653	35	54	2	2	NUM
ejpam-4653	35	55	or	or	CCONJ
ejpam-4653	35	56	f(u	f(u	PROPN
ejpam-4653	35	57	)	)	PUNCT
ejpam-4653	35	58	=	=	SYM
ejpam-4653	36	1	3	3	X
ejpam-4653	36	2	.	.	X
ejpam-4653	36	3	the	the	DET
ejpam-4653	36	4	double	double	ADJ
ejpam-4653	36	5	roman	roman	ADJ
ejpam-4653	36	6	domination	domination	NOUN
ejpam-4653	36	7	number	number	NOUN
ejpam-4653	36	8	of	of	ADP
ejpam-4653	36	9	g	g	NOUN
ejpam-4653	36	10	,	,	PUNCT
ejpam-4653	36	11	denoted	denote	VERB
ejpam-4653	36	12	by	by	ADP
ejpam-4653	36	13	γdr(g	γdr(g	PROPN
ejpam-4653	36	14	)	)	PUNCT
ejpam-4653	36	15	,	,	PUNCT
ejpam-4653	36	16	is	be	AUX
ejpam-4653	36	17	the	the	DET
ejpam-4653	36	18	minimum	minimum	ADJ
ejpam-4653	36	19	weight	weight	NOUN
ejpam-4653	36	20	ωg(f	ωg(f	PRON
ejpam-4653	36	21	)	)	PUNCT
ejpam-4653	36	22	=	=	SYM
ejpam-4653	36	23	∑	∑	PUNCT
ejpam-4653	36	24	v∈v	v∈v	PROPN
ejpam-4653	36	25	(	(	PUNCT
ejpam-4653	36	26	g	g	NOUN
ejpam-4653	36	27	)	)	PUNCT
ejpam-4653	36	28	f(v	f(v	NOUN
ejpam-4653	36	29	)	)	PUNCT
ejpam-4653	36	30	of	of	ADP
ejpam-4653	36	31	a	a	DET
ejpam-4653	36	32	double	double	ADJ
ejpam-4653	36	33	roman	roman	ADJ
ejpam-4653	36	34	dominating	dominating	NOUN
ejpam-4653	36	35	function	function	NOUN
ejpam-4653	36	36	f	f	PROPN
ejpam-4653	36	37	of	of	ADP
ejpam-4653	36	38	g.	g.	PROPN
ejpam-4653	36	39	any	any	DET
ejpam-4653	36	40	f	f	PROPN
ejpam-4653	36	41	∈	∈	PROPN
ejpam-4653	36	42	drd(g	drd(g	PROPN
ejpam-4653	36	43	)	)	PUNCT
ejpam-4653	36	44	of	of	ADP
ejpam-4653	36	45	weight	weight	NOUN
ejpam-4653	36	46	equal	equal	ADJ
ejpam-4653	36	47	to	to	ADP
ejpam-4653	36	48	γdr(g	γdr(g	PROPN
ejpam-4653	36	49	)	)	PUNCT
ejpam-4653	36	50	is	be	AUX
ejpam-4653	36	51	referred	refer	VERB
ejpam-4653	36	52	to	to	ADP
ejpam-4653	36	53	as	as	ADP
ejpam-4653	36	54	γdr	γdr	NOUN
ejpam-4653	36	55	-	-	PUNCT
ejpam-4653	36	56	function	function	NOUN
ejpam-4653	36	57	of	of	ADP
ejpam-4653	36	58	g.	g.	PROPN
ejpam-4653	36	59	the	the	DET
ejpam-4653	36	60	concept	concept	NOUN
ejpam-4653	36	61	of	of	ADP
ejpam-4653	36	62	double	double	ADJ
ejpam-4653	36	63	domination	domination	NOUN
ejpam-4653	36	64	in	in	ADP
ejpam-4653	36	65	graphs	graph	NOUN
ejpam-4653	36	66	was	be	AUX
ejpam-4653	36	67	proposed	propose	VERB
ejpam-4653	36	68	by	by	ADP
ejpam-4653	36	69	beeler	beeler	NOUN
ejpam-4653	36	70	,	,	PUNCT
ejpam-4653	36	71	haynes	hayne	NOUN
ejpam-4653	36	72	and	and	CCONJ
ejpam-4653	36	73	hedetniemi	hedetniemi	X
ejpam-4653	36	74	[	[	X
ejpam-4653	36	75	2	2	X
ejpam-4653	36	76	]	]	PUNCT
ejpam-4653	36	77	in	in	ADP
ejpam-4653	36	78	2016	2016	NUM
ejpam-4653	36	79	.	.	PUNCT
ejpam-4653	37	1	it	it	PRON
ejpam-4653	37	2	is	be	AUX
ejpam-4653	37	3	a	a	DET
ejpam-4653	37	4	stronger	strong	ADJ
ejpam-4653	37	5	version	version	NOUN
ejpam-4653	37	6	of	of	ADP
ejpam-4653	37	7	roman	roman	ADJ
ejpam-4653	37	8	domination	domination	NOUN
ejpam-4653	37	9	.	.	PUNCT
ejpam-4653	38	1	if	if	SCONJ
ejpam-4653	38	2	in	in	ADP
ejpam-4653	38	3	roman	roman	ADJ
ejpam-4653	38	4	domination	domination	NOUN
ejpam-4653	38	5	only	only	ADV
ejpam-4653	38	6	one	one	NUM
ejpam-4653	38	7	legion	legion	NOUN
ejpam-4653	38	8	is	be	AUX
ejpam-4653	38	9	required	require	VERB
ejpam-4653	38	10	to	to	PART
ejpam-4653	38	11	defend	defend	VERB
ejpam-4653	38	12	an	an	DET
ejpam-4653	38	13	attacked	attacked	ADJ
ejpam-4653	38	14	city	city	NOUN
ejpam-4653	38	15	,	,	PUNCT
ejpam-4653	38	16	in	in	ADP
ejpam-4653	38	17	double	double	ADJ
ejpam-4653	38	18	roman	roman	ADJ
ejpam-4653	38	19	domination	domination	NOUN
ejpam-4653	38	20	any	any	DET
ejpam-4653	38	21	attack	attack	NOUN
ejpam-4653	38	22	can	can	AUX
ejpam-4653	38	23	be	be	AUX
ejpam-4653	38	24	defended	defend	VERB
ejpam-4653	38	25	by	by	ADP
ejpam-4653	38	26	at	at	ADV
ejpam-4653	38	27	least	least	ADV
ejpam-4653	38	28	two	two	NUM
ejpam-4653	38	29	legions	legion	NOUN
ejpam-4653	38	30	.	.	PUNCT
ejpam-4653	39	1	double	double	ADJ
ejpam-4653	39	2	roman	roman	ADJ
ejpam-4653	39	3	domination	domination	NOUN
ejpam-4653	39	4	is	be	AUX
ejpam-4653	39	5	further	far	ADV
ejpam-4653	39	6	studied	study	VERB
ejpam-4653	39	7	in	in	ADP
ejpam-4653	39	8	[	[	X
ejpam-4653	39	9	9	9	NUM
ejpam-4653	39	10	,	,	PUNCT
ejpam-4653	39	11	18	18	NUM
ejpam-4653	39	12	,	,	PUNCT
ejpam-4653	39	13	19	19	NUM
ejpam-4653	39	14	]	]	PUNCT
ejpam-4653	39	15	.	.	PUNCT
ejpam-4653	40	1	in	in	ADP
ejpam-4653	40	2	this	this	DET
ejpam-4653	40	3	paper	paper	NOUN
ejpam-4653	40	4	,	,	PUNCT
ejpam-4653	40	5	the	the	DET
ejpam-4653	40	6	double	double	ADJ
ejpam-4653	40	7	roman	roman	ADJ
ejpam-4653	40	8	domination	domination	NOUN
ejpam-4653	40	9	in	in	ADP
ejpam-4653	40	10	graphs	graph	NOUN
ejpam-4653	40	11	is	be	AUX
ejpam-4653	40	12	revisited	revisit	VERB
ejpam-4653	40	13	.	.	PUNCT
ejpam-4653	41	1	the	the	DET
ejpam-4653	41	2	main	main	ADJ
ejpam-4653	41	3	interest	interest	NOUN
ejpam-4653	41	4	is	be	AUX
ejpam-4653	41	5	particularly	particularly	ADV
ejpam-4653	41	6	on	on	ADP
ejpam-4653	41	7	the	the	DET
ejpam-4653	41	8	double	double	ADJ
ejpam-4653	41	9	roman	roman	ADJ
ejpam-4653	41	10	dominating	dominating	NOUN
ejpam-4653	41	11	function	function	NOUN
ejpam-4653	41	12	of	of	ADP
ejpam-4653	41	13	the	the	DET
ejpam-4653	41	14	join	join	NOUN
ejpam-4653	41	15	,	,	PUNCT
ejpam-4653	41	16	corona	corona	PROPN
ejpam-4653	41	17	,	,	PUNCT
ejpam-4653	41	18	complementary	complementary	ADJ
ejpam-4653	41	19	prism	prism	NOUN
ejpam-4653	41	20	and	and	CCONJ
ejpam-4653	41	21	lexicographic	lexicographic	ADJ
ejpam-4653	41	22	product	product	NOUN
ejpam-4653	41	23	of	of	ADP
ejpam-4653	41	24	graphs	graph	NOUN
ejpam-4653	41	25	.	.	PUNCT
ejpam-4653	42	1	the	the	DET
ejpam-4653	42	2	following	follow	VERB
ejpam-4653	42	3	results	result	NOUN
ejpam-4653	42	4	established	establish	VERB
ejpam-4653	42	5	in	in	ADP
ejpam-4653	42	6	the	the	DET
ejpam-4653	42	7	referred	refer	VERB
ejpam-4653	42	8	articles	article	NOUN
ejpam-4653	42	9	are	be	AUX
ejpam-4653	42	10	useful	useful	ADJ
ejpam-4653	42	11	in	in	ADP
ejpam-4653	42	12	this	this	DET
ejpam-4653	42	13	paper	paper	NOUN
ejpam-4653	42	14	.	.	PUNCT
ejpam-4653	43	1	proposition	proposition	NOUN
ejpam-4653	43	2	1	1	NUM
ejpam-4653	43	3	.	.	PUNCT
ejpam-4653	44	1	[	[	X
ejpam-4653	44	2	2	2	X
ejpam-4653	44	3	]	]	PUNCT
ejpam-4653	44	4	in	in	ADP
ejpam-4653	44	5	a	a	DET
ejpam-4653	44	6	double	double	ADJ
ejpam-4653	44	7	roman	roman	ADJ
ejpam-4653	44	8	dominating	dominating	NOUN
ejpam-4653	44	9	function	function	NOUN
ejpam-4653	44	10	of	of	ADP
ejpam-4653	44	11	weight	weight	NOUN
ejpam-4653	44	12	γdr(g	γdr(g	PROPN
ejpam-4653	44	13	)	)	PUNCT
ejpam-4653	44	14	,	,	PUNCT
ejpam-4653	44	15	no	no	DET
ejpam-4653	44	16	vertex	vertex	NOUN
ejpam-4653	44	17	needs	need	VERB
ejpam-4653	44	18	to	to	PART
ejpam-4653	44	19	be	be	AUX
ejpam-4653	44	20	assigned	assign	VERB
ejpam-4653	44	21	the	the	DET
ejpam-4653	44	22	value	value	NOUN
ejpam-4653	44	23	1	1	NUM
ejpam-4653	44	24	.	.	PUNCT
ejpam-4653	44	25	proposition	proposition	NOUN
ejpam-4653	44	26	2	2	NUM
ejpam-4653	44	27	.	.	PUNCT
ejpam-4653	45	1	[	[	X
ejpam-4653	45	2	9	9	NUM
ejpam-4653	45	3	]	]	PUNCT
ejpam-4653	45	4	for	for	ADP
ejpam-4653	45	5	n	n	PRON
ejpam-4653	45	6	≥	≥	NUM
ejpam-4653	45	7	1	1	NUM
ejpam-4653	45	8	,	,	PUNCT
ejpam-4653	45	9	γdr(pn	γdr(pn	NOUN
ejpam-4653	45	10	)	)	PUNCT
ejpam-4653	45	11	=	=	SYM
ejpam-4653	45	12	{	{	PUNCT
ejpam-4653	45	13	n	n	CCONJ
ejpam-4653	45	14	,	,	PUNCT
ejpam-4653	45	15	if	if	SCONJ
ejpam-4653	45	16	n	n	PRON
ejpam-4653	45	17	≡	≡	PROPN
ejpam-4653	45	18	0	0	PUNCT
ejpam-4653	45	19	(	(	PUNCT
ejpam-4653	45	20	mod	mod	NOUN
ejpam-4653	45	21	3	3	NUM
ejpam-4653	45	22	)	)	PUNCT
ejpam-4653	45	23	,	,	PUNCT
ejpam-4653	45	24	n+	n+	PUNCT
ejpam-4653	45	25	1	1	NUM
ejpam-4653	45	26	,	,	PUNCT
ejpam-4653	45	27	if	if	SCONJ
ejpam-4653	45	28	n	n	PRON
ejpam-4653	45	29	≡	≡	PROPN
ejpam-4653	45	30	1	1	NUM
ejpam-4653	45	31	,	,	PUNCT
ejpam-4653	45	32	2	2	NUM
ejpam-4653	45	33	(	(	PUNCT
ejpam-4653	45	34	mod	mod	NOUN
ejpam-4653	45	35	3	3	NUM
ejpam-4653	45	36	)	)	PUNCT
ejpam-4653	45	37	.	.	PUNCT
ejpam-4653	46	1	j.	j.	PROPN
ejpam-4653	46	2	b.g	b.g	PROPN
ejpam-4653	46	3	.	.	PROPN
ejpam-4653	46	4	cariaga	cariaga	PROPN
ejpam-4653	46	5	,	,	PUNCT
ejpam-4653	46	6	f.	f.	PROPN
ejpam-4653	46	7	jamil	jamil	PROPN
ejpam-4653	46	8	/	/	SYM
ejpam-4653	46	9	eur	eur	PROPN
ejpam-4653	46	10	.	.	PUNCT
ejpam-4653	47	1	j.	j.	PROPN
ejpam-4653	47	2	pure	pure	PROPN
ejpam-4653	47	3	appl	appl	PROPN
ejpam-4653	47	4	.	.	PROPN
ejpam-4653	47	5	math	math	PROPN
ejpam-4653	47	6	,	,	PUNCT
ejpam-4653	47	7	16	16	NUM
ejpam-4653	47	8	(	(	PUNCT
ejpam-4653	47	9	2	2	NUM
ejpam-4653	47	10	)	)	PUNCT
ejpam-4653	47	11	(	(	PUNCT
ejpam-4653	47	12	2023	2023	NUM
ejpam-4653	47	13	)	)	PUNCT
ejpam-4653	47	14	,	,	PUNCT
ejpam-4653	47	15	847	847	NUM
ejpam-4653	47	16	-	-	SYM
ejpam-4653	47	17	863	863	NUM
ejpam-4653	47	18	849	849	NUM
ejpam-4653	47	19	proposition	proposition	NOUN
ejpam-4653	47	20	3	3	NUM
ejpam-4653	47	21	.	.	PUNCT
ejpam-4653	48	1	[	[	X
ejpam-4653	48	2	9	9	NUM
ejpam-4653	48	3	]	]	PUNCT
ejpam-4653	48	4	for	for	ADP
ejpam-4653	48	5	n	n	PRON
ejpam-4653	48	6	≥	≥	NUM
ejpam-4653	48	7	3	3	NUM
ejpam-4653	48	8	,	,	PUNCT
ejpam-4653	48	9	γdr(cn	γdr(cn	NOUN
ejpam-4653	48	10	)	)	PUNCT
ejpam-4653	48	11	=	=	PRON
ejpam-4653	48	12	{	{	PUNCT
ejpam-4653	48	13	n	n	CCONJ
ejpam-4653	48	14	,	,	PUNCT
ejpam-4653	48	15	if	if	SCONJ
ejpam-4653	48	16	n	n	PRON
ejpam-4653	48	17	≡	≡	PROPN
ejpam-4653	48	18	0	0	NUM
ejpam-4653	48	19	,	,	PUNCT
ejpam-4653	48	20	2	2	NUM
ejpam-4653	48	21	,	,	PUNCT
ejpam-4653	48	22	3	3	NUM
ejpam-4653	48	23	,	,	PUNCT
ejpam-4653	48	24	4	4	NUM
ejpam-4653	48	25	(	(	PUNCT
ejpam-4653	48	26	mod	mod	PROPN
ejpam-4653	48	27	6	6	NUM
ejpam-4653	48	28	)	)	PUNCT
ejpam-4653	48	29	,	,	PUNCT
ejpam-4653	48	30	n+	n+	PUNCT
ejpam-4653	48	31	1	1	NUM
ejpam-4653	48	32	,	,	PUNCT
ejpam-4653	48	33	if	if	SCONJ
ejpam-4653	48	34	n	n	PRON
ejpam-4653	48	35	≡	≡	PROPN
ejpam-4653	48	36	1	1	NUM
ejpam-4653	48	37	,	,	PUNCT
ejpam-4653	48	38	5	5	NUM
ejpam-4653	48	39	(	(	PUNCT
ejpam-4653	48	40	mod	mod	PROPN
ejpam-4653	48	41	6	6	NUM
ejpam-4653	48	42	)	)	PUNCT
ejpam-4653	48	43	.	.	PUNCT
ejpam-4653	49	1	proposition	proposition	NOUN
ejpam-4653	49	2	4	4	NUM
ejpam-4653	49	3	.	.	PUNCT
ejpam-4653	50	1	[	[	X
ejpam-4653	50	2	2	2	X
ejpam-4653	50	3	]	]	PUNCT
ejpam-4653	50	4	for	for	ADP
ejpam-4653	50	5	any	any	DET
ejpam-4653	50	6	graph	graph	NOUN
ejpam-4653	50	7	g	g	NOUN
ejpam-4653	50	8	,	,	PUNCT
ejpam-4653	50	9	2γ(g	2γ(g	NUM
ejpam-4653	50	10	)	)	PUNCT
ejpam-4653	50	11	≤	≤	PUNCT
ejpam-4653	50	12	γdr(g	γdr(g	X
ejpam-4653	50	13	)	)	PUNCT
ejpam-4653	50	14	≤	≤	NUM
ejpam-4653	50	15	3γ(g	3γ(g	NUM
ejpam-4653	50	16	)	)	PUNCT
ejpam-4653	50	17	.	.	PUNCT
ejpam-4653	51	1	2	2	X
ejpam-4653	51	2	.	.	X
ejpam-4653	51	3	results	result	NOUN
ejpam-4653	51	4	for	for	ADP
ejpam-4653	51	5	a	a	DET
ejpam-4653	51	6	function	function	NOUN
ejpam-4653	51	7	f	f	NOUN
ejpam-4653	51	8	:	:	PUNCT
ejpam-4653	51	9	v	v	X
ejpam-4653	51	10	(	(	PUNCT
ejpam-4653	51	11	g	g	NOUN
ejpam-4653	51	12	)	)	PUNCT
ejpam-4653	51	13	→	→	SYM
ejpam-4653	51	14	{	{	PUNCT
ejpam-4653	51	15	0	0	NUM
ejpam-4653	51	16	,	,	PUNCT
ejpam-4653	51	17	1	1	NUM
ejpam-4653	51	18	,	,	PUNCT
ejpam-4653	51	19	2	2	NUM
ejpam-4653	51	20	,	,	PUNCT
ejpam-4653	51	21	3	3	NUM
ejpam-4653	51	22	}	}	PUNCT
ejpam-4653	51	23	,	,	PUNCT
ejpam-4653	51	24	we	we	PRON
ejpam-4653	51	25	write	write	VERB
ejpam-4653	51	26	f	f	PROPN
ejpam-4653	51	27	=	=	SYM
ejpam-4653	51	28	(	(	PUNCT
ejpam-4653	51	29	v0	v0	PROPN
ejpam-4653	51	30	,	,	PUNCT
ejpam-4653	51	31	v1	v1	NOUN
ejpam-4653	51	32	,	,	PUNCT
ejpam-4653	51	33	v2	v2	PROPN
ejpam-4653	51	34	,	,	PUNCT
ejpam-4653	51	35	v3	v3	PROPN
ejpam-4653	51	36	)	)	PUNCT
ejpam-4653	51	37	,	,	PUNCT
ejpam-4653	51	38	where	where	SCONJ
ejpam-4653	51	39	vi	vi	VERB
ejpam-4653	51	40	=	=	PRON
ejpam-4653	51	41	{	{	PUNCT
ejpam-4653	51	42	v	v	NUM
ejpam-4653	51	43	∈	∈	NOUN
ejpam-4653	51	44	v	v	NOUN
ejpam-4653	51	45	(	(	PUNCT
ejpam-4653	51	46	g	g	NOUN
ejpam-4653	51	47	)	)	PUNCT
ejpam-4653	51	48	:	:	PUNCT
ejpam-4653	51	49	f(v	f(v	NOUN
ejpam-4653	51	50	)	)	PUNCT
ejpam-4653	52	1	=	=	PUNCT
ejpam-4653	52	2	i	i	PROPN
ejpam-4653	52	3	}	}	PUNCT
ejpam-4653	52	4	for	for	ADP
ejpam-4653	52	5	all	all	PRON
ejpam-4653	52	6	i	i	PRON
ejpam-4653	52	7	∈	∈	PROPN
ejpam-4653	52	8	{	{	PUNCT
ejpam-4653	52	9	0	0	NUM
ejpam-4653	52	10	,	,	PUNCT
ejpam-4653	52	11	1	1	NUM
ejpam-4653	52	12	,	,	PUNCT
ejpam-4653	52	13	2	2	NUM
ejpam-4653	52	14	,	,	PUNCT
ejpam-4653	52	15	3	3	NUM
ejpam-4653	52	16	}	}	PUNCT
ejpam-4653	52	17	.	.	PUNCT
ejpam-4653	53	1	hence	hence	ADV
ejpam-4653	53	2	,	,	PUNCT
ejpam-4653	53	3	f	f	PROPN
ejpam-4653	53	4	∈	∈	PROPN
ejpam-4653	53	5	drd(g	drd(g	PROPN
ejpam-4653	53	6	)	)	PUNCT
ejpam-4653	53	7	if	if	SCONJ
ejpam-4653	54	1	and	and	CCONJ
ejpam-4653	54	2	only	only	ADV
ejpam-4653	54	3	if	if	SCONJ
ejpam-4653	54	4	each	each	PRON
ejpam-4653	54	5	of	of	ADP
ejpam-4653	54	6	the	the	DET
ejpam-4653	54	7	following	follow	VERB
ejpam-4653	54	8	holds	hold	VERB
ejpam-4653	54	9	:	:	PUNCT
ejpam-4653	54	10	(	(	PUNCT
ejpam-4653	54	11	1	1	X
ejpam-4653	54	12	)	)	PUNCT
ejpam-4653	54	13	for	for	ADP
ejpam-4653	54	14	each	each	DET
ejpam-4653	54	15	v	v	ADP
ejpam-4653	54	16	∈	∈	PROPN
ejpam-4653	54	17	v0	v0	NOUN
ejpam-4653	54	18	,	,	PUNCT
ejpam-4653	54	19	|v2	|v2	PUNCT
ejpam-4653	55	1	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	55	2	≥	≥	NUM
ejpam-4653	55	3	2	2	NUM
ejpam-4653	55	4	or	or	CCONJ
ejpam-4653	55	5	|v3	|v3	NOUN
ejpam-4653	55	6	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	55	7	≥	≥	NUM
ejpam-4653	55	8	1	1	NUM
ejpam-4653	55	9	;	;	PUNCT
ejpam-4653	55	10	and	and	CCONJ
ejpam-4653	55	11	(	(	PUNCT
ejpam-4653	55	12	2	2	X
ejpam-4653	55	13	)	)	PUNCT
ejpam-4653	55	14	for	for	ADP
ejpam-4653	55	15	each	each	DET
ejpam-4653	55	16	v	v	X
ejpam-4653	55	17	∈	∈	PROPN
ejpam-4653	55	18	v1	v1	NOUN
ejpam-4653	55	19	,	,	PUNCT
ejpam-4653	55	20	either	either	CCONJ
ejpam-4653	55	21	|v2	|v2	ADP
ejpam-4653	55	22	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	55	23	≥	≥	NUM
ejpam-4653	55	24	1	1	NUM
ejpam-4653	55	25	or	or	CCONJ
ejpam-4653	55	26	|v3	|v3	NOUN
ejpam-4653	55	27	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	55	28	≥	≥	NUM
ejpam-4653	55	29	1	1	NUM
ejpam-4653	55	30	.	.	PUNCT
ejpam-4653	56	1	in	in	ADP
ejpam-4653	56	2	view	view	NOUN
ejpam-4653	56	3	of	of	ADP
ejpam-4653	56	4	proposition	proposition	NOUN
ejpam-4653	56	5	1	1	NUM
ejpam-4653	56	6	,	,	PUNCT
ejpam-4653	56	7	we	we	PRON
ejpam-4653	56	8	may	may	AUX
ejpam-4653	56	9	always	always	ADV
ejpam-4653	56	10	assume	assume	VERB
ejpam-4653	56	11	that	that	SCONJ
ejpam-4653	56	12	a	a	DET
ejpam-4653	56	13	γdr	γdr	NOUN
ejpam-4653	56	14	-	-	PUNCT
ejpam-4653	56	15	function	function	NOUN
ejpam-4653	56	16	of	of	ADP
ejpam-4653	56	17	g	g	PROPN
ejpam-4653	56	18	is	be	AUX
ejpam-4653	56	19	of	of	ADP
ejpam-4653	56	20	the	the	DET
ejpam-4653	56	21	form	form	NOUN
ejpam-4653	57	1	f	f	X
ejpam-4653	57	2	=	=	SYM
ejpam-4653	57	3	(	(	PUNCT
ejpam-4653	57	4	v0,∅	v0,∅	PROPN
ejpam-4653	57	5	,	,	PUNCT
ejpam-4653	57	6	v2	v2	PROPN
ejpam-4653	57	7	,	,	PUNCT
ejpam-4653	57	8	v3	v3	PROPN
ejpam-4653	57	9	)	)	PUNCT
ejpam-4653	57	10	.	.	PUNCT
ejpam-4653	58	1	thus	thus	ADV
ejpam-4653	58	2	,	,	PUNCT
ejpam-4653	58	3	γdr(g	γdr(g	X
ejpam-4653	58	4	)	)	PUNCT
ejpam-4653	58	5	≥	≥	NOUN
ejpam-4653	58	6	2	2	NUM
ejpam-4653	58	7	for	for	ADP
ejpam-4653	58	8	all	all	DET
ejpam-4653	58	9	graphs	graph	NOUN
ejpam-4653	58	10	g.	g.	PROPN
ejpam-4653	58	11	more	more	ADV
ejpam-4653	58	12	precisely	precisely	ADV
ejpam-4653	58	13	,	,	PUNCT
ejpam-4653	58	14	γdr(g	γdr(g	ADJ
ejpam-4653	58	15	)	)	PUNCT
ejpam-4653	58	16	=	=	SYM
ejpam-4653	58	17	2	2	NUM
ejpam-4653	59	1	if	if	SCONJ
ejpam-4653	59	2	and	and	CCONJ
ejpam-4653	59	3	only	only	ADV
ejpam-4653	59	4	if	if	SCONJ
ejpam-4653	59	5	g	g	PROPN
ejpam-4653	59	6	=	=	SYM
ejpam-4653	59	7	k1	k1	PROPN
ejpam-4653	59	8	.	.	PUNCT
ejpam-4653	60	1	proposition	proposition	NOUN
ejpam-4653	60	2	5	5	NUM
ejpam-4653	60	3	.	.	PUNCT
ejpam-4653	61	1	let	let	VERB
ejpam-4653	61	2	g	g	PRON
ejpam-4653	61	3	be	be	AUX
ejpam-4653	61	4	a	a	DET
ejpam-4653	61	5	nontrivial	nontrivial	ADJ
ejpam-4653	61	6	connected	connect	VERB
ejpam-4653	61	7	graph	graph	NOUN
ejpam-4653	61	8	.	.	PUNCT
ejpam-4653	62	1	then	then	ADV
ejpam-4653	62	2	(	(	PUNCT
ejpam-4653	62	3	i	i	NOUN
ejpam-4653	62	4	)	)	PUNCT
ejpam-4653	62	5	γdr(g	γdr(g	X
ejpam-4653	62	6	)	)	PUNCT
ejpam-4653	62	7	=	=	SYM
ejpam-4653	62	8	3	3	NUM
ejpam-4653	62	9	if	if	SCONJ
ejpam-4653	62	10	and	and	CCONJ
ejpam-4653	62	11	only	only	ADV
ejpam-4653	62	12	if	if	SCONJ
ejpam-4653	62	13	γ(g	γ(g	NOUN
ejpam-4653	62	14	)	)	PUNCT
ejpam-4653	62	15	=	=	PUNCT
ejpam-4653	62	16	1	1	NUM
ejpam-4653	62	17	;	;	PUNCT
ejpam-4653	62	18	and	and	CCONJ
ejpam-4653	62	19	(	(	PUNCT
ejpam-4653	62	20	ii	ii	NOUN
ejpam-4653	62	21	)	)	PUNCT
ejpam-4653	62	22	γdr(g	γdr(g	PROPN
ejpam-4653	62	23	)	)	PUNCT
ejpam-4653	62	24	=	=	SYM
ejpam-4653	62	25	4	4	NUM
ejpam-4653	62	26	if	if	SCONJ
ejpam-4653	62	27	and	and	CCONJ
ejpam-4653	62	28	only	only	ADV
ejpam-4653	62	29	if	if	SCONJ
ejpam-4653	62	30	γ(g	γ(g	NOUN
ejpam-4653	62	31	)	)	PUNCT
ejpam-4653	62	32	=	=	SYM
ejpam-4653	62	33	2	2	NUM
ejpam-4653	62	34	=	=	SYM
ejpam-4653	62	35	γ2(g	γ2(g	NUM
ejpam-4653	62	36	)	)	PUNCT
ejpam-4653	62	37	.	.	PUNCT
ejpam-4653	63	1	proof	proof	NOUN
ejpam-4653	63	2	.	.	PUNCT
ejpam-4653	64	1	if	if	SCONJ
ejpam-4653	64	2	γdr(g	γdr(g	PROPN
ejpam-4653	64	3	)	)	PUNCT
ejpam-4653	64	4	=	=	SYM
ejpam-4653	64	5	3	3	NUM
ejpam-4653	64	6	and	and	CCONJ
ejpam-4653	64	7	f	f	NOUN
ejpam-4653	64	8	=	=	SYM
ejpam-4653	64	9	(	(	PUNCT
ejpam-4653	64	10	v0,∅	v0,∅	PROPN
ejpam-4653	64	11	,	,	PUNCT
ejpam-4653	64	12	v2	v2	PROPN
ejpam-4653	64	13	,	,	PUNCT
ejpam-4653	64	14	v3	v3	PROPN
ejpam-4653	64	15	)	)	PUNCT
ejpam-4653	64	16	is	be	AUX
ejpam-4653	64	17	a	a	DET
ejpam-4653	64	18	γdr	γdr	NOUN
ejpam-4653	64	19	-	-	PUNCT
ejpam-4653	64	20	function	function	NOUN
ejpam-4653	64	21	of	of	ADP
ejpam-4653	64	22	g	g	NOUN
ejpam-4653	64	23	,	,	PUNCT
ejpam-4653	64	24	then	then	ADV
ejpam-4653	64	25	v2	v2	NOUN
ejpam-4653	64	26	=	=	SYM
ejpam-4653	64	27	∅	∅	NOUN
ejpam-4653	64	28	,	,	PUNCT
ejpam-4653	64	29	|v3|	|v3|	NOUN
ejpam-4653	64	30	=	=	SYM
ejpam-4653	64	31	1	1	NUM
ejpam-4653	64	32	and	and	CCONJ
ejpam-4653	64	33	v0	v0	PROPN
ejpam-4653	64	34	=	=	SYM
ejpam-4653	64	35	v	v	PROPN
ejpam-4653	64	36	(	(	PUNCT
ejpam-4653	64	37	g	g	NOUN
ejpam-4653	64	38	)	)	PUNCT
ejpam-4653	64	39	\	\	PROPN
ejpam-4653	65	1	v3	v3	PROPN
ejpam-4653	65	2	.	.	PUNCT
ejpam-4653	66	1	if	if	SCONJ
ejpam-4653	66	2	v3	v3	PROPN
ejpam-4653	66	3	=	=	SYM
ejpam-4653	66	4	{	{	PUNCT
ejpam-4653	66	5	v	v	NOUN
ejpam-4653	66	6	}	}	PUNCT
ejpam-4653	66	7	,	,	PUNCT
ejpam-4653	66	8	then	then	ADV
ejpam-4653	66	9	ng[v	ng[v	X
ejpam-4653	66	10	]	]	X
ejpam-4653	66	11	=	=	SYM
ejpam-4653	66	12	v0	v0	NOUN
ejpam-4653	66	13	∪	∪	X
ejpam-4653	66	14	{	{	PUNCT
ejpam-4653	66	15	v	v	NOUN
ejpam-4653	66	16	}	}	PUNCT
ejpam-4653	66	17	=	=	SYM
ejpam-4653	66	18	v	v	NOUN
ejpam-4653	66	19	(	(	PUNCT
ejpam-4653	66	20	g	g	NOUN
ejpam-4653	66	21	)	)	PUNCT
ejpam-4653	66	22	.	.	PUNCT
ejpam-4653	67	1	this	this	PRON
ejpam-4653	67	2	means	mean	VERB
ejpam-4653	67	3	that	that	SCONJ
ejpam-4653	67	4	γ(g	γ(g	PROPN
ejpam-4653	67	5	)	)	PUNCT
ejpam-4653	67	6	=	=	PUNCT
ejpam-4653	68	1	1	1	X
ejpam-4653	68	2	.	.	PUNCT
ejpam-4653	69	1	conversely	conversely	ADV
ejpam-4653	69	2	,	,	PUNCT
ejpam-4653	69	3	if	if	SCONJ
ejpam-4653	69	4	γ(g	γ(g	PROPN
ejpam-4653	69	5	)	)	PUNCT
ejpam-4653	69	6	=	=	SYM
ejpam-4653	69	7	1	1	NUM
ejpam-4653	69	8	and	and	CCONJ
ejpam-4653	69	9	{	{	PUNCT
ejpam-4653	69	10	v	v	NOUN
ejpam-4653	69	11	}	}	PUNCT
ejpam-4653	69	12	is	be	AUX
ejpam-4653	69	13	a	a	DET
ejpam-4653	69	14	dominating	dominating	NOUN
ejpam-4653	69	15	set	set	VERB
ejpam-4653	69	16	ofg	ofg	PROPN
ejpam-4653	69	17	,	,	PUNCT
ejpam-4653	69	18	then	then	ADV
ejpam-4653	69	19	f	f	PROPN
ejpam-4653	69	20	=	=	PUNCT
ejpam-4653	69	21	(	(	PUNCT
ejpam-4653	69	22	v	v	NOUN
ejpam-4653	69	23	(	(	PUNCT
ejpam-4653	69	24	g)\{v},∅,∅	g)\{v},∅,∅	NOUN
ejpam-4653	69	25	,	,	PUNCT
ejpam-4653	69	26	{	{	PUNCT
ejpam-4653	69	27	v	v	NOUN
ejpam-4653	69	28	}	}	PUNCT
ejpam-4653	69	29	)	)	PUNCT
ejpam-4653	69	30	∈	∈	PROPN
ejpam-4653	69	31	drd(g	drd(g	PROPN
ejpam-4653	69	32	)	)	PUNCT
ejpam-4653	69	33	so	so	SCONJ
ejpam-4653	69	34	that	that	SCONJ
ejpam-4653	69	35	γdr(g	γdr(g	X
ejpam-4653	69	36	)	)	PUNCT
ejpam-4653	69	37	≤	≤	NOUN
ejpam-4653	69	38	ωg(f	ωg(f	NOUN
ejpam-4653	69	39	)	)	PUNCT
ejpam-4653	69	40	=	=	SYM
ejpam-4653	70	1	3	3	X
ejpam-4653	70	2	.	.	PUNCT
ejpam-4653	70	3	since	since	SCONJ
ejpam-4653	70	4	g	g	PROPN
ejpam-4653	70	5	is	be	AUX
ejpam-4653	70	6	nontrivial	nontrivial	ADJ
ejpam-4653	70	7	,	,	PUNCT
ejpam-4653	70	8	γdr(g	γdr(g	PROPN
ejpam-4653	70	9	)	)	PUNCT
ejpam-4653	70	10	=	=	SYM
ejpam-4653	71	1	3	3	X
ejpam-4653	71	2	.	.	PUNCT
ejpam-4653	72	1	this	this	PRON
ejpam-4653	72	2	proves	prove	VERB
ejpam-4653	72	3	(	(	PUNCT
ejpam-4653	72	4	i	i	NOUN
ejpam-4653	72	5	)	)	PUNCT
ejpam-4653	72	6	.	.	PUNCT
ejpam-4653	73	1	assume	assume	VERB
ejpam-4653	73	2	that	that	SCONJ
ejpam-4653	73	3	γdr(g	γdr(g	X
ejpam-4653	73	4	)	)	PUNCT
ejpam-4653	73	5	=	=	SYM
ejpam-4653	73	6	4	4	NUM
ejpam-4653	73	7	,	,	PUNCT
ejpam-4653	73	8	and	and	CCONJ
ejpam-4653	73	9	let	let	VERB
ejpam-4653	73	10	f	f	PROPN
ejpam-4653	73	11	=	=	SYM
ejpam-4653	73	12	(	(	PUNCT
ejpam-4653	73	13	v0,∅	v0,∅	PROPN
ejpam-4653	73	14	,	,	PUNCT
ejpam-4653	73	15	v2	v2	PROPN
ejpam-4653	73	16	,	,	PUNCT
ejpam-4653	73	17	v3	v3	PROPN
ejpam-4653	73	18	)	)	PUNCT
ejpam-4653	73	19	be	be	VERB
ejpam-4653	73	20	a	a	DET
ejpam-4653	73	21	γdr	γdr	NOUN
ejpam-4653	73	22	-	-	PUNCT
ejpam-4653	73	23	function	function	NOUN
ejpam-4653	73	24	of	of	ADP
ejpam-4653	73	25	g.	g.	PROPN
ejpam-4653	73	26	then	then	ADV
ejpam-4653	73	27	|v2|	|v2|	ADV
ejpam-4653	73	28	=	=	SYM
ejpam-4653	73	29	2	2	X
ejpam-4653	73	30	(	(	PUNCT
ejpam-4653	73	31	say	say	VERB
ejpam-4653	73	32	v2	v2	NOUN
ejpam-4653	73	33	=	=	SYM
ejpam-4653	73	34	{	{	PUNCT
ejpam-4653	73	35	u	u	NOUN
ejpam-4653	73	36	,	,	PUNCT
ejpam-4653	73	37	v	v	NOUN
ejpam-4653	73	38	}	}	PUNCT
ejpam-4653	73	39	)	)	PUNCT
ejpam-4653	73	40	,	,	PUNCT
ejpam-4653	73	41	v3	v3	PROPN
ejpam-4653	73	42	=	=	PUNCT
ejpam-4653	73	43	∅	∅	NOUN
ejpam-4653	73	44	and	and	CCONJ
ejpam-4653	73	45	v0	v0	NOUN
ejpam-4653	73	46	=	=	SYM
ejpam-4653	73	47	v	v	PROPN
ejpam-4653	73	48	(	(	PUNCT
ejpam-4653	73	49	g	g	NOUN
ejpam-4653	73	50	)	)	PUNCT
ejpam-4653	73	51	\	\	NOUN
ejpam-4653	74	1	{	{	PUNCT
ejpam-4653	74	2	u	u	NOUN
ejpam-4653	74	3	,	,	PUNCT
ejpam-4653	74	4	v	v	NOUN
ejpam-4653	74	5	}	}	PUNCT
ejpam-4653	74	6	.	.	PUNCT
ejpam-4653	75	1	thus	thus	ADV
ejpam-4653	75	2	,	,	PUNCT
ejpam-4653	75	3	v2	v2	PROPN
ejpam-4653	75	4	is	be	AUX
ejpam-4653	75	5	a	a	DET
ejpam-4653	75	6	γ2	γ2	NOUN
ejpam-4653	75	7	-	-	PUNCT
ejpam-4653	75	8	set	set	NOUN
ejpam-4653	75	9	so	so	SCONJ
ejpam-4653	75	10	that	that	SCONJ
ejpam-4653	75	11	γ2(g	γ2(g	VERB
ejpam-4653	75	12	)	)	PUNCT
ejpam-4653	75	13	=	=	SYM
ejpam-4653	75	14	2	2	X
ejpam-4653	75	15	.	.	PUNCT
ejpam-4653	75	16	moreover	moreover	ADV
ejpam-4653	75	17	,	,	PUNCT
ejpam-4653	75	18	being	be	AUX
ejpam-4653	75	19	a	a	DET
ejpam-4653	75	20	2	2	NUM
ejpam-4653	75	21	-	-	PUNCT
ejpam-4653	75	22	dominating	dominating	NOUN
ejpam-4653	75	23	set	set	NOUN
ejpam-4653	75	24	,	,	PUNCT
ejpam-4653	75	25	v2	v2	PROPN
ejpam-4653	75	26	is	be	AUX
ejpam-4653	75	27	a	a	DET
ejpam-4653	75	28	dominating	dominating	NOUN
ejpam-4653	75	29	set	set	NOUN
ejpam-4653	75	30	of	of	ADP
ejpam-4653	75	31	g	g	NOUN
ejpam-4653	75	32	so	so	SCONJ
ejpam-4653	75	33	that	that	PRON
ejpam-4653	75	34	γ(g	γ(g	PROPN
ejpam-4653	75	35	)	)	PUNCT
ejpam-4653	75	36	≤	≤	NOUN
ejpam-4653	75	37	2	2	NUM
ejpam-4653	75	38	.	.	PUNCT
ejpam-4653	76	1	by	by	ADP
ejpam-4653	76	2	(	(	PUNCT
ejpam-4653	76	3	i	i	NOUN
ejpam-4653	76	4	)	)	PUNCT
ejpam-4653	76	5	,	,	PUNCT
ejpam-4653	76	6	γ(g	γ(g	PROPN
ejpam-4653	76	7	)	)	PUNCT
ejpam-4653	76	8	=	=	SYM
ejpam-4653	76	9	2	2	X
ejpam-4653	76	10	.	.	X
ejpam-4653	76	11	conversely	conversely	ADV
ejpam-4653	76	12	,	,	PUNCT
ejpam-4653	76	13	let	let	VERB
ejpam-4653	76	14	s	s	PRON
ejpam-4653	76	15	=	=	PUNCT
ejpam-4653	76	16	{	{	PUNCT
ejpam-4653	76	17	u	u	NOUN
ejpam-4653	76	18	,	,	PUNCT
ejpam-4653	76	19	v	v	NOUN
ejpam-4653	76	20	}	}	PUNCT
ejpam-4653	76	21	be	be	AUX
ejpam-4653	76	22	a	a	DET
ejpam-4653	76	23	γ2	γ2	NOUN
ejpam-4653	76	24	-	-	PUNCT
ejpam-4653	76	25	set	set	NOUN
ejpam-4653	76	26	of	of	ADP
ejpam-4653	76	27	g.	g.	PROPN
ejpam-4653	76	28	since	since	SCONJ
ejpam-4653	76	29	f	f	PROPN
ejpam-4653	76	30	=	=	PUNCT
ejpam-4653	76	31	(	(	PUNCT
ejpam-4653	76	32	v	v	NOUN
ejpam-4653	76	33	(	(	PUNCT
ejpam-4653	76	34	g	g	NOUN
ejpam-4653	76	35	)	)	PUNCT
ejpam-4653	76	36	\	\	NOUN
ejpam-4653	76	37	s,∅	s,∅	NOUN
ejpam-4653	76	38	,	,	PUNCT
ejpam-4653	76	39	s,∅	s,∅	NOUN
ejpam-4653	76	40	)	)	PUNCT
ejpam-4653	76	41	∈	∈	PROPN
ejpam-4653	76	42	drd(g	drd(g	PROPN
ejpam-4653	76	43	)	)	PUNCT
ejpam-4653	76	44	,	,	PUNCT
ejpam-4653	76	45	γdr(g	γdr(g	X
ejpam-4653	76	46	)	)	PUNCT
ejpam-4653	76	47	≤	≤	NOUN
ejpam-4653	76	48	ωg(f	ωg(f	NOUN
ejpam-4653	76	49	)	)	PUNCT
ejpam-4653	76	50	=	=	SYM
ejpam-4653	77	1	4	4	X
ejpam-4653	77	2	.	.	PUNCT
ejpam-4653	77	3	because	because	SCONJ
ejpam-4653	77	4	g	g	PROPN
ejpam-4653	77	5	is	be	AUX
ejpam-4653	77	6	nontrivial	nontrivial	ADJ
ejpam-4653	77	7	and	and	CCONJ
ejpam-4653	77	8	γ(g	γ(g	PROPN
ejpam-4653	77	9	)	)	PUNCT
ejpam-4653	77	10	̸=	̸=	PROPN
ejpam-4653	77	11	1	1	NUM
ejpam-4653	77	12	,	,	PUNCT
ejpam-4653	77	13	γdr(g	γdr(g	NUM
ejpam-4653	77	14	)	)	PUNCT
ejpam-4653	77	15	≥	≥	NOUN
ejpam-4653	77	16	4	4	NUM
ejpam-4653	77	17	by	by	ADP
ejpam-4653	77	18	(	(	PUNCT
ejpam-4653	77	19	i	i	NOUN
ejpam-4653	77	20	)	)	PUNCT
ejpam-4653	77	21	.	.	PUNCT
ejpam-4653	78	1	hence	hence	ADV
ejpam-4653	78	2	,	,	PUNCT
ejpam-4653	78	3	γdr(g	γdr(g	PROPN
ejpam-4653	78	4	)	)	PUNCT
ejpam-4653	78	5	=	=	SYM
ejpam-4653	79	1	4	4	X
ejpam-4653	79	2	.	.	PUNCT
ejpam-4653	80	1	this	this	PRON
ejpam-4653	80	2	proves	prove	VERB
ejpam-4653	80	3	(	(	PUNCT
ejpam-4653	80	4	ii	ii	NOUN
ejpam-4653	80	5	)	)	PUNCT
ejpam-4653	80	6	.	.	PUNCT
ejpam-4653	81	1	proposition	proposition	NOUN
ejpam-4653	81	2	6	6	NUM
ejpam-4653	81	3	.	.	PUNCT
ejpam-4653	82	1	for	for	ADP
ejpam-4653	82	2	a	a	DET
ejpam-4653	82	3	nontrivial	nontrivial	ADJ
ejpam-4653	82	4	connected	connect	VERB
ejpam-4653	82	5	graph	graph	NOUN
ejpam-4653	82	6	g	g	PROPN
ejpam-4653	82	7	,	,	PUNCT
ejpam-4653	82	8	γdr(g	γdr(g	X
ejpam-4653	82	9	)	)	PUNCT
ejpam-4653	82	10	=	=	SYM
ejpam-4653	82	11	5	5	NUM
ejpam-4653	82	12	if	if	SCONJ
ejpam-4653	82	13	and	and	CCONJ
ejpam-4653	82	14	only	only	ADV
ejpam-4653	82	15	if	if	SCONJ
ejpam-4653	82	16	γ2(g	γ2(g	NOUN
ejpam-4653	82	17	)	)	PUNCT
ejpam-4653	82	18	≥	≥	NOUN
ejpam-4653	82	19	3	3	NUM
ejpam-4653	82	20	and	and	CCONJ
ejpam-4653	82	21	there	there	PRON
ejpam-4653	82	22	exist	exist	VERB
ejpam-4653	82	23	u	u	NOUN
ejpam-4653	82	24	,	,	PUNCT
ejpam-4653	82	25	v	v	NOUN
ejpam-4653	82	26	∈	∈	PROPN
ejpam-4653	82	27	v	v	NOUN
ejpam-4653	82	28	(	(	PUNCT
ejpam-4653	82	29	g	g	NOUN
ejpam-4653	82	30	)	)	PUNCT
ejpam-4653	82	31	for	for	ADP
ejpam-4653	82	32	which	which	PRON
ejpam-4653	82	33	the	the	DET
ejpam-4653	82	34	following	follow	VERB
ejpam-4653	82	35	holds	hold	VERB
ejpam-4653	82	36	:	:	PUNCT
ejpam-4653	82	37	(	(	PUNCT
ejpam-4653	82	38	i	i	NOUN
ejpam-4653	82	39	)	)	PUNCT
ejpam-4653	82	40	uv	uv	NOUN
ejpam-4653	82	41	/∈	/∈	PUNCT
ejpam-4653	82	42	e(g	e(g	PROPN
ejpam-4653	82	43	)	)	PUNCT
ejpam-4653	82	44	;	;	PUNCT
ejpam-4653	82	45	j.	j.	PROPN
ejpam-4653	82	46	b.g	b.g	PROPN
ejpam-4653	82	47	.	.	PROPN
ejpam-4653	82	48	cariaga	cariaga	PROPN
ejpam-4653	82	49	,	,	PUNCT
ejpam-4653	82	50	f.	f.	PROPN
ejpam-4653	82	51	jamil	jamil	PROPN
ejpam-4653	82	52	/	/	SYM
ejpam-4653	82	53	eur	eur	PROPN
ejpam-4653	82	54	.	.	PUNCT
ejpam-4653	83	1	j.	j.	PROPN
ejpam-4653	83	2	pure	pure	PROPN
ejpam-4653	83	3	appl	appl	PROPN
ejpam-4653	83	4	.	.	PROPN
ejpam-4653	83	5	math	math	PROPN
ejpam-4653	83	6	,	,	PUNCT
ejpam-4653	83	7	16	16	NUM
ejpam-4653	83	8	(	(	PUNCT
ejpam-4653	83	9	2	2	NUM
ejpam-4653	83	10	)	)	PUNCT
ejpam-4653	83	11	(	(	PUNCT
ejpam-4653	83	12	2023	2023	NUM
ejpam-4653	83	13	)	)	PUNCT
ejpam-4653	83	14	,	,	PUNCT
ejpam-4653	83	15	847	847	NUM
ejpam-4653	83	16	-	-	SYM
ejpam-4653	83	17	863	863	NUM
ejpam-4653	83	18	850	850	NUM
ejpam-4653	83	19	(	(	PUNCT
ejpam-4653	83	20	ii	ii	NOUN
ejpam-4653	83	21	)	)	PUNCT
ejpam-4653	83	22	v	v	NOUN
ejpam-4653	83	23	(	(	PUNCT
ejpam-4653	83	24	g	g	NOUN
ejpam-4653	83	25	)	)	PUNCT
ejpam-4653	83	26	\ng[v	\ng[v	NOUN
ejpam-4653	83	27	]	]	PUNCT
ejpam-4653	84	1	=	=	PUNCT
ejpam-4653	84	2	{	{	PUNCT
ejpam-4653	84	3	u	u	NOUN
ejpam-4653	84	4	}	}	PUNCT
ejpam-4653	84	5	and	and	CCONJ
ejpam-4653	84	6	v	v	X
ejpam-4653	84	7	(	(	PUNCT
ejpam-4653	84	8	g	g	NOUN
ejpam-4653	84	9	)	)	PUNCT
ejpam-4653	84	10	\ng[u	\ng[u	NOUN
ejpam-4653	84	11	]	]	PUNCT
ejpam-4653	85	1	̸=	̸=	PROPN
ejpam-4653	85	2	{	{	PUNCT
ejpam-4653	85	3	v	v	NOUN
ejpam-4653	85	4	}	}	PUNCT
ejpam-4653	85	5	.	.	PUNCT
ejpam-4653	86	1	proof	proof	NOUN
ejpam-4653	86	2	.	.	PUNCT
ejpam-4653	87	1	assume	assume	VERB
ejpam-4653	87	2	that	that	SCONJ
ejpam-4653	87	3	γdr(g	γdr(g	X
ejpam-4653	87	4	)	)	PUNCT
ejpam-4653	87	5	=	=	SYM
ejpam-4653	88	1	5	5	X
ejpam-4653	88	2	.	.	X
ejpam-4653	89	1	in	in	ADP
ejpam-4653	89	2	view	view	NOUN
ejpam-4653	89	3	of	of	ADP
ejpam-4653	89	4	proposition	proposition	NOUN
ejpam-4653	89	5	5	5	NUM
ejpam-4653	89	6	,	,	PUNCT
ejpam-4653	89	7	γ2(g	γ2(g	ADJ
ejpam-4653	89	8	)	)	PUNCT
ejpam-4653	89	9	≥	≥	NOUN
ejpam-4653	89	10	3	3	X
ejpam-4653	89	11	.	.	PUNCT
ejpam-4653	90	1	let	let	VERB
ejpam-4653	90	2	f	f	PROPN
ejpam-4653	90	3	=	=	SYM
ejpam-4653	90	4	(	(	PUNCT
ejpam-4653	90	5	v0,∅	v0,∅	PROPN
ejpam-4653	90	6	,	,	PUNCT
ejpam-4653	90	7	v2	v2	PROPN
ejpam-4653	90	8	,	,	PUNCT
ejpam-4653	90	9	v3	v3	PROPN
ejpam-4653	90	10	)	)	PUNCT
ejpam-4653	90	11	be	be	VERB
ejpam-4653	90	12	a	a	DET
ejpam-4653	90	13	γdr	γdr	NOUN
ejpam-4653	90	14	-	-	PUNCT
ejpam-4653	90	15	function	function	NOUN
ejpam-4653	90	16	of	of	ADP
ejpam-4653	90	17	g.	g.	PROPN
ejpam-4653	90	18	then	then	ADV
ejpam-4653	90	19	|v2|	|v2|	ADV
ejpam-4653	90	20	=	=	SYM
ejpam-4653	90	21	|v3|	|v3|	NOUN
ejpam-4653	90	22	=	=	SYM
ejpam-4653	90	23	1	1	X
ejpam-4653	90	24	.	.	PUNCT
ejpam-4653	90	25	let	let	VERB
ejpam-4653	90	26	v2	v2	VERB
ejpam-4653	90	27	=	=	SYM
ejpam-4653	90	28	{	{	PUNCT
ejpam-4653	90	29	u	u	NOUN
ejpam-4653	90	30	}	}	PUNCT
ejpam-4653	90	31	and	and	CCONJ
ejpam-4653	90	32	v3	v3	PROPN
ejpam-4653	90	33	=	=	SYM
ejpam-4653	90	34	{	{	PUNCT
ejpam-4653	90	35	v	v	NOUN
ejpam-4653	90	36	}	}	PUNCT
ejpam-4653	90	37	.	.	PUNCT
ejpam-4653	91	1	since	since	SCONJ
ejpam-4653	91	2	f	f	PROPN
ejpam-4653	91	3	is	be	AUX
ejpam-4653	91	4	a	a	DET
ejpam-4653	91	5	γdr	γdr	NOUN
ejpam-4653	91	6	-	-	PUNCT
ejpam-4653	91	7	function	function	NOUN
ejpam-4653	91	8	of	of	ADP
ejpam-4653	91	9	g	g	NOUN
ejpam-4653	91	10	,	,	PUNCT
ejpam-4653	91	11	ng(v	ng(v	PUNCT
ejpam-4653	91	12	)	)	PUNCT
ejpam-4653	91	13	=	=	SYM
ejpam-4653	91	14	v0	v0	NOUN
ejpam-4653	91	15	.	.	PUNCT
ejpam-4653	92	1	because	because	SCONJ
ejpam-4653	92	2	γ(g	γ(g	PROPN
ejpam-4653	92	3	)	)	PUNCT
ejpam-4653	92	4	≥	≥	NOUN
ejpam-4653	92	5	2	2	NUM
ejpam-4653	92	6	,	,	PUNCT
ejpam-4653	92	7	uv	uv	NOUN
ejpam-4653	92	8	/∈	/∈	PUNCT
ejpam-4653	92	9	e(g	e(g	PROPN
ejpam-4653	92	10	)	)	PUNCT
ejpam-4653	92	11	.	.	PUNCT
ejpam-4653	93	1	hence	hence	ADV
ejpam-4653	93	2	,	,	PUNCT
ejpam-4653	93	3	v	v	X
ejpam-4653	93	4	(	(	PUNCT
ejpam-4653	93	5	g	g	NOUN
ejpam-4653	93	6	)	)	PUNCT
ejpam-4653	93	7	\ng[v	\ng[v	NOUN
ejpam-4653	93	8	]	]	PUNCT
ejpam-4653	94	1	=	=	PUNCT
ejpam-4653	94	2	{	{	PUNCT
ejpam-4653	94	3	u	u	NOUN
ejpam-4653	94	4	}	}	PUNCT
ejpam-4653	94	5	.	.	PUNCT
ejpam-4653	95	1	moreover	moreover	ADV
ejpam-4653	95	2	,	,	PUNCT
ejpam-4653	95	3	since	since	SCONJ
ejpam-4653	95	4	γ2(g	γ2(g	VERB
ejpam-4653	95	5	)	)	PUNCT
ejpam-4653	95	6	≥	≥	NOUN
ejpam-4653	95	7	3	3	NUM
ejpam-4653	95	8	,	,	PUNCT
ejpam-4653	95	9	uw	uw	PROPN
ejpam-4653	95	10	/∈	/∈	PUNCT
ejpam-4653	95	11	e(g	e(g	PROPN
ejpam-4653	95	12	)	)	PUNCT
ejpam-4653	95	13	for	for	ADP
ejpam-4653	95	14	some	some	DET
ejpam-4653	95	15	w	w	PROPN
ejpam-4653	95	16	∈	∈	PROPN
ejpam-4653	95	17	v	v	ADP
ejpam-4653	95	18	(	(	PUNCT
ejpam-4653	95	19	g	g	NOUN
ejpam-4653	95	20	)	)	PUNCT
ejpam-4653	95	21	\	\	NOUN
ejpam-4653	95	22	{	{	PUNCT
ejpam-4653	95	23	u	u	NOUN
ejpam-4653	95	24	,	,	PUNCT
ejpam-4653	95	25	v	v	NOUN
ejpam-4653	95	26	}	}	PUNCT
ejpam-4653	95	27	.	.	PUNCT
ejpam-4653	96	1	therefore	therefore	ADV
ejpam-4653	96	2	,	,	PUNCT
ejpam-4653	96	3	v	v	X
ejpam-4653	96	4	(	(	PUNCT
ejpam-4653	96	5	g	g	NOUN
ejpam-4653	96	6	)	)	PUNCT
ejpam-4653	96	7	\ng[u	\ng[u	NOUN
ejpam-4653	96	8	]	]	PUNCT
ejpam-4653	97	1	̸=	̸=	PROPN
ejpam-4653	97	2	{	{	PUNCT
ejpam-4653	97	3	v	v	NOUN
ejpam-4653	97	4	}	}	PUNCT
ejpam-4653	97	5	.	.	PUNCT
ejpam-4653	98	1	conversely	conversely	ADV
ejpam-4653	98	2	,	,	PUNCT
ejpam-4653	98	3	suppose	suppose	VERB
ejpam-4653	98	4	that	that	SCONJ
ejpam-4653	98	5	γ2(g	γ2(g	VERB
ejpam-4653	98	6	)	)	PUNCT
ejpam-4653	98	7	≥	≥	NOUN
ejpam-4653	98	8	3	3	NUM
ejpam-4653	98	9	,	,	PUNCT
ejpam-4653	98	10	and	and	CCONJ
ejpam-4653	98	11	let	let	VERB
ejpam-4653	98	12	u	u	NOUN
ejpam-4653	98	13	,	,	PUNCT
ejpam-4653	98	14	v	v	PROPN
ejpam-4653	98	15	∈	∈	PROPN
ejpam-4653	98	16	v	v	NOUN
ejpam-4653	98	17	(	(	PUNCT
ejpam-4653	98	18	g	g	NOUN
ejpam-4653	98	19	)	)	PUNCT
ejpam-4653	98	20	such	such	ADJ
ejpam-4653	98	21	that	that	DET
ejpam-4653	98	22	uv	uv	NOUN
ejpam-4653	98	23	/∈	/∈	PUNCT
ejpam-4653	98	24	e(g	e(g	PROPN
ejpam-4653	98	25	)	)	PUNCT
ejpam-4653	98	26	,	,	PUNCT
ejpam-4653	98	27	v	v	X
ejpam-4653	98	28	(	(	PUNCT
ejpam-4653	98	29	g	g	NOUN
ejpam-4653	98	30	)	)	PUNCT
ejpam-4653	98	31	\ng[v	\ng[v	NOUN
ejpam-4653	98	32	]	]	PUNCT
ejpam-4653	99	1	=	=	PUNCT
ejpam-4653	99	2	{	{	PUNCT
ejpam-4653	99	3	u	u	NOUN
ejpam-4653	99	4	}	}	PUNCT
ejpam-4653	99	5	and	and	CCONJ
ejpam-4653	99	6	v	v	X
ejpam-4653	99	7	(	(	PUNCT
ejpam-4653	99	8	g	g	NOUN
ejpam-4653	99	9	)	)	PUNCT
ejpam-4653	99	10	\ng[u	\ng[u	NOUN
ejpam-4653	99	11	]	]	PUNCT
ejpam-4653	100	1	̸=	̸=	PROPN
ejpam-4653	100	2	{	{	PUNCT
ejpam-4653	100	3	v	v	NOUN
ejpam-4653	100	4	}	}	PUNCT
ejpam-4653	100	5	.	.	PUNCT
ejpam-4653	101	1	in	in	ADP
ejpam-4653	101	2	view	view	NOUN
ejpam-4653	101	3	of	of	ADP
ejpam-4653	101	4	proposition	proposition	NOUN
ejpam-4653	101	5	5	5	NUM
ejpam-4653	101	6	,	,	PUNCT
ejpam-4653	101	7	{	{	PUNCT
ejpam-4653	101	8	u	u	NOUN
ejpam-4653	101	9	,	,	PUNCT
ejpam-4653	101	10	v	v	NOUN
ejpam-4653	101	11	}	}	PUNCT
ejpam-4653	101	12	is	be	AUX
ejpam-4653	101	13	a	a	DET
ejpam-4653	101	14	γ	γ	NOUN
ejpam-4653	101	15	-	-	PUNCT
ejpam-4653	101	16	set	set	NOUN
ejpam-4653	101	17	of	of	ADP
ejpam-4653	101	18	g.	g.	PROPN
ejpam-4653	101	19	since	since	SCONJ
ejpam-4653	101	20	f	f	PROPN
ejpam-4653	101	21	=	=	PUNCT
ejpam-4653	101	22	(	(	PUNCT
ejpam-4653	101	23	v	v	NOUN
ejpam-4653	101	24	(	(	PUNCT
ejpam-4653	101	25	g	g	NOUN
ejpam-4653	101	26	)	)	PUNCT
ejpam-4653	101	27	\	\	NOUN
ejpam-4653	102	1	{	{	PUNCT
ejpam-4653	102	2	u	u	NOUN
ejpam-4653	102	3	,	,	PUNCT
ejpam-4653	102	4	v},∅	v},∅	PROPN
ejpam-4653	102	5	,	,	PUNCT
ejpam-4653	102	6	{	{	PUNCT
ejpam-4653	102	7	u	u	NOUN
ejpam-4653	102	8	}	}	PUNCT
ejpam-4653	102	9	,	,	PUNCT
ejpam-4653	102	10	{	{	PUNCT
ejpam-4653	102	11	v	v	NOUN
ejpam-4653	102	12	}	}	PUNCT
ejpam-4653	102	13	)	)	PUNCT
ejpam-4653	102	14	∈	∈	PROPN
ejpam-4653	102	15	drd(g	drd(g	PROPN
ejpam-4653	102	16	)	)	PUNCT
ejpam-4653	102	17	,	,	PUNCT
ejpam-4653	102	18	γdr(g	γdr(g	X
ejpam-4653	102	19	)	)	PUNCT
ejpam-4653	102	20	≤	≤	NOUN
ejpam-4653	102	21	ωg(f	ωg(f	NOUN
ejpam-4653	102	22	)	)	PUNCT
ejpam-4653	102	23	=	=	SYM
ejpam-4653	102	24	5	5	X
ejpam-4653	102	25	.	.	PUNCT
ejpam-4653	102	26	since	since	SCONJ
ejpam-4653	102	27	γ(g	γ(g	PROPN
ejpam-4653	102	28	)	)	PUNCT
ejpam-4653	102	29	=	=	SYM
ejpam-4653	102	30	2	2	NUM
ejpam-4653	102	31	and	and	CCONJ
ejpam-4653	102	32	γ2(g	γ2(g	NOUN
ejpam-4653	102	33	)	)	PUNCT
ejpam-4653	102	34	̸=	̸=	PROPN
ejpam-4653	102	35	2	2	NUM
ejpam-4653	102	36	,	,	PUNCT
ejpam-4653	102	37	γdr(g	γdr(g	X
ejpam-4653	102	38	)	)	PUNCT
ejpam-4653	102	39	=	=	SYM
ejpam-4653	102	40	5	5	NUM
ejpam-4653	102	41	by	by	ADP
ejpam-4653	102	42	proposition	proposition	NOUN
ejpam-4653	102	43	5	5	NUM
ejpam-4653	102	44	.	.	PUNCT
ejpam-4653	102	45	corollary	corollary	ADJ
ejpam-4653	102	46	1	1	NUM
ejpam-4653	102	47	.	.	PUNCT
ejpam-4653	103	1	for	for	ADP
ejpam-4653	103	2	a	a	DET
ejpam-4653	103	3	nontrivial	nontrivial	ADJ
ejpam-4653	103	4	connected	connect	VERB
ejpam-4653	103	5	graph	graph	NOUN
ejpam-4653	103	6	g	g	PROPN
ejpam-4653	103	7	,	,	PUNCT
ejpam-4653	103	8	γdr(g	γdr(g	X
ejpam-4653	103	9	)	)	PUNCT
ejpam-4653	103	10	=	=	SYM
ejpam-4653	103	11	5	5	NUM
ejpam-4653	103	12	if	if	SCONJ
ejpam-4653	103	13	and	and	CCONJ
ejpam-4653	103	14	only	only	ADV
ejpam-4653	103	15	if	if	SCONJ
ejpam-4653	103	16	γ(g	γ(g	NOUN
ejpam-4653	103	17	)	)	PUNCT
ejpam-4653	103	18	=	=	SYM
ejpam-4653	103	19	2	2	NUM
ejpam-4653	103	20	,	,	PUNCT
ejpam-4653	103	21	γ2(g	γ2(g	ADJ
ejpam-4653	103	22	)	)	PUNCT
ejpam-4653	103	23	≥	≥	NOUN
ejpam-4653	103	24	3	3	NUM
ejpam-4653	103	25	and	and	CCONJ
ejpam-4653	103	26	γ(g−	γ(g−	NOUN
ejpam-4653	103	27	v	v	NOUN
ejpam-4653	103	28	)	)	PUNCT
ejpam-4653	103	29	=	=	SYM
ejpam-4653	103	30	1	1	NUM
ejpam-4653	103	31	for	for	ADP
ejpam-4653	103	32	some	some	DET
ejpam-4653	103	33	v	v	ADP
ejpam-4653	103	34	∈	∈	NOUN
ejpam-4653	103	35	v	v	NOUN
ejpam-4653	103	36	(	(	PUNCT
ejpam-4653	103	37	g	g	NOUN
ejpam-4653	103	38	)	)	PUNCT
ejpam-4653	103	39	,	,	PUNCT
ejpam-4653	103	40	where	where	SCONJ
ejpam-4653	103	41	g−	g−	PROPN
ejpam-4653	103	42	v	v	NOUN
ejpam-4653	103	43	is	be	AUX
ejpam-4653	103	44	the	the	DET
ejpam-4653	103	45	resulting	result	VERB
ejpam-4653	103	46	graph	graph	NOUN
ejpam-4653	103	47	after	after	ADP
ejpam-4653	103	48	removing	remove	VERB
ejpam-4653	103	49	the	the	DET
ejpam-4653	103	50	vertex	vertex	NOUN
ejpam-4653	103	51	v.	v.	ADP
ejpam-4653	103	52	proof	proof	NOUN
ejpam-4653	103	53	.	.	PUNCT
ejpam-4653	104	1	assume	assume	VERB
ejpam-4653	104	2	that	that	SCONJ
ejpam-4653	104	3	γdr(g	γdr(g	X
ejpam-4653	104	4	)	)	PUNCT
ejpam-4653	104	5	=	=	SYM
ejpam-4653	105	1	5	5	X
ejpam-4653	105	2	.	.	PUNCT
ejpam-4653	105	3	by	by	ADP
ejpam-4653	105	4	proposition	proposition	NOUN
ejpam-4653	105	5	6	6	NUM
ejpam-4653	105	6	,	,	PUNCT
ejpam-4653	105	7	γ2(g	γ2(g	CCONJ
ejpam-4653	105	8	)	)	PUNCT
ejpam-4653	105	9	≥	≥	NOUN
ejpam-4653	105	10	3	3	NUM
ejpam-4653	105	11	and	and	CCONJ
ejpam-4653	105	12	there	there	PRON
ejpam-4653	105	13	exist	exist	VERB
ejpam-4653	105	14	vertices	vertex	NOUN
ejpam-4653	105	15	u	u	NOUN
ejpam-4653	105	16	,	,	PUNCT
ejpam-4653	105	17	v	v	NOUN
ejpam-4653	105	18	∈	∈	PROPN
ejpam-4653	105	19	v	v	NOUN
ejpam-4653	105	20	(	(	PUNCT
ejpam-4653	105	21	g	g	NOUN
ejpam-4653	105	22	)	)	PUNCT
ejpam-4653	105	23	for	for	ADP
ejpam-4653	105	24	which	which	PRON
ejpam-4653	105	25	uv	uv	NOUN
ejpam-4653	105	26	/∈	/∈	PROPN
ejpam-4653	105	27	e(g	e(g	PROPN
ejpam-4653	105	28	)	)	PUNCT
ejpam-4653	105	29	,	,	PUNCT
ejpam-4653	105	30	v	v	X
ejpam-4653	105	31	(	(	PUNCT
ejpam-4653	105	32	g	g	NOUN
ejpam-4653	105	33	)	)	PUNCT
ejpam-4653	105	34	\	\	PUNCT
ejpam-4653	106	1	ng[u	ng[u	PROPN
ejpam-4653	106	2	]	]	X
ejpam-4653	106	3	=	=	SYM
ejpam-4653	106	4	{	{	PUNCT
ejpam-4653	106	5	v	v	NOUN
ejpam-4653	106	6	}	}	PUNCT
ejpam-4653	106	7	and	and	CCONJ
ejpam-4653	106	8	v	v	X
ejpam-4653	106	9	(	(	PUNCT
ejpam-4653	106	10	g	g	NOUN
ejpam-4653	106	11	)	)	PUNCT
ejpam-4653	106	12	\	\	PUNCT
ejpam-4653	107	1	ng[v	ng[v	ADV
ejpam-4653	107	2	]	]	X
ejpam-4653	107	3	̸=	̸=	PROPN
ejpam-4653	107	4	{	{	PUNCT
ejpam-4653	107	5	u	u	NOUN
ejpam-4653	107	6	}	}	PUNCT
ejpam-4653	107	7	.	.	PUNCT
ejpam-4653	108	1	this	this	PRON
ejpam-4653	108	2	means	mean	VERB
ejpam-4653	108	3	that	that	SCONJ
ejpam-4653	108	4	{	{	PUNCT
ejpam-4653	108	5	u	u	NOUN
ejpam-4653	108	6	,	,	PUNCT
ejpam-4653	108	7	v	v	NOUN
ejpam-4653	108	8	}	}	PUNCT
ejpam-4653	108	9	and	and	CCONJ
ejpam-4653	108	10	{	{	PUNCT
ejpam-4653	108	11	u	u	NOUN
ejpam-4653	108	12	}	}	PUNCT
ejpam-4653	108	13	are	be	AUX
ejpam-4653	108	14	γ	γ	NOUN
ejpam-4653	108	15	-	-	PUNCT
ejpam-4653	108	16	sets	set	NOUN
ejpam-4653	108	17	of	of	ADP
ejpam-4653	108	18	g	g	PROPN
ejpam-4653	108	19	and	and	CCONJ
ejpam-4653	108	20	g	g	PROPN
ejpam-4653	108	21	−	−	PROPN
ejpam-4653	108	22	v	v	NOUN
ejpam-4653	108	23	,	,	PUNCT
ejpam-4653	108	24	respectively	respectively	ADV
ejpam-4653	108	25	.	.	PUNCT
ejpam-4653	109	1	thus	thus	ADV
ejpam-4653	109	2	,	,	PUNCT
ejpam-4653	109	3	γ(g	γ(g	PROPN
ejpam-4653	109	4	)	)	PUNCT
ejpam-4653	109	5	=	=	SYM
ejpam-4653	109	6	2	2	NUM
ejpam-4653	109	7	and	and	CCONJ
ejpam-4653	109	8	γ(g−	γ(g−	NOUN
ejpam-4653	109	9	v	v	NOUN
ejpam-4653	109	10	)	)	PUNCT
ejpam-4653	109	11	=	=	SYM
ejpam-4653	110	1	1	1	X
ejpam-4653	110	2	.	.	PUNCT
ejpam-4653	110	3	conversely	conversely	ADV
ejpam-4653	110	4	,	,	PUNCT
ejpam-4653	110	5	suppose	suppose	VERB
ejpam-4653	110	6	that	that	SCONJ
ejpam-4653	110	7	γ(g	γ(g	PROPN
ejpam-4653	110	8	)	)	PUNCT
ejpam-4653	110	9	=	=	SYM
ejpam-4653	110	10	2	2	NUM
ejpam-4653	110	11	,	,	PUNCT
ejpam-4653	110	12	γ2(g	γ2(g	ADJ
ejpam-4653	110	13	)	)	PUNCT
ejpam-4653	110	14	≥	≥	NOUN
ejpam-4653	110	15	3	3	NUM
ejpam-4653	110	16	and	and	CCONJ
ejpam-4653	110	17	let	let	VERB
ejpam-4653	110	18	u	u	NOUN
ejpam-4653	110	19	,	,	PUNCT
ejpam-4653	110	20	v	v	PROPN
ejpam-4653	110	21	∈	∈	PROPN
ejpam-4653	110	22	v	v	NOUN
ejpam-4653	110	23	(	(	PUNCT
ejpam-4653	110	24	g	g	NOUN
ejpam-4653	110	25	)	)	PUNCT
ejpam-4653	110	26	such	such	ADJ
ejpam-4653	110	27	thatng−v[u	thatng−v[u	NOUN
ejpam-4653	110	28	]	]	PUNCT
ejpam-4653	110	29	=	=	SYM
ejpam-4653	110	30	v	v	X
ejpam-4653	110	31	(	(	PUNCT
ejpam-4653	110	32	g	g	PROPN
ejpam-4653	110	33	−	−	PROPN
ejpam-4653	110	34	v	v	NOUN
ejpam-4653	110	35	)	)	PUNCT
ejpam-4653	110	36	.	.	PUNCT
ejpam-4653	111	1	since	since	SCONJ
ejpam-4653	111	2	γ2(g	γ2(g	ADJ
ejpam-4653	111	3	)	)	PUNCT
ejpam-4653	111	4	̸=	̸=	PROPN
ejpam-4653	111	5	2	2	NUM
ejpam-4653	111	6	,	,	PUNCT
ejpam-4653	111	7	there	there	PRON
ejpam-4653	111	8	exists	exist	VERB
ejpam-4653	111	9	w	w	PROPN
ejpam-4653	111	10	∈	∈	PROPN
ejpam-4653	111	11	v	v	ADP
ejpam-4653	111	12	(	(	PUNCT
ejpam-4653	111	13	g	g	NOUN
ejpam-4653	111	14	)	)	PUNCT
ejpam-4653	111	15	\	\	NOUN
ejpam-4653	111	16	{	{	PUNCT
ejpam-4653	111	17	u	u	NOUN
ejpam-4653	111	18	,	,	PUNCT
ejpam-4653	111	19	v	v	NOUN
ejpam-4653	111	20	}	}	PUNCT
ejpam-4653	111	21	such	such	ADJ
ejpam-4653	111	22	that	that	PRON
ejpam-4653	111	23	vw	vw	PROPN
ejpam-4653	111	24	/∈	/∈	PUNCT
ejpam-4653	111	25	e(g	e(g	PROPN
ejpam-4653	111	26	)	)	PUNCT
ejpam-4653	111	27	.	.	PUNCT
ejpam-4653	112	1	thus	thus	ADV
ejpam-4653	112	2	,	,	PUNCT
ejpam-4653	112	3	w	w	PROPN
ejpam-4653	112	4	∈	∈	PROPN
ejpam-4653	112	5	v	v	ADP
ejpam-4653	112	6	(	(	PUNCT
ejpam-4653	112	7	g	g	NOUN
ejpam-4653	112	8	)	)	PUNCT
ejpam-4653	112	9	\ng[u	\ng[u	NOUN
ejpam-4653	112	10	]	]	PUNCT
ejpam-4653	113	1	so	so	SCONJ
ejpam-4653	113	2	that	that	SCONJ
ejpam-4653	113	3	v	v	NOUN
ejpam-4653	113	4	(	(	PUNCT
ejpam-4653	113	5	g	g	NOUN
ejpam-4653	113	6	)	)	PUNCT
ejpam-4653	113	7	\ng[v	\ng[v	NOUN
ejpam-4653	113	8	]	]	PUNCT
ejpam-4653	114	1	̸=	̸=	PROPN
ejpam-4653	114	2	{	{	PUNCT
ejpam-4653	114	3	u	u	NOUN
ejpam-4653	114	4	}	}	PUNCT
ejpam-4653	114	5	.	.	PUNCT
ejpam-4653	115	1	moreover	moreover	ADV
ejpam-4653	115	2	,	,	PUNCT
ejpam-4653	115	3	since	since	SCONJ
ejpam-4653	115	4	γ(g	γ(g	PROPN
ejpam-4653	115	5	)	)	PUNCT
ejpam-4653	115	6	=	=	SYM
ejpam-4653	115	7	2	2	NUM
ejpam-4653	115	8	,	,	PUNCT
ejpam-4653	115	9	uv	uv	NOUN
ejpam-4653	115	10	/∈	/∈	PUNCT
ejpam-4653	115	11	e(g	e(g	PROPN
ejpam-4653	115	12	)	)	PUNCT
ejpam-4653	116	1	so	so	SCONJ
ejpam-4653	116	2	that	that	SCONJ
ejpam-4653	116	3	v	v	NOUN
ejpam-4653	116	4	(	(	PUNCT
ejpam-4653	116	5	g	g	NOUN
ejpam-4653	116	6	)	)	PUNCT
ejpam-4653	116	7	\ng[u	\ng[u	NOUN
ejpam-4653	116	8	]	]	PUNCT
ejpam-4653	117	1	=	=	PUNCT
ejpam-4653	117	2	{	{	PUNCT
ejpam-4653	117	3	v	v	NOUN
ejpam-4653	117	4	}	}	PUNCT
ejpam-4653	117	5	.	.	PUNCT
ejpam-4653	118	1	by	by	ADP
ejpam-4653	118	2	proposition	proposition	NOUN
ejpam-4653	118	3	6	6	NUM
ejpam-4653	118	4	,	,	PUNCT
ejpam-4653	118	5	γdr(g	γdr(g	PROPN
ejpam-4653	118	6	)	)	PUNCT
ejpam-4653	118	7	=	=	SYM
ejpam-4653	118	8	5	5	X
ejpam-4653	118	9	.	.	X
ejpam-4653	118	10	proposition	proposition	NOUN
ejpam-4653	118	11	7	7	NUM
ejpam-4653	118	12	.	.	X
ejpam-4653	119	1	for	for	ADP
ejpam-4653	119	2	nontrivial	nontrivial	ADJ
ejpam-4653	119	3	connected	connect	VERB
ejpam-4653	119	4	graph	graph	NOUN
ejpam-4653	119	5	g	g	PROPN
ejpam-4653	119	6	,	,	PUNCT
ejpam-4653	119	7	γdr(g	γdr(g	X
ejpam-4653	119	8	)	)	PUNCT
ejpam-4653	119	9	=	=	PUNCT
ejpam-4653	119	10	6	6	NUM
ejpam-4653	119	11	if	if	SCONJ
ejpam-4653	119	12	and	and	CCONJ
ejpam-4653	119	13	only	only	ADV
ejpam-4653	119	14	if	if	SCONJ
ejpam-4653	119	15	one	one	NUM
ejpam-4653	119	16	of	of	ADP
ejpam-4653	119	17	the	the	DET
ejpam-4653	119	18	following	follow	VERB
ejpam-4653	119	19	holds	hold	VERB
ejpam-4653	119	20	:	:	PUNCT
ejpam-4653	119	21	(	(	PUNCT
ejpam-4653	119	22	i	i	NOUN
ejpam-4653	119	23	)	)	PUNCT
ejpam-4653	119	24	γ(g	γ(g	PROPN
ejpam-4653	119	25	)	)	PUNCT
ejpam-4653	120	1	=	=	SYM
ejpam-4653	120	2	2	2	NUM
ejpam-4653	120	3	,	,	PUNCT
ejpam-4653	120	4	γ2(g	γ2(g	ADJ
ejpam-4653	120	5	)	)	PUNCT
ejpam-4653	120	6	≥	≥	NOUN
ejpam-4653	120	7	3	3	NUM
ejpam-4653	120	8	and	and	CCONJ
ejpam-4653	120	9	γ(g−	γ(g−	NOUN
ejpam-4653	120	10	v	v	NOUN
ejpam-4653	120	11	)	)	PUNCT
ejpam-4653	120	12	≥	≥	NOUN
ejpam-4653	120	13	2	2	NUM
ejpam-4653	120	14	for	for	ADP
ejpam-4653	120	15	all	all	PRON
ejpam-4653	120	16	v	v	ADP
ejpam-4653	120	17	∈	∈	NOUN
ejpam-4653	120	18	v	v	NOUN
ejpam-4653	120	19	(	(	PUNCT
ejpam-4653	120	20	g	g	NOUN
ejpam-4653	120	21	)	)	PUNCT
ejpam-4653	120	22	.	.	PUNCT
ejpam-4653	121	1	(	(	PUNCT
ejpam-4653	121	2	ii	ii	X
ejpam-4653	121	3	)	)	PUNCT
ejpam-4653	121	4	γ(g	γ(g	PROPN
ejpam-4653	121	5	)	)	PUNCT
ejpam-4653	121	6	≥	≥	NOUN
ejpam-4653	121	7	2	2	NUM
ejpam-4653	121	8	and	and	CCONJ
ejpam-4653	121	9	γ2(g	γ2(g	NUM
ejpam-4653	121	10	)	)	PUNCT
ejpam-4653	121	11	=	=	SYM
ejpam-4653	121	12	3	3	NUM
ejpam-4653	121	13	and	and	CCONJ
ejpam-4653	121	14	γ(g−	γ(g−	NOUN
ejpam-4653	121	15	v	v	NOUN
ejpam-4653	121	16	)	)	PUNCT
ejpam-4653	121	17	≥	≥	NOUN
ejpam-4653	121	18	2	2	NUM
ejpam-4653	121	19	for	for	ADP
ejpam-4653	121	20	all	all	DET
ejpam-4653	121	21	v	v	ADP
ejpam-4653	121	22	∈	∈	NOUN
ejpam-4653	121	23	v	v	NOUN
ejpam-4653	121	24	(	(	PUNCT
ejpam-4653	121	25	g	g	NOUN
ejpam-4653	121	26	)	)	PUNCT
ejpam-4653	121	27	.	.	PUNCT
ejpam-4653	122	1	proof	proof	NOUN
ejpam-4653	122	2	.	.	PUNCT
ejpam-4653	123	1	let	let	VERB
ejpam-4653	123	2	γdr(g	γdr(g	X
ejpam-4653	123	3	)	)	PUNCT
ejpam-4653	123	4	=	=	SYM
ejpam-4653	124	1	6	6	X
ejpam-4653	124	2	.	.	PUNCT
ejpam-4653	124	3	then	then	ADV
ejpam-4653	124	4	γ(g	γ(g	PROPN
ejpam-4653	124	5	)	)	PUNCT
ejpam-4653	124	6	≥	≥	NOUN
ejpam-4653	124	7	2	2	NUM
ejpam-4653	124	8	by	by	ADP
ejpam-4653	124	9	proposition	proposition	NOUN
ejpam-4653	124	10	5	5	NUM
ejpam-4653	124	11	.	.	PUNCT
ejpam-4653	125	1	let	let	VERB
ejpam-4653	125	2	f	f	PROPN
ejpam-4653	125	3	=	=	SYM
ejpam-4653	125	4	(	(	PUNCT
ejpam-4653	125	5	v0,∅	v0,∅	PROPN
ejpam-4653	125	6	,	,	PUNCT
ejpam-4653	125	7	v2	v2	PROPN
ejpam-4653	125	8	,	,	PUNCT
ejpam-4653	125	9	v3	v3	PROPN
ejpam-4653	125	10	)	)	PUNCT
ejpam-4653	125	11	be	be	VERB
ejpam-4653	125	12	a	a	DET
ejpam-4653	125	13	γdr	γdr	NOUN
ejpam-4653	125	14	-	-	PUNCT
ejpam-4653	125	15	function	function	NOUN
ejpam-4653	125	16	of	of	ADP
ejpam-4653	125	17	g.	g.	PROPN
ejpam-4653	125	18	consider	consider	VERB
ejpam-4653	125	19	the	the	DET
ejpam-4653	125	20	following	follow	VERB
ejpam-4653	125	21	cases	case	NOUN
ejpam-4653	125	22	:	:	PUNCT
ejpam-4653	125	23	case	case	NOUN
ejpam-4653	125	24	1	1	NUM
ejpam-4653	125	25	.	.	PUNCT
ejpam-4653	125	26	suppose	suppose	VERB
ejpam-4653	125	27	that	that	SCONJ
ejpam-4653	125	28	v2	v2	NOUN
ejpam-4653	125	29	=	=	SYM
ejpam-4653	125	30	∅	∅	NOUN
ejpam-4653	125	31	and	and	CCONJ
ejpam-4653	125	32	|v3|	|v3|	NOUN
ejpam-4653	125	33	=	=	SYM
ejpam-4653	125	34	2	2	X
ejpam-4653	125	35	.	.	X
ejpam-4653	125	36	then	then	ADV
ejpam-4653	125	37	v3	v3	PROPN
ejpam-4653	125	38	is	be	AUX
ejpam-4653	125	39	a	a	DET
ejpam-4653	125	40	dominating	dominating	NOUN
ejpam-4653	125	41	set	set	NOUN
ejpam-4653	125	42	of	of	ADP
ejpam-4653	125	43	g	g	PROPN
ejpam-4653	125	44	and	and	CCONJ
ejpam-4653	125	45	so	so	ADV
ejpam-4653	125	46	γ(g	γ(g	PROPN
ejpam-4653	125	47	)	)	PUNCT
ejpam-4653	126	1	=	=	PUNCT
ejpam-4653	126	2	2	2	X
ejpam-4653	126	3	.	.	PUNCT
ejpam-4653	126	4	since	since	SCONJ
ejpam-4653	126	5	γdr(g	γdr(g	X
ejpam-4653	126	6	)	)	PUNCT
ejpam-4653	126	7	̸=	̸=	PROPN
ejpam-4653	126	8	4	4	NUM
ejpam-4653	126	9	,	,	PUNCT
ejpam-4653	126	10	γ2(g	γ2(g	CCONJ
ejpam-4653	126	11	)	)	PUNCT
ejpam-4653	126	12	≥	≥	NOUN
ejpam-4653	126	13	3	3	NUM
ejpam-4653	126	14	by	by	ADP
ejpam-4653	126	15	proposition	proposition	NOUN
ejpam-4653	126	16	5	5	NUM
ejpam-4653	126	17	.	.	PUNCT
ejpam-4653	127	1	moreover	moreover	ADV
ejpam-4653	127	2	,	,	PUNCT
ejpam-4653	127	3	by	by	ADP
ejpam-4653	127	4	proposition	proposition	NOUN
ejpam-4653	127	5	1	1	NUM
ejpam-4653	127	6	,	,	PUNCT
ejpam-4653	127	7	γ(g−	γ(g−	NOUN
ejpam-4653	127	8	v	v	NOUN
ejpam-4653	127	9	)	)	PUNCT
ejpam-4653	127	10	≥	≥	NOUN
ejpam-4653	127	11	2	2	NUM
ejpam-4653	127	12	for	for	ADP
ejpam-4653	127	13	all	all	PRON
ejpam-4653	127	14	v	v	ADP
ejpam-4653	127	15	∈	∈	NOUN
ejpam-4653	127	16	v	v	NOUN
ejpam-4653	127	17	(	(	PUNCT
ejpam-4653	127	18	g	g	NOUN
ejpam-4653	127	19	)	)	PUNCT
ejpam-4653	127	20	.	.	PUNCT
ejpam-4653	128	1	this	this	PRON
ejpam-4653	128	2	proves	prove	VERB
ejpam-4653	128	3	(	(	PUNCT
ejpam-4653	128	4	i	i	NOUN
ejpam-4653	128	5	)	)	PUNCT
ejpam-4653	128	6	.	.	PUNCT
ejpam-4653	129	1	case	case	NOUN
ejpam-4653	129	2	2	2	X
ejpam-4653	129	3	.	.	PUNCT
ejpam-4653	129	4	suppose	suppose	VERB
ejpam-4653	129	5	that	that	SCONJ
ejpam-4653	129	6	|v2|	|v2|	NOUN
ejpam-4653	129	7	=	=	SYM
ejpam-4653	129	8	3	3	NUM
ejpam-4653	129	9	and	and	CCONJ
ejpam-4653	129	10	v3	v3	PROPN
ejpam-4653	129	11	=	=	PUNCT
ejpam-4653	129	12	∅.	∅.	PROPN
ejpam-4653	129	13	then	then	ADV
ejpam-4653	129	14	v2	v2	PROPN
ejpam-4653	129	15	is	be	AUX
ejpam-4653	129	16	a	a	DET
ejpam-4653	129	17	2	2	NUM
ejpam-4653	129	18	-	-	PUNCT
ejpam-4653	129	19	dominating	dominating	NOUN
ejpam-4653	129	20	set	set	NOUN
ejpam-4653	129	21	of	of	ADP
ejpam-4653	129	22	g	g	NOUN
ejpam-4653	129	23	so	so	SCONJ
ejpam-4653	129	24	that	that	SCONJ
ejpam-4653	129	25	γ2(g	γ2(g	ADP
ejpam-4653	129	26	)	)	PUNCT
ejpam-4653	129	27	≤	≤	NUM
ejpam-4653	129	28	3	3	NUM
ejpam-4653	129	29	.	.	PUNCT
ejpam-4653	129	30	since	since	SCONJ
ejpam-4653	129	31	γdr(g	γdr(g	X
ejpam-4653	129	32	)	)	PUNCT
ejpam-4653	129	33	̸=	̸=	PROPN
ejpam-4653	129	34	4	4	NUM
ejpam-4653	129	35	,	,	PUNCT
ejpam-4653	129	36	γ2(g	γ2(g	CCONJ
ejpam-4653	129	37	)	)	PUNCT
ejpam-4653	129	38	=	=	SYM
ejpam-4653	129	39	3	3	NUM
ejpam-4653	129	40	by	by	ADP
ejpam-4653	129	41	proposition	proposition	NOUN
ejpam-4653	129	42	5	5	NUM
ejpam-4653	129	43	.	.	PUNCT
ejpam-4653	130	1	hence	hence	ADV
ejpam-4653	130	2	,	,	PUNCT
ejpam-4653	130	3	(	(	PUNCT
ejpam-4653	130	4	ii	ii	NOUN
ejpam-4653	130	5	)	)	PUNCT
ejpam-4653	130	6	holds	hold	VERB
ejpam-4653	130	7	.	.	PUNCT
ejpam-4653	131	1	conversely	conversely	ADV
ejpam-4653	131	2	,	,	PUNCT
ejpam-4653	131	3	by	by	ADP
ejpam-4653	131	4	proposition	proposition	NOUN
ejpam-4653	131	5	5	5	NUM
ejpam-4653	131	6	,	,	PUNCT
ejpam-4653	131	7	proposition	proposition	NOUN
ejpam-4653	131	8	6	6	NUM
ejpam-4653	131	9	and	and	CCONJ
ejpam-4653	131	10	corollary	corollary	ADJ
ejpam-4653	131	11	1	1	NUM
ejpam-4653	131	12	,	,	PUNCT
ejpam-4653	131	13	γdr(g	γdr(g	NUM
ejpam-4653	131	14	)	)	PUNCT
ejpam-4653	131	15	≥	≥	NOUN
ejpam-4653	131	16	6	6	NUM
ejpam-4653	131	17	.	.	PUNCT
ejpam-4653	132	1	if	if	SCONJ
ejpam-4653	132	2	u	u	PROPN
ejpam-4653	132	3	,	,	PUNCT
ejpam-4653	132	4	v	v	PROPN
ejpam-4653	132	5	∈	∈	PROPN
ejpam-4653	132	6	v	v	NOUN
ejpam-4653	132	7	(	(	PUNCT
ejpam-4653	132	8	g	g	NOUN
ejpam-4653	132	9	)	)	PUNCT
ejpam-4653	132	10	such	such	ADJ
ejpam-4653	132	11	that	that	SCONJ
ejpam-4653	132	12	{	{	PUNCT
ejpam-4653	132	13	u	u	NOUN
ejpam-4653	132	14	,	,	PUNCT
ejpam-4653	132	15	v	v	NOUN
ejpam-4653	132	16	}	}	PUNCT
ejpam-4653	132	17	dominates	dominate	VERB
ejpam-4653	132	18	v	v	ADP
ejpam-4653	132	19	(	(	PUNCT
ejpam-4653	132	20	g	g	NOUN
ejpam-4653	132	21	)	)	PUNCT
ejpam-4653	132	22	,	,	PUNCT
ejpam-4653	132	23	then	then	ADV
ejpam-4653	132	24	f	f	PROPN
ejpam-4653	132	25	=	=	PUNCT
ejpam-4653	132	26	(	(	PUNCT
ejpam-4653	132	27	v	v	NOUN
ejpam-4653	132	28	(	(	PUNCT
ejpam-4653	132	29	g	g	NOUN
ejpam-4653	132	30	)	)	PUNCT
ejpam-4653	132	31	\	\	NOUN
ejpam-4653	133	1	{	{	PUNCT
ejpam-4653	133	2	u	u	PROPN
ejpam-4653	133	3	,	,	PUNCT
ejpam-4653	133	4	v},∅,∅	v},∅,∅	NOUN
ejpam-4653	133	5	,	,	PUNCT
ejpam-4653	133	6	{	{	PUNCT
ejpam-4653	133	7	u	u	NOUN
ejpam-4653	133	8	,	,	PUNCT
ejpam-4653	133	9	v	v	NOUN
ejpam-4653	133	10	}	}	PUNCT
ejpam-4653	133	11	)	)	PUNCT
ejpam-4653	133	12	∈	∈	PROPN
ejpam-4653	133	13	drd(g	drd(g	PROPN
ejpam-4653	133	14	)	)	PUNCT
ejpam-4653	133	15	so	so	SCONJ
ejpam-4653	133	16	that	that	SCONJ
ejpam-4653	133	17	γdr(g	γdr(g	X
ejpam-4653	133	18	)	)	PUNCT
ejpam-4653	133	19	≤	≤	NOUN
ejpam-4653	133	20	ωg(f	ωg(f	NOUN
ejpam-4653	133	21	)	)	PUNCT
ejpam-4653	133	22	=	=	SYM
ejpam-4653	134	1	6	6	X
ejpam-4653	134	2	.	.	PUNCT
ejpam-4653	135	1	on	on	ADP
ejpam-4653	135	2	the	the	DET
ejpam-4653	135	3	other	other	ADJ
ejpam-4653	135	4	hand	hand	NOUN
ejpam-4653	135	5	,	,	PUNCT
ejpam-4653	135	6	if	if	SCONJ
ejpam-4653	135	7	{	{	PUNCT
ejpam-4653	135	8	u	u	NOUN
ejpam-4653	135	9	,	,	PUNCT
ejpam-4653	135	10	v	v	NOUN
ejpam-4653	135	11	,	,	PUNCT
ejpam-4653	135	12	w	w	NOUN
ejpam-4653	135	13	}	}	PUNCT
ejpam-4653	135	14	is	be	AUX
ejpam-4653	135	15	a	a	DET
ejpam-4653	135	16	γ2	γ2	NOUN
ejpam-4653	135	17	-	-	PUNCT
ejpam-4653	135	18	set	set	NOUN
ejpam-4653	135	19	of	of	ADP
ejpam-4653	135	20	g	g	NOUN
ejpam-4653	135	21	,	,	PUNCT
ejpam-4653	135	22	then	then	ADV
ejpam-4653	135	23	f	f	PROPN
ejpam-4653	135	24	=	=	SYM
ejpam-4653	135	25	(	(	PUNCT
ejpam-4653	135	26	v	v	NOUN
ejpam-4653	135	27	(	(	PUNCT
ejpam-4653	135	28	g)\{u	g)\{u	PROPN
ejpam-4653	135	29	,	,	PUNCT
ejpam-4653	135	30	v	v	NOUN
ejpam-4653	135	31	,	,	PUNCT
ejpam-4653	135	32	w},∅	w},∅	PROPN
ejpam-4653	135	33	,	,	PUNCT
ejpam-4653	135	34	{	{	PUNCT
ejpam-4653	135	35	u	u	NOUN
ejpam-4653	135	36	,	,	PUNCT
ejpam-4653	135	37	v	v	NOUN
ejpam-4653	135	38	,	,	PUNCT
ejpam-4653	135	39	w},∅	w},∅	PROPN
ejpam-4653	135	40	)	)	PUNCT
ejpam-4653	135	41	∈	∈	PROPN
ejpam-4653	135	42	drd(g	drd(g	PROPN
ejpam-4653	135	43	)	)	PUNCT
ejpam-4653	136	1	so	so	SCONJ
ejpam-4653	136	2	that	that	SCONJ
ejpam-4653	136	3	γdr(g	γdr(g	X
ejpam-4653	136	4	)	)	PUNCT
ejpam-4653	136	5	≤	≤	NOUN
ejpam-4653	136	6	ωg(f	ωg(f	NOUN
ejpam-4653	136	7	)	)	PUNCT
ejpam-4653	136	8	=	=	SYM
ejpam-4653	136	9	6	6	NUM
ejpam-4653	136	10	.	.	PUNCT
ejpam-4653	137	1	therefore	therefore	ADV
ejpam-4653	137	2	,	,	PUNCT
ejpam-4653	137	3	each	each	PRON
ejpam-4653	137	4	of	of	ADP
ejpam-4653	137	5	(	(	PUNCT
ejpam-4653	137	6	i	i	NOUN
ejpam-4653	137	7	)	)	PUNCT
ejpam-4653	137	8	and	and	CCONJ
ejpam-4653	137	9	(	(	PUNCT
ejpam-4653	137	10	ii	ii	NOUN
ejpam-4653	137	11	)	)	PUNCT
ejpam-4653	137	12	implies	imply	VERB
ejpam-4653	137	13	that	that	SCONJ
ejpam-4653	137	14	γdr(g	γdr(g	X
ejpam-4653	137	15	)	)	PUNCT
ejpam-4653	137	16	=	=	SYM
ejpam-4653	137	17	6	6	X
ejpam-4653	137	18	.	.	PUNCT
ejpam-4653	137	19	j.	j.	PROPN
ejpam-4653	137	20	b.g	b.g	PROPN
ejpam-4653	137	21	.	.	PROPN
ejpam-4653	137	22	cariaga	cariaga	PROPN
ejpam-4653	137	23	,	,	PUNCT
ejpam-4653	137	24	f.	f.	PROPN
ejpam-4653	137	25	jamil	jamil	PROPN
ejpam-4653	137	26	/	/	SYM
ejpam-4653	137	27	eur	eur	PROPN
ejpam-4653	137	28	.	.	PUNCT
ejpam-4653	138	1	j.	j.	PROPN
ejpam-4653	138	2	pure	pure	PROPN
ejpam-4653	138	3	appl	appl	PROPN
ejpam-4653	138	4	.	.	PROPN
ejpam-4653	138	5	math	math	PROPN
ejpam-4653	138	6	,	,	PUNCT
ejpam-4653	138	7	16	16	NUM
ejpam-4653	138	8	(	(	PUNCT
ejpam-4653	138	9	2	2	NUM
ejpam-4653	138	10	)	)	PUNCT
ejpam-4653	138	11	(	(	PUNCT
ejpam-4653	138	12	2023	2023	NUM
ejpam-4653	138	13	)	)	PUNCT
ejpam-4653	138	14	,	,	PUNCT
ejpam-4653	138	15	847	847	NUM
ejpam-4653	138	16	-	-	SYM
ejpam-4653	138	17	863	863	NUM
ejpam-4653	138	18	851	851	NUM
ejpam-4653	138	19	proposition	proposition	NOUN
ejpam-4653	138	20	8	8	NUM
ejpam-4653	138	21	.	.	PUNCT
ejpam-4653	139	1	for	for	ADP
ejpam-4653	139	2	a	a	DET
ejpam-4653	139	3	nontrivial	nontrivial	ADJ
ejpam-4653	139	4	connected	connect	VERB
ejpam-4653	139	5	graph	graph	NOUN
ejpam-4653	139	6	g	g	PROPN
ejpam-4653	139	7	,	,	PUNCT
ejpam-4653	139	8	if	if	SCONJ
ejpam-4653	139	9	γdr(g	γdr(g	ADJ
ejpam-4653	139	10	)	)	PUNCT
ejpam-4653	139	11	=	=	SYM
ejpam-4653	139	12	7	7	NUM
ejpam-4653	139	13	,	,	PUNCT
ejpam-4653	139	14	then	then	ADV
ejpam-4653	139	15	γ(g	γ(g	PROPN
ejpam-4653	139	16	)	)	PUNCT
ejpam-4653	139	17	=	=	SYM
ejpam-4653	139	18	3	3	NUM
ejpam-4653	139	19	and	and	CCONJ
ejpam-4653	139	20	γ2(g	γ2(g	ADP
ejpam-4653	139	21	)	)	PUNCT
ejpam-4653	139	22	≥	≥	NOUN
ejpam-4653	139	23	4	4	NUM
ejpam-4653	139	24	.	.	PUNCT
ejpam-4653	139	25	proof	proof	NOUN
ejpam-4653	139	26	.	.	PUNCT
ejpam-4653	140	1	let	let	VERB
ejpam-4653	140	2	γdr(g	γdr(g	X
ejpam-4653	140	3	)	)	PUNCT
ejpam-4653	140	4	=	=	SYM
ejpam-4653	141	1	7	7	X
ejpam-4653	141	2	.	.	PUNCT
ejpam-4653	141	3	by	by	ADP
ejpam-4653	141	4	proposition	proposition	NOUN
ejpam-4653	141	5	5	5	NUM
ejpam-4653	141	6	,	,	PUNCT
ejpam-4653	141	7	corollary	corollary	ADJ
ejpam-4653	141	8	1	1	NUM
ejpam-4653	141	9	,	,	PUNCT
ejpam-4653	141	10	and	and	CCONJ
ejpam-4653	141	11	proposition	proposition	NOUN
ejpam-4653	141	12	7	7	NUM
ejpam-4653	141	13	,	,	PUNCT
ejpam-4653	141	14	γ(g	γ(g	PROPN
ejpam-4653	141	15	)	)	PUNCT
ejpam-4653	141	16	≥	≥	NOUN
ejpam-4653	141	17	3	3	NUM
ejpam-4653	141	18	and	and	CCONJ
ejpam-4653	141	19	γ2(g	γ2(g	ADP
ejpam-4653	141	20	)	)	PUNCT
ejpam-4653	141	21	≥	≥	NOUN
ejpam-4653	141	22	4	4	NUM
ejpam-4653	141	23	.	.	PUNCT
ejpam-4653	142	1	let	let	VERB
ejpam-4653	142	2	f	f	PROPN
ejpam-4653	142	3	=	=	SYM
ejpam-4653	142	4	(	(	PUNCT
ejpam-4653	142	5	v0,∅	v0,∅	PROPN
ejpam-4653	142	6	,	,	PUNCT
ejpam-4653	142	7	v2	v2	PROPN
ejpam-4653	142	8	,	,	PUNCT
ejpam-4653	142	9	v3	v3	PROPN
ejpam-4653	142	10	)	)	PUNCT
ejpam-4653	142	11	be	be	VERB
ejpam-4653	142	12	a	a	DET
ejpam-4653	142	13	γdr	γdr	NOUN
ejpam-4653	142	14	-	-	PUNCT
ejpam-4653	142	15	function	function	NOUN
ejpam-4653	142	16	of	of	ADP
ejpam-4653	142	17	g.	g.	PROPN
ejpam-4653	142	18	then	then	ADV
ejpam-4653	142	19	|v2|	|v2|	ADV
ejpam-4653	143	1	=	=	SYM
ejpam-4653	143	2	2	2	NUM
ejpam-4653	143	3	and	and	CCONJ
ejpam-4653	143	4	|v3|	|v3|	NOUN
ejpam-4653	143	5	=	=	SYM
ejpam-4653	143	6	1	1	X
ejpam-4653	143	7	.	.	X
ejpam-4653	144	1	write	write	VERB
ejpam-4653	144	2	v2	v2	PROPN
ejpam-4653	144	3	=	=	PUNCT
ejpam-4653	144	4	{	{	PUNCT
ejpam-4653	144	5	u	u	NOUN
ejpam-4653	144	6	,	,	PUNCT
ejpam-4653	144	7	v	v	NOUN
ejpam-4653	144	8	}	}	PUNCT
ejpam-4653	144	9	and	and	CCONJ
ejpam-4653	144	10	v3	v3	PROPN
ejpam-4653	144	11	=	=	SYM
ejpam-4653	144	12	{	{	PUNCT
ejpam-4653	144	13	w	w	NOUN
ejpam-4653	144	14	}	}	PUNCT
ejpam-4653	144	15	.	.	PUNCT
ejpam-4653	145	1	then	then	ADV
ejpam-4653	145	2	{	{	PUNCT
ejpam-4653	145	3	u	u	NOUN
ejpam-4653	145	4	,	,	PUNCT
ejpam-4653	145	5	v	v	NOUN
ejpam-4653	145	6	,	,	PUNCT
ejpam-4653	145	7	w	w	NOUN
ejpam-4653	145	8	}	}	PUNCT
ejpam-4653	145	9	is	be	AUX
ejpam-4653	145	10	a	a	DET
ejpam-4653	145	11	dominating	dominating	NOUN
ejpam-4653	145	12	set	set	NOUN
ejpam-4653	145	13	of	of	ADP
ejpam-4653	145	14	g	g	NOUN
ejpam-4653	145	15	,	,	PUNCT
ejpam-4653	145	16	and	and	CCONJ
ejpam-4653	145	17	so	so	ADV
ejpam-4653	145	18	,	,	PUNCT
ejpam-4653	145	19	γ(g	γ(g	PROPN
ejpam-4653	145	20	)	)	PUNCT
ejpam-4653	145	21	≤	≤	NOUN
ejpam-4653	145	22	3	3	NUM
ejpam-4653	145	23	.	.	PUNCT
ejpam-4653	146	1	hence	hence	ADV
ejpam-4653	146	2	,	,	PUNCT
ejpam-4653	146	3	γ(g	γ(g	PROPN
ejpam-4653	146	4	)	)	PUNCT
ejpam-4653	147	1	=	=	SYM
ejpam-4653	147	2	3	3	X
ejpam-4653	147	3	.	.	PUNCT
ejpam-4653	148	1	the	the	DET
ejpam-4653	148	2	converse	converse	NOUN
ejpam-4653	148	3	of	of	ADP
ejpam-4653	148	4	proposition	proposition	NOUN
ejpam-4653	148	5	8	8	NUM
ejpam-4653	148	6	need	need	AUX
ejpam-4653	148	7	not	not	PART
ejpam-4653	148	8	be	be	AUX
ejpam-4653	148	9	true	true	ADJ
ejpam-4653	148	10	.	.	PUNCT
ejpam-4653	149	1	consider	consider	VERB
ejpam-4653	149	2	for	for	ADP
ejpam-4653	149	3	example	example	NOUN
ejpam-4653	149	4	,	,	PUNCT
ejpam-4653	149	5	the	the	DET
ejpam-4653	149	6	graph	graph	NOUN
ejpam-4653	149	7	g	g	NOUN
ejpam-4653	149	8	in	in	ADP
ejpam-4653	149	9	figure	figure	NOUN
ejpam-4653	149	10	1	1	NUM
ejpam-4653	149	11	obtained	obtain	VERB
ejpam-4653	149	12	from	from	ADP
ejpam-4653	149	13	p9	p9	PROPN
ejpam-4653	149	14	=	=	PUNCT
ejpam-4653	150	1	[	[	X
ejpam-4653	150	2	x1	x1	PROPN
ejpam-4653	150	3	,	,	PUNCT
ejpam-4653	150	4	x2	x2	PROPN
ejpam-4653	150	5	,	,	PUNCT
ejpam-4653	150	6	x3	x3	ADJ
ejpam-4653	150	7	,	,	PUNCT
ejpam-4653	150	8	.	.	PUNCT
ejpam-4653	150	9	.	.	PUNCT
ejpam-4653	150	10	.	.	PUNCT
ejpam-4653	151	1	,	,	PUNCT
ejpam-4653	151	2	x9	x9	PROPN
ejpam-4653	151	3	]	]	PUNCT
ejpam-4653	151	4	by	by	ADP
ejpam-4653	151	5	adding	add	VERB
ejpam-4653	151	6	the	the	DET
ejpam-4653	151	7	edges	edge	NOUN
ejpam-4653	151	8	x3x5	x3x5	X
ejpam-4653	151	9	and	and	CCONJ
ejpam-4653	151	10	x7x5	x7x5	PROPN
ejpam-4653	151	11	.	.	PROPN
ejpam-4653	151	12	observe	observe	VERB
ejpam-4653	151	13	that	that	SCONJ
ejpam-4653	151	14	γ(g	γ(g	PROPN
ejpam-4653	151	15	)	)	PUNCT
ejpam-4653	151	16	=	=	SYM
ejpam-4653	151	17	3	3	NUM
ejpam-4653	151	18	,	,	PUNCT
ejpam-4653	151	19	γ2(g	γ2(g	ADJ
ejpam-4653	151	20	)	)	PUNCT
ejpam-4653	151	21	≥	≥	NOUN
ejpam-4653	151	22	4	4	NUM
ejpam-4653	151	23	but	but	CCONJ
ejpam-4653	151	24	γdr(g	γdr(g	NUM
ejpam-4653	151	25	)	)	PUNCT
ejpam-4653	151	26	=	=	SYM
ejpam-4653	152	1	9	9	NUM
ejpam-4653	152	2	>	>	SYM
ejpam-4653	152	3	7	7	NUM
ejpam-4653	152	4	.	.	PUNCT
ejpam-4653	152	5	....................................	....................................	PUNCT
ejpam-4653	152	6	....................................	....................................	PUNCT
ejpam-4653	153	1	....................................	....................................	PUNCT
ejpam-4653	153	2	....................................	....................................	PUNCT
ejpam-4653	154	1	....................................	....................................	PUNCT
ejpam-4653	154	2	....................................	....................................	PUNCT
ejpam-4653	155	1	....................................	....................................	PUNCT
ejpam-4653	155	2	....................................	....................................	PUNCT
ejpam-4653	156	1	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-4653	156	2	.........	.........	PUNCT
ejpam-4653	156	3	.............	.............	PUNCT
ejpam-4653	156	4	.............	.............	PUNCT
ejpam-4653	156	5	.............	.............	PUNCT
ejpam-4653	156	6	.............	.............	PUNCT
ejpam-4653	156	7	.............	.............	PUNCT
ejpam-4653	156	8	.............	.............	PUNCT
ejpam-4653	156	9	.............	.............	PUNCT
ejpam-4653	156	10	.............	.............	PUNCT
ejpam-4653	156	11	.............	.............	PUNCT
ejpam-4653	156	12	.......	.......	PUNCT
ejpam-4653	157	1	.....................................................................................................................................................................................................	.....................................................................................................................................................................................................	PUNCT
ejpam-4653	157	2	...........	...........	PUNCT
ejpam-4653	158	1	..........	..........	PUNCT
ejpam-4653	158	2	..........	..........	PUNCT
ejpam-4653	159	1	..........	..........	PUNCT
ejpam-4653	159	2	..........	..........	PUNCT
ejpam-4653	160	1	..........	..........	PUNCT
ejpam-4653	160	2	..........	..........	PUNCT
ejpam-4653	161	1	..........	..........	PUNCT
ejpam-4653	161	2	..........	..........	PUNCT
ejpam-4653	162	1	..........	..........	PUNCT
ejpam-4653	162	2	..........	..........	PUNCT
ejpam-4653	163	1	..........	..........	PUNCT
ejpam-4653	163	2	..........	..........	PUNCT
ejpam-4653	164	1	..........	..........	PUNCT
ejpam-4653	164	2	..........	..........	PUNCT
ejpam-4653	165	1	..........	..........	PUNCT
ejpam-4653	165	2	..........	..........	PUNCT
ejpam-4653	166	1	..	..	PUNCT
ejpam-4653	166	2	.............................................................................................................................................................................	.............................................................................................................................................................................	PUNCT
ejpam-4653	166	3	...........	...........	PUNCT
ejpam-4653	167	1	..........	..........	PUNCT
ejpam-4653	167	2	..........	..........	PUNCT
ejpam-4653	168	1	..........	..........	PUNCT
ejpam-4653	168	2	..........	..........	PUNCT
ejpam-4653	169	1	..........	..........	PUNCT
ejpam-4653	169	2	..........	..........	PUNCT
ejpam-4653	170	1	..........	..........	PUNCT
ejpam-4653	170	2	..........	..........	PUNCT
ejpam-4653	171	1	..........	..........	PUNCT
ejpam-4653	171	2	..........	..........	PUNCT
ejpam-4653	172	1	..........	..........	PUNCT
ejpam-4653	172	2	..........	..........	PUNCT
ejpam-4653	173	1	..........	..........	PUNCT
ejpam-4653	173	2	..........	..........	PUNCT
ejpam-4653	174	1	..........	..........	PUNCT
ejpam-4653	174	2	..........	..........	PUNCT
ejpam-4653	175	1	..........	..........	PUNCT
ejpam-4653	175	2	..........	..........	PUNCT
ejpam-4653	176	1	......	......	PUNCT
ejpam-4653	176	2	.....................................................................................................................................	.....................................................................................................................................	PUNCT
ejpam-4653	177	1	........................................................................................................	........................................................................................................	PUNCT
ejpam-4653	177	2	.............................................................................................................................................................................................................................	.............................................................................................................................................................................................................................	PUNCT
ejpam-4653	178	1	................................................................	................................................................	PUNCT
ejpam-4653	178	2	...............................................................	...............................................................	PUNCT
ejpam-4653	179	1	...............................................................	...............................................................	PUNCT
ejpam-4653	179	2	...............................	...............................	PUNCT
ejpam-4653	180	1	•	•	NUM
ejpam-4653	180	2	•	•	NOUN
ejpam-4653	180	3	•	•	NOUN
ejpam-4653	181	1	x1	x1	NOUN
ejpam-4653	182	1	x2	x2	NOUN
ejpam-4653	182	2	x3	x3	PROPN
ejpam-4653	182	3	x4	x4	PROPN
ejpam-4653	182	4	x5	x5	PROPN
ejpam-4653	182	5	x6	x6	PROPN
ejpam-4653	182	6	x7	x7	NOUN
ejpam-4653	182	7	x8	x8	NOUN
ejpam-4653	182	8	x9	x9	NOUN
ejpam-4653	182	9	0	0	NUM
ejpam-4653	182	10	3	3	NUM
ejpam-4653	182	11	0	0	NUM
ejpam-4653	182	12	0	0	NUM
ejpam-4653	182	13	3	3	NUM
ejpam-4653	182	14	0	0	NUM
ejpam-4653	182	15	0	0	NUM
ejpam-4653	182	16	3	3	NUM
ejpam-4653	182	17	0	0	NUM
ejpam-4653	182	18	figure	figure	NOUN
ejpam-4653	182	19	1	1	NUM
ejpam-4653	182	20	:	:	PUNCT
ejpam-4653	182	21	a	a	DET
ejpam-4653	182	22	graph	graph	NOUN
ejpam-4653	182	23	g	g	NOUN
ejpam-4653	182	24	with	with	ADP
ejpam-4653	182	25	γdr(g	γdr(g	PROPN
ejpam-4653	182	26	)	)	PUNCT
ejpam-4653	182	27	>	>	PUNCT
ejpam-4653	182	28	7	7	NUM
ejpam-4653	182	29	proposition	proposition	NOUN
ejpam-4653	182	30	9	9	NUM
ejpam-4653	182	31	.	.	PUNCT
ejpam-4653	183	1	let	let	VERB
ejpam-4653	183	2	g	g	PRON
ejpam-4653	183	3	be	be	AUX
ejpam-4653	183	4	a	a	DET
ejpam-4653	183	5	disconnected	disconnected	ADJ
ejpam-4653	183	6	graph	graph	NOUN
ejpam-4653	183	7	with	with	ADP
ejpam-4653	183	8	components	component	NOUN
ejpam-4653	183	9	c1	c1	PROPN
ejpam-4653	183	10	,	,	PUNCT
ejpam-4653	183	11	c2	c2	PROPN
ejpam-4653	183	12	,	,	PUNCT
ejpam-4653	183	13	.	.	PUNCT
ejpam-4653	183	14	.	.	PUNCT
ejpam-4653	184	1	.	.	PUNCT
ejpam-4653	185	1	,	,	PUNCT
ejpam-4653	185	2	ck	ck	INTJ
ejpam-4653	185	3	.	.	PUNCT
ejpam-4653	186	1	then	then	ADV
ejpam-4653	186	2	γdr(g	γdr(g	X
ejpam-4653	186	3	)	)	PUNCT
ejpam-4653	186	4	=	=	SYM
ejpam-4653	187	1	∑k	∑k	PROPN
ejpam-4653	187	2	j=1	j=1	PROPN
ejpam-4653	187	3	γdr(cj	γdr(cj	NUM
ejpam-4653	187	4	)	)	PUNCT
ejpam-4653	187	5	.	.	PUNCT
ejpam-4653	188	1	in	in	ADP
ejpam-4653	188	2	particular	particular	ADJ
ejpam-4653	188	3	,	,	PUNCT
ejpam-4653	188	4	if	if	SCONJ
ejpam-4653	188	5	g	g	PROPN
ejpam-4653	188	6	=	=	SYM
ejpam-4653	188	7	kn	kn	PROPN
ejpam-4653	188	8	,	,	PUNCT
ejpam-4653	188	9	then	then	ADV
ejpam-4653	188	10	γdr(g	γdr(g	X
ejpam-4653	188	11	)	)	PUNCT
ejpam-4653	188	12	=	=	SYM
ejpam-4653	188	13	2n	2n	X
ejpam-4653	188	14	.	.	PUNCT
ejpam-4653	189	1	proof	proof	NOUN
ejpam-4653	189	2	.	.	PUNCT
ejpam-4653	190	1	if	if	SCONJ
ejpam-4653	190	2	f1	f1	PROPN
ejpam-4653	190	3	,	,	PUNCT
ejpam-4653	190	4	f2	f2	PROPN
ejpam-4653	190	5	,	,	PUNCT
ejpam-4653	190	6	.	.	PUNCT
ejpam-4653	190	7	.	.	PUNCT
ejpam-4653	191	1	.	.	PUNCT
ejpam-4653	192	1	,	,	PUNCT
ejpam-4653	192	2	fk	fk	INTJ
ejpam-4653	192	3	are	be	AUX
ejpam-4653	192	4	γdr	γdr	NOUN
ejpam-4653	192	5	-	-	PUNCT
ejpam-4653	192	6	functions	function	NOUN
ejpam-4653	192	7	of	of	ADP
ejpam-4653	192	8	c1	c1	NOUN
ejpam-4653	192	9	,	,	PUNCT
ejpam-4653	192	10	c2	c2	PROPN
ejpam-4653	192	11	,	,	PUNCT
ejpam-4653	192	12	.	.	PUNCT
ejpam-4653	192	13	.	.	PUNCT
ejpam-4653	193	1	.	.	PUNCT
ejpam-4653	193	2	,	,	PUNCT
ejpam-4653	193	3	ck	ck	INTJ
ejpam-4653	193	4	,	,	PUNCT
ejpam-4653	193	5	respectively	respectively	ADV
ejpam-4653	193	6	,	,	PUNCT
ejpam-4653	193	7	then	then	ADV
ejpam-4653	193	8	the	the	DET
ejpam-4653	193	9	function	function	NOUN
ejpam-4653	193	10	f	f	X
ejpam-4653	194	1	:	:	PUNCT
ejpam-4653	194	2	v	v	X
ejpam-4653	194	3	(	(	PUNCT
ejpam-4653	194	4	g	g	NOUN
ejpam-4653	194	5	)	)	PUNCT
ejpam-4653	194	6	→	→	SYM
ejpam-4653	194	7	{	{	PUNCT
ejpam-4653	194	8	0	0	NUM
ejpam-4653	194	9	,	,	PUNCT
ejpam-4653	194	10	1	1	NUM
ejpam-4653	194	11	,	,	PUNCT
ejpam-4653	194	12	2	2	NUM
ejpam-4653	194	13	,	,	PUNCT
ejpam-4653	194	14	3	3	NUM
ejpam-4653	194	15	}	}	PUNCT
ejpam-4653	194	16	given	give	VERB
ejpam-4653	194	17	by	by	ADP
ejpam-4653	194	18	f(x	f(x	PROPN
ejpam-4653	194	19	)	)	PUNCT
ejpam-4653	194	20	=	=	SYM
ejpam-4653	195	1	fk(x	fk(x	NOUN
ejpam-4653	195	2	)	)	PUNCT
ejpam-4653	195	3	for	for	ADP
ejpam-4653	195	4	all	all	PRON
ejpam-4653	195	5	x	x	SYM
ejpam-4653	195	6	∈	∈	PROPN
ejpam-4653	195	7	v	v	NOUN
ejpam-4653	195	8	(	(	PUNCT
ejpam-4653	195	9	ck	ck	NOUN
ejpam-4653	195	10	)	)	PUNCT
ejpam-4653	195	11	is	be	AUX
ejpam-4653	195	12	a	a	DET
ejpam-4653	195	13	γdr	γdr	NOUN
ejpam-4653	195	14	-	-	PUNCT
ejpam-4653	195	15	function	function	NOUN
ejpam-4653	195	16	of	of	ADP
ejpam-4653	195	17	g.	g.	PROPN
ejpam-4653	195	18	thus	thus	ADV
ejpam-4653	195	19	,	,	PUNCT
ejpam-4653	195	20	γdr(g	γdr(g	X
ejpam-4653	195	21	)	)	PUNCT
ejpam-4653	195	22	≤	≤	PUNCT
ejpam-4653	196	1	∑k	∑k	PROPN
ejpam-4653	196	2	j=1	j=1	PROPN
ejpam-4653	196	3	γdr(cj	γdr(cj	NUM
ejpam-4653	196	4	)	)	PUNCT
ejpam-4653	196	5	.	.	PUNCT
ejpam-4653	197	1	conversely	conversely	ADV
ejpam-4653	197	2	,	,	PUNCT
ejpam-4653	197	3	if	if	SCONJ
ejpam-4653	197	4	f	f	PRON
ejpam-4653	197	5	be	be	VERB
ejpam-4653	197	6	a	a	DET
ejpam-4653	197	7	γdr	γdr	NOUN
ejpam-4653	197	8	-	-	PUNCT
ejpam-4653	197	9	function	function	NOUN
ejpam-4653	197	10	of	of	ADP
ejpam-4653	197	11	g	g	NOUN
ejpam-4653	197	12	,	,	PUNCT
ejpam-4653	197	13	then	then	ADV
ejpam-4653	197	14	the	the	DET
ejpam-4653	197	15	restriction	restriction	NOUN
ejpam-4653	197	16	f	f	PROPN
ejpam-4653	197	17	|cj	|cj	PROPN
ejpam-4653	197	18	of	of	ADP
ejpam-4653	197	19	f	f	PROPN
ejpam-4653	197	20	to	to	PART
ejpam-4653	197	21	cj	cj	PROPN
ejpam-4653	197	22	,	,	PUNCT
ejpam-4653	197	23	for	for	ADP
ejpam-4653	197	24	any	any	DET
ejpam-4653	197	25	j	j	PROPN
ejpam-4653	197	26	=	=	SYM
ejpam-4653	197	27	1	1	NUM
ejpam-4653	197	28	,	,	PUNCT
ejpam-4653	197	29	2	2	NUM
ejpam-4653	197	30	,	,	PUNCT
ejpam-4653	197	31	.	.	PUNCT
ejpam-4653	197	32	.	.	PUNCT
ejpam-4653	197	33	.	.	PUNCT
ejpam-4653	198	1	,	,	PUNCT
ejpam-4653	198	2	k	k	NOUN
ejpam-4653	198	3	,	,	PUNCT
ejpam-4653	198	4	is	be	AUX
ejpam-4653	198	5	a	a	DET
ejpam-4653	198	6	γdr	γdr	NOUN
ejpam-4653	198	7	-	-	PUNCT
ejpam-4653	198	8	function	function	NOUN
ejpam-4653	198	9	of	of	ADP
ejpam-4653	198	10	cj	cj	NOUN
ejpam-4653	198	11	.	.	PUNCT
ejpam-4653	199	1	thus	thus	ADV
ejpam-4653	199	2	,	,	PUNCT
ejpam-4653	199	3	γdr(cj	γdr(cj	NUM
ejpam-4653	199	4	)	)	PUNCT
ejpam-4653	199	5	≤	≤	NUM
ejpam-4653	199	6	ωcj	ωcj	NOUN
ejpam-4653	199	7	(	(	PUNCT
ejpam-4653	199	8	f	f	PROPN
ejpam-4653	199	9	|cj	|cj	PROPN
ejpam-4653	199	10	)	)	PUNCT
ejpam-4653	199	11	for	for	ADP
ejpam-4653	199	12	all	all	DET
ejpam-4653	199	13	j	j	NOUN
ejpam-4653	199	14	=	=	SYM
ejpam-4653	199	15	1	1	NUM
ejpam-4653	199	16	,	,	PUNCT
ejpam-4653	199	17	2	2	NUM
ejpam-4653	199	18	,	,	PUNCT
ejpam-4653	199	19	.	.	PUNCT
ejpam-4653	199	20	.	.	PUNCT
ejpam-4653	199	21	.	.	PUNCT
ejpam-4653	200	1	,	,	PUNCT
ejpam-4653	200	2	k.	k.	PROPN
ejpam-4653	200	3	hence	hence	ADV
ejpam-4653	200	4	,	,	PUNCT
ejpam-4653	200	5	∑k	∑k	PROPN
ejpam-4653	200	6	j=1	j=1	PROPN
ejpam-4653	200	7	γdr(cj	γdr(cj	VERB
ejpam-4653	200	8	)	)	PUNCT
ejpam-4653	200	9	≤	≤	NOUN
ejpam-4653	200	10	γdr(g	γdr(g	X
ejpam-4653	200	11	)	)	PUNCT
ejpam-4653	200	12	.	.	PUNCT
ejpam-4653	201	1	proposition	proposition	NOUN
ejpam-4653	201	2	10	10	NUM
ejpam-4653	201	3	.	.	PUNCT
ejpam-4653	202	1	(	(	PUNCT
ejpam-4653	202	2	i	i	NOUN
ejpam-4653	202	3	)	)	PUNCT
ejpam-4653	202	4	for	for	ADP
ejpam-4653	202	5	any	any	DET
ejpam-4653	202	6	path	path	NOUN
ejpam-4653	202	7	pn	pn	NOUN
ejpam-4653	202	8	of	of	ADP
ejpam-4653	202	9	order	order	NOUN
ejpam-4653	202	10	n	n	CCONJ
ejpam-4653	202	11	,	,	PUNCT
ejpam-4653	202	12	γdr(pn	γdr(pn	NOUN
ejpam-4653	202	13	)	)	PUNCT
ejpam-4653	202	14	=	=	SYM
ejpam-4653	203	1			NOUN
ejpam-4653	203	2	2	2	NUM
ejpam-4653	203	3	,	,	PUNCT
ejpam-4653	203	4	n	n	NOUN
ejpam-4653	203	5	=	=	SYM
ejpam-4653	203	6	1	1	NUM
ejpam-4653	203	7	;	;	PUNCT
ejpam-4653	203	8	4	4	NUM
ejpam-4653	203	9	,	,	PUNCT
ejpam-4653	203	10	n	n	NOUN
ejpam-4653	203	11	=	=	SYM
ejpam-4653	203	12	2	2	NUM
ejpam-4653	203	13	;	;	PUNCT
ejpam-4653	203	14	5	5	NUM
ejpam-4653	203	15	,	,	PUNCT
ejpam-4653	203	16	n	n	PRON
ejpam-4653	203	17	≥	≥	NOUN
ejpam-4653	203	18	3	3	NUM
ejpam-4653	203	19	.	.	PUNCT
ejpam-4653	203	20	(	(	PUNCT
ejpam-4653	203	21	ii	ii	NOUN
ejpam-4653	203	22	)	)	PUNCT
ejpam-4653	203	23	for	for	ADP
ejpam-4653	203	24	any	any	DET
ejpam-4653	203	25	cycle	cycle	NOUN
ejpam-4653	203	26	cn	cn	NOUN
ejpam-4653	203	27	of	of	ADP
ejpam-4653	203	28	order	order	NOUN
ejpam-4653	203	29	n	n	PRON
ejpam-4653	203	30	≥	≥	NOUN
ejpam-4653	203	31	3	3	NUM
ejpam-4653	203	32	,	,	PUNCT
ejpam-4653	203	33	γdr(cn	γdr(cn	NOUN
ejpam-4653	203	34	)	)	PUNCT
ejpam-4653	203	35	=	=	SYM
ejpam-4653	203	36	6	6	X
ejpam-4653	203	37	.	.	PUNCT
ejpam-4653	204	1	proof	proof	NOUN
ejpam-4653	204	2	.	.	PUNCT
ejpam-4653	205	1	for	for	ADP
ejpam-4653	205	2	(	(	PUNCT
ejpam-4653	205	3	i	i	NOUN
ejpam-4653	205	4	):	):	PUNCT
ejpam-4653	205	5	the	the	DET
ejpam-4653	205	6	cases	case	NOUN
ejpam-4653	205	7	where	where	SCONJ
ejpam-4653	205	8	n	n	NOUN
ejpam-4653	205	9	=	=	SYM
ejpam-4653	205	10	1	1	NUM
ejpam-4653	205	11	,	,	PUNCT
ejpam-4653	205	12	2	2	NUM
ejpam-4653	205	13	,	,	PUNCT
ejpam-4653	205	14	3	3	NUM
ejpam-4653	205	15	,	,	PUNCT
ejpam-4653	205	16	4	4	NUM
ejpam-4653	205	17	are	be	AUX
ejpam-4653	205	18	clear	clear	ADJ
ejpam-4653	205	19	.	.	PUNCT
ejpam-4653	206	1	suppose	suppose	VERB
ejpam-4653	206	2	that	that	SCONJ
ejpam-4653	206	3	n	n	PROPN
ejpam-4653	206	4	≥	≥	NUM
ejpam-4653	206	5	5	5	NUM
ejpam-4653	206	6	.	.	PUNCT
ejpam-4653	207	1	let	let	VERB
ejpam-4653	207	2	v	v	X
ejpam-4653	207	3	(	(	PUNCT
ejpam-4653	207	4	pn	pn	NOUN
ejpam-4653	207	5	)	)	PUNCT
ejpam-4653	207	6	=	=	PUNCT
ejpam-4653	208	1	[	[	X
ejpam-4653	208	2	v1	v1	NOUN
ejpam-4653	208	3	,	,	PUNCT
ejpam-4653	208	4	v2	v2	NOUN
ejpam-4653	208	5	,	,	PUNCT
ejpam-4653	208	6	.	.	PUNCT
ejpam-4653	208	7	.	.	PUNCT
ejpam-4653	208	8	.	.	PUNCT
ejpam-4653	209	1	,	,	PUNCT
ejpam-4653	209	2	vn	vn	X
ejpam-4653	209	3	]	]	PUNCT
ejpam-4653	209	4	.	.	PUNCT
ejpam-4653	210	1	then	then	ADV
ejpam-4653	210	2	the	the	DET
ejpam-4653	210	3	sets	set	NOUN
ejpam-4653	210	4	{	{	PUNCT
ejpam-4653	210	5	v1	v1	NOUN
ejpam-4653	210	6	,	,	PUNCT
ejpam-4653	210	7	v2	v2	NOUN
ejpam-4653	210	8	}	}	PUNCT
ejpam-4653	210	9	and	and	CCONJ
ejpam-4653	210	10	{	{	PUNCT
ejpam-4653	210	11	v1	v1	NOUN
ejpam-4653	210	12	,	,	PUNCT
ejpam-4653	210	13	v2	v2	PROPN
ejpam-4653	210	14	,	,	PUNCT
ejpam-4653	210	15	v3	v3	PROPN
ejpam-4653	210	16	}	}	PUNCT
ejpam-4653	210	17	are	be	AUX
ejpam-4653	210	18	γ	γ	X
ejpam-4653	210	19	-	-	PUNCT
ejpam-4653	210	20	set	set	VERB
ejpam-4653	210	21	and	and	CCONJ
ejpam-4653	210	22	γ2	γ2	NOUN
ejpam-4653	210	23	-	-	PUNCT
ejpam-4653	210	24	set	set	NOUN
ejpam-4653	210	25	of	of	ADP
ejpam-4653	210	26	pn	pn	PROPN
ejpam-4653	210	27	,	,	PUNCT
ejpam-4653	210	28	respectively	respectively	ADV
ejpam-4653	210	29	.	.	PUNCT
ejpam-4653	211	1	moreover	moreover	ADV
ejpam-4653	211	2	,	,	PUNCT
ejpam-4653	211	3	γ(pn	γ(pn	PROPN
ejpam-4653	211	4	−	−	NOUN
ejpam-4653	211	5	v2	v2	NOUN
ejpam-4653	211	6	)	)	PUNCT
ejpam-4653	211	7	=	=	SYM
ejpam-4653	212	1	1	1	X
ejpam-4653	212	2	.	.	PUNCT
ejpam-4653	212	3	by	by	ADP
ejpam-4653	212	4	proposition	proposition	NOUN
ejpam-4653	212	5	6	6	NUM
ejpam-4653	212	6	,	,	PUNCT
ejpam-4653	212	7	γdr(pn	γdr(pn	NOUN
ejpam-4653	212	8	)	)	PUNCT
ejpam-4653	212	9	=	=	SYM
ejpam-4653	212	10	5	5	X
ejpam-4653	212	11	.	.	PUNCT
ejpam-4653	212	12	j.	j.	PROPN
ejpam-4653	212	13	b.g	b.g	PROPN
ejpam-4653	212	14	.	.	PROPN
ejpam-4653	212	15	cariaga	cariaga	PROPN
ejpam-4653	212	16	,	,	PUNCT
ejpam-4653	212	17	f.	f.	PROPN
ejpam-4653	212	18	jamil	jamil	PROPN
ejpam-4653	212	19	/	/	SYM
ejpam-4653	212	20	eur	eur	PROPN
ejpam-4653	212	21	.	.	PUNCT
ejpam-4653	213	1	j.	j.	PROPN
ejpam-4653	213	2	pure	pure	PROPN
ejpam-4653	213	3	appl	appl	PROPN
ejpam-4653	213	4	.	.	PROPN
ejpam-4653	213	5	math	math	PROPN
ejpam-4653	213	6	,	,	PUNCT
ejpam-4653	213	7	16	16	NUM
ejpam-4653	213	8	(	(	PUNCT
ejpam-4653	213	9	2	2	NUM
ejpam-4653	213	10	)	)	PUNCT
ejpam-4653	213	11	(	(	PUNCT
ejpam-4653	213	12	2023	2023	NUM
ejpam-4653	213	13	)	)	PUNCT
ejpam-4653	213	14	,	,	PUNCT
ejpam-4653	213	15	847	847	NUM
ejpam-4653	213	16	-	-	SYM
ejpam-4653	213	17	863	863	NUM
ejpam-4653	213	18	852	852	NUM
ejpam-4653	213	19	for	for	ADP
ejpam-4653	213	20	(	(	PUNCT
ejpam-4653	213	21	ii	ii	NOUN
ejpam-4653	213	22	):	):	PUNCT
ejpam-4653	213	23	the	the	DET
ejpam-4653	213	24	case	case	NOUN
ejpam-4653	213	25	where	where	SCONJ
ejpam-4653	213	26	n	n	PROPN
ejpam-4653	213	27	=	=	SYM
ejpam-4653	213	28	3	3	NUM
ejpam-4653	213	29	,	,	PUNCT
ejpam-4653	213	30	4	4	NUM
ejpam-4653	213	31	is	be	AUX
ejpam-4653	213	32	clear	clear	ADJ
ejpam-4653	213	33	.	.	PUNCT
ejpam-4653	213	34	suppose	suppose	VERB
ejpam-4653	213	35	that	that	SCONJ
ejpam-4653	213	36	n	n	PROPN
ejpam-4653	213	37	≥	≥	NUM
ejpam-4653	213	38	5	5	NUM
ejpam-4653	213	39	.	.	PUNCT
ejpam-4653	214	1	let	let	VERB
ejpam-4653	214	2	u	u	NOUN
ejpam-4653	214	3	,	,	PUNCT
ejpam-4653	214	4	w	w	PROPN
ejpam-4653	214	5	,	,	PUNCT
ejpam-4653	214	6	v	v	NOUN
ejpam-4653	214	7	∈	∈	PROPN
ejpam-4653	214	8	v	v	NOUN
ejpam-4653	214	9	(	(	PUNCT
ejpam-4653	214	10	cn	cn	PROPN
ejpam-4653	214	11	)	)	PUNCT
ejpam-4653	214	12	such	such	ADJ
ejpam-4653	214	13	that	that	SCONJ
ejpam-4653	214	14	u	u	NOUN
ejpam-4653	214	15	,	,	PUNCT
ejpam-4653	214	16	w	w	PROPN
ejpam-4653	214	17	∈	∈	PROPN
ejpam-4653	214	18	ncn(v	ncn(v	PROPN
ejpam-4653	214	19	)	)	PUNCT
ejpam-4653	214	20	.	.	PUNCT
ejpam-4653	215	1	then	then	ADV
ejpam-4653	215	2	{	{	PUNCT
ejpam-4653	215	3	u	u	NOUN
ejpam-4653	215	4	,	,	PUNCT
ejpam-4653	215	5	v	v	NOUN
ejpam-4653	215	6	}	}	PUNCT
ejpam-4653	215	7	and	and	CCONJ
ejpam-4653	215	8	{	{	PUNCT
ejpam-4653	215	9	u	u	NOUN
ejpam-4653	215	10	,	,	PUNCT
ejpam-4653	215	11	v	v	NOUN
ejpam-4653	215	12	,	,	PUNCT
ejpam-4653	215	13	w	w	NOUN
ejpam-4653	215	14	}	}	PUNCT
ejpam-4653	215	15	are	be	AUX
ejpam-4653	215	16	γ	γ	X
ejpam-4653	215	17	-	-	PUNCT
ejpam-4653	215	18	set	set	VERB
ejpam-4653	215	19	and	and	CCONJ
ejpam-4653	215	20	γ2	γ2	NOUN
ejpam-4653	215	21	-	-	PUNCT
ejpam-4653	215	22	set	set	NOUN
ejpam-4653	215	23	of	of	ADP
ejpam-4653	215	24	cn	cn	PROPN
ejpam-4653	215	25	,	,	PUNCT
ejpam-4653	215	26	respectively	respectively	ADV
ejpam-4653	215	27	.	.	PUNCT
ejpam-4653	216	1	moreover	moreover	ADV
ejpam-4653	216	2	,	,	PUNCT
ejpam-4653	216	3	γ(cn	γ(cn	PROPN
ejpam-4653	216	4	−	−	PROPN
ejpam-4653	216	5	v	v	NOUN
ejpam-4653	216	6	)	)	PUNCT
ejpam-4653	216	7	=	=	SYM
ejpam-4653	216	8	2	2	NUM
ejpam-4653	216	9	for	for	ADP
ejpam-4653	216	10	all	all	PRON
ejpam-4653	216	11	v	v	ADP
ejpam-4653	216	12	∈	∈	NOUN
ejpam-4653	216	13	v	v	NOUN
ejpam-4653	216	14	(	(	PUNCT
ejpam-4653	216	15	cn	cn	PROPN
ejpam-4653	216	16	)	)	PUNCT
ejpam-4653	216	17	.	.	PUNCT
ejpam-4653	217	1	by	by	ADP
ejpam-4653	217	2	proposition	proposition	NOUN
ejpam-4653	217	3	7	7	NUM
ejpam-4653	217	4	,	,	PUNCT
ejpam-4653	217	5	γdr(cn	γdr(cn	NOUN
ejpam-4653	217	6	)	)	PUNCT
ejpam-4653	217	7	=	=	SYM
ejpam-4653	218	1	6	6	X
ejpam-4653	218	2	.	.	PUNCT
ejpam-4653	219	1	the	the	PRON
ejpam-4653	219	2	(	(	PUNCT
ejpam-4653	219	3	n	n	X
ejpam-4653	219	4	,	,	PUNCT
ejpam-4653	219	5	m)-tadpole	m)-tadpole	PUNCT
ejpam-4653	219	6	graph	graph	NOUN
ejpam-4653	219	7	tn	tn	PROPN
ejpam-4653	219	8	,	,	PUNCT
ejpam-4653	219	9	m	m	VERB
ejpam-4653	219	10	is	be	AUX
ejpam-4653	219	11	obtained	obtain	VERB
ejpam-4653	219	12	by	by	ADP
ejpam-4653	219	13	joining	join	VERB
ejpam-4653	219	14	a	a	DET
ejpam-4653	219	15	cycle	cycle	NOUN
ejpam-4653	219	16	graph	graph	NOUN
ejpam-4653	219	17	cn	cn	PROPN
ejpam-4653	219	18	and	and	CCONJ
ejpam-4653	219	19	a	a	DET
ejpam-4653	219	20	path	path	NOUN
ejpam-4653	219	21	pm	pm	NOUN
ejpam-4653	219	22	with	with	ADP
ejpam-4653	219	23	a	a	DET
ejpam-4653	219	24	bridge	bridge	NOUN
ejpam-4653	219	25	.	.	PUNCT
ejpam-4653	220	1	the	the	DET
ejpam-4653	220	2	graph	graph	NOUN
ejpam-4653	220	3	in	in	ADP
ejpam-4653	220	4	figure	figure	NOUN
ejpam-4653	220	5	2	2	NUM
ejpam-4653	220	6	is	be	AUX
ejpam-4653	220	7	the	the	DET
ejpam-4653	220	8	tadpole	tadpole	NOUN
ejpam-4653	220	9	t6,3	t6,3	PROPN
ejpam-4653	220	10	.	.	PUNCT
ejpam-4653	220	11	....................................	....................................	PUNCT
ejpam-4653	220	12	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-4653	221	1	........................................................................................................	........................................................................................................	PUNCT
ejpam-4653	221	2	....................................	....................................	PUNCT
ejpam-4653	222	1	..........	..........	PUNCT
ejpam-4653	222	2	.........	.........	PUNCT
ejpam-4653	223	1	.........	.........	PUNCT
ejpam-4653	223	2	.........	.........	PUNCT
ejpam-4653	224	1	.........	.........	PUNCT
ejpam-4653	224	2	.........	.........	PUNCT
ejpam-4653	225	1	.........	.........	PUNCT
ejpam-4653	225	2	.........	.........	PUNCT
ejpam-4653	226	1	.........	.........	PUNCT
ejpam-4653	226	2	.........	.........	PUNCT
ejpam-4653	227	1	.........	.........	PUNCT
ejpam-4653	227	2	....	....	PUNCT
ejpam-4653	228	1	....................................	....................................	PUNCT
ejpam-4653	228	2	..........	..........	PUNCT
ejpam-4653	229	1	.........	.........	PUNCT
ejpam-4653	229	2	.........	.........	PUNCT
ejpam-4653	230	1	.........	.........	PUNCT
ejpam-4653	230	2	.........	.........	PUNCT
ejpam-4653	231	1	.........	.........	PUNCT
ejpam-4653	231	2	.........	.........	PUNCT
ejpam-4653	232	1	.........	.........	PUNCT
ejpam-4653	232	2	.........	.........	PUNCT
ejpam-4653	233	1	.........	.........	PUNCT
ejpam-4653	233	2	.........	.........	PUNCT
ejpam-4653	234	1	....	....	PUNCT
ejpam-4653	234	2	....................................	....................................	PUNCT
ejpam-4653	235	1	........................................................................................................	........................................................................................................	PUNCT
ejpam-4653	235	2	............................................................................................................................................	............................................................................................................................................	PUNCT
ejpam-4653	236	1	....................................	....................................	PUNCT
ejpam-4653	236	2	....................................	....................................	PUNCT
ejpam-4653	237	1	....................................	....................................	PUNCT
ejpam-4653	237	2	....................................................................................................................................................	....................................................................................................................................................	PUNCT
ejpam-4653	238	1	................................................................................................................	................................................................................................................	PUNCT
ejpam-4653	238	2	................................................................................................................	................................................................................................................	PUNCT
ejpam-4653	239	1	....................................	....................................	PUNCT
ejpam-4653	239	2	figure	figure	NOUN
ejpam-4653	239	3	2	2	NUM
ejpam-4653	239	4	:	:	PUNCT
ejpam-4653	239	5	the	the	DET
ejpam-4653	239	6	tadpole	tadpole	NOUN
ejpam-4653	239	7	t6,3	t6,3	PROPN
ejpam-4653	239	8	proposition	proposition	NOUN
ejpam-4653	239	9	11	11	NUM
ejpam-4653	239	10	.	.	PUNCT
ejpam-4653	240	1	for	for	ADP
ejpam-4653	240	2	tn,1	tn,1	PROPN
ejpam-4653	240	3	with	with	ADP
ejpam-4653	240	4	n	n	PRON
ejpam-4653	240	5	≥	≥	NUM
ejpam-4653	240	6	3	3	NUM
ejpam-4653	240	7	,	,	PUNCT
ejpam-4653	240	8	n	n	PRON
ejpam-4653	240	9	≤	≤	NOUN
ejpam-4653	240	10	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	240	11	)	)	PUNCT
ejpam-4653	240	12	≤	≤	NOUN
ejpam-4653	240	13	n+	n+	PUNCT
ejpam-4653	241	1	1	1	X
ejpam-4653	241	2	.	.	X
ejpam-4653	241	3	more	more	ADV
ejpam-4653	241	4	precisely	precisely	ADV
ejpam-4653	241	5	,	,	PUNCT
ejpam-4653	241	6	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	241	7	)	)	PUNCT
ejpam-4653	241	8	=	=	PRON
ejpam-4653	241	9	{	{	PUNCT
ejpam-4653	241	10	n	n	CCONJ
ejpam-4653	241	11	,	,	PUNCT
ejpam-4653	241	12	n	n	CCONJ
ejpam-4653	241	13	≡	≡	PROPN
ejpam-4653	241	14	0	0	NUM
ejpam-4653	241	15	,	,	PUNCT
ejpam-4653	241	16	3	3	NUM
ejpam-4653	241	17	(	(	PUNCT
ejpam-4653	241	18	mod	mod	PROPN
ejpam-4653	241	19	6	6	NUM
ejpam-4653	241	20	)	)	PUNCT
ejpam-4653	241	21	;	;	PUNCT
ejpam-4653	241	22	n+	n+	PUNCT
ejpam-4653	241	23	1	1	NUM
ejpam-4653	241	24	,	,	PUNCT
ejpam-4653	241	25	n	n	PRON
ejpam-4653	241	26	≡	≡	PROPN
ejpam-4653	241	27	1	1	NUM
ejpam-4653	241	28	,	,	PUNCT
ejpam-4653	241	29	2	2	NUM
ejpam-4653	241	30	,	,	PUNCT
ejpam-4653	241	31	4	4	NUM
ejpam-4653	241	32	,	,	PUNCT
ejpam-4653	241	33	5	5	NUM
ejpam-4653	241	34	(	(	PUNCT
ejpam-4653	241	35	mod	mod	PROPN
ejpam-4653	241	36	6	6	NUM
ejpam-4653	241	37	)	)	PUNCT
ejpam-4653	241	38	.	.	PUNCT
ejpam-4653	242	1	proof	proof	NOUN
ejpam-4653	242	2	.	.	PUNCT
ejpam-4653	243	1	let	let	VERB
ejpam-4653	243	2	v	v	NUM
ejpam-4653	243	3	∈	∈	PROPN
ejpam-4653	243	4	v	v	NOUN
ejpam-4653	243	5	(	(	PUNCT
ejpam-4653	243	6	cn	cn	PROPN
ejpam-4653	243	7	)	)	PUNCT
ejpam-4653	243	8	be	be	AUX
ejpam-4653	243	9	the	the	DET
ejpam-4653	243	10	vertex	vertex	NOUN
ejpam-4653	243	11	that	that	PRON
ejpam-4653	243	12	connects	connect	VERB
ejpam-4653	243	13	cn	cn	PROPN
ejpam-4653	243	14	to	to	ADP
ejpam-4653	243	15	p1	p1	PROPN
ejpam-4653	243	16	=	=	SYM
ejpam-4653	243	17	{	{	PUNCT
ejpam-4653	243	18	u	u	NOUN
ejpam-4653	243	19	}	}	PUNCT
ejpam-4653	243	20	and	and	CCONJ
ejpam-4653	243	21	let	let	VERB
ejpam-4653	243	22	f	f	PROPN
ejpam-4653	243	23	=	=	SYM
ejpam-4653	243	24	(	(	PUNCT
ejpam-4653	243	25	v0,∅	v0,∅	PROPN
ejpam-4653	243	26	,	,	PUNCT
ejpam-4653	243	27	v2	v2	PROPN
ejpam-4653	243	28	,	,	PUNCT
ejpam-4653	243	29	v3	v3	PROPN
ejpam-4653	243	30	)	)	PUNCT
ejpam-4653	243	31	be	be	VERB
ejpam-4653	243	32	a	a	DET
ejpam-4653	243	33	γdr	γdr	NOUN
ejpam-4653	243	34	-	-	PUNCT
ejpam-4653	243	35	function	function	NOUN
ejpam-4653	243	36	of	of	ADP
ejpam-4653	243	37	cn	cn	PROPN
ejpam-4653	243	38	.	.	PUNCT
ejpam-4653	244	1	we	we	PRON
ejpam-4653	244	2	may	may	AUX
ejpam-4653	244	3	assume	assume	VERB
ejpam-4653	244	4	that	that	SCONJ
ejpam-4653	244	5	v	v	X
ejpam-4653	244	6	/∈	/∈	PROPN
ejpam-4653	244	7	v0	v0	PROPN
ejpam-4653	244	8	.	.	PUNCT
ejpam-4653	245	1	if	if	SCONJ
ejpam-4653	245	2	v	v	NUM
ejpam-4653	245	3	∈	∈	PROPN
ejpam-4653	245	4	v3	v3	PROPN
ejpam-4653	245	5	,	,	PUNCT
ejpam-4653	245	6	then	then	ADV
ejpam-4653	245	7	g	g	PROPN
ejpam-4653	245	8	=	=	SYM
ejpam-4653	245	9	(	(	PUNCT
ejpam-4653	245	10	v0	v0	NOUN
ejpam-4653	245	11	∪	∪	NOUN
ejpam-4653	245	12	{	{	PUNCT
ejpam-4653	245	13	u},∅	u},∅	PROPN
ejpam-4653	245	14	,	,	PUNCT
ejpam-4653	245	15	v2	v2	PROPN
ejpam-4653	245	16	,	,	PUNCT
ejpam-4653	245	17	v3	v3	PROPN
ejpam-4653	245	18	)	)	PUNCT
ejpam-4653	245	19	∈	∈	PROPN
ejpam-4653	245	20	drd(tn,1	drd(tn,1	PROPN
ejpam-4653	245	21	)	)	PUNCT
ejpam-4653	245	22	.	.	PUNCT
ejpam-4653	246	1	if	if	SCONJ
ejpam-4653	246	2	v	v	NUM
ejpam-4653	246	3	∈	∈	PROPN
ejpam-4653	246	4	v2	v2	NOUN
ejpam-4653	246	5	,	,	PUNCT
ejpam-4653	246	6	then	then	ADV
ejpam-4653	246	7	g	g	PROPN
ejpam-4653	246	8	=	=	SYM
ejpam-4653	246	9	(	(	PUNCT
ejpam-4653	246	10	v0	v0	PROPN
ejpam-4653	246	11	,	,	PUNCT
ejpam-4653	246	12	{	{	PUNCT
ejpam-4653	246	13	u	u	NOUN
ejpam-4653	246	14	}	}	PUNCT
ejpam-4653	246	15	,	,	PUNCT
ejpam-4653	246	16	v2	v2	PROPN
ejpam-4653	246	17	,	,	PUNCT
ejpam-4653	246	18	v3	v3	PROPN
ejpam-4653	246	19	)	)	PUNCT
ejpam-4653	246	20	∈	∈	PROPN
ejpam-4653	246	21	drd(tn,1	drd(tn,1	PROPN
ejpam-4653	246	22	)	)	PUNCT
ejpam-4653	246	23	.	.	PUNCT
ejpam-4653	247	1	in	in	ADP
ejpam-4653	247	2	any	any	DET
ejpam-4653	247	3	case	case	NOUN
ejpam-4653	247	4	,	,	PUNCT
ejpam-4653	247	5	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	247	6	)	)	PUNCT
ejpam-4653	247	7	≤	≤	NOUN
ejpam-4653	247	8	ωtn,1(g	ωtn,1(g	NUM
ejpam-4653	247	9	)	)	PUNCT
ejpam-4653	247	10	≤	≤	NUM
ejpam-4653	247	11	1	1	NUM
ejpam-4653	247	12	+	+	NUM
ejpam-4653	247	13	γdr(cn	γdr(cn	NOUN
ejpam-4653	247	14	)	)	PUNCT
ejpam-4653	247	15	.	.	PUNCT
ejpam-4653	248	1	now	now	ADV
ejpam-4653	248	2	,	,	PUNCT
ejpam-4653	248	3	let	let	VERB
ejpam-4653	248	4	f	f	PROPN
ejpam-4653	248	5	=	=	SYM
ejpam-4653	248	6	(	(	PUNCT
ejpam-4653	248	7	v0,∅	v0,∅	PROPN
ejpam-4653	248	8	,	,	PUNCT
ejpam-4653	248	9	v2	v2	PROPN
ejpam-4653	248	10	,	,	PUNCT
ejpam-4653	248	11	v3	v3	PROPN
ejpam-4653	248	12	)	)	PUNCT
ejpam-4653	248	13	be	be	VERB
ejpam-4653	248	14	a	a	DET
ejpam-4653	248	15	γdr	γdr	NOUN
ejpam-4653	248	16	-	-	PUNCT
ejpam-4653	248	17	function	function	NOUN
ejpam-4653	248	18	of	of	ADP
ejpam-4653	248	19	tn,1	tn,1	PROPN
ejpam-4653	248	20	.	.	PUNCT
ejpam-4653	249	1	if	if	SCONJ
ejpam-4653	249	2	v	v	NUM
ejpam-4653	249	3	/∈	/∈	SYM
ejpam-4653	249	4	v0	v0	PROPN
ejpam-4653	249	5	,	,	PUNCT
ejpam-4653	249	6	then	then	ADV
ejpam-4653	249	7	v	v	ADP
ejpam-4653	249	8	∈	∈	PROPN
ejpam-4653	249	9	v3	v3	PROPN
ejpam-4653	249	10	and	and	CCONJ
ejpam-4653	249	11	u	u	PROPN
ejpam-4653	249	12	∈	∈	PROPN
ejpam-4653	249	13	v0	v0	NOUN
ejpam-4653	249	14	so	so	SCONJ
ejpam-4653	249	15	that	that	SCONJ
ejpam-4653	249	16	g	g	NOUN
ejpam-4653	249	17	=	=	SYM
ejpam-4653	249	18	(	(	PUNCT
ejpam-4653	249	19	v0	v0	PROPN
ejpam-4653	249	20	\	\	PROPN
ejpam-4653	249	21	{	{	PUNCT
ejpam-4653	249	22	u},∅	u},∅	PROPN
ejpam-4653	249	23	,	,	PUNCT
ejpam-4653	249	24	v2	v2	PROPN
ejpam-4653	249	25	,	,	PUNCT
ejpam-4653	249	26	v3	v3	PROPN
ejpam-4653	249	27	)	)	PUNCT
ejpam-4653	249	28	∈	∈	PROPN
ejpam-4653	249	29	drd(cn	drd(cn	NOUN
ejpam-4653	249	30	)	)	PUNCT
ejpam-4653	249	31	.	.	PUNCT
ejpam-4653	250	1	if	if	SCONJ
ejpam-4653	250	2	v	v	NUM
ejpam-4653	250	3	∈	∈	PROPN
ejpam-4653	250	4	v0	v0	NOUN
ejpam-4653	250	5	and	and	CCONJ
ejpam-4653	250	6	u	u	NOUN
ejpam-4653	250	7	∈	∈	PROPN
ejpam-4653	250	8	v2	v2	PROPN
ejpam-4653	250	9	,	,	PUNCT
ejpam-4653	250	10	then	then	ADV
ejpam-4653	250	11	g	g	PROPN
ejpam-4653	250	12	=	=	SYM
ejpam-4653	250	13	(	(	PUNCT
ejpam-4653	250	14	v0	v0	PROPN
ejpam-4653	250	15	\	\	PROPN
ejpam-4653	250	16	{	{	PUNCT
ejpam-4653	250	17	v},∅	v},∅	PROPN
ejpam-4653	250	18	,	,	PUNCT
ejpam-4653	250	19	(	(	PUNCT
ejpam-4653	250	20	v2	v2	PROPN
ejpam-4653	250	21	∩	∩	ADJ
ejpam-4653	250	22	v	v	NOUN
ejpam-4653	250	23	(	(	PUNCT
ejpam-4653	250	24	cn	cn	NOUN
ejpam-4653	250	25	)	)	PUNCT
ejpam-4653	250	26	)	)	PUNCT
ejpam-4653	250	27	∪	∪	ADP
ejpam-4653	250	28	{	{	PUNCT
ejpam-4653	250	29	v	v	NOUN
ejpam-4653	250	30	}	}	PUNCT
ejpam-4653	250	31	,	,	PUNCT
ejpam-4653	250	32	v3	v3	PROPN
ejpam-4653	250	33	)	)	PUNCT
ejpam-4653	250	34	∈	∈	PROPN
ejpam-4653	250	35	drd(cn	drd(cn	NOUN
ejpam-4653	250	36	)	)	PUNCT
ejpam-4653	250	37	.	.	PUNCT
ejpam-4653	251	1	and	and	CCONJ
ejpam-4653	251	2	,	,	PUNCT
ejpam-4653	251	3	if	if	SCONJ
ejpam-4653	251	4	v	v	NUM
ejpam-4653	251	5	∈	∈	PROPN
ejpam-4653	251	6	v0	v0	NOUN
ejpam-4653	251	7	and	and	CCONJ
ejpam-4653	251	8	u	u	PROPN
ejpam-4653	251	9	∈	∈	PROPN
ejpam-4653	251	10	v3	v3	PROPN
ejpam-4653	251	11	,	,	PUNCT
ejpam-4653	251	12	then	then	ADV
ejpam-4653	251	13	g	g	PROPN
ejpam-4653	251	14	=	=	SYM
ejpam-4653	251	15	(	(	PUNCT
ejpam-4653	251	16	v0	v0	PROPN
ejpam-4653	251	17	\	\	PROPN
ejpam-4653	251	18	{	{	PUNCT
ejpam-4653	251	19	v},∅	v},∅	PROPN
ejpam-4653	251	20	,	,	PUNCT
ejpam-4653	251	21	v2	v2	PROPN
ejpam-4653	251	22	,	,	PUNCT
ejpam-4653	251	23	(	(	PUNCT
ejpam-4653	251	24	v3	v3	PROPN
ejpam-4653	251	25	∩	∩	PROPN
ejpam-4653	251	26	v	v	X
ejpam-4653	251	27	(	(	PUNCT
ejpam-4653	251	28	cn	cn	NOUN
ejpam-4653	251	29	)	)	PUNCT
ejpam-4653	251	30	)	)	PUNCT
ejpam-4653	251	31	∪	∪	ADP
ejpam-4653	251	32	{	{	PUNCT
ejpam-4653	251	33	v	v	NOUN
ejpam-4653	251	34	}	}	PUNCT
ejpam-4653	251	35	)	)	PUNCT
ejpam-4653	251	36	∈	∈	PROPN
ejpam-4653	251	37	drd(cn	drd(cn	NOUN
ejpam-4653	251	38	)	)	PUNCT
ejpam-4653	251	39	.	.	PUNCT
ejpam-4653	252	1	in	in	ADP
ejpam-4653	252	2	any	any	DET
ejpam-4653	252	3	case	case	NOUN
ejpam-4653	252	4	,	,	PUNCT
ejpam-4653	252	5	γdr(cn	γdr(cn	NOUN
ejpam-4653	252	6	)	)	PUNCT
ejpam-4653	252	7	≤	≤	NOUN
ejpam-4653	252	8	ωcn(g	ωcn(g	X
ejpam-4653	252	9	)	)	PUNCT
ejpam-4653	252	10	=	=	SYM
ejpam-4653	252	11	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	252	12	)	)	PUNCT
ejpam-4653	252	13	.	.	PUNCT
ejpam-4653	253	1	hence	hence	ADV
ejpam-4653	253	2	,	,	PUNCT
ejpam-4653	253	3	the	the	DET
ejpam-4653	253	4	desired	desire	VERB
ejpam-4653	253	5	inequalities	inequality	NOUN
ejpam-4653	253	6	hold	hold	VERB
ejpam-4653	253	7	.	.	PUNCT
ejpam-4653	254	1	suppose	suppose	VERB
ejpam-4653	254	2	that	that	SCONJ
ejpam-4653	254	3	n	n	PROPN
ejpam-4653	254	4	≡	≡	PROPN
ejpam-4653	254	5	0	0	NUM
ejpam-4653	254	6	,	,	PUNCT
ejpam-4653	254	7	3	3	NUM
ejpam-4653	254	8	(	(	PUNCT
ejpam-4653	254	9	mod	mod	PROPN
ejpam-4653	254	10	6	6	NUM
ejpam-4653	254	11	)	)	PUNCT
ejpam-4653	254	12	.	.	PUNCT
ejpam-4653	255	1	then	then	ADV
ejpam-4653	255	2	γdr(cn	γdr(cn	NOUN
ejpam-4653	255	3	)	)	PUNCT
ejpam-4653	255	4	=	=	SYM
ejpam-4653	256	1	n	n	CCONJ
ejpam-4653	256	2	(	(	PUNCT
ejpam-4653	256	3	by	by	ADP
ejpam-4653	256	4	proposition	proposition	NOUN
ejpam-4653	256	5	2.2.3	2.2.3	NUM
ejpam-4653	256	6	)	)	PUNCT
ejpam-4653	257	1	and	and	CCONJ
ejpam-4653	257	2	cn	cn	PROPN
ejpam-4653	257	3	has	have	VERB
ejpam-4653	257	4	a	a	DET
ejpam-4653	257	5	γdr	γdr	NOUN
ejpam-4653	257	6	-	-	PUNCT
ejpam-4653	257	7	function	function	NOUN
ejpam-4653	257	8	f	f	NOUN
ejpam-4653	257	9	=	=	SYM
ejpam-4653	257	10	(	(	PUNCT
ejpam-4653	257	11	v0,∅	v0,∅	PROPN
ejpam-4653	257	12	,	,	PUNCT
ejpam-4653	257	13	v2	v2	PROPN
ejpam-4653	257	14	,	,	PUNCT
ejpam-4653	257	15	v3	v3	PROPN
ejpam-4653	257	16	)	)	PUNCT
ejpam-4653	257	17	with	with	ADP
ejpam-4653	257	18	v3	v3	PROPN
ejpam-4653	257	19	̸=	̸=	PROPN
ejpam-4653	257	20	∅.	∅.	NOUN
ejpam-4653	257	21	by	by	ADP
ejpam-4653	257	22	symmetry	symmetry	NOUN
ejpam-4653	257	23	,	,	PUNCT
ejpam-4653	257	24	we	we	PRON
ejpam-4653	257	25	assume	assume	VERB
ejpam-4653	257	26	that	that	SCONJ
ejpam-4653	257	27	v	v	ADP
ejpam-4653	257	28	∈	∈	PROPN
ejpam-4653	257	29	v3	v3	PROPN
ejpam-4653	257	30	.	.	PUNCT
ejpam-4653	258	1	thus	thus	ADV
ejpam-4653	258	2	,	,	PUNCT
ejpam-4653	258	3	g	g	PROPN
ejpam-4653	258	4	=	=	SYM
ejpam-4653	258	5	(	(	PUNCT
ejpam-4653	258	6	v0	v0	NOUN
ejpam-4653	258	7	∪	∪	NOUN
ejpam-4653	258	8	{	{	PUNCT
ejpam-4653	258	9	u},∅	u},∅	PROPN
ejpam-4653	258	10	,	,	PUNCT
ejpam-4653	258	11	v2	v2	PROPN
ejpam-4653	258	12	,	,	PUNCT
ejpam-4653	258	13	v3	v3	PROPN
ejpam-4653	258	14	)	)	PUNCT
ejpam-4653	258	15	∈	∈	PROPN
ejpam-4653	258	16	drd(tn,1	drd(tn,1	PROPN
ejpam-4653	258	17	)	)	PUNCT
ejpam-4653	258	18	.	.	PUNCT
ejpam-4653	259	1	together	together	ADV
ejpam-4653	259	2	with	with	ADP
ejpam-4653	259	3	the	the	DET
ejpam-4653	259	4	inequality	inequality	NOUN
ejpam-4653	259	5	,	,	PUNCT
ejpam-4653	259	6	n	n	CCONJ
ejpam-4653	259	7	≤	≤	NOUN
ejpam-4653	259	8	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	259	9	)	)	PUNCT
ejpam-4653	259	10	≤	≤	NOUN
ejpam-4653	259	11	ωtn,1(g	ωtn,1(g	NUM
ejpam-4653	259	12	)	)	PUNCT
ejpam-4653	259	13	=	=	SYM
ejpam-4653	259	14	γdr(cn	γdr(cn	NOUN
ejpam-4653	259	15	)	)	PUNCT
ejpam-4653	259	16	=	=	SYM
ejpam-4653	259	17	n.	n.	NOUN
ejpam-4653	259	18	therefore	therefore	ADV
ejpam-4653	259	19	,	,	PUNCT
ejpam-4653	259	20	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	259	21	)	)	PUNCT
ejpam-4653	260	1	=	=	SYM
ejpam-4653	260	2	n.	n.	NOUN
ejpam-4653	260	3	suppose	suppose	VERB
ejpam-4653	260	4	that	that	SCONJ
ejpam-4653	260	5	n	n	X
ejpam-4653	260	6	≡	≡	PROPN
ejpam-4653	260	7	2	2	NUM
ejpam-4653	260	8	,	,	PUNCT
ejpam-4653	260	9	4	4	NUM
ejpam-4653	260	10	(	(	PUNCT
ejpam-4653	260	11	mod	mod	PROPN
ejpam-4653	260	12	6	6	NUM
ejpam-4653	260	13	)	)	PUNCT
ejpam-4653	260	14	.	.	PUNCT
ejpam-4653	261	1	then	then	ADV
ejpam-4653	261	2	γdr(cn	γdr(cn	NOUN
ejpam-4653	261	3	)	)	PUNCT
ejpam-4653	261	4	=	=	SYM
ejpam-4653	261	5	n	n	PROPN
ejpam-4653	261	6	and	and	CCONJ
ejpam-4653	261	7	v3	v3	PROPN
ejpam-4653	261	8	=	=	PUNCT
ejpam-4653	261	9	∅	∅	NOUN
ejpam-4653	261	10	for	for	ADP
ejpam-4653	261	11	all	all	DET
ejpam-4653	261	12	γdr	γdr	NOUN
ejpam-4653	261	13	-	-	PUNCT
ejpam-4653	261	14	functions	function	NOUN
ejpam-4653	261	15	f	f	NOUN
ejpam-4653	261	16	=	=	SYM
ejpam-4653	261	17	(	(	PUNCT
ejpam-4653	261	18	v0,∅	v0,∅	PROPN
ejpam-4653	261	19	,	,	PUNCT
ejpam-4653	261	20	v2	v2	PROPN
ejpam-4653	261	21	,	,	PUNCT
ejpam-4653	261	22	v3	v3	PROPN
ejpam-4653	261	23	)	)	PUNCT
ejpam-4653	261	24	of	of	ADP
ejpam-4653	261	25	cn	cn	PROPN
ejpam-4653	261	26	.	.	PUNCT
ejpam-4653	262	1	with	with	ADP
ejpam-4653	262	2	v	v	NUM
ejpam-4653	262	3	∈	∈	PROPN
ejpam-4653	262	4	v2	v2	NOUN
ejpam-4653	262	5	,	,	PUNCT
ejpam-4653	262	6	g	g	NOUN
ejpam-4653	262	7	=	=	SYM
ejpam-4653	262	8	(	(	PUNCT
ejpam-4653	262	9	v0	v0	PROPN
ejpam-4653	262	10	,	,	PUNCT
ejpam-4653	262	11	{	{	PUNCT
ejpam-4653	262	12	u	u	NOUN
ejpam-4653	262	13	}	}	PUNCT
ejpam-4653	262	14	,	,	PUNCT
ejpam-4653	262	15	v2	v2	PROPN
ejpam-4653	262	16	,	,	PUNCT
ejpam-4653	262	17	v3	v3	PROPN
ejpam-4653	262	18	)	)	PUNCT
ejpam-4653	262	19	is	be	AUX
ejpam-4653	262	20	a	a	DET
ejpam-4653	262	21	γdr	γdr	NOUN
ejpam-4653	262	22	-	-	PUNCT
ejpam-4653	262	23	function	function	NOUN
ejpam-4653	262	24	of	of	ADP
ejpam-4653	262	25	tn,1	tn,1	PROPN
ejpam-4653	262	26	.	.	PUNCT
ejpam-4653	263	1	thus	thus	ADV
ejpam-4653	263	2	,	,	PUNCT
ejpam-4653	263	3	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	263	4	)	)	PUNCT
ejpam-4653	263	5	=	=	PUNCT
ejpam-4653	263	6	n+	n+	PUNCT
ejpam-4653	263	7	1	1	X
ejpam-4653	263	8	.	.	PUNCT
ejpam-4653	263	9	j.	j.	PROPN
ejpam-4653	263	10	b.g	b.g	PROPN
ejpam-4653	263	11	.	.	PROPN
ejpam-4653	263	12	cariaga	cariaga	PROPN
ejpam-4653	263	13	,	,	PUNCT
ejpam-4653	263	14	f.	f.	PROPN
ejpam-4653	263	15	jamil	jamil	PROPN
ejpam-4653	263	16	/	/	SYM
ejpam-4653	263	17	eur	eur	PROPN
ejpam-4653	263	18	.	.	PUNCT
ejpam-4653	264	1	j.	j.	PROPN
ejpam-4653	264	2	pure	pure	PROPN
ejpam-4653	264	3	appl	appl	PROPN
ejpam-4653	264	4	.	.	PROPN
ejpam-4653	264	5	math	math	PROPN
ejpam-4653	264	6	,	,	PUNCT
ejpam-4653	264	7	16	16	NUM
ejpam-4653	264	8	(	(	PUNCT
ejpam-4653	264	9	2	2	NUM
ejpam-4653	264	10	)	)	PUNCT
ejpam-4653	264	11	(	(	PUNCT
ejpam-4653	264	12	2023	2023	NUM
ejpam-4653	264	13	)	)	PUNCT
ejpam-4653	264	14	,	,	PUNCT
ejpam-4653	264	15	847	847	NUM
ejpam-4653	264	16	-	-	SYM
ejpam-4653	264	17	863	863	NUM
ejpam-4653	264	18	853	853	NUM
ejpam-4653	264	19	finally	finally	ADV
ejpam-4653	264	20	,	,	PUNCT
ejpam-4653	264	21	suppose	suppose	VERB
ejpam-4653	264	22	that	that	SCONJ
ejpam-4653	264	23	n	n	X
ejpam-4653	264	24	≡	≡	PROPN
ejpam-4653	264	25	1	1	NUM
ejpam-4653	264	26	,	,	PUNCT
ejpam-4653	264	27	5	5	NUM
ejpam-4653	264	28	(	(	PUNCT
ejpam-4653	264	29	mod	mod	NOUN
ejpam-4653	264	30	3	3	NUM
ejpam-4653	264	31	)	)	PUNCT
ejpam-4653	264	32	.	.	PUNCT
ejpam-4653	265	1	then	then	ADV
ejpam-4653	265	2	γdr(cn	γdr(cn	NOUN
ejpam-4653	265	3	)	)	PUNCT
ejpam-4653	265	4	=	=	SYM
ejpam-4653	266	1	n	n	PROPN
ejpam-4653	266	2	+	+	CCONJ
ejpam-4653	266	3	1	1	NUM
ejpam-4653	267	1	and	and	CCONJ
ejpam-4653	267	2	cn	cn	PROPN
ejpam-4653	267	3	has	have	VERB
ejpam-4653	267	4	a	a	DET
ejpam-4653	267	5	γdrfunction	γdrfunction	NOUN
ejpam-4653	268	1	f	f	PROPN
ejpam-4653	269	1	=	=	SYM
ejpam-4653	269	2	(	(	PUNCT
ejpam-4653	269	3	v0,∅	v0,∅	PROPN
ejpam-4653	269	4	,	,	PUNCT
ejpam-4653	269	5	v2	v2	PROPN
ejpam-4653	269	6	,	,	PUNCT
ejpam-4653	269	7	v3	v3	PROPN
ejpam-4653	269	8	)	)	PUNCT
ejpam-4653	269	9	with	with	ADP
ejpam-4653	269	10	v3	v3	PROPN
ejpam-4653	269	11	̸=	̸=	PROPN
ejpam-4653	269	12	∅.	∅.	ADV
ejpam-4653	269	13	with	with	ADP
ejpam-4653	269	14	v	v	PROPN
ejpam-4653	269	15	∈	∈	PROPN
ejpam-4653	269	16	v3	v3	NOUN
ejpam-4653	269	17	,	,	PUNCT
ejpam-4653	269	18	g	g	PROPN
ejpam-4653	269	19	=	=	SYM
ejpam-4653	269	20	(	(	PUNCT
ejpam-4653	269	21	v0	v0	NOUN
ejpam-4653	269	22	∪	∪	NOUN
ejpam-4653	269	23	{	{	PUNCT
ejpam-4653	269	24	u},∅	u},∅	PROPN
ejpam-4653	269	25	,	,	PUNCT
ejpam-4653	269	26	v2	v2	PROPN
ejpam-4653	269	27	,	,	PUNCT
ejpam-4653	269	28	v3	v3	PROPN
ejpam-4653	269	29	)	)	PUNCT
ejpam-4653	269	30	∈	∈	PROPN
ejpam-4653	269	31	drd(tn,1	drd(tn,1	PROPN
ejpam-4653	269	32	)	)	PUNCT
ejpam-4653	269	33	.	.	PUNCT
ejpam-4653	270	1	thus	thus	ADV
ejpam-4653	270	2	,	,	PUNCT
ejpam-4653	270	3	n+	n+	X
ejpam-4653	270	4	1	1	NUM
ejpam-4653	270	5	=	=	SYM
ejpam-4653	270	6	γdr(cn	γdr(cn	NOUN
ejpam-4653	270	7	)	)	PUNCT
ejpam-4653	270	8	≤	≤	NOUN
ejpam-4653	270	9	γdr(tn,1	γdr(tn,1	PROPN
ejpam-4653	270	10	)	)	PUNCT
ejpam-4653	270	11	≤	≤	NOUN
ejpam-4653	270	12	ωtn,1(g	ωtn,1(g	NUM
ejpam-4653	270	13	)	)	PUNCT
ejpam-4653	270	14	=	=	SYM
ejpam-4653	270	15	ωcn(f	ωcn(f	X
ejpam-4653	270	16	)	)	PUNCT
ejpam-4653	270	17	=	=	PUNCT
ejpam-4653	270	18	n+	n+	PUNCT
ejpam-4653	271	1	1	1	X
ejpam-4653	271	2	.	.	X
ejpam-4653	271	3	proposition	proposition	NOUN
ejpam-4653	271	4	12	12	NUM
ejpam-4653	271	5	.	.	PUNCT
ejpam-4653	272	1	let	let	VERB
ejpam-4653	272	2	n	n	PRON
ejpam-4653	272	3	≥	≥	X
ejpam-4653	272	4	3	3	NUM
ejpam-4653	272	5	and	and	CCONJ
ejpam-4653	272	6	m	m	PROPN
ejpam-4653	272	7	≥	≥	NOUN
ejpam-4653	272	8	2	2	NUM
ejpam-4653	272	9	.	.	PUNCT
ejpam-4653	273	1	(	(	PUNCT
ejpam-4653	273	2	i	i	NOUN
ejpam-4653	273	3	)	)	PUNCT
ejpam-4653	273	4	if	if	SCONJ
ejpam-4653	273	5	n	n	PRON
ejpam-4653	273	6	≡	≡	PROPN
ejpam-4653	273	7	0	0	NUM
ejpam-4653	273	8	,	,	PUNCT
ejpam-4653	273	9	3	3	NUM
ejpam-4653	273	10	(	(	PUNCT
ejpam-4653	273	11	mod	mod	PROPN
ejpam-4653	273	12	6	6	NUM
ejpam-4653	273	13	)	)	PUNCT
ejpam-4653	273	14	,	,	PUNCT
ejpam-4653	273	15	then	then	ADV
ejpam-4653	273	16	γdr(tn	γdr(tn	NOUN
ejpam-4653	273	17	,	,	PUNCT
ejpam-4653	273	18	m	m	NOUN
ejpam-4653	273	19	)	)	PUNCT
ejpam-4653	274	1	=	=	PRON
ejpam-4653	274	2	{	{	PUNCT
ejpam-4653	274	3	n+	n+	NOUN
ejpam-4653	274	4	2	2	NUM
ejpam-4653	274	5	,	,	PUNCT
ejpam-4653	274	6	if	if	SCONJ
ejpam-4653	274	7	m	m	ADJ
ejpam-4653	274	8	=	=	SYM
ejpam-4653	274	9	2	2	NUM
ejpam-4653	274	10	n+	n+	PUNCT
ejpam-4653	274	11	γdr(pm−1	γdr(pm−1	NOUN
ejpam-4653	274	12	)	)	PUNCT
ejpam-4653	274	13	,	,	PUNCT
ejpam-4653	274	14	if	if	SCONJ
ejpam-4653	274	15	m	m	PROPN
ejpam-4653	274	16	≥	≥	NOUN
ejpam-4653	274	17	3	3	NUM
ejpam-4653	274	18	.	.	PUNCT
ejpam-4653	274	19	(	(	PUNCT
ejpam-4653	274	20	ii	ii	NOUN
ejpam-4653	274	21	)	)	PUNCT
ejpam-4653	274	22	if	if	SCONJ
ejpam-4653	274	23	n	n	PRON
ejpam-4653	274	24	≡	≡	PROPN
ejpam-4653	274	25	2	2	NUM
ejpam-4653	274	26	,	,	PUNCT
ejpam-4653	274	27	4	4	NUM
ejpam-4653	274	28	(	(	PUNCT
ejpam-4653	274	29	mod	mod	PROPN
ejpam-4653	274	30	6	6	NUM
ejpam-4653	274	31	)	)	PUNCT
ejpam-4653	274	32	,	,	PUNCT
ejpam-4653	274	33	then	then	ADV
ejpam-4653	274	34	γdr(tn	γdr(tn	NOUN
ejpam-4653	274	35	,	,	PUNCT
ejpam-4653	274	36	m	m	NOUN
ejpam-4653	274	37	)	)	PUNCT
ejpam-4653	275	1	=	=	SYM
ejpam-4653	275	2			PRON
ejpam-4653	275	3	n+	n+	PUNCT
ejpam-4653	275	4	2	2	NUM
ejpam-4653	275	5	,	,	PUNCT
ejpam-4653	275	6	if	if	SCONJ
ejpam-4653	275	7	m	m	ADJ
ejpam-4653	275	8	=	=	SYM
ejpam-4653	275	9	2	2	NUM
ejpam-4653	275	10	n+	n+	X
ejpam-4653	275	11	γdr(pm	γdr(pm	NOUN
ejpam-4653	275	12	)	)	PUNCT
ejpam-4653	275	13	,	,	PUNCT
ejpam-4653	275	14	if	if	SCONJ
ejpam-4653	275	15	m	m	PROPN
ejpam-4653	275	16	≥	≥	NOUN
ejpam-4653	275	17	3	3	NUM
ejpam-4653	275	18	;	;	PUNCT
ejpam-4653	275	19	m	m	NUM
ejpam-4653	275	20	≡	≡	PROPN
ejpam-4653	275	21	0	0	PUNCT
ejpam-4653	276	1	(	(	PUNCT
ejpam-4653	276	2	mod	mod	PROPN
ejpam-4653	276	3	3	3	NUM
ejpam-4653	276	4	)	)	PUNCT
ejpam-4653	276	5	,	,	PUNCT
ejpam-4653	276	6	n+	n+	PUNCT
ejpam-4653	276	7	γdr(pm)−	γdr(pm)−	NOUN
ejpam-4653	276	8	1	1	NUM
ejpam-4653	276	9	,	,	PUNCT
ejpam-4653	276	10	if	if	SCONJ
ejpam-4653	276	11	m	m	PROPN
ejpam-4653	276	12	≥	≥	NOUN
ejpam-4653	276	13	3	3	NUM
ejpam-4653	276	14	;	;	PUNCT
ejpam-4653	276	15	m	m	NUM
ejpam-4653	276	16	≡	≡	PROPN
ejpam-4653	276	17	1	1	NUM
ejpam-4653	276	18	,	,	PUNCT
ejpam-4653	276	19	2	2	NUM
ejpam-4653	276	20	(	(	PUNCT
ejpam-4653	276	21	mod	mod	NOUN
ejpam-4653	276	22	3	3	NUM
ejpam-4653	276	23	)	)	PUNCT
ejpam-4653	276	24	.	.	PUNCT
ejpam-4653	277	1	(	(	PUNCT
ejpam-4653	277	2	iii	iii	X
ejpam-4653	277	3	)	)	PUNCT
ejpam-4653	277	4	if	if	SCONJ
ejpam-4653	277	5	n	n	PRON
ejpam-4653	277	6	≡	≡	PROPN
ejpam-4653	277	7	1	1	NUM
ejpam-4653	277	8	,	,	PUNCT
ejpam-4653	277	9	5	5	NUM
ejpam-4653	277	10	(	(	PUNCT
ejpam-4653	277	11	mod	mod	PROPN
ejpam-4653	277	12	6	6	NUM
ejpam-4653	277	13	)	)	PUNCT
ejpam-4653	277	14	,	,	PUNCT
ejpam-4653	277	15	then	then	ADV
ejpam-4653	277	16	γdr(tn	γdr(tn	NOUN
ejpam-4653	277	17	,	,	PUNCT
ejpam-4653	277	18	m	m	NOUN
ejpam-4653	277	19	)	)	PUNCT
ejpam-4653	278	1	=	=	PRON
ejpam-4653	278	2	{	{	PUNCT
ejpam-4653	278	3	n+	n+	NOUN
ejpam-4653	278	4	3	3	NUM
ejpam-4653	278	5	,	,	PUNCT
ejpam-4653	278	6	if	if	SCONJ
ejpam-4653	278	7	m	m	ADJ
ejpam-4653	278	8	=	=	SYM
ejpam-4653	278	9	2	2	NUM
ejpam-4653	278	10	n+	n+	SYM
ejpam-4653	278	11	1	1	NUM
ejpam-4653	278	12	+	+	CCONJ
ejpam-4653	278	13	γdr(pm−1	γdr(pm−1	NOUN
ejpam-4653	278	14	)	)	PUNCT
ejpam-4653	278	15	,	,	PUNCT
ejpam-4653	278	16	if	if	SCONJ
ejpam-4653	278	17	m	m	PROPN
ejpam-4653	278	18	≥	≥	NOUN
ejpam-4653	278	19	3	3	NUM
ejpam-4653	278	20	.	.	PUNCT
ejpam-4653	279	1	proof	proof	NOUN
ejpam-4653	279	2	.	.	PUNCT
ejpam-4653	280	1	write	write	VERB
ejpam-4653	280	2	pm	pm	NOUN
ejpam-4653	280	3	=	=	PUNCT
ejpam-4653	281	1	[	[	X
ejpam-4653	281	2	v1	v1	NOUN
ejpam-4653	281	3	,	,	PUNCT
ejpam-4653	281	4	v2	v2	NOUN
ejpam-4653	281	5	,	,	PUNCT
ejpam-4653	281	6	.	.	PUNCT
ejpam-4653	281	7	.	.	PUNCT
ejpam-4653	281	8	.	.	PUNCT
ejpam-4653	282	1	,	,	PUNCT
ejpam-4653	282	2	vm	vm	PROPN
ejpam-4653	282	3	]	]	PUNCT
ejpam-4653	282	4	.	.	PUNCT
ejpam-4653	283	1	let	let	VERB
ejpam-4653	283	2	v	v	NUM
ejpam-4653	283	3	∈	∈	PROPN
ejpam-4653	283	4	v	v	NOUN
ejpam-4653	283	5	(	(	PUNCT
ejpam-4653	283	6	cn	cn	PROPN
ejpam-4653	283	7	)	)	PUNCT
ejpam-4653	283	8	be	be	AUX
ejpam-4653	283	9	the	the	DET
ejpam-4653	283	10	vertex	vertex	NOUN
ejpam-4653	283	11	that	that	PRON
ejpam-4653	283	12	connects	connect	VERB
ejpam-4653	283	13	cn	cn	PROPN
ejpam-4653	283	14	to	to	PART
ejpam-4653	283	15	pm	pm	VERB
ejpam-4653	283	16	through	through	ADP
ejpam-4653	283	17	the	the	DET
ejpam-4653	283	18	edge	edge	NOUN
ejpam-4653	283	19	vv1	vv1	NOUN
ejpam-4653	283	20	.	.	PUNCT
ejpam-4653	284	1	we	we	PRON
ejpam-4653	284	2	consider	consider	VERB
ejpam-4653	284	3	the	the	DET
ejpam-4653	284	4	following	follow	VERB
ejpam-4653	284	5	cases	case	NOUN
ejpam-4653	284	6	:	:	PUNCT
ejpam-4653	284	7	case	case	NOUN
ejpam-4653	284	8	1	1	NUM
ejpam-4653	284	9	:	:	PUNCT
ejpam-4653	284	10	suppose	suppose	VERB
ejpam-4653	284	11	that	that	SCONJ
ejpam-4653	284	12	n	n	PROPN
ejpam-4653	284	13	≡	≡	PROPN
ejpam-4653	284	14	0	0	NUM
ejpam-4653	284	15	,	,	PUNCT
ejpam-4653	284	16	3	3	NUM
ejpam-4653	284	17	(	(	PUNCT
ejpam-4653	284	18	mod	mod	PROPN
ejpam-4653	284	19	6	6	NUM
ejpam-4653	284	20	)	)	PUNCT
ejpam-4653	284	21	.	.	PUNCT
ejpam-4653	285	1	let	let	VERB
ejpam-4653	285	2	f	f	PROPN
ejpam-4653	285	3	=	=	SYM
ejpam-4653	285	4	(	(	PUNCT
ejpam-4653	285	5	v0,∅	v0,∅	PROPN
ejpam-4653	285	6	,	,	PUNCT
ejpam-4653	285	7	v2	v2	PROPN
ejpam-4653	285	8	,	,	PUNCT
ejpam-4653	285	9	v3	v3	PROPN
ejpam-4653	285	10	)	)	PUNCT
ejpam-4653	285	11	be	be	VERB
ejpam-4653	285	12	a	a	DET
ejpam-4653	285	13	γdr	γdr	NOUN
ejpam-4653	285	14	-	-	PUNCT
ejpam-4653	285	15	function	function	NOUN
ejpam-4653	285	16	of	of	ADP
ejpam-4653	285	17	cn	cn	PROPN
ejpam-4653	285	18	with	with	ADP
ejpam-4653	285	19	v3	v3	PROPN
ejpam-4653	285	20	̸=	̸=	PROPN
ejpam-4653	286	1	∅.	∅.	ADV
ejpam-4653	286	2	assume	assume	VERB
ejpam-4653	286	3	v	v	ADP
ejpam-4653	286	4	∈	∈	PROPN
ejpam-4653	286	5	v3	v3	PROPN
ejpam-4653	286	6	.	.	PUNCT
ejpam-4653	287	1	if	if	SCONJ
ejpam-4653	287	2	m	m	VERB
ejpam-4653	287	3	=	=	SYM
ejpam-4653	287	4	2	2	NUM
ejpam-4653	287	5	,	,	PUNCT
ejpam-4653	287	6	then	then	ADV
ejpam-4653	287	7	g	g	PROPN
ejpam-4653	287	8	=	=	SYM
ejpam-4653	287	9	(	(	PUNCT
ejpam-4653	287	10	v0	v0	NOUN
ejpam-4653	287	11	∪	∪	NOUN
ejpam-4653	287	12	{	{	PUNCT
ejpam-4653	287	13	v1},∅	v1},∅	NOUN
ejpam-4653	287	14	,	,	PUNCT
ejpam-4653	287	15	v2	v2	NOUN
ejpam-4653	287	16	∪	∪	X
ejpam-4653	287	17	{	{	PUNCT
ejpam-4653	287	18	v2	v2	NOUN
ejpam-4653	287	19	}	}	PUNCT
ejpam-4653	287	20	,	,	PUNCT
ejpam-4653	287	21	v3	v3	PROPN
ejpam-4653	287	22	)	)	PUNCT
ejpam-4653	287	23	is	be	AUX
ejpam-4653	287	24	a	a	DET
ejpam-4653	287	25	γdrfunction	γdrfunction	NOUN
ejpam-4653	287	26	of	of	ADP
ejpam-4653	287	27	tn,2	tn,2	PROPN
ejpam-4653	287	28	.	.	PUNCT
ejpam-4653	288	1	thus	thus	ADV
ejpam-4653	288	2	,	,	PUNCT
ejpam-4653	288	3	γdr(tn,2	γdr(tn,2	NUM
ejpam-4653	288	4	)	)	PUNCT
ejpam-4653	288	5	=	=	SYM
ejpam-4653	289	1	γdr(cn)+2	γdr(cn)+2	PROPN
ejpam-4653	289	2	=	=	SYM
ejpam-4653	289	3	n+2	n+2	PROPN
ejpam-4653	289	4	.	.	PUNCT
ejpam-4653	290	1	assumem	assumem	PROPN
ejpam-4653	290	2	≥	≥	NUM
ejpam-4653	290	3	3	3	NUM
ejpam-4653	290	4	,	,	PUNCT
ejpam-4653	290	5	and	and	CCONJ
ejpam-4653	290	6	letm	letm	NOUN
ejpam-4653	290	7	=	=	SYM
ejpam-4653	290	8	3k+r	3k+r	NUM
ejpam-4653	290	9	,	,	PUNCT
ejpam-4653	290	10	where	where	SCONJ
ejpam-4653	290	11	0	0	NUM
ejpam-4653	290	12	≤	≤	NUM
ejpam-4653	290	13	r	r	NOUN
ejpam-4653	290	14	≤	≤	NUM
ejpam-4653	290	15	2	2	NUM
ejpam-4653	290	16	.	.	PUNCT
ejpam-4653	290	17	put	put	VERB
ejpam-4653	290	18	v	v	NUM
ejpam-4653	290	19	∗	∗	NOUN
ejpam-4653	290	20	3	3	NUM
ejpam-4653	290	21	=	=	SYM
ejpam-4653	290	22	{	{	PUNCT
ejpam-4653	290	23	v3j	v3j	ADV
ejpam-4653	290	24	:	:	PUNCT
ejpam-4653	290	25	j	j	PROPN
ejpam-4653	290	26	∈	∈	PROPN
ejpam-4653	290	27	{	{	PUNCT
ejpam-4653	290	28	1	1	NUM
ejpam-4653	290	29	,	,	PUNCT
ejpam-4653	290	30	2	2	NUM
ejpam-4653	290	31	,	,	PUNCT
ejpam-4653	290	32	.	.	PUNCT
ejpam-4653	290	33	.	.	PUNCT
ejpam-4653	291	1	.	.	PUNCT
ejpam-4653	292	1	,	,	PUNCT
ejpam-4653	292	2	k	k	X
ejpam-4653	292	3	}	}	PUNCT
ejpam-4653	292	4	}	}	PUNCT
ejpam-4653	292	5	.	.	PUNCT
ejpam-4653	293	1	if	if	SCONJ
ejpam-4653	293	2	0	0	NUM
ejpam-4653	293	3	≤	≤	NUM
ejpam-4653	293	4	r	r	NOUN
ejpam-4653	293	5	≤	≤	NUM
ejpam-4653	293	6	1	1	NUM
ejpam-4653	293	7	,	,	PUNCT
ejpam-4653	293	8	put	put	VERB
ejpam-4653	293	9	v	v	NUM
ejpam-4653	293	10	∗	∗	NOUN
ejpam-4653	293	11	0	0	NUM
ejpam-4653	294	1	=	=	SYM
ejpam-4653	294	2	v	v	NOUN
ejpam-4653	294	3	(	(	PUNCT
ejpam-4653	294	4	pm	pm	NOUN
ejpam-4653	294	5	)	)	PUNCT
ejpam-4653	294	6	\	\	PROPN
ejpam-4653	294	7	v	v	ADP
ejpam-4653	294	8	∗	∗	NOUN
ejpam-4653	294	9	3	3	NUM
ejpam-4653	294	10	and	and	CCONJ
ejpam-4653	294	11	v	v	NOUN
ejpam-4653	294	12	∗	∗	NOUN
ejpam-4653	294	13	2	2	NUM
ejpam-4653	294	14	=	=	SYM
ejpam-4653	294	15	∅.	∅.	NOUN
ejpam-4653	294	16	on	on	ADP
ejpam-4653	294	17	the	the	DET
ejpam-4653	294	18	other	other	ADJ
ejpam-4653	294	19	hand	hand	NOUN
ejpam-4653	294	20	,	,	PUNCT
ejpam-4653	294	21	if	if	SCONJ
ejpam-4653	294	22	r	r	NOUN
ejpam-4653	294	23	=	=	SYM
ejpam-4653	294	24	2	2	NUM
ejpam-4653	294	25	,	,	PUNCT
ejpam-4653	294	26	put	put	VERB
ejpam-4653	294	27	v	v	NUM
ejpam-4653	294	28	∗	∗	NOUN
ejpam-4653	294	29	0	0	NUM
ejpam-4653	295	1	=	=	SYM
ejpam-4653	295	2	v	v	NOUN
ejpam-4653	295	3	(	(	PUNCT
ejpam-4653	295	4	pm	pm	NOUN
ejpam-4653	295	5	)	)	PUNCT
ejpam-4653	295	6	\	\	PUNCT
ejpam-4653	296	1	(	(	PUNCT
ejpam-4653	296	2	v	v	NOUN
ejpam-4653	296	3	∗	∗	NOUN
ejpam-4653	296	4	3	3	NUM
ejpam-4653	296	5	∪	∪	X
ejpam-4653	296	6	{	{	PUNCT
ejpam-4653	296	7	v3k+2	v3k+2	NOUN
ejpam-4653	296	8	}	}	PUNCT
ejpam-4653	296	9	)	)	PUNCT
ejpam-4653	296	10	and	and	CCONJ
ejpam-4653	296	11	v	v	ADP
ejpam-4653	296	12	∗	∗	NOUN
ejpam-4653	296	13	2	2	NUM
ejpam-4653	296	14	=	=	SYM
ejpam-4653	296	15	{	{	PUNCT
ejpam-4653	296	16	v3k+2	v3k+2	NOUN
ejpam-4653	296	17	}	}	PUNCT
ejpam-4653	296	18	.	.	PUNCT
ejpam-4653	297	1	since	since	SCONJ
ejpam-4653	297	2	(	(	PUNCT
ejpam-4653	297	3	v	v	NOUN
ejpam-4653	297	4	∗	∗	NOUN
ejpam-4653	297	5	0	0	NUM
ejpam-4653	297	6	\	\	NOUN
ejpam-4653	297	7	{	{	PUNCT
ejpam-4653	297	8	v1},∅	v1},∅	PROPN
ejpam-4653	297	9	,	,	PUNCT
ejpam-4653	297	10	v	v	NOUN
ejpam-4653	297	11	∗	∗	NOUN
ejpam-4653	297	12	2	2	NUM
ejpam-4653	297	13	,	,	PUNCT
ejpam-4653	297	14	v	v	NOUN
ejpam-4653	297	15	∗	∗	X
ejpam-4653	297	16	3	3	NUM
ejpam-4653	297	17	)	)	PUNCT
ejpam-4653	297	18	is	be	AUX
ejpam-4653	297	19	a	a	DET
ejpam-4653	297	20	γdr	γdr	NOUN
ejpam-4653	297	21	-	-	PUNCT
ejpam-4653	297	22	function	function	NOUN
ejpam-4653	297	23	of	of	ADP
ejpam-4653	297	24	pm	pm	NOUN
ejpam-4653	297	25	−	−	PROPN
ejpam-4653	297	26	v1	v1	NOUN
ejpam-4653	297	27	∼=	∼=	PART
ejpam-4653	297	28	pm−1	pm−1	NOUN
ejpam-4653	297	29	,	,	PUNCT
ejpam-4653	297	30	g	g	NOUN
ejpam-4653	297	31	=	=	PUNCT
ejpam-4653	297	32	(	(	PUNCT
ejpam-4653	297	33	v0∪v	v0∪v	ADV
ejpam-4653	297	34	∗	∗	X
ejpam-4653	297	35	0	0	NUM
ejpam-4653	297	36	,	,	PUNCT
ejpam-4653	297	37	∅	∅	NOUN
ejpam-4653	297	38	,	,	PUNCT
ejpam-4653	297	39	v2∪v	v2∪v	PROPN
ejpam-4653	297	40	∗	∗	NOUN
ejpam-4653	297	41	2	2	NUM
ejpam-4653	297	42	,	,	PUNCT
ejpam-4653	297	43	v3∪v	v3∪v	NUM
ejpam-4653	297	44	∗	∗	NOUN
ejpam-4653	297	45	3	3	NUM
ejpam-4653	297	46	)	)	PUNCT
ejpam-4653	297	47	is	be	AUX
ejpam-4653	297	48	a	a	DET
ejpam-4653	297	49	γdr	γdr	NOUN
ejpam-4653	297	50	-	-	PUNCT
ejpam-4653	297	51	function	function	NOUN
ejpam-4653	297	52	of	of	ADP
ejpam-4653	297	53	tn	tn	PROPN
ejpam-4653	297	54	,	,	PUNCT
ejpam-4653	297	55	m.	m.	NOUN
ejpam-4653	297	56	thus	thus	ADV
ejpam-4653	297	57	,	,	PUNCT
ejpam-4653	297	58	γdr(tn	γdr(tn	X
ejpam-4653	297	59	,	,	PUNCT
ejpam-4653	297	60	m	m	NOUN
ejpam-4653	297	61	)	)	PUNCT
ejpam-4653	297	62	=	=	SYM
ejpam-4653	297	63	n+γdr(pm−1	n+γdr(pm−1	NOUN
ejpam-4653	297	64	)	)	PUNCT
ejpam-4653	297	65	.	.	PUNCT
ejpam-4653	298	1	case	case	NOUN
ejpam-4653	298	2	2	2	NUM
ejpam-4653	298	3	:	:	PUNCT
ejpam-4653	298	4	suppose	suppose	VERB
ejpam-4653	298	5	that	that	SCONJ
ejpam-4653	298	6	n	n	X
ejpam-4653	298	7	≡	≡	PROPN
ejpam-4653	298	8	2	2	NUM
ejpam-4653	298	9	,	,	PUNCT
ejpam-4653	298	10	4	4	NUM
ejpam-4653	298	11	(	(	PUNCT
ejpam-4653	298	12	mod	mod	PROPN
ejpam-4653	298	13	6	6	NUM
ejpam-4653	298	14	)	)	PUNCT
ejpam-4653	298	15	.	.	PUNCT
ejpam-4653	299	1	let	let	VERB
ejpam-4653	299	2	f	f	PROPN
ejpam-4653	299	3	=	=	SYM
ejpam-4653	299	4	(	(	PUNCT
ejpam-4653	299	5	v0	v0	PROPN
ejpam-4653	299	6	,	,	PUNCT
ejpam-4653	299	7	v1	v1	NOUN
ejpam-4653	299	8	,	,	PUNCT
ejpam-4653	299	9	v2	v2	PROPN
ejpam-4653	299	10	,	,	PUNCT
ejpam-4653	299	11	v3	v3	PROPN
ejpam-4653	299	12	)	)	PUNCT
ejpam-4653	299	13	be	be	VERB
ejpam-4653	299	14	a	a	DET
ejpam-4653	299	15	γdr	γdr	NOUN
ejpam-4653	299	16	-	-	PUNCT
ejpam-4653	299	17	function	function	NOUN
ejpam-4653	299	18	of	of	ADP
ejpam-4653	299	19	cn	cn	PROPN
ejpam-4653	299	20	.	.	PUNCT
ejpam-4653	299	21	accordingly	accordingly	ADV
ejpam-4653	299	22	,	,	PUNCT
ejpam-4653	299	23	v3	v3	PROPN
ejpam-4653	299	24	=	=	NOUN
ejpam-4653	299	25	∅	∅	NOUN
ejpam-4653	300	1	and	and	CCONJ
ejpam-4653	300	2	we	we	PRON
ejpam-4653	300	3	may	may	AUX
ejpam-4653	300	4	assume	assume	VERB
ejpam-4653	300	5	that	that	SCONJ
ejpam-4653	300	6	v	v	X
ejpam-4653	300	7	∈	∈	PROPN
ejpam-4653	300	8	v2	v2	NOUN
ejpam-4653	300	9	.	.	PUNCT
ejpam-4653	301	1	if	if	SCONJ
ejpam-4653	301	2	m	m	NOUN
ejpam-4653	301	3	=	=	SYM
ejpam-4653	301	4	2	2	NUM
ejpam-4653	301	5	,	,	PUNCT
ejpam-4653	301	6	then	then	ADV
ejpam-4653	301	7	g	g	PROPN
ejpam-4653	301	8	=	=	SYM
ejpam-4653	301	9	(	(	PUNCT
ejpam-4653	301	10	v0	v0	NOUN
ejpam-4653	301	11	∪	∪	NOUN
ejpam-4653	301	12	{	{	PUNCT
ejpam-4653	301	13	v1},∅	v1},∅	NOUN
ejpam-4653	301	14	,	,	PUNCT
ejpam-4653	301	15	v2	v2	NOUN
ejpam-4653	301	16	∪	∪	X
ejpam-4653	301	17	{	{	PUNCT
ejpam-4653	301	18	v2},∅	v2},∅	ADJ
ejpam-4653	301	19	)	)	PUNCT
ejpam-4653	301	20	is	be	AUX
ejpam-4653	301	21	a	a	DET
ejpam-4653	301	22	γdr	γdr	NOUN
ejpam-4653	301	23	-	-	PUNCT
ejpam-4653	301	24	function	function	NOUN
ejpam-4653	301	25	of	of	ADP
ejpam-4653	301	26	tn,2	tn,2	PROPN
ejpam-4653	301	27	.	.	PUNCT
ejpam-4653	302	1	thus	thus	ADV
ejpam-4653	302	2	,	,	PUNCT
ejpam-4653	302	3	γdr(tn,2	γdr(tn,2	NUM
ejpam-4653	302	4	)	)	PUNCT
ejpam-4653	302	5	=	=	PUNCT
ejpam-4653	302	6	n+	n+	PUNCT
ejpam-4653	302	7	2	2	X
ejpam-4653	302	8	.	.	PUNCT
ejpam-4653	302	9	suppose	suppose	VERB
ejpam-4653	302	10	that	that	SCONJ
ejpam-4653	302	11	m	m	PROPN
ejpam-4653	302	12	≥	≥	NOUN
ejpam-4653	302	13	3	3	NUM
ejpam-4653	302	14	,	,	PUNCT
ejpam-4653	302	15	and	and	CCONJ
ejpam-4653	302	16	let	let	VERB
ejpam-4653	302	17	m	m	VERB
ejpam-4653	302	18	=	=	VERB
ejpam-4653	302	19	3k	3k	X
ejpam-4653	302	20	+	+	CCONJ
ejpam-4653	302	21	r	r	NOUN
ejpam-4653	302	22	,	,	PUNCT
ejpam-4653	302	23	where	where	SCONJ
ejpam-4653	302	24	0	0	NUM
ejpam-4653	302	25	≤	≤	NUM
ejpam-4653	302	26	r	r	NOUN
ejpam-4653	302	27	≤	≤	NUM
ejpam-4653	302	28	2	2	NUM
ejpam-4653	302	29	.	.	PUNCT
ejpam-4653	303	1	whenever	whenever	SCONJ
ejpam-4653	303	2	r	r	NOUN
ejpam-4653	303	3	=	=	SYM
ejpam-4653	303	4	0	0	NUM
ejpam-4653	303	5	,	,	PUNCT
ejpam-4653	303	6	put	put	VERB
ejpam-4653	303	7	v	v	NOUN
ejpam-4653	303	8	∗	∗	NOUN
ejpam-4653	303	9	3	3	NUM
ejpam-4653	303	10	=	=	SYM
ejpam-4653	303	11	{	{	PUNCT
ejpam-4653	303	12	v3j−1	v3j−1	PROPN
ejpam-4653	303	13	:	:	PUNCT
ejpam-4653	303	14	j	j	PROPN
ejpam-4653	303	15	∈	∈	PROPN
ejpam-4653	303	16	{	{	PUNCT
ejpam-4653	303	17	1	1	NUM
ejpam-4653	303	18	,	,	PUNCT
ejpam-4653	303	19	2	2	NUM
ejpam-4653	303	20	,	,	PUNCT
ejpam-4653	303	21	.	.	PUNCT
ejpam-4653	303	22	.	.	PUNCT
ejpam-4653	304	1	.	.	PUNCT
ejpam-4653	305	1	,	,	PUNCT
ejpam-4653	305	2	k	k	X
ejpam-4653	305	3	}	}	PUNCT
ejpam-4653	305	4	}	}	PUNCT
ejpam-4653	305	5	,	,	PUNCT
ejpam-4653	305	6	v	v	ADP
ejpam-4653	305	7	∗	∗	NOUN
ejpam-4653	305	8	0	0	NUM
ejpam-4653	305	9	=	=	SYM
ejpam-4653	305	10	v	v	NOUN
ejpam-4653	305	11	(	(	PUNCT
ejpam-4653	305	12	pm	pm	NOUN
ejpam-4653	305	13	)	)	PUNCT
ejpam-4653	305	14	\	\	PROPN
ejpam-4653	305	15	v	v	ADP
ejpam-4653	305	16	∗	∗	NOUN
ejpam-4653	305	17	3	3	NUM
ejpam-4653	305	18	and	and	CCONJ
ejpam-4653	305	19	v	v	NOUN
ejpam-4653	305	20	∗	∗	NOUN
ejpam-4653	305	21	2	2	NUM
ejpam-4653	305	22	=	=	SYM
ejpam-4653	305	23	∅.	∅.	NOUN
ejpam-4653	305	24	then	then	ADV
ejpam-4653	305	25	(	(	PUNCT
ejpam-4653	305	26	v	v	NOUN
ejpam-4653	305	27	∗	∗	NOUN
ejpam-4653	305	28	0	0	NUM
ejpam-4653	305	29	,	,	PUNCT
ejpam-4653	305	30	∅	∅	NOUN
ejpam-4653	305	31	,	,	PUNCT
ejpam-4653	305	32	v	v	NOUN
ejpam-4653	305	33	∗	∗	NOUN
ejpam-4653	305	34	2	2	NUM
ejpam-4653	305	35	,	,	PUNCT
ejpam-4653	305	36	v	v	NOUN
ejpam-4653	305	37	∗	∗	X
ejpam-4653	305	38	3	3	NUM
ejpam-4653	305	39	)	)	PUNCT
ejpam-4653	305	40	is	be	AUX
ejpam-4653	305	41	a	a	DET
ejpam-4653	305	42	γdr	γdr	NOUN
ejpam-4653	305	43	-	-	PUNCT
ejpam-4653	305	44	function	function	NOUN
ejpam-4653	305	45	of	of	ADP
ejpam-4653	305	46	pm	pm	NOUN
ejpam-4653	305	47	.	.	PUNCT
ejpam-4653	306	1	thus	thus	ADV
ejpam-4653	306	2	g	g	PROPN
ejpam-4653	306	3	=	=	SYM
ejpam-4653	306	4	(	(	PUNCT
ejpam-4653	306	5	v0	v0	PROPN
ejpam-4653	306	6	∪	∪	NOUN
ejpam-4653	306	7	v	v	ADP
ejpam-4653	306	8	∗	∗	NOUN
ejpam-4653	306	9	0	0	NUM
ejpam-4653	306	10	,	,	PUNCT
ejpam-4653	306	11	∅	∅	NOUN
ejpam-4653	306	12	,	,	PUNCT
ejpam-4653	306	13	v2	v2	PROPN
ejpam-4653	306	14	∪	∪	NOUN
ejpam-4653	306	15	v	v	NOUN
ejpam-4653	306	16	∗	∗	NOUN
ejpam-4653	306	17	2	2	NUM
ejpam-4653	306	18	,	,	PUNCT
ejpam-4653	306	19	v3	v3	PROPN
ejpam-4653	306	20	∪	∪	NOUN
ejpam-4653	306	21	v	v	ADP
ejpam-4653	306	22	∗	∗	X
ejpam-4653	306	23	3	3	NUM
ejpam-4653	306	24	)	)	PUNCT
ejpam-4653	306	25	is	be	AUX
ejpam-4653	306	26	a	a	DET
ejpam-4653	306	27	γdr	γdr	NOUN
ejpam-4653	306	28	-	-	PUNCT
ejpam-4653	306	29	function	function	NOUN
ejpam-4653	306	30	of	of	ADP
ejpam-4653	306	31	tn	tn	PROPN
ejpam-4653	306	32	,	,	PUNCT
ejpam-4653	306	33	m.	m.	NOUN
ejpam-4653	306	34	consequently	consequently	ADV
ejpam-4653	306	35	,	,	PUNCT
ejpam-4653	306	36	γdr(tn	γdr(tn	PROPN
ejpam-4653	306	37	,	,	PUNCT
ejpam-4653	306	38	m	m	NOUN
ejpam-4653	306	39	)	)	PUNCT
ejpam-4653	307	1	=	=	SYM
ejpam-4653	307	2	n+	n+	NOUN
ejpam-4653	307	3	γdr(pm	γdr(pm	NOUN
ejpam-4653	307	4	)	)	PUNCT
ejpam-4653	307	5	.	.	PUNCT
ejpam-4653	308	1	suppose	suppose	VERB
ejpam-4653	308	2	that	that	SCONJ
ejpam-4653	308	3	r	r	NOUN
ejpam-4653	308	4	=	=	SYM
ejpam-4653	308	5	1	1	X
ejpam-4653	308	6	.	.	PUNCT
ejpam-4653	308	7	let	let	VERB
ejpam-4653	308	8	j	j	PROPN
ejpam-4653	308	9	be	be	AUX
ejpam-4653	308	10	the	the	DET
ejpam-4653	308	11	largest	large	ADJ
ejpam-4653	308	12	positive	positive	ADJ
ejpam-4653	308	13	integer	integer	NOUN
ejpam-4653	308	14	for	for	ADP
ejpam-4653	308	15	which	which	PRON
ejpam-4653	308	16	2j	2j	NUM
ejpam-4653	308	17	≤	≤	NUM
ejpam-4653	308	18	3k	3k	NUM
ejpam-4653	308	19	.	.	PUNCT
ejpam-4653	309	1	if	if	SCONJ
ejpam-4653	309	2	2j	2j	NUM
ejpam-4653	309	3	=	=	SYM
ejpam-4653	309	4	3k	3k	NUM
ejpam-4653	309	5	,	,	PUNCT
ejpam-4653	309	6	put	put	VERB
ejpam-4653	309	7	v	v	NUM
ejpam-4653	309	8	∗	∗	NOUN
ejpam-4653	309	9	2	2	NUM
ejpam-4653	309	10	=	=	SYM
ejpam-4653	309	11	{	{	PUNCT
ejpam-4653	309	12	v2i	v2i	NOUN
ejpam-4653	309	13	:	:	PUNCT
ejpam-4653	309	14	i	i	PRON
ejpam-4653	309	15	∈	∈	PROPN
ejpam-4653	309	16	{	{	PUNCT
ejpam-4653	309	17	1	1	NUM
ejpam-4653	309	18	,	,	PUNCT
ejpam-4653	309	19	2	2	NUM
ejpam-4653	309	20	,	,	PUNCT
ejpam-4653	309	21	.	.	PUNCT
ejpam-4653	309	22	.	.	PUNCT
ejpam-4653	309	23	.	.	PUNCT
ejpam-4653	310	1	,	,	PUNCT
ejpam-4653	310	2	j	j	PROPN
ejpam-4653	310	3	−	−	PROPN
ejpam-4653	310	4	1	1	NUM
ejpam-4653	310	5	}	}	PUNCT
ejpam-4653	310	6	}	}	PUNCT
ejpam-4653	310	7	,	,	PUNCT
ejpam-4653	310	8	v	v	ADP
ejpam-4653	310	9	∗	∗	NOUN
ejpam-4653	310	10	0	0	NUM
ejpam-4653	310	11	=	=	SYM
ejpam-4653	310	12	v	v	NOUN
ejpam-4653	310	13	(	(	PUNCT
ejpam-4653	310	14	pm	pm	NOUN
ejpam-4653	310	15	)	)	PUNCT
ejpam-4653	310	16	\	\	PUNCT
ejpam-4653	311	1	(	(	PUNCT
ejpam-4653	311	2	v	v	NOUN
ejpam-4653	311	3	∗	∗	NOUN
ejpam-4653	311	4	2	2	NUM
ejpam-4653	311	5	∪	∪	X
ejpam-4653	311	6	{	{	PUNCT
ejpam-4653	311	7	v3k	v3k	NOUN
ejpam-4653	311	8	}	}	PUNCT
ejpam-4653	311	9	)	)	PUNCT
ejpam-4653	312	1	and	and	CCONJ
ejpam-4653	312	2	v	v	X
ejpam-4653	312	3	∗	∗	NOUN
ejpam-4653	312	4	3	3	NUM
ejpam-4653	312	5	=	=	SYM
ejpam-4653	312	6	{	{	PUNCT
ejpam-4653	312	7	v3k	v3k	ADV
ejpam-4653	312	8	}	}	PUNCT
ejpam-4653	312	9	.	.	PUNCT
ejpam-4653	313	1	on	on	ADP
ejpam-4653	313	2	the	the	DET
ejpam-4653	313	3	other	other	ADJ
ejpam-4653	313	4	hand	hand	NOUN
ejpam-4653	313	5	,	,	PUNCT
ejpam-4653	313	6	if	if	SCONJ
ejpam-4653	313	7	2j	2j	VERB
ejpam-4653	313	8	<	<	X
ejpam-4653	313	9	3k	3k	X
ejpam-4653	313	10	,	,	PUNCT
ejpam-4653	313	11	put	put	VERB
ejpam-4653	313	12	v	v	NUM
ejpam-4653	313	13	∗	∗	NOUN
ejpam-4653	313	14	2	2	NUM
ejpam-4653	313	15	=	=	SYM
ejpam-4653	313	16	{	{	PUNCT
ejpam-4653	313	17	v2i	v2i	NOUN
ejpam-4653	313	18	:	:	PUNCT
ejpam-4653	313	19	i	i	PRON
ejpam-4653	313	20	∈	∈	PROPN
ejpam-4653	313	21	{	{	PUNCT
ejpam-4653	313	22	1	1	NUM
ejpam-4653	313	23	,	,	PUNCT
ejpam-4653	313	24	2	2	NUM
ejpam-4653	313	25	,	,	PUNCT
ejpam-4653	313	26	.	.	PUNCT
ejpam-4653	313	27	.	.	PUNCT
ejpam-4653	313	28	.	.	PUNCT
ejpam-4653	314	1	,	,	PUNCT
ejpam-4653	314	2	j	j	NOUN
ejpam-4653	314	3	}	}	PUNCT
ejpam-4653	314	4	}	}	PUNCT
ejpam-4653	314	5	∪	∪	ADJ
ejpam-4653	314	6	{	{	PUNCT
ejpam-4653	314	7	v3k+1	v3k+1	ADJ
ejpam-4653	314	8	}	}	PUNCT
ejpam-4653	314	9	,	,	PUNCT
ejpam-4653	314	10	v	v	ADP
ejpam-4653	314	11	∗	∗	NOUN
ejpam-4653	314	12	0	0	NUM
ejpam-4653	315	1	=	=	SYM
ejpam-4653	315	2	v	v	NOUN
ejpam-4653	315	3	(	(	PUNCT
ejpam-4653	315	4	pm	pm	NOUN
ejpam-4653	315	5	)	)	PUNCT
ejpam-4653	315	6	\	\	PROPN
ejpam-4653	315	7	v	v	ADP
ejpam-4653	315	8	∗	∗	X
ejpam-4653	315	9	2	2	NUM
ejpam-4653	315	10	j.	j.	PROPN
ejpam-4653	315	11	b.g	b.g	PROPN
ejpam-4653	315	12	.	.	PROPN
ejpam-4653	315	13	cariaga	cariaga	PROPN
ejpam-4653	315	14	,	,	PUNCT
ejpam-4653	315	15	f.	f.	PROPN
ejpam-4653	315	16	jamil	jamil	PROPN
ejpam-4653	315	17	/	/	SYM
ejpam-4653	315	18	eur	eur	PROPN
ejpam-4653	315	19	.	.	PUNCT
ejpam-4653	316	1	j.	j.	PROPN
ejpam-4653	316	2	pure	pure	PROPN
ejpam-4653	316	3	appl	appl	PROPN
ejpam-4653	316	4	.	.	PROPN
ejpam-4653	316	5	math	math	PROPN
ejpam-4653	316	6	,	,	PUNCT
ejpam-4653	316	7	16	16	NUM
ejpam-4653	316	8	(	(	PUNCT
ejpam-4653	316	9	2	2	NUM
ejpam-4653	316	10	)	)	PUNCT
ejpam-4653	316	11	(	(	PUNCT
ejpam-4653	316	12	2023	2023	NUM
ejpam-4653	316	13	)	)	PUNCT
ejpam-4653	316	14	,	,	PUNCT
ejpam-4653	316	15	847	847	NUM
ejpam-4653	316	16	-	-	SYM
ejpam-4653	316	17	863	863	NUM
ejpam-4653	316	18	854	854	NUM
ejpam-4653	316	19	and	and	CCONJ
ejpam-4653	316	20	v	v	NOUN
ejpam-4653	316	21	∗	∗	NOUN
ejpam-4653	316	22	3	3	NUM
ejpam-4653	316	23	=	=	SYM
ejpam-4653	316	24	∅.	∅.	NOUN
ejpam-4653	316	25	in	in	ADP
ejpam-4653	316	26	either	either	DET
ejpam-4653	316	27	case	case	NOUN
ejpam-4653	316	28	,	,	PUNCT
ejpam-4653	316	29	g	g	NOUN
ejpam-4653	316	30	=	=	SYM
ejpam-4653	316	31	(	(	PUNCT
ejpam-4653	316	32	v0	v0	PROPN
ejpam-4653	316	33	∪	∪	NOUN
ejpam-4653	316	34	v	v	ADP
ejpam-4653	316	35	∗	∗	NOUN
ejpam-4653	316	36	0	0	NUM
ejpam-4653	316	37	,	,	PUNCT
ejpam-4653	316	38	∅	∅	NOUN
ejpam-4653	316	39	,	,	PUNCT
ejpam-4653	316	40	v2	v2	PROPN
ejpam-4653	316	41	∪	∪	NOUN
ejpam-4653	316	42	v	v	NOUN
ejpam-4653	316	43	∗	∗	NOUN
ejpam-4653	316	44	2	2	NUM
ejpam-4653	316	45	,	,	PUNCT
ejpam-4653	316	46	v	v	NOUN
ejpam-4653	316	47	∗	∗	NOUN
ejpam-4653	316	48	3	3	NUM
ejpam-4653	316	49	∪	∪	NOUN
ejpam-4653	316	50	v	v	ADP
ejpam-4653	316	51	∗	∗	X
ejpam-4653	316	52	3	3	NUM
ejpam-4653	316	53	)	)	PUNCT
ejpam-4653	316	54	is	be	AUX
ejpam-4653	316	55	a	a	DET
ejpam-4653	316	56	γdr	γdr	NOUN
ejpam-4653	316	57	-	-	PUNCT
ejpam-4653	316	58	function	function	NOUN
ejpam-4653	316	59	of	of	ADP
ejpam-4653	316	60	tn	tn	PROPN
ejpam-4653	316	61	,	,	PUNCT
ejpam-4653	316	62	m.	m.	NOUN
ejpam-4653	316	63	thus	thus	ADV
ejpam-4653	316	64	,	,	PUNCT
ejpam-4653	316	65	γdr(tn	γdr(tn	X
ejpam-4653	316	66	,	,	PUNCT
ejpam-4653	316	67	m	m	NOUN
ejpam-4653	316	68	)	)	PUNCT
ejpam-4653	316	69	=	=	SYM
ejpam-4653	316	70	n+	n+	PROPN
ejpam-4653	316	71	γdr(pm)−	γdr(pm)−	NOUN
ejpam-4653	316	72	1	1	NUM
ejpam-4653	316	73	.	.	PUNCT
ejpam-4653	316	74	suppose	suppose	VERB
ejpam-4653	316	75	that	that	SCONJ
ejpam-4653	316	76	r	r	NOUN
ejpam-4653	316	77	=	=	SYM
ejpam-4653	316	78	2	2	X
ejpam-4653	316	79	.	.	PUNCT
ejpam-4653	316	80	let	let	VERB
ejpam-4653	316	81	j	j	PROPN
ejpam-4653	316	82	be	be	AUX
ejpam-4653	316	83	the	the	DET
ejpam-4653	316	84	largest	large	ADJ
ejpam-4653	316	85	positive	positive	ADJ
ejpam-4653	316	86	integer	integer	NOUN
ejpam-4653	316	87	for	for	ADP
ejpam-4653	316	88	which	which	PRON
ejpam-4653	316	89	2j	2j	NUM
ejpam-4653	316	90	≤	≤	NUM
ejpam-4653	316	91	3k	3k	NUM
ejpam-4653	316	92	.	.	PUNCT
ejpam-4653	317	1	if	if	SCONJ
ejpam-4653	317	2	2j	2j	NUM
ejpam-4653	317	3	=	=	SYM
ejpam-4653	317	4	3k	3k	NUM
ejpam-4653	317	5	,	,	PUNCT
ejpam-4653	317	6	put	put	VERB
ejpam-4653	317	7	v	v	NUM
ejpam-4653	317	8	∗	∗	NOUN
ejpam-4653	317	9	2	2	NUM
ejpam-4653	317	10	=	=	SYM
ejpam-4653	317	11	{	{	PUNCT
ejpam-4653	317	12	v2i	v2i	NOUN
ejpam-4653	317	13	:	:	PUNCT
ejpam-4653	317	14	i	i	PRON
ejpam-4653	317	15	∈	∈	PROPN
ejpam-4653	317	16	{	{	PUNCT
ejpam-4653	317	17	1	1	NUM
ejpam-4653	317	18	,	,	PUNCT
ejpam-4653	317	19	2	2	NUM
ejpam-4653	317	20	,	,	PUNCT
ejpam-4653	317	21	.	.	PUNCT
ejpam-4653	317	22	.	.	PUNCT
ejpam-4653	318	1	.	.	PUNCT
ejpam-4653	319	1	,	,	PUNCT
ejpam-4653	319	2	j	j	PROPN
ejpam-4653	319	3	+	+	CCONJ
ejpam-4653	319	4	1	1	NUM
ejpam-4653	319	5	}	}	PUNCT
ejpam-4653	319	6	}	}	PUNCT
ejpam-4653	319	7	,	,	PUNCT
ejpam-4653	319	8	v	v	ADP
ejpam-4653	319	9	∗	∗	NOUN
ejpam-4653	319	10	0	0	NUM
ejpam-4653	319	11	=	=	SYM
ejpam-4653	319	12	v	v	NOUN
ejpam-4653	319	13	(	(	PUNCT
ejpam-4653	319	14	pm	pm	NOUN
ejpam-4653	319	15	)	)	PUNCT
ejpam-4653	319	16	\	\	PROPN
ejpam-4653	319	17	v	v	ADP
ejpam-4653	319	18	∗	∗	NOUN
ejpam-4653	319	19	2	2	NUM
ejpam-4653	319	20	and	and	CCONJ
ejpam-4653	319	21	v	v	NOUN
ejpam-4653	319	22	∗	∗	NOUN
ejpam-4653	319	23	3	3	NUM
ejpam-4653	319	24	=	=	SYM
ejpam-4653	319	25	∅.	∅.	NOUN
ejpam-4653	319	26	on	on	ADP
ejpam-4653	319	27	the	the	DET
ejpam-4653	319	28	other	other	ADJ
ejpam-4653	319	29	hand	hand	NOUN
ejpam-4653	319	30	,	,	PUNCT
ejpam-4653	319	31	if	if	SCONJ
ejpam-4653	319	32	2j	2j	VERB
ejpam-4653	319	33	<	<	X
ejpam-4653	319	34	3k	3k	X
ejpam-4653	319	35	,	,	PUNCT
ejpam-4653	319	36	put	put	VERB
ejpam-4653	319	37	v	v	NUM
ejpam-4653	319	38	∗	∗	NOUN
ejpam-4653	319	39	2	2	NUM
ejpam-4653	319	40	=	=	SYM
ejpam-4653	319	41	{	{	PUNCT
ejpam-4653	319	42	v2i	v2i	NOUN
ejpam-4653	319	43	:	:	PUNCT
ejpam-4653	319	44	i	i	PRON
ejpam-4653	319	45	∈	∈	PROPN
ejpam-4653	319	46	{	{	PUNCT
ejpam-4653	319	47	1	1	NUM
ejpam-4653	319	48	,	,	PUNCT
ejpam-4653	319	49	2	2	NUM
ejpam-4653	319	50	,	,	PUNCT
ejpam-4653	319	51	.	.	PUNCT
ejpam-4653	319	52	.	.	PUNCT
ejpam-4653	319	53	.	.	PUNCT
ejpam-4653	320	1	,	,	PUNCT
ejpam-4653	320	2	j	j	NOUN
ejpam-4653	320	3	}	}	PUNCT
ejpam-4653	320	4	}	}	PUNCT
ejpam-4653	320	5	,	,	PUNCT
ejpam-4653	320	6	v	v	ADP
ejpam-4653	320	7	∗	∗	NOUN
ejpam-4653	320	8	0	0	NUM
ejpam-4653	320	9	=	=	SYM
ejpam-4653	320	10	v	v	NOUN
ejpam-4653	320	11	(	(	PUNCT
ejpam-4653	320	12	pm	pm	NOUN
ejpam-4653	320	13	)	)	PUNCT
ejpam-4653	320	14	\	\	PUNCT
ejpam-4653	321	1	(	(	PUNCT
ejpam-4653	321	2	v	v	NOUN
ejpam-4653	321	3	∗	∗	NOUN
ejpam-4653	321	4	2	2	NUM
ejpam-4653	321	5	∪	∪	X
ejpam-4653	321	6	{	{	PUNCT
ejpam-4653	321	7	v3k+1	v3k+1	ADJ
ejpam-4653	321	8	}	}	PUNCT
ejpam-4653	321	9	)	)	PUNCT
ejpam-4653	321	10	and	and	CCONJ
ejpam-4653	321	11	v	v	ADP
ejpam-4653	321	12	∗	∗	NOUN
ejpam-4653	321	13	3	3	NUM
ejpam-4653	321	14	=	=	SYM
ejpam-4653	321	15	{	{	PUNCT
ejpam-4653	321	16	v3k+1	v3k+1	ADJ
ejpam-4653	321	17	}	}	PUNCT
ejpam-4653	321	18	.	.	PUNCT
ejpam-4653	322	1	in	in	ADP
ejpam-4653	322	2	either	either	DET
ejpam-4653	322	3	case	case	NOUN
ejpam-4653	322	4	,	,	PUNCT
ejpam-4653	322	5	g	g	NOUN
ejpam-4653	322	6	=	=	SYM
ejpam-4653	322	7	(	(	PUNCT
ejpam-4653	322	8	v0	v0	PROPN
ejpam-4653	322	9	∪	∪	NOUN
ejpam-4653	322	10	v	v	ADP
ejpam-4653	322	11	∗	∗	NOUN
ejpam-4653	322	12	0	0	NUM
ejpam-4653	322	13	,	,	PUNCT
ejpam-4653	322	14	∅	∅	NOUN
ejpam-4653	322	15	,	,	PUNCT
ejpam-4653	322	16	v2	v2	PROPN
ejpam-4653	322	17	∪	∪	NOUN
ejpam-4653	322	18	v	v	NOUN
ejpam-4653	322	19	∗	∗	NOUN
ejpam-4653	322	20	2	2	NUM
ejpam-4653	322	21	,	,	PUNCT
ejpam-4653	322	22	v	v	NOUN
ejpam-4653	322	23	∗	∗	NOUN
ejpam-4653	322	24	3	3	NUM
ejpam-4653	322	25	∪	∪	NOUN
ejpam-4653	322	26	v	v	ADP
ejpam-4653	322	27	∗	∗	X
ejpam-4653	322	28	3	3	NUM
ejpam-4653	322	29	)	)	PUNCT
ejpam-4653	322	30	is	be	AUX
ejpam-4653	322	31	a	a	DET
ejpam-4653	322	32	γdr	γdr	NOUN
ejpam-4653	322	33	-	-	PUNCT
ejpam-4653	322	34	function	function	NOUN
ejpam-4653	322	35	of	of	ADP
ejpam-4653	322	36	tn	tn	PROPN
ejpam-4653	322	37	,	,	PUNCT
ejpam-4653	322	38	m.	m.	NOUN
ejpam-4653	322	39	thus	thus	ADV
ejpam-4653	322	40	,	,	PUNCT
ejpam-4653	322	41	γdr(tn	γdr(tn	X
ejpam-4653	322	42	,	,	PUNCT
ejpam-4653	322	43	m	m	NOUN
ejpam-4653	322	44	)	)	PUNCT
ejpam-4653	322	45	=	=	SYM
ejpam-4653	322	46	n+	n+	PROPN
ejpam-4653	323	1	γdr(pm)−	γdr(pm)−	NOUN
ejpam-4653	323	2	1	1	NUM
ejpam-4653	323	3	.	.	PUNCT
ejpam-4653	323	4	case	case	NOUN
ejpam-4653	323	5	3	3	X
ejpam-4653	323	6	:	:	PUNCT
ejpam-4653	323	7	suppose	suppose	VERB
ejpam-4653	323	8	that	that	SCONJ
ejpam-4653	323	9	n	n	X
ejpam-4653	323	10	≡	≡	PROPN
ejpam-4653	323	11	1	1	NUM
ejpam-4653	323	12	,	,	PUNCT
ejpam-4653	323	13	5	5	NUM
ejpam-4653	323	14	(	(	PUNCT
ejpam-4653	323	15	mod	mod	PROPN
ejpam-4653	323	16	6	6	NUM
ejpam-4653	323	17	)	)	PUNCT
ejpam-4653	323	18	.	.	PUNCT
ejpam-4653	324	1	let	let	VERB
ejpam-4653	324	2	f	f	PROPN
ejpam-4653	324	3	=	=	SYM
ejpam-4653	324	4	(	(	PUNCT
ejpam-4653	324	5	v0,∅	v0,∅	PROPN
ejpam-4653	324	6	,	,	PUNCT
ejpam-4653	324	7	v2	v2	PROPN
ejpam-4653	324	8	,	,	PUNCT
ejpam-4653	324	9	v3	v3	PROPN
ejpam-4653	324	10	)	)	PUNCT
ejpam-4653	324	11	be	be	VERB
ejpam-4653	324	12	a	a	DET
ejpam-4653	324	13	γdr	γdr	NOUN
ejpam-4653	324	14	-	-	PUNCT
ejpam-4653	324	15	function	function	NOUN
ejpam-4653	324	16	of	of	ADP
ejpam-4653	324	17	cn	cn	PROPN
ejpam-4653	324	18	with	with	ADP
ejpam-4653	324	19	v3	v3	PROPN
ejpam-4653	324	20	̸=	̸=	PROPN
ejpam-4653	324	21	∅	∅	NOUN
ejpam-4653	324	22	and	and	CCONJ
ejpam-4653	324	23	v	v	ADP
ejpam-4653	324	24	∈	∈	PROPN
ejpam-4653	324	25	v3	v3	PROPN
ejpam-4653	324	26	.	.	PUNCT
ejpam-4653	325	1	if	if	SCONJ
ejpam-4653	325	2	m	m	VERB
ejpam-4653	325	3	=	=	SYM
ejpam-4653	325	4	2	2	NUM
ejpam-4653	325	5	,	,	PUNCT
ejpam-4653	325	6	then	then	ADV
ejpam-4653	325	7	g	g	PROPN
ejpam-4653	325	8	=	=	SYM
ejpam-4653	325	9	(	(	PUNCT
ejpam-4653	325	10	v0	v0	NOUN
ejpam-4653	325	11	∪	∪	NOUN
ejpam-4653	325	12	{	{	PUNCT
ejpam-4653	325	13	v1},∅	v1},∅	NOUN
ejpam-4653	325	14	,	,	PUNCT
ejpam-4653	325	15	v2	v2	NOUN
ejpam-4653	325	16	∪	∪	X
ejpam-4653	325	17	{	{	PUNCT
ejpam-4653	325	18	v2	v2	NOUN
ejpam-4653	325	19	}	}	PUNCT
ejpam-4653	325	20	,	,	PUNCT
ejpam-4653	325	21	v3	v3	PROPN
ejpam-4653	325	22	)	)	PUNCT
ejpam-4653	325	23	is	be	AUX
ejpam-4653	325	24	a	a	DET
ejpam-4653	325	25	γdr	γdr	NOUN
ejpam-4653	325	26	-	-	PUNCT
ejpam-4653	325	27	function	function	NOUN
ejpam-4653	325	28	of	of	ADP
ejpam-4653	325	29	tn,2	tn,2	PROPN
ejpam-4653	325	30	.	.	PUNCT
ejpam-4653	326	1	thus	thus	ADV
ejpam-4653	326	2	,	,	PUNCT
ejpam-4653	326	3	γdr(tn,2	γdr(tn,2	NUM
ejpam-4653	326	4	)	)	PUNCT
ejpam-4653	326	5	=	=	SYM
ejpam-4653	326	6	γdr(cn	γdr(cn	NOUN
ejpam-4653	326	7	)	)	PUNCT
ejpam-4653	327	1	+	+	CCONJ
ejpam-4653	327	2	2	2	NUM
ejpam-4653	327	3	=	=	SYM
ejpam-4653	327	4	n+	n+	ADP
ejpam-4653	327	5	1	1	NUM
ejpam-4653	327	6	+	+	NUM
ejpam-4653	327	7	2	2	X
ejpam-4653	327	8	.	.	PUNCT
ejpam-4653	327	9	suppose	suppose	VERB
ejpam-4653	327	10	that	that	SCONJ
ejpam-4653	327	11	m	m	PROPN
ejpam-4653	327	12	≥	≥	NOUN
ejpam-4653	327	13	3	3	NUM
ejpam-4653	327	14	,	,	PUNCT
ejpam-4653	327	15	and	and	CCONJ
ejpam-4653	327	16	let	let	VERB
ejpam-4653	327	17	m	m	VERB
ejpam-4653	327	18	=	=	VERB
ejpam-4653	327	19	3k	3k	X
ejpam-4653	327	20	+	+	CCONJ
ejpam-4653	327	21	r	r	NOUN
ejpam-4653	327	22	,	,	PUNCT
ejpam-4653	327	23	where	where	SCONJ
ejpam-4653	327	24	0	0	NUM
ejpam-4653	327	25	≤	≤	NUM
ejpam-4653	327	26	r	r	NOUN
ejpam-4653	327	27	≤	≤	NUM
ejpam-4653	327	28	2	2	NUM
ejpam-4653	327	29	.	.	PUNCT
ejpam-4653	328	1	if	if	SCONJ
ejpam-4653	328	2	r	r	NOUN
ejpam-4653	328	3	=	=	SYM
ejpam-4653	328	4	0	0	NUM
ejpam-4653	328	5	,	,	PUNCT
ejpam-4653	328	6	put	put	VERB
ejpam-4653	328	7	v	v	NOUN
ejpam-4653	328	8	∗	∗	NOUN
ejpam-4653	328	9	3	3	NUM
ejpam-4653	328	10	=	=	SYM
ejpam-4653	328	11	{	{	PUNCT
ejpam-4653	328	12	v3j	v3j	ADV
ejpam-4653	328	13	:	:	PUNCT
ejpam-4653	328	14	j	j	PROPN
ejpam-4653	328	15	∈	∈	PROPN
ejpam-4653	328	16	{	{	PUNCT
ejpam-4653	328	17	1	1	NUM
ejpam-4653	328	18	,	,	PUNCT
ejpam-4653	328	19	2	2	NUM
ejpam-4653	328	20	,	,	PUNCT
ejpam-4653	328	21	.	.	PUNCT
ejpam-4653	328	22	.	.	PUNCT
ejpam-4653	329	1	.	.	PUNCT
ejpam-4653	330	1	,	,	PUNCT
ejpam-4653	330	2	k	k	X
ejpam-4653	330	3	}	}	PUNCT
ejpam-4653	330	4	}	}	PUNCT
ejpam-4653	330	5	,	,	PUNCT
ejpam-4653	330	6	v	v	ADP
ejpam-4653	330	7	∗	∗	NOUN
ejpam-4653	330	8	0	0	NUM
ejpam-4653	330	9	=	=	SYM
ejpam-4653	330	10	v	v	NOUN
ejpam-4653	330	11	(	(	PUNCT
ejpam-4653	330	12	pm	pm	NOUN
ejpam-4653	330	13	)	)	PUNCT
ejpam-4653	330	14	\	\	PROPN
ejpam-4653	330	15	v	v	ADP
ejpam-4653	330	16	∗	∗	NOUN
ejpam-4653	330	17	3	3	NUM
ejpam-4653	330	18	and	and	CCONJ
ejpam-4653	330	19	v	v	NOUN
ejpam-4653	330	20	∗	∗	NOUN
ejpam-4653	330	21	2	2	NUM
ejpam-4653	330	22	=	=	PUNCT
ejpam-4653	330	23	∅.	∅.	NOUN
ejpam-4653	330	24	if	if	SCONJ
ejpam-4653	330	25	r	r	NOUN
ejpam-4653	330	26	=	=	SYM
ejpam-4653	330	27	1	1	NUM
ejpam-4653	330	28	,	,	PUNCT
ejpam-4653	330	29	put	put	VERB
ejpam-4653	330	30	v	v	NOUN
ejpam-4653	330	31	∗	∗	NOUN
ejpam-4653	330	32	3	3	NUM
ejpam-4653	330	33	=	=	SYM
ejpam-4653	330	34	{	{	PUNCT
ejpam-4653	330	35	v3j	v3j	ADV
ejpam-4653	330	36	:	:	PUNCT
ejpam-4653	330	37	j	j	PROPN
ejpam-4653	330	38	∈	∈	PROPN
ejpam-4653	330	39	{	{	PUNCT
ejpam-4653	330	40	1	1	NUM
ejpam-4653	330	41	,	,	PUNCT
ejpam-4653	330	42	2	2	NUM
ejpam-4653	330	43	,	,	PUNCT
ejpam-4653	330	44	.	.	PUNCT
ejpam-4653	330	45	.	.	PUNCT
ejpam-4653	330	46	.	.	PUNCT
ejpam-4653	331	1	,	,	PUNCT
ejpam-4653	331	2	k	k	X
ejpam-4653	331	3	}	}	PUNCT
ejpam-4653	331	4	}	}	PUNCT
ejpam-4653	331	5	,	,	PUNCT
ejpam-4653	331	6	v	v	ADP
ejpam-4653	331	7	∗	∗	NOUN
ejpam-4653	331	8	0	0	NUM
ejpam-4653	331	9	=	=	SYM
ejpam-4653	331	10	v	v	NOUN
ejpam-4653	331	11	(	(	PUNCT
ejpam-4653	331	12	pm	pm	NOUN
ejpam-4653	331	13	)	)	PUNCT
ejpam-4653	331	14	\	\	PROPN
ejpam-4653	331	15	v	v	ADP
ejpam-4653	331	16	∗	∗	NOUN
ejpam-4653	331	17	3	3	NUM
ejpam-4653	331	18	and	and	CCONJ
ejpam-4653	331	19	v	v	NOUN
ejpam-4653	331	20	∗	∗	NOUN
ejpam-4653	331	21	2	2	NUM
ejpam-4653	331	22	=	=	SYM
ejpam-4653	331	23	∅.	∅.	NOUN
ejpam-4653	331	24	and	and	CCONJ
ejpam-4653	331	25	if	if	SCONJ
ejpam-4653	331	26	r	r	NOUN
ejpam-4653	331	27	=	=	SYM
ejpam-4653	331	28	2	2	NUM
ejpam-4653	331	29	,	,	PUNCT
ejpam-4653	331	30	put	put	VERB
ejpam-4653	331	31	v	v	NOUN
ejpam-4653	331	32	∗	∗	NOUN
ejpam-4653	331	33	3	3	NUM
ejpam-4653	331	34	=	=	SYM
ejpam-4653	331	35	{	{	PUNCT
ejpam-4653	331	36	v3j	v3j	ADV
ejpam-4653	331	37	:	:	PUNCT
ejpam-4653	331	38	j	j	PROPN
ejpam-4653	331	39	∈	∈	PROPN
ejpam-4653	331	40	{	{	PUNCT
ejpam-4653	331	41	1	1	NUM
ejpam-4653	331	42	,	,	PUNCT
ejpam-4653	331	43	2	2	NUM
ejpam-4653	331	44	,	,	PUNCT
ejpam-4653	331	45	.	.	PUNCT
ejpam-4653	331	46	.	.	PUNCT
ejpam-4653	332	1	.	.	PUNCT
ejpam-4653	333	1	,	,	PUNCT
ejpam-4653	333	2	k	k	X
ejpam-4653	333	3	}	}	PUNCT
ejpam-4653	333	4	}	}	PUNCT
ejpam-4653	333	5	,	,	PUNCT
ejpam-4653	333	6	v	v	ADP
ejpam-4653	333	7	∗	∗	NOUN
ejpam-4653	333	8	0	0	NUM
ejpam-4653	333	9	=	=	SYM
ejpam-4653	333	10	v	v	NOUN
ejpam-4653	333	11	(	(	PUNCT
ejpam-4653	333	12	pm)\(v	pm)\(v	INTJ
ejpam-4653	333	13	∗	∗	NOUN
ejpam-4653	333	14	3	3	NUM
ejpam-4653	333	15	∪	∪	X
ejpam-4653	333	16	{	{	PUNCT
ejpam-4653	333	17	v3k+2	v3k+2	NOUN
ejpam-4653	333	18	}	}	PUNCT
ejpam-4653	333	19	)	)	PUNCT
ejpam-4653	333	20	and	and	CCONJ
ejpam-4653	333	21	v	v	ADP
ejpam-4653	333	22	∗	∗	NOUN
ejpam-4653	333	23	2	2	NUM
ejpam-4653	333	24	=	=	SYM
ejpam-4653	333	25	{	{	PUNCT
ejpam-4653	333	26	v3k+2	v3k+2	NOUN
ejpam-4653	333	27	}	}	PUNCT
ejpam-4653	333	28	.	.	PUNCT
ejpam-4653	334	1	since	since	SCONJ
ejpam-4653	334	2	(	(	PUNCT
ejpam-4653	334	3	v	v	NOUN
ejpam-4653	334	4	∗	∗	NOUN
ejpam-4653	334	5	0	0	NUM
ejpam-4653	334	6	\{v1},∅	\{v1},∅	NOUN
ejpam-4653	334	7	,	,	PUNCT
ejpam-4653	334	8	v	v	NOUN
ejpam-4653	334	9	∗	∗	NOUN
ejpam-4653	334	10	2	2	NUM
ejpam-4653	334	11	,	,	PUNCT
ejpam-4653	334	12	v	v	NOUN
ejpam-4653	334	13	∗	∗	X
ejpam-4653	334	14	3	3	NUM
ejpam-4653	334	15	)	)	PUNCT
ejpam-4653	334	16	is	be	AUX
ejpam-4653	334	17	a	a	DET
ejpam-4653	334	18	γdr	γdr	NOUN
ejpam-4653	334	19	-	-	PUNCT
ejpam-4653	334	20	function	function	NOUN
ejpam-4653	334	21	of	of	ADP
ejpam-4653	334	22	pm	pm	NOUN
ejpam-4653	334	23	−	−	PROPN
ejpam-4653	334	24	v1	v1	PROPN
ejpam-4653	334	25	≡	≡	PROPN
ejpam-4653	334	26	pm−1	pm−1	NOUN
ejpam-4653	334	27	,	,	PUNCT
ejpam-4653	334	28	g	g	NOUN
ejpam-4653	334	29	=	=	SYM
ejpam-4653	334	30	(	(	PUNCT
ejpam-4653	334	31	v0	v0	PROPN
ejpam-4653	334	32	∪	∪	NOUN
ejpam-4653	334	33	v	v	ADP
ejpam-4653	334	34	∗	∗	NOUN
ejpam-4653	334	35	0	0	NUM
ejpam-4653	334	36	,	,	PUNCT
ejpam-4653	334	37	∅	∅	NOUN
ejpam-4653	334	38	,	,	PUNCT
ejpam-4653	334	39	v2	v2	PROPN
ejpam-4653	334	40	∪	∪	NOUN
ejpam-4653	334	41	v	v	NOUN
ejpam-4653	334	42	∗	∗	NOUN
ejpam-4653	334	43	2	2	NUM
ejpam-4653	334	44	,	,	PUNCT
ejpam-4653	334	45	v3	v3	PROPN
ejpam-4653	334	46	∪	∪	NOUN
ejpam-4653	334	47	v	v	ADP
ejpam-4653	334	48	∗	∗	X
ejpam-4653	334	49	3	3	NUM
ejpam-4653	334	50	)	)	PUNCT
ejpam-4653	334	51	is	be	AUX
ejpam-4653	334	52	a	a	DET
ejpam-4653	334	53	γdr	γdr	NOUN
ejpam-4653	334	54	-	-	PUNCT
ejpam-4653	334	55	function	function	NOUN
ejpam-4653	334	56	of	of	ADP
ejpam-4653	334	57	tn	tn	PROPN
ejpam-4653	334	58	,	,	PUNCT
ejpam-4653	334	59	m.	m.	NOUN
ejpam-4653	334	60	thus	thus	ADV
ejpam-4653	334	61	,	,	PUNCT
ejpam-4653	334	62	γdr(tn	γdr(tn	X
ejpam-4653	334	63	,	,	PUNCT
ejpam-4653	334	64	m	m	NOUN
ejpam-4653	334	65	)	)	PUNCT
ejpam-4653	334	66	=	=	SYM
ejpam-4653	335	1	n	n	PROPN
ejpam-4653	335	2	+	+	CCONJ
ejpam-4653	335	3	1	1	NUM
ejpam-4653	335	4	+	+	CCONJ
ejpam-4653	335	5	γdr(pm−1	γdr(pm−1	NOUN
ejpam-4653	335	6	)	)	PUNCT
ejpam-4653	335	7	.	.	PUNCT
ejpam-4653	336	1	let	let	VERB
ejpam-4653	336	2	g	g	NOUN
ejpam-4653	336	3	and	and	CCONJ
ejpam-4653	336	4	h	h	NOUN
ejpam-4653	336	5	be	be	AUX
ejpam-4653	336	6	graphs	graph	NOUN
ejpam-4653	336	7	with	with	ADP
ejpam-4653	336	8	disjoint	disjoint	ADJ
ejpam-4653	336	9	vertex	vertex	NOUN
ejpam-4653	336	10	sets	set	NOUN
ejpam-4653	336	11	.	.	PUNCT
ejpam-4653	337	1	the	the	DET
ejpam-4653	337	2	join	join	NOUN
ejpam-4653	337	3	of	of	ADP
ejpam-4653	337	4	g	g	PROPN
ejpam-4653	337	5	and	and	CCONJ
ejpam-4653	337	6	h	h	NOUN
ejpam-4653	337	7	is	be	AUX
ejpam-4653	337	8	the	the	DET
ejpam-4653	337	9	graph	graph	NOUN
ejpam-4653	337	10	g	g	PROPN
ejpam-4653	337	11	+	+	CCONJ
ejpam-4653	337	12	h	h	NOUN
ejpam-4653	337	13	with	with	ADP
ejpam-4653	337	14	v	v	NOUN
ejpam-4653	337	15	(	(	PUNCT
ejpam-4653	337	16	g	g	PROPN
ejpam-4653	337	17	+	+	NOUN
ejpam-4653	337	18	h	h	NOUN
ejpam-4653	337	19	)	)	PUNCT
ejpam-4653	337	20	=	=	NOUN
ejpam-4653	337	21	v	v	X
ejpam-4653	337	22	(	(	PUNCT
ejpam-4653	337	23	g	g	NOUN
ejpam-4653	337	24	)	)	PUNCT
ejpam-4653	337	25	∪	∪	NOUN
ejpam-4653	337	26	v	v	NOUN
ejpam-4653	337	27	(	(	PUNCT
ejpam-4653	337	28	h	h	NOUN
ejpam-4653	337	29	)	)	PUNCT
ejpam-4653	337	30	and	and	CCONJ
ejpam-4653	337	31	e(g	e(g	PROPN
ejpam-4653	337	32	+	+	CCONJ
ejpam-4653	337	33	h	h	NOUN
ejpam-4653	337	34	)	)	PUNCT
ejpam-4653	337	35	=	=	SYM
ejpam-4653	337	36	e(g	e(g	PROPN
ejpam-4653	337	37	)	)	PUNCT
ejpam-4653	337	38	∪	∪	ADP
ejpam-4653	337	39	e(h	e(h	PROPN
ejpam-4653	337	40	)	)	PUNCT
ejpam-4653	337	41	∪	∪	NOUN
ejpam-4653	337	42	{	{	PUNCT
ejpam-4653	337	43	uv	uv	NOUN
ejpam-4653	337	44	:	:	PUNCT
ejpam-4653	337	45	u	u	PROPN
ejpam-4653	337	46	∈	∈	PROPN
ejpam-4653	337	47	v	v	ADP
ejpam-4653	337	48	(	(	PUNCT
ejpam-4653	337	49	g	g	NOUN
ejpam-4653	337	50	)	)	PUNCT
ejpam-4653	337	51	,	,	PUNCT
ejpam-4653	337	52	v	v	X
ejpam-4653	337	53	∈	∈	PROPN
ejpam-4653	337	54	v	v	NOUN
ejpam-4653	337	55	(	(	PUNCT
ejpam-4653	337	56	h	h	NOUN
ejpam-4653	337	57	)	)	PUNCT
ejpam-4653	337	58	}	}	PUNCT
ejpam-4653	337	59	.	.	PUNCT
ejpam-4653	338	1	proposition	proposition	NOUN
ejpam-4653	338	2	13	13	NUM
ejpam-4653	338	3	.	.	PUNCT
ejpam-4653	339	1	(	(	PUNCT
ejpam-4653	339	2	join	join	VERB
ejpam-4653	339	3	of	of	ADP
ejpam-4653	339	4	graphs	graph	NOUN
ejpam-4653	339	5	)	)	PUNCT
ejpam-4653	339	6	let	let	VERB
ejpam-4653	339	7	g	g	NOUN
ejpam-4653	339	8	and	and	CCONJ
ejpam-4653	339	9	h	h	NOUN
ejpam-4653	339	10	be	be	AUX
ejpam-4653	339	11	nontrivial	nontrivial	ADJ
ejpam-4653	339	12	graphs	graph	NOUN
ejpam-4653	339	13	.	.	PUNCT
ejpam-4653	340	1	then	then	ADV
ejpam-4653	340	2	3	3	NUM
ejpam-4653	340	3	≤	≤	NUM
ejpam-4653	340	4	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	340	5	)	)	PUNCT
ejpam-4653	340	6	≤	≤	NUM
ejpam-4653	340	7	6	6	NUM
ejpam-4653	340	8	.	.	PUNCT
ejpam-4653	341	1	(	(	PUNCT
ejpam-4653	341	2	1	1	X
ejpam-4653	341	3	)	)	PUNCT
ejpam-4653	341	4	more	more	ADV
ejpam-4653	341	5	precisely	precisely	ADV
ejpam-4653	341	6	,	,	PUNCT
ejpam-4653	341	7	(	(	PUNCT
ejpam-4653	341	8	i	i	NOUN
ejpam-4653	341	9	)	)	PUNCT
ejpam-4653	341	10	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	341	11	)	)	PUNCT
ejpam-4653	341	12	=	=	SYM
ejpam-4653	341	13	3	3	NUM
ejpam-4653	341	14	if	if	SCONJ
ejpam-4653	341	15	and	and	CCONJ
ejpam-4653	341	16	only	only	ADV
ejpam-4653	341	17	if	if	SCONJ
ejpam-4653	341	18	γdr(g	γdr(g	X
ejpam-4653	341	19	)	)	PUNCT
ejpam-4653	342	1	=	=	SYM
ejpam-4653	342	2	3	3	NUM
ejpam-4653	342	3	or	or	CCONJ
ejpam-4653	342	4	γdr(h	γdr(h	NUM
ejpam-4653	342	5	)	)	PUNCT
ejpam-4653	342	6	=	=	SYM
ejpam-4653	343	1	3	3	X
ejpam-4653	343	2	.	.	PUNCT
ejpam-4653	343	3	(	(	PUNCT
ejpam-4653	343	4	ii	ii	NOUN
ejpam-4653	343	5	)	)	PUNCT
ejpam-4653	343	6	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	343	7	)	)	PUNCT
ejpam-4653	343	8	=	=	SYM
ejpam-4653	344	1	4	4	NUM
ejpam-4653	345	1	if	if	SCONJ
ejpam-4653	345	2	and	and	CCONJ
ejpam-4653	345	3	only	only	ADV
ejpam-4653	345	4	if	if	SCONJ
ejpam-4653	345	5	min{γdr(g	min{γdr(g	PROPN
ejpam-4653	345	6	)	)	PUNCT
ejpam-4653	345	7	,	,	PUNCT
ejpam-4653	345	8	γdr(h	γdr(h	NUM
ejpam-4653	345	9	)	)	PUNCT
ejpam-4653	345	10	}	}	PUNCT
ejpam-4653	345	11	=	=	SYM
ejpam-4653	345	12	4	4	X
ejpam-4653	345	13	.	.	PUNCT
ejpam-4653	345	14	(	(	PUNCT
ejpam-4653	345	15	iii	iii	NOUN
ejpam-4653	345	16	)	)	PUNCT
ejpam-4653	345	17	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	345	18	)	)	PUNCT
ejpam-4653	345	19	=	=	SYM
ejpam-4653	345	20	5	5	NUM
ejpam-4653	345	21	if	if	SCONJ
ejpam-4653	345	22	and	and	CCONJ
ejpam-4653	345	23	only	only	ADV
ejpam-4653	345	24	if	if	SCONJ
ejpam-4653	345	25	min{γdr(g	min{γdr(g	PROPN
ejpam-4653	345	26	)	)	PUNCT
ejpam-4653	345	27	,	,	PUNCT
ejpam-4653	345	28	γdr(h	γdr(h	NUM
ejpam-4653	345	29	)	)	PUNCT
ejpam-4653	345	30	}	}	PUNCT
ejpam-4653	346	1	=	=	SYM
ejpam-4653	346	2	5	5	X
ejpam-4653	346	3	.	.	PUNCT
ejpam-4653	346	4	(	(	PUNCT
ejpam-4653	346	5	iv	iv	X
ejpam-4653	346	6	)	)	PUNCT
ejpam-4653	346	7	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	346	8	)	)	PUNCT
ejpam-4653	346	9	=	=	PUNCT
ejpam-4653	347	1	6	6	NUM
ejpam-4653	347	2	if	if	SCONJ
ejpam-4653	347	3	and	and	CCONJ
ejpam-4653	347	4	only	only	ADV
ejpam-4653	347	5	if	if	SCONJ
ejpam-4653	347	6	γdr(g	γdr(g	X
ejpam-4653	347	7	)	)	PUNCT
ejpam-4653	347	8	≥	≥	NOUN
ejpam-4653	347	9	6	6	NUM
ejpam-4653	347	10	and	and	CCONJ
ejpam-4653	347	11	γdr(h	γdr(h	NUM
ejpam-4653	347	12	)	)	PUNCT
ejpam-4653	347	13	≥	≥	NOUN
ejpam-4653	347	14	6	6	NUM
ejpam-4653	347	15	.	.	PUNCT
ejpam-4653	348	1	proof	proof	NOUN
ejpam-4653	348	2	.	.	PUNCT
ejpam-4653	349	1	since	since	SCONJ
ejpam-4653	349	2	g+h	g+h	PROPN
ejpam-4653	349	3	is	be	AUX
ejpam-4653	349	4	nontrivial	nontrivial	ADJ
ejpam-4653	349	5	,	,	PUNCT
ejpam-4653	349	6	γdr(g+h	γdr(g+h	PROPN
ejpam-4653	349	7	)	)	PUNCT
ejpam-4653	349	8	≥	≥	NOUN
ejpam-4653	350	1	3	3	NUM
ejpam-4653	350	2	.	.	PUNCT
ejpam-4653	351	1	now	now	ADV
ejpam-4653	351	2	,	,	PUNCT
ejpam-4653	351	3	let	let	VERB
ejpam-4653	351	4	u	u	PRON
ejpam-4653	351	5	∈	∈	PROPN
ejpam-4653	351	6	v	v	ADP
ejpam-4653	351	7	(	(	PUNCT
ejpam-4653	351	8	g	g	NOUN
ejpam-4653	351	9	)	)	PUNCT
ejpam-4653	351	10	and	and	CCONJ
ejpam-4653	351	11	v	v	ADP
ejpam-4653	351	12	∈	∈	NOUN
ejpam-4653	351	13	v	v	NOUN
ejpam-4653	351	14	(	(	PUNCT
ejpam-4653	351	15	g	g	NOUN
ejpam-4653	351	16	)	)	PUNCT
ejpam-4653	351	17	.	.	PUNCT
ejpam-4653	352	1	then	then	ADV
ejpam-4653	352	2	f	f	PROPN
ejpam-4653	352	3	=	=	PUNCT
ejpam-4653	352	4	(	(	PUNCT
ejpam-4653	352	5	v	v	NOUN
ejpam-4653	352	6	(	(	PUNCT
ejpam-4653	352	7	g	g	PROPN
ejpam-4653	352	8	+	+	NOUN
ejpam-4653	352	9	h	h	NOUN
ejpam-4653	352	10	)	)	PUNCT
ejpam-4653	352	11	\	\	NOUN
ejpam-4653	353	1	{	{	PUNCT
ejpam-4653	353	2	u	u	PROPN
ejpam-4653	353	3	,	,	PUNCT
ejpam-4653	353	4	v},∅,∅	v},∅,∅	NOUN
ejpam-4653	353	5	,	,	PUNCT
ejpam-4653	353	6	{	{	PUNCT
ejpam-4653	353	7	u	u	NOUN
ejpam-4653	353	8	,	,	PUNCT
ejpam-4653	353	9	v	v	NOUN
ejpam-4653	353	10	}	}	PUNCT
ejpam-4653	353	11	)	)	PUNCT
ejpam-4653	353	12	∈	∈	PROPN
ejpam-4653	353	13	drd(g	drd(g	VERB
ejpam-4653	353	14	+	+	NUM
ejpam-4653	353	15	h	h	NOUN
ejpam-4653	353	16	)	)	PUNCT
ejpam-4653	353	17	.	.	PUNCT
ejpam-4653	354	1	thus	thus	ADV
ejpam-4653	354	2	,	,	PUNCT
ejpam-4653	354	3	γdr(g	γdr(g	PROPN
ejpam-4653	354	4	+	+	NUM
ejpam-4653	354	5	h	h	NOUN
ejpam-4653	354	6	)	)	PUNCT
ejpam-4653	354	7	≤	≤	NUM
ejpam-4653	354	8	ωg+h(f	ωg+h(f	PROPN
ejpam-4653	354	9	)	)	PUNCT
ejpam-4653	354	10	=	=	SYM
ejpam-4653	354	11	6	6	X
ejpam-4653	354	12	.	.	PUNCT
ejpam-4653	354	13	to	to	PART
ejpam-4653	354	14	prove	prove	VERB
ejpam-4653	354	15	(	(	PUNCT
ejpam-4653	354	16	i	i	NOUN
ejpam-4653	354	17	)	)	PUNCT
ejpam-4653	354	18	,	,	PUNCT
ejpam-4653	354	19	we	we	PRON
ejpam-4653	354	20	have	have	VERB
ejpam-4653	354	21	from	from	ADP
ejpam-4653	354	22	proposition	proposition	NOUN
ejpam-4653	354	23	5	5	NUM
ejpam-4653	354	24	,	,	PUNCT
ejpam-4653	354	25	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	354	26	)	)	PUNCT
ejpam-4653	355	1	=	=	SYM
ejpam-4653	355	2	3	3	NUM
ejpam-4653	355	3	⇐	⇐	ADJ
ejpam-4653	355	4	⇒	⇒	NOUN
ejpam-4653	355	5	γ(g+h	γ(g+h	PROPN
ejpam-4653	355	6	)	)	PUNCT
ejpam-4653	356	1	=	=	SYM
ejpam-4653	356	2	1	1	NUM
ejpam-4653	356	3	⇐	⇐	ADJ
ejpam-4653	356	4	⇒	⇒	PROPN
ejpam-4653	356	5	γ(g	γ(g	PROPN
ejpam-4653	356	6	)	)	PUNCT
ejpam-4653	356	7	=	=	SYM
ejpam-4653	356	8	1	1	NUM
ejpam-4653	356	9	or	or	CCONJ
ejpam-4653	356	10	γ(h	γ(h	NOUN
ejpam-4653	356	11	)	)	PUNCT
ejpam-4653	356	12	=	=	SYM
ejpam-4653	356	13	1	1	NUM
ejpam-4653	356	14	j.	j.	PROPN
ejpam-4653	356	15	b.g	b.g	PROPN
ejpam-4653	356	16	.	.	PROPN
ejpam-4653	356	17	cariaga	cariaga	PROPN
ejpam-4653	356	18	,	,	PUNCT
ejpam-4653	356	19	f.	f.	PROPN
ejpam-4653	356	20	jamil	jamil	PROPN
ejpam-4653	356	21	/	/	SYM
ejpam-4653	356	22	eur	eur	PROPN
ejpam-4653	356	23	.	.	PUNCT
ejpam-4653	357	1	j.	j.	PROPN
ejpam-4653	357	2	pure	pure	PROPN
ejpam-4653	357	3	appl	appl	PROPN
ejpam-4653	357	4	.	.	PROPN
ejpam-4653	357	5	math	math	PROPN
ejpam-4653	357	6	,	,	PUNCT
ejpam-4653	357	7	16	16	NUM
ejpam-4653	357	8	(	(	PUNCT
ejpam-4653	357	9	2	2	NUM
ejpam-4653	357	10	)	)	PUNCT
ejpam-4653	357	11	(	(	PUNCT
ejpam-4653	357	12	2023	2023	NUM
ejpam-4653	357	13	)	)	PUNCT
ejpam-4653	357	14	,	,	PUNCT
ejpam-4653	357	15	847	847	NUM
ejpam-4653	357	16	-	-	SYM
ejpam-4653	357	17	863	863	NUM
ejpam-4653	357	18	855	855	NUM
ejpam-4653	357	19	⇐	⇐	ADJ
ejpam-4653	357	20	⇒	⇒	NOUN
ejpam-4653	357	21	γdr(g	γdr(g	PROPN
ejpam-4653	357	22	)	)	PUNCT
ejpam-4653	357	23	=	=	SYM
ejpam-4653	357	24	3	3	NUM
ejpam-4653	357	25	or	or	CCONJ
ejpam-4653	357	26	γdr(h	γdr(h	NUM
ejpam-4653	357	27	)	)	PUNCT
ejpam-4653	357	28	=	=	PUNCT
ejpam-4653	358	1	3	3	X
ejpam-4653	358	2	.	.	X
ejpam-4653	358	3	for	for	ADP
ejpam-4653	358	4	(	(	PUNCT
ejpam-4653	358	5	ii)-(iii	ii)-(iii	NOUN
ejpam-4653	358	6	)	)	PUNCT
ejpam-4653	358	7	,	,	PUNCT
ejpam-4653	358	8	put	put	VERB
ejpam-4653	358	9	α	α	PROPN
ejpam-4653	358	10	=	=	SYM
ejpam-4653	358	11	min{γdr(g	min{γdr(g	PROPN
ejpam-4653	358	12	)	)	PUNCT
ejpam-4653	358	13	,	,	PUNCT
ejpam-4653	358	14	γdr(h	γdr(h	PROPN
ejpam-4653	358	15	)	)	PUNCT
ejpam-4653	358	16	}	}	PUNCT
ejpam-4653	358	17	.	.	PUNCT
ejpam-4653	359	1	suppose	suppose	VERB
ejpam-4653	359	2	that	that	SCONJ
ejpam-4653	359	3	γdr(g	γdr(g	PROPN
ejpam-4653	359	4	+	+	NUM
ejpam-4653	359	5	h	h	NOUN
ejpam-4653	359	6	)	)	PUNCT
ejpam-4653	359	7	=	=	SYM
ejpam-4653	359	8	4	4	X
ejpam-4653	359	9	.	.	PUNCT
ejpam-4653	359	10	by	by	ADP
ejpam-4653	359	11	proposition	proposition	NOUN
ejpam-4653	359	12	5	5	NUM
ejpam-4653	359	13	,	,	PUNCT
ejpam-4653	359	14	γ(g	γ(g	PUNCT
ejpam-4653	360	1	+	+	CCONJ
ejpam-4653	360	2	h	h	X
ejpam-4653	360	3	)	)	PUNCT
ejpam-4653	360	4	=	=	PUNCT
ejpam-4653	361	1	γ2(g	γ2(g	VERB
ejpam-4653	361	2	+	+	NUM
ejpam-4653	361	3	h	h	NOUN
ejpam-4653	361	4	)	)	PUNCT
ejpam-4653	361	5	=	=	SYM
ejpam-4653	362	1	2	2	X
ejpam-4653	362	2	.	.	X
ejpam-4653	362	3	let	let	VERB
ejpam-4653	362	4	s	s	AUX
ejpam-4653	362	5	=	=	PUNCT
ejpam-4653	362	6	{	{	PUNCT
ejpam-4653	362	7	u	u	NOUN
ejpam-4653	362	8	,	,	PUNCT
ejpam-4653	362	9	v	v	NOUN
ejpam-4653	362	10	}	}	PUNCT
ejpam-4653	362	11	be	be	AUX
ejpam-4653	362	12	a	a	DET
ejpam-4653	362	13	γ2	γ2	NOUN
ejpam-4653	362	14	-	-	PUNCT
ejpam-4653	362	15	set	set	NOUN
ejpam-4653	362	16	of	of	ADP
ejpam-4653	362	17	g	g	PROPN
ejpam-4653	362	18	+	+	CCONJ
ejpam-4653	362	19	h.	h.	NOUN
ejpam-4653	362	20	if	if	SCONJ
ejpam-4653	362	21	u	u	PROPN
ejpam-4653	362	22	∈	∈	PROPN
ejpam-4653	362	23	v	v	ADP
ejpam-4653	362	24	(	(	PUNCT
ejpam-4653	362	25	g	g	NOUN
ejpam-4653	362	26	)	)	PUNCT
ejpam-4653	362	27	and	and	CCONJ
ejpam-4653	362	28	v	v	ADP
ejpam-4653	362	29	∈	∈	PROPN
ejpam-4653	362	30	v	v	NOUN
ejpam-4653	362	31	(	(	PUNCT
ejpam-4653	362	32	h	h	NOUN
ejpam-4653	362	33	)	)	PUNCT
ejpam-4653	362	34	,	,	PUNCT
ejpam-4653	362	35	then	then	ADV
ejpam-4653	362	36	γ(g	γ(g	PROPN
ejpam-4653	362	37	)	)	PUNCT
ejpam-4653	363	1	=	=	SYM
ejpam-4653	363	2	1	1	NUM
ejpam-4653	363	3	and	and	CCONJ
ejpam-4653	363	4	γ(h	γ(h	NOUN
ejpam-4653	363	5	)	)	PUNCT
ejpam-4653	363	6	=	=	SYM
ejpam-4653	364	1	1	1	NUM
ejpam-4653	364	2	,	,	PUNCT
ejpam-4653	364	3	a	a	DET
ejpam-4653	364	4	contradiction	contradiction	NOUN
ejpam-4653	364	5	.	.	PUNCT
ejpam-4653	365	1	thus	thus	ADV
ejpam-4653	365	2	,	,	PUNCT
ejpam-4653	365	3	s	s	VERB
ejpam-4653	365	4	⊆	⊆	NUM
ejpam-4653	365	5	v	v	NOUN
ejpam-4653	365	6	(	(	PUNCT
ejpam-4653	365	7	g	g	NOUN
ejpam-4653	365	8	)	)	PUNCT
ejpam-4653	365	9	or	or	CCONJ
ejpam-4653	365	10	s	s	PRON
ejpam-4653	365	11	⊆	⊆	NUM
ejpam-4653	365	12	v	v	NOUN
ejpam-4653	365	13	(	(	PUNCT
ejpam-4653	365	14	h	h	NOUN
ejpam-4653	365	15	)	)	PUNCT
ejpam-4653	365	16	.	.	PUNCT
ejpam-4653	366	1	consequently	consequently	ADV
ejpam-4653	366	2	,	,	PUNCT
ejpam-4653	366	3	γ(g	γ(g	PROPN
ejpam-4653	366	4	)	)	PUNCT
ejpam-4653	366	5	=	=	PUNCT
ejpam-4653	366	6	γ2(g	γ2(g	VERB
ejpam-4653	366	7	)	)	PUNCT
ejpam-4653	366	8	=	=	SYM
ejpam-4653	366	9	2	2	NUM
ejpam-4653	366	10	or	or	CCONJ
ejpam-4653	366	11	γ(h	γ(h	NOUN
ejpam-4653	366	12	)	)	PUNCT
ejpam-4653	366	13	=	=	PUNCT
ejpam-4653	367	1	γ2(h	γ2(h	X
ejpam-4653	367	2	)	)	PUNCT
ejpam-4653	367	3	=	=	SYM
ejpam-4653	367	4	2	2	X
ejpam-4653	367	5	.	.	PUNCT
ejpam-4653	367	6	by	by	ADP
ejpam-4653	367	7	proposition	proposition	NOUN
ejpam-4653	367	8	5	5	NUM
ejpam-4653	367	9	,	,	PUNCT
ejpam-4653	367	10	γdr(g	γdr(g	X
ejpam-4653	367	11	)	)	PUNCT
ejpam-4653	367	12	=	=	SYM
ejpam-4653	367	13	4	4	NUM
ejpam-4653	367	14	or	or	CCONJ
ejpam-4653	367	15	γdr(h	γdr(h	NUM
ejpam-4653	367	16	)	)	PUNCT
ejpam-4653	367	17	=	=	PUNCT
ejpam-4653	368	1	4	4	X
ejpam-4653	368	2	.	.	PUNCT
ejpam-4653	368	3	by	by	ADP
ejpam-4653	368	4	(	(	PUNCT
ejpam-4653	368	5	i	i	NOUN
ejpam-4653	368	6	)	)	PUNCT
ejpam-4653	368	7	,	,	PUNCT
ejpam-4653	368	8	α	α	X
ejpam-4653	368	9	=	=	VERB
ejpam-4653	368	10	4	4	X
ejpam-4653	368	11	.	.	PUNCT
ejpam-4653	368	12	conversely	conversely	ADV
ejpam-4653	368	13	,	,	PUNCT
ejpam-4653	368	14	suppose	suppose	VERB
ejpam-4653	368	15	that	that	SCONJ
ejpam-4653	368	16	α	α	NOUN
ejpam-4653	368	17	=	=	NOUN
ejpam-4653	368	18	4	4	NUM
ejpam-4653	368	19	,	,	PUNCT
ejpam-4653	368	20	and	and	CCONJ
ejpam-4653	368	21	let	let	VERB
ejpam-4653	368	22	f	f	PRON
ejpam-4653	368	23	be	be	AUX
ejpam-4653	368	24	a	a	DET
ejpam-4653	368	25	γdr	γdr	NOUN
ejpam-4653	368	26	-	-	PUNCT
ejpam-4653	368	27	function	function	NOUN
ejpam-4653	368	28	of	of	ADP
ejpam-4653	368	29	g.	g.	PROPN
ejpam-4653	368	30	then	then	ADV
ejpam-4653	368	31	g	g	PROPN
ejpam-4653	368	32	=	=	SYM
ejpam-4653	368	33	(	(	PUNCT
ejpam-4653	368	34	v0	v0	PROPN
ejpam-4653	368	35	∪	∪	X
ejpam-4653	368	36	v	v	NOUN
ejpam-4653	368	37	(	(	PUNCT
ejpam-4653	368	38	h	h	NOUN
ejpam-4653	368	39	)	)	PUNCT
ejpam-4653	368	40	,	,	PUNCT
ejpam-4653	368	41	v1	v1	NOUN
ejpam-4653	368	42	,	,	PUNCT
ejpam-4653	368	43	v2	v2	PROPN
ejpam-4653	368	44	,	,	PUNCT
ejpam-4653	368	45	v3	v3	PROPN
ejpam-4653	368	46	)	)	PUNCT
ejpam-4653	368	47	∈	∈	PROPN
ejpam-4653	368	48	drd(g+h	drd(g+h	PROPN
ejpam-4653	368	49	)	)	PUNCT
ejpam-4653	368	50	with	with	ADP
ejpam-4653	368	51	ωg+h(g	ωg+h(g	NOUN
ejpam-4653	368	52	)	)	PUNCT
ejpam-4653	368	53	=	=	SYM
ejpam-4653	368	54	4	4	X
ejpam-4653	368	55	.	.	X
ejpam-4653	368	56	hence	hence	ADV
ejpam-4653	368	57	,	,	PUNCT
ejpam-4653	368	58	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	368	59	)	)	PUNCT
ejpam-4653	368	60	≤	≤	NOUN
ejpam-4653	368	61	ωg+h(g	ωg+h(g	NOUN
ejpam-4653	368	62	)	)	PUNCT
ejpam-4653	368	63	=	=	SYM
ejpam-4653	368	64	ωg(f	ωg(f	X
ejpam-4653	368	65	)	)	PUNCT
ejpam-4653	368	66	=	=	SYM
ejpam-4653	368	67	4	4	X
ejpam-4653	368	68	.	.	PUNCT
ejpam-4653	368	69	by	by	ADP
ejpam-4653	368	70	(	(	PUNCT
ejpam-4653	368	71	i	i	NOUN
ejpam-4653	368	72	)	)	PUNCT
ejpam-4653	368	73	,	,	PUNCT
ejpam-4653	368	74	γdr(g+h	γdr(g+h	PUNCT
ejpam-4653	368	75	)	)	PUNCT
ejpam-4653	368	76	=	=	SYM
ejpam-4653	368	77	4	4	X
ejpam-4653	368	78	.	.	PUNCT
ejpam-4653	368	79	suppose	suppose	VERB
ejpam-4653	368	80	that	that	SCONJ
ejpam-4653	368	81	the	the	DET
ejpam-4653	368	82	γdr(g+h	γdr(g+h	NOUN
ejpam-4653	368	83	)	)	PUNCT
ejpam-4653	368	84	=	=	SYM
ejpam-4653	369	1	5	5	X
ejpam-4653	369	2	.	.	PUNCT
ejpam-4653	369	3	by	by	ADP
ejpam-4653	369	4	(	(	PUNCT
ejpam-4653	369	5	i	i	NOUN
ejpam-4653	369	6	)	)	PUNCT
ejpam-4653	369	7	and	and	CCONJ
ejpam-4653	369	8	(	(	PUNCT
ejpam-4653	369	9	ii	ii	NOUN
ejpam-4653	369	10	)	)	PUNCT
ejpam-4653	369	11	,	,	PUNCT
ejpam-4653	369	12	α	α	PRON
ejpam-4653	369	13	≥	≥	NUM
ejpam-4653	369	14	5	5	NUM
ejpam-4653	369	15	.	.	PUNCT
ejpam-4653	370	1	it	it	PRON
ejpam-4653	370	2	follows	follow	VERB
ejpam-4653	370	3	from	from	ADP
ejpam-4653	370	4	proposition	proposition	NOUN
ejpam-4653	370	5	6	6	NUM
ejpam-4653	370	6	that	that	SCONJ
ejpam-4653	370	7	γ(g+h	γ(g+h	ADP
ejpam-4653	370	8	)	)	PUNCT
ejpam-4653	370	9	=	=	SYM
ejpam-4653	370	10	2	2	NUM
ejpam-4653	370	11	,	,	PUNCT
ejpam-4653	370	12	γ2(g+h	γ2(g+h	NOUN
ejpam-4653	370	13	)	)	PUNCT
ejpam-4653	370	14	≥	≥	NOUN
ejpam-4653	370	15	3	3	NUM
ejpam-4653	370	16	and	and	CCONJ
ejpam-4653	370	17	there	there	PRON
ejpam-4653	370	18	exists	exist	VERB
ejpam-4653	370	19	v	v	ADP
ejpam-4653	370	20	∈	∈	PROPN
ejpam-4653	370	21	v	v	NOUN
ejpam-4653	370	22	(	(	PUNCT
ejpam-4653	370	23	g+h	g+h	NOUN
ejpam-4653	370	24	)	)	PUNCT
ejpam-4653	370	25	for	for	ADP
ejpam-4653	370	26	which	which	PRON
ejpam-4653	370	27	γ((g+h)−v	γ((g+h)−v	PUNCT
ejpam-4653	370	28	)	)	PUNCT
ejpam-4653	371	1	=	=	SYM
ejpam-4653	371	2	1	1	X
ejpam-4653	371	3	.	.	X
ejpam-4653	371	4	wlog	wlog	NOUN
ejpam-4653	371	5	,	,	PUNCT
ejpam-4653	371	6	assume	assume	VERB
ejpam-4653	371	7	that	that	SCONJ
ejpam-4653	371	8	v	v	X
ejpam-4653	371	9	∈	∈	PROPN
ejpam-4653	371	10	v	v	NOUN
ejpam-4653	371	11	(	(	PUNCT
ejpam-4653	371	12	g	g	NOUN
ejpam-4653	371	13	)	)	PUNCT
ejpam-4653	371	14	.	.	PUNCT
ejpam-4653	372	1	then	then	ADV
ejpam-4653	372	2	γ(g	γ(g	VERB
ejpam-4653	372	3	−	−	PROPN
ejpam-4653	372	4	v	v	NOUN
ejpam-4653	372	5	)	)	PUNCT
ejpam-4653	372	6	=	=	SYM
ejpam-4653	372	7	1	1	NUM
ejpam-4653	372	8	and	and	CCONJ
ejpam-4653	372	9	,	,	PUNCT
ejpam-4653	372	10	consequently	consequently	ADV
ejpam-4653	372	11	,	,	PUNCT
ejpam-4653	372	12	γ(g	γ(g	PROPN
ejpam-4653	372	13	)	)	PUNCT
ejpam-4653	372	14	=	=	PUNCT
ejpam-4653	373	1	2	2	X
ejpam-4653	373	2	.	.	X
ejpam-4653	373	3	if	if	SCONJ
ejpam-4653	373	4	γ2(g	γ2(g	VERB
ejpam-4653	373	5	)	)	PUNCT
ejpam-4653	373	6	=	=	SYM
ejpam-4653	373	7	2	2	NUM
ejpam-4653	373	8	,	,	PUNCT
ejpam-4653	373	9	then	then	ADV
ejpam-4653	373	10	γdr(g	γdr(g	X
ejpam-4653	373	11	)	)	PUNCT
ejpam-4653	373	12	=	=	SYM
ejpam-4653	373	13	4	4	NUM
ejpam-4653	373	14	by	by	ADP
ejpam-4653	373	15	proposition	proposition	NOUN
ejpam-4653	373	16	5	5	NUM
ejpam-4653	373	17	,	,	PUNCT
ejpam-4653	373	18	a	a	DET
ejpam-4653	373	19	contradiction	contradiction	NOUN
ejpam-4653	373	20	by	by	ADP
ejpam-4653	373	21	(	(	PUNCT
ejpam-4653	373	22	ii	ii	NOUN
ejpam-4653	373	23	)	)	PUNCT
ejpam-4653	373	24	.	.	PUNCT
ejpam-4653	374	1	thus	thus	ADV
ejpam-4653	374	2	,	,	PUNCT
ejpam-4653	374	3	γ2(g	γ2(g	ADP
ejpam-4653	374	4	)	)	PUNCT
ejpam-4653	374	5	≥	≥	NOUN
ejpam-4653	374	6	3	3	NUM
ejpam-4653	374	7	so	so	SCONJ
ejpam-4653	374	8	that	that	SCONJ
ejpam-4653	374	9	γdr(g	γdr(g	X
ejpam-4653	374	10	)	)	PUNCT
ejpam-4653	374	11	=	=	SYM
ejpam-4653	374	12	5	5	X
ejpam-4653	374	13	.	.	PUNCT
ejpam-4653	375	1	thus	thus	ADV
ejpam-4653	375	2	,	,	PUNCT
ejpam-4653	375	3	α	α	PROPN
ejpam-4653	375	4	≤	≤	ADJ
ejpam-4653	375	5	5	5	NUM
ejpam-4653	375	6	.	.	PUNCT
ejpam-4653	376	1	conversely	conversely	ADV
ejpam-4653	376	2	,	,	PUNCT
ejpam-4653	376	3	assume	assume	VERB
ejpam-4653	376	4	α	α	X
ejpam-4653	376	5	=	=	SYM
ejpam-4653	376	6	γdr(g	γdr(g	PROPN
ejpam-4653	376	7	)	)	PUNCT
ejpam-4653	376	8	=	=	SYM
ejpam-4653	376	9	5	5	NUM
ejpam-4653	376	10	,	,	PUNCT
ejpam-4653	376	11	and	and	CCONJ
ejpam-4653	376	12	let	let	VERB
ejpam-4653	376	13	f	f	PROPN
ejpam-4653	376	14	=	=	SYM
ejpam-4653	376	15	(	(	PUNCT
ejpam-4653	376	16	v0	v0	PROPN
ejpam-4653	376	17	,	,	PUNCT
ejpam-4653	376	18	v1	v1	NOUN
ejpam-4653	376	19	,	,	PUNCT
ejpam-4653	376	20	v2	v2	PROPN
ejpam-4653	376	21	,	,	PUNCT
ejpam-4653	376	22	v3	v3	PROPN
ejpam-4653	376	23	)	)	PUNCT
ejpam-4653	376	24	be	be	VERB
ejpam-4653	376	25	a	a	DET
ejpam-4653	376	26	γdr	γdr	NOUN
ejpam-4653	376	27	-	-	PUNCT
ejpam-4653	376	28	function	function	NOUN
ejpam-4653	376	29	of	of	ADP
ejpam-4653	376	30	g.	g.	PROPN
ejpam-4653	377	1	then	then	ADV
ejpam-4653	377	2	g	g	PROPN
ejpam-4653	377	3	=	=	SYM
ejpam-4653	377	4	(	(	PUNCT
ejpam-4653	377	5	v0	v0	PROPN
ejpam-4653	377	6	∪	∪	X
ejpam-4653	377	7	v	v	NOUN
ejpam-4653	377	8	(	(	PUNCT
ejpam-4653	377	9	h	h	NOUN
ejpam-4653	377	10	)	)	PUNCT
ejpam-4653	377	11	,	,	PUNCT
ejpam-4653	377	12	v1	v1	NOUN
ejpam-4653	377	13	,	,	PUNCT
ejpam-4653	377	14	v2	v2	PROPN
ejpam-4653	377	15	,	,	PUNCT
ejpam-4653	377	16	v3	v3	PROPN
ejpam-4653	377	17	)	)	PUNCT
ejpam-4653	377	18	∈	∈	PROPN
ejpam-4653	377	19	drd(g+h	drd(g+h	PROPN
ejpam-4653	377	20	)	)	PUNCT
ejpam-4653	377	21	.	.	PUNCT
ejpam-4653	378	1	hence	hence	ADV
ejpam-4653	378	2	,	,	PUNCT
ejpam-4653	378	3	γdr(g	γdr(g	PROPN
ejpam-4653	378	4	+	+	NOUN
ejpam-4653	378	5	h	h	NOUN
ejpam-4653	378	6	)	)	PUNCT
ejpam-4653	378	7	≤	≤	NOUN
ejpam-4653	378	8	ωg+h(g	ωg+h(g	NOUN
ejpam-4653	378	9	)	)	PUNCT
ejpam-4653	378	10	=	=	SYM
ejpam-4653	378	11	ωg(f	ωg(f	X
ejpam-4653	378	12	)	)	PUNCT
ejpam-4653	378	13	=	=	SYM
ejpam-4653	378	14	5	5	NUM
ejpam-4653	378	15	=	=	SYM
ejpam-4653	378	16	α	α	NOUN
ejpam-4653	378	17	.	.	PUNCT
ejpam-4653	379	1	but	but	CCONJ
ejpam-4653	379	2	by	by	ADP
ejpam-4653	379	3	(	(	PUNCT
ejpam-4653	379	4	i	i	NOUN
ejpam-4653	379	5	)	)	PUNCT
ejpam-4653	379	6	and	and	CCONJ
ejpam-4653	379	7	(	(	PUNCT
ejpam-4653	379	8	ii	ii	NOUN
ejpam-4653	379	9	)	)	PUNCT
ejpam-4653	379	10	,	,	PUNCT
ejpam-4653	380	1	γdr(g	γdr(g	PROPN
ejpam-4653	380	2	+	+	NOUN
ejpam-4653	380	3	h	h	NOUN
ejpam-4653	380	4	)	)	PUNCT
ejpam-4653	380	5	≥	≥	NOUN
ejpam-4653	380	6	5	5	NUM
ejpam-4653	380	7	.	.	PUNCT
ejpam-4653	380	8	therefore	therefore	ADV
ejpam-4653	380	9	,	,	PUNCT
ejpam-4653	380	10	γdr(g+h	γdr(g+h	PUNCT
ejpam-4653	380	11	)	)	PUNCT
ejpam-4653	380	12	=	=	SYM
ejpam-4653	381	1	5	5	X
ejpam-4653	381	2	.	.	PUNCT
ejpam-4653	381	3	finally	finally	ADV
ejpam-4653	381	4	,	,	PUNCT
ejpam-4653	381	5	(	(	PUNCT
ejpam-4653	381	6	iv	iv	X
ejpam-4653	381	7	)	)	PUNCT
ejpam-4653	381	8	follows	follow	VERB
ejpam-4653	381	9	immediately	immediately	ADV
ejpam-4653	381	10	from	from	ADP
ejpam-4653	381	11	equation	equation	NOUN
ejpam-4653	381	12	1	1	NUM
ejpam-4653	381	13	and	and	CCONJ
ejpam-4653	381	14	statements	statement	NOUN
ejpam-4653	381	15	(	(	PUNCT
ejpam-4653	381	16	i	i	NOUN
ejpam-4653	381	17	)	)	PUNCT
ejpam-4653	381	18	,	,	PUNCT
ejpam-4653	381	19	(	(	PUNCT
ejpam-4653	381	20	ii	ii	NOUN
ejpam-4653	381	21	)	)	PUNCT
ejpam-4653	381	22	and	and	CCONJ
ejpam-4653	381	23	(	(	PUNCT
ejpam-4653	381	24	iii	iii	NOUN
ejpam-4653	381	25	)	)	PUNCT
ejpam-4653	381	26	.	.	PUNCT
ejpam-4653	382	1	the	the	DET
ejpam-4653	382	2	complementary	complementary	ADJ
ejpam-4653	382	3	prism	prism	NOUN
ejpam-4653	382	4	is	be	AUX
ejpam-4653	382	5	the	the	DET
ejpam-4653	382	6	graph	graph	NOUN
ejpam-4653	382	7	gg	gg	NOUN
ejpam-4653	382	8	formed	form	VERB
ejpam-4653	382	9	from	from	ADP
ejpam-4653	382	10	g	g	PROPN
ejpam-4653	382	11	and	and	CCONJ
ejpam-4653	382	12	its	its	PRON
ejpam-4653	382	13	complement	complement	NOUN
ejpam-4653	382	14	g	g	NOUN
ejpam-4653	382	15	by	by	ADP
ejpam-4653	382	16	adding	add	VERB
ejpam-4653	382	17	a	a	DET
ejpam-4653	382	18	perfect	perfect	ADJ
ejpam-4653	382	19	matching	matching	NOUN
ejpam-4653	382	20	between	between	ADP
ejpam-4653	382	21	corresponding	corresponding	ADJ
ejpam-4653	382	22	vertices	vertex	NOUN
ejpam-4653	382	23	of	of	ADP
ejpam-4653	382	24	g	g	PROPN
ejpam-4653	382	25	and	and	CCONJ
ejpam-4653	382	26	g.	g.	PROPN
ejpam-4653	382	27	if	if	SCONJ
ejpam-4653	382	28	for	for	ADP
ejpam-4653	382	29	each	each	DET
ejpam-4653	382	30	v	v	NUM
ejpam-4653	382	31	∈	∈	PROPN
ejpam-4653	382	32	v	v	NOUN
ejpam-4653	382	33	(	(	PUNCT
ejpam-4653	382	34	g	g	NOUN
ejpam-4653	382	35	)	)	PUNCT
ejpam-4653	382	36	,	,	PUNCT
ejpam-4653	382	37	v	v	NOUN
ejpam-4653	382	38	is	be	AUX
ejpam-4653	382	39	the	the	DET
ejpam-4653	382	40	vertex	vertex	NOUN
ejpam-4653	382	41	in	in	ADP
ejpam-4653	382	42	g	g	PROPN
ejpam-4653	382	43	corresponding	correspond	VERB
ejpam-4653	382	44	to	to	ADP
ejpam-4653	382	45	v	v	NOUN
ejpam-4653	382	46	,	,	PUNCT
ejpam-4653	382	47	then	then	ADV
ejpam-4653	382	48	gg	gg	PROPN
ejpam-4653	382	49	is	be	AUX
ejpam-4653	382	50	formed	form	VERB
ejpam-4653	382	51	by	by	ADP
ejpam-4653	382	52	adding	add	VERB
ejpam-4653	382	53	the	the	DET
ejpam-4653	382	54	edge	edge	NOUN
ejpam-4653	382	55	vv	vv	NOUN
ejpam-4653	382	56	for	for	ADP
ejpam-4653	382	57	every	every	DET
ejpam-4653	382	58	v	v	NUM
ejpam-4653	382	59	∈	∈	NOUN
ejpam-4653	382	60	v	v	NOUN
ejpam-4653	382	61	(	(	PUNCT
ejpam-4653	382	62	g	g	NOUN
ejpam-4653	382	63	)	)	PUNCT
ejpam-4653	382	64	.	.	PUNCT
ejpam-4653	383	1	remark	remark	PROPN
ejpam-4653	383	2	1	1	NUM
ejpam-4653	383	3	.	.	PUNCT
ejpam-4653	384	1	(	(	PUNCT
ejpam-4653	384	2	i	i	NOUN
ejpam-4653	384	3	)	)	PUNCT
ejpam-4653	384	4	for	for	ADP
ejpam-4653	384	5	any	any	DET
ejpam-4653	384	6	path	path	NOUN
ejpam-4653	384	7	pn	pn	NOUN
ejpam-4653	384	8	of	of	ADP
ejpam-4653	384	9	order	order	NOUN
ejpam-4653	384	10	n	n	PRON
ejpam-4653	384	11	≥	≥	NUM
ejpam-4653	384	12	3	3	NUM
ejpam-4653	384	13	,	,	PUNCT
ejpam-4653	384	14	γdr(pnpn	γdr(pnpn	VERB
ejpam-4653	384	15	)	)	PUNCT
ejpam-4653	384	16	=	=	PRON
ejpam-4653	384	17	{	{	PUNCT
ejpam-4653	384	18	3	3	NUM
ejpam-4653	384	19	+	+	SYM
ejpam-4653	384	20	n	n	CCONJ
ejpam-4653	384	21	,	,	PUNCT
ejpam-4653	385	1	if	if	SCONJ
ejpam-4653	385	2	n	n	PRON
ejpam-4653	385	3	≡	≡	PROPN
ejpam-4653	385	4	0	0	PUNCT
ejpam-4653	385	5	(	(	PUNCT
ejpam-4653	385	6	mod	mod	NOUN
ejpam-4653	385	7	3	3	NUM
ejpam-4653	385	8	)	)	PUNCT
ejpam-4653	385	9	,	,	PUNCT
ejpam-4653	385	10	3	3	NUM
ejpam-4653	385	11	+	+	CCONJ
ejpam-4653	385	12	(	(	PUNCT
ejpam-4653	385	13	n+	n+	NOUN
ejpam-4653	385	14	1	1	NUM
ejpam-4653	385	15	)	)	PUNCT
ejpam-4653	385	16	,	,	PUNCT
ejpam-4653	385	17	if	if	SCONJ
ejpam-4653	385	18	n	n	PRON
ejpam-4653	385	19	≡	≡	PROPN
ejpam-4653	385	20	1	1	NUM
ejpam-4653	385	21	,	,	PUNCT
ejpam-4653	385	22	2	2	NUM
ejpam-4653	385	23	(	(	PUNCT
ejpam-4653	385	24	mod	mod	NOUN
ejpam-4653	385	25	3	3	NUM
ejpam-4653	385	26	)	)	PUNCT
ejpam-4653	385	27	.	.	PUNCT
ejpam-4653	386	1	(	(	PUNCT
ejpam-4653	386	2	ii	ii	NOUN
ejpam-4653	386	3	)	)	PUNCT
ejpam-4653	386	4	for	for	ADP
ejpam-4653	386	5	any	any	DET
ejpam-4653	386	6	cycle	cycle	NOUN
ejpam-4653	386	7	cn	cn	NOUN
ejpam-4653	386	8	of	of	ADP
ejpam-4653	386	9	order	order	NOUN
ejpam-4653	386	10	n	n	PRON
ejpam-4653	386	11	≥	≥	NOUN
ejpam-4653	386	12	3	3	NUM
ejpam-4653	386	13	,	,	PUNCT
ejpam-4653	386	14	γdr(cncn	γdr(cncn	PUNCT
ejpam-4653	386	15	)	)	PUNCT
ejpam-4653	386	16	=	=	PRON
ejpam-4653	386	17	{	{	PUNCT
ejpam-4653	387	1	4	4	NUM
ejpam-4653	387	2	+	+	SYM
ejpam-4653	387	3	n	n	CCONJ
ejpam-4653	387	4	,	,	PUNCT
ejpam-4653	387	5	if	if	SCONJ
ejpam-4653	387	6	n	n	PRON
ejpam-4653	387	7	≡	≡	PROPN
ejpam-4653	387	8	1	1	NUM
ejpam-4653	387	9	,	,	PUNCT
ejpam-4653	387	10	2	2	NUM
ejpam-4653	387	11	,	,	PUNCT
ejpam-4653	387	12	3	3	NUM
ejpam-4653	387	13	,	,	PUNCT
ejpam-4653	387	14	5	5	NUM
ejpam-4653	387	15	(	(	PUNCT
ejpam-4653	387	16	mod	mod	PROPN
ejpam-4653	387	17	6	6	NUM
ejpam-4653	387	18	)	)	PUNCT
ejpam-4653	387	19	;	;	PUNCT
ejpam-4653	387	20	5	5	NUM
ejpam-4653	387	21	+	+	SYM
ejpam-4653	387	22	n	n	CCONJ
ejpam-4653	387	23	,	,	PUNCT
ejpam-4653	388	1	if	if	SCONJ
ejpam-4653	388	2	n	n	PRON
ejpam-4653	388	3	≡	≡	PROPN
ejpam-4653	388	4	0	0	NUM
ejpam-4653	388	5	,	,	PUNCT
ejpam-4653	388	6	4	4	NUM
ejpam-4653	388	7	(	(	PUNCT
ejpam-4653	388	8	mod	mod	PROPN
ejpam-4653	388	9	6	6	NUM
ejpam-4653	388	10	)	)	PUNCT
ejpam-4653	389	1	.	.	PUNCT
ejpam-4653	390	1	the	the	DET
ejpam-4653	390	2	following	follow	VERB
ejpam-4653	390	3	lemma	lemma	PROPN
ejpam-4653	390	4	is	be	AUX
ejpam-4653	390	5	obvious	obvious	ADJ
ejpam-4653	390	6	.	.	PUNCT
ejpam-4653	391	1	lemma	lemma	PROPN
ejpam-4653	391	2	1	1	NUM
ejpam-4653	391	3	.	.	PUNCT
ejpam-4653	392	1	for	for	ADP
ejpam-4653	392	2	any	any	DET
ejpam-4653	392	3	graph	graph	NOUN
ejpam-4653	392	4	g	g	NOUN
ejpam-4653	392	5	,	,	PUNCT
ejpam-4653	392	6	γ(gg	γ(gg	NUM
ejpam-4653	392	7	)	)	PUNCT
ejpam-4653	392	8	=	=	SYM
ejpam-4653	392	9	1	1	NUM
ejpam-4653	392	10	if	if	SCONJ
ejpam-4653	392	11	and	and	CCONJ
ejpam-4653	392	12	only	only	ADV
ejpam-4653	392	13	if	if	SCONJ
ejpam-4653	392	14	g	g	PROPN
ejpam-4653	392	15	=	=	SYM
ejpam-4653	392	16	k1	k1	PROPN
ejpam-4653	392	17	.	.	PUNCT
ejpam-4653	393	1	proposition	proposition	NOUN
ejpam-4653	393	2	14	14	NUM
ejpam-4653	393	3	.	.	PUNCT
ejpam-4653	394	1	let	let	VERB
ejpam-4653	394	2	g	g	PRON
ejpam-4653	394	3	be	be	AUX
ejpam-4653	394	4	a	a	DET
ejpam-4653	394	5	nontrivial	nontrivial	ADJ
ejpam-4653	394	6	graph	graph	NOUN
ejpam-4653	394	7	.	.	PUNCT
ejpam-4653	395	1	then	then	ADV
ejpam-4653	395	2	(	(	PUNCT
ejpam-4653	395	3	i	i	NOUN
ejpam-4653	395	4	)	)	PUNCT
ejpam-4653	395	5	γdr(g	γdr(g	X
ejpam-4653	395	6	)	)	PUNCT
ejpam-4653	395	7	̸=	̸=	NOUN
ejpam-4653	395	8	4	4	NUM
ejpam-4653	395	9	;	;	PUNCT
ejpam-4653	395	10	(	(	PUNCT
ejpam-4653	395	11	ii	ii	NOUN
ejpam-4653	395	12	)	)	PUNCT
ejpam-4653	395	13	γdr(gg	γdr(gg	NOUN
ejpam-4653	395	14	)	)	PUNCT
ejpam-4653	395	15	=	=	SYM
ejpam-4653	395	16	3	3	NUM
ejpam-4653	395	17	if	if	SCONJ
ejpam-4653	395	18	and	and	CCONJ
ejpam-4653	395	19	only	only	ADV
ejpam-4653	395	20	if	if	SCONJ
ejpam-4653	395	21	g	g	PROPN
ejpam-4653	395	22	=	=	SYM
ejpam-4653	395	23	k1	k1	X
ejpam-4653	395	24	;	;	PUNCT
ejpam-4653	395	25	and	and	CCONJ
ejpam-4653	395	26	j.	j.	PROPN
ejpam-4653	395	27	b.g	b.g	PROPN
ejpam-4653	395	28	.	.	PROPN
ejpam-4653	395	29	cariaga	cariaga	PROPN
ejpam-4653	395	30	,	,	PUNCT
ejpam-4653	395	31	f.	f.	PROPN
ejpam-4653	395	32	jamil	jamil	PROPN
ejpam-4653	395	33	/	/	SYM
ejpam-4653	395	34	eur	eur	PROPN
ejpam-4653	395	35	.	.	PUNCT
ejpam-4653	396	1	j.	j.	PROPN
ejpam-4653	396	2	pure	pure	PROPN
ejpam-4653	396	3	appl	appl	PROPN
ejpam-4653	396	4	.	.	PROPN
ejpam-4653	396	5	math	math	PROPN
ejpam-4653	396	6	,	,	PUNCT
ejpam-4653	396	7	16	16	NUM
ejpam-4653	396	8	(	(	PUNCT
ejpam-4653	396	9	2	2	NUM
ejpam-4653	396	10	)	)	PUNCT
ejpam-4653	396	11	(	(	PUNCT
ejpam-4653	396	12	2023	2023	NUM
ejpam-4653	396	13	)	)	PUNCT
ejpam-4653	396	14	,	,	PUNCT
ejpam-4653	396	15	847	847	NUM
ejpam-4653	396	16	-	-	SYM
ejpam-4653	396	17	863	863	NUM
ejpam-4653	396	18	856	856	NUM
ejpam-4653	396	19	(	(	PUNCT
ejpam-4653	396	20	iii	iii	NOUN
ejpam-4653	396	21	)	)	PUNCT
ejpam-4653	396	22	γdr(gg	γdr(gg	NOUN
ejpam-4653	396	23	)	)	PUNCT
ejpam-4653	396	24	=	=	SYM
ejpam-4653	396	25	5	5	NUM
ejpam-4653	396	26	if	if	SCONJ
ejpam-4653	396	27	and	and	CCONJ
ejpam-4653	396	28	only	only	ADV
ejpam-4653	396	29	if	if	SCONJ
ejpam-4653	396	30	g	g	PROPN
ejpam-4653	396	31	=	=	SYM
ejpam-4653	396	32	{	{	PUNCT
ejpam-4653	396	33	k2,k2	k2,k2	PROPN
ejpam-4653	396	34	}	}	PUNCT
ejpam-4653	396	35	.	.	PUNCT
ejpam-4653	397	1	proof	proof	NOUN
ejpam-4653	397	2	.	.	PUNCT
ejpam-4653	398	1	to	to	PART
ejpam-4653	398	2	prove	prove	VERB
ejpam-4653	398	3	(	(	PUNCT
ejpam-4653	398	4	i	i	NOUN
ejpam-4653	398	5	)	)	PUNCT
ejpam-4653	398	6	,	,	PUNCT
ejpam-4653	398	7	we	we	PRON
ejpam-4653	398	8	claim	claim	VERB
ejpam-4653	398	9	that	that	SCONJ
ejpam-4653	398	10	γ2(gg	γ2(gg	NUM
ejpam-4653	398	11	)	)	PUNCT
ejpam-4653	398	12	̸=	̸=	PROPN
ejpam-4653	398	13	2	2	NUM
ejpam-4653	398	14	.	.	PUNCT
ejpam-4653	398	15	suppose	suppose	VERB
ejpam-4653	398	16	,	,	PUNCT
ejpam-4653	398	17	in	in	ADP
ejpam-4653	398	18	the	the	DET
ejpam-4653	398	19	contrary	contrary	NOUN
ejpam-4653	398	20	,	,	PUNCT
ejpam-4653	398	21	that	that	SCONJ
ejpam-4653	398	22	there	there	PRON
ejpam-4653	398	23	exist	exist	VERB
ejpam-4653	398	24	u	u	NOUN
ejpam-4653	398	25	,	,	PUNCT
ejpam-4653	398	26	v	v	PROPN
ejpam-4653	398	27	∈	∈	PROPN
ejpam-4653	398	28	v	v	NOUN
ejpam-4653	398	29	(	(	PUNCT
ejpam-4653	398	30	gg	gg	NOUN
ejpam-4653	398	31	)	)	PUNCT
ejpam-4653	398	32	such	such	ADJ
ejpam-4653	398	33	that	that	PRON
ejpam-4653	398	34	s	s	PART
ejpam-4653	398	35	=	=	PUNCT
ejpam-4653	398	36	{	{	PUNCT
ejpam-4653	398	37	u	u	NOUN
ejpam-4653	398	38	,	,	PUNCT
ejpam-4653	398	39	v	v	NOUN
ejpam-4653	398	40	}	}	PUNCT
ejpam-4653	398	41	is	be	AUX
ejpam-4653	398	42	a	a	DET
ejpam-4653	398	43	γ2	γ2	NOUN
ejpam-4653	398	44	-	-	PUNCT
ejpam-4653	398	45	set	set	NOUN
ejpam-4653	398	46	of	of	ADP
ejpam-4653	398	47	gg	gg	PROPN
ejpam-4653	398	48	.	.	PUNCT
ejpam-4653	399	1	if	if	SCONJ
ejpam-4653	399	2	u	u	NOUN
ejpam-4653	399	3	,	,	PUNCT
ejpam-4653	399	4	v	v	PROPN
ejpam-4653	399	5	∈	∈	PROPN
ejpam-4653	399	6	v	v	NOUN
ejpam-4653	399	7	(	(	PUNCT
ejpam-4653	399	8	g	g	NOUN
ejpam-4653	399	9	)	)	PUNCT
ejpam-4653	399	10	,	,	PUNCT
ejpam-4653	399	11	then	then	ADV
ejpam-4653	399	12	uv	uv	INTJ
ejpam-4653	399	13	/∈	/∈	PUNCT
ejpam-4653	399	14	e(gg	e(gg	PROPN
ejpam-4653	399	15	)	)	PUNCT
ejpam-4653	399	16	,	,	PUNCT
ejpam-4653	399	17	a	a	DET
ejpam-4653	399	18	contradiction	contradiction	NOUN
ejpam-4653	399	19	.	.	PUNCT
ejpam-4653	400	1	similar	similar	ADJ
ejpam-4653	400	2	contradiction	contradiction	NOUN
ejpam-4653	400	3	is	be	AUX
ejpam-4653	400	4	attained	attain	VERB
ejpam-4653	400	5	if	if	SCONJ
ejpam-4653	400	6	u	u	NOUN
ejpam-4653	400	7	,	,	PUNCT
ejpam-4653	400	8	v	v	PROPN
ejpam-4653	400	9	∈	∈	PROPN
ejpam-4653	400	10	v	v	NOUN
ejpam-4653	400	11	(	(	PUNCT
ejpam-4653	400	12	g	g	NOUN
ejpam-4653	400	13	)	)	PUNCT
ejpam-4653	400	14	.	.	PUNCT
ejpam-4653	401	1	assume	assume	VERB
ejpam-4653	401	2	v	v	ADP
ejpam-4653	401	3	∈	∈	PROPN
ejpam-4653	401	4	v	v	NOUN
ejpam-4653	401	5	(	(	PUNCT
ejpam-4653	401	6	g	g	NOUN
ejpam-4653	401	7	)	)	PUNCT
ejpam-4653	401	8	and	and	CCONJ
ejpam-4653	401	9	u	u	PROPN
ejpam-4653	401	10	∈	∈	PROPN
ejpam-4653	401	11	v	v	NOUN
ejpam-4653	401	12	(	(	PUNCT
ejpam-4653	401	13	g	g	NOUN
ejpam-4653	401	14	)	)	PUNCT
ejpam-4653	401	15	.	.	PUNCT
ejpam-4653	402	1	if	if	SCONJ
ejpam-4653	402	2	u	u	PROPN
ejpam-4653	402	3	=	=	PROPN
ejpam-4653	402	4	v	v	NOUN
ejpam-4653	402	5	,	,	PUNCT
ejpam-4653	402	6	then	then	ADV
ejpam-4653	402	7	for	for	ADP
ejpam-4653	402	8	each	each	DET
ejpam-4653	402	9	w	w	PROPN
ejpam-4653	402	10	∈	∈	PROPN
ejpam-4653	402	11	v	v	ADP
ejpam-4653	402	12	(	(	PUNCT
ejpam-4653	402	13	g	g	NOUN
ejpam-4653	402	14	)	)	PUNCT
ejpam-4653	402	15	\	\	NOUN
ejpam-4653	402	16	{	{	PUNCT
ejpam-4653	402	17	v	v	NOUN
ejpam-4653	402	18	}	}	PUNCT
ejpam-4653	402	19	either	either	CCONJ
ejpam-4653	402	20	wv	wv	PROPN
ejpam-4653	402	21	/∈	/∈	PUNCT
ejpam-4653	402	22	e(gg	e(gg	PROPN
ejpam-4653	402	23	)	)	PUNCT
ejpam-4653	402	24	or	or	CCONJ
ejpam-4653	402	25	uw	uw	PROPN
ejpam-4653	402	26	/∈	/∈	PUNCT
ejpam-4653	402	27	e(gg	e(gg	PROPN
ejpam-4653	402	28	)	)	PUNCT
ejpam-4653	402	29	,	,	PUNCT
ejpam-4653	402	30	a	a	DET
ejpam-4653	402	31	contradiction	contradiction	NOUN
ejpam-4653	402	32	.	.	PUNCT
ejpam-4653	402	33	suppose	suppose	VERB
ejpam-4653	402	34	that	that	SCONJ
ejpam-4653	402	35	u	u	PRON
ejpam-4653	402	36	̸=	̸=	PROPN
ejpam-4653	402	37	v.	v.	ADP
ejpam-4653	402	38	a	a	DET
ejpam-4653	402	39	contradiction	contradiction	NOUN
ejpam-4653	402	40	is	be	AUX
ejpam-4653	402	41	already	already	ADV
ejpam-4653	402	42	attained	attain	VERB
ejpam-4653	402	43	if	if	SCONJ
ejpam-4653	402	44	uv	uv	PROPN
ejpam-4653	402	45	/∈	/∈	PUNCT
ejpam-4653	402	46	e(g	e(g	PROPN
ejpam-4653	402	47	)	)	PUNCT
ejpam-4653	402	48	.	.	PUNCT
ejpam-4653	403	1	however	however	ADV
ejpam-4653	403	2	,	,	PUNCT
ejpam-4653	403	3	if	if	SCONJ
ejpam-4653	403	4	uv	uv	PROPN
ejpam-4653	403	5	∈	∈	PROPN
ejpam-4653	403	6	e(g	e(g	PROPN
ejpam-4653	403	7	)	)	PUNCT
ejpam-4653	403	8	,	,	PUNCT
ejpam-4653	403	9	then	then	ADV
ejpam-4653	403	10	uv	uv	PROPN
ejpam-4653	403	11	/∈	/∈	PUNCT
ejpam-4653	403	12	e(g	e(g	PROPN
ejpam-4653	403	13	)	)	PUNCT
ejpam-4653	403	14	,	,	PUNCT
ejpam-4653	403	15	a	a	DET
ejpam-4653	403	16	contradiction	contradiction	NOUN
ejpam-4653	403	17	.	.	PUNCT
ejpam-4653	404	1	therefore	therefore	ADV
ejpam-4653	404	2	,	,	PUNCT
ejpam-4653	404	3	γ2(gg	γ2(gg	NUM
ejpam-4653	404	4	)	)	PUNCT
ejpam-4653	404	5	̸=	̸=	PROPN
ejpam-4653	404	6	2	2	NUM
ejpam-4653	404	7	.	.	PUNCT
ejpam-4653	404	8	by	by	ADP
ejpam-4653	404	9	proposition	proposition	NOUN
ejpam-4653	404	10	5	5	NUM
ejpam-4653	404	11	,	,	PUNCT
ejpam-4653	404	12	γdr(gg	γdr(gg	NOUN
ejpam-4653	404	13	)	)	PUNCT
ejpam-4653	404	14	̸=	̸=	PROPN
ejpam-4653	404	15	4	4	NUM
ejpam-4653	404	16	.	.	PUNCT
ejpam-4653	405	1	clearly	clearly	ADV
ejpam-4653	405	2	,	,	PUNCT
ejpam-4653	405	3	if	if	SCONJ
ejpam-4653	405	4	g	g	PROPN
ejpam-4653	405	5	=	=	SYM
ejpam-4653	405	6	k1	k1	PROPN
ejpam-4653	405	7	,	,	PUNCT
ejpam-4653	405	8	then	then	ADV
ejpam-4653	405	9	γdr(gg	γdr(gg	NOUN
ejpam-4653	405	10	)	)	PUNCT
ejpam-4653	405	11	=	=	SYM
ejpam-4653	405	12	3	3	X
ejpam-4653	405	13	.	.	PUNCT
ejpam-4653	405	14	suppose	suppose	VERB
ejpam-4653	405	15	that	that	SCONJ
ejpam-4653	405	16	γdr(gg	γdr(gg	NOUN
ejpam-4653	405	17	)	)	PUNCT
ejpam-4653	405	18	=	=	SYM
ejpam-4653	406	1	3	3	X
ejpam-4653	406	2	.	.	PUNCT
ejpam-4653	406	3	then	then	ADV
ejpam-4653	406	4	γ(gg	γ(gg	NUM
ejpam-4653	406	5	)	)	PUNCT
ejpam-4653	406	6	=	=	SYM
ejpam-4653	407	1	1	1	X
ejpam-4653	407	2	,	,	PUNCT
ejpam-4653	407	3	by	by	ADP
ejpam-4653	407	4	proposition	proposition	NOUN
ejpam-4653	407	5	5	5	NUM
ejpam-4653	407	6	.	.	PUNCT
ejpam-4653	407	7	thus	thus	ADV
ejpam-4653	407	8	,	,	PUNCT
ejpam-4653	407	9	by	by	ADP
ejpam-4653	407	10	lemma	lemma	PROPN
ejpam-4653	407	11	1	1	NUM
ejpam-4653	407	12	,	,	PUNCT
ejpam-4653	407	13	g	g	PROPN
ejpam-4653	407	14	=	=	SYM
ejpam-4653	407	15	k1	k1	PROPN
ejpam-4653	407	16	.	.	PUNCT
ejpam-4653	408	1	this	this	PRON
ejpam-4653	408	2	proves	prove	VERB
ejpam-4653	408	3	(	(	PUNCT
ejpam-4653	408	4	ii	ii	NOUN
ejpam-4653	408	5	)	)	PUNCT
ejpam-4653	408	6	.	.	PUNCT
ejpam-4653	409	1	now	now	ADV
ejpam-4653	409	2	,	,	PUNCT
ejpam-4653	409	3	we	we	PRON
ejpam-4653	409	4	prove	prove	VERB
ejpam-4653	409	5	(	(	PUNCT
ejpam-4653	409	6	iii	iii	NOUN
ejpam-4653	409	7	)	)	PUNCT
ejpam-4653	409	8	.	.	PUNCT
ejpam-4653	410	1	if	if	SCONJ
ejpam-4653	410	2	g	g	PROPN
ejpam-4653	410	3	∈	∈	PROPN
ejpam-4653	410	4	{	{	PUNCT
ejpam-4653	410	5	k2,k2	k2,k2	PROPN
ejpam-4653	410	6	}	}	PUNCT
ejpam-4653	410	7	,	,	PUNCT
ejpam-4653	410	8	then	then	ADV
ejpam-4653	410	9	gg	gg	VERB
ejpam-4653	410	10	∼=	∼=	NOUN
ejpam-4653	410	11	p4	p4	ADJ
ejpam-4653	410	12	so	so	SCONJ
ejpam-4653	410	13	that	that	SCONJ
ejpam-4653	410	14	γdr(gg	γdr(gg	NOUN
ejpam-4653	410	15	)	)	PUNCT
ejpam-4653	410	16	=	=	SYM
ejpam-4653	411	1	5	5	X
ejpam-4653	411	2	.	.	PUNCT
ejpam-4653	411	3	conversely	conversely	ADV
ejpam-4653	411	4	,	,	PUNCT
ejpam-4653	411	5	assume	assume	VERB
ejpam-4653	411	6	γdr(gg	γdr(gg	NUM
ejpam-4653	411	7	)	)	PUNCT
ejpam-4653	411	8	=	=	SYM
ejpam-4653	411	9	5	5	X
ejpam-4653	411	10	.	.	PUNCT
ejpam-4653	412	1	by	by	ADP
ejpam-4653	412	2	(	(	PUNCT
ejpam-4653	412	3	ii	ii	NOUN
ejpam-4653	412	4	)	)	PUNCT
ejpam-4653	412	5	,	,	PUNCT
ejpam-4653	412	6	g	g	PROPN
ejpam-4653	412	7	̸=	̸=	PROPN
ejpam-4653	412	8	k1	k1	PROPN
ejpam-4653	412	9	.	.	PUNCT
ejpam-4653	413	1	suppose	suppose	VERB
ejpam-4653	413	2	that	that	SCONJ
ejpam-4653	413	3	g	g	PROPN
ejpam-4653	413	4	/∈	/∈	PUNCT
ejpam-4653	413	5	{	{	PUNCT
ejpam-4653	413	6	k2,k2	k2,k2	PROPN
ejpam-4653	413	7	}	}	PUNCT
ejpam-4653	413	8	.	.	PUNCT
ejpam-4653	414	1	let	let	VERB
ejpam-4653	414	2	u	u	NOUN
ejpam-4653	414	3	,	,	PUNCT
ejpam-4653	414	4	v	v	NOUN
ejpam-4653	414	5	and	and	CCONJ
ejpam-4653	414	6	w	w	NOUN
ejpam-4653	414	7	be	be	AUX
ejpam-4653	414	8	distinct	distinct	ADJ
ejpam-4653	414	9	vertices	vertex	NOUN
ejpam-4653	414	10	of	of	ADP
ejpam-4653	414	11	g.	g.	PROPN
ejpam-4653	414	12	then	then	ADV
ejpam-4653	414	13	u	u	PROPN
ejpam-4653	414	14	,	,	PUNCT
ejpam-4653	414	15	w	w	PROPN
ejpam-4653	414	16	∈	∈	PROPN
ejpam-4653	414	17	v	v	NOUN
ejpam-4653	414	18	(	(	PUNCT
ejpam-4653	414	19	gg)\ngg[v	gg)\ngg[v	PROPN
ejpam-4653	414	20	]	]	PUNCT
ejpam-4653	414	21	.	.	PUNCT
ejpam-4653	415	1	this	this	PRON
ejpam-4653	415	2	means	mean	VERB
ejpam-4653	415	3	that	that	SCONJ
ejpam-4653	415	4	|v	|v	PROPN
ejpam-4653	415	5	(	(	PUNCT
ejpam-4653	415	6	gg)\ngg[v]|	gg)\ngg[v]|	X
ejpam-4653	415	7	≥	≥	NUM
ejpam-4653	415	8	2	2	NUM
ejpam-4653	415	9	for	for	ADP
ejpam-4653	415	10	all	all	PRON
ejpam-4653	415	11	v	v	ADP
ejpam-4653	415	12	∈	∈	NOUN
ejpam-4653	415	13	v	v	NOUN
ejpam-4653	415	14	(	(	PUNCT
ejpam-4653	415	15	g	g	NOUN
ejpam-4653	415	16	)	)	PUNCT
ejpam-4653	415	17	.	.	PUNCT
ejpam-4653	416	1	similarly	similarly	ADV
ejpam-4653	416	2	,	,	PUNCT
ejpam-4653	416	3	|v	|v	PROPN
ejpam-4653	416	4	(	(	PUNCT
ejpam-4653	416	5	gg	gg	NOUN
ejpam-4653	416	6	)	)	PUNCT
ejpam-4653	416	7	\	\	PROPN
ejpam-4653	416	8	ngg[v]|	ngg[v]|	ADJ
ejpam-4653	416	9	≥	≥	NOUN
ejpam-4653	416	10	2	2	NUM
ejpam-4653	416	11	for	for	ADP
ejpam-4653	416	12	all	all	DET
ejpam-4653	416	13	v	v	ADP
ejpam-4653	416	14	∈	∈	NOUN
ejpam-4653	416	15	v	v	NOUN
ejpam-4653	416	16	(	(	PUNCT
ejpam-4653	416	17	g	g	NOUN
ejpam-4653	416	18	)	)	PUNCT
ejpam-4653	416	19	.	.	PUNCT
ejpam-4653	417	1	therefore	therefore	ADV
ejpam-4653	417	2	,	,	PUNCT
ejpam-4653	417	3	|v	|v	PROPN
ejpam-4653	417	4	(	(	PUNCT
ejpam-4653	417	5	gg	gg	PROPN
ejpam-4653	417	6	)	)	PUNCT
ejpam-4653	417	7	\	\	PROPN
ejpam-4653	417	8	ngg[v]|	ngg[v]|	ADJ
ejpam-4653	417	9	≥	≥	NOUN
ejpam-4653	417	10	2	2	NUM
ejpam-4653	417	11	for	for	ADP
ejpam-4653	417	12	all	all	DET
ejpam-4653	417	13	v	v	ADP
ejpam-4653	417	14	∈	∈	NOUN
ejpam-4653	417	15	v	v	NOUN
ejpam-4653	417	16	(	(	PUNCT
ejpam-4653	417	17	gg	gg	PROPN
ejpam-4653	417	18	)	)	PUNCT
ejpam-4653	417	19	.	.	PUNCT
ejpam-4653	418	1	this	this	PRON
ejpam-4653	418	2	is	be	AUX
ejpam-4653	418	3	a	a	DET
ejpam-4653	418	4	contradiction	contradiction	NOUN
ejpam-4653	418	5	to	to	PART
ejpam-4653	418	6	proposition	proposition	VERB
ejpam-4653	418	7	6	6	NUM
ejpam-4653	418	8	.	.	PUNCT
ejpam-4653	419	1	therefore	therefore	ADV
ejpam-4653	419	2	,	,	PUNCT
ejpam-4653	419	3	g	g	PROPN
ejpam-4653	419	4	∈	∈	PROPN
ejpam-4653	419	5	{	{	PUNCT
ejpam-4653	419	6	k2,k2	k2,k2	PROPN
ejpam-4653	419	7	}	}	PUNCT
ejpam-4653	419	8	.	.	PUNCT
ejpam-4653	420	1	theorem	theorem	NOUN
ejpam-4653	420	2	1	1	NUM
ejpam-4653	420	3	.	.	PUNCT
ejpam-4653	420	4	(	(	PUNCT
ejpam-4653	420	5	complementary	complementary	ADJ
ejpam-4653	420	6	prism	prism	NOUN
ejpam-4653	420	7	)	)	PUNCT
ejpam-4653	420	8	let	let	VERB
ejpam-4653	420	9	g	g	NOUN
ejpam-4653	420	10	be	be	AUX
ejpam-4653	420	11	a	a	DET
ejpam-4653	420	12	graph	graph	NOUN
ejpam-4653	420	13	of	of	ADP
ejpam-4653	420	14	order	order	NOUN
ejpam-4653	420	15	n	n	PRON
ejpam-4653	420	16	≥	≥	NOUN
ejpam-4653	420	17	3	3	X
ejpam-4653	420	18	.	.	PUNCT
ejpam-4653	420	19	assume	assume	VERB
ejpam-4653	420	20	γdr(g	γdr(g	X
ejpam-4653	420	21	)	)	PUNCT
ejpam-4653	420	22	≤	≤	NOUN
ejpam-4653	421	1	γdr(g	γdr(g	X
ejpam-4653	421	2	)	)	PUNCT
ejpam-4653	421	3	.	.	PUNCT
ejpam-4653	422	1	then	then	ADV
ejpam-4653	422	2	1	1	NUM
ejpam-4653	422	3	+	+	CCONJ
ejpam-4653	422	4	γdr(g	γdr(g	X
ejpam-4653	422	5	)	)	PUNCT
ejpam-4653	422	6	≤	≤	NUM
ejpam-4653	422	7	γdr(gg	γdr(gg	NOUN
ejpam-4653	422	8	)	)	PUNCT
ejpam-4653	422	9	≤	≤	NOUN
ejpam-4653	422	10	ρ	ρ	PROPN
ejpam-4653	422	11	,	,	PUNCT
ejpam-4653	422	12	where	where	SCONJ
ejpam-4653	422	13	ρ	ρ	PROPN
ejpam-4653	422	14	=	=	SYM
ejpam-4653	422	15	min{ωg(f	min{ωg(f	NOUN
ejpam-4653	422	16	)	)	PUNCT
ejpam-4653	422	17	+	+	CCONJ
ejpam-4653	422	18	2	2	NUM
ejpam-4653	422	19	(	(	PUNCT
ejpam-4653	422	20	n−	n−	NOUN
ejpam-4653	422	21	|v3|)−	|v3|)−	PROPN
ejpam-4653	422	22	|v2|	|v2|	ADV
ejpam-4653	422	23	:	:	PUNCT
ejpam-4653	423	1	f	f	X
ejpam-4653	423	2	=	=	SYM
ejpam-4653	423	3	(	(	PUNCT
ejpam-4653	423	4	v0	v0	PROPN
ejpam-4653	423	5	,	,	PUNCT
ejpam-4653	423	6	v1	v1	NOUN
ejpam-4653	423	7	,	,	PUNCT
ejpam-4653	423	8	v2	v2	PROPN
ejpam-4653	423	9	,	,	PUNCT
ejpam-4653	423	10	v3	v3	PROPN
ejpam-4653	423	11	)	)	PUNCT
ejpam-4653	423	12	∈	∈	PROPN
ejpam-4653	423	13	drd(g	drd(g	PROPN
ejpam-4653	423	14	)	)	PUNCT
ejpam-4653	423	15	∪drd(g	∪drd(g	PROPN
ejpam-4653	423	16	)	)	PUNCT
ejpam-4653	423	17	}	}	PUNCT
ejpam-4653	423	18	.	.	PUNCT
ejpam-4653	424	1	moreover	moreover	ADV
ejpam-4653	424	2	,	,	PUNCT
ejpam-4653	424	3	these	these	DET
ejpam-4653	424	4	bounds	bound	NOUN
ejpam-4653	424	5	are	be	AUX
ejpam-4653	424	6	sharp	sharp	ADJ
ejpam-4653	424	7	.	.	PUNCT
ejpam-4653	425	1	proof	proof	NOUN
ejpam-4653	425	2	.	.	PUNCT
ejpam-4653	426	1	let	let	VERB
ejpam-4653	426	2	f	f	PROPN
ejpam-4653	426	3	=	=	SYM
ejpam-4653	426	4	(	(	PUNCT
ejpam-4653	426	5	v0	v0	PROPN
ejpam-4653	426	6	,	,	PUNCT
ejpam-4653	426	7	v1	v1	NOUN
ejpam-4653	426	8	,	,	PUNCT
ejpam-4653	426	9	v2	v2	PROPN
ejpam-4653	426	10	,	,	PUNCT
ejpam-4653	426	11	v3	v3	PROPN
ejpam-4653	426	12	)	)	PUNCT
ejpam-4653	426	13	∈	∈	PROPN
ejpam-4653	426	14	drd(g	drd(g	PROPN
ejpam-4653	426	15	)	)	PUNCT
ejpam-4653	426	16	.	.	PUNCT
ejpam-4653	427	1	extend	extend	VERB
ejpam-4653	427	2	f	f	PROPN
ejpam-4653	427	3	to	to	ADP
ejpam-4653	427	4	a	a	DET
ejpam-4653	427	5	function	function	NOUN
ejpam-4653	427	6	on	on	ADP
ejpam-4653	427	7	v	v	PROPN
ejpam-4653	427	8	(	(	PUNCT
ejpam-4653	427	9	gg	gg	NOUN
ejpam-4653	427	10	)	)	PUNCT
ejpam-4653	427	11	by	by	ADP
ejpam-4653	427	12	defining	define	VERB
ejpam-4653	427	13	f(v	f(v	NOUN
ejpam-4653	427	14	)	)	PUNCT
ejpam-4653	428	1	=	=	SYM
ejpam-4653	428	2			NOUN
ejpam-4653	428	3	0	0	NUM
ejpam-4653	428	4	,	,	PUNCT
ejpam-4653	428	5	if	if	SCONJ
ejpam-4653	428	6	v	v	ADP
ejpam-4653	428	7	∈	∈	PROPN
ejpam-4653	428	8	v3	v3	NOUN
ejpam-4653	428	9	;	;	PUNCT
ejpam-4653	428	10	1	1	NUM
ejpam-4653	428	11	,	,	PUNCT
ejpam-4653	428	12	if	if	SCONJ
ejpam-4653	428	13	v	v	ADP
ejpam-4653	428	14	∈	∈	PROPN
ejpam-4653	428	15	v2	v2	NOUN
ejpam-4653	428	16	;	;	PUNCT
ejpam-4653	428	17	2	2	NUM
ejpam-4653	428	18	,	,	PUNCT
ejpam-4653	428	19	if	if	SCONJ
ejpam-4653	428	20	v	v	NUM
ejpam-4653	428	21	∈	∈	PROPN
ejpam-4653	428	22	v0	v0	NOUN
ejpam-4653	428	23	∪	∪	X
ejpam-4653	428	24	v1	v1	PROPN
ejpam-4653	428	25	.	.	PUNCT
ejpam-4653	429	1	then	then	ADV
ejpam-4653	429	2	f	f	PROPN
ejpam-4653	429	3	∈	∈	PROPN
ejpam-4653	429	4	drd(gg	drd(gg	PROPN
ejpam-4653	429	5	)	)	PUNCT
ejpam-4653	429	6	so	so	SCONJ
ejpam-4653	429	7	that	that	SCONJ
ejpam-4653	429	8	γdr(gg	γdr(gg	NOUN
ejpam-4653	429	9	)	)	PUNCT
ejpam-4653	429	10	≤	≤	NUM
ejpam-4653	429	11	ωg(f)+	ωg(f)+	SYM
ejpam-4653	429	12	2	2	NUM
ejpam-4653	429	13	(	(	PUNCT
ejpam-4653	429	14	n−	n−	NOUN
ejpam-4653	429	15	|v3|)−	|v3|)−	PROPN
ejpam-4653	429	16	|v2|	|v2|	ADV
ejpam-4653	429	17	.	.	PUNCT
ejpam-4653	430	1	thus	thus	ADV
ejpam-4653	430	2	,	,	PUNCT
ejpam-4653	430	3	γdr(gg	γdr(gg	PROPN
ejpam-4653	430	4	)	)	PUNCT
ejpam-4653	430	5	≤	≤	NOUN
ejpam-4653	430	6	ρ	ρ	NOUN
ejpam-4653	430	7	.	.	PUNCT
ejpam-4653	431	1	now	now	ADV
ejpam-4653	431	2	,	,	PUNCT
ejpam-4653	431	3	we	we	PRON
ejpam-4653	431	4	show	show	VERB
ejpam-4653	431	5	the	the	DET
ejpam-4653	431	6	left	left	ADJ
ejpam-4653	431	7	-	-	PUNCT
ejpam-4653	431	8	hand	hand	NOUN
ejpam-4653	431	9	inequality	inequality	NOUN
ejpam-4653	431	10	.	.	PUNCT
ejpam-4653	432	1	let	let	VERB
ejpam-4653	432	2	f	f	PROPN
ejpam-4653	432	3	=	=	SYM
ejpam-4653	432	4	(	(	PUNCT
ejpam-4653	432	5	v0,∅	v0,∅	PROPN
ejpam-4653	432	6	,	,	PUNCT
ejpam-4653	432	7	v2	v2	PROPN
ejpam-4653	432	8	,	,	PUNCT
ejpam-4653	432	9	v3	v3	PROPN
ejpam-4653	432	10	)	)	PUNCT
ejpam-4653	432	11	be	be	VERB
ejpam-4653	432	12	a	a	DET
ejpam-4653	432	13	γdr	γdr	NOUN
ejpam-4653	432	14	-	-	PUNCT
ejpam-4653	432	15	function	function	NOUN
ejpam-4653	432	16	of	of	ADP
ejpam-4653	432	17	gg	gg	PROPN
ejpam-4653	432	18	.	.	PUNCT
ejpam-4653	433	1	if	if	SCONJ
ejpam-4653	433	2	v	v	INTJ
ejpam-4653	433	3	(	(	PUNCT
ejpam-4653	433	4	g	g	NOUN
ejpam-4653	433	5	)	)	PUNCT
ejpam-4653	433	6	⊆	⊆	NUM
ejpam-4653	433	7	v0	v0	NOUN
ejpam-4653	433	8	,	,	PUNCT
ejpam-4653	433	9	then	then	ADV
ejpam-4653	433	10	v3	v3	PROPN
ejpam-4653	433	11	=	=	SYM
ejpam-4653	433	12	v	v	PROPN
ejpam-4653	433	13	(	(	PUNCT
ejpam-4653	433	14	g	g	NOUN
ejpam-4653	433	15	)	)	PUNCT
ejpam-4653	433	16	so	so	SCONJ
ejpam-4653	433	17	that	that	SCONJ
ejpam-4653	433	18	γdr(gg	γdr(gg	NOUN
ejpam-4653	433	19	)	)	PUNCT
ejpam-4653	433	20	=	=	SYM
ejpam-4653	434	1	3|v3|	3|v3|	NUM
ejpam-4653	434	2	=	=	SYM
ejpam-4653	434	3	3n	3n	NUM
ejpam-4653	434	4	≥	≥	NOUN
ejpam-4653	434	5	1	1	NUM
ejpam-4653	434	6	+	+	CCONJ
ejpam-4653	434	7	γdr(g	γdr(g	X
ejpam-4653	434	8	)	)	PUNCT
ejpam-4653	434	9	.	.	PUNCT
ejpam-4653	435	1	suppose	suppose	VERB
ejpam-4653	435	2	that	that	SCONJ
ejpam-4653	435	3	v	v	INTJ
ejpam-4653	435	4	(	(	PUNCT
ejpam-4653	435	5	g	g	NOUN
ejpam-4653	435	6	)	)	PUNCT
ejpam-4653	435	7	∩	∩	NOUN
ejpam-4653	435	8	(	(	PUNCT
ejpam-4653	435	9	v2	v2	PROPN
ejpam-4653	435	10	∪	∪	X
ejpam-4653	435	11	v3	v3	PROPN
ejpam-4653	435	12	)	)	PUNCT
ejpam-4653	435	13	̸=	̸=	PROPN
ejpam-4653	435	14	∅.	∅.	ADV
ejpam-4653	435	15	let	let	VERB
ejpam-4653	435	16	a	a	DET
ejpam-4653	435	17	=	=	X
ejpam-4653	435	18	{	{	PUNCT
ejpam-4653	435	19	v	v	NOUN
ejpam-4653	435	20	∈	∈	PROPN
ejpam-4653	435	21	v0	v0	NOUN
ejpam-4653	435	22	∩	∩	X
ejpam-4653	435	23	v	v	X
ejpam-4653	435	24	(	(	PUNCT
ejpam-4653	435	25	g	g	NOUN
ejpam-4653	435	26	)	)	PUNCT
ejpam-4653	435	27	:	:	PUNCT
ejpam-4653	435	28	v3	v3	PROPN
ejpam-4653	435	29	∩	∩	ADJ
ejpam-4653	435	30	ngg(v	ngg(v	PROPN
ejpam-4653	435	31	)	)	PUNCT
ejpam-4653	435	32	=	=	PRON
ejpam-4653	435	33	{	{	PUNCT
ejpam-4653	435	34	v	v	NOUN
ejpam-4653	435	35	}	}	PUNCT
ejpam-4653	435	36	}	}	PUNCT
ejpam-4653	435	37	,	,	PUNCT
ejpam-4653	435	38	b	b	X
ejpam-4653	435	39	=	=	PRON
ejpam-4653	435	40	{	{	PUNCT
ejpam-4653	435	41	v	v	NOUN
ejpam-4653	435	42	∈	∈	PROPN
ejpam-4653	435	43	v0	v0	NOUN
ejpam-4653	435	44	∩	∩	X
ejpam-4653	435	45	v	v	X
ejpam-4653	435	46	(	(	PUNCT
ejpam-4653	435	47	g	g	NOUN
ejpam-4653	435	48	)	)	PUNCT
ejpam-4653	435	49	:	:	PUNCT
ejpam-4653	435	50	v	v	X
ejpam-4653	435	51	∈	∈	PROPN
ejpam-4653	435	52	v2	v2	PROPN
ejpam-4653	435	53	and	and	CCONJ
ejpam-4653	435	54	|v2	|v2	NOUN
ejpam-4653	435	55	∩	∩	PROPN
ejpam-4653	435	56	ngg(v)|	ngg(v)|	PROPN
ejpam-4653	435	57	=	=	SYM
ejpam-4653	435	58	2	2	NUM
ejpam-4653	435	59	}	}	PUNCT
ejpam-4653	435	60	and	and	CCONJ
ejpam-4653	435	61	c	c	X
ejpam-4653	435	62	=	=	PRON
ejpam-4653	435	63	{	{	PUNCT
ejpam-4653	435	64	v	v	NOUN
ejpam-4653	435	65	∈	∈	PROPN
ejpam-4653	435	66	v1	v1	NOUN
ejpam-4653	435	67	∩	∩	ADJ
ejpam-4653	435	68	v	v	NOUN
ejpam-4653	435	69	(	(	PUNCT
ejpam-4653	435	70	g	g	NOUN
ejpam-4653	435	71	)	)	PUNCT
ejpam-4653	435	72	:	:	PUNCT
ejpam-4653	435	73	(	(	PUNCT
ejpam-4653	435	74	v2	v2	NOUN
ejpam-4653	435	75	∪	∪	X
ejpam-4653	435	76	v3)∩ngg(v	v3)∩ngg(v	X
ejpam-4653	435	77	)	)	PUNCT
ejpam-4653	435	78	=	=	SYM
ejpam-4653	435	79	{	{	PUNCT
ejpam-4653	435	80	v	v	NOUN
ejpam-4653	435	81	}	}	PUNCT
ejpam-4653	435	82	.	.	PUNCT
ejpam-4653	436	1	define	define	VERB
ejpam-4653	436	2	g	g	PROPN
ejpam-4653	436	3	=	=	SYM
ejpam-4653	436	4	(	(	PUNCT
ejpam-4653	436	5	v	v	NOUN
ejpam-4653	436	6	∗	∗	NOUN
ejpam-4653	436	7	0	0	NUM
ejpam-4653	436	8	,	,	PUNCT
ejpam-4653	436	9	v	v	NOUN
ejpam-4653	436	10	∗	∗	NOUN
ejpam-4653	436	11	1	1	NUM
ejpam-4653	436	12	,	,	PUNCT
ejpam-4653	436	13	v	v	NOUN
ejpam-4653	436	14	∗	∗	NOUN
ejpam-4653	436	15	2	2	NUM
ejpam-4653	436	16	,	,	PUNCT
ejpam-4653	436	17	v	v	NOUN
ejpam-4653	436	18	∗	∗	X
ejpam-4653	436	19	3	3	NUM
ejpam-4653	436	20	)	)	PUNCT
ejpam-4653	436	21	on	on	ADP
ejpam-4653	436	22	v	v	ADP
ejpam-4653	436	23	(	(	PUNCT
ejpam-4653	436	24	g	g	NOUN
ejpam-4653	436	25	)	)	PUNCT
ejpam-4653	436	26	by	by	ADP
ejpam-4653	436	27	g(x	g(x	NOUN
ejpam-4653	436	28	)	)	PUNCT
ejpam-4653	437	1	=	=	PUNCT
ejpam-4653	438	1			PROPN
ejpam-4653	438	2	f(x	f(x	PROPN
ejpam-4653	438	3	)	)	PUNCT
ejpam-4653	438	4	,	,	PUNCT
ejpam-4653	438	5	if	if	SCONJ
ejpam-4653	438	6	x	x	SYM
ejpam-4653	438	7	∈	∈	PROPN
ejpam-4653	438	8	v	v	X
ejpam-4653	438	9	(	(	PUNCT
ejpam-4653	438	10	g	g	NOUN
ejpam-4653	438	11	)	)	PUNCT
ejpam-4653	438	12	\	\	PUNCT
ejpam-4653	439	1	(	(	PUNCT
ejpam-4653	439	2	a	a	DET
ejpam-4653	439	3	∪b	∪b	PUNCT
ejpam-4653	439	4	∪	∪	PROPN
ejpam-4653	439	5	c	c	NOUN
ejpam-4653	439	6	)	)	PUNCT
ejpam-4653	439	7	;	;	PUNCT
ejpam-4653	439	8	2	2	X
ejpam-4653	439	9	,	,	PUNCT
ejpam-4653	439	10	if	if	SCONJ
ejpam-4653	439	11	x	x	PROPN
ejpam-4653	439	12	∈	∈	PROPN
ejpam-4653	439	13	a	a	DET
ejpam-4653	439	14	∪	∪	NOUN
ejpam-4653	439	15	c	c	NOUN
ejpam-4653	439	16	;	;	PUNCT
ejpam-4653	439	17	1	1	NUM
ejpam-4653	439	18	,	,	PUNCT
ejpam-4653	439	19	if	if	SCONJ
ejpam-4653	439	20	x	x	PROPN
ejpam-4653	439	21	∈	∈	PROPN
ejpam-4653	439	22	b.	b.	PROPN
ejpam-4653	439	23	j.	j.	PROPN
ejpam-4653	439	24	b.g	b.g	PROPN
ejpam-4653	439	25	.	.	PROPN
ejpam-4653	439	26	cariaga	cariaga	PROPN
ejpam-4653	439	27	,	,	PUNCT
ejpam-4653	439	28	f.	f.	PROPN
ejpam-4653	439	29	jamil	jamil	PROPN
ejpam-4653	439	30	/	/	SYM
ejpam-4653	439	31	eur	eur	PROPN
ejpam-4653	439	32	.	.	PUNCT
ejpam-4653	440	1	j.	j.	PROPN
ejpam-4653	440	2	pure	pure	PROPN
ejpam-4653	440	3	appl	appl	PROPN
ejpam-4653	440	4	.	.	PROPN
ejpam-4653	440	5	math	math	PROPN
ejpam-4653	440	6	,	,	PUNCT
ejpam-4653	440	7	16	16	NUM
ejpam-4653	440	8	(	(	PUNCT
ejpam-4653	440	9	2	2	NUM
ejpam-4653	440	10	)	)	PUNCT
ejpam-4653	440	11	(	(	PUNCT
ejpam-4653	440	12	2023	2023	NUM
ejpam-4653	440	13	)	)	PUNCT
ejpam-4653	440	14	,	,	PUNCT
ejpam-4653	440	15	847	847	NUM
ejpam-4653	440	16	-	-	SYM
ejpam-4653	440	17	863	863	NUM
ejpam-4653	440	18	857	857	NUM
ejpam-4653	440	19	it	it	PRON
ejpam-4653	440	20	follows	follow	VERB
ejpam-4653	440	21	that	that	SCONJ
ejpam-4653	440	22	g	g	PROPN
ejpam-4653	440	23	∈	∈	PROPN
ejpam-4653	440	24	drd(g	drd(g	PROPN
ejpam-4653	440	25	)	)	PUNCT
ejpam-4653	440	26	with	with	ADP
ejpam-4653	440	27	v	v	NOUN
ejpam-4653	440	28	∗	∗	NOUN
ejpam-4653	440	29	0	0	NUM
ejpam-4653	441	1	=	=	SYM
ejpam-4653	441	2	(	(	PUNCT
ejpam-4653	441	3	v0∩v	v0∩v	X
ejpam-4653	441	4	(	(	PUNCT
ejpam-4653	441	5	g))\(a	g))\(a	NOUN
ejpam-4653	441	6	∪b	∪b	NOUN
ejpam-4653	441	7	)	)	PUNCT
ejpam-4653	441	8	,	,	PUNCT
ejpam-4653	441	9	v	v	X
ejpam-4653	441	10	∗	∗	X
ejpam-4653	441	11	1	1	NUM
ejpam-4653	441	12	=	=	SYM
ejpam-4653	441	13	b∪	b∪	PROPN
ejpam-4653	442	1	[	[	X
ejpam-4653	442	2	(	(	PUNCT
ejpam-4653	442	3	v1	v1	NOUN
ejpam-4653	442	4	∩	∩	ADJ
ejpam-4653	442	5	v	v	NOUN
ejpam-4653	442	6	(	(	PUNCT
ejpam-4653	442	7	g	g	NOUN
ejpam-4653	442	8	)	)	PUNCT
ejpam-4653	442	9	)	)	PUNCT
ejpam-4653	442	10	\	\	PUNCT
ejpam-4653	443	1	c	c	X
ejpam-4653	443	2	]	]	X
ejpam-4653	443	3	,	,	PUNCT
ejpam-4653	443	4	v	v	X
ejpam-4653	443	5	∗	∗	NOUN
ejpam-4653	443	6	2	2	NUM
ejpam-4653	443	7	=	=	SYM
ejpam-4653	444	1	[	[	X
ejpam-4653	444	2	v2	v2	PROPN
ejpam-4653	444	3	∩	∩	ADJ
ejpam-4653	444	4	v	v	NOUN
ejpam-4653	444	5	(	(	PUNCT
ejpam-4653	444	6	g	g	NOUN
ejpam-4653	444	7	)	)	PUNCT
ejpam-4653	444	8	]	]	PUNCT
ejpam-4653	444	9	∪a	∪a	X
ejpam-4653	444	10	∪	∪	ADP
ejpam-4653	444	11	c	c	NOUN
ejpam-4653	444	12	and	and	CCONJ
ejpam-4653	444	13	v	v	NOUN
ejpam-4653	444	14	∗	∗	NOUN
ejpam-4653	444	15	3	3	NUM
ejpam-4653	444	16	=	=	SYM
ejpam-4653	444	17	v3	v3	PROPN
ejpam-4653	444	18	∩	∩	PROPN
ejpam-4653	444	19	v	v	X
ejpam-4653	444	20	(	(	PUNCT
ejpam-4653	444	21	g	g	NOUN
ejpam-4653	444	22	)	)	PUNCT
ejpam-4653	444	23	.	.	PUNCT
ejpam-4653	445	1	moreover	moreover	ADV
ejpam-4653	445	2	,	,	PUNCT
ejpam-4653	445	3	γdr(gg	γdr(gg	PROPN
ejpam-4653	445	4	)	)	PUNCT
ejpam-4653	445	5	=	=	PUNCT
ejpam-4653	445	6	ωg(g	ωg(g	NUM
ejpam-4653	445	7	)	)	PUNCT
ejpam-4653	446	1	+	+	CCONJ
ejpam-4653	446	2	∑	∑	PROPN
ejpam-4653	446	3	x∈v	x∈v	PROPN
ejpam-4653	446	4	(	(	PUNCT
ejpam-4653	446	5	g	g	NOUN
ejpam-4653	446	6	)	)	PUNCT
ejpam-4653	446	7	f(x)−	f(x)−	PROPN
ejpam-4653	446	8	2|a|	2|a|	NUM
ejpam-4653	446	9	−	−	NOUN
ejpam-4653	446	10	|b|	|b|	PROPN
ejpam-4653	446	11	−	−	PROPN
ejpam-4653	446	12	|c|	|c|	PROPN
ejpam-4653	446	13	≥	≥	NOUN
ejpam-4653	446	14	ωg(g	ωg(g	PUNCT
ejpam-4653	446	15	)	)	PUNCT
ejpam-4653	446	16	+	+	CCONJ
ejpam-4653	446	17	1	1	NUM
ejpam-4653	446	18	≥	≥	NOUN
ejpam-4653	446	19	γdr(g	γdr(g	X
ejpam-4653	446	20	)	)	PUNCT
ejpam-4653	446	21	+	+	CCONJ
ejpam-4653	447	1	1	1	X
ejpam-4653	447	2	.	.	X
ejpam-4653	447	3	to	to	PART
ejpam-4653	447	4	show	show	VERB
ejpam-4653	447	5	sharpness	sharpness	NOUN
ejpam-4653	447	6	of	of	ADP
ejpam-4653	447	7	the	the	DET
ejpam-4653	447	8	lower	lower	ADV
ejpam-4653	447	9	bound	bind	VERB
ejpam-4653	447	10	,	,	PUNCT
ejpam-4653	447	11	note	note	VERB
ejpam-4653	447	12	that	that	SCONJ
ejpam-4653	447	13	by	by	ADP
ejpam-4653	447	14	proposition	proposition	NOUN
ejpam-4653	447	15	14	14	NUM
ejpam-4653	447	16	,	,	PUNCT
ejpam-4653	447	17	γdr(k1k1	γdr(k1k1	NOUN
ejpam-4653	447	18	)	)	PUNCT
ejpam-4653	447	19	=	=	SYM
ejpam-4653	447	20	3	3	NUM
ejpam-4653	447	21	=	=	SYM
ejpam-4653	447	22	1	1	NUM
ejpam-4653	447	23	+	+	NUM
ejpam-4653	447	24	γdr(k1	γdr(k1	NOUN
ejpam-4653	447	25	)	)	PUNCT
ejpam-4653	447	26	.	.	PUNCT
ejpam-4653	448	1	for	for	ADP
ejpam-4653	448	2	the	the	DET
ejpam-4653	448	3	upper	upper	ADJ
ejpam-4653	448	4	bound	bind	VERB
ejpam-4653	448	5	,	,	PUNCT
ejpam-4653	448	6	pick	pick	VERB
ejpam-4653	448	7	g	g	PROPN
ejpam-4653	448	8	=	=	SYM
ejpam-4653	448	9	kn	kn	PROPN
ejpam-4653	448	10	,	,	PUNCT
ejpam-4653	448	11	n	n	PRON
ejpam-4653	448	12	≥	≥	NOUN
ejpam-4653	448	13	3	3	NUM
ejpam-4653	448	14	.	.	X
ejpam-4653	448	15	observe	observe	VERB
ejpam-4653	448	16	that	that	SCONJ
ejpam-4653	448	17	γdr(gg	γdr(gg	NOUN
ejpam-4653	448	18	)	)	PUNCT
ejpam-4653	448	19	=	=	SYM
ejpam-4653	449	1	3	3	NUM
ejpam-4653	449	2	+	+	CCONJ
ejpam-4653	449	3	2(n−	2(n−	NUM
ejpam-4653	449	4	1	1	NUM
ejpam-4653	449	5	)	)	PUNCT
ejpam-4653	449	6	=	=	SYM
ejpam-4653	449	7	ρ	ρ	PROPN
ejpam-4653	449	8	.	.	PUNCT
ejpam-4653	449	9	corollary	corollary	ADJ
ejpam-4653	449	10	2	2	NUM
ejpam-4653	449	11	.	.	PUNCT
ejpam-4653	450	1	let	let	VERB
ejpam-4653	450	2	g	g	PRON
ejpam-4653	450	3	be	be	AUX
ejpam-4653	450	4	a	a	DET
ejpam-4653	450	5	nontrivial	nontrivial	ADJ
ejpam-4653	450	6	graph	graph	NOUN
ejpam-4653	450	7	with	with	ADP
ejpam-4653	450	8	isolated	isolated	ADJ
ejpam-4653	450	9	vertex	vertex	NOUN
ejpam-4653	450	10	v.	v.	ADP
ejpam-4653	450	11	then	then	ADV
ejpam-4653	450	12	γdr(gg	γdr(gg	NUM
ejpam-4653	450	13	)	)	PUNCT
ejpam-4653	450	14	=	=	SYM
ejpam-4653	451	1	3	3	NUM
ejpam-4653	451	2	+	+	CCONJ
ejpam-4653	451	3	γdr(g−	γdr(g−	NOUN
ejpam-4653	451	4	v	v	NOUN
ejpam-4653	451	5	)	)	PUNCT
ejpam-4653	451	6	.	.	PUNCT
ejpam-4653	452	1	proof	proof	NOUN
ejpam-4653	452	2	.	.	PUNCT
ejpam-4653	453	1	let	let	VERB
ejpam-4653	453	2	f	f	PRON
ejpam-4653	453	3	be	be	AUX
ejpam-4653	453	4	a	a	DET
ejpam-4653	453	5	γdr	γdr	NOUN
ejpam-4653	453	6	-	-	PUNCT
ejpam-4653	453	7	function	function	NOUN
ejpam-4653	453	8	of	of	ADP
ejpam-4653	453	9	g−v	g−v	NOUN
ejpam-4653	453	10	.	.	PUNCT
ejpam-4653	454	1	extend	extend	VERB
ejpam-4653	454	2	f	f	PROPN
ejpam-4653	454	3	to	to	ADP
ejpam-4653	454	4	a	a	DET
ejpam-4653	454	5	function	function	NOUN
ejpam-4653	454	6	on	on	ADP
ejpam-4653	454	7	v	v	PROPN
ejpam-4653	454	8	(	(	PUNCT
ejpam-4653	454	9	gg	gg	NOUN
ejpam-4653	454	10	)	)	PUNCT
ejpam-4653	454	11	by	by	ADP
ejpam-4653	454	12	defining	define	VERB
ejpam-4653	454	13	f(v	f(v	NOUN
ejpam-4653	454	14	)	)	PUNCT
ejpam-4653	454	15	=	=	SYM
ejpam-4653	454	16	3	3	NUM
ejpam-4653	454	17	and	and	CCONJ
ejpam-4653	454	18	f(x	f(x	PROPN
ejpam-4653	454	19	)	)	PUNCT
ejpam-4653	455	1	=	=	SYM
ejpam-4653	455	2	0	0	NUM
ejpam-4653	456	1	for	for	ADP
ejpam-4653	456	2	all	all	PRON
ejpam-4653	456	3	x	x	SYM
ejpam-4653	456	4	∈	∈	PROPN
ejpam-4653	456	5	v	v	NOUN
ejpam-4653	456	6	(	(	PUNCT
ejpam-4653	456	7	g	g	NOUN
ejpam-4653	456	8	)	)	PUNCT
ejpam-4653	456	9	∪	∪	NOUN
ejpam-4653	456	10	{	{	PUNCT
ejpam-4653	456	11	v	v	NOUN
ejpam-4653	456	12	}	}	PUNCT
ejpam-4653	456	13	.	.	PUNCT
ejpam-4653	457	1	since	since	SCONJ
ejpam-4653	457	2	f	f	PROPN
ejpam-4653	457	3	∈	∈	PROPN
ejpam-4653	457	4	drd(gg	drd(gg	PROPN
ejpam-4653	457	5	)	)	PUNCT
ejpam-4653	457	6	,	,	PUNCT
ejpam-4653	457	7	γdr(gg	γdr(gg	NOUN
ejpam-4653	457	8	)	)	PUNCT
ejpam-4653	457	9	≤	≤	NOUN
ejpam-4653	457	10	3	3	NUM
ejpam-4653	458	1	+	+	CCONJ
ejpam-4653	458	2	γdr(g−	γdr(g−	NOUN
ejpam-4653	458	3	v	v	NOUN
ejpam-4653	458	4	)	)	PUNCT
ejpam-4653	458	5	.	.	PUNCT
ejpam-4653	459	1	on	on	ADP
ejpam-4653	459	2	the	the	DET
ejpam-4653	459	3	other	other	ADJ
ejpam-4653	459	4	hand	hand	NOUN
ejpam-4653	459	5	,	,	PUNCT
ejpam-4653	459	6	by	by	ADP
ejpam-4653	459	7	proposition	proposition	NOUN
ejpam-4653	459	8	9	9	NUM
ejpam-4653	459	9	,	,	PUNCT
ejpam-4653	459	10	γdr(g	γdr(g	X
ejpam-4653	459	11	)	)	PUNCT
ejpam-4653	459	12	=	=	SYM
ejpam-4653	459	13	2+γdr(g−v	2+γdr(g−v	NOUN
ejpam-4653	459	14	)	)	PUNCT
ejpam-4653	459	15	.	.	PUNCT
ejpam-4653	460	1	thus	thus	ADV
ejpam-4653	460	2	,	,	PUNCT
ejpam-4653	460	3	3+γdr(g−v	3+γdr(g−v	NOUN
ejpam-4653	460	4	)	)	PUNCT
ejpam-4653	460	5	=	=	SYM
ejpam-4653	461	1	1	1	NUM
ejpam-4653	461	2	+	+	CCONJ
ejpam-4653	461	3	γdr(g	γdr(g	X
ejpam-4653	461	4	)	)	PUNCT
ejpam-4653	461	5	≤	≤	NUM
ejpam-4653	461	6	γdr(gg	γdr(gg	NOUN
ejpam-4653	461	7	)	)	PUNCT
ejpam-4653	461	8	by	by	ADP
ejpam-4653	461	9	theorem	theorem	NOUN
ejpam-4653	461	10	1	1	NUM
ejpam-4653	461	11	.	.	PUNCT
ejpam-4653	462	1	let	let	VERB
ejpam-4653	462	2	g	g	NOUN
ejpam-4653	463	1	and	and	CCONJ
ejpam-4653	463	2	h	h	NOUN
ejpam-4653	463	3	be	be	AUX
ejpam-4653	463	4	graphs	graph	NOUN
ejpam-4653	463	5	with	with	ADP
ejpam-4653	463	6	disjoint	disjoint	ADJ
ejpam-4653	463	7	vertex	vertex	NOUN
ejpam-4653	463	8	sets	set	NOUN
ejpam-4653	463	9	.	.	PUNCT
ejpam-4653	464	1	the	the	DET
ejpam-4653	464	2	corona	corona	NOUN
ejpam-4653	464	3	of	of	ADP
ejpam-4653	464	4	g	g	PROPN
ejpam-4653	464	5	and	and	CCONJ
ejpam-4653	464	6	h	h	NOUN
ejpam-4653	464	7	is	be	AUX
ejpam-4653	464	8	the	the	DET
ejpam-4653	464	9	graph	graph	NOUN
ejpam-4653	464	10	g	g	PROPN
ejpam-4653	464	11	◦	◦	NOUN
ejpam-4653	464	12	h	h	NOUN
ejpam-4653	464	13	obtained	obtain	VERB
ejpam-4653	464	14	by	by	ADP
ejpam-4653	464	15	taking	take	VERB
ejpam-4653	464	16	one	one	NUM
ejpam-4653	464	17	copy	copy	NOUN
ejpam-4653	464	18	of	of	ADP
ejpam-4653	464	19	g	g	PROPN
ejpam-4653	464	20	and	and	CCONJ
ejpam-4653	464	21	|v	|v	PROPN
ejpam-4653	464	22	(	(	PUNCT
ejpam-4653	464	23	g)|	g)|	NOUN
ejpam-4653	464	24	copies	copy	NOUN
ejpam-4653	464	25	of	of	ADP
ejpam-4653	464	26	h	h	NOUN
ejpam-4653	464	27	,	,	PUNCT
ejpam-4653	464	28	and	and	CCONJ
ejpam-4653	464	29	then	then	ADV
ejpam-4653	464	30	joining	join	VERB
ejpam-4653	464	31	the	the	DET
ejpam-4653	464	32	ith	ith	PROPN
ejpam-4653	464	33	vertex	vertex	NOUN
ejpam-4653	464	34	of	of	ADP
ejpam-4653	464	35	g	g	NOUN
ejpam-4653	464	36	to	to	ADP
ejpam-4653	464	37	every	every	DET
ejpam-4653	464	38	vertex	vertex	NOUN
ejpam-4653	464	39	of	of	ADP
ejpam-4653	464	40	the	the	DET
ejpam-4653	464	41	ith	ith	PROPN
ejpam-4653	464	42	copy	copy	NOUN
ejpam-4653	464	43	of	of	ADP
ejpam-4653	464	44	h.	h.	PROPN
ejpam-4653	464	45	for	for	ADP
ejpam-4653	464	46	convenience	convenience	NOUN
ejpam-4653	464	47	,	,	PUNCT
ejpam-4653	464	48	we	we	PRON
ejpam-4653	464	49	write	write	VERB
ejpam-4653	464	50	hv	hv	PROPN
ejpam-4653	464	51	to	to	PART
ejpam-4653	464	52	denote	denote	VERB
ejpam-4653	464	53	the	the	DET
ejpam-4653	464	54	copy	copy	NOUN
ejpam-4653	464	55	of	of	ADP
ejpam-4653	464	56	h	h	NOUN
ejpam-4653	464	57	joined	join	VERB
ejpam-4653	464	58	to	to	ADP
ejpam-4653	464	59	v	v	VERB
ejpam-4653	464	60	and	and	CCONJ
ejpam-4653	464	61	write	write	VERB
ejpam-4653	464	62	hv+v	hv+v	PROPN
ejpam-4653	464	63	=	=	PUNCT
ejpam-4653	464	64	hv+⟨{v}⟩.	hv+⟨{v}⟩.	PROPN
ejpam-4653	464	65	if	if	SCONJ
ejpam-4653	464	66	h	h	PRON
ejpam-4653	464	67	=	=	PRON
ejpam-4653	464	68	{	{	PUNCT
ejpam-4653	464	69	u	u	NOUN
ejpam-4653	464	70	}	}	PUNCT
ejpam-4653	464	71	,	,	PUNCT
ejpam-4653	464	72	then	then	ADV
ejpam-4653	464	73	v	v	X
ejpam-4653	464	74	(	(	PUNCT
ejpam-4653	464	75	hv	hv	PROPN
ejpam-4653	464	76	)	)	PUNCT
ejpam-4653	464	77	=	=	PRON
ejpam-4653	464	78	{	{	PUNCT
ejpam-4653	464	79	uv	uv	NOUN
ejpam-4653	464	80	}	}	PUNCT
ejpam-4653	464	81	.	.	PUNCT
ejpam-4653	465	1	given	give	VERB
ejpam-4653	465	2	a	a	DET
ejpam-4653	465	3	function	function	NOUN
ejpam-4653	465	4	f	f	NOUN
ejpam-4653	465	5	=	=	SYM
ejpam-4653	465	6	(	(	PUNCT
ejpam-4653	465	7	v0	v0	PROPN
ejpam-4653	465	8	,	,	PUNCT
ejpam-4653	465	9	v1	v1	NOUN
ejpam-4653	465	10	,	,	PUNCT
ejpam-4653	465	11	v2	v2	PROPN
ejpam-4653	465	12	,	,	PUNCT
ejpam-4653	465	13	v3	v3	PROPN
ejpam-4653	465	14	)	)	PUNCT
ejpam-4653	465	15	on	on	ADP
ejpam-4653	465	16	v	v	NUM
ejpam-4653	465	17	(	(	PUNCT
ejpam-4653	465	18	g	g	PROPN
ejpam-4653	465	19	◦	◦	NOUN
ejpam-4653	465	20	h	h	NOUN
ejpam-4653	465	21	)	)	PUNCT
ejpam-4653	465	22	,	,	PUNCT
ejpam-4653	465	23	we	we	PRON
ejpam-4653	465	24	write	write	VERB
ejpam-4653	465	25	for	for	ADP
ejpam-4653	465	26	each	each	PRON
ejpam-4653	465	27	v	v	NUM
ejpam-4653	465	28	∈	∈	PROPN
ejpam-4653	465	29	v	v	NOUN
ejpam-4653	465	30	(	(	PUNCT
ejpam-4653	465	31	g	g	NOUN
ejpam-4653	465	32	)	)	PUNCT
ejpam-4653	465	33	,	,	PUNCT
ejpam-4653	465	34	v	v	NOUN
ejpam-4653	465	35	v	v	ADP
ejpam-4653	465	36	i	i	NOUN
ejpam-4653	465	37	=	=	SYM
ejpam-4653	465	38	vi	vi	PROPN
ejpam-4653	465	39	∩	∩	ADJ
ejpam-4653	465	40	v	v	X
ejpam-4653	465	41	(	(	PUNCT
ejpam-4653	465	42	hv	hv	PROPN
ejpam-4653	465	43	)	)	PUNCT
ejpam-4653	465	44	for	for	ADP
ejpam-4653	465	45	all	all	DET
ejpam-4653	465	46	i	i	PRON
ejpam-4653	465	47	=	=	NOUN
ejpam-4653	465	48	0	0	NUM
ejpam-4653	465	49	,	,	PUNCT
ejpam-4653	465	50	1	1	NUM
ejpam-4653	465	51	,	,	PUNCT
ejpam-4653	465	52	2	2	NUM
ejpam-4653	465	53	,	,	PUNCT
ejpam-4653	465	54	3	3	NUM
ejpam-4653	465	55	.	.	X
ejpam-4653	465	56	observe	observe	VERB
ejpam-4653	465	57	also	also	ADV
ejpam-4653	465	58	that	that	SCONJ
ejpam-4653	465	59	ωg	ωg	PART
ejpam-4653	465	60	◦	◦	VERB
ejpam-4653	465	61	h(f	h(f	NOUN
ejpam-4653	465	62	)	)	PUNCT
ejpam-4653	465	63	=	=	PUNCT
ejpam-4653	465	64	∑	∑	PUNCT
ejpam-4653	465	65	v∈v	v∈v	NOUN
ejpam-4653	465	66	(	(	PUNCT
ejpam-4653	465	67	g	g	NOUN
ejpam-4653	465	68	)	)	PUNCT
ejpam-4653	465	69	ωhv+v(f	ωhv+v(f	NUM
ejpam-4653	465	70	|hv+v	|hv+v	NOUN
ejpam-4653	465	71	)	)	PUNCT
ejpam-4653	465	72	.	.	PUNCT
ejpam-4653	466	1	proposition	proposition	NOUN
ejpam-4653	466	2	15	15	NUM
ejpam-4653	466	3	.	.	PUNCT
ejpam-4653	467	1	let	let	VERB
ejpam-4653	467	2	g	g	PRON
ejpam-4653	467	3	be	be	AUX
ejpam-4653	467	4	a	a	DET
ejpam-4653	467	5	nontrivial	nontrivial	ADJ
ejpam-4653	467	6	connected	connect	VERB
ejpam-4653	467	7	graph	graph	NOUN
ejpam-4653	467	8	and	and	CCONJ
ejpam-4653	467	9	h	h	NOUN
ejpam-4653	467	10	any	any	DET
ejpam-4653	467	11	graph	graph	NOUN
ejpam-4653	467	12	,	,	PUNCT
ejpam-4653	467	13	and	and	CCONJ
ejpam-4653	467	14	let	let	VERB
ejpam-4653	467	15	f	f	PROPN
ejpam-4653	467	16	=	=	SYM
ejpam-4653	467	17	(	(	PUNCT
ejpam-4653	467	18	v0	v0	PROPN
ejpam-4653	467	19	,	,	PUNCT
ejpam-4653	467	20	v1	v1	NOUN
ejpam-4653	467	21	,	,	PUNCT
ejpam-4653	467	22	v2	v2	PROPN
ejpam-4653	467	23	,	,	PUNCT
ejpam-4653	467	24	v3	v3	PROPN
ejpam-4653	467	25	)	)	PUNCT
ejpam-4653	467	26	be	be	VERB
ejpam-4653	467	27	a	a	DET
ejpam-4653	467	28	function	function	NOUN
ejpam-4653	467	29	on	on	ADP
ejpam-4653	467	30	v	v	NOUN
ejpam-4653	467	31	(	(	PUNCT
ejpam-4653	467	32	g	g	PROPN
ejpam-4653	467	33	◦	◦	NOUN
ejpam-4653	467	34	h	h	NOUN
ejpam-4653	467	35	)	)	PUNCT
ejpam-4653	467	36	.	.	PUNCT
ejpam-4653	468	1	then	then	ADV
ejpam-4653	468	2	f	f	PROPN
ejpam-4653	468	3	∈	∈	PROPN
ejpam-4653	468	4	drd(g	drd(g	VERB
ejpam-4653	468	5	◦	◦	NOUN
ejpam-4653	468	6	h	h	NOUN
ejpam-4653	468	7	)	)	PUNCT
ejpam-4653	469	1	if	if	SCONJ
ejpam-4653	469	2	and	and	CCONJ
ejpam-4653	469	3	only	only	ADV
ejpam-4653	469	4	if	if	SCONJ
ejpam-4653	469	5	each	each	PRON
ejpam-4653	469	6	of	of	ADP
ejpam-4653	469	7	the	the	DET
ejpam-4653	469	8	following	following	NOUN
ejpam-4653	469	9	holds	hold	VERB
ejpam-4653	469	10	for	for	ADP
ejpam-4653	469	11	f	f	PROPN
ejpam-4653	469	12	:	:	PUNCT
ejpam-4653	469	13	(	(	PUNCT
ejpam-4653	469	14	i	i	NOUN
ejpam-4653	469	15	)	)	PUNCT
ejpam-4653	469	16	for	for	ADP
ejpam-4653	469	17	each	each	DET
ejpam-4653	469	18	v	v	X
ejpam-4653	469	19	∈	∈	PROPN
ejpam-4653	469	20	(	(	PUNCT
ejpam-4653	469	21	v0	v0	NOUN
ejpam-4653	469	22	∪	∪	NOUN
ejpam-4653	469	23	v1)∩v	v1)∩v	X
ejpam-4653	469	24	(	(	PUNCT
ejpam-4653	469	25	g	g	NOUN
ejpam-4653	469	26	)	)	PUNCT
ejpam-4653	469	27	,	,	PUNCT
ejpam-4653	469	28	f	f	PROPN
ejpam-4653	469	29	|hv	|hv	NUM
ejpam-4653	469	30	∈	∈	PROPN
ejpam-4653	469	31	drd(hv	drd(hv	PROPN
ejpam-4653	469	32	)	)	PUNCT
ejpam-4653	469	33	.	.	PUNCT
ejpam-4653	470	1	moreover	moreover	ADV
ejpam-4653	470	2	,	,	PUNCT
ejpam-4653	470	3	for	for	ADP
ejpam-4653	470	4	each	each	DET
ejpam-4653	470	5	v	v	NUM
ejpam-4653	470	6	∈	∈	NOUN
ejpam-4653	470	7	v0∩v	v0∩v	ADP
ejpam-4653	470	8	(	(	PUNCT
ejpam-4653	470	9	g	g	NOUN
ejpam-4653	470	10	)	)	PUNCT
ejpam-4653	470	11	,	,	PUNCT
ejpam-4653	470	12	if	if	SCONJ
ejpam-4653	470	13	|v	|v	PROPN
ejpam-4653	470	14	v	v	ADP
ejpam-4653	470	15	2	2	NUM
ejpam-4653	470	16	|	|	NOUN
ejpam-4653	470	17	=	=	SYM
ejpam-4653	470	18	1	1	NUM
ejpam-4653	470	19	and	and	CCONJ
ejpam-4653	470	20	|v	|v	X
ejpam-4653	470	21	v	v	ADP
ejpam-4653	470	22	3	3	NUM
ejpam-4653	470	23	|	|	ADV
ejpam-4653	470	24	=	=	SYM
ejpam-4653	470	25	0	0	NUM
ejpam-4653	470	26	,	,	PUNCT
ejpam-4653	470	27	then	then	ADV
ejpam-4653	470	28	|	|	INTJ
ejpam-4653	470	29	(	(	PUNCT
ejpam-4653	470	30	v2	v2	PROPN
ejpam-4653	470	31	∪	∪	X
ejpam-4653	470	32	v3	v3	PROPN
ejpam-4653	470	33	)	)	PUNCT
ejpam-4653	470	34	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	470	35	≥	≥	NUM
ejpam-4653	470	36	1	1	NUM
ejpam-4653	470	37	.	.	PUNCT
ejpam-4653	470	38	(	(	PUNCT
ejpam-4653	470	39	ii	ii	NOUN
ejpam-4653	470	40	)	)	PUNCT
ejpam-4653	470	41	for	for	ADP
ejpam-4653	470	42	each	each	DET
ejpam-4653	470	43	v	v	NUM
ejpam-4653	470	44	∈	∈	PROPN
ejpam-4653	470	45	v2	v2	NOUN
ejpam-4653	470	46	∩	∩	ADJ
ejpam-4653	470	47	v	v	NOUN
ejpam-4653	470	48	(	(	PUNCT
ejpam-4653	470	49	g	g	NOUN
ejpam-4653	470	50	)	)	PUNCT
ejpam-4653	470	51	,	,	PUNCT
ejpam-4653	470	52	v	v	NOUN
ejpam-4653	470	53	v	v	ADP
ejpam-4653	470	54	2	2	NUM
ejpam-4653	470	55	∪	∪	NOUN
ejpam-4653	470	56	v	v	ADP
ejpam-4653	470	57	v	v	NUM
ejpam-4653	470	58	3	3	NUM
ejpam-4653	470	59	dominates	dominate	VERB
ejpam-4653	470	60	v	v	ADP
ejpam-4653	470	61	v	v	NOUN
ejpam-4653	470	62	0	0	NUM
ejpam-4653	470	63	.	.	PUNCT
ejpam-4653	471	1	proof	proof	NOUN
ejpam-4653	471	2	.	.	PUNCT
ejpam-4653	472	1	assume	assume	VERB
ejpam-4653	472	2	that	that	SCONJ
ejpam-4653	472	3	f	f	PROPN
ejpam-4653	472	4	∈	∈	PROPN
ejpam-4653	472	5	drd(g	drd(g	VERB
ejpam-4653	472	6	◦	◦	NOUN
ejpam-4653	472	7	h	h	NOUN
ejpam-4653	472	8	)	)	PUNCT
ejpam-4653	472	9	and	and	CCONJ
ejpam-4653	472	10	let	let	VERB
ejpam-4653	472	11	v	v	X
ejpam-4653	472	12	∈	∈	PROPN
ejpam-4653	472	13	(	(	PUNCT
ejpam-4653	472	14	v0	v0	NOUN
ejpam-4653	472	15	∪	∪	X
ejpam-4653	472	16	v1	v1	NOUN
ejpam-4653	472	17	)	)	PUNCT
ejpam-4653	472	18	∩	∩	ADJ
ejpam-4653	472	19	v	v	X
ejpam-4653	472	20	(	(	PUNCT
ejpam-4653	472	21	g	g	NOUN
ejpam-4653	472	22	)	)	PUNCT
ejpam-4653	472	23	.	.	PUNCT
ejpam-4653	473	1	to	to	PART
ejpam-4653	473	2	show	show	VERB
ejpam-4653	473	3	that	that	SCONJ
ejpam-4653	473	4	f	f	PROPN
ejpam-4653	473	5	|hv	|hv	NUM
ejpam-4653	473	6	∈	∈	PROPN
ejpam-4653	473	7	drd(hv	drd(hv	PROPN
ejpam-4653	473	8	)	)	PUNCT
ejpam-4653	473	9	,	,	PUNCT
ejpam-4653	473	10	first	first	ADV
ejpam-4653	473	11	let	let	VERB
ejpam-4653	473	12	u	u	PRON
ejpam-4653	473	13	∈	∈	PROPN
ejpam-4653	473	14	v	v	ADP
ejpam-4653	473	15	v	v	NOUN
ejpam-4653	473	16	0	0	NUM
ejpam-4653	473	17	.	.	PUNCT
ejpam-4653	474	1	note	note	VERB
ejpam-4653	474	2	that	that	SCONJ
ejpam-4653	474	3	ng	ng	PROPN
ejpam-4653	474	4	◦	◦	PROPN
ejpam-4653	474	5	h(u	h(u	PROPN
ejpam-4653	474	6	)	)	PUNCT
ejpam-4653	475	1	=	=	PRON
ejpam-4653	475	2	{	{	PUNCT
ejpam-4653	475	3	v	v	NOUN
ejpam-4653	475	4	}	}	PUNCT
ejpam-4653	475	5	∪	∪	ADP
ejpam-4653	475	6	nhv(u	nhv(u	PROPN
ejpam-4653	475	7	)	)	PUNCT
ejpam-4653	475	8	.	.	PUNCT
ejpam-4653	476	1	if	if	SCONJ
ejpam-4653	476	2	|v2	|v2	NOUN
ejpam-4653	476	3	∩ng	∩ng	VERB
ejpam-4653	476	4	◦	◦	PROPN
ejpam-4653	476	5	h(u)|	h(u)|	PROPN
ejpam-4653	476	6	≥	≥	NUM
ejpam-4653	476	7	2	2	NUM
ejpam-4653	476	8	,	,	PUNCT
ejpam-4653	476	9	then	then	ADV
ejpam-4653	476	10	|v	|v	VERB
ejpam-4653	476	11	v	v	ADP
ejpam-4653	476	12	2	2	NUM
ejpam-4653	476	13	∩nhv(u)|	∩nhv(u)|	NOUN
ejpam-4653	476	14	≥	≥	NOUN
ejpam-4653	476	15	2	2	NUM
ejpam-4653	476	16	.	.	PUNCT
ejpam-4653	477	1	on	on	ADP
ejpam-4653	477	2	the	the	DET
ejpam-4653	477	3	other	other	ADJ
ejpam-4653	477	4	hand	hand	NOUN
ejpam-4653	477	5	,	,	PUNCT
ejpam-4653	477	6	if	if	SCONJ
ejpam-4653	477	7	|v3	|v3	NOUN
ejpam-4653	477	8	∩ng	∩ng	VERB
ejpam-4653	477	9	◦	◦	NOUN
ejpam-4653	477	10	h(u)|	h(u)|	PROPN
ejpam-4653	477	11	≥	≥	NUM
ejpam-4653	477	12	1	1	NUM
ejpam-4653	477	13	,	,	PUNCT
ejpam-4653	477	14	then	then	ADV
ejpam-4653	477	15	|v	|v	VERB
ejpam-4653	477	16	v	v	ADP
ejpam-4653	477	17	3	3	NUM
ejpam-4653	477	18	∩	∩	PROPN
ejpam-4653	477	19	nhv(u)|	nhv(u)|	PROPN
ejpam-4653	477	20	≥	≥	NUM
ejpam-4653	477	21	1	1	NUM
ejpam-4653	477	22	.	.	PUNCT
ejpam-4653	478	1	next	next	ADV
ejpam-4653	478	2	,	,	PUNCT
ejpam-4653	478	3	let	let	VERB
ejpam-4653	478	4	u	u	PRON
ejpam-4653	478	5	∈	∈	PROPN
ejpam-4653	478	6	v	v	ADP
ejpam-4653	478	7	v	v	ADP
ejpam-4653	478	8	1	1	NUM
ejpam-4653	478	9	.	.	PUNCT
ejpam-4653	479	1	then	then	ADV
ejpam-4653	479	2	there	there	PRON
ejpam-4653	479	3	exists	exist	VERB
ejpam-4653	479	4	w	w	PROPN
ejpam-4653	479	5	∈	∈	PROPN
ejpam-4653	479	6	v2	v2	PROPN
ejpam-4653	479	7	∪	∪	X
ejpam-4653	479	8	v3	v3	PROPN
ejpam-4653	479	9	such	such	ADJ
ejpam-4653	479	10	that	that	SCONJ
ejpam-4653	479	11	w	w	PROPN
ejpam-4653	479	12	∈	∈	PROPN
ejpam-4653	479	13	ng	ng	PROPN
ejpam-4653	479	14	◦	◦	NOUN
ejpam-4653	479	15	h(u	h(u	PROPN
ejpam-4653	479	16	)	)	PUNCT
ejpam-4653	479	17	.	.	PUNCT
ejpam-4653	480	1	necessarily	necessarily	ADV
ejpam-4653	480	2	,	,	PUNCT
ejpam-4653	480	3	w	w	PROPN
ejpam-4653	480	4	∈	∈	PROPN
ejpam-4653	480	5	v	v	ADP
ejpam-4653	480	6	v	v	ADP
ejpam-4653	480	7	2	2	NUM
ejpam-4653	480	8	∪	∪	NOUN
ejpam-4653	480	9	v	v	NUM
ejpam-4653	480	10	v	v	ADP
ejpam-4653	480	11	3	3	NUM
ejpam-4653	480	12	and	and	CCONJ
ejpam-4653	480	13	w	w	PROPN
ejpam-4653	480	14	∈	∈	PROPN
ejpam-4653	480	15	nhv(u	nhv(u	PROPN
ejpam-4653	480	16	)	)	PUNCT
ejpam-4653	480	17	.	.	PUNCT
ejpam-4653	481	1	therefore	therefore	ADV
ejpam-4653	481	2	,	,	PUNCT
ejpam-4653	481	3	f	f	PROPN
ejpam-4653	481	4	|hv	|hv	NUM
ejpam-4653	481	5	∈	∈	PROPN
ejpam-4653	481	6	drd(hv	drd(hv	PROPN
ejpam-4653	481	7	)	)	PUNCT
ejpam-4653	481	8	.	.	PUNCT
ejpam-4653	482	1	j.	j.	PROPN
ejpam-4653	482	2	b.g	b.g	PROPN
ejpam-4653	482	3	.	.	PROPN
ejpam-4653	482	4	cariaga	cariaga	PROPN
ejpam-4653	482	5	,	,	PUNCT
ejpam-4653	482	6	f.	f.	PROPN
ejpam-4653	482	7	jamil	jamil	PROPN
ejpam-4653	482	8	/	/	SYM
ejpam-4653	482	9	eur	eur	PROPN
ejpam-4653	482	10	.	.	PUNCT
ejpam-4653	483	1	j.	j.	PROPN
ejpam-4653	483	2	pure	pure	PROPN
ejpam-4653	483	3	appl	appl	PROPN
ejpam-4653	483	4	.	.	PROPN
ejpam-4653	483	5	math	math	PROPN
ejpam-4653	483	6	,	,	PUNCT
ejpam-4653	483	7	16	16	NUM
ejpam-4653	483	8	(	(	PUNCT
ejpam-4653	483	9	2	2	NUM
ejpam-4653	483	10	)	)	PUNCT
ejpam-4653	483	11	(	(	PUNCT
ejpam-4653	483	12	2023	2023	NUM
ejpam-4653	483	13	)	)	PUNCT
ejpam-4653	483	14	,	,	PUNCT
ejpam-4653	483	15	847	847	NUM
ejpam-4653	483	16	-	-	SYM
ejpam-4653	483	17	863	863	NUM
ejpam-4653	483	18	858	858	NUM
ejpam-4653	483	19	now	now	ADV
ejpam-4653	483	20	,	,	PUNCT
ejpam-4653	483	21	let	let	VERB
ejpam-4653	483	22	v	v	PRON
ejpam-4653	483	23	∈	∈	PROPN
ejpam-4653	483	24	v0	v0	NOUN
ejpam-4653	483	25	∩	∩	X
ejpam-4653	483	26	v	v	X
ejpam-4653	483	27	(	(	PUNCT
ejpam-4653	483	28	g	g	NOUN
ejpam-4653	483	29	)	)	PUNCT
ejpam-4653	483	30	and	and	CCONJ
ejpam-4653	483	31	,	,	PUNCT
ejpam-4653	483	32	suppose	suppose	VERB
ejpam-4653	483	33	that	that	SCONJ
ejpam-4653	483	34	|v	|v	PROPN
ejpam-4653	483	35	v	v	ADP
ejpam-4653	483	36	2	2	NUM
ejpam-4653	483	37	|	|	NOUN
ejpam-4653	483	38	=	=	SYM
ejpam-4653	483	39	1	1	NUM
ejpam-4653	483	40	and	and	CCONJ
ejpam-4653	483	41	v	v	NOUN
ejpam-4653	483	42	v	v	ADP
ejpam-4653	483	43	3	3	NUM
ejpam-4653	483	44	=	=	SYM
ejpam-4653	483	45	∅.	∅.	NOUN
ejpam-4653	483	46	if	if	SCONJ
ejpam-4653	483	47	u	u	PROPN
ejpam-4653	483	48	∈	∈	PROPN
ejpam-4653	483	49	v3	v3	PROPN
ejpam-4653	483	50	∩ng	∩ng	PROPN
ejpam-4653	483	51	◦	◦	PROPN
ejpam-4653	483	52	h(v	h(v	PROPN
ejpam-4653	483	53	)	)	PUNCT
ejpam-4653	483	54	,	,	PUNCT
ejpam-4653	483	55	then	then	ADV
ejpam-4653	483	56	u	u	PROPN
ejpam-4653	483	57	∈	∈	PROPN
ejpam-4653	483	58	v3	v3	PROPN
ejpam-4653	483	59	∩ng(v	∩ng(v	PROPN
ejpam-4653	483	60	)	)	PUNCT
ejpam-4653	483	61	.	.	PUNCT
ejpam-4653	484	1	suppose	suppose	VERB
ejpam-4653	484	2	that	that	SCONJ
ejpam-4653	484	3	|v2	|v2	PROPN
ejpam-4653	484	4	∩ng	∩ng	VERB
ejpam-4653	484	5	◦	◦	NOUN
ejpam-4653	484	6	h(v)|	h(v)|	NOUN
ejpam-4653	484	7	≥	≥	NOUN
ejpam-4653	484	8	2	2	NUM
ejpam-4653	484	9	.	.	PUNCT
ejpam-4653	485	1	since	since	SCONJ
ejpam-4653	485	2	|v	|v	PROPN
ejpam-4653	485	3	v	v	ADP
ejpam-4653	485	4	2	2	NUM
ejpam-4653	485	5	|	|	NOUN
ejpam-4653	485	6	=	=	NOUN
ejpam-4653	485	7	1	1	NUM
ejpam-4653	485	8	,	,	PUNCT
ejpam-4653	485	9	|v2	|v2	NOUN
ejpam-4653	485	10	∩ng(v)|	∩ng(v)|	PROPN
ejpam-4653	485	11	≥	≥	NUM
ejpam-4653	485	12	1	1	NUM
ejpam-4653	485	13	.	.	PUNCT
ejpam-4653	485	14	this	this	PRON
ejpam-4653	485	15	completely	completely	ADV
ejpam-4653	485	16	proves	prove	VERB
ejpam-4653	485	17	(	(	PUNCT
ejpam-4653	485	18	i	i	NOUN
ejpam-4653	485	19	)	)	PUNCT
ejpam-4653	485	20	.	.	PUNCT
ejpam-4653	486	1	to	to	PART
ejpam-4653	486	2	prove	prove	VERB
ejpam-4653	486	3	(	(	PUNCT
ejpam-4653	486	4	ii	ii	NOUN
ejpam-4653	486	5	)	)	PUNCT
ejpam-4653	486	6	,	,	PUNCT
ejpam-4653	486	7	let	let	VERB
ejpam-4653	486	8	v	v	NUM
ejpam-4653	486	9	∈	∈	PROPN
ejpam-4653	486	10	v2	v2	PROPN
ejpam-4653	486	11	∩	∩	ADJ
ejpam-4653	486	12	v	v	NOUN
ejpam-4653	486	13	(	(	PUNCT
ejpam-4653	486	14	g	g	NOUN
ejpam-4653	486	15	)	)	PUNCT
ejpam-4653	486	16	and	and	CCONJ
ejpam-4653	486	17	u	u	PROPN
ejpam-4653	486	18	∈	∈	PROPN
ejpam-4653	486	19	v	v	ADP
ejpam-4653	486	20	v	v	NOUN
ejpam-4653	486	21	0	0	NUM
ejpam-4653	486	22	.	.	PUNCT
ejpam-4653	487	1	suppose	suppose	VERB
ejpam-4653	487	2	there	there	PRON
ejpam-4653	487	3	exists	exist	VERB
ejpam-4653	487	4	{	{	PUNCT
ejpam-4653	487	5	w	w	NOUN
ejpam-4653	487	6	,	,	PUNCT
ejpam-4653	487	7	z	z	NOUN
ejpam-4653	487	8	}	}	PUNCT
ejpam-4653	487	9	⊆	⊆	NUM
ejpam-4653	487	10	v2	v2	PROPN
ejpam-4653	487	11	∩	∩	ADJ
ejpam-4653	487	12	ng	ng	PROPN
ejpam-4653	487	13	◦	◦	NOUN
ejpam-4653	487	14	h(u	h(u	PROPN
ejpam-4653	487	15	)	)	PUNCT
ejpam-4653	487	16	.	.	PUNCT
ejpam-4653	488	1	if	if	SCONJ
ejpam-4653	488	2	w	w	PROPN
ejpam-4653	488	3	=	=	SYM
ejpam-4653	488	4	v	v	NOUN
ejpam-4653	488	5	,	,	PUNCT
ejpam-4653	488	6	then	then	ADV
ejpam-4653	488	7	z	z	PROPN
ejpam-4653	488	8	∈	∈	PROPN
ejpam-4653	488	9	v	v	ADP
ejpam-4653	488	10	v	v	NOUN
ejpam-4653	488	11	2	2	NUM
ejpam-4653	488	12	and	and	CCONJ
ejpam-4653	488	13	zu	zu	NUM
ejpam-4653	488	14	∈	∈	PROPN
ejpam-4653	488	15	e(hv	e(hv	PROPN
ejpam-4653	488	16	)	)	PUNCT
ejpam-4653	488	17	.	.	PUNCT
ejpam-4653	488	18	suppose	suppose	VERB
ejpam-4653	488	19	there	there	PRON
ejpam-4653	488	20	exists	exist	VERB
ejpam-4653	488	21	w	w	PROPN
ejpam-4653	488	22	∈	∈	PROPN
ejpam-4653	488	23	v3∩ng	v3∩ng	PROPN
ejpam-4653	488	24	◦	◦	PROPN
ejpam-4653	488	25	h(u	h(u	PROPN
ejpam-4653	488	26	)	)	PUNCT
ejpam-4653	488	27	.	.	PUNCT
ejpam-4653	489	1	then	then	ADV
ejpam-4653	489	2	as	as	ADP
ejpam-4653	489	3	w	w	PROPN
ejpam-4653	489	4	̸=	̸=	PROPN
ejpam-4653	489	5	v	v	NOUN
ejpam-4653	489	6	,	,	PUNCT
ejpam-4653	489	7	w	w	PROPN
ejpam-4653	489	8	∈	∈	PROPN
ejpam-4653	489	9	v	v	ADP
ejpam-4653	489	10	v	v	NOUN
ejpam-4653	489	11	3	3	NUM
ejpam-4653	489	12	and	and	CCONJ
ejpam-4653	489	13	wu	wu	PROPN
ejpam-4653	489	14	∈	∈	PROPN
ejpam-4653	489	15	e(hv	e(hv	PROPN
ejpam-4653	489	16	)	)	PUNCT
ejpam-4653	489	17	.	.	PUNCT
ejpam-4653	490	1	this	this	PRON
ejpam-4653	490	2	means	mean	VERB
ejpam-4653	490	3	that	that	SCONJ
ejpam-4653	490	4	v	v	X
ejpam-4653	490	5	v	v	ADP
ejpam-4653	490	6	2	2	NUM
ejpam-4653	490	7	∪	∪	NOUN
ejpam-4653	490	8	v	v	ADP
ejpam-4653	490	9	v	v	NUM
ejpam-4653	490	10	3	3	NUM
ejpam-4653	490	11	dominates	dominate	VERB
ejpam-4653	490	12	v	v	ADP
ejpam-4653	490	13	v	v	NOUN
ejpam-4653	490	14	0	0	NUM
ejpam-4653	490	15	.	.	PUNCT
ejpam-4653	491	1	conversely	conversely	ADV
ejpam-4653	491	2	,	,	PUNCT
ejpam-4653	491	3	assume	assume	VERB
ejpam-4653	491	4	that	that	SCONJ
ejpam-4653	491	5	(	(	PUNCT
ejpam-4653	491	6	i	i	NOUN
ejpam-4653	491	7	)	)	PUNCT
ejpam-4653	491	8	and	and	CCONJ
ejpam-4653	491	9	(	(	PUNCT
ejpam-4653	491	10	ii	ii	NOUN
ejpam-4653	491	11	)	)	PUNCT
ejpam-4653	491	12	all	all	PRON
ejpam-4653	491	13	hold	hold	VERB
ejpam-4653	491	14	for	for	ADP
ejpam-4653	491	15	f	f	PROPN
ejpam-4653	491	16	.	.	PUNCT
ejpam-4653	492	1	let	let	VERB
ejpam-4653	492	2	u	u	PRON
ejpam-4653	492	3	∈	∈	PROPN
ejpam-4653	492	4	v0	v0	NOUN
ejpam-4653	492	5	,	,	PUNCT
ejpam-4653	492	6	and	and	CCONJ
ejpam-4653	492	7	let	let	VERB
ejpam-4653	492	8	v	v	NUM
ejpam-4653	492	9	∈	∈	PROPN
ejpam-4653	492	10	v	v	NOUN
ejpam-4653	492	11	(	(	PUNCT
ejpam-4653	492	12	g	g	NOUN
ejpam-4653	492	13	)	)	PUNCT
ejpam-4653	492	14	for	for	ADP
ejpam-4653	492	15	which	which	PRON
ejpam-4653	492	16	u	u	PROPN
ejpam-4653	492	17	∈	∈	PROPN
ejpam-4653	492	18	v	v	NOUN
ejpam-4653	492	19	(	(	PUNCT
ejpam-4653	492	20	hv	hv	PROPN
ejpam-4653	492	21	+	+	PROPN
ejpam-4653	492	22	v	v	NOUN
ejpam-4653	492	23	)	)	PUNCT
ejpam-4653	492	24	.	.	PUNCT
ejpam-4653	493	1	first	first	ADV
ejpam-4653	493	2	,	,	PUNCT
ejpam-4653	493	3	suppose	suppose	VERB
ejpam-4653	493	4	that	that	SCONJ
ejpam-4653	493	5	u	u	PRON
ejpam-4653	494	1	=	=	NOUN
ejpam-4653	494	2	v.	v.	INTJ
ejpam-4653	494	3	if	if	SCONJ
ejpam-4653	494	4	v	v	NOUN
ejpam-4653	494	5	v	v	VERB
ejpam-4653	494	6	3	3	NUM
ejpam-4653	494	7	̸=	̸=	PROPN
ejpam-4653	494	8	∅	∅	NOUN
ejpam-4653	494	9	and	and	CCONJ
ejpam-4653	494	10	w	w	NOUN
ejpam-4653	494	11	∈	∈	PROPN
ejpam-4653	494	12	v	v	ADP
ejpam-4653	494	13	v	v	NOUN
ejpam-4653	494	14	3	3	NUM
ejpam-4653	494	15	,	,	PUNCT
ejpam-4653	494	16	then	then	ADV
ejpam-4653	494	17	w	w	PROPN
ejpam-4653	494	18	∈	∈	PROPN
ejpam-4653	494	19	v3	v3	PROPN
ejpam-4653	494	20	∩ng	∩ng	PROPN
ejpam-4653	494	21	◦	◦	PROPN
ejpam-4653	494	22	h(u	h(u	PROPN
ejpam-4653	494	23	)	)	PUNCT
ejpam-4653	494	24	.	.	PUNCT
ejpam-4653	495	1	suppose	suppose	VERB
ejpam-4653	495	2	that	that	SCONJ
ejpam-4653	495	3	v	v	X
ejpam-4653	495	4	v	v	ADP
ejpam-4653	495	5	3	3	NUM
ejpam-4653	495	6	=	=	SYM
ejpam-4653	495	7	∅.	∅.	NOUN
ejpam-4653	495	8	since	since	SCONJ
ejpam-4653	495	9	f	f	PROPN
ejpam-4653	495	10	|v	|v	PROPN
ejpam-4653	495	11	∈	∈	PROPN
ejpam-4653	495	12	drd(hv	drd(hv	NOUN
ejpam-4653	495	13	)	)	PUNCT
ejpam-4653	495	14	,	,	PUNCT
ejpam-4653	495	15	v	v	X
ejpam-4653	495	16	v	v	PRON
ejpam-4653	495	17	2	2	NUM
ejpam-4653	495	18	̸=	̸=	PROPN
ejpam-4653	495	19	∅.	∅.	ADV
ejpam-4653	495	20	if	if	SCONJ
ejpam-4653	495	21	|v	|v	PROPN
ejpam-4653	495	22	v	v	ADP
ejpam-4653	495	23	2	2	NUM
ejpam-4653	495	24	|	|	CCONJ
ejpam-4653	495	25	≥	≥	NOUN
ejpam-4653	495	26	2	2	NUM
ejpam-4653	495	27	,	,	PUNCT
ejpam-4653	495	28	then	then	ADV
ejpam-4653	495	29	|v2∩ng	|v2∩ng	VERB
ejpam-4653	495	30	◦	◦	NOUN
ejpam-4653	495	31	h(u)|	h(u)|	PROPN
ejpam-4653	495	32	≥	≥	NOUN
ejpam-4653	495	33	2	2	NUM
ejpam-4653	495	34	.	.	PUNCT
ejpam-4653	495	35	suppose	suppose	VERB
ejpam-4653	495	36	that	that	SCONJ
ejpam-4653	495	37	|v	|v	PROPN
ejpam-4653	495	38	v	v	ADP
ejpam-4653	495	39	2	2	NUM
ejpam-4653	495	40	|	|	NOUN
ejpam-4653	495	41	=	=	NOUN
ejpam-4653	495	42	1	1	X
ejpam-4653	495	43	.	.	PUNCT
ejpam-4653	495	44	by	by	ADP
ejpam-4653	495	45	condition	condition	NOUN
ejpam-4653	495	46	(	(	PUNCT
ejpam-4653	495	47	i	i	NOUN
ejpam-4653	495	48	)	)	PUNCT
ejpam-4653	495	49	,	,	PUNCT
ejpam-4653	496	1	|	|	ADV
ejpam-4653	496	2	(	(	PUNCT
ejpam-4653	496	3	v2	v2	VERB
ejpam-4653	496	4	∪	∪	ADP
ejpam-4653	496	5	v3)∩ng(v)|	v3)∩ng(v)|	NOUN
ejpam-4653	496	6	≥	≥	NOUN
ejpam-4653	496	7	1	1	NUM
ejpam-4653	496	8	.	.	PUNCT
ejpam-4653	497	1	this	this	PRON
ejpam-4653	497	2	means	mean	VERB
ejpam-4653	497	3	that	that	SCONJ
ejpam-4653	497	4	|v2	|v2	PROPN
ejpam-4653	497	5	∩ng	∩ng	VERB
ejpam-4653	497	6	◦	◦	PROPN
ejpam-4653	497	7	h(u)|	h(u)|	PROPN
ejpam-4653	497	8	≥	≥	NUM
ejpam-4653	497	9	2	2	NUM
ejpam-4653	497	10	or	or	CCONJ
ejpam-4653	497	11	|v3	|v3	NOUN
ejpam-4653	497	12	∩ng	∩ng	VERB
ejpam-4653	497	13	◦	◦	NOUN
ejpam-4653	497	14	h(u)|	h(u)|	PROPN
ejpam-4653	497	15	≥	≥	NUM
ejpam-4653	497	16	1	1	NUM
ejpam-4653	497	17	.	.	PUNCT
ejpam-4653	498	1	next	next	ADV
ejpam-4653	498	2	,	,	PUNCT
ejpam-4653	498	3	suppose	suppose	VERB
ejpam-4653	498	4	that	that	SCONJ
ejpam-4653	498	5	u	u	PROPN
ejpam-4653	498	6	∈	∈	PROPN
ejpam-4653	498	7	v	v	ADP
ejpam-4653	498	8	v	v	NOUN
ejpam-4653	498	9	0	0	NUM
ejpam-4653	498	10	.	.	PUNCT
ejpam-4653	499	1	if	if	SCONJ
ejpam-4653	499	2	v	v	NUM
ejpam-4653	499	3	∈	∈	PROPN
ejpam-4653	499	4	v3	v3	PROPN
ejpam-4653	499	5	,	,	PUNCT
ejpam-4653	499	6	then	then	ADV
ejpam-4653	499	7	|v3	|v3	NOUN
ejpam-4653	499	8	∩	∩	PROPN
ejpam-4653	499	9	ng	ng	PROPN
ejpam-4653	499	10	◦	◦	PROPN
ejpam-4653	499	11	h(u)|	h(u)|	PROPN
ejpam-4653	499	12	≥	≥	NOUN
ejpam-4653	499	13	1	1	NUM
ejpam-4653	499	14	.	.	PUNCT
ejpam-4653	500	1	if	if	SCONJ
ejpam-4653	500	2	v	v	NUM
ejpam-4653	500	3	∈	∈	PROPN
ejpam-4653	500	4	v0	v0	NOUN
ejpam-4653	500	5	∪	∪	X
ejpam-4653	500	6	v1	v1	NOUN
ejpam-4653	500	7	,	,	PUNCT
ejpam-4653	500	8	then	then	ADV
ejpam-4653	500	9	by	by	ADP
ejpam-4653	500	10	condition	condition	NOUN
ejpam-4653	500	11	(	(	PUNCT
ejpam-4653	500	12	i	i	NOUN
ejpam-4653	500	13	)	)	PUNCT
ejpam-4653	500	14	,	,	PUNCT
ejpam-4653	500	15	|v	|v	PROPN
ejpam-4653	500	16	v	v	ADP
ejpam-4653	500	17	2	2	NUM
ejpam-4653	500	18	∩	∩	NOUN
ejpam-4653	500	19	nhv(u)|	nhv(u)|	PROPN
ejpam-4653	500	20	≥	≥	NUM
ejpam-4653	500	21	2	2	NUM
ejpam-4653	500	22	or	or	CCONJ
ejpam-4653	500	23	|v	|v	PRON
ejpam-4653	500	24	v	v	ADP
ejpam-4653	500	25	3	3	NUM
ejpam-4653	500	26	∩	∩	PROPN
ejpam-4653	500	27	nhv(u)|	nhv(u)|	PROPN
ejpam-4653	500	28	≥	≥	NUM
ejpam-4653	500	29	1	1	NUM
ejpam-4653	500	30	.	.	PUNCT
ejpam-4653	501	1	this	this	PRON
ejpam-4653	501	2	means	mean	VERB
ejpam-4653	501	3	that	that	SCONJ
ejpam-4653	501	4	|v2	|v2	PROPN
ejpam-4653	501	5	∩ng	∩ng	VERB
ejpam-4653	501	6	◦	◦	PROPN
ejpam-4653	501	7	h(u)|	h(u)|	PROPN
ejpam-4653	501	8	≥	≥	NUM
ejpam-4653	501	9	2	2	NUM
ejpam-4653	501	10	or	or	CCONJ
ejpam-4653	501	11	|v3	|v3	NOUN
ejpam-4653	501	12	∩ng	∩ng	VERB
ejpam-4653	501	13	◦	◦	NOUN
ejpam-4653	501	14	h(u)|	h(u)|	PROPN
ejpam-4653	501	15	≥	≥	NOUN
ejpam-4653	501	16	1	1	NUM
ejpam-4653	501	17	.	.	PUNCT
ejpam-4653	502	1	now	now	ADV
ejpam-4653	502	2	,	,	PUNCT
ejpam-4653	502	3	suppose	suppose	VERB
ejpam-4653	502	4	that	that	SCONJ
ejpam-4653	502	5	v	v	ADP
ejpam-4653	502	6	∈	∈	PROPN
ejpam-4653	502	7	v2	v2	NOUN
ejpam-4653	502	8	.	.	PUNCT
ejpam-4653	503	1	by	by	ADP
ejpam-4653	503	2	condition	condition	NOUN
ejpam-4653	503	3	(	(	PUNCT
ejpam-4653	503	4	ii	ii	NOUN
ejpam-4653	503	5	)	)	PUNCT
ejpam-4653	503	6	,	,	PUNCT
ejpam-4653	503	7	there	there	PRON
ejpam-4653	503	8	exists	exist	VERB
ejpam-4653	503	9	w	w	PROPN
ejpam-4653	503	10	∈	∈	PROPN
ejpam-4653	503	11	v	v	ADP
ejpam-4653	503	12	v	v	ADP
ejpam-4653	503	13	2	2	NUM
ejpam-4653	503	14	∪	∪	NOUN
ejpam-4653	503	15	v	v	NUM
ejpam-4653	503	16	v	v	ADP
ejpam-4653	503	17	3	3	NUM
ejpam-4653	503	18	for	for	ADP
ejpam-4653	503	19	which	which	PRON
ejpam-4653	503	20	w	w	PROPN
ejpam-4653	503	21	∈	∈	PROPN
ejpam-4653	503	22	nhv(u	nhv(u	PROPN
ejpam-4653	503	23	)	)	PUNCT
ejpam-4653	503	24	.	.	PUNCT
ejpam-4653	504	1	if	if	SCONJ
ejpam-4653	504	2	w	w	PROPN
ejpam-4653	504	3	∈	∈	PROPN
ejpam-4653	504	4	v	v	ADP
ejpam-4653	504	5	v	v	NOUN
ejpam-4653	504	6	3	3	NUM
ejpam-4653	504	7	,	,	PUNCT
ejpam-4653	504	8	then	then	ADV
ejpam-4653	504	9	|v3	|v3	NOUN
ejpam-4653	504	10	∩ng	∩ng	VERB
ejpam-4653	504	11	◦	◦	NOUN
ejpam-4653	504	12	h(v)|	h(v)|	NOUN
ejpam-4653	504	13	≥	≥	NOUN
ejpam-4653	504	14	1	1	NUM
ejpam-4653	504	15	.	.	PUNCT
ejpam-4653	505	1	if	if	SCONJ
ejpam-4653	505	2	w	w	PROPN
ejpam-4653	505	3	∈	∈	PROPN
ejpam-4653	505	4	v	v	ADP
ejpam-4653	505	5	v	v	NOUN
ejpam-4653	505	6	2	2	NUM
ejpam-4653	505	7	,	,	PUNCT
ejpam-4653	505	8	then	then	ADV
ejpam-4653	505	9	{	{	PUNCT
ejpam-4653	505	10	w	w	PROPN
ejpam-4653	505	11	,	,	PUNCT
ejpam-4653	505	12	v	v	NOUN
ejpam-4653	505	13	}	}	PUNCT
ejpam-4653	505	14	⊆	⊆	NUM
ejpam-4653	505	15	v2	v2	PROPN
ejpam-4653	505	16	∩ng	∩ng	NOUN
ejpam-4653	505	17	◦	◦	NOUN
ejpam-4653	505	18	h(u	h(u	PROPN
ejpam-4653	505	19	)	)	PUNCT
ejpam-4653	505	20	.	.	PUNCT
ejpam-4653	506	1	finally	finally	ADV
ejpam-4653	506	2	,	,	PUNCT
ejpam-4653	506	3	let	let	VERB
ejpam-4653	506	4	u	u	PRON
ejpam-4653	506	5	∈	∈	PROPN
ejpam-4653	506	6	v1	v1	NOUN
ejpam-4653	506	7	.	.	PUNCT
ejpam-4653	507	1	if	if	SCONJ
ejpam-4653	507	2	u	u	PROPN
ejpam-4653	507	3	∈	∈	PROPN
ejpam-4653	507	4	v	v	X
ejpam-4653	507	5	(	(	PUNCT
ejpam-4653	507	6	g	g	NOUN
ejpam-4653	507	7	)	)	PUNCT
ejpam-4653	507	8	,	,	PUNCT
ejpam-4653	507	9	then	then	ADV
ejpam-4653	507	10	since	since	SCONJ
ejpam-4653	507	11	f	f	PROPN
ejpam-4653	507	12	|hv	|hv	NUM
ejpam-4653	507	13	∈	∈	PROPN
ejpam-4653	507	14	drd(hu	drd(hu	NOUN
ejpam-4653	507	15	)	)	PUNCT
ejpam-4653	507	16	(	(	PUNCT
ejpam-4653	507	17	by	by	ADP
ejpam-4653	507	18	(	(	PUNCT
ejpam-4653	507	19	i	i	NOUN
ejpam-4653	507	20	)	)	PUNCT
ejpam-4653	507	21	)	)	PUNCT
ejpam-4653	507	22	,	,	PUNCT
ejpam-4653	507	23	v	v	X
ejpam-4653	507	24	u	u	NOUN
ejpam-4653	507	25	2	2	NUM
ejpam-4653	507	26	∪	∪	X
ejpam-4653	507	27	v	v	NUM
ejpam-4653	507	28	u	u	NOUN
ejpam-4653	507	29	3	3	NUM
ejpam-4653	507	30	̸=	̸=	PROPN
ejpam-4653	507	31	∅	∅	NOUN
ejpam-4653	507	32	,	,	PUNCT
ejpam-4653	507	33	say	say	VERB
ejpam-4653	507	34	w	w	PROPN
ejpam-4653	507	35	∈	∈	PROPN
ejpam-4653	507	36	v	v	ADP
ejpam-4653	507	37	u	u	NOUN
ejpam-4653	507	38	2	2	NUM
ejpam-4653	507	39	∪	∪	X
ejpam-4653	507	40	v	v	NUM
ejpam-4653	507	41	u	u	NOUN
ejpam-4653	507	42	2	2	NUM
ejpam-4653	507	43	.	.	PUNCT
ejpam-4653	508	1	then	then	ADV
ejpam-4653	508	2	w	w	PROPN
ejpam-4653	508	3	∈	∈	PROPN
ejpam-4653	508	4	(	(	PUNCT
ejpam-4653	508	5	v1	v1	NOUN
ejpam-4653	508	6	∪	∪	NOUN
ejpam-4653	508	7	v2	v2	NOUN
ejpam-4653	508	8	)	)	PUNCT
ejpam-4653	508	9	∩	∩	PROPN
ejpam-4653	508	10	ng	ng	PROPN
ejpam-4653	508	11	◦	◦	PROPN
ejpam-4653	508	12	h(u	h(u	PROPN
ejpam-4653	508	13	)	)	PUNCT
ejpam-4653	508	14	.	.	PUNCT
ejpam-4653	509	1	suppose	suppose	VERB
ejpam-4653	509	2	that	that	SCONJ
ejpam-4653	509	3	u	u	PROPN
ejpam-4653	509	4	∈	∈	PROPN
ejpam-4653	509	5	v	v	ADP
ejpam-4653	509	6	(	(	PUNCT
ejpam-4653	509	7	hv	hv	PROPN
ejpam-4653	509	8	)	)	PUNCT
ejpam-4653	509	9	for	for	ADP
ejpam-4653	509	10	some	some	DET
ejpam-4653	509	11	v	v	ADP
ejpam-4653	509	12	∈	∈	PROPN
ejpam-4653	509	13	v	v	NOUN
ejpam-4653	509	14	(	(	PUNCT
ejpam-4653	509	15	g	g	NOUN
ejpam-4653	509	16	)	)	PUNCT
ejpam-4653	509	17	.	.	PUNCT
ejpam-4653	510	1	if	if	SCONJ
ejpam-4653	510	2	v	v	NUM
ejpam-4653	510	3	∈	∈	PROPN
ejpam-4653	510	4	v2	v2	PROPN
ejpam-4653	510	5	∪	∪	X
ejpam-4653	510	6	v3	v3	PROPN
ejpam-4653	510	7	,	,	PUNCT
ejpam-4653	510	8	then	then	ADV
ejpam-4653	510	9	v	v	X
ejpam-4653	510	10	∈	∈	PROPN
ejpam-4653	510	11	(	(	PUNCT
ejpam-4653	510	12	v1	v1	NOUN
ejpam-4653	510	13	∪	∪	NOUN
ejpam-4653	510	14	v2	v2	NOUN
ejpam-4653	510	15	)	)	PUNCT
ejpam-4653	510	16	∩	∩	PROPN
ejpam-4653	510	17	ng	ng	PROPN
ejpam-4653	510	18	◦	◦	PROPN
ejpam-4653	510	19	h(u	h(u	PROPN
ejpam-4653	510	20	)	)	PUNCT
ejpam-4653	510	21	.	.	PUNCT
ejpam-4653	511	1	if	if	SCONJ
ejpam-4653	511	2	v	v	NUM
ejpam-4653	511	3	∈	∈	PROPN
ejpam-4653	511	4	v0	v0	NOUN
ejpam-4653	511	5	∪	∪	X
ejpam-4653	511	6	v1	v1	PROPN
ejpam-4653	511	7	,	,	PUNCT
ejpam-4653	511	8	then	then	ADV
ejpam-4653	511	9	as	as	ADP
ejpam-4653	511	10	f	f	PROPN
ejpam-4653	511	11	|v	|v	PROPN
ejpam-4653	511	12	∈	∈	PROPN
ejpam-4653	511	13	drd(hv	drd(hv	NOUN
ejpam-4653	511	14	)	)	PUNCT
ejpam-4653	511	15	(	(	PUNCT
ejpam-4653	511	16	by	by	ADP
ejpam-4653	511	17	(	(	PUNCT
ejpam-4653	511	18	i	i	NOUN
ejpam-4653	511	19	)	)	PUNCT
ejpam-4653	511	20	)	)	PUNCT
ejpam-4653	511	21	,	,	PUNCT
ejpam-4653	511	22	there	there	PRON
ejpam-4653	511	23	exists	exist	VERB
ejpam-4653	511	24	w	w	PROPN
ejpam-4653	511	25	∈	∈	PROPN
ejpam-4653	511	26	v	v	ADP
ejpam-4653	511	27	v	v	ADP
ejpam-4653	511	28	2	2	NUM
ejpam-4653	511	29	∪	∪	NOUN
ejpam-4653	511	30	v	v	NUM
ejpam-4653	511	31	v	v	ADP
ejpam-4653	511	32	3	3	NUM
ejpam-4653	511	33	such	such	ADJ
ejpam-4653	511	34	that	that	DET
ejpam-4653	511	35	w	w	PROPN
ejpam-4653	511	36	∈	∈	PROPN
ejpam-4653	511	37	nhv(u	nhv(u	PROPN
ejpam-4653	511	38	)	)	PUNCT
ejpam-4653	511	39	.	.	PUNCT
ejpam-4653	512	1	this	this	PRON
ejpam-4653	512	2	means	mean	VERB
ejpam-4653	512	3	that	that	SCONJ
ejpam-4653	512	4	w	w	PROPN
ejpam-4653	512	5	∈	∈	PROPN
ejpam-4653	512	6	v2	v2	NOUN
ejpam-4653	512	7	∪	∪	X
ejpam-4653	512	8	v3	v3	PROPN
ejpam-4653	512	9	and	and	CCONJ
ejpam-4653	512	10	w	w	PROPN
ejpam-4653	512	11	∈	∈	PROPN
ejpam-4653	512	12	ng	ng	PROPN
ejpam-4653	512	13	◦	◦	NOUN
ejpam-4653	512	14	h(u	h(u	PROPN
ejpam-4653	512	15	)	)	PUNCT
ejpam-4653	512	16	.	.	PUNCT
ejpam-4653	513	1	therefore	therefore	ADV
ejpam-4653	513	2	,	,	PUNCT
ejpam-4653	513	3	f	f	PROPN
ejpam-4653	513	4	∈	∈	PROPN
ejpam-4653	513	5	drd(g	drd(g	VERB
ejpam-4653	513	6	◦	◦	NOUN
ejpam-4653	513	7	h	h	NOUN
ejpam-4653	513	8	)	)	PUNCT
ejpam-4653	513	9	.	.	PUNCT
ejpam-4653	514	1	corollary	corollary	ADJ
ejpam-4653	514	2	3	3	X
ejpam-4653	514	3	.	.	PUNCT
ejpam-4653	515	1	let	let	VERB
ejpam-4653	515	2	g	g	PRON
ejpam-4653	515	3	be	be	AUX
ejpam-4653	515	4	a	a	DET
ejpam-4653	515	5	nontrivial	nontrivial	ADJ
ejpam-4653	515	6	connected	connect	VERB
ejpam-4653	515	7	graph	graph	NOUN
ejpam-4653	515	8	of	of	ADP
ejpam-4653	515	9	order	order	NOUN
ejpam-4653	516	1	n.	n.	NOUN
ejpam-4653	516	2	then	then	ADV
ejpam-4653	516	3	(	(	PUNCT
ejpam-4653	516	4	i	i	NOUN
ejpam-4653	516	5	)	)	PUNCT
ejpam-4653	516	6	γdr(g	γdr(g	PROPN
ejpam-4653	516	7	◦	◦	NOUN
ejpam-4653	516	8	k1	k1	NOUN
ejpam-4653	516	9	)	)	PUNCT
ejpam-4653	516	10	=	=	SYM
ejpam-4653	517	1	3n−max{|v0|	3n−max{|v0|	NOUN
ejpam-4653	517	2	:	:	PUNCT
ejpam-4653	517	3	f	f	X
ejpam-4653	517	4	=	=	SYM
ejpam-4653	517	5	(	(	PUNCT
ejpam-4653	517	6	v0	v0	PROPN
ejpam-4653	517	7	,	,	PUNCT
ejpam-4653	517	8	v1	v1	NOUN
ejpam-4653	517	9	,	,	PUNCT
ejpam-4653	517	10	v2.v3	v2.v3	NOUN
ejpam-4653	517	11	)	)	PUNCT
ejpam-4653	517	12	∈	∈	PROPN
ejpam-4653	517	13	drd(g	drd(g	PROPN
ejpam-4653	517	14	)	)	PUNCT
ejpam-4653	517	15	}	}	PUNCT
ejpam-4653	517	16	.	.	PUNCT
ejpam-4653	518	1	(	(	PUNCT
ejpam-4653	518	2	ii	ii	NOUN
ejpam-4653	518	3	)	)	PUNCT
ejpam-4653	518	4	γdr(g	γdr(g	PROPN
ejpam-4653	518	5	◦	◦	NOUN
ejpam-4653	518	6	h	h	NOUN
ejpam-4653	518	7	)	)	PUNCT
ejpam-4653	518	8	=	=	NOUN
ejpam-4653	518	9	3n	3n	NOUN
ejpam-4653	518	10	for	for	ADP
ejpam-4653	518	11	all	all	DET
ejpam-4653	518	12	nontrivial	nontrivial	ADJ
ejpam-4653	518	13	graphs	graph	NOUN
ejpam-4653	518	14	h.	h.	NOUN
ejpam-4653	518	15	proof	proof	NOUN
ejpam-4653	518	16	.	.	PUNCT
ejpam-4653	519	1	for	for	ADP
ejpam-4653	519	2	(	(	PUNCT
ejpam-4653	519	3	i	i	NOUN
ejpam-4653	519	4	):	):	PUNCT
ejpam-4653	519	5	let	let	VERB
ejpam-4653	519	6	α	α	NOUN
ejpam-4653	519	7	=	=	NOUN
ejpam-4653	519	8	3n	3n	NUM
ejpam-4653	519	9	−	−	PROPN
ejpam-4653	519	10	max{|v0|	max{|v0|	NOUN
ejpam-4653	519	11	:	:	PUNCT
ejpam-4653	519	12	f	f	X
ejpam-4653	519	13	=	=	SYM
ejpam-4653	519	14	(	(	PUNCT
ejpam-4653	519	15	v0	v0	PROPN
ejpam-4653	519	16	,	,	PUNCT
ejpam-4653	519	17	v1	v1	NOUN
ejpam-4653	519	18	,	,	PUNCT
ejpam-4653	519	19	v2.v3	v2.v3	NOUN
ejpam-4653	519	20	)	)	PUNCT
ejpam-4653	519	21	∈	∈	PROPN
ejpam-4653	519	22	drd(g	drd(g	PROPN
ejpam-4653	519	23	)	)	PUNCT
ejpam-4653	519	24	}	}	PUNCT
ejpam-4653	519	25	and	and	CCONJ
ejpam-4653	519	26	put	put	VERB
ejpam-4653	519	27	v	v	NOUN
ejpam-4653	519	28	(	(	PUNCT
ejpam-4653	519	29	k1	k1	NOUN
ejpam-4653	519	30	)	)	PUNCT
ejpam-4653	519	31	=	=	SYM
ejpam-4653	519	32	{	{	PUNCT
ejpam-4653	519	33	u	u	NOUN
ejpam-4653	519	34	}	}	PUNCT
ejpam-4653	519	35	.	.	PUNCT
ejpam-4653	520	1	let	let	VERB
ejpam-4653	520	2	f	f	PROPN
ejpam-4653	520	3	=	=	SYM
ejpam-4653	520	4	(	(	PUNCT
ejpam-4653	520	5	v0	v0	PROPN
ejpam-4653	520	6	,	,	PUNCT
ejpam-4653	520	7	v1	v1	NOUN
ejpam-4653	520	8	,	,	PUNCT
ejpam-4653	520	9	v2	v2	PROPN
ejpam-4653	520	10	,	,	PUNCT
ejpam-4653	520	11	v3	v3	PROPN
ejpam-4653	520	12	)	)	PUNCT
ejpam-4653	520	13	∈	∈	PROPN
ejpam-4653	520	14	drd(g	drd(g	PROPN
ejpam-4653	520	15	)	)	PUNCT
ejpam-4653	520	16	for	for	ADP
ejpam-4653	520	17	which	which	PRON
ejpam-4653	520	18	|v0|	|v0|	NOUN
ejpam-4653	520	19	is	be	AUX
ejpam-4653	520	20	maximum	maximum	ADJ
ejpam-4653	520	21	.	.	PUNCT
ejpam-4653	521	1	define	define	VERB
ejpam-4653	521	2	v	v	ADP
ejpam-4653	521	3	∗	∗	NOUN
ejpam-4653	521	4	0	0	NUM
ejpam-4653	522	1	=	=	SYM
ejpam-4653	522	2	v0	v0	NOUN
ejpam-4653	522	3	∪	∪	NOUN
ejpam-4653	522	4	{	{	PUNCT
ejpam-4653	522	5	uv	uv	NOUN
ejpam-4653	522	6	:	:	PUNCT
ejpam-4653	522	7	v	v	NUM
ejpam-4653	522	8	∈	∈	PROPN
ejpam-4653	522	9	v3	v3	PROPN
ejpam-4653	522	10	}	}	PUNCT
ejpam-4653	522	11	,	,	PUNCT
ejpam-4653	522	12	v	v	X
ejpam-4653	522	13	∗	∗	NOUN
ejpam-4653	522	14	1	1	NUM
ejpam-4653	522	15	=	=	SYM
ejpam-4653	522	16	v1	v1	NOUN
ejpam-4653	522	17	∪	∪	X
ejpam-4653	522	18	{	{	PUNCT
ejpam-4653	522	19	uv	uv	NOUN
ejpam-4653	522	20	:	:	PUNCT
ejpam-4653	522	21	v	v	NUM
ejpam-4653	522	22	∈	∈	PROPN
ejpam-4653	522	23	v2	v2	PROPN
ejpam-4653	522	24	}	}	PUNCT
ejpam-4653	522	25	,	,	PUNCT
ejpam-4653	522	26	v	v	NOUN
ejpam-4653	522	27	∗	∗	NOUN
ejpam-4653	522	28	2	2	NUM
ejpam-4653	522	29	=	=	SYM
ejpam-4653	522	30	v2	v2	NOUN
ejpam-4653	522	31	∪	∪	X
ejpam-4653	522	32	{	{	PUNCT
ejpam-4653	522	33	uv	uv	NOUN
ejpam-4653	522	34	:	:	PUNCT
ejpam-4653	522	35	v	v	NUM
ejpam-4653	522	36	∈	∈	PROPN
ejpam-4653	522	37	v0	v0	NOUN
ejpam-4653	522	38	∪	∪	X
ejpam-4653	522	39	v1	v1	PROPN
ejpam-4653	522	40	}	}	PUNCT
ejpam-4653	522	41	and	and	CCONJ
ejpam-4653	522	42	v	v	ADP
ejpam-4653	522	43	∗	∗	NOUN
ejpam-4653	522	44	3	3	NUM
ejpam-4653	522	45	=	=	SYM
ejpam-4653	522	46	v3	v3	PROPN
ejpam-4653	522	47	.	.	PUNCT
ejpam-4653	523	1	by	by	ADP
ejpam-4653	523	2	proposition	proposition	NOUN
ejpam-4653	523	3	15	15	NUM
ejpam-4653	523	4	,	,	PUNCT
ejpam-4653	523	5	g	g	NOUN
ejpam-4653	523	6	=	=	PUNCT
ejpam-4653	523	7	(	(	PUNCT
ejpam-4653	523	8	v	v	NOUN
ejpam-4653	523	9	∗	∗	NOUN
ejpam-4653	523	10	0	0	NUM
ejpam-4653	523	11	,	,	PUNCT
ejpam-4653	523	12	v	v	NOUN
ejpam-4653	523	13	∗	∗	NOUN
ejpam-4653	523	14	1	1	NUM
ejpam-4653	523	15	,	,	PUNCT
ejpam-4653	523	16	v	v	NOUN
ejpam-4653	523	17	∗	∗	NOUN
ejpam-4653	523	18	2	2	NUM
ejpam-4653	523	19	,	,	PUNCT
ejpam-4653	523	20	v	v	NOUN
ejpam-4653	523	21	∗	∗	X
ejpam-4653	523	22	3	3	NUM
ejpam-4653	523	23	)	)	PUNCT
ejpam-4653	523	24	∈	∈	PROPN
ejpam-4653	523	25	drd(g	drd(g	VERB
ejpam-4653	523	26	◦	◦	NOUN
ejpam-4653	523	27	k1	k1	NOUN
ejpam-4653	523	28	)	)	PUNCT
ejpam-4653	523	29	.	.	PUNCT
ejpam-4653	524	1	thus	thus	ADV
ejpam-4653	524	2	,	,	PUNCT
ejpam-4653	524	3	γdr(g	γdr(g	PROPN
ejpam-4653	524	4	◦	◦	NOUN
ejpam-4653	524	5	k1	k1	NOUN
ejpam-4653	524	6	)	)	PUNCT
ejpam-4653	524	7	≤	≤	NOUN
ejpam-4653	524	8	3(n−	3(n−	NUM
ejpam-4653	524	9	|v0|	|v0|	NOUN
ejpam-4653	524	10	)	)	PUNCT
ejpam-4653	525	1	+	+	NUM
ejpam-4653	525	2	2|v0|	2|v0|	NUM
ejpam-4653	525	3	=	=	SYM
ejpam-4653	525	4	3n−	3n−	PROPN
ejpam-4653	525	5	|v0|	|v0|	NOUN
ejpam-4653	525	6	=	=	SYM
ejpam-4653	525	7	α	α	NOUN
ejpam-4653	525	8	.	.	PUNCT
ejpam-4653	525	9	to	to	PART
ejpam-4653	525	10	get	get	VERB
ejpam-4653	525	11	the	the	DET
ejpam-4653	525	12	other	other	ADJ
ejpam-4653	525	13	inequality	inequality	NOUN
ejpam-4653	525	14	,	,	PUNCT
ejpam-4653	525	15	let	let	VERB
ejpam-4653	525	16	f	f	PROPN
ejpam-4653	525	17	=	=	SYM
ejpam-4653	525	18	(	(	PUNCT
ejpam-4653	525	19	v0,∅	v0,∅	PROPN
ejpam-4653	525	20	,	,	PUNCT
ejpam-4653	525	21	v2	v2	PROPN
ejpam-4653	525	22	,	,	PUNCT
ejpam-4653	525	23	v3	v3	PROPN
ejpam-4653	525	24	)	)	PUNCT
ejpam-4653	525	25	be	be	VERB
ejpam-4653	525	26	a	a	DET
ejpam-4653	525	27	γdr	γdr	NOUN
ejpam-4653	525	28	-	-	PUNCT
ejpam-4653	525	29	function	function	NOUN
ejpam-4653	525	30	of	of	ADP
ejpam-4653	525	31	g	g	NOUN
ejpam-4653	525	32	◦	◦	NOUN
ejpam-4653	525	33	k1	k1	NOUN
ejpam-4653	525	34	.	.	PUNCT
ejpam-4653	526	1	first	first	ADV
ejpam-4653	526	2	,	,	PUNCT
ejpam-4653	526	3	we	we	PRON
ejpam-4653	526	4	claim	claim	VERB
ejpam-4653	526	5	that	that	SCONJ
ejpam-4653	526	6	v2∩v	v2∩v	NOUN
ejpam-4653	526	7	(	(	PUNCT
ejpam-4653	526	8	g	g	NOUN
ejpam-4653	526	9	)	)	PUNCT
ejpam-4653	526	10	=	=	VERB
ejpam-4653	526	11	∅.	∅.	NOUN
ejpam-4653	526	12	suppose	suppose	VERB
ejpam-4653	526	13	not	not	PART
ejpam-4653	526	14	,	,	PUNCT
ejpam-4653	526	15	and	and	CCONJ
ejpam-4653	526	16	let	let	VERB
ejpam-4653	526	17	w	w	PROPN
ejpam-4653	526	18	∈	∈	NOUN
ejpam-4653	526	19	v2∩v	v2∩v	NOUN
ejpam-4653	526	20	(	(	PUNCT
ejpam-4653	526	21	g	g	NOUN
ejpam-4653	526	22	)	)	PUNCT
ejpam-4653	526	23	.	.	PUNCT
ejpam-4653	527	1	since	since	SCONJ
ejpam-4653	527	2	f	f	PROPN
ejpam-4653	527	3	is	be	AUX
ejpam-4653	527	4	a	a	DET
ejpam-4653	527	5	γdr	γdr	NOUN
ejpam-4653	527	6	-	-	PUNCT
ejpam-4653	527	7	function	function	NOUN
ejpam-4653	527	8	,	,	PUNCT
ejpam-4653	527	9	uw	uw	PROPN
ejpam-4653	527	10	∈	∈	PROPN
ejpam-4653	527	11	v1	v1	NOUN
ejpam-4653	527	12	,	,	PUNCT
ejpam-4653	527	13	a	a	DET
ejpam-4653	527	14	contradiction	contradiction	NOUN
ejpam-4653	527	15	to	to	ADP
ejpam-4653	527	16	the	the	DET
ejpam-4653	527	17	choice	choice	NOUN
ejpam-4653	527	18	of	of	ADP
ejpam-4653	527	19	f	f	PROPN
ejpam-4653	527	20	.	.	PUNCT
ejpam-4653	528	1	next	next	ADV
ejpam-4653	528	2	,	,	PUNCT
ejpam-4653	528	3	we	we	PRON
ejpam-4653	528	4	claim	claim	VERB
ejpam-4653	528	5	that	that	SCONJ
ejpam-4653	528	6	f	f	PROPN
ejpam-4653	528	7	|g	|g	PROPN
ejpam-4653	528	8	∈	∈	PROPN
ejpam-4653	528	9	drd(g	drd(g	PROPN
ejpam-4653	528	10	)	)	PUNCT
ejpam-4653	528	11	.	.	PUNCT
ejpam-4653	529	1	let	let	VERB
ejpam-4653	529	2	v	v	NUM
ejpam-4653	529	3	∈	∈	PROPN
ejpam-4653	529	4	v0	v0	NOUN
ejpam-4653	529	5	∩	∩	X
ejpam-4653	529	6	v	v	X
ejpam-4653	529	7	(	(	PUNCT
ejpam-4653	529	8	g	g	NOUN
ejpam-4653	529	9	)	)	PUNCT
ejpam-4653	529	10	.	.	PUNCT
ejpam-4653	530	1	if	if	SCONJ
ejpam-4653	530	2	uv	uv	PROPN
ejpam-4653	530	3	∈	∈	PROPN
ejpam-4653	530	4	v3	v3	PROPN
ejpam-4653	530	5	,	,	PUNCT
ejpam-4653	530	6	then	then	ADV
ejpam-4653	530	7	g	g	PROPN
ejpam-4653	530	8	=	=	SYM
ejpam-4653	530	9	(	(	PUNCT
ejpam-4653	530	10	v0	v0	PROPN
ejpam-4653	530	11	\	\	PUNCT
ejpam-4653	530	12	{	{	PUNCT
ejpam-4653	530	13	v	v	NOUN
ejpam-4653	530	14	}	}	PUNCT
ejpam-4653	530	15	,	,	PUNCT
ejpam-4653	530	16	{	{	PUNCT
ejpam-4653	530	17	v	v	NOUN
ejpam-4653	530	18	,	,	PUNCT
ejpam-4653	530	19	uv	uv	NOUN
ejpam-4653	530	20	}	}	PUNCT
ejpam-4653	530	21	,	,	PUNCT
ejpam-4653	530	22	v2	v2	PROPN
ejpam-4653	530	23	,	,	PUNCT
ejpam-4653	530	24	v3	v3	PROPN
ejpam-4653	530	25	\	\	PROPN
ejpam-4653	530	26	{	{	PUNCT
ejpam-4653	530	27	uv	uv	NOUN
ejpam-4653	530	28	}	}	PUNCT
ejpam-4653	530	29	)	)	PUNCT
ejpam-4653	530	30	∈	∈	PROPN
ejpam-4653	530	31	drd(g	drd(g	VERB
ejpam-4653	530	32	◦	◦	NOUN
ejpam-4653	530	33	k1	k1	NOUN
ejpam-4653	530	34	)	)	PUNCT
ejpam-4653	530	35	with	with	ADP
ejpam-4653	530	36	ωg	ωg	NOUN
ejpam-4653	530	37	◦	◦	NOUN
ejpam-4653	530	38	k1(g	k1(g	PROPN
ejpam-4653	530	39	)	)	PUNCT
ejpam-4653	530	40	=	=	SYM
ejpam-4653	530	41	ωg	ωg	PROPN
ejpam-4653	530	42	◦	◦	NOUN
ejpam-4653	530	43	k1(f	k1(f	NOUN
ejpam-4653	530	44	)	)	PUNCT
ejpam-4653	530	45	−	−	PROPN
ejpam-4653	530	46	1	1	NUM
ejpam-4653	530	47	,	,	PUNCT
ejpam-4653	530	48	a	a	DET
ejpam-4653	530	49	contradiction	contradiction	NOUN
ejpam-4653	530	50	.	.	PUNCT
ejpam-4653	531	1	thus	thus	ADV
ejpam-4653	531	2	,	,	PUNCT
ejpam-4653	531	3	uv	uv	PROPN
ejpam-4653	531	4	∈	∈	PROPN
ejpam-4653	531	5	v2	v2	PROPN
ejpam-4653	531	6	.	.	PUNCT
ejpam-4653	532	1	since	since	SCONJ
ejpam-4653	532	2	v2	v2	PROPN
ejpam-4653	532	3	∩	∩	NOUN
ejpam-4653	532	4	v	v	NOUN
ejpam-4653	532	5	(	(	PUNCT
ejpam-4653	532	6	g	g	NOUN
ejpam-4653	532	7	)	)	PUNCT
ejpam-4653	532	8	=	=	NOUN
ejpam-4653	532	9	∅	∅	NOUN
ejpam-4653	532	10	,	,	PUNCT
ejpam-4653	532	11	there	there	PRON
ejpam-4653	532	12	exists	exist	VERB
ejpam-4653	532	13	w	w	PROPN
ejpam-4653	532	14	∈	∈	PROPN
ejpam-4653	532	15	v3∩v	v3∩v	X
ejpam-4653	532	16	(	(	PUNCT
ejpam-4653	532	17	g	g	NOUN
ejpam-4653	532	18	)	)	PUNCT
ejpam-4653	532	19	for	for	ADP
ejpam-4653	532	20	which	which	PRON
ejpam-4653	532	21	vw	vw	PROPN
ejpam-4653	532	22	∈	∈	PROPN
ejpam-4653	532	23	e(g	e(g	PROPN
ejpam-4653	532	24	)	)	PUNCT
ejpam-4653	532	25	.	.	PUNCT
ejpam-4653	533	1	since	since	SCONJ
ejpam-4653	533	2	v1∩v	v1∩v	NOUN
ejpam-4653	533	3	(	(	PUNCT
ejpam-4653	533	4	g	g	NOUN
ejpam-4653	533	5	)	)	PUNCT
ejpam-4653	533	6	=	=	NOUN
ejpam-4653	533	7	∅	∅	NOUN
ejpam-4653	533	8	,	,	PUNCT
ejpam-4653	533	9	f	f	PROPN
ejpam-4653	533	10	|g	|g	X
ejpam-4653	533	11	=	=	SYM
ejpam-4653	533	12	(	(	PUNCT
ejpam-4653	533	13	v	v	NOUN
ejpam-4653	533	14	∗	∗	NOUN
ejpam-4653	533	15	0	0	NUM
ejpam-4653	533	16	,	,	PUNCT
ejpam-4653	533	17	v	v	NOUN
ejpam-4653	533	18	∗	∗	NOUN
ejpam-4653	533	19	1	1	NUM
ejpam-4653	533	20	,	,	PUNCT
ejpam-4653	533	21	v	v	NOUN
ejpam-4653	533	22	∗	∗	NOUN
ejpam-4653	533	23	2	2	NUM
ejpam-4653	533	24	,	,	PUNCT
ejpam-4653	533	25	v	v	NOUN
ejpam-4653	533	26	∗	∗	X
ejpam-4653	533	27	3	3	NUM
ejpam-4653	533	28	)	)	PUNCT
ejpam-4653	533	29	∈	∈	PROPN
ejpam-4653	533	30	drd(g	drd(g	PROPN
ejpam-4653	533	31	)	)	PUNCT
ejpam-4653	533	32	with	with	ADP
ejpam-4653	533	33	v	v	NOUN
ejpam-4653	533	34	∗	∗	NOUN
ejpam-4653	533	35	0	0	NUM
ejpam-4653	533	36	=	=	SYM
ejpam-4653	533	37	v0	v0	NOUN
ejpam-4653	533	38	∩	∩	X
ejpam-4653	533	39	v	v	X
ejpam-4653	533	40	(	(	PUNCT
ejpam-4653	533	41	g	g	NOUN
ejpam-4653	533	42	)	)	PUNCT
ejpam-4653	533	43	,	,	PUNCT
ejpam-4653	533	44	v	v	X
ejpam-4653	533	45	∗	∗	NOUN
ejpam-4653	533	46	1	1	NUM
ejpam-4653	533	47	=	=	SYM
ejpam-4653	533	48	v	v	NOUN
ejpam-4653	533	49	∗	∗	NOUN
ejpam-4653	533	50	2	2	NUM
ejpam-4653	533	51	=	=	NOUN
ejpam-4653	533	52	∅	∅	NOUN
ejpam-4653	533	53	and	and	CCONJ
ejpam-4653	533	54	v	v	NOUN
ejpam-4653	533	55	∗	∗	NOUN
ejpam-4653	533	56	3	3	NUM
ejpam-4653	533	57	=	=	SYM
ejpam-4653	533	58	v3	v3	PROPN
ejpam-4653	533	59	.	.	PUNCT
ejpam-4653	534	1	observe	observe	VERB
ejpam-4653	534	2	also	also	ADV
ejpam-4653	534	3	that	that	SCONJ
ejpam-4653	534	4	for	for	ADP
ejpam-4653	534	5	each	each	DET
ejpam-4653	534	6	j.	j.	PROPN
ejpam-4653	534	7	b.g	b.g	PROPN
ejpam-4653	534	8	.	.	PROPN
ejpam-4653	534	9	cariaga	cariaga	PROPN
ejpam-4653	534	10	,	,	PUNCT
ejpam-4653	534	11	f.	f.	PROPN
ejpam-4653	534	12	jamil	jamil	PROPN
ejpam-4653	534	13	/	/	SYM
ejpam-4653	534	14	eur	eur	PROPN
ejpam-4653	534	15	.	.	PUNCT
ejpam-4653	535	1	j.	j.	PROPN
ejpam-4653	535	2	pure	pure	PROPN
ejpam-4653	535	3	appl	appl	PROPN
ejpam-4653	535	4	.	.	PROPN
ejpam-4653	535	5	math	math	PROPN
ejpam-4653	535	6	,	,	PUNCT
ejpam-4653	535	7	16	16	NUM
ejpam-4653	535	8	(	(	PUNCT
ejpam-4653	535	9	2	2	NUM
ejpam-4653	535	10	)	)	PUNCT
ejpam-4653	535	11	(	(	PUNCT
ejpam-4653	535	12	2023	2023	NUM
ejpam-4653	535	13	)	)	PUNCT
ejpam-4653	535	14	,	,	PUNCT
ejpam-4653	535	15	847	847	NUM
ejpam-4653	535	16	-	-	SYM
ejpam-4653	535	17	863	863	NUM
ejpam-4653	535	18	859	859	NUM
ejpam-4653	535	19	v	v	NOUN
ejpam-4653	535	20	∈	∈	PROPN
ejpam-4653	535	21	v	v	NOUN
ejpam-4653	535	22	(	(	PUNCT
ejpam-4653	535	23	g	g	NOUN
ejpam-4653	535	24	)	)	PUNCT
ejpam-4653	535	25	,	,	PUNCT
ejpam-4653	535	26	either	either	CCONJ
ejpam-4653	535	27	uv	uv	PROPN
ejpam-4653	535	28	∈	∈	PROPN
ejpam-4653	535	29	v0	v0	NOUN
ejpam-4653	535	30	or	or	CCONJ
ejpam-4653	535	31	uv	uv	NOUN
ejpam-4653	535	32	∈	∈	PROPN
ejpam-4653	535	33	v2	v2	NOUN
ejpam-4653	535	34	.	.	PUNCT
ejpam-4653	536	1	more	more	ADV
ejpam-4653	536	2	precisely	precisely	ADV
ejpam-4653	536	3	,	,	PUNCT
ejpam-4653	536	4	uv	uv	PROPN
ejpam-4653	536	5	∈	∈	PROPN
ejpam-4653	536	6	v0	v0	NOUN
ejpam-4653	536	7	if	if	SCONJ
ejpam-4653	536	8	and	and	CCONJ
ejpam-4653	536	9	only	only	ADV
ejpam-4653	536	10	if	if	SCONJ
ejpam-4653	536	11	v	v	PROPN
ejpam-4653	536	12	∈	∈	PROPN
ejpam-4653	536	13	v3	v3	PROPN
ejpam-4653	536	14	and	and	CCONJ
ejpam-4653	536	15	uv	uv	NOUN
ejpam-4653	536	16	∈	∈	PROPN
ejpam-4653	536	17	v2	v2	NOUN
ejpam-4653	536	18	if	if	SCONJ
ejpam-4653	536	19	and	and	CCONJ
ejpam-4653	536	20	only	only	ADV
ejpam-4653	536	21	if	if	SCONJ
ejpam-4653	536	22	v	v	PROPN
ejpam-4653	536	23	∈	∈	PROPN
ejpam-4653	536	24	v0	v0	NOUN
ejpam-4653	536	25	.	.	PUNCT
ejpam-4653	537	1	thus	thus	ADV
ejpam-4653	537	2	γdr(g	γdr(g	PROPN
ejpam-4653	537	3	◦	◦	NOUN
ejpam-4653	537	4	k1	k1	NOUN
ejpam-4653	537	5	)	)	PUNCT
ejpam-4653	538	1	=	=	SYM
ejpam-4653	538	2	ωg	ωg	PROPN
ejpam-4653	538	3	◦	◦	NOUN
ejpam-4653	538	4	k1(f	k1(f	NOUN
ejpam-4653	538	5	)	)	PUNCT
ejpam-4653	538	6	=	=	SYM
ejpam-4653	539	1	3|v3|+	3|v3|+	NUM
ejpam-4653	539	2	2|v0	2|v0	NUM
ejpam-4653	539	3	∩	∩	ADJ
ejpam-4653	539	4	v	v	X
ejpam-4653	539	5	(	(	PUNCT
ejpam-4653	539	6	g)|	g)|	NOUN
ejpam-4653	539	7	=	=	SYM
ejpam-4653	539	8	3|v3|+	3|v3|+	PROPN
ejpam-4653	539	9	3|v0	3|v0	NUM
ejpam-4653	539	10	∩	∩	PROPN
ejpam-4653	539	11	v	v	X
ejpam-4653	539	12	(	(	PUNCT
ejpam-4653	539	13	g)|	g)|	PROPN
ejpam-4653	539	14	−	−	PROPN
ejpam-4653	539	15	|v0	|v0	PROPN
ejpam-4653	539	16	∩	∩	PROPN
ejpam-4653	539	17	v	v	X
ejpam-4653	539	18	(	(	PUNCT
ejpam-4653	539	19	g)|	g)|	NOUN
ejpam-4653	539	20	=	=	SYM
ejpam-4653	539	21	3n−	3n−	PROPN
ejpam-4653	539	22	|v	|v	NOUN
ejpam-4653	539	23	∗	∗	NOUN
ejpam-4653	539	24	0	0	NUM
ejpam-4653	540	1	|	|	CCONJ
ejpam-4653	540	2	≥	≥	NOUN
ejpam-4653	540	3	α	α	X
ejpam-4653	540	4	.	.	PUNCT
ejpam-4653	541	1	for	for	ADP
ejpam-4653	541	2	(	(	PUNCT
ejpam-4653	541	3	ii	ii	NOUN
ejpam-4653	541	4	):	):	PUNCT
ejpam-4653	541	5	by	by	ADP
ejpam-4653	541	6	proposition	proposition	NOUN
ejpam-4653	541	7	15	15	NUM
ejpam-4653	541	8	,	,	PUNCT
ejpam-4653	541	9	f	f	X
ejpam-4653	541	10	=	=	PUNCT
ejpam-4653	541	11	(	(	PUNCT
ejpam-4653	541	12	∪v∈v	∪v∈v	X
ejpam-4653	541	13	(	(	PUNCT
ejpam-4653	541	14	g)v	g)v	X
ejpam-4653	541	15	(	(	PUNCT
ejpam-4653	541	16	hv),∅,∅	hv),∅,∅	NOUN
ejpam-4653	541	17	,	,	PUNCT
ejpam-4653	541	18	v	v	NOUN
ejpam-4653	541	19	(	(	PUNCT
ejpam-4653	541	20	g	g	NOUN
ejpam-4653	541	21	)	)	PUNCT
ejpam-4653	541	22	)	)	PUNCT
ejpam-4653	541	23	∈	∈	PROPN
ejpam-4653	541	24	drd(g	drd(g	VERB
ejpam-4653	541	25	◦	◦	NOUN
ejpam-4653	541	26	h	h	NOUN
ejpam-4653	541	27	)	)	PUNCT
ejpam-4653	541	28	.	.	PUNCT
ejpam-4653	542	1	thus	thus	ADV
ejpam-4653	542	2	,	,	PUNCT
ejpam-4653	542	3	γdr(g	γdr(g	PROPN
ejpam-4653	542	4	◦	◦	NOUN
ejpam-4653	542	5	h	h	NOUN
ejpam-4653	542	6	)	)	PUNCT
ejpam-4653	542	7	≤	≤	NOUN
ejpam-4653	542	8	3|v	3|v	NUM
ejpam-4653	542	9	(	(	PUNCT
ejpam-4653	542	10	g)|	g)|	NOUN
ejpam-4653	542	11	=	=	ADJ
ejpam-4653	542	12	3n	3n	NUM
ejpam-4653	542	13	.	.	PUNCT
ejpam-4653	543	1	on	on	ADP
ejpam-4653	543	2	the	the	DET
ejpam-4653	543	3	other	other	ADJ
ejpam-4653	543	4	hand	hand	NOUN
ejpam-4653	543	5	,	,	PUNCT
ejpam-4653	543	6	if	if	SCONJ
ejpam-4653	543	7	f	f	PROPN
ejpam-4653	543	8	=	=	SYM
ejpam-4653	543	9	(	(	PUNCT
ejpam-4653	543	10	v0	v0	PROPN
ejpam-4653	543	11	,	,	PUNCT
ejpam-4653	543	12	v1	v1	NOUN
ejpam-4653	543	13	,	,	PUNCT
ejpam-4653	543	14	v2	v2	PROPN
ejpam-4653	543	15	,	,	PUNCT
ejpam-4653	543	16	v3	v3	PROPN
ejpam-4653	543	17	)	)	PUNCT
ejpam-4653	543	18	∈	∈	PROPN
ejpam-4653	543	19	drd(g	drd(g	VERB
ejpam-4653	543	20	◦	◦	NOUN
ejpam-4653	543	21	h	h	NOUN
ejpam-4653	543	22	)	)	PUNCT
ejpam-4653	543	23	,	,	PUNCT
ejpam-4653	543	24	then	then	ADV
ejpam-4653	543	25	ωhv+v(f	ωhv+v(f	PRON
ejpam-4653	543	26	|hv+v	|hv+v	NOUN
ejpam-4653	543	27	)	)	PUNCT
ejpam-4653	543	28	≥	≥	NOUN
ejpam-4653	543	29	3	3	NUM
ejpam-4653	543	30	for	for	ADP
ejpam-4653	543	31	each	each	DET
ejpam-4653	543	32	v	v	NUM
ejpam-4653	543	33	∈	∈	PROPN
ejpam-4653	543	34	v	v	NOUN
ejpam-4653	543	35	(	(	PUNCT
ejpam-4653	543	36	g	g	NOUN
ejpam-4653	543	37	)	)	PUNCT
ejpam-4653	543	38	.	.	PUNCT
ejpam-4653	544	1	thus	thus	ADV
ejpam-4653	544	2	,	,	PUNCT
ejpam-4653	544	3	γdr(g	γdr(g	PROPN
ejpam-4653	544	4	◦	◦	NOUN
ejpam-4653	544	5	h	h	NOUN
ejpam-4653	544	6	)	)	PUNCT
ejpam-4653	544	7	=	=	SYM
ejpam-4653	544	8	ωg	ωg	PART
ejpam-4653	544	9	◦	◦	NOUN
ejpam-4653	544	10	h(f	h(f	NOUN
ejpam-4653	544	11	)	)	PUNCT
ejpam-4653	545	1	=	=	PUNCT
ejpam-4653	545	2	∑	∑	PUNCT
ejpam-4653	545	3	v∈v	v∈v	NOUN
ejpam-4653	545	4	(	(	PUNCT
ejpam-4653	545	5	g	g	NOUN
ejpam-4653	545	6	)	)	PUNCT
ejpam-4653	545	7	ωhv+v(f	ωhv+v(f	NUM
ejpam-4653	545	8	|hv+v	|hv+v	NOUN
ejpam-4653	545	9	)	)	PUNCT
ejpam-4653	545	10	≥	≥	NOUN
ejpam-4653	545	11	3n	3n	NUM
ejpam-4653	545	12	.	.	PUNCT
ejpam-4653	546	1	the	the	DET
ejpam-4653	546	2	succeeding	succeed	VERB
ejpam-4653	546	3	corollary	corollary	NOUN
ejpam-4653	546	4	,	,	PUNCT
ejpam-4653	546	5	which	which	PRON
ejpam-4653	546	6	are	be	AUX
ejpam-4653	546	7	found	find	VERB
ejpam-4653	546	8	in	in	ADP
ejpam-4653	546	9	[	[	X
ejpam-4653	546	10	19	19	NUM
ejpam-4653	546	11	]	]	PUNCT
ejpam-4653	546	12	,	,	PUNCT
ejpam-4653	546	13	are	be	AUX
ejpam-4653	546	14	immediate	immediate	ADJ
ejpam-4653	546	15	consequences	consequence	NOUN
ejpam-4653	546	16	of	of	ADP
ejpam-4653	546	17	corollary	corollary	ADJ
ejpam-4653	546	18	3(i	3(i	NUM
ejpam-4653	546	19	)	)	PUNCT
ejpam-4653	546	20	.	.	PUNCT
ejpam-4653	547	1	corollary	corollary	ADJ
ejpam-4653	547	2	4	4	NUM
ejpam-4653	547	3	.	.	PUNCT
ejpam-4653	548	1	[	[	X
ejpam-4653	548	2	19	19	NUM
ejpam-4653	548	3	]	]	PUNCT
ejpam-4653	548	4	(	(	PUNCT
ejpam-4653	548	5	i	i	NOUN
ejpam-4653	548	6	)	)	PUNCT
ejpam-4653	548	7	γdr(pn	γdr(pn	PROPN
ejpam-4653	548	8	◦	◦	NOUN
ejpam-4653	548	9	k1	k1	NOUN
ejpam-4653	548	10	)	)	PUNCT
ejpam-4653	548	11	=	=	PUNCT
ejpam-4653	549	1			PRON
ejpam-4653	549	2	7n	7n	NOUN
ejpam-4653	549	3	3	3	NUM
ejpam-4653	549	4	,	,	PUNCT
ejpam-4653	549	5	if	if	SCONJ
ejpam-4653	549	6	n	n	NOUN
ejpam-4653	549	7	=	=	SYM
ejpam-4653	549	8	3k	3k	NUM
ejpam-4653	549	9	,	,	PUNCT
ejpam-4653	549	10	7n+2	7n+2	NOUN
ejpam-4653	549	11	3	3	NUM
ejpam-4653	549	12	,	,	PUNCT
ejpam-4653	549	13	if	if	SCONJ
ejpam-4653	549	14	n	n	ADV
ejpam-4653	549	15	=	=	SYM
ejpam-4653	549	16	3k	3k	X
ejpam-4653	550	1	+	+	CCONJ
ejpam-4653	550	2	1	1	NUM
ejpam-4653	550	3	,	,	PUNCT
ejpam-4653	550	4	7n+1	7n+1	NOUN
ejpam-4653	550	5	3	3	NUM
ejpam-4653	550	6	,	,	PUNCT
ejpam-4653	550	7	if	if	SCONJ
ejpam-4653	550	8	n	n	ADV
ejpam-4653	550	9	=	=	SYM
ejpam-4653	550	10	3k	3k	X
ejpam-4653	551	1	+	+	CCONJ
ejpam-4653	551	2	2	2	X
ejpam-4653	551	3	.	.	PUNCT
ejpam-4653	551	4	(	(	PUNCT
ejpam-4653	551	5	ii	ii	NOUN
ejpam-4653	551	6	)	)	PUNCT
ejpam-4653	551	7	γdr(cn	γdr(cn	ADP
ejpam-4653	551	8	◦	◦	NOUN
ejpam-4653	551	9	k1	k1	NOUN
ejpam-4653	551	10	)	)	PUNCT
ejpam-4653	551	11	=	=	PUNCT
ejpam-4653	552	1			PRON
ejpam-4653	552	2	7n	7n	NOUN
ejpam-4653	552	3	3	3	NUM
ejpam-4653	552	4	,	,	PUNCT
ejpam-4653	552	5	if	if	SCONJ
ejpam-4653	552	6	n	n	NOUN
ejpam-4653	552	7	=	=	SYM
ejpam-4653	552	8	3k	3k	NUM
ejpam-4653	552	9	,	,	PUNCT
ejpam-4653	552	10	7n+2	7n+2	NOUN
ejpam-4653	552	11	3	3	NUM
ejpam-4653	552	12	,	,	PUNCT
ejpam-4653	552	13	if	if	SCONJ
ejpam-4653	552	14	n	n	ADV
ejpam-4653	552	15	=	=	SYM
ejpam-4653	552	16	3k	3k	X
ejpam-4653	553	1	+	+	CCONJ
ejpam-4653	553	2	1	1	NUM
ejpam-4653	553	3	,	,	PUNCT
ejpam-4653	553	4	7n+1	7n+1	NOUN
ejpam-4653	553	5	3	3	NUM
ejpam-4653	553	6	,	,	PUNCT
ejpam-4653	553	7	if	if	SCONJ
ejpam-4653	553	8	n	n	ADV
ejpam-4653	553	9	=	=	SYM
ejpam-4653	553	10	3k	3k	X
ejpam-4653	554	1	+	+	CCONJ
ejpam-4653	554	2	2	2	X
ejpam-4653	554	3	.	.	X
ejpam-4653	554	4	(	(	PUNCT
ejpam-4653	554	5	iii	iii	NOUN
ejpam-4653	554	6	)	)	PUNCT
ejpam-4653	554	7	γdr(kn	γdr(kn	NOUN
ejpam-4653	554	8	◦	◦	NOUN
ejpam-4653	554	9	k1	k1	NOUN
ejpam-4653	554	10	)	)	PUNCT
ejpam-4653	554	11	=	=	SYM
ejpam-4653	555	1	2n+	2n+	NUM
ejpam-4653	555	2	1	1	NUM
ejpam-4653	555	3	.	.	PUNCT
ejpam-4653	555	4	(	(	PUNCT
ejpam-4653	555	5	iv	iv	X
ejpam-4653	555	6	)	)	PUNCT
ejpam-4653	555	7	γdr(kp	γdr(kp	PROPN
ejpam-4653	555	8	,	,	PUNCT
ejpam-4653	555	9	q	q	NOUN
ejpam-4653	555	10	◦	◦	NOUN
ejpam-4653	555	11	k1	k1	NOUN
ejpam-4653	555	12	)	)	PUNCT
ejpam-4653	555	13	=	=	PRON
ejpam-4653	555	14	{	{	PUNCT
ejpam-4653	555	15	2(p+	2(p+	NUM
ejpam-4653	555	16	q	q	NOUN
ejpam-4653	555	17	)	)	PUNCT
ejpam-4653	556	1	+	+	NUM
ejpam-4653	556	2	1	1	NUM
ejpam-4653	556	3	,	,	PUNCT
ejpam-4653	556	4	if	if	SCONJ
ejpam-4653	556	5	p	p	NOUN
ejpam-4653	556	6	=	=	NOUN
ejpam-4653	556	7	1	1	NUM
ejpam-4653	556	8	or	or	CCONJ
ejpam-4653	556	9	q	q	ADJ
ejpam-4653	556	10	=	=	SYM
ejpam-4653	556	11	1	1	NUM
ejpam-4653	556	12	,	,	PUNCT
ejpam-4653	556	13	2(p+	2(p+	NUM
ejpam-4653	556	14	q	q	NOUN
ejpam-4653	557	1	+	+	NUM
ejpam-4653	557	2	1	1	NUM
ejpam-4653	557	3	)	)	PUNCT
ejpam-4653	557	4	,	,	PUNCT
ejpam-4653	557	5	otherwise	otherwise	ADV
ejpam-4653	557	6	.	.	PUNCT
ejpam-4653	558	1	the	the	DET
ejpam-4653	558	2	lexicographic	lexicographic	ADJ
ejpam-4653	558	3	product	product	NOUN
ejpam-4653	558	4	of	of	ADP
ejpam-4653	558	5	graphs	graph	NOUN
ejpam-4653	558	6	g	g	NOUN
ejpam-4653	558	7	and	and	CCONJ
ejpam-4653	558	8	h	h	NOUN
ejpam-4653	558	9	is	be	AUX
ejpam-4653	558	10	the	the	DET
ejpam-4653	558	11	graph	graph	NOUN
ejpam-4653	558	12	g[h	g[h	PROPN
ejpam-4653	558	13	]	]	PUNCT
ejpam-4653	558	14	with	with	ADP
ejpam-4653	558	15	v	v	NOUN
ejpam-4653	558	16	(	(	PUNCT
ejpam-4653	558	17	g[h])=	g[h])=	NOUN
ejpam-4653	558	18	v	v	NOUN
ejpam-4653	558	19	(	(	PUNCT
ejpam-4653	558	20	g	g	NOUN
ejpam-4653	558	21	)	)	PUNCT
ejpam-4653	558	22	×	×	NOUN
ejpam-4653	558	23	v	v	NOUN
ejpam-4653	558	24	(	(	PUNCT
ejpam-4653	558	25	h	h	NOUN
ejpam-4653	558	26	)	)	PUNCT
ejpam-4653	558	27	and	and	CCONJ
ejpam-4653	558	28	(	(	PUNCT
ejpam-4653	558	29	u1	u1	NOUN
ejpam-4653	558	30	,	,	PUNCT
ejpam-4653	558	31	u2)(v1	u2)(v1	NOUN
ejpam-4653	558	32	,	,	PUNCT
ejpam-4653	558	33	v2	v2	NOUN
ejpam-4653	558	34	)	)	PUNCT
ejpam-4653	558	35	∈	∈	NOUN
ejpam-4653	558	36	e(g[h	e(g[h	NOUN
ejpam-4653	558	37	]	]	PUNCT
ejpam-4653	558	38	)	)	PUNCT
ejpam-4653	558	39	if	if	SCONJ
ejpam-4653	558	40	and	and	CCONJ
ejpam-4653	558	41	only	only	ADV
ejpam-4653	558	42	if	if	SCONJ
ejpam-4653	558	43	either	either	CCONJ
ejpam-4653	558	44	u1v1	u1v1	PROPN
ejpam-4653	558	45	∈	∈	PROPN
ejpam-4653	558	46	e(g	e(g	PROPN
ejpam-4653	558	47	)	)	PUNCT
ejpam-4653	558	48	or	or	CCONJ
ejpam-4653	558	49	u1	u1	NOUN
ejpam-4653	558	50	=	=	SYM
ejpam-4653	558	51	v1	v1	NOUN
ejpam-4653	558	52	and	and	CCONJ
ejpam-4653	558	53	u2v2	u2v2	ADJ
ejpam-4653	558	54	∈	∈	PROPN
ejpam-4653	558	55	e(h	e(h	PROPN
ejpam-4653	558	56	)	)	PUNCT
ejpam-4653	558	57	.	.	PUNCT
ejpam-4653	559	1	j.	j.	PROPN
ejpam-4653	559	2	b.g	b.g	PROPN
ejpam-4653	559	3	.	.	PROPN
ejpam-4653	559	4	cariaga	cariaga	PROPN
ejpam-4653	559	5	,	,	PUNCT
ejpam-4653	559	6	f.	f.	PROPN
ejpam-4653	559	7	jamil	jamil	PROPN
ejpam-4653	559	8	/	/	SYM
ejpam-4653	559	9	eur	eur	PROPN
ejpam-4653	559	10	.	.	PUNCT
ejpam-4653	560	1	j.	j.	PROPN
ejpam-4653	560	2	pure	pure	PROPN
ejpam-4653	560	3	appl	appl	PROPN
ejpam-4653	560	4	.	.	PROPN
ejpam-4653	560	5	math	math	PROPN
ejpam-4653	560	6	,	,	PUNCT
ejpam-4653	560	7	16	16	NUM
ejpam-4653	560	8	(	(	PUNCT
ejpam-4653	560	9	2	2	NUM
ejpam-4653	560	10	)	)	PUNCT
ejpam-4653	560	11	(	(	PUNCT
ejpam-4653	560	12	2023	2023	NUM
ejpam-4653	560	13	)	)	PUNCT
ejpam-4653	560	14	,	,	PUNCT
ejpam-4653	560	15	847	847	NUM
ejpam-4653	560	16	-	-	SYM
ejpam-4653	560	17	863	863	NUM
ejpam-4653	560	18	860	860	NUM
ejpam-4653	560	19	for	for	ADP
ejpam-4653	560	20	s	s	PROPN
ejpam-4653	560	21	⊆	⊆	NUM
ejpam-4653	560	22	v	v	NOUN
ejpam-4653	560	23	(	(	PUNCT
ejpam-4653	560	24	g[h	g[h	PROPN
ejpam-4653	560	25	]	]	PUNCT
ejpam-4653	560	26	)	)	PUNCT
ejpam-4653	560	27	,	,	PUNCT
ejpam-4653	560	28	we	we	PRON
ejpam-4653	560	29	write	write	VERB
ejpam-4653	560	30	sg	sg	X
ejpam-4653	560	31	=	=	PUNCT
ejpam-4653	560	32	{	{	PUNCT
ejpam-4653	560	33	x	x	PUNCT
ejpam-4653	560	34	∈	∈	PROPN
ejpam-4653	560	35	v	v	NOUN
ejpam-4653	560	36	(	(	PUNCT
ejpam-4653	560	37	g	g	NOUN
ejpam-4653	560	38	)	)	PUNCT
ejpam-4653	560	39	:	:	PUNCT
ejpam-4653	560	40	(	(	PUNCT
ejpam-4653	560	41	x	x	X
ejpam-4653	560	42	,	,	PUNCT
ejpam-4653	560	43	y	y	PROPN
ejpam-4653	560	44	)	)	PUNCT
ejpam-4653	560	45	∈	∈	PROPN
ejpam-4653	560	46	s	s	PART
ejpam-4653	560	47	for	for	ADP
ejpam-4653	560	48	some	some	DET
ejpam-4653	560	49	y	y	PROPN
ejpam-4653	560	50	∈	∈	PROPN
ejpam-4653	560	51	v	v	ADP
ejpam-4653	560	52	(	(	PUNCT
ejpam-4653	560	53	h	h	NOUN
ejpam-4653	560	54	)	)	PUNCT
ejpam-4653	560	55	}	}	PUNCT
ejpam-4653	560	56	.	.	PUNCT
ejpam-4653	561	1	sg	sg	PROPN
ejpam-4653	561	2	is	be	AUX
ejpam-4653	561	3	referred	refer	VERB
ejpam-4653	561	4	to	to	ADP
ejpam-4653	561	5	as	as	ADP
ejpam-4653	561	6	the	the	DET
ejpam-4653	561	7	g	g	NOUN
ejpam-4653	561	8	-	-	PUNCT
ejpam-4653	561	9	projection	projection	NOUN
ejpam-4653	561	10	of	of	ADP
ejpam-4653	561	11	s	s	PRON
ejpam-4653	561	12	in	in	ADP
ejpam-4653	561	13	g[h	g[h	NOUN
ejpam-4653	561	14	]	]	PUNCT
ejpam-4653	561	15	.	.	PUNCT
ejpam-4653	562	1	for	for	ADP
ejpam-4653	562	2	a	a	DET
ejpam-4653	562	3	graph	graph	NOUN
ejpam-4653	562	4	g	g	NOUN
ejpam-4653	562	5	,	,	PUNCT
ejpam-4653	562	6	we	we	PRON
ejpam-4653	562	7	define	define	VERB
ejpam-4653	562	8	cg	cg	NOUN
ejpam-4653	562	9	=	=	PUNCT
ejpam-4653	562	10	{	{	PUNCT
ejpam-4653	562	11	f	f	NOUN
ejpam-4653	562	12	=	=	SYM
ejpam-4653	562	13	(	(	PUNCT
ejpam-4653	562	14	v0,∅	v0,∅	PROPN
ejpam-4653	562	15	,	,	PUNCT
ejpam-4653	562	16	v2	v2	PROPN
ejpam-4653	562	17	,	,	PUNCT
ejpam-4653	562	18	v3	v3	PROPN
ejpam-4653	562	19	)	)	PUNCT
ejpam-4653	562	20	∈	∈	PROPN
ejpam-4653	562	21	drd(g	drd(g	PROPN
ejpam-4653	562	22	)	)	PUNCT
ejpam-4653	562	23	:	:	PUNCT
ejpam-4653	563	1	v2	v2	VERB
ejpam-4653	563	2	\ng(v2	\ng(v2	NUM
ejpam-4653	563	3	∪	∪	PROPN
ejpam-4653	563	4	v3	v3	PROPN
ejpam-4653	563	5	)	)	PUNCT
ejpam-4653	563	6	=	=	SYM
ejpam-4653	563	7	∅	∅	NOUN
ejpam-4653	563	8	}	}	PUNCT
ejpam-4653	563	9	.	.	PUNCT
ejpam-4653	564	1	since	since	SCONJ
ejpam-4653	564	2	f	f	PROPN
ejpam-4653	564	3	=	=	PUNCT
ejpam-4653	564	4	(	(	PUNCT
ejpam-4653	564	5	∅,∅	∅,∅	PROPN
ejpam-4653	564	6	,	,	PUNCT
ejpam-4653	564	7	v	v	NOUN
ejpam-4653	564	8	(	(	PUNCT
ejpam-4653	564	9	g),∅	g),∅	NOUN
ejpam-4653	564	10	)	)	PUNCT
ejpam-4653	564	11	∈	∈	PROPN
ejpam-4653	564	12	cg	cg	NOUN
ejpam-4653	564	13	,	,	PUNCT
ejpam-4653	564	14	cg	cg	NOUN
ejpam-4653	564	15	̸=	̸=	PROPN
ejpam-4653	564	16	∅.	∅.	PRON
ejpam-4653	564	17	proposition	proposition	NOUN
ejpam-4653	564	18	16	16	NUM
ejpam-4653	564	19	.	.	PUNCT
ejpam-4653	565	1	let	let	VERB
ejpam-4653	565	2	g	g	PRON
ejpam-4653	565	3	be	be	AUX
ejpam-4653	565	4	a	a	DET
ejpam-4653	565	5	connected	connected	ADJ
ejpam-4653	565	6	noncomplete	noncomplete	ADJ
ejpam-4653	565	7	graph	graph	NOUN
ejpam-4653	565	8	and	and	CCONJ
ejpam-4653	565	9	h	h	NOUN
ejpam-4653	565	10	any	any	DET
ejpam-4653	565	11	nontrivial	nontrivial	ADJ
ejpam-4653	565	12	graph	graph	NOUN
ejpam-4653	565	13	with	with	ADP
ejpam-4653	565	14	γ(h	γ(h	NOUN
ejpam-4653	565	15	)	)	PUNCT
ejpam-4653	565	16	=	=	SYM
ejpam-4653	566	1	1	1	X
ejpam-4653	566	2	.	.	PUNCT
ejpam-4653	566	3	then	then	ADV
ejpam-4653	566	4	γdr(g[h	γdr(g[h	NUM
ejpam-4653	566	5	]	]	PUNCT
ejpam-4653	566	6	)	)	PUNCT
ejpam-4653	566	7	≤	≤	NUM
ejpam-4653	566	8	min{ωg(f	min{ωg(f	NOUN
ejpam-4653	566	9	)	)	PUNCT
ejpam-4653	566	10	:	:	PUNCT
ejpam-4653	567	1	f	f	X
ejpam-4653	567	2	=	=	SYM
ejpam-4653	567	3	(	(	PUNCT
ejpam-4653	567	4	v0,∅	v0,∅	PROPN
ejpam-4653	567	5	,	,	PUNCT
ejpam-4653	567	6	v2	v2	PROPN
ejpam-4653	567	7	,	,	PUNCT
ejpam-4653	567	8	v3	v3	PROPN
ejpam-4653	567	9	)	)	PUNCT
ejpam-4653	567	10	∈	∈	PROPN
ejpam-4653	567	11	cg	cg	NOUN
ejpam-4653	567	12	}	}	PUNCT
ejpam-4653	567	13	.	.	PUNCT
ejpam-4653	568	1	moreover	moreover	ADV
ejpam-4653	568	2	,	,	PUNCT
ejpam-4653	568	3	this	this	DET
ejpam-4653	568	4	upper	upper	ADJ
ejpam-4653	568	5	bound	bind	VERB
ejpam-4653	568	6	is	be	AUX
ejpam-4653	568	7	sharp	sharp	ADJ
ejpam-4653	568	8	.	.	PUNCT
ejpam-4653	569	1	proof	proof	NOUN
ejpam-4653	569	2	.	.	PUNCT
ejpam-4653	570	1	put	put	VERB
ejpam-4653	570	2	α	α	NOUN
ejpam-4653	570	3	=	=	SYM
ejpam-4653	570	4	min{ωg(f	min{ωg(f	PROPN
ejpam-4653	570	5	)	)	PUNCT
ejpam-4653	570	6	:	:	PUNCT
ejpam-4653	571	1	f	f	X
ejpam-4653	571	2	=	=	SYM
ejpam-4653	571	3	(	(	PUNCT
ejpam-4653	571	4	v0,∅	v0,∅	PROPN
ejpam-4653	571	5	,	,	PUNCT
ejpam-4653	571	6	v2	v2	PROPN
ejpam-4653	571	7	,	,	PUNCT
ejpam-4653	571	8	v3	v3	PROPN
ejpam-4653	571	9	)	)	PUNCT
ejpam-4653	571	10	∈	∈	PROPN
ejpam-4653	571	11	cg	cg	NOUN
ejpam-4653	571	12	}	}	PUNCT
ejpam-4653	571	13	,	,	PUNCT
ejpam-4653	571	14	and	and	CCONJ
ejpam-4653	571	15	let	let	VERB
ejpam-4653	571	16	v	v	NUM
ejpam-4653	571	17	∈	∈	PROPN
ejpam-4653	571	18	v	v	NOUN
ejpam-4653	571	19	(	(	PUNCT
ejpam-4653	571	20	h	h	NOUN
ejpam-4653	571	21	)	)	PUNCT
ejpam-4653	571	22	for	for	ADP
ejpam-4653	571	23	which	which	PRON
ejpam-4653	571	24	nh	nh	NOUN
ejpam-4653	572	1	[	[	X
ejpam-4653	572	2	v	v	X
ejpam-4653	572	3	]	]	X
ejpam-4653	572	4	=	=	SYM
ejpam-4653	572	5	v	v	ADJ
ejpam-4653	572	6	(	(	PUNCT
ejpam-4653	572	7	h	h	NOUN
ejpam-4653	572	8	)	)	PUNCT
ejpam-4653	572	9	.	.	PUNCT
ejpam-4653	573	1	let	let	VERB
ejpam-4653	573	2	f	f	PROPN
ejpam-4653	573	3	=	=	SYM
ejpam-4653	573	4	(	(	PUNCT
ejpam-4653	573	5	v0,∅	v0,∅	PROPN
ejpam-4653	573	6	,	,	PUNCT
ejpam-4653	573	7	v2	v2	PROPN
ejpam-4653	573	8	,	,	PUNCT
ejpam-4653	573	9	v3	v3	PROPN
ejpam-4653	573	10	)	)	PUNCT
ejpam-4653	573	11	∈	∈	PROPN
ejpam-4653	573	12	cg	cg	NOUN
ejpam-4653	573	13	.	.	PUNCT
ejpam-4653	574	1	put	put	VERB
ejpam-4653	574	2	v	v	NUM
ejpam-4653	574	3	∗	∗	NOUN
ejpam-4653	574	4	1	1	NUM
ejpam-4653	574	5	=	=	NOUN
ejpam-4653	574	6	∅	∅	NOUN
ejpam-4653	574	7	,	,	PUNCT
ejpam-4653	574	8	v	v	NOUN
ejpam-4653	574	9	∗	∗	NOUN
ejpam-4653	574	10	2	2	NUM
ejpam-4653	574	11	=	=	SYM
ejpam-4653	574	12	v2	v2	PROPN
ejpam-4653	574	13	×	×	NOUN
ejpam-4653	574	14	{	{	PUNCT
ejpam-4653	574	15	v	v	NOUN
ejpam-4653	574	16	}	}	PUNCT
ejpam-4653	574	17	,	,	PUNCT
ejpam-4653	574	18	v	v	X
ejpam-4653	574	19	∗	∗	X
ejpam-4653	574	20	3	3	NUM
ejpam-4653	574	21	=	=	SYM
ejpam-4653	574	22	v3	v3	PROPN
ejpam-4653	574	23	×	×	PROPN
ejpam-4653	574	24	{	{	PUNCT
ejpam-4653	574	25	v	v	NOUN
ejpam-4653	574	26	}	}	PUNCT
ejpam-4653	574	27	and	and	CCONJ
ejpam-4653	574	28	v	v	ADP
ejpam-4653	574	29	∗	∗	NOUN
ejpam-4653	574	30	0	0	NUM
ejpam-4653	574	31	=	=	SYM
ejpam-4653	574	32	v	v	NOUN
ejpam-4653	574	33	(	(	PUNCT
ejpam-4653	574	34	g[h])\(v	g[h])\(v	NOUN
ejpam-4653	574	35	∗	∗	NOUN
ejpam-4653	574	36	2	2	NUM
ejpam-4653	574	37	∪	∪	NOUN
ejpam-4653	574	38	v	v	NOUN
ejpam-4653	574	39	∗	∗	NOUN
ejpam-4653	574	40	3	3	NUM
ejpam-4653	574	41	)	)	PUNCT
ejpam-4653	574	42	.	.	PUNCT
ejpam-4653	575	1	let	let	VERB
ejpam-4653	575	2	(	(	PUNCT
ejpam-4653	575	3	x	x	NOUN
ejpam-4653	575	4	,	,	PUNCT
ejpam-4653	575	5	y	y	NOUN
ejpam-4653	575	6	)	)	PUNCT
ejpam-4653	575	7	∈	∈	PROPN
ejpam-4653	575	8	v	v	ADP
ejpam-4653	575	9	∗	∗	NOUN
ejpam-4653	575	10	0	0	NUM
ejpam-4653	575	11	.	.	PUNCT
ejpam-4653	576	1	if	if	SCONJ
ejpam-4653	576	2	x	x	PROPN
ejpam-4653	576	3	∈	∈	PROPN
ejpam-4653	576	4	v3	v3	PROPN
ejpam-4653	576	5	,	,	PUNCT
ejpam-4653	576	6	then	then	ADV
ejpam-4653	576	7	(	(	PUNCT
ejpam-4653	576	8	x	x	NOUN
ejpam-4653	576	9	,	,	PUNCT
ejpam-4653	576	10	v	v	NOUN
ejpam-4653	576	11	)	)	PUNCT
ejpam-4653	576	12	∈	∈	NOUN
ejpam-4653	576	13	v	v	ADP
ejpam-4653	576	14	∗	∗	NOUN
ejpam-4653	576	15	3	3	NUM
ejpam-4653	576	16	∩ng[h]((x	∩ng[h]((x	NOUN
ejpam-4653	576	17	,	,	PUNCT
ejpam-4653	576	18	y	y	NOUN
ejpam-4653	576	19	)	)	PUNCT
ejpam-4653	576	20	)	)	PUNCT
ejpam-4653	576	21	.	.	PUNCT
ejpam-4653	577	1	suppose	suppose	VERB
ejpam-4653	577	2	that	that	SCONJ
ejpam-4653	577	3	x	x	PUNCT
ejpam-4653	577	4	∈	∈	PROPN
ejpam-4653	577	5	v2	v2	PROPN
ejpam-4653	577	6	.	.	PUNCT
ejpam-4653	578	1	then	then	ADV
ejpam-4653	578	2	y	y	PROPN
ejpam-4653	578	3	̸=	̸=	PROPN
ejpam-4653	578	4	v.	v.	ADV
ejpam-4653	578	5	since	since	SCONJ
ejpam-4653	578	6	f	f	PROPN
ejpam-4653	578	7	∈	∈	PROPN
ejpam-4653	578	8	cg	cg	NOUN
ejpam-4653	578	9	,	,	PUNCT
ejpam-4653	578	10	there	there	PRON
ejpam-4653	578	11	exists	exist	VERB
ejpam-4653	578	12	w	w	PROPN
ejpam-4653	578	13	∈	∈	PROPN
ejpam-4653	578	14	v2	v2	NOUN
ejpam-4653	578	15	∩	∩	NOUN
ejpam-4653	578	16	ng(x	ng(x	NUM
ejpam-4653	578	17	)	)	PUNCT
ejpam-4653	578	18	or	or	CCONJ
ejpam-4653	578	19	there	there	PRON
ejpam-4653	578	20	exists	exist	VERB
ejpam-4653	578	21	z	z	PROPN
ejpam-4653	578	22	∈	∈	PROPN
ejpam-4653	578	23	v3	v3	PROPN
ejpam-4653	578	24	∩	∩	NOUN
ejpam-4653	578	25	ng(x	ng(x	NUM
ejpam-4653	578	26	)	)	PUNCT
ejpam-4653	578	27	.	.	PUNCT
ejpam-4653	579	1	the	the	DET
ejpam-4653	579	2	former	former	ADJ
ejpam-4653	579	3	implies	imply	VERB
ejpam-4653	579	4	that	that	SCONJ
ejpam-4653	579	5	(	(	PUNCT
ejpam-4653	579	6	x	x	NOUN
ejpam-4653	579	7	,	,	PUNCT
ejpam-4653	579	8	v	v	NOUN
ejpam-4653	579	9	)	)	PUNCT
ejpam-4653	579	10	,	,	PUNCT
ejpam-4653	579	11	(	(	PUNCT
ejpam-4653	579	12	w	w	NOUN
ejpam-4653	579	13	,	,	PUNCT
ejpam-4653	579	14	v	v	NOUN
ejpam-4653	579	15	)	)	PUNCT
ejpam-4653	579	16	∈	∈	NOUN
ejpam-4653	579	17	v	v	ADP
ejpam-4653	579	18	∗	∗	X
ejpam-4653	579	19	2	2	NUM
ejpam-4653	579	20	∩ng[kp]((x	∩ng[kp]((x	X
ejpam-4653	579	21	,	,	PUNCT
ejpam-4653	579	22	y	y	NOUN
ejpam-4653	579	23	)	)	PUNCT
ejpam-4653	579	24	)	)	PUNCT
ejpam-4653	579	25	.	.	PUNCT
ejpam-4653	580	1	the	the	DET
ejpam-4653	580	2	latter	latter	ADJ
ejpam-4653	580	3	,	,	PUNCT
ejpam-4653	580	4	on	on	ADP
ejpam-4653	580	5	the	the	DET
ejpam-4653	580	6	other	other	ADJ
ejpam-4653	580	7	hand	hand	NOUN
ejpam-4653	580	8	,	,	PUNCT
ejpam-4653	580	9	implies	imply	VERB
ejpam-4653	580	10	that	that	SCONJ
ejpam-4653	580	11	(	(	PUNCT
ejpam-4653	580	12	z	z	NOUN
ejpam-4653	580	13	,	,	PUNCT
ejpam-4653	580	14	v	v	NOUN
ejpam-4653	580	15	)	)	PUNCT
ejpam-4653	580	16	∈	∈	NOUN
ejpam-4653	580	17	v	v	ADP
ejpam-4653	580	18	∗	∗	X
ejpam-4653	580	19	3	3	NUM
ejpam-4653	580	20	∩ng[kp]((x	∩ng[kp]((x	NOUN
ejpam-4653	580	21	,	,	PUNCT
ejpam-4653	580	22	y	y	NOUN
ejpam-4653	580	23	)	)	PUNCT
ejpam-4653	580	24	)	)	PUNCT
ejpam-4653	580	25	.	.	PUNCT
ejpam-4653	581	1	finally	finally	ADV
ejpam-4653	581	2	,	,	PUNCT
ejpam-4653	581	3	suppose	suppose	VERB
ejpam-4653	581	4	that	that	SCONJ
ejpam-4653	581	5	x	x	PROPN
ejpam-4653	581	6	∈	∈	PROPN
ejpam-4653	581	7	v0	v0	NOUN
ejpam-4653	581	8	.	.	PUNCT
ejpam-4653	582	1	since	since	SCONJ
ejpam-4653	582	2	f	f	PROPN
ejpam-4653	582	3	∈	∈	PROPN
ejpam-4653	582	4	drd(g	drd(g	PROPN
ejpam-4653	582	5	)	)	PUNCT
ejpam-4653	582	6	,	,	PUNCT
ejpam-4653	582	7	there	there	PRON
ejpam-4653	582	8	exists	exist	VERB
ejpam-4653	582	9	u	u	PROPN
ejpam-4653	582	10	∈	∈	PROPN
ejpam-4653	582	11	v3∩ng(x	v3∩ng(x	NUM
ejpam-4653	582	12	)	)	PUNCT
ejpam-4653	582	13	or	or	CCONJ
ejpam-4653	582	14	there	there	PRON
ejpam-4653	582	15	exist	exist	VERB
ejpam-4653	582	16	distinct	distinct	ADJ
ejpam-4653	582	17	w	w	NOUN
ejpam-4653	582	18	,	,	PUNCT
ejpam-4653	582	19	z	z	PROPN
ejpam-4653	582	20	∈	∈	PROPN
ejpam-4653	582	21	v2	v2	PROPN
ejpam-4653	582	22	∩ng(x	∩ng(x	NOUN
ejpam-4653	582	23	)	)	PUNCT
ejpam-4653	582	24	.	.	PUNCT
ejpam-4653	583	1	this	this	PRON
ejpam-4653	583	2	means	mean	VERB
ejpam-4653	583	3	that	that	SCONJ
ejpam-4653	583	4	(	(	PUNCT
ejpam-4653	583	5	u	u	NOUN
ejpam-4653	583	6	,	,	PUNCT
ejpam-4653	583	7	v	v	NOUN
ejpam-4653	583	8	)	)	PUNCT
ejpam-4653	583	9	∈	∈	NOUN
ejpam-4653	583	10	v	v	ADP
ejpam-4653	583	11	∗	∗	X
ejpam-4653	583	12	3	3	NUM
ejpam-4653	583	13	∩ng[kp]((x	∩ng[kp]((x	NOUN
ejpam-4653	583	14	,	,	PUNCT
ejpam-4653	583	15	y	y	NOUN
ejpam-4653	583	16	)	)	PUNCT
ejpam-4653	583	17	)	)	PUNCT
ejpam-4653	583	18	or	or	CCONJ
ejpam-4653	583	19	we	we	PRON
ejpam-4653	583	20	have	have	VERB
ejpam-4653	583	21	distinct	distinct	ADJ
ejpam-4653	583	22	(	(	PUNCT
ejpam-4653	583	23	w	w	PROPN
ejpam-4653	583	24	,	,	PUNCT
ejpam-4653	583	25	v	v	NOUN
ejpam-4653	583	26	)	)	PUNCT
ejpam-4653	583	27	,	,	PUNCT
ejpam-4653	583	28	(	(	PUNCT
ejpam-4653	583	29	z	z	NOUN
ejpam-4653	583	30	,	,	PUNCT
ejpam-4653	583	31	v	v	NOUN
ejpam-4653	583	32	)	)	PUNCT
ejpam-4653	583	33	∈	∈	PROPN
ejpam-4653	583	34	ng[h]((x	ng[h]((x	NOUN
ejpam-4653	583	35	,	,	PUNCT
ejpam-4653	583	36	y	y	NOUN
ejpam-4653	583	37	)	)	PUNCT
ejpam-4653	583	38	)	)	PUNCT
ejpam-4653	583	39	.	.	PUNCT
ejpam-4653	584	1	accordingly	accordingly	ADV
ejpam-4653	584	2	,	,	PUNCT
ejpam-4653	584	3	g	g	PROPN
ejpam-4653	584	4	=	=	SYM
ejpam-4653	584	5	(	(	PUNCT
ejpam-4653	584	6	v	v	NOUN
ejpam-4653	584	7	∗	∗	NOUN
ejpam-4653	584	8	0	0	NUM
ejpam-4653	584	9	,	,	PUNCT
ejpam-4653	584	10	v	v	NOUN
ejpam-4653	584	11	∗	∗	NOUN
ejpam-4653	584	12	1	1	NUM
ejpam-4653	584	13	,	,	PUNCT
ejpam-4653	584	14	v	v	NOUN
ejpam-4653	584	15	∗	∗	NOUN
ejpam-4653	584	16	2	2	NUM
ejpam-4653	584	17	,	,	PUNCT
ejpam-4653	584	18	v	v	NOUN
ejpam-4653	584	19	∗	∗	X
ejpam-4653	584	20	3	3	NUM
ejpam-4653	584	21	)	)	PUNCT
ejpam-4653	584	22	∈	∈	PROPN
ejpam-4653	584	23	drd(g[h	drd(g[h	NUM
ejpam-4653	584	24	]	]	PUNCT
ejpam-4653	584	25	)	)	PUNCT
ejpam-4653	584	26	.	.	PUNCT
ejpam-4653	585	1	moreover	moreover	ADV
ejpam-4653	585	2	,	,	PUNCT
ejpam-4653	585	3	ωg[h](g	ωg[h](g	NOUN
ejpam-4653	585	4	)	)	PUNCT
ejpam-4653	585	5	=	=	PUNCT
ejpam-4653	586	1	2|v	2|v	PROPN
ejpam-4653	586	2	∗	∗	NOUN
ejpam-4653	586	3	2	2	NUM
ejpam-4653	586	4	|+	|+	NOUN
ejpam-4653	586	5	3|v	3|v	NUM
ejpam-4653	586	6	∗	∗	NOUN
ejpam-4653	586	7	3	3	NUM
ejpam-4653	586	8	|	|	NOUN
ejpam-4653	586	9	=	=	SYM
ejpam-4653	586	10	2|v2|+	2|v2|+	NUM
ejpam-4653	586	11	3|v3|	3|v3|	NUM
ejpam-4653	586	12	=	=	SYM
ejpam-4653	586	13	ωg(f	ωg(f	NOUN
ejpam-4653	586	14	)	)	PUNCT
ejpam-4653	586	15	.	.	PUNCT
ejpam-4653	587	1	therefore	therefore	ADV
ejpam-4653	587	2	,	,	PUNCT
ejpam-4653	587	3	γdr(g[h	γdr(g[h	NUM
ejpam-4653	587	4	]	]	PUNCT
ejpam-4653	587	5	)	)	PUNCT
ejpam-4653	587	6	≤	≤	NOUN
ejpam-4653	587	7	ωg(f	ωg(f	NUM
ejpam-4653	587	8	)	)	PUNCT
ejpam-4653	587	9	.	.	PUNCT
ejpam-4653	588	1	since	since	SCONJ
ejpam-4653	588	2	f	f	PROPN
ejpam-4653	588	3	is	be	AUX
ejpam-4653	588	4	arbitrary	arbitrary	ADJ
ejpam-4653	588	5	,	,	PUNCT
ejpam-4653	588	6	γdr(g[h	γdr(g[h	NUM
ejpam-4653	588	7	]	]	PUNCT
ejpam-4653	588	8	)	)	PUNCT
ejpam-4653	588	9	≤	≤	NUM
ejpam-4653	589	1	α	α	X
ejpam-4653	589	2	.	.	PUNCT
ejpam-4653	590	1	to	to	PART
ejpam-4653	590	2	show	show	VERB
ejpam-4653	590	3	sharpness	sharpness	NOUN
ejpam-4653	590	4	,	,	PUNCT
ejpam-4653	590	5	consider	consider	VERB
ejpam-4653	590	6	the	the	DET
ejpam-4653	590	7	lexicographic	lexicographic	ADJ
ejpam-4653	590	8	product	product	NOUN
ejpam-4653	590	9	of	of	ADP
ejpam-4653	590	10	g	g	NOUN
ejpam-4653	590	11	=	=	SYM
ejpam-4653	590	12	p4	p4	NOUN
ejpam-4653	590	13	=	=	PUNCT
ejpam-4653	591	1	[	[	X
ejpam-4653	591	2	v1	v1	NOUN
ejpam-4653	591	3	,	,	PUNCT
ejpam-4653	591	4	v2	v2	PROPN
ejpam-4653	591	5	,	,	PUNCT
ejpam-4653	591	6	v3	v3	PROPN
ejpam-4653	591	7	,	,	PUNCT
ejpam-4653	591	8	v4	v4	NOUN
ejpam-4653	591	9	]	]	PUNCT
ejpam-4653	591	10	and	and	CCONJ
ejpam-4653	591	11	h	h	NOUN
ejpam-4653	591	12	=	=	PROPN
ejpam-4653	591	13	p3	p3	PROPN
ejpam-4653	591	14	as	as	SCONJ
ejpam-4653	591	15	shown	show	VERB
ejpam-4653	591	16	in	in	ADP
ejpam-4653	591	17	figure	figure	NOUN
ejpam-4653	591	18	3	3	X
ejpam-4653	591	19	.	.	PUNCT
ejpam-4653	592	1	we	we	PRON
ejpam-4653	592	2	have	have	VERB
ejpam-4653	592	3	for	for	ADP
ejpam-4653	592	4	this	this	DET
ejpam-4653	592	5	case	case	NOUN
ejpam-4653	592	6	,	,	PUNCT
ejpam-4653	592	7	γdr(g[h	γdr(g[h	NUM
ejpam-4653	592	8	]	]	PUNCT
ejpam-4653	592	9	)	)	PUNCT
ejpam-4653	592	10	=	=	SYM
ejpam-4653	592	11	6	6	NUM
ejpam-4653	592	12	=	=	PUNCT
ejpam-4653	592	13	ω(f	ω(f	NUM
ejpam-4653	592	14	)	)	PUNCT
ejpam-4653	592	15	,	,	PUNCT
ejpam-4653	592	16	where	where	SCONJ
ejpam-4653	592	17	f	f	PROPN
ejpam-4653	592	18	=	=	PRON
ejpam-4653	592	19	(	(	PUNCT
ejpam-4653	592	20	{	{	PUNCT
ejpam-4653	592	21	v2	v2	NOUN
ejpam-4653	592	22	,	,	PUNCT
ejpam-4653	592	23	v3},∅,∅	v3},∅,∅	PROPN
ejpam-4653	592	24	,	,	PUNCT
ejpam-4653	592	25	{	{	PUNCT
ejpam-4653	592	26	v1	v1	NOUN
ejpam-4653	592	27	,	,	PUNCT
ejpam-4653	592	28	v4	v4	NOUN
ejpam-4653	592	29	}	}	PUNCT
ejpam-4653	592	30	)	)	PUNCT
ejpam-4653	592	31	.	.	PUNCT
ejpam-4653	593	1	the	the	DET
ejpam-4653	593	2	example	example	NOUN
ejpam-4653	593	3	presented	present	VERB
ejpam-4653	593	4	in	in	ADP
ejpam-4653	593	5	the	the	DET
ejpam-4653	593	6	proof	proof	NOUN
ejpam-4653	593	7	of	of	ADP
ejpam-4653	593	8	proposition	proposition	NOUN
ejpam-4653	593	9	16	16	NUM
ejpam-4653	593	10	also	also	ADV
ejpam-4653	593	11	shows	show	VERB
ejpam-4653	593	12	that	that	SCONJ
ejpam-4653	593	13	min{ωg(f	min{ωg(f	NOUN
ejpam-4653	593	14	)	)	PUNCT
ejpam-4653	593	15	:	:	PUNCT
ejpam-4653	594	1	f	f	X
ejpam-4653	594	2	=	=	SYM
ejpam-4653	594	3	(	(	PUNCT
ejpam-4653	594	4	v0,∅	v0,∅	PROPN
ejpam-4653	594	5	,	,	PUNCT
ejpam-4653	594	6	v2	v2	PROPN
ejpam-4653	594	7	,	,	PUNCT
ejpam-4653	594	8	v3	v3	PROPN
ejpam-4653	594	9	)	)	PUNCT
ejpam-4653	594	10	∈	∈	PROPN
ejpam-4653	594	11	cg	cg	NOUN
ejpam-4653	594	12	}	}	PUNCT
ejpam-4653	594	13	need	need	AUX
ejpam-4653	594	14	not	not	PART
ejpam-4653	594	15	be	be	AUX
ejpam-4653	594	16	determined	determine	VERB
ejpam-4653	594	17	by	by	ADP
ejpam-4653	594	18	a	a	DET
ejpam-4653	594	19	γdr	γdr	NOUN
ejpam-4653	594	20	-	-	PUNCT
ejpam-4653	594	21	function	function	NOUN
ejpam-4653	594	22	f	f	PROPN
ejpam-4653	594	23	of	of	ADP
ejpam-4653	594	24	g.	g.	PROPN
ejpam-4653	594	25	j.	j.	PROPN
ejpam-4653	594	26	b.g	b.g	PROPN
ejpam-4653	594	27	.	.	PROPN
ejpam-4653	594	28	cariaga	cariaga	PROPN
ejpam-4653	594	29	,	,	PUNCT
ejpam-4653	594	30	f.	f.	PROPN
ejpam-4653	594	31	jamil	jamil	PROPN
ejpam-4653	594	32	/	/	SYM
ejpam-4653	594	33	eur	eur	PROPN
ejpam-4653	594	34	.	.	PUNCT
ejpam-4653	595	1	j.	j.	PROPN
ejpam-4653	595	2	pure	pure	PROPN
ejpam-4653	595	3	appl	appl	PROPN
ejpam-4653	595	4	.	.	PROPN
ejpam-4653	595	5	math	math	PROPN
ejpam-4653	595	6	,	,	PUNCT
ejpam-4653	595	7	16	16	NUM
ejpam-4653	595	8	(	(	PUNCT
ejpam-4653	595	9	2	2	NUM
ejpam-4653	595	10	)	)	PUNCT
ejpam-4653	595	11	(	(	PUNCT
ejpam-4653	595	12	2023	2023	NUM
ejpam-4653	595	13	)	)	PUNCT
ejpam-4653	595	14	,	,	PUNCT
ejpam-4653	595	15	847	847	NUM
ejpam-4653	595	16	-	-	SYM
ejpam-4653	595	17	863	863	NUM
ejpam-4653	595	18	861	861	NUM
ejpam-4653	595	19	....................................	....................................	PUNCT
ejpam-4653	595	20	....................................	....................................	PUNCT
ejpam-4653	596	1	....................................	....................................	PUNCT
ejpam-4653	596	2	....................................	....................................	PUNCT
ejpam-4653	597	1	....................................	....................................	PUNCT
ejpam-4653	597	2	....................................	....................................	PUNCT
ejpam-4653	598	1	....................................	....................................	PUNCT
ejpam-4653	598	2	....................................	....................................	PUNCT
ejpam-4653	599	1	....................................	....................................	PUNCT
ejpam-4653	599	2	....................................	....................................	PUNCT
ejpam-4653	600	1	....................................	....................................	PUNCT
ejpam-4653	600	2	....................................	....................................	PUNCT
ejpam-4653	601	1	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PUNCT
ejpam-4653	601	2	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PUNCT
ejpam-4653	602	1	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PROPN
ejpam-4653	602	2	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PROPN
ejpam-4653	603	1	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PROPN
ejpam-4653	603	2	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PROPN
ejpam-4653	604	1	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PROPN
ejpam-4653	604	2	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PUNCT
ejpam-4653	604	3	...............................................................................................................................................................................................	...............................................................................................................................................................................................	PUNCT
ejpam-4653	605	1	.........	.........	PUNCT
ejpam-4653	605	2	........	........	PUNCT
ejpam-4653	605	3	........	........	PUNCT
ejpam-4653	605	4	........	........	PUNCT
ejpam-4653	605	5	........	........	PUNCT
ejpam-4653	605	6	........	........	PUNCT
ejpam-4653	605	7	........	........	PUNCT
ejpam-4653	605	8	........	........	PUNCT
ejpam-4653	605	9	........	........	PUNCT
ejpam-4653	605	10	........	........	PUNCT
ejpam-4653	605	11	........	........	PUNCT
ejpam-4653	605	12	........	........	PUNCT
ejpam-4653	605	13	........	........	PUNCT
ejpam-4653	605	14	........	........	PUNCT
ejpam-4653	605	15	........	........	PUNCT
ejpam-4653	606	1	........	........	PUNCT
ejpam-4653	606	2	....	....	PUNCT
ejpam-4653	607	1	.........	.........	PUNCT
ejpam-4653	607	2	........	........	PUNCT
ejpam-4653	607	3	........	........	PUNCT
ejpam-4653	607	4	........	........	PUNCT
ejpam-4653	607	5	........	........	PUNCT
ejpam-4653	607	6	........	........	PUNCT
ejpam-4653	607	7	........	........	PUNCT
ejpam-4653	607	8	........	........	PUNCT
ejpam-4653	607	9	........	........	PUNCT
ejpam-4653	607	10	........	........	PUNCT
ejpam-4653	607	11	........	........	PUNCT
ejpam-4653	607	12	........	........	PUNCT
ejpam-4653	607	13	........	........	PUNCT
ejpam-4653	607	14	........	........	PUNCT
ejpam-4653	607	15	........	........	PUNCT
ejpam-4653	608	1	........	........	PUNCT
ejpam-4653	608	2	....	....	PUNCT
ejpam-4653	609	1	.........	.........	PUNCT
ejpam-4653	609	2	........	........	PUNCT
ejpam-4653	609	3	........	........	PUNCT
ejpam-4653	609	4	........	........	PUNCT
ejpam-4653	609	5	........	........	PUNCT
ejpam-4653	609	6	........	........	PUNCT
ejpam-4653	609	7	........	........	PUNCT
ejpam-4653	609	8	........	........	PUNCT
ejpam-4653	609	9	........	........	PUNCT
ejpam-4653	609	10	........	........	PUNCT
ejpam-4653	609	11	........	........	PUNCT
ejpam-4653	609	12	........	........	PUNCT
ejpam-4653	609	13	........	........	PUNCT
ejpam-4653	609	14	........	........	PUNCT
ejpam-4653	609	15	........	........	PUNCT
ejpam-4653	610	1	........	........	PUNCT
ejpam-4653	610	2	....	....	PUNCT
ejpam-4653	611	1	.........	.........	PUNCT
ejpam-4653	611	2	........	........	PUNCT
ejpam-4653	611	3	........	........	PUNCT
ejpam-4653	611	4	........	........	PUNCT
ejpam-4653	611	5	........	........	PUNCT
ejpam-4653	611	6	........	........	PUNCT
ejpam-4653	611	7	........	........	PUNCT
ejpam-4653	611	8	........	........	PUNCT
ejpam-4653	611	9	........	........	PUNCT
ejpam-4653	611	10	........	........	PUNCT
ejpam-4653	611	11	........	........	PUNCT
ejpam-4653	611	12	........	........	PUNCT
ejpam-4653	611	13	........	........	PUNCT
ejpam-4653	611	14	........	........	PUNCT
ejpam-4653	611	15	........	........	PUNCT
ejpam-4653	612	1	........	........	PUNCT
ejpam-4653	612	2	....	....	PUNCT
ejpam-4653	613	1	.........	.........	PUNCT
ejpam-4653	613	2	........	........	PUNCT
ejpam-4653	613	3	........	........	PUNCT
ejpam-4653	613	4	........	........	PUNCT
ejpam-4653	613	5	........	........	PUNCT
ejpam-4653	613	6	........	........	PUNCT
ejpam-4653	613	7	........	........	PUNCT
ejpam-4653	613	8	........	........	PUNCT
ejpam-4653	613	9	........	........	PUNCT
ejpam-4653	613	10	........	........	PUNCT
ejpam-4653	613	11	........	........	PUNCT
ejpam-4653	613	12	........	........	PUNCT
ejpam-4653	613	13	........	........	PUNCT
ejpam-4653	613	14	........	........	PUNCT
ejpam-4653	613	15	........	........	PUNCT
ejpam-4653	614	1	........	........	PUNCT
ejpam-4653	614	2	....	....	PUNCT
ejpam-4653	615	1	.........	.........	PUNCT
ejpam-4653	615	2	........	........	PUNCT
ejpam-4653	615	3	........	........	PUNCT
ejpam-4653	615	4	........	........	PUNCT
ejpam-4653	615	5	........	........	PUNCT
ejpam-4653	615	6	........	........	PUNCT
ejpam-4653	615	7	........	........	PUNCT
ejpam-4653	615	8	........	........	PUNCT
ejpam-4653	615	9	........	........	PUNCT
ejpam-4653	615	10	........	........	PUNCT
ejpam-4653	615	11	........	........	PUNCT
ejpam-4653	615	12	........	........	PUNCT
ejpam-4653	615	13	........	........	PUNCT
ejpam-4653	615	14	........	........	PUNCT
ejpam-4653	615	15	........	........	PUNCT
ejpam-4653	616	1	........	........	PUNCT
ejpam-4653	616	2	....	....	PUNCT
ejpam-4653	617	1	.........	.........	PUNCT
ejpam-4653	617	2	........	........	PUNCT
ejpam-4653	617	3	........	........	PUNCT
ejpam-4653	617	4	........	........	PUNCT
ejpam-4653	617	5	........	........	PUNCT
ejpam-4653	617	6	........	........	PUNCT
ejpam-4653	617	7	........	........	PUNCT
ejpam-4653	617	8	........	........	PUNCT
ejpam-4653	617	9	........	........	PUNCT
ejpam-4653	617	10	........	........	PUNCT
ejpam-4653	617	11	........	........	PUNCT
ejpam-4653	617	12	........	........	PUNCT
ejpam-4653	617	13	........	........	PUNCT
ejpam-4653	617	14	........	........	PUNCT
ejpam-4653	617	15	........	........	PUNCT
ejpam-4653	618	1	........	........	PUNCT
ejpam-4653	618	2	....	....	PUNCT
ejpam-4653	619	1	.........	.........	PUNCT
ejpam-4653	619	2	........	........	PUNCT
ejpam-4653	619	3	........	........	PUNCT
ejpam-4653	619	4	........	........	PUNCT
ejpam-4653	619	5	........	........	PUNCT
ejpam-4653	619	6	........	........	PUNCT
ejpam-4653	619	7	........	........	PUNCT
ejpam-4653	619	8	........	........	PUNCT
ejpam-4653	619	9	........	........	PUNCT
ejpam-4653	619	10	........	........	PUNCT
ejpam-4653	619	11	........	........	PUNCT
ejpam-4653	619	12	........	........	PUNCT
ejpam-4653	619	13	........	........	PUNCT
ejpam-4653	619	14	........	........	PUNCT
ejpam-4653	619	15	........	........	PUNCT
ejpam-4653	620	1	........	........	PUNCT
ejpam-4653	620	2	....	....	PUNCT
ejpam-4653	621	1	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	621	2	...............	...............	PUNCT
ejpam-4653	622	1	..............	..............	PUNCT
ejpam-4653	622	2	..............	..............	PUNCT
ejpam-4653	623	1	..............	..............	PUNCT
ejpam-4653	623	2	..............	..............	PUNCT
ejpam-4653	624	1	..............	..............	PUNCT
ejpam-4653	624	2	..............	..............	PUNCT
ejpam-4653	625	1	..............	..............	PUNCT
ejpam-4653	625	2	..............	..............	PUNCT
ejpam-4653	626	1	..............	..............	PUNCT
ejpam-4653	626	2	..............	..............	PUNCT
ejpam-4653	627	1	..............	..............	PUNCT
ejpam-4653	627	2	..............	..............	PUNCT
ejpam-4653	628	1	..............	..............	PUNCT
ejpam-4653	628	2	..............	..............	PUNCT
ejpam-4653	629	1	..............	..............	PUNCT
ejpam-4653	629	2	............	............	PUNCT
ejpam-4653	629	3	...............	...............	PUNCT
ejpam-4653	629	4	..............	..............	PUNCT
ejpam-4653	629	5	..............	..............	PUNCT
ejpam-4653	630	1	..............	..............	PUNCT
ejpam-4653	630	2	..............	..............	PUNCT
ejpam-4653	631	1	..............	..............	PUNCT
ejpam-4653	631	2	..............	..............	PUNCT
ejpam-4653	632	1	..............	..............	PUNCT
ejpam-4653	632	2	..............	..............	PUNCT
ejpam-4653	633	1	..............	..............	PUNCT
ejpam-4653	633	2	..............	..............	PUNCT
ejpam-4653	634	1	..............	..............	PUNCT
ejpam-4653	634	2	..............	..............	PUNCT
ejpam-4653	635	1	..............	..............	PUNCT
ejpam-4653	635	2	..............	..............	PUNCT
ejpam-4653	636	1	..............	..............	PUNCT
ejpam-4653	636	2	............	............	PUNCT
ejpam-4653	637	1	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	637	2	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	637	3	...............	...............	PUNCT
ejpam-4653	638	1	..............	..............	PUNCT
ejpam-4653	638	2	..............	..............	PUNCT
ejpam-4653	639	1	..............	..............	PUNCT
ejpam-4653	639	2	..............	..............	PUNCT
ejpam-4653	640	1	..............	..............	PUNCT
ejpam-4653	640	2	..............	..............	PUNCT
ejpam-4653	641	1	..............	..............	PUNCT
ejpam-4653	641	2	..............	..............	PUNCT
ejpam-4653	642	1	..............	..............	PUNCT
ejpam-4653	642	2	..............	..............	PUNCT
ejpam-4653	643	1	..............	..............	PUNCT
ejpam-4653	643	2	..............	..............	PUNCT
ejpam-4653	644	1	..............	..............	PUNCT
ejpam-4653	644	2	..............	..............	PUNCT
ejpam-4653	645	1	..............	..............	PUNCT
ejpam-4653	645	2	............	............	PUNCT
ejpam-4653	645	3	...............	...............	PUNCT
ejpam-4653	645	4	..............	..............	PUNCT
ejpam-4653	645	5	..............	..............	PUNCT
ejpam-4653	646	1	..............	..............	PUNCT
ejpam-4653	646	2	..............	..............	PUNCT
ejpam-4653	647	1	..............	..............	PUNCT
ejpam-4653	647	2	..............	..............	PUNCT
ejpam-4653	648	1	..............	..............	PUNCT
ejpam-4653	648	2	..............	..............	PUNCT
ejpam-4653	649	1	..............	..............	PUNCT
ejpam-4653	649	2	..............	..............	PUNCT
ejpam-4653	650	1	..............	..............	PUNCT
ejpam-4653	650	2	..............	..............	PUNCT
ejpam-4653	651	1	..............	..............	PUNCT
ejpam-4653	651	2	..............	..............	PUNCT
ejpam-4653	652	1	..............	..............	PUNCT
ejpam-4653	652	2	............	............	PUNCT
ejpam-4653	652	3	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	652	4	...............	...............	PUNCT
ejpam-4653	652	5	..............	..............	PUNCT
ejpam-4653	653	1	..............	..............	PUNCT
ejpam-4653	653	2	..............	..............	PUNCT
ejpam-4653	654	1	..............	..............	PUNCT
ejpam-4653	654	2	..............	..............	PUNCT
ejpam-4653	655	1	..............	..............	PUNCT
ejpam-4653	655	2	..............	..............	PUNCT
ejpam-4653	656	1	..............	..............	PUNCT
ejpam-4653	656	2	..............	..............	PUNCT
ejpam-4653	657	1	..............	..............	PUNCT
ejpam-4653	657	2	..............	..............	PUNCT
ejpam-4653	658	1	..............	..............	PUNCT
ejpam-4653	658	2	..............	..............	PUNCT
ejpam-4653	659	1	..............	..............	PUNCT
ejpam-4653	659	2	..............	..............	PUNCT
ejpam-4653	659	3	............	............	PUNCT
ejpam-4653	660	1	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	660	2	...............	...............	PUNCT
ejpam-4653	660	3	..............	..............	PUNCT
ejpam-4653	661	1	..............	..............	PUNCT
ejpam-4653	661	2	..............	..............	PUNCT
ejpam-4653	662	1	..............	..............	PUNCT
ejpam-4653	662	2	..............	..............	PUNCT
ejpam-4653	663	1	..............	..............	PUNCT
ejpam-4653	663	2	..............	..............	PUNCT
ejpam-4653	664	1	..............	..............	PUNCT
ejpam-4653	664	2	..............	..............	PUNCT
ejpam-4653	665	1	..............	..............	PUNCT
ejpam-4653	665	2	..............	..............	PUNCT
ejpam-4653	666	1	..............	..............	PUNCT
ejpam-4653	666	2	..............	..............	PUNCT
ejpam-4653	667	1	..............	..............	PUNCT
ejpam-4653	667	2	..............	..............	PUNCT
ejpam-4653	667	3	............	............	PUNCT
ejpam-4653	668	1	.............................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................	PUNCT
ejpam-4653	668	2	...........	...........	PUNCT
ejpam-4653	668	3	..........	..........	PUNCT
ejpam-4653	669	1	..........	..........	PUNCT
ejpam-4653	669	2	..........	..........	PUNCT
ejpam-4653	670	1	..........	..........	PUNCT
ejpam-4653	670	2	..........	..........	PUNCT
ejpam-4653	671	1	..........	..........	PUNCT
ejpam-4653	671	2	..........	..........	PUNCT
ejpam-4653	672	1	..........	..........	PUNCT
ejpam-4653	672	2	..........	..........	PUNCT
ejpam-4653	673	1	..........	..........	PUNCT
ejpam-4653	673	2	..........	..........	PUNCT
ejpam-4653	674	1	..........	..........	PUNCT
ejpam-4653	674	2	..........	..........	PUNCT
ejpam-4653	675	1	..........	..........	PUNCT
ejpam-4653	675	2	..........	..........	PUNCT
ejpam-4653	676	1	..........	..........	PUNCT
ejpam-4653	676	2	..........	..........	PUNCT
ejpam-4653	677	1	..........	..........	PUNCT
ejpam-4653	677	2	..........	..........	PUNCT
ejpam-4653	678	1	..........	..........	PUNCT
ejpam-4653	678	2	..........	..........	PUNCT
ejpam-4653	679	1	..........	..........	PUNCT
ejpam-4653	679	2	..........	..........	PUNCT
ejpam-4653	680	1	..........	..........	PUNCT
ejpam-4653	680	2	..........	..........	PUNCT
ejpam-4653	681	1	..........	..........	PUNCT
ejpam-4653	681	2	..........	..........	PUNCT
ejpam-4653	682	1	..........	..........	PUNCT
ejpam-4653	682	2	..........	..........	PUNCT
ejpam-4653	683	1	..........	..........	PUNCT
ejpam-4653	683	2	..........	..........	PUNCT
ejpam-4653	684	1	..........	..........	PUNCT
ejpam-4653	684	2	...........................................................................................................................................................................................................................................................................................................................................................	...........................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-4653	685	1	...........	...........	PUNCT
ejpam-4653	685	2	..........	..........	PUNCT
ejpam-4653	686	1	..........	..........	PUNCT
ejpam-4653	686	2	..........	..........	PUNCT
ejpam-4653	687	1	..........	..........	PUNCT
ejpam-4653	687	2	..........	..........	PUNCT
ejpam-4653	688	1	..........	..........	PUNCT
ejpam-4653	688	2	..........	..........	PUNCT
ejpam-4653	689	1	..........	..........	PUNCT
ejpam-4653	689	2	..........	..........	PUNCT
ejpam-4653	690	1	..........	..........	PUNCT
ejpam-4653	690	2	..........	..........	PUNCT
ejpam-4653	691	1	..........	..........	PUNCT
ejpam-4653	691	2	..........	..........	PUNCT
ejpam-4653	692	1	..........	..........	PUNCT
ejpam-4653	692	2	..........	..........	PUNCT
ejpam-4653	693	1	..........	..........	PUNCT
ejpam-4653	693	2	..........	..........	PUNCT
ejpam-4653	694	1	..........	..........	PUNCT
ejpam-4653	694	2	..........	..........	PUNCT
ejpam-4653	695	1	..........	..........	PUNCT
ejpam-4653	695	2	..........	..........	PUNCT
ejpam-4653	696	1	..........	..........	PUNCT
ejpam-4653	696	2	..........	..........	PUNCT
ejpam-4653	697	1	..........	..........	PUNCT
ejpam-4653	697	2	..........	..........	PUNCT
ejpam-4653	698	1	..........	..........	PUNCT
ejpam-4653	698	2	..........	..........	PUNCT
ejpam-4653	699	1	..........	..........	PUNCT
ejpam-4653	699	2	..........	..........	PUNCT
ejpam-4653	700	1	..........	..........	PUNCT
ejpam-4653	700	2	..........	..........	PUNCT
ejpam-4653	701	1	..........	..........	PUNCT
ejpam-4653	701	2	...........................................................................................................................................................................................................................................................................................................................................................	...........................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-4653	702	1	..............................................................................................................................................................................................................................................................................................................................................................	..............................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-4653	702	2	..........	..........	PUNCT
ejpam-4653	703	1	..........	..........	PUNCT
ejpam-4653	703	2	..........	..........	PUNCT
ejpam-4653	704	1	..........	..........	PUNCT
ejpam-4653	704	2	..........	..........	PUNCT
ejpam-4653	705	1	..........	..........	PUNCT
ejpam-4653	705	2	..........	..........	PUNCT
ejpam-4653	706	1	..........	..........	PUNCT
ejpam-4653	706	2	..........	..........	PUNCT
ejpam-4653	707	1	..........	..........	PUNCT
ejpam-4653	707	2	..........	..........	PUNCT
ejpam-4653	708	1	..........	..........	PUNCT
ejpam-4653	708	2	..........	..........	PUNCT
ejpam-4653	709	1	..........	..........	PUNCT
ejpam-4653	709	2	..........	..........	PUNCT
ejpam-4653	710	1	..........	..........	PUNCT
ejpam-4653	710	2	..........	..........	PUNCT
ejpam-4653	711	1	..........	..........	PUNCT
ejpam-4653	711	2	..........	..........	PUNCT
ejpam-4653	712	1	..........	..........	PUNCT
ejpam-4653	712	2	..........	..........	PUNCT
ejpam-4653	713	1	..........	..........	PUNCT
ejpam-4653	713	2	..........	..........	PUNCT
ejpam-4653	714	1	..........	..........	PUNCT
ejpam-4653	714	2	..........	..........	PUNCT
ejpam-4653	715	1	..........	..........	PUNCT
ejpam-4653	715	2	..........	..........	PUNCT
ejpam-4653	716	1	..........	..........	PUNCT
ejpam-4653	716	2	..........	..........	PUNCT
ejpam-4653	717	1	..........	..........	PUNCT
ejpam-4653	717	2	..........	..........	PUNCT
ejpam-4653	718	1	..........	..........	PUNCT
ejpam-4653	719	1	........	........	PUNCT
ejpam-4653	720	1	•	•	NUM
ejpam-4653	720	2	•	•	NOUN
ejpam-4653	720	3	0	0	NUM
ejpam-4653	720	4	0	0	NUM
ejpam-4653	720	5	0	0	NUM
ejpam-4653	720	6	0	0	NUM
ejpam-4653	720	7	0	0	NUM
ejpam-4653	720	8	0	0	NUM
ejpam-4653	720	9	0	0	NUM
ejpam-4653	720	10	0	0	NUM
ejpam-4653	720	11	0	0	NUM
ejpam-4653	720	12	0	0	NUM
ejpam-4653	720	13	3	3	NUM
ejpam-4653	720	14	3	3	NUM
ejpam-4653	720	15	g[h	g[h	NOUN
ejpam-4653	720	16	]	]	PUNCT
ejpam-4653	720	17	figure	figure	NOUN
ejpam-4653	720	18	3	3	NUM
ejpam-4653	720	19	:	:	PUNCT
ejpam-4653	720	20	graph	graph	NOUN
ejpam-4653	720	21	g[h	g[h	PROPN
ejpam-4653	720	22	]	]	PUNCT
ejpam-4653	720	23	where	where	SCONJ
ejpam-4653	720	24	g	g	NOUN
ejpam-4653	720	25	=	=	SYM
ejpam-4653	720	26	p4	p4	ADJ
ejpam-4653	720	27	and	and	CCONJ
ejpam-4653	720	28	h	h	NOUN
ejpam-4653	720	29	=	=	SYM
ejpam-4653	720	30	p3	p3	PROPN
ejpam-4653	720	31	proposition	proposition	NOUN
ejpam-4653	720	32	17	17	NUM
ejpam-4653	720	33	.	.	PUNCT
ejpam-4653	721	1	let	let	VERB
ejpam-4653	721	2	g	g	PRON
ejpam-4653	721	3	be	be	AUX
ejpam-4653	721	4	a	a	DET
ejpam-4653	721	5	connected	connected	ADJ
ejpam-4653	721	6	noncomplete	noncomplete	ADJ
ejpam-4653	721	7	graph	graph	NOUN
ejpam-4653	721	8	and	and	CCONJ
ejpam-4653	721	9	p	p	PRON
ejpam-4653	721	10	≥	≥	NUM
ejpam-4653	721	11	2	2	NUM
ejpam-4653	721	12	.	.	PUNCT
ejpam-4653	722	1	then	then	ADV
ejpam-4653	722	2	γdr(g[kp	γdr(g[kp	PROPN
ejpam-4653	722	3	]	]	X
ejpam-4653	722	4	)	)	PUNCT
ejpam-4653	722	5	=	=	SYM
ejpam-4653	722	6	min{ωg(f	min{ωg(f	PROPN
ejpam-4653	722	7	)	)	PUNCT
ejpam-4653	722	8	:	:	PUNCT
ejpam-4653	723	1	f	f	X
ejpam-4653	723	2	=	=	SYM
ejpam-4653	723	3	(	(	PUNCT
ejpam-4653	723	4	v0,∅	v0,∅	PROPN
ejpam-4653	723	5	,	,	PUNCT
ejpam-4653	723	6	v2	v2	PROPN
ejpam-4653	723	7	,	,	PUNCT
ejpam-4653	723	8	v3	v3	PROPN
ejpam-4653	723	9	)	)	PUNCT
ejpam-4653	723	10	∈	∈	PROPN
ejpam-4653	723	11	cg	cg	NOUN
ejpam-4653	723	12	}	}	PUNCT
ejpam-4653	723	13	.	.	PUNCT
ejpam-4653	724	1	proof	proof	NOUN
ejpam-4653	724	2	.	.	PUNCT
ejpam-4653	725	1	put	put	VERB
ejpam-4653	725	2	α	α	NOUN
ejpam-4653	725	3	=	=	SYM
ejpam-4653	725	4	min{ωg(f	min{ωg(f	PROPN
ejpam-4653	725	5	)	)	PUNCT
ejpam-4653	725	6	:	:	PUNCT
ejpam-4653	726	1	f	f	X
ejpam-4653	726	2	=	=	SYM
ejpam-4653	726	3	(	(	PUNCT
ejpam-4653	726	4	v0,∅	v0,∅	PROPN
ejpam-4653	726	5	,	,	PUNCT
ejpam-4653	726	6	v2	v2	PROPN
ejpam-4653	726	7	,	,	PUNCT
ejpam-4653	726	8	v3	v3	PROPN
ejpam-4653	726	9	)	)	PUNCT
ejpam-4653	726	10	∈	∈	PROPN
ejpam-4653	726	11	cg	cg	NOUN
ejpam-4653	726	12	}	}	PUNCT
ejpam-4653	726	13	,	,	PUNCT
ejpam-4653	726	14	and	and	CCONJ
ejpam-4653	726	15	let	let	VERB
ejpam-4653	726	16	v	v	NUM
ejpam-4653	726	17	∈	∈	PROPN
ejpam-4653	726	18	v	v	NOUN
ejpam-4653	726	19	(	(	PUNCT
ejpam-4653	726	20	kp	kp	PROPN
ejpam-4653	726	21	)	)	PUNCT
ejpam-4653	726	22	.	.	PUNCT
ejpam-4653	727	1	by	by	ADP
ejpam-4653	727	2	proposition	proposition	NOUN
ejpam-4653	727	3	16	16	NUM
ejpam-4653	727	4	,	,	PUNCT
ejpam-4653	727	5	γdr(g[kp	γdr(g[kp	PROPN
ejpam-4653	727	6	]	]	X
ejpam-4653	727	7	)	)	PUNCT
ejpam-4653	727	8	≤	≤	NUM
ejpam-4653	727	9	α	α	X
ejpam-4653	727	10	.	.	PUNCT
ejpam-4653	728	1	to	to	PART
ejpam-4653	728	2	get	get	VERB
ejpam-4653	728	3	the	the	DET
ejpam-4653	728	4	other	other	ADJ
ejpam-4653	728	5	inequality	inequality	NOUN
ejpam-4653	728	6	,	,	PUNCT
ejpam-4653	728	7	let	let	VERB
ejpam-4653	728	8	f	f	PROPN
ejpam-4653	728	9	=	=	SYM
ejpam-4653	728	10	(	(	PUNCT
ejpam-4653	728	11	v0	v0	PROPN
ejpam-4653	728	12	,	,	PUNCT
ejpam-4653	728	13	v1	v1	NOUN
ejpam-4653	728	14	,	,	PUNCT
ejpam-4653	728	15	v2	v2	PROPN
ejpam-4653	728	16	,	,	PUNCT
ejpam-4653	728	17	v3	v3	PROPN
ejpam-4653	728	18	)	)	PUNCT
ejpam-4653	728	19	be	be	VERB
ejpam-4653	728	20	a	a	DET
ejpam-4653	728	21	γdr	γdr	NOUN
ejpam-4653	728	22	-	-	PUNCT
ejpam-4653	728	23	function	function	NOUN
ejpam-4653	728	24	of	of	ADP
ejpam-4653	728	25	g[kp	g[kp	NOUN
ejpam-4653	728	26	]	]	PUNCT
ejpam-4653	728	27	.	.	PUNCT
ejpam-4653	729	1	we	we	PRON
ejpam-4653	729	2	assume	assume	VERB
ejpam-4653	729	3	that	that	SCONJ
ejpam-4653	729	4	v1	v1	NOUN
ejpam-4653	729	5	=	=	PUNCT
ejpam-4653	729	6	∅.	∅.	VERB
ejpam-4653	729	7	first	first	ADV
ejpam-4653	729	8	,	,	PUNCT
ejpam-4653	729	9	we	we	PRON
ejpam-4653	729	10	claim	claim	VERB
ejpam-4653	729	11	the	the	DET
ejpam-4653	729	12	following	following	NOUN
ejpam-4653	729	13	:	:	PUNCT
ejpam-4653	729	14	(	(	PUNCT
ejpam-4653	729	15	i	i	NOUN
ejpam-4653	729	16	)	)	PUNCT
ejpam-4653	729	17	(	(	PUNCT
ejpam-4653	729	18	v2)g	v2)g	PROPN
ejpam-4653	729	19	∩	∩	NOUN
ejpam-4653	729	20	(	(	PUNCT
ejpam-4653	729	21	v3)g	v3)g	NOUN
ejpam-4653	729	22	=	=	SYM
ejpam-4653	729	23	∅	∅	NOUN
ejpam-4653	729	24	;	;	PUNCT
ejpam-4653	729	25	(	(	PUNCT
ejpam-4653	729	26	ii	ii	NOUN
ejpam-4653	729	27	)	)	PUNCT
ejpam-4653	729	28	for	for	ADP
ejpam-4653	729	29	each	each	DET
ejpam-4653	729	30	x	x	SYM
ejpam-4653	729	31	∈	∈	PROPN
ejpam-4653	729	32	(	(	PUNCT
ejpam-4653	729	33	v2)g	v2)g	NOUN
ejpam-4653	729	34	,	,	PUNCT
ejpam-4653	729	35	|{y	|{y	PUNCT
ejpam-4653	729	36	:	:	PUNCT
ejpam-4653	729	37	(	(	PUNCT
ejpam-4653	729	38	x	x	X
ejpam-4653	729	39	,	,	PUNCT
ejpam-4653	729	40	y	y	NOUN
ejpam-4653	729	41	)	)	PUNCT
ejpam-4653	729	42	∈	∈	PROPN
ejpam-4653	729	43	v2}|	v2}|	PROPN
ejpam-4653	729	44	=	=	SYM
ejpam-4653	729	45	1	1	NUM
ejpam-4653	729	46	;	;	PUNCT
ejpam-4653	729	47	and	and	CCONJ
ejpam-4653	729	48	(	(	PUNCT
ejpam-4653	729	49	iii	iii	NOUN
ejpam-4653	729	50	)	)	PUNCT
ejpam-4653	729	51	for	for	ADP
ejpam-4653	729	52	each	each	DET
ejpam-4653	729	53	x	x	SYM
ejpam-4653	729	54	∈	∈	PROPN
ejpam-4653	729	55	(	(	PUNCT
ejpam-4653	729	56	v3)g	v3)g	NOUN
ejpam-4653	729	57	,	,	PUNCT
ejpam-4653	729	58	|{y	|{y	PROPN
ejpam-4653	729	59	:	:	PUNCT
ejpam-4653	729	60	(	(	PUNCT
ejpam-4653	729	61	x	x	X
ejpam-4653	729	62	,	,	PUNCT
ejpam-4653	729	63	y	y	NOUN
ejpam-4653	729	64	)	)	PUNCT
ejpam-4653	729	65	∈	∈	PROPN
ejpam-4653	729	66	v3}|	v3}|	NOUN
ejpam-4653	729	67	=	=	NOUN
ejpam-4653	729	68	1	1	X
ejpam-4653	729	69	.	.	X
ejpam-4653	729	70	for	for	AUX
ejpam-4653	729	71	suppose	suppose	VERB
ejpam-4653	729	72	that	that	SCONJ
ejpam-4653	729	73	u	u	PROPN
ejpam-4653	729	74	∈	∈	PROPN
ejpam-4653	729	75	(	(	PUNCT
ejpam-4653	729	76	v2)g	v2)g	PROPN
ejpam-4653	729	77	∩	∩	NOUN
ejpam-4653	729	78	(	(	PUNCT
ejpam-4653	729	79	v3)g	v3)g	NOUN
ejpam-4653	729	80	,	,	PUNCT
ejpam-4653	729	81	and	and	CCONJ
ejpam-4653	729	82	let	let	VERB
ejpam-4653	729	83	w	w	PROPN
ejpam-4653	729	84	∈	∈	PROPN
ejpam-4653	729	85	v	v	X
ejpam-4653	729	86	(	(	PUNCT
ejpam-4653	729	87	kp	kp	PROPN
ejpam-4653	729	88	)	)	PUNCT
ejpam-4653	729	89	for	for	ADP
ejpam-4653	729	90	which	which	PRON
ejpam-4653	729	91	(	(	PUNCT
ejpam-4653	729	92	u	u	NOUN
ejpam-4653	729	93	,	,	PUNCT
ejpam-4653	729	94	w	w	NOUN
ejpam-4653	729	95	)	)	PUNCT
ejpam-4653	729	96	∈	∈	PROPN
ejpam-4653	729	97	v2	v2	NOUN
ejpam-4653	729	98	.	.	PUNCT
ejpam-4653	730	1	then	then	ADV
ejpam-4653	730	2	f∗	f∗	PROPN
ejpam-4653	730	3	∈	∈	PROPN
ejpam-4653	730	4	drd(g[kp	drd(g[kp	PROPN
ejpam-4653	730	5	]	]	X
ejpam-4653	730	6	)	)	PUNCT
ejpam-4653	730	7	,	,	PUNCT
ejpam-4653	730	8	where	where	SCONJ
ejpam-4653	730	9	f∗	f∗	NOUN
ejpam-4653	730	10	is	be	AUX
ejpam-4653	730	11	defined	define	VERB
ejpam-4653	730	12	on	on	ADP
ejpam-4653	730	13	v	v	PROPN
ejpam-4653	730	14	(	(	PUNCT
ejpam-4653	730	15	g[kp	g[kp	NOUN
ejpam-4653	730	16	]	]	PUNCT
ejpam-4653	730	17	)	)	PUNCT
ejpam-4653	730	18	by	by	ADP
ejpam-4653	730	19	f∗((x	f∗((x	PROPN
ejpam-4653	730	20	,	,	PUNCT
ejpam-4653	730	21	y	y	NOUN
ejpam-4653	730	22	)	)	PUNCT
ejpam-4653	730	23	)	)	PUNCT
ejpam-4653	731	1	=	=	SYM
ejpam-4653	731	2	f((x	f((x	NOUN
ejpam-4653	731	3	,	,	PUNCT
ejpam-4653	731	4	y	y	NOUN
ejpam-4653	731	5	)	)	PUNCT
ejpam-4653	731	6	)	)	PUNCT
ejpam-4653	731	7	for	for	ADP
ejpam-4653	731	8	all	all	DET
ejpam-4653	731	9	(	(	PUNCT
ejpam-4653	731	10	x	x	NOUN
ejpam-4653	731	11	,	,	PUNCT
ejpam-4653	731	12	y	y	NOUN
ejpam-4653	731	13	)	)	PUNCT
ejpam-4653	731	14	∈	∈	PROPN
ejpam-4653	731	15	v	v	NOUN
ejpam-4653	731	16	(	(	PUNCT
ejpam-4653	731	17	g[kp])\{(u	g[kp])\{(u	NOUN
ejpam-4653	731	18	,	,	PUNCT
ejpam-4653	731	19	w	w	NOUN
ejpam-4653	731	20	)	)	PUNCT
ejpam-4653	731	21	}	}	PUNCT
ejpam-4653	731	22	and	and	CCONJ
ejpam-4653	731	23	f∗((u	f∗((u	NOUN
ejpam-4653	731	24	,	,	PUNCT
ejpam-4653	731	25	w	w	NOUN
ejpam-4653	731	26	)	)	PUNCT
ejpam-4653	731	27	)	)	PUNCT
ejpam-4653	732	1	=	=	PUNCT
ejpam-4653	732	2	0	0	X
ejpam-4653	732	3	.	.	PUNCT
ejpam-4653	733	1	this	this	PRON
ejpam-4653	733	2	is	be	AUX
ejpam-4653	733	3	a	a	DET
ejpam-4653	733	4	contradiction	contradiction	NOUN
ejpam-4653	733	5	since	since	SCONJ
ejpam-4653	733	6	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	733	7	∗	∗	NOUN
ejpam-4653	733	8	)	)	PUNCT
ejpam-4653	733	9	<	<	X
ejpam-4653	733	10	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	733	11	)	)	PUNCT
ejpam-4653	733	12	and	and	CCONJ
ejpam-4653	733	13	f	f	PROPN
ejpam-4653	733	14	is	be	AUX
ejpam-4653	733	15	a	a	DET
ejpam-4653	733	16	γdr	γdr	NOUN
ejpam-4653	733	17	-	-	PUNCT
ejpam-4653	733	18	function	function	NOUN
ejpam-4653	733	19	.	.	PUNCT
ejpam-4653	734	1	this	this	PRON
ejpam-4653	734	2	proves	prove	VERB
ejpam-4653	734	3	claim	claim	NOUN
ejpam-4653	734	4	(	(	PUNCT
ejpam-4653	734	5	i	i	NOUN
ejpam-4653	734	6	)	)	PUNCT
ejpam-4653	734	7	.	.	PUNCT
ejpam-4653	735	1	to	to	PART
ejpam-4653	735	2	prove	prove	VERB
ejpam-4653	735	3	(	(	PUNCT
ejpam-4653	735	4	ii	ii	NOUN
ejpam-4653	735	5	)	)	PUNCT
ejpam-4653	735	6	,	,	PUNCT
ejpam-4653	735	7	suppose	suppose	VERB
ejpam-4653	735	8	that	that	SCONJ
ejpam-4653	735	9	for	for	ADP
ejpam-4653	735	10	some	some	DET
ejpam-4653	735	11	u	u	NOUN
ejpam-4653	735	12	∈	∈	PROPN
ejpam-4653	735	13	(	(	PUNCT
ejpam-4653	735	14	v2)g	v2)g	NOUN
ejpam-4653	735	15	,	,	PUNCT
ejpam-4653	735	16	we	we	PRON
ejpam-4653	735	17	have	have	VERB
ejpam-4653	735	18	(	(	PUNCT
ejpam-4653	735	19	u	u	NOUN
ejpam-4653	735	20	,	,	PUNCT
ejpam-4653	735	21	w	w	NOUN
ejpam-4653	735	22	)	)	PUNCT
ejpam-4653	735	23	,	,	PUNCT
ejpam-4653	735	24	(	(	PUNCT
ejpam-4653	735	25	u	u	NOUN
ejpam-4653	735	26	,	,	PUNCT
ejpam-4653	735	27	t	t	PROPN
ejpam-4653	735	28	)	)	PUNCT
ejpam-4653	735	29	∈	∈	PROPN
ejpam-4653	735	30	v2	v2	PROPN
ejpam-4653	735	31	.	.	PUNCT
ejpam-4653	736	1	then	then	ADV
ejpam-4653	736	2	f∗	f∗	PROPN
ejpam-4653	736	3	∈	∈	PROPN
ejpam-4653	736	4	drd(g[kp	drd(g[kp	PROPN
ejpam-4653	736	5	]	]	X
ejpam-4653	736	6	)	)	PUNCT
ejpam-4653	736	7	,	,	PUNCT
ejpam-4653	736	8	where	where	SCONJ
ejpam-4653	736	9	f∗	f∗	NOUN
ejpam-4653	736	10	is	be	AUX
ejpam-4653	736	11	defined	define	VERB
ejpam-4653	736	12	on	on	ADP
ejpam-4653	736	13	v	v	PROPN
ejpam-4653	736	14	(	(	PUNCT
ejpam-4653	736	15	g[kp	g[kp	NOUN
ejpam-4653	736	16	]	]	PUNCT
ejpam-4653	736	17	)	)	PUNCT
ejpam-4653	736	18	by	by	ADP
ejpam-4653	736	19	f∗((u	f∗((u	NOUN
ejpam-4653	736	20	,	,	PUNCT
ejpam-4653	736	21	w	w	NOUN
ejpam-4653	736	22	)	)	PUNCT
ejpam-4653	736	23	)	)	PUNCT
ejpam-4653	737	1	=	=	SYM
ejpam-4653	737	2	3	3	X
ejpam-4653	737	3	,	,	PUNCT
ejpam-4653	737	4	f∗((u	f∗((u	NOUN
ejpam-4653	737	5	,	,	PUNCT
ejpam-4653	737	6	t	t	PROPN
ejpam-4653	737	7	)	)	PUNCT
ejpam-4653	737	8	)	)	PUNCT
ejpam-4653	738	1	=	=	SYM
ejpam-4653	738	2	0	0	NUM
ejpam-4653	738	3	and	and	CCONJ
ejpam-4653	738	4	f∗((x	f∗((x	PROPN
ejpam-4653	738	5	,	,	PUNCT
ejpam-4653	738	6	y	y	NOUN
ejpam-4653	738	7	)	)	PUNCT
ejpam-4653	738	8	)	)	PUNCT
ejpam-4653	739	1	=	=	SYM
ejpam-4653	739	2	f((x	f((x	NOUN
ejpam-4653	739	3	,	,	PUNCT
ejpam-4653	739	4	y	y	NOUN
ejpam-4653	739	5	)	)	PUNCT
ejpam-4653	739	6	)	)	PUNCT
ejpam-4653	739	7	for	for	ADP
ejpam-4653	739	8	all	all	DET
ejpam-4653	739	9	(	(	PUNCT
ejpam-4653	739	10	x	x	NOUN
ejpam-4653	739	11	,	,	PUNCT
ejpam-4653	739	12	y	y	NOUN
ejpam-4653	739	13	)	)	PUNCT
ejpam-4653	739	14	∈	∈	NOUN
ejpam-4653	739	15	v	v	X
ejpam-4653	739	16	(	(	PUNCT
ejpam-4653	739	17	g[kp	g[kp	NOUN
ejpam-4653	739	18	]	]	PUNCT
ejpam-4653	739	19	)	)	PUNCT
ejpam-4653	739	20	\	\	PUNCT
ejpam-4653	739	21	{	{	PUNCT
ejpam-4653	739	22	(	(	PUNCT
ejpam-4653	739	23	u	u	NOUN
ejpam-4653	739	24	,	,	PUNCT
ejpam-4653	739	25	w	w	NOUN
ejpam-4653	739	26	)	)	PUNCT
ejpam-4653	739	27	,	,	PUNCT
ejpam-4653	739	28	(	(	PUNCT
ejpam-4653	739	29	u	u	NOUN
ejpam-4653	739	30	,	,	PUNCT
ejpam-4653	739	31	t	t	PROPN
ejpam-4653	739	32	)	)	PUNCT
ejpam-4653	739	33	}	}	PUNCT
ejpam-4653	739	34	.	.	PUNCT
ejpam-4653	740	1	since	since	SCONJ
ejpam-4653	740	2	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	740	3	∗	∗	NOUN
ejpam-4653	740	4	)	)	PUNCT
ejpam-4653	740	5	<	<	X
ejpam-4653	740	6	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	740	7	)	)	PUNCT
ejpam-4653	740	8	,	,	PUNCT
ejpam-4653	740	9	this	this	PRON
ejpam-4653	740	10	is	be	AUX
ejpam-4653	740	11	a	a	DET
ejpam-4653	740	12	contradiction	contradiction	NOUN
ejpam-4653	740	13	.	.	PUNCT
ejpam-4653	741	1	claim	claim	NOUN
ejpam-4653	741	2	(	(	PUNCT
ejpam-4653	741	3	iii	iii	NOUN
ejpam-4653	741	4	)	)	PUNCT
ejpam-4653	741	5	is	be	AUX
ejpam-4653	741	6	clear	clear	ADJ
ejpam-4653	741	7	.	.	PUNCT
ejpam-4653	742	1	let	let	VERB
ejpam-4653	742	2	a	a	PRON
ejpam-4653	742	3	=	=	SYM
ejpam-4653	742	4	(	(	PUNCT
ejpam-4653	742	5	v2)g	v2)g	PROPN
ejpam-4653	742	6	,	,	PUNCT
ejpam-4653	742	7	b	b	NOUN
ejpam-4653	742	8	=	=	SYM
ejpam-4653	742	9	(	(	PUNCT
ejpam-4653	742	10	v3)g	v3)g	NOUN
ejpam-4653	742	11	and	and	CCONJ
ejpam-4653	742	12	c	c	NOUN
ejpam-4653	742	13	=	=	SYM
ejpam-4653	742	14	v	v	PROPN
ejpam-4653	742	15	(	(	PUNCT
ejpam-4653	742	16	g	g	NOUN
ejpam-4653	742	17	)	)	PUNCT
ejpam-4653	742	18	\	\	PUNCT
ejpam-4653	743	1	(	(	PUNCT
ejpam-4653	743	2	a	a	DET
ejpam-4653	743	3	∪b	∪b	NOUN
ejpam-4653	743	4	)	)	PUNCT
ejpam-4653	743	5	,	,	PUNCT
ejpam-4653	743	6	and	and	CCONJ
ejpam-4653	743	7	define	define	VERB
ejpam-4653	743	8	v	v	NUM
ejpam-4653	743	9	∗	∗	NOUN
ejpam-4653	743	10	0	0	NUM
ejpam-4653	744	1	=	=	SYM
ejpam-4653	744	2	v	v	X
ejpam-4653	744	3	(	(	PUNCT
ejpam-4653	744	4	g[kp	g[kp	NOUN
ejpam-4653	744	5	]	]	X
ejpam-4653	744	6	)	)	PUNCT
ejpam-4653	744	7	\	\	PUNCT
ejpam-4653	745	1	(	(	PUNCT
ejpam-4653	745	2	(	(	PUNCT
ejpam-4653	745	3	a	a	DET
ejpam-4653	745	4	∪b)×	∪b)×	ADV
ejpam-4653	745	5	{	{	PUNCT
ejpam-4653	745	6	v	v	NOUN
ejpam-4653	745	7	}	}	PUNCT
ejpam-4653	745	8	)	)	PUNCT
ejpam-4653	745	9	,	,	PUNCT
ejpam-4653	745	10	v	v	X
ejpam-4653	745	11	∗	∗	NOUN
ejpam-4653	745	12	1	1	NUM
ejpam-4653	745	13	=	=	NOUN
ejpam-4653	745	14	∅	∅	NOUN
ejpam-4653	745	15	,	,	PUNCT
ejpam-4653	745	16	v	v	NOUN
ejpam-4653	745	17	∗	∗	NOUN
ejpam-4653	745	18	2	2	NUM
ejpam-4653	745	19	=	=	SYM
ejpam-4653	745	20	a	a	DET
ejpam-4653	745	21	×	×	NOUN
ejpam-4653	745	22	{	{	PUNCT
ejpam-4653	745	23	v	v	NOUN
ejpam-4653	745	24	}	}	PUNCT
ejpam-4653	745	25	and	and	CCONJ
ejpam-4653	745	26	v	v	ADP
ejpam-4653	745	27	∗	∗	NOUN
ejpam-4653	745	28	3	3	NUM
ejpam-4653	745	29	=	=	SYM
ejpam-4653	745	30	b	b	SYM
ejpam-4653	745	31	×	×	NOUN
ejpam-4653	745	32	{	{	PUNCT
ejpam-4653	745	33	v	v	NOUN
ejpam-4653	745	34	}	}	PUNCT
ejpam-4653	745	35	.	.	PUNCT
ejpam-4653	746	1	define	define	VERB
ejpam-4653	746	2	the	the	DET
ejpam-4653	746	3	function	function	NOUN
ejpam-4653	746	4	g	g	NOUN
ejpam-4653	746	5	=	=	SYM
ejpam-4653	746	6	(	(	PUNCT
ejpam-4653	746	7	v	v	NOUN
ejpam-4653	746	8	∗	∗	NOUN
ejpam-4653	746	9	0	0	NUM
ejpam-4653	746	10	,	,	PUNCT
ejpam-4653	746	11	∅	∅	NOUN
ejpam-4653	746	12	,	,	PUNCT
ejpam-4653	746	13	v	v	NOUN
ejpam-4653	746	14	∗	∗	NOUN
ejpam-4653	746	15	2	2	NUM
ejpam-4653	746	16	,	,	PUNCT
ejpam-4653	746	17	v	v	NOUN
ejpam-4653	746	18	∗	∗	X
ejpam-4653	746	19	3	3	NUM
ejpam-4653	746	20	)	)	PUNCT
ejpam-4653	746	21	on	on	ADP
ejpam-4653	746	22	v	v	PROPN
ejpam-4653	746	23	(	(	PUNCT
ejpam-4653	746	24	g[kp	g[kp	NOUN
ejpam-4653	746	25	]	]	PUNCT
ejpam-4653	746	26	)	)	PUNCT
ejpam-4653	746	27	.	.	PUNCT
ejpam-4653	747	1	more	more	ADV
ejpam-4653	747	2	specifically	specifically	ADV
ejpam-4653	747	3	,	,	PUNCT
ejpam-4653	747	4	g((x	g((x	NOUN
ejpam-4653	747	5	,	,	PUNCT
ejpam-4653	747	6	y	y	NOUN
ejpam-4653	747	7	)	)	PUNCT
ejpam-4653	747	8	)	)	PUNCT
ejpam-4653	748	1	=	=	PUNCT
ejpam-4653	749	1			NOUN
ejpam-4653	749	2	3	3	NUM
ejpam-4653	749	3	,	,	PUNCT
ejpam-4653	749	4	if	if	SCONJ
ejpam-4653	749	5	x	x	PROPN
ejpam-4653	749	6	∈	∈	PROPN
ejpam-4653	749	7	b	b	PROPN
ejpam-4653	749	8	and	and	CCONJ
ejpam-4653	749	9	y	y	PROPN
ejpam-4653	749	10	=	=	SYM
ejpam-4653	749	11	v	v	PROPN
ejpam-4653	749	12	,	,	PUNCT
ejpam-4653	749	13	2	2	NUM
ejpam-4653	749	14	,	,	PUNCT
ejpam-4653	749	15	if	if	SCONJ
ejpam-4653	749	16	x	x	PROPN
ejpam-4653	749	17	∈	∈	PROPN
ejpam-4653	749	18	a	a	PRON
ejpam-4653	749	19	and	and	CCONJ
ejpam-4653	749	20	y	y	PROPN
ejpam-4653	749	21	=	=	PUNCT
ejpam-4653	749	22	v	v	PROPN
ejpam-4653	749	23	,	,	PUNCT
ejpam-4653	749	24	0	0	NUM
ejpam-4653	749	25	,	,	PUNCT
ejpam-4653	749	26	else	else	ADV
ejpam-4653	749	27	references	reference	NOUN
ejpam-4653	749	28	862	862	NUM
ejpam-4653	749	29	let	let	VERB
ejpam-4653	749	30	(	(	PUNCT
ejpam-4653	749	31	x	x	NOUN
ejpam-4653	749	32	,	,	PUNCT
ejpam-4653	749	33	y	y	NOUN
ejpam-4653	749	34	)	)	PUNCT
ejpam-4653	749	35	∈	∈	PROPN
ejpam-4653	749	36	v	v	ADP
ejpam-4653	749	37	∗	∗	NOUN
ejpam-4653	749	38	0	0	NUM
ejpam-4653	749	39	.	.	PUNCT
ejpam-4653	750	1	we	we	PRON
ejpam-4653	750	2	consider	consider	VERB
ejpam-4653	750	3	the	the	DET
ejpam-4653	750	4	following	follow	VERB
ejpam-4653	750	5	cases	case	NOUN
ejpam-4653	750	6	:	:	PUNCT
ejpam-4653	750	7	case	case	NOUN
ejpam-4653	750	8	1	1	NUM
ejpam-4653	750	9	:	:	PUNCT
ejpam-4653	750	10	assume	assume	VERB
ejpam-4653	750	11	x	x	SYM
ejpam-4653	750	12	∈	∈	PROPN
ejpam-4653	750	13	a	a	PRON
ejpam-4653	750	14	and	and	CCONJ
ejpam-4653	750	15	y	y	PROPN
ejpam-4653	750	16	̸=	̸=	PROPN
ejpam-4653	750	17	v.	v.	CCONJ
ejpam-4653	750	18	since	since	SCONJ
ejpam-4653	750	19	p	p	PROPN
ejpam-4653	750	20	≥	≥	NUM
ejpam-4653	750	21	2	2	NUM
ejpam-4653	750	22	,	,	PUNCT
ejpam-4653	750	23	claim	claim	NOUN
ejpam-4653	750	24	(	(	PUNCT
ejpam-4653	750	25	ii	ii	NOUN
ejpam-4653	750	26	)	)	PUNCT
ejpam-4653	750	27	implies	imply	VERB
ejpam-4653	750	28	that	that	SCONJ
ejpam-4653	750	29	there	there	PRON
ejpam-4653	750	30	exists	exist	VERB
ejpam-4653	750	31	w	w	PROPN
ejpam-4653	750	32	∈	∈	PROPN
ejpam-4653	750	33	v	v	NOUN
ejpam-4653	750	34	(	(	PUNCT
ejpam-4653	750	35	kp	kp	PROPN
ejpam-4653	750	36	)	)	PUNCT
ejpam-4653	750	37	for	for	ADP
ejpam-4653	750	38	which	which	PRON
ejpam-4653	750	39	(	(	PUNCT
ejpam-4653	750	40	x	x	NOUN
ejpam-4653	750	41	,	,	PUNCT
ejpam-4653	750	42	w	w	NOUN
ejpam-4653	750	43	)	)	PUNCT
ejpam-4653	750	44	∈	∈	PROPN
ejpam-4653	750	45	v0	v0	NOUN
ejpam-4653	750	46	.	.	PUNCT
ejpam-4653	751	1	thus	thus	ADV
ejpam-4653	751	2	,	,	PUNCT
ejpam-4653	751	3	there	there	PRON
ejpam-4653	751	4	exists	exist	VERB
ejpam-4653	751	5	(	(	PUNCT
ejpam-4653	751	6	a	a	DET
ejpam-4653	751	7	,	,	PUNCT
ejpam-4653	751	8	b	b	NOUN
ejpam-4653	751	9	)	)	PUNCT
ejpam-4653	751	10	∈	∈	PROPN
ejpam-4653	751	11	v3	v3	PROPN
ejpam-4653	751	12	∩	∩	PROPN
ejpam-4653	751	13	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	751	14	,	,	PUNCT
ejpam-4653	751	15	w	w	NOUN
ejpam-4653	751	16	)	)	PUNCT
ejpam-4653	751	17	)	)	PUNCT
ejpam-4653	751	18	or	or	CCONJ
ejpam-4653	751	19	there	there	PRON
ejpam-4653	751	20	exist	exist	VERB
ejpam-4653	751	21	distinct	distinct	ADJ
ejpam-4653	751	22	(	(	PUNCT
ejpam-4653	751	23	c	c	NOUN
ejpam-4653	751	24	,	,	PUNCT
ejpam-4653	751	25	d	d	NOUN
ejpam-4653	751	26	)	)	PUNCT
ejpam-4653	751	27	,	,	PUNCT
ejpam-4653	751	28	(	(	PUNCT
ejpam-4653	751	29	e	e	NOUN
ejpam-4653	751	30	,	,	PUNCT
ejpam-4653	751	31	f	f	X
ejpam-4653	751	32	)	)	PUNCT
ejpam-4653	751	33	∈	∈	PROPN
ejpam-4653	751	34	v2	v2	PROPN
ejpam-4653	751	35	∩	∩	NOUN
ejpam-4653	751	36	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	751	37	,	,	PUNCT
ejpam-4653	751	38	w	w	NOUN
ejpam-4653	751	39	)	)	PUNCT
ejpam-4653	751	40	)	)	PUNCT
ejpam-4653	751	41	.	.	PUNCT
ejpam-4653	752	1	if	if	SCONJ
ejpam-4653	752	2	the	the	DET
ejpam-4653	752	3	former	former	ADJ
ejpam-4653	752	4	holds	hold	VERB
ejpam-4653	752	5	,	,	PUNCT
ejpam-4653	752	6	then	then	ADV
ejpam-4653	752	7	(	(	PUNCT
ejpam-4653	752	8	a	a	PRON
ejpam-4653	752	9	,	,	PUNCT
ejpam-4653	752	10	v	v	NOUN
ejpam-4653	752	11	)	)	PUNCT
ejpam-4653	752	12	∈	∈	NOUN
ejpam-4653	752	13	v	v	ADP
ejpam-4653	752	14	∗	∗	X
ejpam-4653	752	15	3	3	NUM
ejpam-4653	752	16	∩ng[kp]((x	∩ng[kp]((x	NOUN
ejpam-4653	752	17	,	,	PUNCT
ejpam-4653	752	18	y	y	NOUN
ejpam-4653	752	19	)	)	PUNCT
ejpam-4653	752	20	)	)	PUNCT
ejpam-4653	752	21	.	.	PUNCT
ejpam-4653	753	1	suppose	suppose	VERB
ejpam-4653	753	2	the	the	DET
ejpam-4653	753	3	latter	latter	ADJ
ejpam-4653	753	4	holds	hold	VERB
ejpam-4653	753	5	.	.	PUNCT
ejpam-4653	754	1	by	by	ADP
ejpam-4653	754	2	claim	claim	NOUN
ejpam-4653	754	3	(	(	PUNCT
ejpam-4653	754	4	ii	ii	NOUN
ejpam-4653	754	5	)	)	PUNCT
ejpam-4653	754	6	,	,	PUNCT
ejpam-4653	754	7	c	c	PROPN
ejpam-4653	754	8	̸=	̸=	PROPN
ejpam-4653	754	9	e	e	NOUN
ejpam-4653	754	10	so	so	SCONJ
ejpam-4653	754	11	that	that	SCONJ
ejpam-4653	754	12	we	we	PRON
ejpam-4653	754	13	have	have	VERB
ejpam-4653	754	14	distinct	distinct	ADJ
ejpam-4653	754	15	points	point	NOUN
ejpam-4653	754	16	(	(	PUNCT
ejpam-4653	754	17	c	c	X
ejpam-4653	754	18	,	,	PUNCT
ejpam-4653	754	19	v	v	NOUN
ejpam-4653	754	20	)	)	PUNCT
ejpam-4653	754	21	,	,	PUNCT
ejpam-4653	754	22	(	(	PUNCT
ejpam-4653	754	23	e	e	NOUN
ejpam-4653	754	24	,	,	PUNCT
ejpam-4653	754	25	v	v	NOUN
ejpam-4653	754	26	)	)	PUNCT
ejpam-4653	754	27	∈	∈	NOUN
ejpam-4653	754	28	v	v	ADP
ejpam-4653	754	29	∗	∗	X
ejpam-4653	754	30	2	2	NUM
ejpam-4653	754	31	∩ng[kp]((x	∩ng[kp]((x	X
ejpam-4653	754	32	,	,	PUNCT
ejpam-4653	754	33	y	y	NOUN
ejpam-4653	754	34	)	)	PUNCT
ejpam-4653	754	35	)	)	PUNCT
ejpam-4653	754	36	.	.	PUNCT
ejpam-4653	755	1	we	we	PRON
ejpam-4653	755	2	note	note	VERB
ejpam-4653	755	3	here	here	ADV
ejpam-4653	755	4	that	that	SCONJ
ejpam-4653	755	5	it	it	PRON
ejpam-4653	755	6	is	be	AUX
ejpam-4653	755	7	possible	possible	ADJ
ejpam-4653	755	8	to	to	PART
ejpam-4653	755	9	have	have	VERB
ejpam-4653	755	10	x	x	X
ejpam-4653	755	11	=	=	SYM
ejpam-4653	755	12	c	c	PROPN
ejpam-4653	755	13	or	or	CCONJ
ejpam-4653	755	14	x	x	X
ejpam-4653	755	15	=	=	PUNCT
ejpam-4653	755	16	e.	e.	PROPN
ejpam-4653	755	17	case	case	NOUN
ejpam-4653	755	18	2	2	NUM
ejpam-4653	755	19	:	:	PUNCT
ejpam-4653	755	20	assume	assume	VERB
ejpam-4653	755	21	x	x	X
ejpam-4653	755	22	∈	∈	PROPN
ejpam-4653	755	23	b	b	PROPN
ejpam-4653	755	24	and	and	CCONJ
ejpam-4653	755	25	y	y	PROPN
ejpam-4653	755	26	̸=	̸=	PROPN
ejpam-4653	755	27	v.	v.	CCONJ
ejpam-4653	755	28	then	then	ADV
ejpam-4653	755	29	(	(	PUNCT
ejpam-4653	755	30	x	x	NOUN
ejpam-4653	755	31	,	,	PUNCT
ejpam-4653	755	32	v	v	NOUN
ejpam-4653	755	33	)	)	PUNCT
ejpam-4653	755	34	∈	∈	NOUN
ejpam-4653	755	35	v	v	ADP
ejpam-4653	755	36	∗	∗	X
ejpam-4653	755	37	3	3	NUM
ejpam-4653	755	38	∩ng[kp]((x	∩ng[kp]((x	NOUN
ejpam-4653	755	39	,	,	PUNCT
ejpam-4653	755	40	y	y	NOUN
ejpam-4653	755	41	)	)	PUNCT
ejpam-4653	755	42	)	)	PUNCT
ejpam-4653	755	43	.	.	PUNCT
ejpam-4653	756	1	case	case	NOUN
ejpam-4653	756	2	3	3	X
ejpam-4653	756	3	:	:	PUNCT
ejpam-4653	756	4	assume	assume	VERB
ejpam-4653	756	5	x	x	X
ejpam-4653	756	6	∈	∈	PROPN
ejpam-4653	756	7	(	(	PUNCT
ejpam-4653	756	8	v0)g	v0)g	NOUN
ejpam-4653	756	9	\	\	PROPN
ejpam-4653	756	10	(	(	PUNCT
ejpam-4653	756	11	a	a	DET
ejpam-4653	756	12	∪	∪	X
ejpam-4653	756	13	b	b	NOUN
ejpam-4653	756	14	)	)	PUNCT
ejpam-4653	756	15	.	.	PUNCT
ejpam-4653	757	1	then	then	ADV
ejpam-4653	757	2	(	(	PUNCT
ejpam-4653	757	3	x	x	NOUN
ejpam-4653	757	4	,	,	PUNCT
ejpam-4653	757	5	w	w	NOUN
ejpam-4653	757	6	)	)	PUNCT
ejpam-4653	757	7	∈	∈	PROPN
ejpam-4653	757	8	v0	v0	NOUN
ejpam-4653	757	9	for	for	ADP
ejpam-4653	757	10	all	all	DET
ejpam-4653	757	11	w	w	PROPN
ejpam-4653	757	12	∈	∈	PROPN
ejpam-4653	757	13	v	v	NOUN
ejpam-4653	757	14	(	(	PUNCT
ejpam-4653	757	15	kp	kp	PROPN
ejpam-4653	757	16	)	)	PUNCT
ejpam-4653	757	17	.	.	PUNCT
ejpam-4653	758	1	since	since	SCONJ
ejpam-4653	758	2	f	f	PROPN
ejpam-4653	758	3	∈	∈	PROPN
ejpam-4653	758	4	drd(g[kp	drd(g[kp	PROPN
ejpam-4653	758	5	]	]	X
ejpam-4653	758	6	)	)	PUNCT
ejpam-4653	758	7	,	,	PUNCT
ejpam-4653	758	8	there	there	PRON
ejpam-4653	758	9	exists	exist	VERB
ejpam-4653	758	10	(	(	PUNCT
ejpam-4653	758	11	a	a	DET
ejpam-4653	758	12	,	,	PUNCT
ejpam-4653	758	13	b	b	NOUN
ejpam-4653	758	14	)	)	PUNCT
ejpam-4653	758	15	∈	∈	PROPN
ejpam-4653	758	16	v3	v3	PROPN
ejpam-4653	758	17	∩ng[kp]((x	∩ng[kp]((x	PROPN
ejpam-4653	758	18	,	,	PUNCT
ejpam-4653	758	19	y	y	NOUN
ejpam-4653	758	20	)	)	PUNCT
ejpam-4653	758	21	)	)	PUNCT
ejpam-4653	758	22	or	or	CCONJ
ejpam-4653	758	23	there	there	PRON
ejpam-4653	758	24	exist	exist	VERB
ejpam-4653	758	25	distinct	distinct	ADJ
ejpam-4653	758	26	(	(	PUNCT
ejpam-4653	758	27	c	c	NOUN
ejpam-4653	758	28	,	,	PUNCT
ejpam-4653	758	29	d	d	NOUN
ejpam-4653	758	30	)	)	PUNCT
ejpam-4653	758	31	,	,	PUNCT
ejpam-4653	758	32	(	(	PUNCT
ejpam-4653	758	33	e	e	NOUN
ejpam-4653	758	34	,	,	PUNCT
ejpam-4653	758	35	f	f	X
ejpam-4653	758	36	)	)	PUNCT
ejpam-4653	758	37	∈	∈	PROPN
ejpam-4653	758	38	v2	v2	PROPN
ejpam-4653	758	39	∩	∩	NOUN
ejpam-4653	758	40	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	758	41	,	,	PUNCT
ejpam-4653	758	42	y	y	NOUN
ejpam-4653	758	43	)	)	PUNCT
ejpam-4653	758	44	)	)	PUNCT
ejpam-4653	758	45	.	.	PUNCT
ejpam-4653	759	1	if	if	SCONJ
ejpam-4653	759	2	the	the	DET
ejpam-4653	759	3	former	former	ADJ
ejpam-4653	759	4	holds	hold	VERB
ejpam-4653	759	5	,	,	PUNCT
ejpam-4653	759	6	then	then	ADV
ejpam-4653	759	7	(	(	PUNCT
ejpam-4653	759	8	a	a	PRON
ejpam-4653	759	9	,	,	PUNCT
ejpam-4653	759	10	v	v	NOUN
ejpam-4653	759	11	)	)	PUNCT
ejpam-4653	759	12	∈	∈	NOUN
ejpam-4653	759	13	v	v	ADP
ejpam-4653	759	14	∗	∗	X
ejpam-4653	759	15	3	3	NUM
ejpam-4653	759	16	∩	∩	NOUN
ejpam-4653	759	17	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	759	18	,	,	PUNCT
ejpam-4653	759	19	y	y	NOUN
ejpam-4653	759	20	)	)	PUNCT
ejpam-4653	759	21	)	)	PUNCT
ejpam-4653	759	22	.	.	PUNCT
ejpam-4653	760	1	suppose	suppose	VERB
ejpam-4653	760	2	the	the	DET
ejpam-4653	760	3	latter	latter	ADJ
ejpam-4653	760	4	holds	hold	VERB
ejpam-4653	760	5	.	.	PUNCT
ejpam-4653	761	1	by	by	ADP
ejpam-4653	761	2	claim	claim	NOUN
ejpam-4653	761	3	(	(	PUNCT
ejpam-4653	761	4	ii	ii	NOUN
ejpam-4653	761	5	)	)	PUNCT
ejpam-4653	761	6	,	,	PUNCT
ejpam-4653	761	7	x	x	X
ejpam-4653	761	8	,	,	PUNCT
ejpam-4653	761	9	c	c	PROPN
ejpam-4653	761	10	and	and	CCONJ
ejpam-4653	761	11	e	e	PROPN
ejpam-4653	761	12	are	be	AUX
ejpam-4653	761	13	distinct	distinct	ADJ
ejpam-4653	761	14	vertices	vertex	NOUN
ejpam-4653	761	15	of	of	ADP
ejpam-4653	761	16	g	g	PROPN
ejpam-4653	761	17	and	and	CCONJ
ejpam-4653	761	18	(	(	PUNCT
ejpam-4653	761	19	c	c	NOUN
ejpam-4653	761	20	,	,	PUNCT
ejpam-4653	761	21	v	v	NOUN
ejpam-4653	761	22	)	)	PUNCT
ejpam-4653	761	23	,	,	PUNCT
ejpam-4653	761	24	(	(	PUNCT
ejpam-4653	761	25	e	e	NOUN
ejpam-4653	761	26	,	,	PUNCT
ejpam-4653	761	27	v	v	NOUN
ejpam-4653	761	28	)	)	PUNCT
ejpam-4653	761	29	∈	∈	NOUN
ejpam-4653	761	30	v	v	ADP
ejpam-4653	761	31	∗	∗	X
ejpam-4653	761	32	2	2	NUM
ejpam-4653	761	33	∩ng[kp]((x	∩ng[kp]((x	X
ejpam-4653	761	34	,	,	PUNCT
ejpam-4653	761	35	y	y	NOUN
ejpam-4653	761	36	)	)	PUNCT
ejpam-4653	761	37	)	)	PUNCT
ejpam-4653	761	38	.	.	PUNCT
ejpam-4653	762	1	all	all	PRON
ejpam-4653	762	2	of	of	ADP
ejpam-4653	762	3	the	the	DET
ejpam-4653	762	4	above	above	ADJ
ejpam-4653	762	5	imply	imply	VERB
ejpam-4653	762	6	that	that	SCONJ
ejpam-4653	762	7	g	g	PROPN
ejpam-4653	762	8	∈	∈	PROPN
ejpam-4653	762	9	drd(g[kp	drd(g[kp	PROPN
ejpam-4653	762	10	]	]	PUNCT
ejpam-4653	762	11	)	)	PUNCT
ejpam-4653	762	12	.	.	PUNCT
ejpam-4653	763	1	since	since	SCONJ
ejpam-4653	763	2	f	f	PROPN
ejpam-4653	763	3	is	be	AUX
ejpam-4653	763	4	a	a	DET
ejpam-4653	763	5	γdr	γdr	NOUN
ejpam-4653	763	6	-	-	PUNCT
ejpam-4653	763	7	function	function	NOUN
ejpam-4653	763	8	,	,	PUNCT
ejpam-4653	763	9	ωg[kp](g	ωg[kp](g	NOUN
ejpam-4653	763	10	)	)	PUNCT
ejpam-4653	763	11	=	=	SYM
ejpam-4653	763	12	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	763	13	)	)	PUNCT
ejpam-4653	763	14	.	.	PUNCT
ejpam-4653	764	1	thus	thus	ADV
ejpam-4653	764	2	,	,	PUNCT
ejpam-4653	764	3	ωg[kp](f	ωg[kp](f	NOUN
ejpam-4653	764	4	)	)	PUNCT
ejpam-4653	764	5	≥	≥	NOUN
ejpam-4653	764	6	ωg[kp](g	ωg[kp](g	NOUN
ejpam-4653	764	7	)	)	PUNCT
ejpam-4653	764	8	=	=	SYM
ejpam-4653	765	1	2|a|+	2|a|+	NUM
ejpam-4653	765	2	3|b|	3|b|	NUM
ejpam-4653	765	3	.	.	PUNCT
ejpam-4653	766	1	now	now	ADV
ejpam-4653	766	2	consider	consider	VERB
ejpam-4653	766	3	the	the	DET
ejpam-4653	766	4	function	function	NOUN
ejpam-4653	766	5	h	h	NOUN
ejpam-4653	766	6	=	=	SYM
ejpam-4653	766	7	(	(	PUNCT
ejpam-4653	766	8	c,∅	c,∅	PROPN
ejpam-4653	766	9	,	,	PUNCT
ejpam-4653	766	10	a	a	DET
ejpam-4653	766	11	,	,	PUNCT
ejpam-4653	766	12	b	b	NOUN
ejpam-4653	766	13	)	)	PUNCT
ejpam-4653	766	14	on	on	ADP
ejpam-4653	766	15	v	v	ADP
ejpam-4653	766	16	(	(	PUNCT
ejpam-4653	766	17	g	g	NOUN
ejpam-4653	766	18	)	)	PUNCT
ejpam-4653	766	19	.	.	PUNCT
ejpam-4653	767	1	let	let	VERB
ejpam-4653	767	2	x	x	SYM
ejpam-4653	767	3	∈	∈	PROPN
ejpam-4653	767	4	c.	c.	NOUN
ejpam-4653	767	5	then	then	ADV
ejpam-4653	767	6	,	,	PUNCT
ejpam-4653	767	7	in	in	ADP
ejpam-4653	767	8	particular	particular	ADJ
ejpam-4653	767	9	,	,	PUNCT
ejpam-4653	767	10	(	(	PUNCT
ejpam-4653	767	11	x	x	NOUN
ejpam-4653	767	12	,	,	PUNCT
ejpam-4653	767	13	v	v	NOUN
ejpam-4653	767	14	)	)	PUNCT
ejpam-4653	767	15	∈	∈	NOUN
ejpam-4653	767	16	v	v	ADP
ejpam-4653	767	17	∗	∗	NOUN
ejpam-4653	767	18	0	0	NUM
ejpam-4653	767	19	.	.	PUNCT
ejpam-4653	768	1	thus	thus	ADV
ejpam-4653	768	2	,	,	PUNCT
ejpam-4653	768	3	there	there	PRON
ejpam-4653	768	4	exists	exist	VERB
ejpam-4653	768	5	u	u	PROPN
ejpam-4653	768	6	∈	∈	PROPN
ejpam-4653	768	7	b	b	PROPN
ejpam-4653	768	8	such	such	ADJ
ejpam-4653	768	9	that	that	PRON
ejpam-4653	768	10	(	(	PUNCT
ejpam-4653	768	11	u	u	NOUN
ejpam-4653	768	12	,	,	PUNCT
ejpam-4653	768	13	v	v	NOUN
ejpam-4653	768	14	)	)	PUNCT
ejpam-4653	768	15	∈	∈	PROPN
ejpam-4653	768	16	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	768	17	,	,	PUNCT
ejpam-4653	768	18	v	v	NOUN
ejpam-4653	768	19	)	)	PUNCT
ejpam-4653	768	20	)	)	PUNCT
ejpam-4653	768	21	or	or	CCONJ
ejpam-4653	768	22	there	there	PRON
ejpam-4653	768	23	exist	exist	VERB
ejpam-4653	768	24	distinct	distinct	ADJ
ejpam-4653	768	25	w	w	NOUN
ejpam-4653	768	26	,	,	PUNCT
ejpam-4653	768	27	z	z	PROPN
ejpam-4653	768	28	∈	∈	PROPN
ejpam-4653	768	29	a	a	PRON
ejpam-4653	768	30	for	for	ADP
ejpam-4653	768	31	which	which	PRON
ejpam-4653	768	32	(	(	PUNCT
ejpam-4653	768	33	w	w	PROPN
ejpam-4653	768	34	,	,	PUNCT
ejpam-4653	768	35	v	v	NOUN
ejpam-4653	768	36	)	)	PUNCT
ejpam-4653	768	37	,	,	PUNCT
ejpam-4653	768	38	(	(	PUNCT
ejpam-4653	768	39	z	z	NOUN
ejpam-4653	768	40	,	,	PUNCT
ejpam-4653	768	41	v	v	NOUN
ejpam-4653	768	42	)	)	PUNCT
ejpam-4653	768	43	∈	∈	PROPN
ejpam-4653	768	44	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	768	45	,	,	PUNCT
ejpam-4653	768	46	v	v	NOUN
ejpam-4653	768	47	)	)	PUNCT
ejpam-4653	768	48	)	)	PUNCT
ejpam-4653	768	49	.	.	PUNCT
ejpam-4653	769	1	this	this	PRON
ejpam-4653	769	2	means	mean	VERB
ejpam-4653	769	3	that	that	SCONJ
ejpam-4653	769	4	there	there	PRON
ejpam-4653	769	5	exists	exist	VERB
ejpam-4653	769	6	u	u	PROPN
ejpam-4653	769	7	∈	∈	PROPN
ejpam-4653	769	8	b	b	PROPN
ejpam-4653	769	9	∩	∩	NOUN
ejpam-4653	769	10	ng(x	ng(x	NUM
ejpam-4653	769	11	)	)	PUNCT
ejpam-4653	769	12	or	or	CCONJ
ejpam-4653	769	13	there	there	PRON
ejpam-4653	769	14	exist	exist	VERB
ejpam-4653	769	15	distinct	distinct	ADJ
ejpam-4653	769	16	w	w	NOUN
ejpam-4653	769	17	,	,	PUNCT
ejpam-4653	769	18	z	z	PROPN
ejpam-4653	769	19	∈	∈	PROPN
ejpam-4653	769	20	a	a	DET
ejpam-4653	769	21	∩	∩	NOUN
ejpam-4653	769	22	ng(x	ng(x	NUM
ejpam-4653	769	23	)	)	PUNCT
ejpam-4653	769	24	.	.	PUNCT
ejpam-4653	770	1	therefore	therefore	ADV
ejpam-4653	770	2	,	,	PUNCT
ejpam-4653	770	3	h	h	PROPN
ejpam-4653	770	4	∈	∈	PROPN
ejpam-4653	770	5	drd(g	drd(g	PROPN
ejpam-4653	770	6	)	)	PUNCT
ejpam-4653	770	7	with	with	ADP
ejpam-4653	770	8	ωg(h	ωg(h	NOUN
ejpam-4653	770	9	)	)	PUNCT
ejpam-4653	770	10	=	=	SYM
ejpam-4653	771	1	2|a|+3|b|	2|a|+3|b|	X
ejpam-4653	771	2	.	.	PUNCT
ejpam-4653	772	1	let	let	VERB
ejpam-4653	772	2	x	x	X
ejpam-4653	772	3	∈	∈	PROPN
ejpam-4653	772	4	a\ng(a∪b	a\ng(a∪b	NUM
ejpam-4653	772	5	)	)	PUNCT
ejpam-4653	772	6	,	,	PUNCT
ejpam-4653	772	7	and	and	CCONJ
ejpam-4653	772	8	pick	pick	VERB
ejpam-4653	772	9	y	y	PROPN
ejpam-4653	772	10	∈	∈	PROPN
ejpam-4653	772	11	v	v	PROPN
ejpam-4653	772	12	(	(	PUNCT
ejpam-4653	772	13	kp)\{v	kp)\{v	NOUN
ejpam-4653	772	14	}	}	PUNCT
ejpam-4653	772	15	.	.	PUNCT
ejpam-4653	773	1	then	then	ADV
ejpam-4653	773	2	(	(	PUNCT
ejpam-4653	773	3	x	x	X
ejpam-4653	773	4	,	,	PUNCT
ejpam-4653	773	5	y	y	NOUN
ejpam-4653	773	6	)	)	PUNCT
ejpam-4653	773	7	∈	∈	PROPN
ejpam-4653	773	8	v	v	ADP
ejpam-4653	773	9	∗	∗	NOUN
ejpam-4653	773	10	0	0	NUM
ejpam-4653	773	11	.	.	PUNCT
ejpam-4653	774	1	in	in	ADP
ejpam-4653	774	2	view	view	NOUN
ejpam-4653	774	3	of	of	ADP
ejpam-4653	774	4	claim(iii	claim(iii	PROPN
ejpam-4653	774	5	)	)	PUNCT
ejpam-4653	774	6	,	,	PUNCT
ejpam-4653	774	7	there	there	PRON
ejpam-4653	774	8	exists	exist	VERB
ejpam-4653	774	9	(	(	PUNCT
ejpam-4653	774	10	w	w	PROPN
ejpam-4653	774	11	,	,	PUNCT
ejpam-4653	774	12	z	z	NOUN
ejpam-4653	774	13	)	)	PUNCT
ejpam-4653	774	14	∈	∈	NOUN
ejpam-4653	774	15	v	v	ADP
ejpam-4653	774	16	∗	∗	NOUN
ejpam-4653	774	17	2	2	NUM
ejpam-4653	774	18	∩	∩	NOUN
ejpam-4653	774	19	ng[kp]((x	ng[kp]((x	NUM
ejpam-4653	774	20	,	,	PUNCT
ejpam-4653	774	21	y	y	NOUN
ejpam-4653	774	22	)	)	PUNCT
ejpam-4653	774	23	)	)	PUNCT
ejpam-4653	774	24	.	.	PUNCT
ejpam-4653	775	1	this	this	PRON
ejpam-4653	775	2	means	mean	VERB
ejpam-4653	775	3	that	that	SCONJ
ejpam-4653	775	4	either	either	CCONJ
ejpam-4653	775	5	w	w	X
ejpam-4653	775	6	=	=	PUNCT
ejpam-4653	775	7	x	x	X
ejpam-4653	775	8	or	or	CCONJ
ejpam-4653	775	9	w	w	PROPN
ejpam-4653	775	10	∈	∈	PROPN
ejpam-4653	775	11	ng(x	ng(x	NUM
ejpam-4653	775	12	)	)	PUNCT
ejpam-4653	775	13	,	,	PUNCT
ejpam-4653	775	14	a	a	DET
ejpam-4653	775	15	contradiction	contradiction	NOUN
ejpam-4653	775	16	.	.	PUNCT
ejpam-4653	776	1	thus	thus	ADV
ejpam-4653	776	2	,	,	PUNCT
ejpam-4653	776	3	a	a	DET
ejpam-4653	776	4	\	\	PROPN
ejpam-4653	776	5	ng(a	ng(a	NOUN
ejpam-4653	776	6	∪	∪	X
ejpam-4653	776	7	b	b	NOUN
ejpam-4653	776	8	)	)	PUNCT
ejpam-4653	776	9	=	=	NOUN
ejpam-4653	776	10	∅	∅	NOUN
ejpam-4653	776	11	and	and	CCONJ
ejpam-4653	776	12	h	h	NOUN
ejpam-4653	776	13	∈	∈	PROPN
ejpam-4653	776	14	cg	cg	NOUN
ejpam-4653	776	15	.	.	PUNCT
ejpam-4653	777	1	finally	finally	ADV
ejpam-4653	777	2	,	,	PUNCT
ejpam-4653	777	3	therefore	therefore	ADV
ejpam-4653	777	4	,	,	PUNCT
ejpam-4653	777	5	γdr(g[kp	γdr(g[kp	PROPN
ejpam-4653	777	6	]	]	PUNCT
ejpam-4653	777	7	)	)	PUNCT
ejpam-4653	777	8	≥	≥	NOUN
ejpam-4653	777	9	ωg(h	ωg(h	ADV
ejpam-4653	777	10	)	)	PUNCT
ejpam-4653	777	11	≥	≥	PROPN
ejpam-4653	777	12	α	α	X
ejpam-4653	777	13	.	.	PUNCT
ejpam-4653	778	1	acknowledgements	acknowledgement	NOUN
ejpam-4653	778	2	the	the	DET
ejpam-4653	778	3	authors	author	NOUN
ejpam-4653	778	4	would	would	AUX
ejpam-4653	778	5	like	like	VERB
ejpam-4653	778	6	to	to	PART
ejpam-4653	778	7	thank	thank	VERB
ejpam-4653	778	8	the	the	DET
ejpam-4653	778	9	referees	referee	NOUN
ejpam-4653	778	10	for	for	ADP
ejpam-4653	778	11	the	the	DET
ejpam-4653	778	12	invaluable	invaluable	ADJ
ejpam-4653	778	13	assistance	assistance	NOUN
ejpam-4653	778	14	they	they	PRON
ejpam-4653	778	15	gave	give	VERB
ejpam-4653	778	16	us	we	PRON
ejpam-4653	778	17	through	through	ADP
ejpam-4653	778	18	their	their	PRON
ejpam-4653	778	19	comments	comment	NOUN
ejpam-4653	778	20	and	and	CCONJ
ejpam-4653	778	21	suggestions	suggestion	NOUN
ejpam-4653	778	22	which	which	PRON
ejpam-4653	778	23	led	lead	VERB
ejpam-4653	778	24	to	to	ADP
ejpam-4653	778	25	the	the	DET
ejpam-4653	778	26	improvement	improvement	NOUN
ejpam-4653	778	27	of	of	ADP
ejpam-4653	778	28	the	the	DET
ejpam-4653	778	29	paper	paper	NOUN
ejpam-4653	778	30	.	.	PUNCT
ejpam-4653	779	1	also	also	ADV
ejpam-4653	779	2	,	,	PUNCT
ejpam-4653	779	3	the	the	DET
ejpam-4653	779	4	authors	author	NOUN
ejpam-4653	779	5	would	would	AUX
ejpam-4653	779	6	like	like	VERB
ejpam-4653	779	7	to	to	PART
ejpam-4653	779	8	thank	thank	VERB
ejpam-4653	779	9	the	the	DET
ejpam-4653	779	10	department	department	NOUN
ejpam-4653	779	11	of	of	ADP
ejpam-4653	779	12	science	science	NOUN
ejpam-4653	779	13	and	and	CCONJ
ejpam-4653	779	14	technology	technology	NOUN
ejpam-4653	779	15	accelerated	accelerate	VERB
ejpam-4653	779	16	science	science	NOUN
ejpam-4653	779	17	and	and	CCONJ
ejpam-4653	779	18	technology	technology	NOUN
ejpam-4653	779	19	human	human	ADJ
ejpam-4653	779	20	resource	resource	NOUN
ejpam-4653	779	21	development	development	NOUN
ejpam-4653	779	22	program	program	NOUN
ejpam-4653	779	23	(	(	PUNCT
ejpam-4653	779	24	dost	dost	NOUN
ejpam-4653	779	25	-	-	PUNCT
ejpam-4653	779	26	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-4653	779	27	,	,	PUNCT
ejpam-4653	779	28	and	and	CCONJ
ejpam-4653	779	29	msu	msu	PROPN
ejpam-4653	779	30	-	-	PUNCT
ejpam-4653	779	31	iligan	iligan	PROPN
ejpam-4653	779	32	institute	institute	PROPN
ejpam-4653	779	33	of	of	ADP
ejpam-4653	779	34	technology	technology	NOUN
ejpam-4653	779	35	for	for	ADP
ejpam-4653	779	36	funding	fund	VERB
ejpam-4653	779	37	this	this	DET
ejpam-4653	779	38	research	research	NOUN
ejpam-4653	779	39	.	.	PUNCT
ejpam-4653	780	1	references	reference	NOUN
ejpam-4653	780	2	[	[	X
ejpam-4653	780	3	1	1	X
ejpam-4653	780	4	]	]	PUNCT
ejpam-4653	780	5	s.	s.	PROPN
ejpam-4653	780	6	arumugam	arumugam	PROPN
ejpam-4653	780	7	and	and	CCONJ
ejpam-4653	780	8	k.	k.	PROPN
ejpam-4653	780	9	karuppasamy	karuppasamy	PROPN
ejpam-4653	780	10	.	.	PUNCT
ejpam-4653	781	1	fractional	fractional	ADJ
ejpam-4653	781	2	global	global	ADJ
ejpam-4653	781	3	domination	domination	NOUN
ejpam-4653	781	4	in	in	ADP
ejpam-4653	781	5	graphs	graph	NOUN
ejpam-4653	781	6	.	.	PUNCT
ejpam-4653	782	1	discussiones	discussione	NOUN
ejpam-4653	782	2	mathematicae	mathematicae	PROPN
ejpam-4653	782	3	graph	graph	NOUN
ejpam-4653	782	4	theory	theory	NOUN
ejpam-4653	782	5	.	.	PUNCT
ejpam-4653	782	6	,	,	PUNCT
ejpam-4653	782	7	30:33–34	30:33–34	NUM
ejpam-4653	782	8	,	,	PUNCT
ejpam-4653	782	9	2010	2010	NUM
ejpam-4653	782	10	.	.	PUNCT
ejpam-4653	783	1	[	[	X
ejpam-4653	783	2	2	2	NUM
ejpam-4653	783	3	]	]	X
ejpam-4653	783	4	r.a	r.a	PROPN
ejpam-4653	783	5	.	.	PROPN
ejpam-4653	783	6	beeler	beeler	PROPN
ejpam-4653	783	7	,	,	PUNCT
ejpam-4653	783	8	t.w	t.w	PROPN
ejpam-4653	783	9	.	.	PROPN
ejpam-4653	783	10	haynes	haynes	PROPN
ejpam-4653	783	11	,	,	PUNCT
ejpam-4653	783	12	and	and	CCONJ
ejpam-4653	783	13	s.t	s.t	PROPN
ejpam-4653	783	14	.	.	PROPN
ejpam-4653	783	15	hedetniemi	hedetniemi	PROPN
ejpam-4653	783	16	.	.	PUNCT
ejpam-4653	784	1	double	double	ADJ
ejpam-4653	784	2	roman	roman	ADJ
ejpam-4653	784	3	domination	domination	NOUN
ejpam-4653	784	4	.	.	PUNCT
ejpam-4653	785	1	discrete	discrete	ADJ
ejpam-4653	785	2	appl	appl	PROPN
ejpam-4653	785	3	.	.	PUNCT
ejpam-4653	785	4	math	math	PROPN
ejpam-4653	785	5	.	.	PUNCT
ejpam-4653	785	6	,	,	PUNCT
ejpam-4653	786	1	211:23–29	211:23–29	NUM
ejpam-4653	786	2	,	,	PUNCT
ejpam-4653	786	3	2016	2016	NUM
ejpam-4653	786	4	.	.	PUNCT
ejpam-4653	787	1	references	reference	NOUN
ejpam-4653	787	2	863	863	NUM
ejpam-4653	787	3	[	[	X
ejpam-4653	787	4	3	3	NUM
ejpam-4653	787	5	]	]	PUNCT
ejpam-4653	787	6	c.	c.	PROPN
ejpam-4653	787	7	berge	berge	PROPN
ejpam-4653	787	8	.	.	PUNCT
ejpam-4653	788	1	theory	theory	NOUN
ejpam-4653	788	2	of	of	ADP
ejpam-4653	788	3	graphs	graph	NOUN
ejpam-4653	788	4	and	and	CCONJ
ejpam-4653	788	5	its	its	PRON
ejpam-4653	788	6	applications	application	NOUN
ejpam-4653	788	7	.	.	PUNCT
ejpam-4653	789	1	discrete	discrete	ADJ
ejpam-4653	789	2	applied	apply	VERB
ejpam-4653	789	3	mathematics	mathematic	NOUN
ejpam-4653	789	4	.	.	PUNCT
ejpam-4653	789	5	,	,	PUNCT
ejpam-4653	789	6	1962	1962	NUM
ejpam-4653	789	7	.	.	PUNCT
ejpam-4653	790	1	[	[	X
ejpam-4653	790	2	4	4	NUM
ejpam-4653	790	3	]	]	X
ejpam-4653	790	4	f.	f.	PROPN
ejpam-4653	790	5	buckley	buckley	PROPN
ejpam-4653	790	6	and	and	CCONJ
ejpam-4653	790	7	f.	f.	PROPN
ejpam-4653	790	8	harary	harary	PROPN
ejpam-4653	790	9	.	.	PUNCT
ejpam-4653	791	1	distance	distance	NOUN
ejpam-4653	791	2	in	in	ADP
ejpam-4653	791	3	graphs	graph	NOUN
ejpam-4653	791	4	.	.	PUNCT
ejpam-4653	792	1	redwood	redwood	NOUN
ejpam-4653	792	2	city	city	NOUN
ejpam-4653	792	3	,	,	PUNCT
ejpam-4653	792	4	ca	ca	PROPN
ejpam-4653	792	5	:	:	PUNCT
ejpam-4653	792	6	addison	addison	PROPN
ejpam-4653	792	7	-	-	PUNCT
ejpam-4653	792	8	wesley	wesley	PROPN
ejpam-4653	792	9	.	.	PUNCT
ejpam-4653	792	10	,	,	PUNCT
ejpam-4653	792	11	1990	1990	NUM
ejpam-4653	792	12	.	.	PUNCT
ejpam-4653	793	1	[	[	X
ejpam-4653	793	2	5	5	X
ejpam-4653	793	3	]	]	PUNCT
ejpam-4653	793	4	e.	e.	PROPN
ejpam-4653	793	5	j.	j.	PROPN
ejpam-4653	793	6	cockayne	cockayne	PROPN
ejpam-4653	793	7	and	and	CCONJ
ejpam-4653	793	8	s.	s.	PROPN
ejpam-4653	793	9	t.	t.	PROPN
ejpam-4653	793	10	hedetniemi	hedetniemi	PROPN
ejpam-4653	793	11	.	.	PUNCT
ejpam-4653	794	1	towards	towards	ADP
ejpam-4653	794	2	a	a	DET
ejpam-4653	794	3	theory	theory	NOUN
ejpam-4653	794	4	of	of	ADP
ejpam-4653	794	5	domination	domination	NOUN
ejpam-4653	794	6	in	in	ADP
ejpam-4653	794	7	graphs	graph	NOUN
ejpam-4653	794	8	.	.	PUNCT
ejpam-4653	795	1	networks	network	NOUN
ejpam-4653	795	2	.	.	PUNCT
ejpam-4653	795	3	,	,	PUNCT
ejpam-4653	795	4	7:247–261	7:247–261	NUM
ejpam-4653	795	5	,	,	PUNCT
ejpam-4653	795	6	1997	1997	NUM
ejpam-4653	795	7	.	.	PUNCT
ejpam-4653	796	1	[	[	X
ejpam-4653	796	2	6	6	NUM
ejpam-4653	796	3	]	]	SYM
ejpam-4653	796	4	e.j	e.j	PROPN
ejpam-4653	796	5	.	.	PROPN
ejpam-4653	796	6	cockayne	cockayne	PROPN
ejpam-4653	796	7	,	,	PUNCT
ejpam-4653	796	8	p.m.	p.m.	NOUN
ejpam-4653	796	9	dreyer	dreyer	PROPN
ejpam-4653	796	10	sr	sr	PROPN
ejpam-4653	796	11	.	.	PROPN
ejpam-4653	796	12	,	,	PUNCT
ejpam-4653	796	13	s.m	s.m	PROPN
ejpam-4653	796	14	.	.	PROPN
ejpam-4653	796	15	hedetniemi	hedetniemi	PROPN
ejpam-4653	796	16	,	,	PUNCT
ejpam-4653	796	17	and	and	CCONJ
ejpam-4653	796	18	s.t	s.t	PROPN
ejpam-4653	796	19	.	.	PROPN
ejpam-4653	796	20	hedetniemi	hedetniemi	PROPN
ejpam-4653	796	21	.	.	PUNCT
ejpam-4653	797	1	roman	roman	ADJ
ejpam-4653	797	2	domination	domination	NOUN
ejpam-4653	797	3	in	in	ADP
ejpam-4653	797	4	graphs	graph	NOUN
ejpam-4653	797	5	.	.	PUNCT
ejpam-4653	798	1	discrete	discrete	ADJ
ejpam-4653	798	2	math	math	NOUN
ejpam-4653	798	3	.	.	PUNCT
ejpam-4653	798	4	,	,	PUNCT
ejpam-4653	798	5	278:11–22	278:11–22	NUM
ejpam-4653	798	6	,	,	PUNCT
ejpam-4653	798	7	2004	2004	NUM
ejpam-4653	798	8	.	.	PUNCT
ejpam-4653	799	1	[	[	X
ejpam-4653	799	2	7	7	X
ejpam-4653	799	3	]	]	X
ejpam-4653	799	4	j.f	j.f	PROPN
ejpam-4653	799	5	.	.	PROPN
ejpam-4653	799	6	fink	fink	PROPN
ejpam-4653	799	7	,	,	PUNCT
ejpam-4653	799	8	m.s	m.s	PROPN
ejpam-4653	799	9	.	.	PROPN
ejpam-4653	799	10	jacobson	jacobson	PROPN
ejpam-4653	799	11	,	,	PUNCT
ejpam-4653	799	12	and	and	CCONJ
ejpam-4653	799	13	in	in	ADP
ejpam-4653	799	14	:	:	PUNCT
ejpam-4653	799	15	y.	y.	PROPN
ejpam-4653	799	16	alavi	alavi	PROPN
ejpam-4653	799	17	et	et	PROPN
ejpam-4653	799	18	al	al	PROPN
ejpam-4653	799	19	.	.	PROPN
ejpam-4653	800	1	(	(	PUNCT
ejpam-4653	800	2	eds	ed	NOUN
ejpam-4653	800	3	.	.	PUNCT
ejpam-4653	800	4	)	)	PUNCT
ejpam-4653	801	1	n	n	CCONJ
ejpam-4653	801	2	-	-	PUNCT
ejpam-4653	801	3	domination	domination	NOUN
ejpam-4653	801	4	in	in	ADP
ejpam-4653	801	5	graphs	graph	NOUN
ejpam-4653	801	6	.	.	PUNCT
ejpam-4653	802	1	graph	graph	NOUN
ejpam-4653	802	2	theory	theory	NOUN
ejpam-4653	802	3	with	with	ADP
ejpam-4653	802	4	applications	application	NOUN
ejpam-4653	802	5	to	to	ADP
ejpam-4653	802	6	algorithms	algorithm	NOUN
ejpam-4653	802	7	and	and	CCONJ
ejpam-4653	802	8	computer	computer	NOUN
ejpam-4653	802	9	science	science	NOUN
ejpam-4653	802	10	.	.	PUNCT
ejpam-4653	803	1	wiley	wiley	PROPN
ejpam-4653	803	2	,	,	PUNCT
ejpam-4653	803	3	new	new	PROPN
ejpam-4653	803	4	york	york	PROPN
ejpam-4653	803	5	.	.	PROPN
ejpam-4653	803	6	,	,	PUNCT
ejpam-4653	803	7	pages	page	NOUN
ejpam-4653	803	8	283–300	283–300	NUM
ejpam-4653	803	9	,	,	PUNCT
ejpam-4653	803	10	1985	1985	NUM
ejpam-4653	803	11	.	.	PUNCT
ejpam-4653	804	1	[	[	X
ejpam-4653	804	2	8	8	NUM
ejpam-4653	804	3	]	]	PUNCT
ejpam-4653	804	4	b.	b.	PROPN
ejpam-4653	804	5	gayathri	gayathri	PROPN
ejpam-4653	804	6	and	and	CCONJ
ejpam-4653	804	7	s.	s.	PROPN
ejpam-4653	804	8	kaspar	kaspar	PROPN
ejpam-4653	804	9	.	.	PUNCT
ejpam-4653	805	1	connected	connect	VERB
ejpam-4653	805	2	co	co	ADJ
ejpam-4653	805	3	-	-	ADJ
ejpam-4653	805	4	independent	independent	ADJ
ejpam-4653	805	5	domination	domination	NOUN
ejpam-4653	805	6	of	of	ADP
ejpam-4653	805	7	a	a	DET
ejpam-4653	805	8	graph	graph	NOUN
ejpam-4653	805	9	.	.	PUNCT
ejpam-4653	806	1	int	int	NOUN
ejpam-4653	806	2	.	.	PUNCT
ejpam-4653	807	1	j.	j.	PROPN
ejpam-4653	807	2	contemp	contemp	PROPN
ejpam-4653	807	3	.	.	PUNCT
ejpam-4653	808	1	math	math	NOUN
ejpam-4653	808	2	.	.	PUNCT
ejpam-4653	809	1	sciences	sciences	PROPN
ejpam-4653	809	2	.	.	PUNCT
ejpam-4653	809	3	,	,	PUNCT
ejpam-4653	809	4	6:423–429	6:423–429	PROPN
ejpam-4653	809	5	,	,	PUNCT
ejpam-4653	809	6	2011	2011	NUM
ejpam-4653	809	7	.	.	PUNCT
ejpam-4653	810	1	[	[	X
ejpam-4653	810	2	9	9	NUM
ejpam-4653	810	3	]	]	SYM
ejpam-4653	810	4	s.m	s.m	PROPN
ejpam-4653	810	5	.	.	PROPN
ejpam-4653	810	6	sheikholeslami	sheikholeslami	PROPN
ejpam-4653	810	7	h.	h.	PROPN
ejpam-4653	810	8	abdollahzadeh	abdollahzadeh	PROPN
ejpam-4653	810	9	ahangar	ahangar	NOUN
ejpam-4653	810	10	,	,	PUNCT
ejpam-4653	810	11	m.	m.	NOUN
ejpam-4653	810	12	chellali	chellali	PROPN
ejpam-4653	810	13	.	.	PUNCT
ejpam-4653	811	1	on	on	ADP
ejpam-4653	811	2	the	the	DET
ejpam-4653	811	3	double	double	ADJ
ejpam-4653	811	4	roman	roman	ADJ
ejpam-4653	811	5	domination	domination	NOUN
ejpam-4653	811	6	in	in	ADP
ejpam-4653	811	7	graphs	graph	NOUN
ejpam-4653	811	8	.	.	PUNCT
ejpam-4653	812	1	discrete	discrete	ADJ
ejpam-4653	812	2	appl	appl	PROPN
ejpam-4653	812	3	.	.	PUNCT
ejpam-4653	812	4	math	math	PROPN
ejpam-4653	812	5	.	.	PUNCT
ejpam-4653	813	1	,	,	PUNCT
ejpam-4653	813	2	pages	page	NOUN
ejpam-4653	813	3	1–7	1–7	NUM
ejpam-4653	813	4	,	,	PUNCT
ejpam-4653	813	5	2017	2017	NUM
ejpam-4653	813	6	.	.	PUNCT
ejpam-4653	814	1	[	[	X
ejpam-4653	814	2	10	10	NUM
ejpam-4653	814	3	]	]	PUNCT
ejpam-4653	814	4	a.	a.	NOUN
ejpam-4653	814	5	hansberg	hansberg	PROPN
ejpam-4653	814	6	and	and	CCONJ
ejpam-4653	814	7	l.	l.	PROPN
ejpam-4653	814	8	volkmann	volkmann	PROPN
ejpam-4653	814	9	.	.	PUNCT
ejpam-4653	815	1	on	on	ADP
ejpam-4653	815	2	graphs	graph	NOUN
ejpam-4653	815	3	with	with	ADP
ejpam-4653	815	4	equal	equal	ADJ
ejpam-4653	815	5	domination	domination	NOUN
ejpam-4653	815	6	and	and	CCONJ
ejpam-4653	815	7	2	2	NUM
ejpam-4653	815	8	-	-	PUNCT
ejpam-4653	815	9	domination	domination	NOUN
ejpam-4653	815	10	numbers	number	NOUN
ejpam-4653	815	11	.	.	PUNCT
ejpam-4653	816	1	discrete	discrete	ADJ
ejpam-4653	816	2	mathematics	mathematic	NOUN
ejpam-4653	816	3	.	.	PUNCT
ejpam-4653	816	4	,	,	PUNCT
ejpam-4653	816	5	308(11):2277–2281	308(11):2277–2281	NUM
ejpam-4653	816	6	,	,	PUNCT
ejpam-4653	816	7	2008	2008	NUM
ejpam-4653	816	8	.	.	PUNCT
ejpam-4653	817	1	[	[	X
ejpam-4653	817	2	11	11	NUM
ejpam-4653	817	3	]	]	X
ejpam-4653	817	4	f.	f.	PROPN
ejpam-4653	817	5	harary	harary	PROPN
ejpam-4653	817	6	.	.	PUNCT
ejpam-4653	818	1	graph	graph	NOUN
ejpam-4653	818	2	theory	theory	NOUN
ejpam-4653	818	3	.	.	PUNCT
ejpam-4653	819	1	addison	addison	PROPN
ejpam-4653	819	2	-	-	PUNCT
ejpam-4653	819	3	wesley	wesley	PROPN
ejpam-4653	819	4	publication	publication	PROPN
ejpam-4653	819	5	company	company	PROPN
ejpam-4653	819	6	,	,	PUNCT
ejpam-4653	819	7	inc	inc	PROPN
ejpam-4653	819	8	.	.	PROPN
ejpam-4653	819	9	,	,	PUNCT
ejpam-4653	819	10	massachusetts	massachusetts	PROPN
ejpam-4653	819	11	.	.	PROPN
ejpam-4653	819	12	,	,	PUNCT
ejpam-4653	819	13	1969	1969	NUM
ejpam-4653	819	14	.	.	PUNCT
ejpam-4653	820	1	[	[	X
ejpam-4653	820	2	12	12	NUM
ejpam-4653	820	3	]	]	X
ejpam-4653	820	4	f.	f.	PROPN
ejpam-4653	820	5	harary	harary	PROPN
ejpam-4653	820	6	and	and	CCONJ
ejpam-4653	820	7	t.w.haynes	t.w.hayne	NOUN
ejpam-4653	820	8	.	.	PUNCT
ejpam-4653	821	1	double	double	ADJ
ejpam-4653	821	2	domination	domination	NOUN
ejpam-4653	821	3	in	in	ADP
ejpam-4653	821	4	graphs	graph	NOUN
ejpam-4653	821	5	.	.	PUNCT
ejpam-4653	822	1	ars	ars	PROPN
ejpam-4653	822	2	combis	combis	PROPN
ejpam-4653	822	3	.	.	PUNCT
ejpam-4653	822	4	,	,	PUNCT
ejpam-4653	822	5	55:201–213	55:201–213	NUM
ejpam-4653	822	6	,	,	PUNCT
ejpam-4653	822	7	2000	2000	NUM
ejpam-4653	822	8	.	.	PUNCT
ejpam-4653	823	1	[	[	X
ejpam-4653	823	2	13	13	NUM
ejpam-4653	823	3	]	]	X
ejpam-4653	823	4	m.a	m.a	PROPN
ejpam-4653	823	5	.	.	PROPN
ejpam-4653	823	6	henning	henning	PROPN
ejpam-4653	823	7	and	and	CCONJ
ejpam-4653	823	8	s.t	s.t	PROPN
ejpam-4653	823	9	.	.	PROPN
ejpam-4653	823	10	hedetniemi	hedetniemi	PROPN
ejpam-4653	823	11	.	.	PUNCT
ejpam-4653	824	1	defending	defend	VERB
ejpam-4653	824	2	the	the	DET
ejpam-4653	824	3	roman	roman	ADJ
ejpam-4653	824	4	empire	empire	NOUN
ejpam-4653	824	5	—	—	PUNCT
ejpam-4653	824	6	a	a	DET
ejpam-4653	824	7	new	new	ADJ
ejpam-4653	824	8	strategy	strategy	NOUN
ejpam-4653	824	9	.	.	PUNCT
ejpam-4653	825	1	discrete	discrete	ADJ
ejpam-4653	825	2	math	math	NOUN
ejpam-4653	825	3	.	.	PUNCT
ejpam-4653	825	4	,	,	PUNCT
ejpam-4653	825	5	266:239–251	266:239–251	NUM
ejpam-4653	825	6	,	,	PUNCT
ejpam-4653	825	7	2003	2003	NUM
ejpam-4653	825	8	.	.	PUNCT
ejpam-4653	826	1	[	[	X
ejpam-4653	826	2	14	14	NUM
ejpam-4653	826	3	]	]	X
ejpam-4653	826	4	o.	o.	PROPN
ejpam-4653	826	5	ore	ore	PROPN
ejpam-4653	826	6	.	.	PUNCT
ejpam-4653	827	1	theory	theory	NOUN
ejpam-4653	827	2	of	of	ADP
ejpam-4653	827	3	graphs	graph	NOUN
ejpam-4653	827	4	.	.	PUNCT
ejpam-4653	828	1	amer	amer	PROPN
ejpam-4653	828	2	.	.	PUNCT
ejpam-4653	828	3	math	math	PROPN
ejpam-4653	828	4	.	.	PUNCT
ejpam-4653	829	1	soc	soc	PROPN
ejpam-4653	829	2	.	.	PUNCT
ejpam-4653	830	1	colloq	colloq	PROPN
ejpam-4653	830	2	.	.	PUNCT
ejpam-4653	831	1	publ	publ	PROPN
ejpam-4653	831	2	.	.	PROPN
ejpam-4653	831	3	,	,	PUNCT
ejpam-4653	831	4	1962	1962	NUM
ejpam-4653	831	5	.	.	PUNCT
ejpam-4653	832	1	[	[	X
ejpam-4653	832	2	15	15	NUM
ejpam-4653	832	3	]	]	X
ejpam-4653	832	4	l.	l.	PROPN
ejpam-4653	832	5	paleta	paleta	PROPN
ejpam-4653	832	6	and	and	CCONJ
ejpam-4653	832	7	f.	f.	PROPN
ejpam-4653	832	8	jamil	jamil	PROPN
ejpam-4653	832	9	.	.	PUNCT
ejpam-4653	833	1	more	more	ADJ
ejpam-4653	833	2	on	on	ADP
ejpam-4653	833	3	perfect	perfect	ADJ
ejpam-4653	833	4	roman	roman	ADJ
ejpam-4653	833	5	domination	domination	NOUN
ejpam-4653	833	6	in	in	ADP
ejpam-4653	833	7	graphs	graph	NOUN
ejpam-4653	833	8	.	.	PUNCT
ejpam-4653	834	1	european	european	ADJ
ejpam-4653	834	2	journal	journal	PROPN
ejpam-4653	834	3	of	of	ADP
ejpam-4653	834	4	pure	pure	ADJ
ejpam-4653	834	5	and	and	CCONJ
ejpam-4653	834	6	applied	applied	ADJ
ejpam-4653	834	7	mathematics	mathematic	NOUN
ejpam-4653	834	8	.	.	PUNCT
ejpam-4653	835	1	,	,	PUNCT
ejpam-4653	835	2	13(3):529–548	13(3):529–548	NOUN
ejpam-4653	835	3	,	,	PUNCT
ejpam-4653	835	4	2020	2020	NUM
ejpam-4653	835	5	.	.	PUNCT
ejpam-4653	836	1	[	[	X
ejpam-4653	836	2	16	16	NUM
ejpam-4653	836	3	]	]	X
ejpam-4653	836	4	c.s	c.s	PROPN
ejpam-4653	836	5	.	.	PROPN
ejpam-4653	836	6	revelle	revelle	PROPN
ejpam-4653	836	7	and	and	CCONJ
ejpam-4653	836	8	k.e	k.e	PROPN
ejpam-4653	836	9	.	.	PUNCT
ejpam-4653	837	1	rosing	rosing	PROPN
ejpam-4653	837	2	.	.	PUNCT
ejpam-4653	838	1	defendens	defenden	VERB
ejpam-4653	838	2	imperium	imperium	NOUN
ejpam-4653	838	3	romanum	romanum	NOUN
ejpam-4653	838	4	:	:	PUNCT
ejpam-4653	838	5	a	a	DET
ejpam-4653	838	6	classical	classical	ADJ
ejpam-4653	838	7	problem	problem	NOUN
ejpam-4653	838	8	in	in	ADP
ejpam-4653	838	9	military	military	ADJ
ejpam-4653	838	10	strategy	strategy	NOUN
ejpam-4653	838	11	.	.	PUNCT
ejpam-4653	839	1	amer	amer	PROPN
ejpam-4653	839	2	.	.	PUNCT
ejpam-4653	839	3	math	math	PROPN
ejpam-4653	839	4	.	.	PUNCT
ejpam-4653	840	1	monthly	monthly	ADV
ejpam-4653	840	2	.	.	PUNCT
ejpam-4653	840	3	,	,	PUNCT
ejpam-4653	841	1	107(7):585–594	107(7):585–594	PROPN
ejpam-4653	841	2	,	,	PUNCT
ejpam-4653	841	3	2000	2000	NUM
ejpam-4653	841	4	.	.	PUNCT
ejpam-4653	842	1	[	[	X
ejpam-4653	842	2	17	17	NUM
ejpam-4653	842	3	]	]	X
ejpam-4653	842	4	i.	i.	PROPN
ejpam-4653	842	5	stewart	stewart	PROPN
ejpam-4653	842	6	.	.	PUNCT
ejpam-4653	843	1	defend	defend	VERB
ejpam-4653	843	2	the	the	DET
ejpam-4653	843	3	roman	roman	ADJ
ejpam-4653	843	4	empire	empire	NOUN
ejpam-4653	843	5	!	!	PUNCT
ejpam-4653	843	6	.	.	PUNCT
ejpam-4653	844	1	sci	sci	PROPN
ejpam-4653	844	2	.	.	PROPN
ejpam-4653	844	3	amer	amer	PROPN
ejpam-4653	844	4	.	.	PROPN
ejpam-4653	844	5	,	,	PUNCT
ejpam-4653	844	6	281(6):136–139	281(6):136–139	NUM
ejpam-4653	844	7	,	,	PUNCT
ejpam-4653	844	8	1999	1999	NUM
ejpam-4653	844	9	.	.	PUNCT
ejpam-4653	845	1	[	[	X
ejpam-4653	845	2	18	18	NUM
ejpam-4653	845	3	]	]	X
ejpam-4653	845	4	anu	anu	PROPN
ejpam-4653	845	5	v.	v.	PROPN
ejpam-4653	845	6	and	and	CCONJ
ejpam-4653	845	7	aparna	aparna	PROPN
ejpam-4653	845	8	lakshmanan	lakshmanan	PROPN
ejpam-4653	845	9	s.	s.	PROPN
ejpam-4653	845	10	double	double	PROPN
ejpam-4653	845	11	roman	roman	ADJ
ejpam-4653	845	12	domination	domination	NOUN
ejpam-4653	845	13	number	number	NOUN
ejpam-4653	845	14	.	.	PUNCT
ejpam-4653	846	1	discrete	discrete	ADJ
ejpam-4653	846	2	appl	appl	PROPN
ejpam-4653	846	3	.	.	PUNCT
ejpam-4653	846	4	math	math	PROPN
ejpam-4653	846	5	.	.	PUNCT
ejpam-4653	846	6	,	,	PUNCT
ejpam-4653	846	7	244:198–204	244:198–204	NUM
ejpam-4653	846	8	,	,	PUNCT
ejpam-4653	846	9	2018	2018	NUM
ejpam-4653	846	10	.	.	PUNCT
ejpam-4653	847	1	[	[	X
ejpam-4653	847	2	19	19	NUM
ejpam-4653	847	3	]	]	X
ejpam-4653	847	4	anu	anu	PROPN
ejpam-4653	847	5	v.	v.	PROPN
ejpam-4653	847	6	and	and	CCONJ
ejpam-4653	847	7	aparna	aparna	PROPN
ejpam-4653	847	8	lakshmanan	lakshmanan	PROPN
ejpam-4653	847	9	s.	s.	PROPN
ejpam-4653	847	10	impact	impact	NOUN
ejpam-4653	847	11	of	of	ADP
ejpam-4653	847	12	some	some	DET
ejpam-4653	847	13	graph	graph	NOUN
ejpam-4653	847	14	operations	operation	NOUN
ejpam-4653	847	15	on	on	ADP
ejpam-4653	847	16	double	double	ADJ
ejpam-4653	847	17	roman	roman	ADJ
ejpam-4653	847	18	domination	domination	NOUN
ejpam-4653	847	19	number	number	NOUN
ejpam-4653	847	20	.	.	PUNCT
ejpam-4653	848	1	2018	2018	NUM
ejpam-4653	848	2	.	.	PUNCT