id	sid	tid	token	lemma	pos
ejpam-5252	1	1	european	european	PROPN
ejpam-5252	1	2	journal	journal	PROPN
ejpam-5252	1	3	of	of	ADP
ejpam-5252	1	4	pure	pure	ADJ
ejpam-5252	1	5	and	and	CCONJ
ejpam-5252	1	6	applied	applied	ADJ
ejpam-5252	1	7	mathematics	mathematic	NOUN
ejpam-5252	1	8	2025	2025	NUM
ejpam-5252	1	9	,	,	PUNCT
ejpam-5252	1	10	vol	vol	NOUN
ejpam-5252	1	11	.	.	PROPN
ejpam-5252	1	12	18	18	NUM
ejpam-5252	1	13	,	,	PUNCT
ejpam-5252	1	14	issue	issue	NOUN
ejpam-5252	1	15	1	1	NUM
ejpam-5252	1	16	,	,	PUNCT
ejpam-5252	1	17	article	article	NOUN
ejpam-5252	1	18	number	number	NOUN
ejpam-5252	1	19	5252	5252	NUM
ejpam-5252	1	20	issn	issn	PROPN
ejpam-5252	1	21	1307	1307	NUM
ejpam-5252	1	22	-	-	SYM
ejpam-5252	1	23	5543	5543	NUM
ejpam-5252	1	24	–	–	PUNCT
ejpam-5252	1	25	ejpam.com	ejpam.com	X
ejpam-5252	1	26	published	publish	VERB
ejpam-5252	1	27	by	by	ADP
ejpam-5252	1	28	new	new	PROPN
ejpam-5252	1	29	york	york	PROPN
ejpam-5252	1	30	business	business	PROPN
ejpam-5252	1	31	global	global	ADJ
ejpam-5252	1	32	modern	modern	ADJ
ejpam-5252	1	33	roman	roman	ADJ
ejpam-5252	1	34	dominating	dominating	NOUN
ejpam-5252	1	35	functions	function	NOUN
ejpam-5252	1	36	in	in	ADP
ejpam-5252	1	37	graphs	graph	NOUN
ejpam-5252	1	38	sherihatha	sherihatha	PROPN
ejpam-5252	1	39	r.	r.	PROPN
ejpam-5252	1	40	ahamad1,2,∗	ahamad1,2,∗	PROPN
ejpam-5252	1	41	,	,	PUNCT
ejpam-5252	1	42	jerry	jerry	NOUN
ejpam-5252	1	43	boy	boy	NOUN
ejpam-5252	1	44	g.	g.	PROPN
ejpam-5252	1	45	cariaga1,2	cariaga1,2	PROPN
ejpam-5252	1	46	,	,	PUNCT
ejpam-5252	1	47	sheila	sheila	PROPN
ejpam-5252	1	48	m.	m.	NOUN
ejpam-5252	1	49	menchavez1,2	menchavez1,2	PROPN
ejpam-5252	1	50	1	1	NUM
ejpam-5252	1	51	department	department	NOUN
ejpam-5252	1	52	of	of	ADP
ejpam-5252	1	53	mathematics	mathematic	NOUN
ejpam-5252	1	54	and	and	CCONJ
ejpam-5252	1	55	statistics	statistic	NOUN
ejpam-5252	1	56	,	,	PUNCT
ejpam-5252	1	57	college	college	NOUN
ejpam-5252	1	58	of	of	ADP
ejpam-5252	1	59	science	science	NOUN
ejpam-5252	1	60	and	and	CCONJ
ejpam-5252	1	61	mathematics	mathematic	NOUN
ejpam-5252	1	62	,	,	PUNCT
ejpam-5252	1	63	msu	msu	PROPN
ejpam-5252	1	64	-	-	PUNCT
ejpam-5252	1	65	iligan	iligan	PROPN
ejpam-5252	1	66	institute	institute	PROPN
ejpam-5252	1	67	of	of	ADP
ejpam-5252	1	68	technology	technology	PROPN
ejpam-5252	1	69	,	,	PUNCT
ejpam-5252	1	70	9200	9200	NUM
ejpam-5252	1	71	iligan	iligan	ADJ
ejpam-5252	1	72	city	city	NOUN
ejpam-5252	1	73	,	,	PUNCT
ejpam-5252	1	74	philippines	philippine	NOUN
ejpam-5252	1	75	2	2	NUM
ejpam-5252	1	76	cmtps	cmtps	NOUN
ejpam-5252	1	77	,	,	PUNCT
ejpam-5252	1	78	premier	premier	PROPN
ejpam-5252	1	79	research	research	PROPN
ejpam-5252	1	80	institute	institute	PROPN
ejpam-5252	1	81	of	of	ADP
ejpam-5252	1	82	science	science	NOUN
ejpam-5252	1	83	and	and	CCONJ
ejpam-5252	1	84	mathematics	mathematic	NOUN
ejpam-5252	1	85	,	,	PUNCT
ejpam-5252	1	86	msu	msu	PROPN
ejpam-5252	1	87	-	-	PUNCT
ejpam-5252	1	88	iligan	iligan	PROPN
ejpam-5252	1	89	institute	institute	PROPN
ejpam-5252	1	90	of	of	ADP
ejpam-5252	1	91	technology	technology	PROPN
ejpam-5252	1	92	,	,	PUNCT
ejpam-5252	1	93	9200	9200	NUM
ejpam-5252	1	94	iligan	iligan	ADJ
ejpam-5252	1	95	city	city	NOUN
ejpam-5252	1	96	,	,	PUNCT
ejpam-5252	1	97	philippines	philippine	NOUN
ejpam-5252	1	98	abstract	abstract	ADJ
ejpam-5252	1	99	.	.	PUNCT
ejpam-5252	2	1	let	let	VERB
ejpam-5252	2	2	g	g	PROPN
ejpam-5252	2	3	=	=	SYM
ejpam-5252	2	4	(	(	PUNCT
ejpam-5252	2	5	v	v	NOUN
ejpam-5252	2	6	(	(	PUNCT
ejpam-5252	2	7	g	g	NOUN
ejpam-5252	2	8	)	)	PUNCT
ejpam-5252	2	9	,	,	PUNCT
ejpam-5252	2	10	e(g	e(g	PROPN
ejpam-5252	2	11	)	)	PUNCT
ejpam-5252	2	12	)	)	PUNCT
ejpam-5252	2	13	be	be	AUX
ejpam-5252	2	14	any	any	DET
ejpam-5252	2	15	connected	connected	ADJ
ejpam-5252	2	16	graph	graph	NOUN
ejpam-5252	2	17	.	.	PUNCT
ejpam-5252	3	1	a	a	DET
ejpam-5252	3	2	function	function	NOUN
ejpam-5252	3	3	f	f	NOUN
ejpam-5252	3	4	:	:	PUNCT
ejpam-5252	3	5	v	v	X
ejpam-5252	3	6	(	(	PUNCT
ejpam-5252	3	7	g	g	NOUN
ejpam-5252	3	8	)	)	PUNCT
ejpam-5252	3	9	→	→	SYM
ejpam-5252	3	10	{	{	PUNCT
ejpam-5252	3	11	0	0	NUM
ejpam-5252	3	12	,	,	PUNCT
ejpam-5252	3	13	1	1	NUM
ejpam-5252	3	14	,	,	PUNCT
ejpam-5252	3	15	2	2	NUM
ejpam-5252	3	16	,	,	PUNCT
ejpam-5252	3	17	3	3	NUM
ejpam-5252	3	18	}	}	PUNCT
ejpam-5252	3	19	is	be	AUX
ejpam-5252	3	20	a	a	DET
ejpam-5252	3	21	modern	modern	ADJ
ejpam-5252	3	22	roman	roman	ADJ
ejpam-5252	3	23	dominating	dominating	NOUN
ejpam-5252	3	24	function	function	NOUN
ejpam-5252	3	25	of	of	ADP
ejpam-5252	3	26	g	g	PROPN
ejpam-5252	3	27	if	if	SCONJ
ejpam-5252	3	28	for	for	ADP
ejpam-5252	3	29	each	each	PRON
ejpam-5252	3	30	v	v	NUM
ejpam-5252	3	31	∈	∈	PROPN
ejpam-5252	3	32	v	v	NOUN
ejpam-5252	3	33	(	(	PUNCT
ejpam-5252	3	34	g	g	NOUN
ejpam-5252	3	35	)	)	PUNCT
ejpam-5252	3	36	with	with	ADP
ejpam-5252	3	37	f(v	f(v	NOUN
ejpam-5252	3	38	)	)	PUNCT
ejpam-5252	3	39	=	=	SYM
ejpam-5252	3	40	0	0	NUM
ejpam-5252	3	41	,	,	PUNCT
ejpam-5252	3	42	there	there	PRON
ejpam-5252	3	43	exist	exist	VERB
ejpam-5252	3	44	u	u	NOUN
ejpam-5252	3	45	,	,	PUNCT
ejpam-5252	3	46	w	w	PROPN
ejpam-5252	3	47	∈	∈	PROPN
ejpam-5252	3	48	ng(v	ng(v	PUNCT
ejpam-5252	3	49	)	)	PUNCT
ejpam-5252	3	50	such	such	ADJ
ejpam-5252	3	51	that	that	DET
ejpam-5252	3	52	f(u	f(u	PROPN
ejpam-5252	3	53	)	)	PUNCT
ejpam-5252	3	54	=	=	SYM
ejpam-5252	3	55	2	2	NUM
ejpam-5252	3	56	and	and	CCONJ
ejpam-5252	3	57	f(w	f(w	NUM
ejpam-5252	3	58	)	)	PUNCT
ejpam-5252	3	59	=	=	SYM
ejpam-5252	3	60	3	3	NUM
ejpam-5252	3	61	;	;	PUNCT
ejpam-5252	3	62	and	and	CCONJ
ejpam-5252	3	63	for	for	ADP
ejpam-5252	3	64	each	each	PRON
ejpam-5252	3	65	v	v	NUM
ejpam-5252	3	66	∈	∈	PROPN
ejpam-5252	3	67	v	v	NOUN
ejpam-5252	3	68	(	(	PUNCT
ejpam-5252	3	69	g	g	NOUN
ejpam-5252	3	70	)	)	PUNCT
ejpam-5252	3	71	with	with	ADP
ejpam-5252	3	72	f(v	f(v	NOUN
ejpam-5252	3	73	)	)	PUNCT
ejpam-5252	3	74	=	=	SYM
ejpam-5252	3	75	1	1	NUM
ejpam-5252	3	76	,	,	PUNCT
ejpam-5252	3	77	there	there	PRON
ejpam-5252	3	78	exists	exist	VERB
ejpam-5252	3	79	u	u	PROPN
ejpam-5252	3	80	∈	∈	PROPN
ejpam-5252	3	81	ng(v	ng(v	PUNCT
ejpam-5252	3	82	)	)	PUNCT
ejpam-5252	3	83	such	such	ADJ
ejpam-5252	3	84	that	that	DET
ejpam-5252	3	85	f(u	f(u	PROPN
ejpam-5252	3	86	)	)	PUNCT
ejpam-5252	4	1	=	=	SYM
ejpam-5252	4	2	2	2	NUM
ejpam-5252	4	3	or	or	CCONJ
ejpam-5252	4	4	f(w	f(w	PROPN
ejpam-5252	4	5	)	)	PUNCT
ejpam-5252	4	6	=	=	PUNCT
ejpam-5252	4	7	3	3	X
ejpam-5252	4	8	.	.	PUNCT
ejpam-5252	5	1	the	the	DET
ejpam-5252	5	2	weight	weight	NOUN
ejpam-5252	5	3	of	of	ADP
ejpam-5252	5	4	a	a	DET
ejpam-5252	5	5	modern	modern	ADJ
ejpam-5252	5	6	roman	roman	ADJ
ejpam-5252	5	7	dominating	dominating	NOUN
ejpam-5252	5	8	function	function	NOUN
ejpam-5252	5	9	f	f	PROPN
ejpam-5252	5	10	of	of	ADP
ejpam-5252	5	11	g	g	PROPN
ejpam-5252	5	12	is	be	AUX
ejpam-5252	5	13	the	the	DET
ejpam-5252	5	14	sum	sum	NOUN
ejpam-5252	5	15	ωmr	ωmr	NOUN
ejpam-5252	5	16	g	g	PROPN
ejpam-5252	5	17	(	(	PUNCT
ejpam-5252	5	18	f	f	X
ejpam-5252	5	19	)	)	PUNCT
ejpam-5252	5	20	=	=	SYM
ejpam-5252	5	21	∑	∑	PUNCT
ejpam-5252	5	22	v∈v	v∈v	PROPN
ejpam-5252	5	23	(	(	PUNCT
ejpam-5252	5	24	g	g	NOUN
ejpam-5252	5	25	)	)	PUNCT
ejpam-5252	5	26	f(v	f(v	NOUN
ejpam-5252	5	27	)	)	PUNCT
ejpam-5252	5	28	and	and	CCONJ
ejpam-5252	5	29	the	the	DET
ejpam-5252	5	30	minimum	minimum	ADJ
ejpam-5252	5	31	weight	weight	NOUN
ejpam-5252	5	32	among	among	ADP
ejpam-5252	5	33	all	all	PRON
ejpam-5252	5	34	of	of	ADP
ejpam-5252	5	35	the	the	DET
ejpam-5252	5	36	modern	modern	ADJ
ejpam-5252	5	37	dominating	dominating	NOUN
ejpam-5252	5	38	functions	function	NOUN
ejpam-5252	5	39	on	on	ADP
ejpam-5252	5	40	g	g	PROPN
ejpam-5252	5	41	is	be	AUX
ejpam-5252	5	42	called	call	VERB
ejpam-5252	5	43	the	the	DET
ejpam-5252	5	44	modern	modern	ADJ
ejpam-5252	5	45	roman	roman	ADJ
ejpam-5252	5	46	domination	domination	NOUN
ejpam-5252	5	47	number	number	NOUN
ejpam-5252	5	48	γmr(g	γmr(g	PROPN
ejpam-5252	5	49	)	)	PUNCT
ejpam-5252	5	50	of	of	ADP
ejpam-5252	5	51	g.	g.	PROPN
ejpam-5252	5	52	in	in	ADP
ejpam-5252	5	53	this	this	DET
ejpam-5252	5	54	paper	paper	NOUN
ejpam-5252	5	55	,	,	PUNCT
ejpam-5252	5	56	we	we	PRON
ejpam-5252	5	57	characterize	characterize	VERB
ejpam-5252	5	58	graphs	graph	NOUN
ejpam-5252	5	59	with	with	ADP
ejpam-5252	5	60	smaller	small	ADJ
ejpam-5252	5	61	modern	modern	ADJ
ejpam-5252	5	62	roman	roman	ADJ
ejpam-5252	5	63	domination	domination	NOUN
ejpam-5252	5	64	number	number	NOUN
ejpam-5252	5	65	and	and	CCONJ
ejpam-5252	5	66	obtain	obtain	VERB
ejpam-5252	5	67	the	the	DET
ejpam-5252	5	68	γmr(g	γmr(g	NOUN
ejpam-5252	5	69	)	)	PUNCT
ejpam-5252	5	70	of	of	ADP
ejpam-5252	5	71	some	some	DET
ejpam-5252	5	72	special	special	ADJ
ejpam-5252	5	73	graphs	graph	NOUN
ejpam-5252	5	74	.	.	PUNCT
ejpam-5252	6	1	moreover	moreover	ADV
ejpam-5252	6	2	,	,	PUNCT
ejpam-5252	6	3	we	we	PRON
ejpam-5252	6	4	investigate	investigate	VERB
ejpam-5252	6	5	and	and	CCONJ
ejpam-5252	6	6	characterize	characterize	VERB
ejpam-5252	6	7	the	the	DET
ejpam-5252	6	8	modern	modern	ADJ
ejpam-5252	6	9	roman	roman	ADJ
ejpam-5252	6	10	domination	domination	NOUN
ejpam-5252	6	11	of	of	ADP
ejpam-5252	6	12	the	the	DET
ejpam-5252	6	13	join	join	NOUN
ejpam-5252	6	14	and	and	CCONJ
ejpam-5252	6	15	corona	corona	NOUN
ejpam-5252	6	16	of	of	ADP
ejpam-5252	6	17	graphs	graph	NOUN
ejpam-5252	6	18	.	.	PUNCT
ejpam-5252	7	1	2020	2020	NUM
ejpam-5252	7	2	mathematics	mathematic	NOUN
ejpam-5252	7	3	subject	subject	NOUN
ejpam-5252	7	4	classifications	classification	NOUN
ejpam-5252	7	5	:	:	PUNCT
ejpam-5252	7	6	05c69	05c69	X
ejpam-5252	7	7	key	key	ADJ
ejpam-5252	7	8	words	word	NOUN
ejpam-5252	7	9	and	and	CCONJ
ejpam-5252	7	10	phrases	phrase	NOUN
ejpam-5252	7	11	:	:	PUNCT
ejpam-5252	7	12	dominating	dominate	VERB
ejpam-5252	7	13	set	set	NOUN
ejpam-5252	7	14	,	,	PUNCT
ejpam-5252	7	15	domination	domination	NOUN
ejpam-5252	7	16	number	number	NOUN
ejpam-5252	7	17	,	,	PUNCT
ejpam-5252	7	18	modern	modern	ADJ
ejpam-5252	7	19	roman	roman	ADJ
ejpam-5252	7	20	dominating	dominating	NOUN
ejpam-5252	7	21	function	function	NOUN
ejpam-5252	7	22	,	,	PUNCT
ejpam-5252	7	23	and	and	CCONJ
ejpam-5252	7	24	modern	modern	ADJ
ejpam-5252	7	25	roman	roman	ADJ
ejpam-5252	7	26	domination	domination	NOUN
ejpam-5252	7	27	number	number	NOUN
ejpam-5252	7	28	.	.	PUNCT
ejpam-5252	8	1	1	1	X
ejpam-5252	8	2	.	.	X
ejpam-5252	8	3	introduction	introduction	NOUN
ejpam-5252	8	4	the	the	DET
ejpam-5252	8	5	concept	concept	NOUN
ejpam-5252	8	6	of	of	ADP
ejpam-5252	8	7	roman	roman	ADJ
ejpam-5252	8	8	domination	domination	NOUN
ejpam-5252	8	9	is	be	AUX
ejpam-5252	8	10	introduced	introduce	VERB
ejpam-5252	8	11	in	in	ADP
ejpam-5252	8	12	2004	2004	NUM
ejpam-5252	9	1	[	[	X
ejpam-5252	9	2	6	6	NUM
ejpam-5252	9	3	]	]	PUNCT
ejpam-5252	9	4	.	.	PUNCT
ejpam-5252	10	1	it	it	PRON
ejpam-5252	10	2	is	be	AUX
ejpam-5252	10	3	inspired	inspire	VERB
ejpam-5252	10	4	by	by	ADP
ejpam-5252	10	5	the	the	DET
ejpam-5252	10	6	strategies	strategy	NOUN
ejpam-5252	10	7	for	for	ADP
ejpam-5252	10	8	defending	defend	VERB
ejpam-5252	10	9	the	the	DET
ejpam-5252	10	10	roman	roman	ADJ
ejpam-5252	10	11	empire	empire	NOUN
ejpam-5252	10	12	presented	present	VERB
ejpam-5252	10	13	in	in	ADP
ejpam-5252	10	14	the	the	DET
ejpam-5252	10	15	work	work	NOUN
ejpam-5252	10	16	of	of	ADP
ejpam-5252	10	17	revelle	revelle	NOUN
ejpam-5252	10	18	and	and	CCONJ
ejpam-5252	10	19	rosing	rose	VERB
ejpam-5252	10	20	in	in	ADP
ejpam-5252	10	21	[	[	X
ejpam-5252	10	22	13	13	NUM
ejpam-5252	10	23	]	]	PUNCT
ejpam-5252	10	24	and	and	CCONJ
ejpam-5252	10	25	stewart	stewart	PROPN
ejpam-5252	10	26	,	,	PUNCT
ejpam-5252	10	27	cockayne	cockayne	PROPN
ejpam-5252	10	28	,	,	PUNCT
ejpam-5252	10	29	et	et	PROPN
ejpam-5252	10	30	al	al	PROPN
ejpam-5252	10	31	.	.	PUNCT
ejpam-5252	11	1	in	in	ADP
ejpam-5252	11	2	[	[	X
ejpam-5252	11	3	15	15	NUM
ejpam-5252	11	4	]	]	PUNCT
ejpam-5252	11	5	.	.	PUNCT
ejpam-5252	12	1	since	since	SCONJ
ejpam-5252	12	2	then	then	ADV
ejpam-5252	12	3	,	,	PUNCT
ejpam-5252	12	4	it	it	PRON
ejpam-5252	12	5	has	have	AUX
ejpam-5252	12	6	emerged	emerge	VERB
ejpam-5252	12	7	as	as	ADP
ejpam-5252	12	8	an	an	DET
ejpam-5252	12	9	active	active	ADJ
ejpam-5252	12	10	research	research	NOUN
ejpam-5252	12	11	field	field	NOUN
ejpam-5252	12	12	in	in	ADP
ejpam-5252	12	13	graph	graph	NOUN
ejpam-5252	12	14	theory	theory	NOUN
ejpam-5252	12	15	(	(	PUNCT
ejpam-5252	12	16	see	see	VERB
ejpam-5252	12	17	[	[	X
ejpam-5252	12	18	10],[8],[1],[4],[3],[12],[9],[7],[14],[11	10],[8],[1],[4],[3],[12],[9],[7],[14],[11	NOUN
ejpam-5252	12	19	]	]	X
ejpam-5252	12	20	)	)	PUNCT
ejpam-5252	12	21	.	.	PUNCT
ejpam-5252	13	1	a	a	DET
ejpam-5252	13	2	new	new	ADJ
ejpam-5252	13	3	model	model	NOUN
ejpam-5252	13	4	of	of	ADP
ejpam-5252	13	5	graph	graph	NOUN
ejpam-5252	13	6	domination	domination	NOUN
ejpam-5252	13	7	based	base	VERB
ejpam-5252	13	8	on	on	ADP
ejpam-5252	13	9	roman	roman	ADJ
ejpam-5252	13	10	domination	domination	NOUN
ejpam-5252	13	11	is	be	AUX
ejpam-5252	13	12	introduced	introduce	VERB
ejpam-5252	13	13	in	in	ADP
ejpam-5252	13	14	[	[	X
ejpam-5252	13	15	8	8	NUM
ejpam-5252	13	16	]	]	PUNCT
ejpam-5252	13	17	,	,	PUNCT
ejpam-5252	13	18	called	call	VERB
ejpam-5252	13	19	modern	modern	ADJ
ejpam-5252	13	20	roman	roman	ADJ
ejpam-5252	13	21	domination	domination	NOUN
ejpam-5252	13	22	.	.	PUNCT
ejpam-5252	14	1	studies	study	NOUN
ejpam-5252	14	2	and	and	CCONJ
ejpam-5252	14	3	exploration	exploration	NOUN
ejpam-5252	14	4	on	on	ADP
ejpam-5252	14	5	this	this	DET
ejpam-5252	14	6	variant	variant	NOUN
ejpam-5252	14	7	can	can	AUX
ejpam-5252	14	8	be	be	AUX
ejpam-5252	14	9	found	find	VERB
ejpam-5252	14	10	in	in	ADP
ejpam-5252	14	11	[	[	X
ejpam-5252	14	12	1	1	NUM
ejpam-5252	14	13	,	,	PUNCT
ejpam-5252	14	14	11	11	NUM
ejpam-5252	14	15	,	,	PUNCT
ejpam-5252	14	16	14	14	NUM
ejpam-5252	14	17	]	]	PUNCT
ejpam-5252	14	18	.	.	PUNCT
ejpam-5252	15	1	explicity	explicity	NOUN
ejpam-5252	15	2	,	,	PUNCT
ejpam-5252	15	3	a	a	DET
ejpam-5252	15	4	function	function	NOUN
ejpam-5252	15	5	f	f	NOUN
ejpam-5252	15	6	:	:	PUNCT
ejpam-5252	15	7	v	v	X
ejpam-5252	15	8	(	(	PUNCT
ejpam-5252	15	9	g	g	NOUN
ejpam-5252	15	10	)	)	PUNCT
ejpam-5252	15	11	→	→	SYM
ejpam-5252	15	12	{	{	PUNCT
ejpam-5252	15	13	0	0	NUM
ejpam-5252	15	14	,	,	PUNCT
ejpam-5252	15	15	1	1	NUM
ejpam-5252	15	16	,	,	PUNCT
ejpam-5252	15	17	2	2	NUM
ejpam-5252	15	18	,	,	PUNCT
ejpam-5252	15	19	3	3	NUM
ejpam-5252	15	20	}	}	PUNCT
ejpam-5252	15	21	is	be	AUX
ejpam-5252	15	22	a	a	DET
ejpam-5252	15	23	modern	modern	ADJ
ejpam-5252	15	24	roman	roman	ADJ
ejpam-5252	15	25	dominating	dominating	NOUN
ejpam-5252	15	26	function	function	NOUN
ejpam-5252	15	27	(	(	PUNCT
ejpam-5252	15	28	mrdf	mrdf	NOUN
ejpam-5252	15	29	)	)	PUNCT
ejpam-5252	15	30	of	of	ADP
ejpam-5252	15	31	g	g	PROPN
ejpam-5252	15	32	if	if	SCONJ
ejpam-5252	15	33	for	for	ADP
ejpam-5252	15	34	each	each	PRON
ejpam-5252	15	35	v	v	NUM
ejpam-5252	15	36	∈	∈	PROPN
ejpam-5252	15	37	v	v	NOUN
ejpam-5252	15	38	(	(	PUNCT
ejpam-5252	15	39	g	g	NOUN
ejpam-5252	15	40	)	)	PUNCT
ejpam-5252	15	41	with	with	ADP
ejpam-5252	15	42	f(v	f(v	NOUN
ejpam-5252	15	43	)	)	PUNCT
ejpam-5252	15	44	=	=	SYM
ejpam-5252	16	1	0	0	NUM
ejpam-5252	16	2	,	,	PUNCT
ejpam-5252	16	3	there	there	PRON
ejpam-5252	16	4	exist	exist	VERB
ejpam-5252	16	5	u	u	NOUN
ejpam-5252	16	6	,	,	PUNCT
ejpam-5252	16	7	w	w	PROPN
ejpam-5252	16	8	∈	∈	PROPN
ejpam-5252	16	9	ng(v	ng(v	PUNCT
ejpam-5252	16	10	)	)	PUNCT
ejpam-5252	16	11	such	such	ADJ
ejpam-5252	16	12	that	that	DET
ejpam-5252	16	13	f(u	f(u	PROPN
ejpam-5252	16	14	)	)	PUNCT
ejpam-5252	16	15	=	=	SYM
ejpam-5252	16	16	2	2	NUM
ejpam-5252	16	17	and	and	CCONJ
ejpam-5252	16	18	f(w	f(w	NUM
ejpam-5252	16	19	)	)	PUNCT
ejpam-5252	16	20	=	=	SYM
ejpam-5252	17	1	3	3	NUM
ejpam-5252	17	2	;	;	PUNCT
ejpam-5252	17	3	and	and	CCONJ
ejpam-5252	17	4	for	for	ADP
ejpam-5252	17	5	each	each	PRON
ejpam-5252	17	6	v	v	NUM
ejpam-5252	17	7	∈	∈	PROPN
ejpam-5252	17	8	v	v	NOUN
ejpam-5252	17	9	(	(	PUNCT
ejpam-5252	17	10	g	g	NOUN
ejpam-5252	17	11	)	)	PUNCT
ejpam-5252	17	12	with	with	ADP
ejpam-5252	17	13	f(v	f(v	NOUN
ejpam-5252	17	14	)	)	PUNCT
ejpam-5252	17	15	=	=	SYM
ejpam-5252	17	16	1	1	NUM
ejpam-5252	17	17	,	,	PUNCT
ejpam-5252	17	18	there	there	PRON
ejpam-5252	17	19	exists	exist	VERB
ejpam-5252	17	20	u	u	PROPN
ejpam-5252	17	21	∈	∈	PROPN
ejpam-5252	17	22	ng(v	ng(v	PUNCT
ejpam-5252	17	23	)	)	PUNCT
ejpam-5252	17	24	such	such	ADJ
ejpam-5252	17	25	that	that	DET
ejpam-5252	17	26	f(u	f(u	PROPN
ejpam-5252	17	27	)	)	PUNCT
ejpam-5252	17	28	=	=	SYM
ejpam-5252	17	29	2	2	NUM
ejpam-5252	17	30	or	or	CCONJ
ejpam-5252	17	31	f(u	f(u	PROPN
ejpam-5252	17	32	)	)	PUNCT
ejpam-5252	17	33	=	=	SYM
ejpam-5252	18	1	3	3	X
ejpam-5252	18	2	.	.	PUNCT
ejpam-5252	18	3	the	the	DET
ejpam-5252	18	4	minimum	minimum	ADJ
ejpam-5252	18	5	weight	weight	NOUN
ejpam-5252	18	6	among	among	ADP
ejpam-5252	18	7	all	all	PRON
ejpam-5252	18	8	of	of	ADP
ejpam-5252	18	9	the	the	DET
ejpam-5252	18	10	mrdf	mrdf	NOUN
ejpam-5252	18	11	is	be	AUX
ejpam-5252	18	12	called	call	VERB
ejpam-5252	18	13	the	the	DET
ejpam-5252	18	14	modern	modern	ADJ
ejpam-5252	18	15	roman	roman	NOUN
ejpam-5252	18	16	∗corresponding	∗corresponde	VERB
ejpam-5252	18	17	author	author	NOUN
ejpam-5252	18	18	.	.	PUNCT
ejpam-5252	19	1	doi	doi	PROPN
ejpam-5252	19	2	:	:	PUNCT
ejpam-5252	19	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5252	https://doi.org/10.29020/nybg.ejpam.v18i1.5252	PROPN
ejpam-5252	19	4	email	email	NOUN
ejpam-5252	19	5	addresses	address	NOUN
ejpam-5252	19	6	:	:	PUNCT
ejpam-5252	19	7	sherihatha.ahamad@g.msuiit.edu.ph	sherihatha.ahamad@g.msuiit.edu.ph	PROPN
ejpam-5252	19	8	(	(	PUNCT
ejpam-5252	19	9	s.	s.	PROPN
ejpam-5252	19	10	ahamad	ahamad	PROPN
ejpam-5252	19	11	)	)	PUNCT
ejpam-5252	19	12	,	,	PUNCT
ejpam-5252	19	13	jerryboy.cariaga@g.msuiit.edu.ph	jerryboy.cariaga@g.msuiit.edu.ph	PROPN
ejpam-5252	19	14	(	(	PUNCT
ejpam-5252	19	15	j.	j.	PROPN
ejpam-5252	19	16	cariaga	cariaga	PROPN
ejpam-5252	19	17	)	)	PUNCT
ejpam-5252	19	18	,	,	PUNCT
ejpam-5252	19	19	sheila.menchavez@g.msuiit.edu.ph	sheila.menchavez@g.msuiit.edu.ph	PROPN
ejpam-5252	19	20	(	(	PUNCT
ejpam-5252	19	21	s.	s.	PROPN
ejpam-5252	19	22	menchavez	menchavez	PROPN
ejpam-5252	19	23	)	)	PUNCT
ejpam-5252	19	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5252	20	1	1	1	NUM
ejpam-5252	20	2	copyright	copyright	NOUN
ejpam-5252	20	3	:	:	PUNCT
ejpam-5252	20	4	©	©	PROPN
ejpam-5252	20	5	2025	2025	NUM
ejpam-5252	20	6	the	the	DET
ejpam-5252	20	7	author(s	author(s	NOUN
ejpam-5252	20	8	)	)	PUNCT
ejpam-5252	20	9	.	.	PUNCT
ejpam-5252	21	1	(	(	PUNCT
ejpam-5252	21	2	cc	cc	NOUN
ejpam-5252	21	3	by	by	ADP
ejpam-5252	21	4	-	-	PUNCT
ejpam-5252	21	5	nc	nc	PROPN
ejpam-5252	21	6	4.0	4.0	NUM
ejpam-5252	21	7	)	)	PUNCT
ejpam-5252	21	8	s.	s.	PROPN
ejpam-5252	21	9	ahamad	ahamad	PROPN
ejpam-5252	21	10	,	,	PUNCT
ejpam-5252	21	11	j.	j.	PROPN
ejpam-5252	21	12	cariaga	cariaga	PROPN
ejpam-5252	21	13	,	,	PUNCT
ejpam-5252	21	14	s.	s.	PROPN
ejpam-5252	21	15	menchavez	menchavez	PROPN
ejpam-5252	21	16	/	/	PUNCT
ejpam-5252	21	17	eur	eur	PROPN
ejpam-5252	21	18	.	.	PUNCT
ejpam-5252	22	1	j.	j.	PROPN
ejpam-5252	22	2	pure	pure	PROPN
ejpam-5252	22	3	appl	appl	PROPN
ejpam-5252	22	4	.	.	PROPN
ejpam-5252	22	5	math	math	PROPN
ejpam-5252	22	6	,	,	PUNCT
ejpam-5252	22	7	18	18	NUM
ejpam-5252	22	8	(	(	PUNCT
ejpam-5252	22	9	1	1	NUM
ejpam-5252	22	10	)	)	PUNCT
ejpam-5252	22	11	(	(	PUNCT
ejpam-5252	22	12	2025	2025	NUM
ejpam-5252	22	13	)	)	PUNCT
ejpam-5252	22	14	,	,	PUNCT
ejpam-5252	22	15	5252	5252	NUM
ejpam-5252	22	16	2	2	NUM
ejpam-5252	22	17	of	of	ADP
ejpam-5252	22	18	18	18	NUM
ejpam-5252	22	19	domination	domination	NOUN
ejpam-5252	22	20	number	number	NOUN
ejpam-5252	22	21	and	and	CCONJ
ejpam-5252	22	22	is	be	AUX
ejpam-5252	22	23	denoted	denote	VERB
ejpam-5252	22	24	by	by	ADP
ejpam-5252	22	25	γmr(g	γmr(g	PROPN
ejpam-5252	22	26	)	)	PUNCT
ejpam-5252	22	27	.	.	PUNCT
ejpam-5252	23	1	in	in	ADP
ejpam-5252	23	2	this	this	DET
ejpam-5252	23	3	model	model	NOUN
ejpam-5252	23	4	,	,	PUNCT
ejpam-5252	23	5	the	the	DET
ejpam-5252	23	6	label	label	NOUN
ejpam-5252	23	7	of	of	ADP
ejpam-5252	23	8	a	a	DET
ejpam-5252	23	9	vertex	vertex	NOUN
ejpam-5252	23	10	under	under	ADP
ejpam-5252	23	11	the	the	DET
ejpam-5252	23	12	function	function	NOUN
ejpam-5252	23	13	f	f	PROPN
ejpam-5252	23	14	represents	represent	VERB
ejpam-5252	23	15	a	a	DET
ejpam-5252	23	16	type	type	NOUN
ejpam-5252	23	17	of	of	ADP
ejpam-5252	23	18	weapon	weapon	NOUN
ejpam-5252	23	19	in	in	ADP
ejpam-5252	23	20	a	a	DET
ejpam-5252	23	21	war	war	NOUN
ejpam-5252	23	22	zone	zone	NOUN
ejpam-5252	23	23	.	.	PUNCT
ejpam-5252	24	1	the	the	DET
ejpam-5252	24	2	four	four	NUM
ejpam-5252	24	3	defensive	defensive	ADJ
ejpam-5252	24	4	weapon	weapon	NOUN
ejpam-5252	24	5	types	type	NOUN
ejpam-5252	24	6	are	be	AUX
ejpam-5252	24	7	represented	represent	VERB
ejpam-5252	24	8	by	by	ADP
ejpam-5252	24	9	the	the	DET
ejpam-5252	24	10	set	set	NOUN
ejpam-5252	24	11	of	of	ADP
ejpam-5252	24	12	weights	weight	NOUN
ejpam-5252	24	13	{	{	PUNCT
ejpam-5252	24	14	0	0	NUM
ejpam-5252	24	15	,	,	PUNCT
ejpam-5252	24	16	1	1	NUM
ejpam-5252	24	17	,	,	PUNCT
ejpam-5252	24	18	2	2	NUM
ejpam-5252	24	19	,	,	PUNCT
ejpam-5252	24	20	3	3	NUM
ejpam-5252	24	21	}	}	PUNCT
ejpam-5252	24	22	under	under	ADP
ejpam-5252	24	23	the	the	DET
ejpam-5252	24	24	function	function	NOUN
ejpam-5252	25	1	f	f	PROPN
ejpam-5252	25	2	.	.	PUNCT
ejpam-5252	25	3	weapon	weapon	NOUN
ejpam-5252	25	4	types	type	NOUN
ejpam-5252	25	5	are	be	AUX
ejpam-5252	25	6	given	give	VERB
ejpam-5252	25	7	ascending	ascend	VERB
ejpam-5252	25	8	weights	weight	NOUN
ejpam-5252	25	9	:	:	PUNCT
ejpam-5252	25	10	light	light	ADJ
ejpam-5252	25	11	,	,	PUNCT
ejpam-5252	25	12	medium	medium	ADJ
ejpam-5252	25	13	,	,	PUNCT
ejpam-5252	25	14	heavy	heavy	ADJ
ejpam-5252	25	15	,	,	PUNCT
ejpam-5252	25	16	and	and	CCONJ
ejpam-5252	25	17	air	air	NOUN
ejpam-5252	25	18	force	force	NOUN
ejpam-5252	25	19	.	.	PUNCT
ejpam-5252	26	1	light	light	ADJ
ejpam-5252	26	2	weapons	weapon	NOUN
ejpam-5252	26	3	are	be	AUX
ejpam-5252	26	4	for	for	ADP
ejpam-5252	26	5	pedestrians	pedestrian	NOUN
ejpam-5252	26	6	;	;	PUNCT
ejpam-5252	26	7	heavy	heavy	ADJ
ejpam-5252	26	8	weapons	weapon	NOUN
ejpam-5252	26	9	can	can	AUX
ejpam-5252	26	10	be	be	AUX
ejpam-5252	26	11	tanks	tank	NOUN
ejpam-5252	26	12	and	and	CCONJ
ejpam-5252	26	13	rockets	rocket	NOUN
ejpam-5252	26	14	.	.	PUNCT
ejpam-5252	27	1	the	the	DET
ejpam-5252	27	2	defense	defense	NOUN
ejpam-5252	27	3	strategy	strategy	NOUN
ejpam-5252	27	4	of	of	ADP
ejpam-5252	27	5	modern	modern	ADJ
ejpam-5252	27	6	roman	roman	ADJ
ejpam-5252	27	7	domination	domination	NOUN
ejpam-5252	27	8	relies	rely	VERB
ejpam-5252	27	9	on	on	ADP
ejpam-5252	27	10	a	a	DET
ejpam-5252	27	11	support	support	NOUN
ejpam-5252	27	12	system	system	NOUN
ejpam-5252	27	13	of	of	ADP
ejpam-5252	27	14	heavy	heavy	ADJ
ejpam-5252	27	15	weapons	weapon	NOUN
ejpam-5252	27	16	and	and	CCONJ
ejpam-5252	27	17	air	air	NOUN
ejpam-5252	27	18	forces	force	NOUN
ejpam-5252	27	19	to	to	PART
ejpam-5252	27	20	back	back	VERB
ejpam-5252	27	21	up	up	ADP
ejpam-5252	27	22	the	the	DET
ejpam-5252	27	23	light	light	ADJ
ejpam-5252	27	24	and	and	CCONJ
ejpam-5252	27	25	medium	medium	ADJ
ejpam-5252	27	26	weapons	weapon	NOUN
ejpam-5252	27	27	.	.	PUNCT
ejpam-5252	28	1	[	[	X
ejpam-5252	28	2	1	1	NUM
ejpam-5252	28	3	]	]	PUNCT
ejpam-5252	28	4	.	.	PUNCT
ejpam-5252	29	1	this	this	DET
ejpam-5252	29	2	study	study	NOUN
ejpam-5252	29	3	explores	explore	VERB
ejpam-5252	29	4	further	far	ADV
ejpam-5252	29	5	the	the	DET
ejpam-5252	29	6	concept	concept	NOUN
ejpam-5252	29	7	of	of	ADP
ejpam-5252	29	8	modern	modern	ADJ
ejpam-5252	29	9	roman	roman	ADJ
ejpam-5252	29	10	domination	domination	NOUN
ejpam-5252	29	11	in	in	ADP
ejpam-5252	29	12	graphs	graph	NOUN
ejpam-5252	29	13	.	.	PUNCT
ejpam-5252	30	1	it	it	PRON
ejpam-5252	30	2	focuses	focus	VERB
ejpam-5252	30	3	on	on	ADP
ejpam-5252	30	4	providing	provide	VERB
ejpam-5252	30	5	the	the	DET
ejpam-5252	30	6	modern	modern	ADJ
ejpam-5252	30	7	roman	roman	ADJ
ejpam-5252	30	8	domination	domination	NOUN
ejpam-5252	30	9	number	number	NOUN
ejpam-5252	30	10	of	of	ADP
ejpam-5252	30	11	some	some	DET
ejpam-5252	30	12	specials	special	NOUN
ejpam-5252	30	13	graphs	graph	NOUN
ejpam-5252	30	14	and	and	CCONJ
ejpam-5252	30	15	some	some	DET
ejpam-5252	30	16	characterizations	characterization	NOUN
ejpam-5252	30	17	for	for	ADP
ejpam-5252	30	18	the	the	DET
ejpam-5252	30	19	modern	modern	ADJ
ejpam-5252	30	20	roman	roman	ADJ
ejpam-5252	30	21	dominating	dominating	NOUN
ejpam-5252	30	22	functions	function	NOUN
ejpam-5252	30	23	of	of	ADP
ejpam-5252	30	24	the	the	DET
ejpam-5252	30	25	join	join	NOUN
ejpam-5252	30	26	and	and	CCONJ
ejpam-5252	30	27	corona	corona	NOUN
ejpam-5252	30	28	of	of	ADP
ejpam-5252	30	29	graphs	graph	NOUN
ejpam-5252	30	30	.	.	PUNCT
ejpam-5252	31	1	2	2	X
ejpam-5252	31	2	.	.	X
ejpam-5252	31	3	terminology	terminology	NOUN
ejpam-5252	31	4	and	and	CCONJ
ejpam-5252	31	5	notation	notation	NOUN
ejpam-5252	31	6	the	the	DET
ejpam-5252	31	7	symbols	symbol	NOUN
ejpam-5252	31	8	v	v	ADP
ejpam-5252	31	9	(	(	PUNCT
ejpam-5252	31	10	g	g	NOUN
ejpam-5252	31	11	)	)	PUNCT
ejpam-5252	31	12	and	and	CCONJ
ejpam-5252	31	13	e(g	e(g	PROPN
ejpam-5252	31	14	)	)	PUNCT
ejpam-5252	31	15	denote	denote	VERB
ejpam-5252	31	16	the	the	DET
ejpam-5252	31	17	vertex	vertex	NOUN
ejpam-5252	31	18	set	set	NOUN
ejpam-5252	31	19	and	and	CCONJ
ejpam-5252	31	20	edge	edge	NOUN
ejpam-5252	31	21	set	set	NOUN
ejpam-5252	31	22	,	,	PUNCT
ejpam-5252	31	23	respectively	respectively	ADV
ejpam-5252	31	24	,	,	PUNCT
ejpam-5252	31	25	of	of	ADP
ejpam-5252	31	26	a	a	DET
ejpam-5252	31	27	graph	graph	NOUN
ejpam-5252	31	28	g.	g.	NOUN
ejpam-5252	31	29	for	for	ADP
ejpam-5252	31	30	s	s	PROPN
ejpam-5252	31	31	⊆	⊆	NUM
ejpam-5252	31	32	v	v	NOUN
ejpam-5252	31	33	(	(	PUNCT
ejpam-5252	31	34	g	g	NOUN
ejpam-5252	31	35	)	)	PUNCT
ejpam-5252	31	36	,	,	PUNCT
ejpam-5252	31	37	|s|	|s|	PROPN
ejpam-5252	31	38	is	be	AUX
ejpam-5252	31	39	the	the	DET
ejpam-5252	31	40	cardinality	cardinality	NOUN
ejpam-5252	31	41	of	of	ADP
ejpam-5252	31	42	s.	s.	PROPN
ejpam-5252	31	43	in	in	ADP
ejpam-5252	31	44	particular	particular	ADJ
ejpam-5252	31	45	,	,	PUNCT
ejpam-5252	31	46	|v	|v	PROPN
ejpam-5252	31	47	(	(	PUNCT
ejpam-5252	31	48	g)|	g)|	NOUN
ejpam-5252	31	49	and	and	CCONJ
ejpam-5252	31	50	|e(g)|	|e(g)|	PROPN
ejpam-5252	31	51	are	be	AUX
ejpam-5252	31	52	the	the	DET
ejpam-5252	31	53	order	order	NOUN
ejpam-5252	31	54	and	and	CCONJ
ejpam-5252	31	55	size	size	NOUN
ejpam-5252	31	56	,	,	PUNCT
ejpam-5252	31	57	respectively	respectively	ADV
ejpam-5252	31	58	,	,	PUNCT
ejpam-5252	31	59	of	of	ADP
ejpam-5252	31	60	g.	g.	PROPN
ejpam-5252	31	61	all	all	DET
ejpam-5252	31	62	graph	graph	NOUN
ejpam-5252	31	63	terminologies	terminology	NOUN
ejpam-5252	31	64	that	that	PRON
ejpam-5252	31	65	are	be	AUX
ejpam-5252	31	66	not	not	PART
ejpam-5252	31	67	introduced	introduce	VERB
ejpam-5252	31	68	but	but	CCONJ
ejpam-5252	31	69	are	be	AUX
ejpam-5252	31	70	being	be	AUX
ejpam-5252	31	71	used	use	VERB
ejpam-5252	31	72	here	here	ADV
ejpam-5252	31	73	are	be	AUX
ejpam-5252	31	74	adapted	adapt	VERB
ejpam-5252	31	75	from	from	ADP
ejpam-5252	31	76	[	[	X
ejpam-5252	31	77	2	2	NUM
ejpam-5252	31	78	]	]	PUNCT
ejpam-5252	31	79	.	.	PUNCT
ejpam-5252	32	1	the	the	DET
ejpam-5252	32	2	set	set	NOUN
ejpam-5252	32	3	of	of	ADP
ejpam-5252	32	4	neighbors	neighbor	NOUN
ejpam-5252	32	5	of	of	ADP
ejpam-5252	32	6	a	a	DET
ejpam-5252	32	7	vertex	vertex	NOUN
ejpam-5252	32	8	u	u	NOUN
ejpam-5252	32	9	in	in	ADP
ejpam-5252	32	10	g	g	NOUN
ejpam-5252	32	11	,	,	PUNCT
ejpam-5252	32	12	denoted	denote	VERB
ejpam-5252	32	13	by	by	ADP
ejpam-5252	32	14	ng(u	ng(u	NOUN
ejpam-5252	32	15	)	)	PUNCT
ejpam-5252	32	16	,	,	PUNCT
ejpam-5252	32	17	is	be	AUX
ejpam-5252	32	18	called	call	VERB
ejpam-5252	32	19	the	the	DET
ejpam-5252	32	20	open	open	ADJ
ejpam-5252	32	21	neighborhood	neighborhood	NOUN
ejpam-5252	32	22	of	of	ADP
ejpam-5252	32	23	u	u	PROPN
ejpam-5252	32	24	in	in	ADP
ejpam-5252	32	25	g.	g.	PROPN
ejpam-5252	32	26	the	the	DET
ejpam-5252	32	27	closed	close	VERB
ejpam-5252	32	28	neighborhood	neighborhood	NOUN
ejpam-5252	32	29	of	of	ADP
ejpam-5252	32	30	u	u	NOUN
ejpam-5252	32	31	in	in	ADP
ejpam-5252	32	32	g	g	PROPN
ejpam-5252	32	33	is	be	AUX
ejpam-5252	32	34	the	the	DET
ejpam-5252	32	35	set	set	NOUN
ejpam-5252	32	36	ng[u	ng[u	PROPN
ejpam-5252	32	37	]	]	X
ejpam-5252	32	38	=	=	SYM
ejpam-5252	32	39	ng(u	ng(u	PROPN
ejpam-5252	32	40	)	)	PUNCT
ejpam-5252	32	41	∪	∪	NOUN
ejpam-5252	32	42	{	{	PUNCT
ejpam-5252	32	43	u	u	NOUN
ejpam-5252	32	44	}	}	PUNCT
ejpam-5252	32	45	.	.	PUNCT
ejpam-5252	33	1	if	if	SCONJ
ejpam-5252	33	2	s	s	VERB
ejpam-5252	33	3	⊆	⊆	NUM
ejpam-5252	33	4	v	v	NOUN
ejpam-5252	33	5	(	(	PUNCT
ejpam-5252	33	6	g	g	NOUN
ejpam-5252	33	7	)	)	PUNCT
ejpam-5252	33	8	,	,	PUNCT
ejpam-5252	33	9	the	the	DET
ejpam-5252	33	10	open	open	ADJ
ejpam-5252	33	11	neighborhood	neighborhood	NOUN
ejpam-5252	33	12	of	of	ADP
ejpam-5252	33	13	s	s	NOUN
ejpam-5252	33	14	in	in	ADP
ejpam-5252	33	15	g	g	PROPN
ejpam-5252	33	16	is	be	AUX
ejpam-5252	33	17	the	the	DET
ejpam-5252	33	18	set	set	NOUN
ejpam-5252	33	19	ng(s	ng(s	NOUN
ejpam-5252	33	20	)	)	PUNCT
ejpam-5252	33	21	=	=	SYM
ejpam-5252	34	1	⋃	⋃	NOUN
ejpam-5252	34	2	u∈s	u∈s	NOUN
ejpam-5252	34	3	ng(u	ng(u	NUM
ejpam-5252	34	4	)	)	PUNCT
ejpam-5252	34	5	.	.	PUNCT
ejpam-5252	35	1	the	the	DET
ejpam-5252	35	2	closed	closed	ADJ
ejpam-5252	35	3	neighborhood	neighborhood	NOUN
ejpam-5252	35	4	of	of	ADP
ejpam-5252	35	5	s	s	NOUN
ejpam-5252	35	6	in	in	ADP
ejpam-5252	35	7	g	g	PROPN
ejpam-5252	35	8	is	be	AUX
ejpam-5252	35	9	the	the	DET
ejpam-5252	35	10	set	set	VERB
ejpam-5252	35	11	ng[s	ng[	NOUN
ejpam-5252	35	12	]	]	PUNCT
ejpam-5252	35	13	=	=	PUNCT
ejpam-5252	35	14	ng(s)∪s	ng(s)∪s	PROPN
ejpam-5252	35	15	.	.	PUNCT
ejpam-5252	36	1	for	for	ADP
ejpam-5252	36	2	s	s	PROPN
ejpam-5252	36	3	⊆	⊆	NUM
ejpam-5252	36	4	v	v	NOUN
ejpam-5252	36	5	(	(	PUNCT
ejpam-5252	36	6	g	g	NOUN
ejpam-5252	36	7	)	)	PUNCT
ejpam-5252	36	8	of	of	ADP
ejpam-5252	36	9	a	a	DET
ejpam-5252	36	10	connected	connected	ADJ
ejpam-5252	36	11	graph	graph	NOUN
ejpam-5252	36	12	g	g	NOUN
ejpam-5252	36	13	,	,	PUNCT
ejpam-5252	36	14	ng(s	ng(s	NUM
ejpam-5252	36	15	)	)	PUNCT
ejpam-5252	37	1	=	=	SYM
ejpam-5252	38	1	⋃	⋃	ADP
ejpam-5252	38	2	v∈s	v∈s	NOUN
ejpam-5252	38	3	ng(v	ng(v	NOUN
ejpam-5252	38	4	)	)	PUNCT
ejpam-5252	38	5	and	and	CCONJ
ejpam-5252	38	6	ng[s	ng[s	PROPN
ejpam-5252	38	7	]	]	PUNCT
ejpam-5252	38	8	=	=	SYM
ejpam-5252	38	9	s∪ng(s	s∪ng(s	PROPN
ejpam-5252	38	10	)	)	PUNCT
ejpam-5252	38	11	.	.	PUNCT
ejpam-5252	39	1	a	a	DET
ejpam-5252	39	2	graph	graph	NOUN
ejpam-5252	39	3	whose	whose	DET
ejpam-5252	39	4	edge	edge	NOUN
ejpam-5252	39	5	-	-	PUNCT
ejpam-5252	39	6	set	set	NOUN
ejpam-5252	39	7	is	be	AUX
ejpam-5252	39	8	empty	empty	ADJ
ejpam-5252	39	9	is	be	AUX
ejpam-5252	39	10	called	call	VERB
ejpam-5252	39	11	an	an	DET
ejpam-5252	39	12	empty	empty	ADJ
ejpam-5252	39	13	graph	graph	NOUN
ejpam-5252	39	14	(	(	PUNCT
ejpam-5252	39	15	also	also	ADV
ejpam-5252	39	16	called	call	VERB
ejpam-5252	39	17	null	null	ADJ
ejpam-5252	39	18	graph	graph	NOUN
ejpam-5252	39	19	or	or	CCONJ
ejpam-5252	39	20	totally	totally	ADV
ejpam-5252	39	21	disconnected	disconnected	ADJ
ejpam-5252	39	22	graph	graph	NOUN
ejpam-5252	39	23	)	)	PUNCT
ejpam-5252	39	24	.	.	PUNCT
ejpam-5252	40	1	an	an	DET
ejpam-5252	40	2	empty	empty	ADJ
ejpam-5252	40	3	graph	graph	NOUN
ejpam-5252	40	4	of	of	ADP
ejpam-5252	40	5	order	order	NOUN
ejpam-5252	40	6	n	n	NOUN
ejpam-5252	40	7	is	be	AUX
ejpam-5252	40	8	denoted	denote	VERB
ejpam-5252	40	9	by	by	ADP
ejpam-5252	40	10	kn	kn	PROPN
ejpam-5252	40	11	.	.	PUNCT
ejpam-5252	41	1	a	a	DET
ejpam-5252	41	2	set	set	NOUN
ejpam-5252	41	3	s	s	NOUN
ejpam-5252	41	4	⊆	⊆	NUM
ejpam-5252	41	5	v	v	NOUN
ejpam-5252	41	6	(	(	PUNCT
ejpam-5252	41	7	g	g	NOUN
ejpam-5252	41	8	)	)	PUNCT
ejpam-5252	41	9	is	be	AUX
ejpam-5252	41	10	a	a	DET
ejpam-5252	41	11	dominating	dominating	NOUN
ejpam-5252	41	12	set	set	VERB
ejpam-5252	41	13	in	in	ADP
ejpam-5252	41	14	g	g	PROPN
ejpam-5252	41	15	if	if	SCONJ
ejpam-5252	41	16	ng[s	ng[	NOUN
ejpam-5252	41	17	]	]	PUNCT
ejpam-5252	41	18	=	=	SYM
ejpam-5252	41	19	v	v	NOUN
ejpam-5252	41	20	(	(	PUNCT
ejpam-5252	41	21	g	g	NOUN
ejpam-5252	41	22	)	)	PUNCT
ejpam-5252	41	23	.	.	PUNCT
ejpam-5252	42	1	thus	thus	ADV
ejpam-5252	42	2	,	,	PUNCT
ejpam-5252	42	3	s	s	VERB
ejpam-5252	42	4	is	be	AUX
ejpam-5252	42	5	a	a	DET
ejpam-5252	42	6	dominating	dominating	NOUN
ejpam-5252	42	7	set	set	VERB
ejpam-5252	42	8	in	in	ADP
ejpam-5252	42	9	g	g	PROPN
ejpam-5252	42	10	if	if	SCONJ
ejpam-5252	42	11	and	and	CCONJ
ejpam-5252	42	12	only	only	ADV
ejpam-5252	42	13	if	if	SCONJ
ejpam-5252	42	14	for	for	ADP
ejpam-5252	42	15	each	each	PRON
ejpam-5252	42	16	v	v	NUM
ejpam-5252	42	17	∈	∈	PROPN
ejpam-5252	42	18	v	v	NOUN
ejpam-5252	42	19	(	(	PUNCT
ejpam-5252	42	20	g	g	NOUN
ejpam-5252	42	21	)	)	PUNCT
ejpam-5252	42	22	\	\	PROPN
ejpam-5252	43	1	s	s	X
ejpam-5252	43	2	,	,	PUNCT
ejpam-5252	43	3	there	there	PRON
ejpam-5252	43	4	exists	exist	VERB
ejpam-5252	43	5	u	u	PROPN
ejpam-5252	43	6	∈	∈	PROPN
ejpam-5252	43	7	s	s	VERB
ejpam-5252	43	8	such	such	ADJ
ejpam-5252	43	9	that	that	DET
ejpam-5252	43	10	uv	uv	PROPN
ejpam-5252	43	11	∈	∈	PROPN
ejpam-5252	43	12	e(g	e(g	PROPN
ejpam-5252	43	13	)	)	PUNCT
ejpam-5252	43	14	.	.	PUNCT
ejpam-5252	44	1	the	the	DET
ejpam-5252	44	2	minimum	minimum	ADJ
ejpam-5252	44	3	cardinality	cardinality	NOUN
ejpam-5252	44	4	of	of	ADP
ejpam-5252	44	5	a	a	DET
ejpam-5252	44	6	dominating	dominating	NOUN
ejpam-5252	44	7	set	set	NOUN
ejpam-5252	44	8	in	in	ADP
ejpam-5252	44	9	g	g	NOUN
ejpam-5252	44	10	,	,	PUNCT
ejpam-5252	44	11	denoted	denote	VERB
ejpam-5252	44	12	by	by	ADP
ejpam-5252	44	13	γ(g	γ(g	PROPN
ejpam-5252	44	14	)	)	PUNCT
ejpam-5252	44	15	,	,	PUNCT
ejpam-5252	44	16	is	be	AUX
ejpam-5252	44	17	the	the	DET
ejpam-5252	44	18	domination	domination	NOUN
ejpam-5252	44	19	number	number	NOUN
ejpam-5252	44	20	of	of	ADP
ejpam-5252	44	21	g.	g.	PROPN
ejpam-5252	44	22	a	a	DET
ejpam-5252	44	23	dominating	dominating	NOUN
ejpam-5252	44	24	set	set	NOUN
ejpam-5252	44	25	s	s	NOUN
ejpam-5252	44	26	of	of	ADP
ejpam-5252	44	27	g	g	NOUN
ejpam-5252	44	28	with	with	ADP
ejpam-5252	44	29	|s|	|s|	PROPN
ejpam-5252	44	30	=	=	SYM
ejpam-5252	44	31	γ(g	γ(g	PROPN
ejpam-5252	44	32	)	)	PUNCT
ejpam-5252	44	33	is	be	AUX
ejpam-5252	44	34	called	call	VERB
ejpam-5252	44	35	a	a	DET
ejpam-5252	44	36	γ	γ	NOUN
ejpam-5252	44	37	set	set	NOUN
ejpam-5252	44	38	of	of	ADP
ejpam-5252	44	39	g.	g.	PROPN
ejpam-5252	44	40	readers	reader	NOUN
ejpam-5252	44	41	may	may	AUX
ejpam-5252	44	42	refer	refer	VERB
ejpam-5252	44	43	to	to	ADP
ejpam-5252	44	44	[	[	X
ejpam-5252	44	45	5	5	NUM
ejpam-5252	44	46	]	]	PUNCT
ejpam-5252	44	47	for	for	ADP
ejpam-5252	44	48	the	the	DET
ejpam-5252	44	49	introduction	introduction	NOUN
ejpam-5252	44	50	and	and	CCONJ
ejpam-5252	44	51	more	more	ADV
ejpam-5252	44	52	comprehensive	comprehensive	ADJ
ejpam-5252	44	53	discussion	discussion	NOUN
ejpam-5252	44	54	of	of	ADP
ejpam-5252	44	55	the	the	DET
ejpam-5252	44	56	development	development	NOUN
ejpam-5252	44	57	of	of	ADP
ejpam-5252	44	58	the	the	DET
ejpam-5252	44	59	concept	concept	NOUN
ejpam-5252	44	60	of	of	ADP
ejpam-5252	44	61	domination	domination	NOUN
ejpam-5252	44	62	in	in	ADP
ejpam-5252	44	63	graphs	graph	NOUN
ejpam-5252	44	64	.	.	PUNCT
ejpam-5252	45	1	for	for	ADP
ejpam-5252	45	2	a	a	DET
ejpam-5252	45	3	positive	positive	ADJ
ejpam-5252	45	4	integer	integer	NOUN
ejpam-5252	45	5	k	k	PROPN
ejpam-5252	45	6	,	,	PUNCT
ejpam-5252	45	7	a	a	DET
ejpam-5252	45	8	set	set	NOUN
ejpam-5252	45	9	d	d	NOUN
ejpam-5252	45	10	⊆	⊆	NUM
ejpam-5252	45	11	v	v	ADP
ejpam-5252	45	12	(	(	PUNCT
ejpam-5252	45	13	g	g	NOUN
ejpam-5252	45	14	)	)	PUNCT
ejpam-5252	45	15	is	be	AUX
ejpam-5252	45	16	called	call	VERB
ejpam-5252	45	17	a	a	DET
ejpam-5252	45	18	k	k	ADJ
ejpam-5252	45	19	-	-	PUNCT
ejpam-5252	45	20	dominating	dominating	NOUN
ejpam-5252	45	21	set	set	NOUN
ejpam-5252	45	22	if	if	SCONJ
ejpam-5252	45	23	each	each	DET
ejpam-5252	45	24	x	x	SYM
ejpam-5252	45	25	∈	∈	PROPN
ejpam-5252	45	26	v	v	ADP
ejpam-5252	45	27	(	(	PUNCT
ejpam-5252	45	28	g	g	NOUN
ejpam-5252	45	29	)	)	PUNCT
ejpam-5252	45	30	\d	\d	NOUN
ejpam-5252	45	31	is	be	AUX
ejpam-5252	45	32	adjacent	adjacent	ADJ
ejpam-5252	45	33	to	to	ADP
ejpam-5252	45	34	at	at	ADP
ejpam-5252	45	35	least	least	ADJ
ejpam-5252	45	36	k	k	X
ejpam-5252	45	37	vertices	vertice	VERB
ejpam-5252	45	38	in	in	ADP
ejpam-5252	45	39	d.	d.	PROPN
ejpam-5252	45	40	the	the	DET
ejpam-5252	45	41	k	k	ADJ
ejpam-5252	45	42	-	-	PUNCT
ejpam-5252	45	43	domination	domination	NOUN
ejpam-5252	45	44	number	number	NOUN
ejpam-5252	45	45	γk(g	γk(g	PUNCT
ejpam-5252	45	46	)	)	PUNCT
ejpam-5252	45	47	is	be	AUX
ejpam-5252	45	48	then	then	ADV
ejpam-5252	45	49	defined	define	VERB
ejpam-5252	45	50	to	to	PART
ejpam-5252	45	51	be	be	AUX
ejpam-5252	45	52	the	the	DET
ejpam-5252	45	53	smallest	small	ADJ
ejpam-5252	45	54	cardinality	cardinality	NOUN
ejpam-5252	45	55	of	of	ADP
ejpam-5252	45	56	a	a	DET
ejpam-5252	45	57	k	k	ADV
ejpam-5252	45	58	-	-	PUNCT
ejpam-5252	45	59	dominating	dominating	ADJ
ejpam-5252	45	60	set	set	NOUN
ejpam-5252	45	61	of	of	ADP
ejpam-5252	45	62	g.	g.	PROPN
ejpam-5252	45	63	a	a	DET
ejpam-5252	45	64	roman	roman	ADJ
ejpam-5252	45	65	dominating	dominating	NOUN
ejpam-5252	45	66	function	function	NOUN
ejpam-5252	45	67	(	(	PUNCT
ejpam-5252	45	68	rdf	rdf	NOUN
ejpam-5252	45	69	)	)	PUNCT
ejpam-5252	45	70	on	on	ADP
ejpam-5252	45	71	g	g	PROPN
ejpam-5252	45	72	is	be	AUX
ejpam-5252	45	73	a	a	DET
ejpam-5252	45	74	function	function	NOUN
ejpam-5252	45	75	f	f	NOUN
ejpam-5252	45	76	:	:	PUNCT
ejpam-5252	45	77	v	v	X
ejpam-5252	45	78	(	(	PUNCT
ejpam-5252	45	79	g	g	NOUN
ejpam-5252	45	80	)	)	PUNCT
ejpam-5252	45	81	→	→	SYM
ejpam-5252	45	82	{	{	PUNCT
ejpam-5252	45	83	0	0	NUM
ejpam-5252	45	84	,	,	PUNCT
ejpam-5252	45	85	1	1	NUM
ejpam-5252	45	86	,	,	PUNCT
ejpam-5252	45	87	2	2	NUM
ejpam-5252	45	88	}	}	PUNCT
ejpam-5252	45	89	such	such	ADJ
ejpam-5252	45	90	that	that	SCONJ
ejpam-5252	45	91	every	every	DET
ejpam-5252	45	92	vertex	vertex	NOUN
ejpam-5252	45	93	u	u	NOUN
ejpam-5252	45	94	∈	∈	PROPN
ejpam-5252	45	95	v	v	ADP
ejpam-5252	45	96	(	(	PUNCT
ejpam-5252	45	97	g	g	NOUN
ejpam-5252	45	98	)	)	PUNCT
ejpam-5252	45	99	for	for	ADP
ejpam-5252	45	100	which	which	PRON
ejpam-5252	45	101	f(u	f(u	PROPN
ejpam-5252	45	102	)	)	PUNCT
ejpam-5252	46	1	=	=	SYM
ejpam-5252	46	2	0	0	NUM
ejpam-5252	46	3	is	be	AUX
ejpam-5252	46	4	adjacent	adjacent	ADJ
ejpam-5252	46	5	to	to	ADP
ejpam-5252	46	6	at	at	ADV
ejpam-5252	46	7	least	least	ADV
ejpam-5252	46	8	one	one	NUM
ejpam-5252	46	9	vertex	vertex	NOUN
ejpam-5252	46	10	v	v	NOUN
ejpam-5252	46	11	for	for	ADP
ejpam-5252	46	12	which	which	PRON
ejpam-5252	46	13	f(v	f(v	NOUN
ejpam-5252	46	14	)	)	PUNCT
ejpam-5252	46	15	=	=	SYM
ejpam-5252	47	1	2	2	X
ejpam-5252	47	2	.	.	PUNCT
ejpam-5252	47	3	the	the	DET
ejpam-5252	47	4	weight	weight	NOUN
ejpam-5252	47	5	of	of	ADP
ejpam-5252	47	6	an	an	DET
ejpam-5252	47	7	rdf	rdf	NOUN
ejpam-5252	47	8	is	be	AUX
ejpam-5252	47	9	the	the	DET
ejpam-5252	47	10	value	value	NOUN
ejpam-5252	47	11	ωg(f	ωg(f	PRON
ejpam-5252	47	12	)	)	PUNCT
ejpam-5252	47	13	=	=	SYM
ejpam-5252	47	14	∑	∑	PUNCT
ejpam-5252	47	15	u∈v	u∈v	NOUN
ejpam-5252	47	16	(	(	PUNCT
ejpam-5252	47	17	g	g	NOUN
ejpam-5252	47	18	)	)	PUNCT
ejpam-5252	47	19	f(u	f(u	PROPN
ejpam-5252	47	20	)	)	PUNCT
ejpam-5252	47	21	.	.	PUNCT
ejpam-5252	48	1	the	the	DET
ejpam-5252	48	2	roman	roman	ADJ
ejpam-5252	48	3	domination	domination	NOUN
ejpam-5252	48	4	number	number	NOUN
ejpam-5252	48	5	γr(g	γr(g	PROPN
ejpam-5252	48	6	)	)	PUNCT
ejpam-5252	48	7	is	be	AUX
ejpam-5252	48	8	the	the	DET
ejpam-5252	48	9	minimum	minimum	ADJ
ejpam-5252	48	10	weight	weight	NOUN
ejpam-5252	48	11	among	among	ADP
ejpam-5252	48	12	all	all	PRON
ejpam-5252	48	13	of	of	ADP
ejpam-5252	48	14	the	the	DET
ejpam-5252	48	15	rdf	rdf	NOUN
ejpam-5252	48	16	on	on	ADP
ejpam-5252	48	17	g.	g.	PROPN
ejpam-5252	48	18	an	an	DET
ejpam-5252	48	19	rdf	rdf	NOUN
ejpam-5252	48	20	with	with	ADP
ejpam-5252	48	21	ωg(f	ωg(f	NOUN
ejpam-5252	48	22	)	)	PUNCT
ejpam-5252	48	23	=	=	SYM
ejpam-5252	48	24	γr(g	γr(g	NOUN
ejpam-5252	48	25	)	)	PUNCT
ejpam-5252	48	26	is	be	AUX
ejpam-5252	48	27	referred	refer	VERB
ejpam-5252	48	28	to	to	ADP
ejpam-5252	48	29	as	as	ADP
ejpam-5252	48	30	a	a	DET
ejpam-5252	48	31	γr	γr	NOUN
ejpam-5252	48	32	-	-	NOUN
ejpam-5252	48	33	function	function	NOUN
ejpam-5252	48	34	[	[	X
ejpam-5252	48	35	3	3	NUM
ejpam-5252	48	36	]	]	PUNCT
ejpam-5252	48	37	.	.	PUNCT
ejpam-5252	49	1	s.	s.	PROPN
ejpam-5252	49	2	ahamad	ahamad	PROPN
ejpam-5252	49	3	,	,	PUNCT
ejpam-5252	49	4	j.	j.	PROPN
ejpam-5252	49	5	cariaga	cariaga	PROPN
ejpam-5252	49	6	,	,	PUNCT
ejpam-5252	49	7	s.	s.	PROPN
ejpam-5252	49	8	menchavez	menchavez	PROPN
ejpam-5252	49	9	/	/	PUNCT
ejpam-5252	49	10	eur	eur	PROPN
ejpam-5252	49	11	.	.	PUNCT
ejpam-5252	50	1	j.	j.	PROPN
ejpam-5252	50	2	pure	pure	PROPN
ejpam-5252	50	3	appl	appl	PROPN
ejpam-5252	50	4	.	.	PROPN
ejpam-5252	50	5	math	math	PROPN
ejpam-5252	50	6	,	,	PUNCT
ejpam-5252	50	7	18	18	NUM
ejpam-5252	50	8	(	(	PUNCT
ejpam-5252	50	9	1	1	NUM
ejpam-5252	50	10	)	)	PUNCT
ejpam-5252	50	11	(	(	PUNCT
ejpam-5252	50	12	2025	2025	NUM
ejpam-5252	50	13	)	)	PUNCT
ejpam-5252	50	14	,	,	PUNCT
ejpam-5252	50	15	5252	5252	NUM
ejpam-5252	50	16	3	3	NUM
ejpam-5252	50	17	of	of	ADP
ejpam-5252	50	18	18	18	NUM
ejpam-5252	50	19	a	a	DET
ejpam-5252	50	20	function	function	NOUN
ejpam-5252	51	1	f	f	NOUN
ejpam-5252	51	2	:	:	PUNCT
ejpam-5252	51	3	v	v	X
ejpam-5252	51	4	(	(	PUNCT
ejpam-5252	51	5	g	g	NOUN
ejpam-5252	51	6	)	)	PUNCT
ejpam-5252	51	7	→	→	SYM
ejpam-5252	51	8	{	{	PUNCT
ejpam-5252	51	9	0	0	NUM
ejpam-5252	51	10	,	,	PUNCT
ejpam-5252	51	11	1	1	NUM
ejpam-5252	51	12	,	,	PUNCT
ejpam-5252	51	13	2	2	NUM
ejpam-5252	51	14	,	,	PUNCT
ejpam-5252	51	15	3	3	NUM
ejpam-5252	51	16	}	}	PUNCT
ejpam-5252	51	17	is	be	AUX
ejpam-5252	51	18	a	a	DET
ejpam-5252	51	19	double	double	ADJ
ejpam-5252	51	20	roman	roman	ADJ
ejpam-5252	51	21	dominating	dominating	NOUN
ejpam-5252	51	22	function	function	NOUN
ejpam-5252	51	23	of	of	ADP
ejpam-5252	51	24	g	g	NOUN
ejpam-5252	51	25	,	,	PUNCT
ejpam-5252	51	26	written	write	VERB
ejpam-5252	51	27	f	f	PROPN
ejpam-5252	51	28	∈	∈	PROPN
ejpam-5252	51	29	drd(g	drd(g	PROPN
ejpam-5252	51	30	)	)	PUNCT
ejpam-5252	51	31	,	,	PUNCT
ejpam-5252	51	32	if	if	SCONJ
ejpam-5252	51	33	each	each	PRON
ejpam-5252	51	34	of	of	ADP
ejpam-5252	51	35	the	the	DET
ejpam-5252	51	36	following	follow	VERB
ejpam-5252	51	37	holds	hold	VERB
ejpam-5252	51	38	:	:	PUNCT
ejpam-5252	51	39	(	(	PUNCT
ejpam-5252	51	40	1	1	X
ejpam-5252	51	41	)	)	PUNCT
ejpam-5252	51	42	for	for	ADP
ejpam-5252	51	43	each	each	DET
ejpam-5252	51	44	v	v	NUM
ejpam-5252	51	45	∈	∈	PROPN
ejpam-5252	51	46	v	v	NOUN
ejpam-5252	51	47	(	(	PUNCT
ejpam-5252	51	48	g	g	NOUN
ejpam-5252	51	49	)	)	PUNCT
ejpam-5252	51	50	with	with	ADP
ejpam-5252	51	51	f(v	f(v	NOUN
ejpam-5252	51	52	)	)	PUNCT
ejpam-5252	52	1	=	=	SYM
ejpam-5252	52	2	0	0	PUNCT
ejpam-5252	52	3	at	at	ADV
ejpam-5252	52	4	least	least	ADJ
ejpam-5252	52	5	one	one	NUM
ejpam-5252	52	6	of	of	ADP
ejpam-5252	52	7	the	the	DET
ejpam-5252	52	8	following	following	NOUN
ejpam-5252	52	9	holds	hold	VERB
ejpam-5252	52	10	:	:	PUNCT
ejpam-5252	52	11	(	(	PUNCT
ejpam-5252	52	12	a	a	X
ejpam-5252	52	13	)	)	PUNCT
ejpam-5252	52	14	v	v	NOUN
ejpam-5252	52	15	has	have	VERB
ejpam-5252	52	16	two	two	NUM
ejpam-5252	52	17	adjacent	adjacent	ADJ
ejpam-5252	52	18	vertices	vertex	NOUN
ejpam-5252	52	19	u	u	NOUN
ejpam-5252	52	20	and	and	CCONJ
ejpam-5252	52	21	w	w	NOUN
ejpam-5252	52	22	for	for	ADP
ejpam-5252	52	23	which	which	PRON
ejpam-5252	52	24	f(u	f(u	PROPN
ejpam-5252	52	25	)	)	PUNCT
ejpam-5252	52	26	=	=	PUNCT
ejpam-5252	53	1	f(w	f(w	PROPN
ejpam-5252	53	2	)	)	PUNCT
ejpam-5252	53	3	=	=	SYM
ejpam-5252	53	4	2	2	NUM
ejpam-5252	53	5	;	;	PUNCT
ejpam-5252	53	6	or	or	CCONJ
ejpam-5252	53	7	(	(	PUNCT
ejpam-5252	53	8	b	b	X
ejpam-5252	53	9	)	)	PUNCT
ejpam-5252	53	10	v	v	NOUN
ejpam-5252	53	11	has	have	VERB
ejpam-5252	53	12	an	an	DET
ejpam-5252	53	13	adjacent	adjacent	ADJ
ejpam-5252	53	14	vertex	vertex	NOUN
ejpam-5252	53	15	u	u	NOUN
ejpam-5252	53	16	for	for	ADP
ejpam-5252	53	17	which	which	PRON
ejpam-5252	53	18	f(u	f(u	PROPN
ejpam-5252	53	19	)	)	PUNCT
ejpam-5252	53	20	=	=	SYM
ejpam-5252	53	21	3	3	NUM
ejpam-5252	53	22	,	,	PUNCT
ejpam-5252	53	23	and	and	CCONJ
ejpam-5252	53	24	(	(	PUNCT
ejpam-5252	53	25	2	2	X
ejpam-5252	53	26	)	)	PUNCT
ejpam-5252	53	27	for	for	ADP
ejpam-5252	53	28	each	each	DET
ejpam-5252	53	29	v	v	NUM
ejpam-5252	53	30	∈	∈	PROPN
ejpam-5252	53	31	v	v	NOUN
ejpam-5252	53	32	(	(	PUNCT
ejpam-5252	53	33	g	g	NOUN
ejpam-5252	53	34	)	)	PUNCT
ejpam-5252	53	35	with	with	ADP
ejpam-5252	53	36	f(v	f(v	NOUN
ejpam-5252	53	37	)	)	PUNCT
ejpam-5252	53	38	=	=	SYM
ejpam-5252	53	39	1	1	NUM
ejpam-5252	53	40	,	,	PUNCT
ejpam-5252	53	41	v	v	NOUN
ejpam-5252	53	42	is	be	AUX
ejpam-5252	53	43	adjacent	adjacent	ADJ
ejpam-5252	53	44	to	to	ADP
ejpam-5252	53	45	a	a	DET
ejpam-5252	53	46	vertex	vertex	NOUN
ejpam-5252	53	47	u	u	NOUN
ejpam-5252	53	48	for	for	ADP
ejpam-5252	53	49	which	which	PRON
ejpam-5252	53	50	either	either	CCONJ
ejpam-5252	53	51	f(u	f(u	PROPN
ejpam-5252	53	52	)	)	PUNCT
ejpam-5252	53	53	=	=	SYM
ejpam-5252	53	54	2	2	NUM
ejpam-5252	53	55	or	or	CCONJ
ejpam-5252	53	56	f(u	f(u	PROPN
ejpam-5252	53	57	)	)	PUNCT
ejpam-5252	53	58	=	=	SYM
ejpam-5252	54	1	3	3	X
ejpam-5252	54	2	.	.	X
ejpam-5252	54	3	the	the	DET
ejpam-5252	54	4	double	double	ADJ
ejpam-5252	54	5	roman	roman	ADJ
ejpam-5252	54	6	domination	domination	NOUN
ejpam-5252	54	7	number	number	NOUN
ejpam-5252	54	8	of	of	ADP
ejpam-5252	54	9	g	g	NOUN
ejpam-5252	54	10	denoted	denote	VERB
ejpam-5252	54	11	by	by	ADP
ejpam-5252	54	12	γdr(g	γdr(g	PROPN
ejpam-5252	54	13	)	)	PUNCT
ejpam-5252	54	14	,	,	PUNCT
ejpam-5252	54	15	is	be	AUX
ejpam-5252	54	16	the	the	DET
ejpam-5252	54	17	minimum	minimum	ADJ
ejpam-5252	54	18	weight	weight	NOUN
ejpam-5252	54	19	ωg(f	ωg(f	PRON
ejpam-5252	54	20	)	)	PUNCT
ejpam-5252	54	21	=	=	SYM
ejpam-5252	54	22	∑	∑	PUNCT
ejpam-5252	54	23	v∈v	v∈v	PROPN
ejpam-5252	54	24	(	(	PUNCT
ejpam-5252	54	25	g	g	NOUN
ejpam-5252	54	26	)	)	PUNCT
ejpam-5252	54	27	f(v	f(v	NOUN
ejpam-5252	54	28	)	)	PUNCT
ejpam-5252	54	29	of	of	ADP
ejpam-5252	54	30	all	all	DET
ejpam-5252	54	31	the	the	DET
ejpam-5252	54	32	double	double	ADJ
ejpam-5252	54	33	roman	roman	ADJ
ejpam-5252	54	34	dominating	dominating	NOUN
ejpam-5252	54	35	functions	function	NOUN
ejpam-5252	54	36	f	f	PROPN
ejpam-5252	54	37	of	of	ADP
ejpam-5252	54	38	g.	g.	PROPN
ejpam-5252	55	1	any	any	DET
ejpam-5252	55	2	f	f	PROPN
ejpam-5252	55	3	∈	∈	PROPN
ejpam-5252	55	4	drd(g	drd(g	PROPN
ejpam-5252	55	5	)	)	PUNCT
ejpam-5252	55	6	of	of	ADP
ejpam-5252	55	7	weight	weight	NOUN
ejpam-5252	55	8	equal	equal	ADJ
ejpam-5252	55	9	to	to	ADP
ejpam-5252	55	10	γdr(g	γdr(g	PROPN
ejpam-5252	55	11	)	)	PUNCT
ejpam-5252	55	12	is	be	AUX
ejpam-5252	55	13	referred	refer	VERB
ejpam-5252	55	14	to	to	ADP
ejpam-5252	55	15	as	as	ADP
ejpam-5252	55	16	γdr	γdr	NOUN
ejpam-5252	55	17	-	-	PUNCT
ejpam-5252	55	18	function	function	NOUN
ejpam-5252	55	19	of	of	ADP
ejpam-5252	55	20	g	g	NOUN
ejpam-5252	55	21	[	[	X
ejpam-5252	55	22	3	3	NUM
ejpam-5252	55	23	]	]	PUNCT
ejpam-5252	55	24	.	.	PUNCT
ejpam-5252	56	1	a	a	DET
ejpam-5252	56	2	modern	modern	ADJ
ejpam-5252	56	3	roman	roman	ADJ
ejpam-5252	56	4	dominating	dominating	NOUN
ejpam-5252	56	5	function	function	NOUN
ejpam-5252	56	6	(	(	PUNCT
ejpam-5252	56	7	mrdf	mrdf	NOUN
ejpam-5252	56	8	)	)	PUNCT
ejpam-5252	56	9	of	of	ADP
ejpam-5252	56	10	g	g	PROPN
ejpam-5252	56	11	is	be	AUX
ejpam-5252	56	12	a	a	DET
ejpam-5252	56	13	function	function	NOUN
ejpam-5252	56	14	f	f	NOUN
ejpam-5252	56	15	:	:	PUNCT
ejpam-5252	56	16	v	v	X
ejpam-5252	56	17	(	(	PUNCT
ejpam-5252	56	18	g	g	NOUN
ejpam-5252	56	19	)	)	PUNCT
ejpam-5252	56	20	→	→	SYM
ejpam-5252	56	21	{	{	PUNCT
ejpam-5252	56	22	0	0	NUM
ejpam-5252	56	23	,	,	PUNCT
ejpam-5252	56	24	1	1	NUM
ejpam-5252	56	25	,	,	PUNCT
ejpam-5252	56	26	2	2	NUM
ejpam-5252	56	27	,	,	PUNCT
ejpam-5252	56	28	3	3	NUM
ejpam-5252	56	29	}	}	PUNCT
ejpam-5252	56	30	if	if	SCONJ
ejpam-5252	56	31	(	(	PUNCT
ejpam-5252	56	32	p1	p1	NOUN
ejpam-5252	56	33	)	)	PUNCT
ejpam-5252	56	34	for	for	ADP
ejpam-5252	56	35	each	each	PRON
ejpam-5252	56	36	v	v	NUM
ejpam-5252	56	37	∈	∈	PROPN
ejpam-5252	56	38	v	v	NOUN
ejpam-5252	56	39	(	(	PUNCT
ejpam-5252	56	40	g	g	NOUN
ejpam-5252	56	41	)	)	PUNCT
ejpam-5252	56	42	with	with	ADP
ejpam-5252	56	43	f(v	f(v	NOUN
ejpam-5252	56	44	)	)	PUNCT
ejpam-5252	56	45	=	=	SYM
ejpam-5252	56	46	0	0	NUM
ejpam-5252	56	47	,	,	PUNCT
ejpam-5252	56	48	there	there	PRON
ejpam-5252	56	49	exist	exist	VERB
ejpam-5252	56	50	u	u	NOUN
ejpam-5252	56	51	,	,	PUNCT
ejpam-5252	56	52	w	w	PROPN
ejpam-5252	56	53	∈	∈	PROPN
ejpam-5252	56	54	ng(v	ng(v	PUNCT
ejpam-5252	56	55	)	)	PUNCT
ejpam-5252	56	56	such	such	ADJ
ejpam-5252	56	57	that	that	DET
ejpam-5252	56	58	f(u	f(u	PROPN
ejpam-5252	56	59	)	)	PUNCT
ejpam-5252	56	60	=	=	SYM
ejpam-5252	56	61	2	2	NUM
ejpam-5252	56	62	and	and	CCONJ
ejpam-5252	56	63	f(w	f(w	NUM
ejpam-5252	56	64	)	)	PUNCT
ejpam-5252	56	65	=	=	SYM
ejpam-5252	57	1	3	3	NUM
ejpam-5252	57	2	;	;	PUNCT
ejpam-5252	57	3	and	and	CCONJ
ejpam-5252	57	4	(	(	PUNCT
ejpam-5252	57	5	p2	p2	PROPN
ejpam-5252	57	6	)	)	PUNCT
ejpam-5252	57	7	for	for	ADP
ejpam-5252	57	8	each	each	DET
ejpam-5252	57	9	v	v	NUM
ejpam-5252	57	10	∈	∈	PROPN
ejpam-5252	57	11	v	v	NOUN
ejpam-5252	57	12	(	(	PUNCT
ejpam-5252	57	13	g	g	NOUN
ejpam-5252	57	14	)	)	PUNCT
ejpam-5252	57	15	with	with	ADP
ejpam-5252	57	16	f(v	f(v	NOUN
ejpam-5252	57	17	)	)	PUNCT
ejpam-5252	57	18	=	=	SYM
ejpam-5252	57	19	1	1	NUM
ejpam-5252	58	1	,	,	PUNCT
ejpam-5252	58	2	there	there	PRON
ejpam-5252	58	3	exists	exist	VERB
ejpam-5252	58	4	u	u	PROPN
ejpam-5252	58	5	∈	∈	PROPN
ejpam-5252	58	6	ng(v	ng(v	PUNCT
ejpam-5252	58	7	)	)	PUNCT
ejpam-5252	58	8	such	such	ADJ
ejpam-5252	58	9	that	that	DET
ejpam-5252	58	10	f(u	f(u	PROPN
ejpam-5252	58	11	)	)	PUNCT
ejpam-5252	58	12	=	=	SYM
ejpam-5252	58	13	2	2	NUM
ejpam-5252	58	14	or	or	CCONJ
ejpam-5252	58	15	f(u	f(u	PROPN
ejpam-5252	58	16	)	)	PUNCT
ejpam-5252	58	17	=	=	SYM
ejpam-5252	59	1	3	3	X
ejpam-5252	59	2	.	.	PUNCT
ejpam-5252	59	3	the	the	DET
ejpam-5252	59	4	weight	weight	NOUN
ejpam-5252	59	5	of	of	ADP
ejpam-5252	59	6	a	a	DET
ejpam-5252	59	7	modern	modern	ADJ
ejpam-5252	59	8	roman	roman	ADJ
ejpam-5252	59	9	dominating	dominating	NOUN
ejpam-5252	59	10	function	function	NOUN
ejpam-5252	59	11	f	f	PROPN
ejpam-5252	59	12	of	of	ADP
ejpam-5252	59	13	g	g	PROPN
ejpam-5252	59	14	is	be	AUX
ejpam-5252	60	1	the	the	DET
ejpam-5252	60	2	sum	sum	NOUN
ejpam-5252	60	3	ωmr	ωmr	NOUN
ejpam-5252	60	4	g	g	PROPN
ejpam-5252	60	5	(	(	PUNCT
ejpam-5252	60	6	f	f	X
ejpam-5252	60	7	)	)	PUNCT
ejpam-5252	60	8	=	=	NOUN
ejpam-5252	60	9	∑	∑	SYM
ejpam-5252	60	10	v∈v	v∈v	NOUN
ejpam-5252	60	11	(	(	PUNCT
ejpam-5252	60	12	g	g	NOUN
ejpam-5252	60	13	)	)	PUNCT
ejpam-5252	60	14	f(v	f(v	NOUN
ejpam-5252	60	15	)	)	PUNCT
ejpam-5252	60	16	and	and	CCONJ
ejpam-5252	60	17	its	its	PRON
ejpam-5252	60	18	minimum	minimum	ADJ
ejpam-5252	60	19	weight	weight	NOUN
ejpam-5252	60	20	among	among	ADP
ejpam-5252	60	21	all	all	PRON
ejpam-5252	60	22	of	of	ADP
ejpam-5252	60	23	the	the	DET
ejpam-5252	60	24	modern	modern	ADJ
ejpam-5252	60	25	roman	roman	ADJ
ejpam-5252	60	26	dominating	dominating	NOUN
ejpam-5252	60	27	function	function	NOUN
ejpam-5252	60	28	is	be	AUX
ejpam-5252	60	29	called	call	VERB
ejpam-5252	60	30	the	the	DET
ejpam-5252	60	31	modern	modern	ADJ
ejpam-5252	60	32	roman	roman	ADJ
ejpam-5252	60	33	domination	domination	NOUN
ejpam-5252	60	34	number	number	NOUN
ejpam-5252	60	35	γmr(g	γmr(g	PROPN
ejpam-5252	60	36	)	)	PUNCT
ejpam-5252	60	37	of	of	ADP
ejpam-5252	60	38	g.	g.	PROPN
ejpam-5252	60	39	a	a	DET
ejpam-5252	60	40	modern	modern	ADJ
ejpam-5252	60	41	roman	roman	ADJ
ejpam-5252	60	42	dominating	dominating	NOUN
ejpam-5252	60	43	function	function	NOUN
ejpam-5252	60	44	of	of	ADP
ejpam-5252	60	45	g	g	NOUN
ejpam-5252	60	46	with	with	ADP
ejpam-5252	60	47	weight	weight	NOUN
ejpam-5252	60	48	ωmr	ωmr	NOUN
ejpam-5252	60	49	g	g	PROPN
ejpam-5252	60	50	(	(	PUNCT
ejpam-5252	60	51	f	f	X
ejpam-5252	60	52	)	)	PUNCT
ejpam-5252	60	53	=	=	SYM
ejpam-5252	60	54	γmr(g	γmr(g	PROPN
ejpam-5252	60	55	)	)	PUNCT
ejpam-5252	60	56	is	be	AUX
ejpam-5252	60	57	called	call	VERB
ejpam-5252	60	58	a	a	DET
ejpam-5252	60	59	γmr	γmr	ADJ
ejpam-5252	60	60	-	-	PUNCT
ejpam-5252	60	61	function	function	NOUN
ejpam-5252	60	62	of	of	ADP
ejpam-5252	60	63	g	g	NOUN
ejpam-5252	60	64	[	[	X
ejpam-5252	60	65	8	8	NUM
ejpam-5252	60	66	]	]	PUNCT
ejpam-5252	60	67	.	.	PUNCT
ejpam-5252	61	1	for	for	ADP
ejpam-5252	61	2	a	a	DET
ejpam-5252	61	3	function	function	NOUN
ejpam-5252	61	4	f	f	NOUN
ejpam-5252	61	5	:	:	PUNCT
ejpam-5252	61	6	v	v	X
ejpam-5252	61	7	(	(	PUNCT
ejpam-5252	61	8	g	g	NOUN
ejpam-5252	61	9	)	)	PUNCT
ejpam-5252	61	10	→	→	SYM
ejpam-5252	61	11	{	{	PUNCT
ejpam-5252	61	12	0	0	NUM
ejpam-5252	61	13	,	,	PUNCT
ejpam-5252	61	14	1	1	NUM
ejpam-5252	61	15	,	,	PUNCT
ejpam-5252	61	16	2	2	NUM
ejpam-5252	61	17	,	,	PUNCT
ejpam-5252	61	18	3	3	NUM
ejpam-5252	61	19	}	}	PUNCT
ejpam-5252	61	20	on	on	ADP
ejpam-5252	61	21	a	a	DET
ejpam-5252	61	22	graph	graph	NOUN
ejpam-5252	61	23	g	g	NOUN
ejpam-5252	61	24	,	,	PUNCT
ejpam-5252	61	25	let	let	VERB
ejpam-5252	61	26	(	(	PUNCT
ejpam-5252	61	27	v0	v0	NOUN
ejpam-5252	61	28	,	,	PUNCT
ejpam-5252	61	29	v1	v1	NOUN
ejpam-5252	61	30	,	,	PUNCT
ejpam-5252	61	31	v2	v2	PROPN
ejpam-5252	61	32	,	,	PUNCT
ejpam-5252	61	33	v3	v3	PROPN
ejpam-5252	61	34	)	)	PUNCT
ejpam-5252	61	35	be	be	VERB
ejpam-5252	61	36	the	the	DET
ejpam-5252	61	37	ordered	order	VERB
ejpam-5252	61	38	partition	partition	NOUN
ejpam-5252	61	39	induced	induce	VERB
ejpam-5252	61	40	by	by	ADP
ejpam-5252	61	41	f	f	PROPN
ejpam-5252	61	42	,	,	PUNCT
ejpam-5252	61	43	where	where	SCONJ
ejpam-5252	61	44	vi	vi	VERB
ejpam-5252	61	45	=	=	PRON
ejpam-5252	61	46	{	{	PUNCT
ejpam-5252	61	47	v	v	NUM
ejpam-5252	61	48	∈	∈	NOUN
ejpam-5252	61	49	v	v	NOUN
ejpam-5252	61	50	(	(	PUNCT
ejpam-5252	61	51	g	g	NOUN
ejpam-5252	61	52	)	)	PUNCT
ejpam-5252	61	53	:	:	PUNCT
ejpam-5252	61	54	f(v	f(v	NOUN
ejpam-5252	61	55	)	)	PUNCT
ejpam-5252	62	1	=	=	PUNCT
ejpam-5252	62	2	i	i	PROPN
ejpam-5252	62	3	}	}	PUNCT
ejpam-5252	62	4	for	for	ADP
ejpam-5252	62	5	i	i	PROPN
ejpam-5252	62	6	∈	∈	PROPN
ejpam-5252	62	7	{	{	PUNCT
ejpam-5252	62	8	0	0	NUM
ejpam-5252	62	9	,	,	PUNCT
ejpam-5252	62	10	1	1	NUM
ejpam-5252	62	11	,	,	PUNCT
ejpam-5252	62	12	2	2	NUM
ejpam-5252	62	13	,	,	PUNCT
ejpam-5252	62	14	3	3	NUM
ejpam-5252	62	15	}	}	PUNCT
ejpam-5252	62	16	.	.	PUNCT
ejpam-5252	63	1	then	then	ADV
ejpam-5252	63	2	we	we	PRON
ejpam-5252	63	3	can	can	AUX
ejpam-5252	63	4	write	write	VERB
ejpam-5252	63	5	f	f	PROPN
ejpam-5252	63	6	=	=	SYM
ejpam-5252	63	7	(	(	PUNCT
ejpam-5252	63	8	v0	v0	PROPN
ejpam-5252	63	9	,	,	PUNCT
ejpam-5252	63	10	v1	v1	NOUN
ejpam-5252	63	11	,	,	PUNCT
ejpam-5252	63	12	v2	v2	PROPN
ejpam-5252	63	13	,	,	PUNCT
ejpam-5252	63	14	v3	v3	PROPN
ejpam-5252	63	15	)	)	PUNCT
ejpam-5252	63	16	.	.	PUNCT
ejpam-5252	64	1	the	the	DET
ejpam-5252	64	2	weight	weight	NOUN
ejpam-5252	64	3	of	of	ADP
ejpam-5252	64	4	f	f	PROPN
ejpam-5252	64	5	is	be	AUX
ejpam-5252	64	6	defined	define	VERB
ejpam-5252	64	7	by	by	ADP
ejpam-5252	64	8	ωg(f	ωg(f	NOUN
ejpam-5252	64	9	)	)	PUNCT
ejpam-5252	64	10	=	=	PUNCT
ejpam-5252	64	11	|v1|+	|v1|+	ADP
ejpam-5252	64	12	2|v2|+	2|v2|+	NUM
ejpam-5252	64	13	3|v3|	3|v3|	NUM
ejpam-5252	64	14	.	.	PUNCT
ejpam-5252	65	1	example	example	NOUN
ejpam-5252	65	2	1	1	NUM
ejpam-5252	65	3	.	.	X
ejpam-5252	66	1	consider	consider	VERB
ejpam-5252	66	2	the	the	DET
ejpam-5252	66	3	given	give	VERB
ejpam-5252	66	4	graph	graph	NOUN
ejpam-5252	66	5	g	g	NOUN
ejpam-5252	66	6	with	with	ADP
ejpam-5252	66	7	v	v	NOUN
ejpam-5252	66	8	(	(	PUNCT
ejpam-5252	66	9	g	g	NOUN
ejpam-5252	66	10	)	)	PUNCT
ejpam-5252	66	11	=	=	NOUN
ejpam-5252	66	12	{	{	PUNCT
ejpam-5252	66	13	a	a	PRON
ejpam-5252	66	14	,	,	PUNCT
ejpam-5252	66	15	b	b	NOUN
ejpam-5252	66	16	,	,	PUNCT
ejpam-5252	66	17	c	c	NOUN
ejpam-5252	66	18	,	,	PUNCT
ejpam-5252	66	19	d	d	NOUN
ejpam-5252	66	20	,	,	PUNCT
ejpam-5252	66	21	e	e	NOUN
ejpam-5252	66	22	,	,	PUNCT
ejpam-5252	66	23	g	g	PROPN
ejpam-5252	66	24	,	,	PUNCT
ejpam-5252	66	25	h	h	NOUN
ejpam-5252	66	26	}	}	PUNCT
ejpam-5252	66	27	in	in	ADP
ejpam-5252	66	28	figure	figure	NOUN
ejpam-5252	66	29	1	1	NUM
ejpam-5252	66	30	.	.	PUNCT
ejpam-5252	67	1	the	the	DET
ejpam-5252	67	2	function	function	NOUN
ejpam-5252	67	3	f	f	X
ejpam-5252	67	4	:	:	PUNCT
ejpam-5252	67	5	v	v	X
ejpam-5252	67	6	(	(	PUNCT
ejpam-5252	67	7	g	g	NOUN
ejpam-5252	67	8	)	)	PUNCT
ejpam-5252	67	9	→	→	SYM
ejpam-5252	67	10	{	{	PUNCT
ejpam-5252	67	11	0	0	NUM
ejpam-5252	67	12	,	,	PUNCT
ejpam-5252	67	13	1	1	NUM
ejpam-5252	67	14	,	,	PUNCT
ejpam-5252	67	15	2	2	NUM
ejpam-5252	67	16	,	,	PUNCT
ejpam-5252	67	17	3	3	NUM
ejpam-5252	67	18	}	}	PUNCT
ejpam-5252	67	19	given	give	VERB
ejpam-5252	67	20	by	by	ADP
ejpam-5252	67	21	f(v	f(v	NOUN
ejpam-5252	67	22	)	)	PUNCT
ejpam-5252	67	23	=	=	SYM
ejpam-5252	67	24			NOUN
ejpam-5252	67	25	3	3	NUM
ejpam-5252	67	26	,	,	PUNCT
ejpam-5252	67	27	if	if	SCONJ
ejpam-5252	67	28	v	v	NOUN
ejpam-5252	67	29	=	=	NOUN
ejpam-5252	67	30	a.	a.	NOUN
ejpam-5252	67	31	2	2	NUM
ejpam-5252	67	32	,	,	PUNCT
ejpam-5252	67	33	if	if	SCONJ
ejpam-5252	67	34	v	v	NOUN
ejpam-5252	67	35	=	=	PUNCT
ejpam-5252	67	36	g.	g.	NOUN
ejpam-5252	67	37	0	0	NUM
ejpam-5252	67	38	,	,	PUNCT
ejpam-5252	67	39	otherwise	otherwise	ADV
ejpam-5252	67	40	.	.	PUNCT
ejpam-5252	67	41	is	be	AUX
ejpam-5252	67	42	a	a	DET
ejpam-5252	67	43	modern	modern	ADJ
ejpam-5252	67	44	roman	roman	ADJ
ejpam-5252	67	45	dominating	dominating	NOUN
ejpam-5252	67	46	function	function	NOUN
ejpam-5252	67	47	of	of	ADP
ejpam-5252	67	48	g.	g.	PROPN
ejpam-5252	67	49	it	it	PRON
ejpam-5252	67	50	can	can	AUX
ejpam-5252	67	51	be	be	AUX
ejpam-5252	67	52	verified	verify	VERB
ejpam-5252	67	53	that	that	SCONJ
ejpam-5252	67	54	the	the	DET
ejpam-5252	67	55	γmr(g	γmr(g	NOUN
ejpam-5252	67	56	)	)	PUNCT
ejpam-5252	67	57	=	=	SYM
ejpam-5252	67	58	5	5	X
ejpam-5252	67	59	.	.	PUNCT
ejpam-5252	68	1	s.	s.	PROPN
ejpam-5252	68	2	ahamad	ahamad	PROPN
ejpam-5252	68	3	,	,	PUNCT
ejpam-5252	68	4	j.	j.	PROPN
ejpam-5252	68	5	cariaga	cariaga	PROPN
ejpam-5252	68	6	,	,	PUNCT
ejpam-5252	68	7	s.	s.	PROPN
ejpam-5252	68	8	menchavez	menchavez	PROPN
ejpam-5252	68	9	/	/	PUNCT
ejpam-5252	68	10	eur	eur	PROPN
ejpam-5252	68	11	.	.	PUNCT
ejpam-5252	69	1	j.	j.	PROPN
ejpam-5252	69	2	pure	pure	PROPN
ejpam-5252	69	3	appl	appl	PROPN
ejpam-5252	69	4	.	.	PROPN
ejpam-5252	69	5	math	math	PROPN
ejpam-5252	69	6	,	,	PUNCT
ejpam-5252	69	7	18	18	NUM
ejpam-5252	69	8	(	(	PUNCT
ejpam-5252	69	9	1	1	NUM
ejpam-5252	69	10	)	)	PUNCT
ejpam-5252	69	11	(	(	PUNCT
ejpam-5252	69	12	2025	2025	NUM
ejpam-5252	69	13	)	)	PUNCT
ejpam-5252	69	14	,	,	PUNCT
ejpam-5252	69	15	5252	5252	NUM
ejpam-5252	69	16	4	4	NUM
ejpam-5252	69	17	of	of	ADP
ejpam-5252	69	18	18	18	NUM
ejpam-5252	69	19	0b	0b	NOUN
ejpam-5252	69	20	0	0	NUM
ejpam-5252	70	1	d	d	NOUN
ejpam-5252	70	2	0e	0e	PROPN
ejpam-5252	70	3	0c	0c	NOUN
ejpam-5252	70	4	0h	0h	X
ejpam-5252	70	5	3a	3a	NUM
ejpam-5252	70	6	2	2	NUM
ejpam-5252	70	7	g	g	NOUN
ejpam-5252	70	8	g	g	NOUN
ejpam-5252	70	9	:	:	PUNCT
ejpam-5252	70	10	figure	figure	NOUN
ejpam-5252	70	11	1	1	NUM
ejpam-5252	70	12	:	:	PUNCT
ejpam-5252	70	13	graph	graph	VERB
ejpam-5252	70	14	g	g	NOUN
ejpam-5252	70	15	of	of	ADP
ejpam-5252	70	16	order	order	NOUN
ejpam-5252	70	17	7	7	NUM
ejpam-5252	70	18	with	with	ADP
ejpam-5252	70	19	γmr(g	γmr(g	PROPN
ejpam-5252	70	20	)	)	PUNCT
ejpam-5252	70	21	=	=	SYM
ejpam-5252	71	1	5	5	NUM
ejpam-5252	71	2	.	.	NOUN
ejpam-5252	71	3	3	3	NUM
ejpam-5252	71	4	.	.	X
ejpam-5252	71	5	known	know	VERB
ejpam-5252	71	6	results	result	NOUN
ejpam-5252	71	7	we	we	PRON
ejpam-5252	71	8	make	make	VERB
ejpam-5252	71	9	use	use	NOUN
ejpam-5252	71	10	of	of	ADP
ejpam-5252	71	11	the	the	DET
ejpam-5252	71	12	following	follow	VERB
ejpam-5252	71	13	known	know	VERB
ejpam-5252	71	14	results	result	NOUN
ejpam-5252	71	15	from	from	ADP
ejpam-5252	71	16	[	[	X
ejpam-5252	71	17	8	8	NUM
ejpam-5252	71	18	]	]	PUNCT
ejpam-5252	71	19	.	.	PUNCT
ejpam-5252	72	1	proposition	proposition	NOUN
ejpam-5252	72	2	1	1	NUM
ejpam-5252	72	3	.	.	PUNCT
ejpam-5252	73	1	let	let	VERB
ejpam-5252	73	2	g	g	PRON
ejpam-5252	73	3	be	be	AUX
ejpam-5252	73	4	a	a	DET
ejpam-5252	73	5	graph	graph	NOUN
ejpam-5252	73	6	of	of	ADP
ejpam-5252	73	7	order	order	NOUN
ejpam-5252	73	8	n	n	NOUN
ejpam-5252	73	9	and	and	CCONJ
ejpam-5252	73	10	let	let	VERB
ejpam-5252	73	11	f	f	PROPN
ejpam-5252	73	12	=	=	SYM
ejpam-5252	73	13	(	(	PUNCT
ejpam-5252	73	14	v0	v0	PROPN
ejpam-5252	73	15	,	,	PUNCT
ejpam-5252	73	16	v1	v1	NOUN
ejpam-5252	73	17	,	,	PUNCT
ejpam-5252	73	18	v2	v2	PROPN
ejpam-5252	73	19	,	,	PUNCT
ejpam-5252	73	20	v3	v3	PROPN
ejpam-5252	73	21	)	)	PUNCT
ejpam-5252	73	22	be	be	VERB
ejpam-5252	73	23	a	a	DET
ejpam-5252	73	24	γmr	γmr	ADJ
ejpam-5252	73	25	-	-	PUNCT
ejpam-5252	73	26	function	function	NOUN
ejpam-5252	73	27	on	on	ADP
ejpam-5252	73	28	g.	g.	PROPN
ejpam-5252	73	29	then	then	ADV
ejpam-5252	73	30	each	each	PRON
ejpam-5252	73	31	of	of	ADP
ejpam-5252	73	32	the	the	DET
ejpam-5252	73	33	following	following	ADJ
ejpam-5252	73	34	statements	statement	NOUN
ejpam-5252	73	35	holds	hold	VERB
ejpam-5252	73	36	:	:	PUNCT
ejpam-5252	73	37	(	(	PUNCT
ejpam-5252	73	38	i	i	NOUN
ejpam-5252	73	39	)	)	PUNCT
ejpam-5252	73	40	if	if	SCONJ
ejpam-5252	73	41	n	n	PRON
ejpam-5252	73	42	≥	≥	NOUN
ejpam-5252	73	43	4	4	NUM
ejpam-5252	73	44	,	,	PUNCT
ejpam-5252	73	45	then	then	ADV
ejpam-5252	73	46	5	5	NUM
ejpam-5252	73	47	≤	≤	NUM
ejpam-5252	73	48	γmr(g	γmr(g	PROPN
ejpam-5252	73	49	)	)	PUNCT
ejpam-5252	73	50	≤	≤	NUM
ejpam-5252	73	51	2n	2n	NUM
ejpam-5252	73	52	.	.	PUNCT
ejpam-5252	74	1	(	(	PUNCT
ejpam-5252	74	2	ii	ii	NOUN
ejpam-5252	74	3	)	)	PUNCT
ejpam-5252	74	4	if	if	SCONJ
ejpam-5252	74	5	there	there	PRON
ejpam-5252	74	6	are	be	VERB
ejpam-5252	74	7	two	two	NUM
ejpam-5252	74	8	vertices	vertex	NOUN
ejpam-5252	74	9	that	that	PRON
ejpam-5252	74	10	are	be	AUX
ejpam-5252	74	11	adjacent	adjacent	ADJ
ejpam-5252	74	12	to	to	ADP
ejpam-5252	74	13	all	all	DET
ejpam-5252	74	14	other	other	ADJ
ejpam-5252	74	15	vertices	vertex	NOUN
ejpam-5252	74	16	in	in	ADP
ejpam-5252	74	17	g	g	NOUN
ejpam-5252	74	18	,	,	PUNCT
ejpam-5252	74	19	then	then	ADV
ejpam-5252	74	20	γmr	γmr	NUM
ejpam-5252	74	21	=	=	SYM
ejpam-5252	74	22	5	5	X
ejpam-5252	74	23	.	.	PUNCT
ejpam-5252	74	24	(	(	PUNCT
ejpam-5252	74	25	iii	iii	X
ejpam-5252	74	26	)	)	PUNCT
ejpam-5252	74	27	if	if	SCONJ
ejpam-5252	74	28	g	g	PROPN
ejpam-5252	74	29	is	be	AUX
ejpam-5252	74	30	empty	empty	ADJ
ejpam-5252	74	31	graph	graph	NOUN
ejpam-5252	74	32	,	,	PUNCT
ejpam-5252	74	33	then	then	ADV
ejpam-5252	74	34	2γ(g	2γ(g	NUM
ejpam-5252	74	35	)	)	PUNCT
ejpam-5252	75	1	=	=	PUNCT
ejpam-5252	75	2	γmr(g	γmr(g	PROPN
ejpam-5252	75	3	)	)	PUNCT
ejpam-5252	75	4	.	.	PUNCT
ejpam-5252	76	1	(	(	PUNCT
ejpam-5252	76	2	iv	iv	X
ejpam-5252	76	3	)	)	PUNCT
ejpam-5252	76	4	v2	v2	PROPN
ejpam-5252	76	5	̸=	̸=	PROPN
ejpam-5252	76	6	∅	∅	NOUN
ejpam-5252	76	7	(	(	PUNCT
ejpam-5252	76	8	v	v	NOUN
ejpam-5252	76	9	)	)	PUNCT
ejpam-5252	76	10	v2	v2	PROPN
ejpam-5252	76	11	∪	∪	X
ejpam-5252	76	12	v3	v3	PROPN
ejpam-5252	76	13	is	be	AUX
ejpam-5252	76	14	a	a	DET
ejpam-5252	76	15	dominating	dominating	NOUN
ejpam-5252	76	16	set	set	NOUN
ejpam-5252	76	17	of	of	ADP
ejpam-5252	76	18	g.	g.	PROPN
ejpam-5252	76	19	moreover	moreover	ADV
ejpam-5252	76	20	,	,	PUNCT
ejpam-5252	76	21	it	it	PRON
ejpam-5252	76	22	is	be	AUX
ejpam-5252	76	23	a	a	DET
ejpam-5252	76	24	2	2	NUM
ejpam-5252	76	25	-	-	PUNCT
ejpam-5252	76	26	dominating	dominating	NOUN
ejpam-5252	76	27	set	set	NOUN
ejpam-5252	76	28	of	of	ADP
ejpam-5252	76	29	g	g	PROPN
ejpam-5252	76	30	[	[	X
ejpam-5252	76	31	v0	v0	X
ejpam-5252	76	32	]	]	X
ejpam-5252	76	33	(	(	PUNCT
ejpam-5252	76	34	vi	vi	NOUN
ejpam-5252	76	35	)	)	PUNCT
ejpam-5252	76	36	if	if	SCONJ
ejpam-5252	76	37	v	v	NOUN
ejpam-5252	76	38	is	be	AUX
ejpam-5252	76	39	a	a	DET
ejpam-5252	76	40	pendant	pendant	ADJ
ejpam-5252	76	41	vertex	vertex	NOUN
ejpam-5252	76	42	,	,	PUNCT
ejpam-5252	76	43	then	then	ADV
ejpam-5252	76	44	f(v	f(v	NOUN
ejpam-5252	76	45	)	)	PUNCT
ejpam-5252	76	46	̸=	̸=	PROPN
ejpam-5252	76	47	0	0	NUM
ejpam-5252	76	48	.	.	PUNCT
ejpam-5252	77	1	(	(	PUNCT
ejpam-5252	77	2	vii	vii	PROPN
ejpam-5252	77	3	)	)	PUNCT
ejpam-5252	77	4	if	if	SCONJ
ejpam-5252	77	5	v	v	NOUN
ejpam-5252	77	6	is	be	AUX
ejpam-5252	77	7	an	an	DET
ejpam-5252	77	8	isolated	isolated	ADJ
ejpam-5252	77	9	vertex	vertex	NOUN
ejpam-5252	77	10	,	,	PUNCT
ejpam-5252	77	11	then	then	ADV
ejpam-5252	77	12	f(v	f(v	NOUN
ejpam-5252	77	13	)	)	PUNCT
ejpam-5252	77	14	=	=	SYM
ejpam-5252	77	15	2	2	X
ejpam-5252	77	16	.	.	X
ejpam-5252	77	17	proposition	proposition	NOUN
ejpam-5252	77	18	2	2	NUM
ejpam-5252	77	19	.	.	X
ejpam-5252	78	1	for	for	ADP
ejpam-5252	78	2	path	path	NOUN
ejpam-5252	78	3	pn	pn	PROPN
ejpam-5252	78	4	,	,	PUNCT
ejpam-5252	78	5	n	n	X
ejpam-5252	78	6	≥	≥	NUM
ejpam-5252	78	7	1	1	NUM
ejpam-5252	78	8	,	,	PUNCT
ejpam-5252	78	9	γmr(pn	γmr(pn	NOUN
ejpam-5252	78	10	)	)	PUNCT
ejpam-5252	78	11	=	=	SYM
ejpam-5252	78	12	n+	n+	X
ejpam-5252	78	13	⌈n	⌈n	NOUN
ejpam-5252	78	14	3	3	NUM
ejpam-5252	78	15	⌉	⌉	PRON
ejpam-5252	78	16	proposition	proposition	NOUN
ejpam-5252	78	17	3	3	NUM
ejpam-5252	78	18	.	.	X
ejpam-5252	79	1	for	for	ADP
ejpam-5252	79	2	cycle	cycle	NOUN
ejpam-5252	79	3	cn	cn	PROPN
ejpam-5252	79	4	,	,	PUNCT
ejpam-5252	79	5	n	n	PRON
ejpam-5252	79	6	≥	≥	NOUN
ejpam-5252	79	7	3	3	NUM
ejpam-5252	79	8	,	,	PUNCT
ejpam-5252	79	9	γmr(cn	γmr(cn	NOUN
ejpam-5252	79	10	)	)	PUNCT
ejpam-5252	79	11	=	=	NOUN
ejpam-5252	79	12	{	{	PUNCT
ejpam-5252	79	13	5	5	NUM
ejpam-5252	79	14	,	,	PUNCT
ejpam-5252	79	15	if	if	SCONJ
ejpam-5252	79	16	n	n	NOUN
ejpam-5252	79	17	=	=	SYM
ejpam-5252	79	18	4	4	NUM
ejpam-5252	79	19	n+	n+	SYM
ejpam-5252	79	20	⌈	⌈	NOUN
ejpam-5252	79	21	n	n	CCONJ
ejpam-5252	79	22	3	3	NUM
ejpam-5252	79	23	⌉	⌉	NOUN
ejpam-5252	79	24	,	,	PUNCT
ejpam-5252	79	25	if	if	SCONJ
ejpam-5252	79	26	n	n	PRON
ejpam-5252	79	27	̸=	̸=	PROPN
ejpam-5252	79	28	4	4	NUM
ejpam-5252	79	29	4	4	NUM
ejpam-5252	79	30	.	.	PUNCT
ejpam-5252	79	31	main	main	ADJ
ejpam-5252	79	32	results	result	NOUN
ejpam-5252	79	33	this	this	DET
ejpam-5252	79	34	section	section	NOUN
ejpam-5252	79	35	begins	begin	VERB
ejpam-5252	79	36	with	with	ADP
ejpam-5252	79	37	the	the	DET
ejpam-5252	79	38	general	general	ADJ
ejpam-5252	79	39	and	and	CCONJ
ejpam-5252	79	40	useful	useful	ADJ
ejpam-5252	79	41	properties	property	NOUN
ejpam-5252	79	42	of	of	ADP
ejpam-5252	79	43	modern	modern	ADJ
ejpam-5252	79	44	roman	roman	ADJ
ejpam-5252	79	45	dominating	dominating	NOUN
ejpam-5252	79	46	functions	function	NOUN
ejpam-5252	79	47	.	.	PUNCT
ejpam-5252	80	1	it	it	PRON
ejpam-5252	80	2	also	also	ADV
ejpam-5252	80	3	presents	present	VERB
ejpam-5252	80	4	the	the	DET
ejpam-5252	80	5	characterizations	characterization	NOUN
ejpam-5252	80	6	of	of	ADP
ejpam-5252	80	7	some	some	DET
ejpam-5252	80	8	graphs	graph	NOUN
ejpam-5252	80	9	g	g	NOUN
ejpam-5252	80	10	with	with	ADP
ejpam-5252	80	11	γmr(g	γmr(g	PROPN
ejpam-5252	80	12	)	)	PUNCT
ejpam-5252	80	13	∈	∈	NOUN
ejpam-5252	80	14	{	{	PUNCT
ejpam-5252	80	15	2	2	NUM
ejpam-5252	80	16	,	,	PUNCT
ejpam-5252	80	17	3	3	NUM
ejpam-5252	80	18	,	,	PUNCT
ejpam-5252	80	19	4	4	NUM
ejpam-5252	80	20	,	,	PUNCT
ejpam-5252	80	21	5	5	NUM
ejpam-5252	80	22	}	}	PUNCT
ejpam-5252	80	23	and	and	CCONJ
ejpam-5252	80	24	the	the	DET
ejpam-5252	80	25	modern	modern	ADJ
ejpam-5252	80	26	roman	roman	ADJ
ejpam-5252	80	27	domination	domination	NOUN
ejpam-5252	80	28	number	number	NOUN
ejpam-5252	80	29	of	of	ADP
ejpam-5252	80	30	the	the	DET
ejpam-5252	80	31	n	n	CCONJ
ejpam-5252	80	32	-	-	PUNCT
ejpam-5252	80	33	barbell	barbell	NOUN
ejpam-5252	80	34	graph	graph	NOUN
ejpam-5252	80	35	bn	bn	NOUN
ejpam-5252	80	36	,	,	PUNCT
ejpam-5252	80	37	windmill	windmill	NOUN
ejpam-5252	80	38	graph	graph	NOUN
ejpam-5252	80	39	wd(k	wd(k	PROPN
ejpam-5252	80	40	,	,	PUNCT
ejpam-5252	80	41	n	n	CCONJ
ejpam-5252	80	42	)	)	PUNCT
ejpam-5252	80	43	,	,	PUNCT
ejpam-5252	80	44	friendship	friendship	NOUN
ejpam-5252	80	45	graph	graph	NOUN
ejpam-5252	80	46	gn	gn	PROPN
ejpam-5252	80	47	3	3	NUM
ejpam-5252	80	48	,	,	PUNCT
ejpam-5252	80	49	butterfly	butterfly	NOUN
ejpam-5252	80	50	graph	graph	NOUN
ejpam-5252	80	51	g2	g2	PROPN
ejpam-5252	80	52	3	3	NUM
ejpam-5252	80	53	,	,	PUNCT
ejpam-5252	80	54	complete	complete	ADJ
ejpam-5252	80	55	bipartite	bipartite	PROPN
ejpam-5252	80	56	graph	graph	NOUN
ejpam-5252	80	57	s.	s.	PROPN
ejpam-5252	80	58	ahamad	ahamad	PROPN
ejpam-5252	80	59	,	,	PUNCT
ejpam-5252	80	60	j.	j.	PROPN
ejpam-5252	80	61	cariaga	cariaga	PROPN
ejpam-5252	80	62	,	,	PUNCT
ejpam-5252	80	63	s.	s.	PROPN
ejpam-5252	80	64	menchavez	menchavez	PROPN
ejpam-5252	80	65	/	/	PUNCT
ejpam-5252	80	66	eur	eur	PROPN
ejpam-5252	80	67	.	.	PUNCT
ejpam-5252	81	1	j.	j.	PROPN
ejpam-5252	81	2	pure	pure	PROPN
ejpam-5252	81	3	appl	appl	PROPN
ejpam-5252	81	4	.	.	PROPN
ejpam-5252	81	5	math	math	PROPN
ejpam-5252	81	6	,	,	PUNCT
ejpam-5252	81	7	18	18	NUM
ejpam-5252	81	8	(	(	PUNCT
ejpam-5252	81	9	1	1	NUM
ejpam-5252	81	10	)	)	PUNCT
ejpam-5252	81	11	(	(	PUNCT
ejpam-5252	81	12	2025	2025	NUM
ejpam-5252	81	13	)	)	PUNCT
ejpam-5252	81	14	,	,	PUNCT
ejpam-5252	81	15	5252	5252	NUM
ejpam-5252	81	16	5	5	NUM
ejpam-5252	81	17	of	of	ADP
ejpam-5252	81	18	18	18	NUM
ejpam-5252	81	19	km	km	NOUN
ejpam-5252	81	20	,	,	PUNCT
ejpam-5252	81	21	n	n	CCONJ
ejpam-5252	81	22	,	,	PUNCT
ejpam-5252	81	23	star	star	NOUN
ejpam-5252	81	24	graph	graph	NOUN
ejpam-5252	81	25	sn	sn	PROPN
ejpam-5252	81	26	and	and	CCONJ
ejpam-5252	81	27	fan	fan	NOUN
ejpam-5252	81	28	graph	graph	NOUN
ejpam-5252	81	29	fn	fn	PROPN
ejpam-5252	81	30	.	.	PUNCT
ejpam-5252	82	1	for	for	ADP
ejpam-5252	82	2	simplicity	simplicity	NOUN
ejpam-5252	82	3	,	,	PUNCT
ejpam-5252	82	4	we	we	PRON
ejpam-5252	82	5	denote	denote	VERB
ejpam-5252	82	6	by	by	ADP
ejpam-5252	82	7	mrdf	mrdf	NOUN
ejpam-5252	82	8	(	(	PUNCT
ejpam-5252	82	9	g	g	NOUN
ejpam-5252	82	10	)	)	PUNCT
ejpam-5252	82	11	the	the	DET
ejpam-5252	82	12	set	set	NOUN
ejpam-5252	82	13	of	of	ADP
ejpam-5252	82	14	all	all	DET
ejpam-5252	82	15	modern	modern	ADJ
ejpam-5252	82	16	roman	roman	ADJ
ejpam-5252	82	17	dominating	dominating	NOUN
ejpam-5252	82	18	functions	function	NOUN
ejpam-5252	82	19	on	on	ADP
ejpam-5252	82	20	a	a	DET
ejpam-5252	82	21	graph	graph	NOUN
ejpam-5252	82	22	g.	g.	NOUN
ejpam-5252	82	23	remark	remark	NOUN
ejpam-5252	82	24	1	1	NUM
ejpam-5252	82	25	.	.	PUNCT
ejpam-5252	83	1	if	if	SCONJ
ejpam-5252	83	2	f	f	PROPN
ejpam-5252	83	3	=	=	SYM
ejpam-5252	83	4	(	(	PUNCT
ejpam-5252	83	5	v0	v0	PROPN
ejpam-5252	83	6	,	,	PUNCT
ejpam-5252	83	7	v1	v1	NOUN
ejpam-5252	83	8	,	,	PUNCT
ejpam-5252	83	9	v2	v2	PROPN
ejpam-5252	83	10	,	,	PUNCT
ejpam-5252	83	11	v3	v3	PROPN
ejpam-5252	83	12	)	)	PUNCT
ejpam-5252	83	13	is	be	AUX
ejpam-5252	83	14	a	a	DET
ejpam-5252	83	15	γmr	γmr	ADJ
ejpam-5252	83	16	-	-	PUNCT
ejpam-5252	83	17	function	function	NOUN
ejpam-5252	83	18	of	of	ADP
ejpam-5252	83	19	g	g	PROPN
ejpam-5252	83	20	and	and	CCONJ
ejpam-5252	83	21	v	v	ADP
ejpam-5252	83	22	∈	∈	PROPN
ejpam-5252	83	23	v1	v1	NOUN
ejpam-5252	83	24	,	,	PUNCT
ejpam-5252	83	25	then	then	ADV
ejpam-5252	83	26	v	v	NOUN
ejpam-5252	83	27	need	need	AUX
ejpam-5252	83	28	not	not	PART
ejpam-5252	83	29	be	be	AUX
ejpam-5252	83	30	in	in	ADP
ejpam-5252	83	31	ng(v2	ng(v2	NOUN
ejpam-5252	83	32	)	)	PUNCT
ejpam-5252	83	33	∩ng(v3	∩ng(v3	NOUN
ejpam-5252	83	34	)	)	PUNCT
ejpam-5252	83	35	.	.	PUNCT
ejpam-5252	84	1	proposition	proposition	NOUN
ejpam-5252	84	2	4	4	NUM
ejpam-5252	84	3	.	.	PUNCT
ejpam-5252	85	1	let	let	VERB
ejpam-5252	85	2	g	g	NOUN
ejpam-5252	85	3	be	be	AUX
ejpam-5252	85	4	any	any	DET
ejpam-5252	85	5	graph	graph	NOUN
ejpam-5252	85	6	with	with	ADP
ejpam-5252	85	7	no	no	DET
ejpam-5252	85	8	isolated	isolated	ADJ
ejpam-5252	85	9	vertex	vertex	NOUN
ejpam-5252	85	10	.	.	PUNCT
ejpam-5252	86	1	if	if	SCONJ
ejpam-5252	86	2	f	f	PROPN
ejpam-5252	86	3	=	=	SYM
ejpam-5252	86	4	(	(	PUNCT
ejpam-5252	86	5	v0	v0	PROPN
ejpam-5252	86	6	,	,	PUNCT
ejpam-5252	86	7	v1	v1	NOUN
ejpam-5252	86	8	,	,	PUNCT
ejpam-5252	86	9	v2	v2	PROPN
ejpam-5252	86	10	,	,	PUNCT
ejpam-5252	86	11	v3	v3	PROPN
ejpam-5252	86	12	)	)	PUNCT
ejpam-5252	86	13	a	a	DET
ejpam-5252	86	14	γmr	γmr	ADJ
ejpam-5252	86	15	-	-	PUNCT
ejpam-5252	86	16	function	function	NOUN
ejpam-5252	86	17	of	of	ADP
ejpam-5252	86	18	g	g	NOUN
ejpam-5252	86	19	,	,	PUNCT
ejpam-5252	86	20	then	then	ADV
ejpam-5252	86	21	the	the	DET
ejpam-5252	86	22	following	follow	VERB
ejpam-5252	86	23	holds	hold	VERB
ejpam-5252	86	24	:	:	PUNCT
ejpam-5252	86	25	(	(	PUNCT
ejpam-5252	86	26	i	i	NOUN
ejpam-5252	86	27	)	)	PUNCT
ejpam-5252	86	28	v0	v0	NOUN
ejpam-5252	86	29	=	=	SYM
ejpam-5252	86	30	∅	∅	NOUN
ejpam-5252	86	31	if	if	SCONJ
ejpam-5252	86	32	and	and	CCONJ
ejpam-5252	86	33	only	only	ADV
ejpam-5252	86	34	if	if	SCONJ
ejpam-5252	86	35	v3	v3	PROPN
ejpam-5252	86	36	=	=	SYM
ejpam-5252	86	37	∅	∅	NOUN
ejpam-5252	86	38	and	and	CCONJ
ejpam-5252	86	39	v2	v2	PROPN
ejpam-5252	86	40	is	be	AUX
ejpam-5252	86	41	a	a	DET
ejpam-5252	86	42	γ	γ	NOUN
ejpam-5252	86	43	-	-	PUNCT
ejpam-5252	86	44	set	set	NOUN
ejpam-5252	86	45	of	of	ADP
ejpam-5252	86	46	g.	g.	PROPN
ejpam-5252	86	47	moreover	moreover	ADV
ejpam-5252	86	48	,	,	PUNCT
ejpam-5252	86	49	γmr(g	γmr(g	PROPN
ejpam-5252	86	50	)	)	PUNCT
ejpam-5252	87	1	=	=	SYM
ejpam-5252	87	2	|v	|v	PROPN
ejpam-5252	87	3	(	(	PUNCT
ejpam-5252	87	4	g)|+	g)|+	PROPN
ejpam-5252	87	5	γ(g	γ(g	PROPN
ejpam-5252	87	6	)	)	PUNCT
ejpam-5252	87	7	.	.	PUNCT
ejpam-5252	88	1	(	(	PUNCT
ejpam-5252	88	2	ii	ii	NOUN
ejpam-5252	88	3	)	)	PUNCT
ejpam-5252	88	4	v1	v1	NOUN
ejpam-5252	88	5	=	=	SYM
ejpam-5252	88	6	∅	∅	NOUN
ejpam-5252	88	7	if	if	SCONJ
ejpam-5252	88	8	and	and	CCONJ
ejpam-5252	88	9	only	only	ADV
ejpam-5252	88	10	if	if	SCONJ
ejpam-5252	88	11	v2	v2	PROPN
ejpam-5252	88	12	∪	∪	X
ejpam-5252	88	13	v3	v3	PROPN
ejpam-5252	88	14	is	be	AUX
ejpam-5252	88	15	a	a	DET
ejpam-5252	88	16	2	2	NUM
ejpam-5252	88	17	-	-	PUNCT
ejpam-5252	88	18	dominating	dominating	NOUN
ejpam-5252	88	19	set	set	NOUN
ejpam-5252	88	20	of	of	ADP
ejpam-5252	88	21	g.	g.	PROPN
ejpam-5252	88	22	moreover	moreover	ADV
ejpam-5252	88	23	,	,	PUNCT
ejpam-5252	88	24	if	if	SCONJ
ejpam-5252	88	25	v1	v1	NOUN
ejpam-5252	88	26	=	=	SYM
ejpam-5252	88	27	∅	∅	NOUN
ejpam-5252	88	28	,	,	PUNCT
ejpam-5252	88	29	⟨v2	⟨v2	ADJ
ejpam-5252	88	30	∪	∪	ADJ
ejpam-5252	88	31	v3⟩	v3⟩	PRON
ejpam-5252	88	32	is	be	AUX
ejpam-5252	88	33	connected	connect	VERB
ejpam-5252	88	34	and	and	CCONJ
ejpam-5252	88	35	v3	v3	PROPN
ejpam-5252	88	36	is	be	AUX
ejpam-5252	88	37	a	a	DET
ejpam-5252	88	38	γ	γ	NOUN
ejpam-5252	88	39	-	-	PUNCT
ejpam-5252	88	40	set	set	NOUN
ejpam-5252	88	41	of	of	ADP
ejpam-5252	88	42	g	g	NOUN
ejpam-5252	88	43	,	,	PUNCT
ejpam-5252	88	44	then	then	ADV
ejpam-5252	88	45	γmr(g	γmr(g	NUM
ejpam-5252	88	46	)	)	PUNCT
ejpam-5252	88	47	≥	≥	PROPN
ejpam-5252	88	48	γ(g	γ(g	PROPN
ejpam-5252	88	49	)	)	PUNCT
ejpam-5252	89	1	+	+	CCONJ
ejpam-5252	89	2	2γ2(g	2γ2(g	NUM
ejpam-5252	89	3	)	)	PUNCT
ejpam-5252	89	4	.	.	PUNCT
ejpam-5252	90	1	proof	proof	NOUN
ejpam-5252	90	2	.	.	PUNCT
ejpam-5252	91	1	clearly	clearly	ADV
ejpam-5252	91	2	,	,	PUNCT
ejpam-5252	91	3	v0	v0	NOUN
ejpam-5252	91	4	=	=	SYM
ejpam-5252	91	5	∅	∅	NOUN
ejpam-5252	91	6	if	if	SCONJ
ejpam-5252	92	1	and	and	CCONJ
ejpam-5252	92	2	only	only	ADV
ejpam-5252	92	3	if	if	SCONJ
ejpam-5252	92	4	v3	v3	PROPN
ejpam-5252	92	5	=	=	PUNCT
ejpam-5252	92	6	∅.	∅.	ADV
ejpam-5252	92	7	suppose	suppose	VERB
ejpam-5252	92	8	v0	v0	NOUN
ejpam-5252	92	9	=	=	PUNCT
ejpam-5252	92	10	∅.	∅.	NOUN
ejpam-5252	92	11	since	since	SCONJ
ejpam-5252	92	12	v2	v2	PROPN
ejpam-5252	92	13	∪	∪	X
ejpam-5252	92	14	v3	v3	PROPN
ejpam-5252	92	15	is	be	AUX
ejpam-5252	92	16	a	a	DET
ejpam-5252	92	17	dominating	dominating	NOUN
ejpam-5252	92	18	set	set	NOUN
ejpam-5252	92	19	of	of	ADP
ejpam-5252	92	20	g	g	PROPN
ejpam-5252	92	21	and	and	CCONJ
ejpam-5252	92	22	v3	v3	PROPN
ejpam-5252	92	23	=	=	SYM
ejpam-5252	92	24	∅	∅	NOUN
ejpam-5252	92	25	,	,	PUNCT
ejpam-5252	92	26	it	it	PRON
ejpam-5252	92	27	follows	follow	VERB
ejpam-5252	92	28	that	that	SCONJ
ejpam-5252	92	29	v2	v2	PROPN
ejpam-5252	92	30	is	be	AUX
ejpam-5252	92	31	a	a	DET
ejpam-5252	92	32	dominating	dominating	NOUN
ejpam-5252	92	33	set	set	NOUN
ejpam-5252	92	34	of	of	ADP
ejpam-5252	92	35	g.	g.	PROPN
ejpam-5252	92	36	suppose	suppose	VERB
ejpam-5252	92	37	v2	v2	NOUN
ejpam-5252	92	38	is	be	AUX
ejpam-5252	92	39	not	not	PART
ejpam-5252	92	40	a	a	DET
ejpam-5252	92	41	γ	γ	NOUN
ejpam-5252	92	42	-	-	PUNCT
ejpam-5252	92	43	set	set	NOUN
ejpam-5252	92	44	of	of	ADP
ejpam-5252	92	45	g.	g.	PROPN
ejpam-5252	92	46	let	let	VERB
ejpam-5252	92	47	s	s	PRON
ejpam-5252	92	48	be	be	AUX
ejpam-5252	92	49	a	a	DET
ejpam-5252	92	50	γ	γ	NOUN
ejpam-5252	92	51	-	-	PUNCT
ejpam-5252	92	52	set	set	NOUN
ejpam-5252	92	53	of	of	ADP
ejpam-5252	92	54	g	g	NOUN
ejpam-5252	92	55	and	and	CCONJ
ejpam-5252	92	56	define	define	VERB
ejpam-5252	92	57	g	g	PROPN
ejpam-5252	92	58	=	=	PUNCT
ejpam-5252	92	59	(	(	PUNCT
ejpam-5252	92	60	v	v	NUM
ejpam-5252	92	61	′	′	NUM
ejpam-5252	92	62	0	0	NUM
ejpam-5252	92	63	,	,	PUNCT
ejpam-5252	92	64	v	v	NOUN
ejpam-5252	92	65	′	′	NUM
ejpam-5252	92	66	1	1	NUM
ejpam-5252	92	67	,	,	PUNCT
ejpam-5252	92	68	v	v	NOUN
ejpam-5252	92	69	′	′	NUM
ejpam-5252	92	70	2	2	NUM
ejpam-5252	92	71	,	,	PUNCT
ejpam-5252	92	72	v	v	NOUN
ejpam-5252	92	73	′	′	NUM
ejpam-5252	92	74	3	3	NUM
ejpam-5252	92	75	)	)	PUNCT
ejpam-5252	93	1	where	where	SCONJ
ejpam-5252	93	2	v	v	X
ejpam-5252	93	3	′	′	NOUN
ejpam-5252	93	4	0	0	NUM
ejpam-5252	94	1	=	=	SYM
ejpam-5252	94	2	v	v	NUM
ejpam-5252	94	3	′	′	NUM
ejpam-5252	94	4	3	3	NUM
ejpam-5252	94	5	=	=	NOUN
ejpam-5252	94	6	∅	∅	NOUN
ejpam-5252	94	7	,	,	PUNCT
ejpam-5252	94	8	v	v	NOUN
ejpam-5252	94	9	′	′	NOUN
ejpam-5252	94	10	1	1	NUM
ejpam-5252	94	11	=	=	SYM
ejpam-5252	94	12	v	v	NOUN
ejpam-5252	94	13	(	(	PUNCT
ejpam-5252	94	14	g)\s	g)\s	NOUN
ejpam-5252	94	15	,	,	PUNCT
ejpam-5252	94	16	and	and	CCONJ
ejpam-5252	94	17	v	v	X
ejpam-5252	94	18	′	′	NUM
ejpam-5252	94	19	2	2	NUM
ejpam-5252	94	20	=	=	PUNCT
ejpam-5252	94	21	s.	s.	PROPN
ejpam-5252	94	22	then	then	ADV
ejpam-5252	94	23	there	there	PRON
ejpam-5252	94	24	exists	exist	VERB
ejpam-5252	94	25	v	v	ADP
ejpam-5252	94	26	∗	∗	NOUN
ejpam-5252	94	27	2	2	NUM
ejpam-5252	94	28	⊆	⊆	NUM
ejpam-5252	94	29	v	v	NOUN
ejpam-5252	94	30	(	(	PUNCT
ejpam-5252	94	31	g	g	NOUN
ejpam-5252	94	32	)	)	PUNCT
ejpam-5252	94	33	such	such	ADJ
ejpam-5252	94	34	that	that	PRON
ejpam-5252	94	35	v	v	NOUN
ejpam-5252	94	36	∗	∗	NOUN
ejpam-5252	94	37	2	2	NUM
ejpam-5252	94	38	is	be	AUX
ejpam-5252	94	39	a	a	DET
ejpam-5252	94	40	γ	γ	NOUN
ejpam-5252	94	41	-	-	PUNCT
ejpam-5252	94	42	set	set	NOUN
ejpam-5252	94	43	of	of	ADP
ejpam-5252	94	44	g.	g.	PROPN
ejpam-5252	94	45	let	let	VERB
ejpam-5252	94	46	v	v	NOUN
ejpam-5252	94	47	′	′	NOUN
ejpam-5252	94	48	2	2	NUM
ejpam-5252	94	49	=	=	SYM
ejpam-5252	94	50	v	v	NOUN
ejpam-5252	94	51	∗	∗	NOUN
ejpam-5252	94	52	2	2	NUM
ejpam-5252	94	53	,	,	PUNCT
ejpam-5252	94	54	v	v	NOUN
ejpam-5252	94	55	′	′	NOUN
ejpam-5252	94	56	0	0	NUM
ejpam-5252	95	1	=	=	SYM
ejpam-5252	95	2	v	v	NUM
ejpam-5252	95	3	′	′	NUM
ejpam-5252	95	4	3	3	NUM
ejpam-5252	95	5	=	=	NOUN
ejpam-5252	95	6	∅	∅	NOUN
ejpam-5252	95	7	and	and	CCONJ
ejpam-5252	95	8	v	v	NOUN
ejpam-5252	95	9	′	′	NUM
ejpam-5252	95	10	1	1	NUM
ejpam-5252	95	11	=	=	SYM
ejpam-5252	95	12	v	v	NOUN
ejpam-5252	95	13	(	(	PUNCT
ejpam-5252	95	14	g)\v	g)\v	NOUN
ejpam-5252	95	15	∗	∗	NOUN
ejpam-5252	95	16	2	2	NUM
ejpam-5252	95	17	.	.	PUNCT
ejpam-5252	96	1	thus	thus	ADV
ejpam-5252	96	2	,	,	PUNCT
ejpam-5252	96	3	g	g	PROPN
ejpam-5252	96	4	=	=	PUNCT
ejpam-5252	96	5	(	(	PUNCT
ejpam-5252	96	6	v	v	NUM
ejpam-5252	96	7	′	′	NUM
ejpam-5252	96	8	0	0	NUM
ejpam-5252	96	9	,	,	PUNCT
ejpam-5252	96	10	v	v	NOUN
ejpam-5252	96	11	′	′	NUM
ejpam-5252	96	12	1	1	NUM
ejpam-5252	96	13	,	,	PUNCT
ejpam-5252	96	14	v	v	NOUN
ejpam-5252	96	15	′	′	NUM
ejpam-5252	96	16	2	2	NUM
ejpam-5252	96	17	,	,	PUNCT
ejpam-5252	96	18	v	v	NOUN
ejpam-5252	96	19	′	′	NUM
ejpam-5252	96	20	3	3	NUM
ejpam-5252	96	21	)	)	PUNCT
ejpam-5252	96	22	∈	∈	NOUN
ejpam-5252	96	23	mrdf	mrdf	NOUN
ejpam-5252	96	24	(	(	PUNCT
ejpam-5252	96	25	g	g	NOUN
ejpam-5252	96	26	)	)	PUNCT
ejpam-5252	96	27	,	,	PUNCT
ejpam-5252	96	28	and	and	CCONJ
ejpam-5252	96	29	so	so	ADV
ejpam-5252	96	30	,	,	PUNCT
ejpam-5252	96	31	ωmr	ωmr	NOUN
ejpam-5252	96	32	g	g	PROPN
ejpam-5252	96	33	(	(	PUNCT
ejpam-5252	96	34	g	g	NOUN
ejpam-5252	96	35	)	)	PUNCT
ejpam-5252	96	36	<	<	X
ejpam-5252	96	37	ωmr	ωmr	PROPN
ejpam-5252	96	38	g	g	PROPN
ejpam-5252	96	39	(	(	PUNCT
ejpam-5252	96	40	f	f	PROPN
ejpam-5252	96	41	)	)	PUNCT
ejpam-5252	96	42	,	,	PUNCT
ejpam-5252	96	43	a	a	DET
ejpam-5252	96	44	contradiction	contradiction	NOUN
ejpam-5252	96	45	.	.	PUNCT
ejpam-5252	97	1	hence	hence	ADV
ejpam-5252	97	2	,	,	PUNCT
ejpam-5252	97	3	v2	v2	PROPN
ejpam-5252	97	4	is	be	AUX
ejpam-5252	97	5	a	a	DET
ejpam-5252	97	6	γ	γ	NOUN
ejpam-5252	97	7	-	-	PUNCT
ejpam-5252	97	8	set	set	NOUN
ejpam-5252	97	9	of	of	ADP
ejpam-5252	97	10	g.	g.	PROPN
ejpam-5252	97	11	furthermore	furthermore	ADV
ejpam-5252	97	12	,	,	PUNCT
ejpam-5252	97	13	γmr(g	γmr(g	PROPN
ejpam-5252	97	14	)	)	PUNCT
ejpam-5252	98	1	=	=	SYM
ejpam-5252	98	2	|v1|+2|v2|	|v1|+2|v2|	NUM
ejpam-5252	98	3	=	=	SYM
ejpam-5252	98	4	|v	|v	X
ejpam-5252	98	5	(	(	PUNCT
ejpam-5252	98	6	g)\v2|+2|v2|	g)\v2|+2|v2|	NOUN
ejpam-5252	98	7	=	=	SYM
ejpam-5252	98	8	|v	|v	X
ejpam-5252	98	9	(	(	PUNCT
ejpam-5252	98	10	g)\v2|+2γ(g	g)\v2|+2γ(g	PROPN
ejpam-5252	98	11	)	)	PUNCT
ejpam-5252	99	1	=	=	PUNCT
ejpam-5252	99	2	|v	|v	PROPN
ejpam-5252	99	3	(	(	PUNCT
ejpam-5252	99	4	g)|−γ(g)+2γ(g	g)|−γ(g)+2γ(g	PROPN
ejpam-5252	99	5	)	)	PUNCT
ejpam-5252	100	1	=	=	SYM
ejpam-5252	100	2	|v	|v	PROPN
ejpam-5252	100	3	(	(	PUNCT
ejpam-5252	100	4	g)|+	g)|+	PROPN
ejpam-5252	100	5	γ(g	γ(g	PROPN
ejpam-5252	100	6	)	)	PUNCT
ejpam-5252	100	7	.	.	PUNCT
ejpam-5252	101	1	this	this	PRON
ejpam-5252	101	2	proves	prove	VERB
ejpam-5252	101	3	(	(	PUNCT
ejpam-5252	101	4	i	i	NOUN
ejpam-5252	101	5	)	)	PUNCT
ejpam-5252	101	6	.	.	PUNCT
ejpam-5252	102	1	now	now	ADV
ejpam-5252	102	2	we	we	PRON
ejpam-5252	102	3	prove	prove	VERB
ejpam-5252	102	4	(	(	PUNCT
ejpam-5252	102	5	ii	ii	NOUN
ejpam-5252	102	6	)	)	PUNCT
ejpam-5252	102	7	.	.	PUNCT
ejpam-5252	103	1	suppose	suppose	VERB
ejpam-5252	103	2	v1	v1	NOUN
ejpam-5252	103	3	=	=	SYM
ejpam-5252	103	4	∅.	∅.	NOUN
ejpam-5252	103	5	then	then	ADV
ejpam-5252	103	6	by	by	ADP
ejpam-5252	103	7	proposition	proposition	NOUN
ejpam-5252	103	8	1	1	NUM
ejpam-5252	103	9	,	,	PUNCT
ejpam-5252	103	10	v2	v2	PROPN
ejpam-5252	103	11	∪v3	∪v3	NOUN
ejpam-5252	103	12	is	be	AUX
ejpam-5252	103	13	a	a	DET
ejpam-5252	103	14	2	2	NUM
ejpam-5252	103	15	-	-	PUNCT
ejpam-5252	103	16	dominating	dominating	NOUN
ejpam-5252	103	17	set	set	NOUN
ejpam-5252	103	18	of	of	ADP
ejpam-5252	103	19	g.	g.	PROPN
ejpam-5252	103	20	conversely	conversely	ADV
ejpam-5252	103	21	,	,	PUNCT
ejpam-5252	103	22	suppose	suppose	VERB
ejpam-5252	103	23	that	that	SCONJ
ejpam-5252	103	24	v1	v1	VERB
ejpam-5252	103	25	̸=	̸=	PROPN
ejpam-5252	103	26	∅	∅	NOUN
ejpam-5252	103	27	and	and	CCONJ
ejpam-5252	103	28	take	take	VERB
ejpam-5252	103	29	{	{	PUNCT
ejpam-5252	103	30	v	v	NOUN
ejpam-5252	103	31	}	}	PUNCT
ejpam-5252	103	32	∈	∈	NOUN
ejpam-5252	103	33	v1	v1	NOUN
ejpam-5252	103	34	.	.	PUNCT
ejpam-5252	104	1	then	then	ADV
ejpam-5252	104	2	by	by	ADP
ejpam-5252	104	3	remark	remark	NOUN
ejpam-5252	104	4	1	1	NUM
ejpam-5252	104	5	,	,	PUNCT
ejpam-5252	104	6	v	v	NOUN
ejpam-5252	104	7	need	need	AUX
ejpam-5252	104	8	not	not	PART
ejpam-5252	104	9	be	be	AUX
ejpam-5252	104	10	in	in	ADP
ejpam-5252	104	11	ng(v2	ng(v2	NOUN
ejpam-5252	104	12	)	)	PUNCT
ejpam-5252	104	13	∩	∩	ADJ
ejpam-5252	104	14	ng(v3	ng(v3	NOUN
ejpam-5252	104	15	)	)	PUNCT
ejpam-5252	104	16	,	,	PUNCT
ejpam-5252	104	17	which	which	PRON
ejpam-5252	104	18	is	be	AUX
ejpam-5252	104	19	a	a	DET
ejpam-5252	104	20	contradiction	contradiction	NOUN
ejpam-5252	104	21	.	.	PUNCT
ejpam-5252	105	1	hence	hence	ADV
ejpam-5252	105	2	,	,	PUNCT
ejpam-5252	105	3	the	the	DET
ejpam-5252	105	4	assertion	assertion	NOUN
ejpam-5252	105	5	follows	follow	VERB
ejpam-5252	105	6	.	.	PUNCT
ejpam-5252	106	1	moreover	moreover	ADV
ejpam-5252	106	2	,	,	PUNCT
ejpam-5252	106	3	assume	assume	VERB
ejpam-5252	106	4	that	that	SCONJ
ejpam-5252	106	5	⟨v2	⟨v2	ADJ
ejpam-5252	106	6	∪	∪	ADJ
ejpam-5252	106	7	v3⟩	v3⟩	PRON
ejpam-5252	106	8	is	be	AUX
ejpam-5252	106	9	connected	connect	VERB
ejpam-5252	106	10	and	and	CCONJ
ejpam-5252	106	11	let	let	VERB
ejpam-5252	106	12	v3	v3	PROPN
ejpam-5252	106	13	be	be	AUX
ejpam-5252	106	14	a	a	DET
ejpam-5252	106	15	γ	γ	NOUN
ejpam-5252	106	16	-	-	PUNCT
ejpam-5252	106	17	set	set	NOUN
ejpam-5252	106	18	of	of	ADP
ejpam-5252	106	19	g.	g.	PROPN
ejpam-5252	106	20	since	since	SCONJ
ejpam-5252	106	21	v1	v1	NOUN
ejpam-5252	106	22	=	=	SYM
ejpam-5252	106	23	∅	∅	NOUN
ejpam-5252	106	24	,	,	PUNCT
ejpam-5252	106	25	γmr(g	γmr(g	PROPN
ejpam-5252	106	26	)	)	PUNCT
ejpam-5252	106	27	=	=	SYM
ejpam-5252	107	1	2|v2|+	2|v2|+	NUM
ejpam-5252	107	2	3|v3|	3|v3|	NUM
ejpam-5252	107	3	=	=	SYM
ejpam-5252	107	4	2|v2	2|v2	NOUN
ejpam-5252	107	5	∪	∪	VERB
ejpam-5252	107	6	v3|+	v3|+	PROPN
ejpam-5252	107	7	|v3|	|v3|	NOUN
ejpam-5252	107	8	≥	≥	NUM
ejpam-5252	107	9	2γ2(g	2γ2(g	NUM
ejpam-5252	107	10	)	)	PUNCT
ejpam-5252	107	11	+	+	NUM
ejpam-5252	107	12	γ(g	γ(g	PROPN
ejpam-5252	107	13	)	)	PUNCT
ejpam-5252	107	14	.	.	PUNCT
ejpam-5252	108	1	proposition	proposition	NOUN
ejpam-5252	108	2	5	5	NUM
ejpam-5252	108	3	.	.	PUNCT
ejpam-5252	109	1	let	let	VERB
ejpam-5252	109	2	g	g	PRON
ejpam-5252	109	3	be	be	AUX
ejpam-5252	109	4	a	a	DET
ejpam-5252	109	5	connected	connected	ADJ
ejpam-5252	109	6	graph	graph	NOUN
ejpam-5252	109	7	.	.	PUNCT
ejpam-5252	110	1	then	then	ADV
ejpam-5252	110	2	(	(	PUNCT
ejpam-5252	110	3	i	i	NOUN
ejpam-5252	110	4	)	)	PUNCT
ejpam-5252	110	5	γmr(g	γmr(g	PROPN
ejpam-5252	110	6	)	)	PUNCT
ejpam-5252	110	7	=	=	SYM
ejpam-5252	110	8	2	2	NUM
ejpam-5252	110	9	if	if	SCONJ
ejpam-5252	110	10	and	and	CCONJ
ejpam-5252	110	11	only	only	ADV
ejpam-5252	110	12	if	if	SCONJ
ejpam-5252	110	13	g	g	PROPN
ejpam-5252	110	14	=	=	PROPN
ejpam-5252	110	15	k1	k1	PROPN
ejpam-5252	110	16	.	.	PUNCT
ejpam-5252	110	17	(	(	PUNCT
ejpam-5252	110	18	ii	ii	NOUN
ejpam-5252	110	19	)	)	PUNCT
ejpam-5252	110	20	γmr(g	γmr(g	PROPN
ejpam-5252	110	21	)	)	PUNCT
ejpam-5252	111	1	=	=	SYM
ejpam-5252	111	2	3	3	NUM
ejpam-5252	111	3	if	if	SCONJ
ejpam-5252	111	4	and	and	CCONJ
ejpam-5252	111	5	only	only	ADV
ejpam-5252	111	6	if	if	SCONJ
ejpam-5252	111	7	g	g	PROPN
ejpam-5252	111	8	=	=	SYM
ejpam-5252	111	9	k2	k2	PROPN
ejpam-5252	111	10	.	.	PUNCT
ejpam-5252	112	1	(	(	PUNCT
ejpam-5252	112	2	iii	iii	NOUN
ejpam-5252	112	3	)	)	PUNCT
ejpam-5252	112	4	γmr(g	γmr(g	PROPN
ejpam-5252	112	5	)	)	PUNCT
ejpam-5252	113	1	=	=	PUNCT
ejpam-5252	113	2	4	4	NUM
ejpam-5252	113	3	if	if	SCONJ
ejpam-5252	113	4	and	and	CCONJ
ejpam-5252	113	5	only	only	ADV
ejpam-5252	113	6	if	if	SCONJ
ejpam-5252	113	7	g	g	PROPN
ejpam-5252	113	8	∈	∈	PROPN
ejpam-5252	113	9	{	{	PUNCT
ejpam-5252	113	10	k3	k3	PROPN
ejpam-5252	113	11	,	,	PUNCT
ejpam-5252	113	12	p3	p3	PROPN
ejpam-5252	113	13	}	}	PUNCT
ejpam-5252	113	14	.	.	PUNCT
ejpam-5252	114	1	(	(	PUNCT
ejpam-5252	114	2	iv	iv	X
ejpam-5252	114	3	)	)	PUNCT
ejpam-5252	114	4	γmr(g	γmr(g	PROPN
ejpam-5252	114	5	)	)	PUNCT
ejpam-5252	114	6	=	=	PUNCT
ejpam-5252	114	7	5	5	NUM
ejpam-5252	114	8	if	if	SCONJ
ejpam-5252	114	9	and	and	CCONJ
ejpam-5252	114	10	only	only	ADV
ejpam-5252	114	11	if	if	SCONJ
ejpam-5252	114	12	|v	|v	PROPN
ejpam-5252	114	13	(	(	PUNCT
ejpam-5252	114	14	g)|	g)|	NOUN
ejpam-5252	114	15	=	=	SYM
ejpam-5252	114	16	4	4	NUM
ejpam-5252	114	17	and	and	CCONJ
ejpam-5252	114	18	γ(g	γ(g	PROPN
ejpam-5252	114	19	)	)	PUNCT
ejpam-5252	115	1	=	=	SYM
ejpam-5252	115	2	1	1	NUM
ejpam-5252	115	3	or	or	CCONJ
ejpam-5252	115	4	γ2(g	γ2(g	VERB
ejpam-5252	115	5	)	)	PUNCT
ejpam-5252	115	6	=	=	SYM
ejpam-5252	115	7	2	2	NUM
ejpam-5252	115	8	and	and	CCONJ
ejpam-5252	115	9	|v	|v	PROPN
ejpam-5252	115	10	(	(	PUNCT
ejpam-5252	115	11	g)|	g)|	X
ejpam-5252	115	12	≥	≥	NOUN
ejpam-5252	115	13	4	4	NUM
ejpam-5252	115	14	.	.	PUNCT
ejpam-5252	116	1	proof	proof	NOUN
ejpam-5252	116	2	.	.	PUNCT
ejpam-5252	117	1	(	(	PUNCT
ejpam-5252	117	2	i	i	NOUN
ejpam-5252	117	3	)	)	PUNCT
ejpam-5252	117	4	suppose	suppose	VERB
ejpam-5252	117	5	γmr(g	γmr(g	X
ejpam-5252	117	6	)	)	PUNCT
ejpam-5252	117	7	=	=	SYM
ejpam-5252	117	8	2	2	X
ejpam-5252	117	9	,	,	PUNCT
ejpam-5252	117	10	say	say	VERB
ejpam-5252	117	11	f	f	PROPN
ejpam-5252	117	12	=	=	SYM
ejpam-5252	117	13	(	(	PUNCT
ejpam-5252	117	14	v0	v0	PROPN
ejpam-5252	117	15	,	,	PUNCT
ejpam-5252	117	16	v1	v1	NOUN
ejpam-5252	117	17	,	,	PUNCT
ejpam-5252	117	18	v2	v2	PROPN
ejpam-5252	117	19	,	,	PUNCT
ejpam-5252	117	20	v3	v3	PROPN
ejpam-5252	117	21	)	)	PUNCT
ejpam-5252	117	22	is	be	AUX
ejpam-5252	117	23	a	a	DET
ejpam-5252	117	24	γmr	γmr	ADJ
ejpam-5252	117	25	-	-	PUNCT
ejpam-5252	117	26	function	function	NOUN
ejpam-5252	117	27	on	on	ADP
ejpam-5252	117	28	g.	g.	PROPN
ejpam-5252	117	29	by	by	ADP
ejpam-5252	117	30	proposition	proposition	NOUN
ejpam-5252	117	31	1(iv	1(iv	NUM
ejpam-5252	117	32	)	)	PUNCT
ejpam-5252	117	33	,	,	PUNCT
ejpam-5252	117	34	v2	v2	NOUN
ejpam-5252	117	35	=	=	SYM
ejpam-5252	117	36	{	{	PUNCT
ejpam-5252	117	37	v	v	NOUN
ejpam-5252	117	38	}	}	PUNCT
ejpam-5252	117	39	.	.	PUNCT
ejpam-5252	118	1	hence	hence	ADV
ejpam-5252	118	2	,	,	PUNCT
ejpam-5252	118	3	v0	v0	NOUN
ejpam-5252	118	4	=	=	SYM
ejpam-5252	118	5	v1	v1	PROPN
ejpam-5252	118	6	=	=	SYM
ejpam-5252	118	7	v3	v3	PROPN
ejpam-5252	118	8	=	=	PUNCT
ejpam-5252	118	9	∅.	∅.	PRON
ejpam-5252	118	10	the	the	DET
ejpam-5252	118	11	converse	converse	NOUN
ejpam-5252	118	12	is	be	AUX
ejpam-5252	118	13	clear	clear	ADJ
ejpam-5252	118	14	.	.	PUNCT
ejpam-5252	119	1	(	(	PUNCT
ejpam-5252	119	2	ii	ii	NOUN
ejpam-5252	119	3	)	)	PUNCT
ejpam-5252	119	4	suppose	suppose	VERB
ejpam-5252	119	5	γmr(g	γmr(g	X
ejpam-5252	119	6	)	)	PUNCT
ejpam-5252	119	7	=	=	SYM
ejpam-5252	119	8	3	3	X
ejpam-5252	119	9	,	,	PUNCT
ejpam-5252	119	10	say	say	VERB
ejpam-5252	119	11	f	f	PROPN
ejpam-5252	119	12	=	=	SYM
ejpam-5252	119	13	(	(	PUNCT
ejpam-5252	119	14	v0	v0	PROPN
ejpam-5252	119	15	,	,	PUNCT
ejpam-5252	119	16	v1	v1	NOUN
ejpam-5252	119	17	,	,	PUNCT
ejpam-5252	119	18	v2	v2	PROPN
ejpam-5252	119	19	,	,	PUNCT
ejpam-5252	119	20	v3	v3	PROPN
ejpam-5252	119	21	)	)	PUNCT
ejpam-5252	119	22	is	be	AUX
ejpam-5252	119	23	a	a	DET
ejpam-5252	119	24	γmr	γmr	ADJ
ejpam-5252	119	25	-	-	PUNCT
ejpam-5252	119	26	function	function	NOUN
ejpam-5252	119	27	on	on	ADP
ejpam-5252	119	28	g.	g.	PROPN
ejpam-5252	119	29	by	by	ADP
ejpam-5252	119	30	(	(	PUNCT
ejpam-5252	119	31	i	i	NOUN
ejpam-5252	119	32	)	)	PUNCT
ejpam-5252	119	33	,	,	PUNCT
ejpam-5252	119	34	|v2|	|v2|	X
ejpam-5252	119	35	≥	≥	NOUN
ejpam-5252	119	36	2	2	NUM
ejpam-5252	119	37	.	.	PUNCT
ejpam-5252	119	38	by	by	ADP
ejpam-5252	119	39	proposition	proposition	NOUN
ejpam-5252	119	40	1(iv	1(iv	NUM
ejpam-5252	119	41	)	)	PUNCT
ejpam-5252	119	42	,	,	PUNCT
ejpam-5252	119	43	and	and	CCONJ
ejpam-5252	119	44	the	the	DET
ejpam-5252	119	45	assumption	assumption	NOUN
ejpam-5252	119	46	that	that	SCONJ
ejpam-5252	119	47	γmr(g	γmr(g	X
ejpam-5252	119	48	)	)	PUNCT
ejpam-5252	119	49	=	=	SYM
ejpam-5252	119	50	3	3	NUM
ejpam-5252	119	51	,	,	PUNCT
ejpam-5252	119	52	|v2|	|v2|	NOUN
ejpam-5252	119	53	=	=	SYM
ejpam-5252	119	54	1	1	NUM
ejpam-5252	119	55	,	,	PUNCT
ejpam-5252	119	56	|v1|	|v1|	NOUN
ejpam-5252	119	57	=	=	SYM
ejpam-5252	119	58	1	1	NUM
ejpam-5252	119	59	and	and	CCONJ
ejpam-5252	119	60	v0	v0	PROPN
ejpam-5252	119	61	=	=	SYM
ejpam-5252	119	62	v3	v3	PROPN
ejpam-5252	119	63	=	=	PUNCT
ejpam-5252	119	64	∅.	∅.	VERB
ejpam-5252	119	65	therefore	therefore	ADV
ejpam-5252	119	66	,	,	PUNCT
ejpam-5252	119	67	|v	|v	PROPN
ejpam-5252	119	68	(	(	PUNCT
ejpam-5252	119	69	g)|	g)|	NOUN
ejpam-5252	119	70	=	=	SYM
ejpam-5252	119	71	2	2	NUM
ejpam-5252	119	72	.	.	PUNCT
ejpam-5252	119	73	since	since	SCONJ
ejpam-5252	119	74	g	g	PROPN
ejpam-5252	119	75	is	be	AUX
ejpam-5252	119	76	connected	connect	VERB
ejpam-5252	119	77	,	,	PUNCT
ejpam-5252	119	78	g	g	PROPN
ejpam-5252	119	79	=	=	SYM
ejpam-5252	119	80	k2	k2	PROPN
ejpam-5252	119	81	.	.	PUNCT
ejpam-5252	120	1	conversely	conversely	ADV
ejpam-5252	120	2	,	,	PUNCT
ejpam-5252	120	3	suppose	suppose	VERB
ejpam-5252	120	4	that	that	SCONJ
ejpam-5252	120	5	g	g	PROPN
ejpam-5252	120	6	=	=	SYM
ejpam-5252	120	7	k2	k2	PROPN
ejpam-5252	120	8	,	,	PUNCT
ejpam-5252	120	9	say	say	VERB
ejpam-5252	120	10	v	v	INTJ
ejpam-5252	120	11	(	(	PUNCT
ejpam-5252	120	12	g	g	NOUN
ejpam-5252	120	13	)	)	PUNCT
ejpam-5252	120	14	=	=	SYM
ejpam-5252	120	15	{	{	PUNCT
ejpam-5252	120	16	x	x	NOUN
ejpam-5252	120	17	,	,	PUNCT
ejpam-5252	120	18	y	y	PROPN
ejpam-5252	120	19	}	}	PUNCT
ejpam-5252	120	20	.	.	PUNCT
ejpam-5252	121	1	then	then	ADV
ejpam-5252	121	2	g	g	PROPN
ejpam-5252	121	3	=	=	SYM
ejpam-5252	121	4	{	{	PUNCT
ejpam-5252	121	5	∅	∅	NOUN
ejpam-5252	121	6	,	,	PUNCT
ejpam-5252	121	7	{	{	PUNCT
ejpam-5252	121	8	x	x	X
ejpam-5252	121	9	}	}	PUNCT
ejpam-5252	121	10	,	,	PUNCT
ejpam-5252	121	11	{	{	PUNCT
ejpam-5252	121	12	y},∅	y},∅	NOUN
ejpam-5252	121	13	}	}	PUNCT
ejpam-5252	121	14	∈	∈	PROPN
ejpam-5252	121	15	mrdf	mrdf	NOUN
ejpam-5252	121	16	(	(	PUNCT
ejpam-5252	121	17	g	g	NOUN
ejpam-5252	121	18	)	)	PUNCT
ejpam-5252	121	19	and	and	CCONJ
ejpam-5252	121	20	s.	s.	PROPN
ejpam-5252	121	21	ahamad	ahamad	PROPN
ejpam-5252	121	22	,	,	PUNCT
ejpam-5252	121	23	j.	j.	PROPN
ejpam-5252	121	24	cariaga	cariaga	PROPN
ejpam-5252	121	25	,	,	PUNCT
ejpam-5252	121	26	s.	s.	PROPN
ejpam-5252	121	27	menchavez	menchavez	PROPN
ejpam-5252	121	28	/	/	PUNCT
ejpam-5252	121	29	eur	eur	PROPN
ejpam-5252	121	30	.	.	PUNCT
ejpam-5252	122	1	j.	j.	PROPN
ejpam-5252	122	2	pure	pure	PROPN
ejpam-5252	122	3	appl	appl	PROPN
ejpam-5252	122	4	.	.	PROPN
ejpam-5252	122	5	math	math	PROPN
ejpam-5252	122	6	,	,	PUNCT
ejpam-5252	122	7	18	18	NUM
ejpam-5252	122	8	(	(	PUNCT
ejpam-5252	122	9	1	1	NUM
ejpam-5252	122	10	)	)	PUNCT
ejpam-5252	122	11	(	(	PUNCT
ejpam-5252	122	12	2025	2025	NUM
ejpam-5252	122	13	)	)	PUNCT
ejpam-5252	122	14	,	,	PUNCT
ejpam-5252	122	15	5252	5252	NUM
ejpam-5252	122	16	6	6	NUM
ejpam-5252	122	17	of	of	ADP
ejpam-5252	122	18	18	18	NUM
ejpam-5252	122	19	ωmr	ωmr	NOUN
ejpam-5252	122	20	g	g	PROPN
ejpam-5252	122	21	(	(	PUNCT
ejpam-5252	122	22	g	g	NOUN
ejpam-5252	122	23	)	)	PUNCT
ejpam-5252	122	24	=	=	SYM
ejpam-5252	123	1	3	3	X
ejpam-5252	123	2	.	.	PUNCT
ejpam-5252	123	3	since	since	SCONJ
ejpam-5252	123	4	γmr(g	γmr(g	PROPN
ejpam-5252	123	5	)	)	PUNCT
ejpam-5252	123	6	≥	≥	NOUN
ejpam-5252	123	7	2	2	NUM
ejpam-5252	123	8	,	,	PUNCT
ejpam-5252	123	9	it	it	PRON
ejpam-5252	123	10	follows	follow	VERB
ejpam-5252	123	11	that	that	PRON
ejpam-5252	123	12	γmr(g	γmr(g	PUNCT
ejpam-5252	123	13	)	)	PUNCT
ejpam-5252	123	14	=	=	SYM
ejpam-5252	124	1	3	3	X
ejpam-5252	124	2	.	.	PUNCT
ejpam-5252	124	3	(	(	PUNCT
ejpam-5252	124	4	iii	iii	NOUN
ejpam-5252	124	5	)	)	PUNCT
ejpam-5252	124	6	note	note	NOUN
ejpam-5252	124	7	that	that	SCONJ
ejpam-5252	124	8	if	if	SCONJ
ejpam-5252	124	9	γmr(g	γmr(g	PROPN
ejpam-5252	124	10	)	)	PUNCT
ejpam-5252	124	11	=	=	SYM
ejpam-5252	125	1	4	4	NUM
ejpam-5252	125	2	,	,	PUNCT
ejpam-5252	125	3	then	then	ADV
ejpam-5252	125	4	1	1	NUM
ejpam-5252	125	5	≤	≤	NOUN
ejpam-5252	125	6	|v2|	|v2|	NOUN
ejpam-5252	125	7	≤	≤	NUM
ejpam-5252	125	8	2	2	NUM
ejpam-5252	125	9	by	by	ADP
ejpam-5252	125	10	proposition	proposition	NOUN
ejpam-5252	125	11	1(iv	1(iv	NUM
ejpam-5252	125	12	)	)	PUNCT
ejpam-5252	125	13	.	.	PUNCT
ejpam-5252	126	1	hence	hence	ADV
ejpam-5252	126	2	,	,	PUNCT
ejpam-5252	126	3	there	there	PRON
ejpam-5252	126	4	are	be	VERB
ejpam-5252	126	5	only	only	ADV
ejpam-5252	126	6	two	two	NUM
ejpam-5252	126	7	cases	case	NOUN
ejpam-5252	126	8	to	to	PART
ejpam-5252	126	9	consider	consider	VERB
ejpam-5252	126	10	,	,	PUNCT
ejpam-5252	126	11	namely	namely	ADV
ejpam-5252	126	12	,	,	PUNCT
ejpam-5252	126	13	|v2|	|v2|	NOUN
ejpam-5252	126	14	=	=	SYM
ejpam-5252	126	15	1	1	NUM
ejpam-5252	126	16	and	and	CCONJ
ejpam-5252	126	17	|v2|	|v2|	NOUN
ejpam-5252	126	18	=	=	SYM
ejpam-5252	126	19	2	2	X
ejpam-5252	126	20	.	.	X
ejpam-5252	127	1	if	if	SCONJ
ejpam-5252	127	2	|v2|	|v2|	NOUN
ejpam-5252	127	3	=	=	SYM
ejpam-5252	127	4	2	2	NUM
ejpam-5252	127	5	,	,	PUNCT
ejpam-5252	127	6	then	then	ADV
ejpam-5252	127	7	|v1|	|v1|	VERB
ejpam-5252	127	8	=	=	SYM
ejpam-5252	127	9	0	0	X
ejpam-5252	127	10	.	.	PUNCT
ejpam-5252	128	1	by	by	ADP
ejpam-5252	128	2	(	(	PUNCT
ejpam-5252	128	3	p2	p2	PROPN
ejpam-5252	128	4	)	)	PUNCT
ejpam-5252	128	5	,	,	PUNCT
ejpam-5252	128	6	this	this	DET
ejpam-5252	128	7	cases	case	NOUN
ejpam-5252	128	8	is	be	AUX
ejpam-5252	128	9	not	not	PART
ejpam-5252	128	10	possible	possible	ADJ
ejpam-5252	128	11	.	.	PUNCT
ejpam-5252	129	1	so	so	ADV
ejpam-5252	129	2	if	if	SCONJ
ejpam-5252	129	3	|v2|	|v2|	NOUN
ejpam-5252	129	4	=	=	SYM
ejpam-5252	129	5	1	1	X
ejpam-5252	129	6	,	,	PUNCT
ejpam-5252	129	7	we	we	PRON
ejpam-5252	129	8	have	have	VERB
ejpam-5252	129	9	|v1|	|v1|	NOUN
ejpam-5252	129	10	=	=	SYM
ejpam-5252	129	11	2	2	X
ejpam-5252	129	12	.	.	PUNCT
ejpam-5252	130	1	by	by	ADP
ejpam-5252	130	2	(	(	PUNCT
ejpam-5252	130	3	p2	p2	PROPN
ejpam-5252	130	4	)	)	PUNCT
ejpam-5252	130	5	,	,	PUNCT
ejpam-5252	130	6	⟨v1	⟨v1	PROPN
ejpam-5252	130	7	∪	∪	ADP
ejpam-5252	130	8	v2⟩	v2⟩	PROPN
ejpam-5252	130	9	must	must	AUX
ejpam-5252	130	10	be	be	AUX
ejpam-5252	130	11	connected	connect	VERB
ejpam-5252	130	12	.	.	PUNCT
ejpam-5252	131	1	thus	thus	ADV
ejpam-5252	131	2	,	,	PUNCT
ejpam-5252	131	3	the	the	DET
ejpam-5252	131	4	result	result	NOUN
ejpam-5252	131	5	follows	follow	VERB
ejpam-5252	131	6	.	.	PUNCT
ejpam-5252	132	1	the	the	DET
ejpam-5252	132	2	converse	converse	NOUN
ejpam-5252	132	3	follows	follow	VERB
ejpam-5252	132	4	directly	directly	ADV
ejpam-5252	132	5	from	from	ADP
ejpam-5252	132	6	propositions	proposition	NOUN
ejpam-5252	132	7	2	2	NUM
ejpam-5252	132	8	and	and	CCONJ
ejpam-5252	132	9	3	3	NUM
ejpam-5252	132	10	.	.	PUNCT
ejpam-5252	132	11	(	(	PUNCT
ejpam-5252	132	12	iv	iv	X
ejpam-5252	132	13	)	)	PUNCT
ejpam-5252	132	14	if	if	SCONJ
ejpam-5252	132	15	γmr(g	γmr(g	PROPN
ejpam-5252	132	16	)	)	PUNCT
ejpam-5252	132	17	=	=	SYM
ejpam-5252	132	18	5	5	NUM
ejpam-5252	132	19	,	,	PUNCT
ejpam-5252	132	20	then	then	ADV
ejpam-5252	132	21	|v3|	|v3|	VERB
ejpam-5252	132	22	≤	≤	NUM
ejpam-5252	132	23	1	1	NUM
ejpam-5252	132	24	and	and	CCONJ
ejpam-5252	132	25	1	1	NUM
ejpam-5252	132	26	≤	≤	NOUN
ejpam-5252	132	27	|v2|	|v2|	NOUN
ejpam-5252	132	28	≤	≤	NOUN
ejpam-5252	132	29	2	2	NUM
ejpam-5252	132	30	.	.	PUNCT
ejpam-5252	133	1	also	also	ADV
ejpam-5252	133	2	,	,	PUNCT
ejpam-5252	133	3	by	by	ADP
ejpam-5252	133	4	(	(	PUNCT
ejpam-5252	133	5	iii	iii	NOUN
ejpam-5252	133	6	)	)	PUNCT
ejpam-5252	133	7	,	,	PUNCT
ejpam-5252	133	8	|v	|v	PROPN
ejpam-5252	133	9	(	(	PUNCT
ejpam-5252	133	10	g)|	g)|	X
ejpam-5252	133	11	≥	≥	NOUN
ejpam-5252	133	12	4	4	NUM
ejpam-5252	133	13	.	.	PUNCT
ejpam-5252	134	1	now	now	ADV
ejpam-5252	134	2	,	,	PUNCT
ejpam-5252	134	3	if	if	SCONJ
ejpam-5252	134	4	|v3|	|v3|	NOUN
ejpam-5252	134	5	=	=	SYM
ejpam-5252	134	6	0	0	PROPN
ejpam-5252	134	7	,	,	PUNCT
ejpam-5252	134	8	then	then	ADV
ejpam-5252	134	9	|v0|	|v0|	NOUN
ejpam-5252	134	10	=	=	SYM
ejpam-5252	134	11	0	0	X
ejpam-5252	134	12	.	.	PUNCT
ejpam-5252	135	1	hence	hence	ADV
ejpam-5252	135	2	,	,	PUNCT
ejpam-5252	135	3	there	there	PRON
ejpam-5252	135	4	are	be	VERB
ejpam-5252	135	5	only	only	ADV
ejpam-5252	135	6	two	two	NUM
ejpam-5252	135	7	cases	case	NOUN
ejpam-5252	135	8	to	to	PART
ejpam-5252	135	9	consider	consider	VERB
ejpam-5252	135	10	,	,	PUNCT
ejpam-5252	135	11	namely	namely	ADV
ejpam-5252	135	12	,	,	PUNCT
ejpam-5252	135	13	|v2|	|v2|	NOUN
ejpam-5252	135	14	=	=	SYM
ejpam-5252	135	15	1	1	NUM
ejpam-5252	135	16	and	and	CCONJ
ejpam-5252	135	17	|v2|	|v2|	ADV
ejpam-5252	135	18	≤	≤	NOUN
ejpam-5252	135	19	2	2	NUM
ejpam-5252	135	20	.	.	PUNCT
ejpam-5252	136	1	if	if	SCONJ
ejpam-5252	136	2	|v2|	|v2|	NOUN
ejpam-5252	136	3	=	=	SYM
ejpam-5252	136	4	2	2	NUM
ejpam-5252	136	5	,	,	PUNCT
ejpam-5252	136	6	then	then	ADV
ejpam-5252	136	7	|v1|	|v1|	NOUN
ejpam-5252	136	8	=	=	SYM
ejpam-5252	136	9	1	1	X
ejpam-5252	136	10	.	.	PUNCT
ejpam-5252	136	11	therefore	therefore	ADV
ejpam-5252	136	12	,	,	PUNCT
ejpam-5252	136	13	|v	|v	PROPN
ejpam-5252	136	14	(	(	PUNCT
ejpam-5252	136	15	g)|	g)|	NOUN
ejpam-5252	136	16	=	=	SYM
ejpam-5252	136	17	3	3	NUM
ejpam-5252	136	18	which	which	PRON
ejpam-5252	136	19	is	be	AUX
ejpam-5252	136	20	not	not	PART
ejpam-5252	136	21	possible	possible	ADJ
ejpam-5252	136	22	by	by	ADP
ejpam-5252	136	23	(	(	PUNCT
ejpam-5252	136	24	iii	iii	NOUN
ejpam-5252	136	25	)	)	PUNCT
ejpam-5252	136	26	.	.	PUNCT
ejpam-5252	137	1	if	if	SCONJ
ejpam-5252	137	2	|v2|	|v2|	NOUN
ejpam-5252	137	3	=	=	SYM
ejpam-5252	137	4	1	1	NUM
ejpam-5252	137	5	,	,	PUNCT
ejpam-5252	137	6	then	then	ADV
ejpam-5252	137	7	|v1|	|v1|	NOUN
ejpam-5252	137	8	=	=	SYM
ejpam-5252	137	9	3	3	X
ejpam-5252	137	10	.	.	PUNCT
ejpam-5252	137	11	by	by	ADP
ejpam-5252	137	12	(	(	PUNCT
ejpam-5252	137	13	p2	p2	PROPN
ejpam-5252	137	14	)	)	PUNCT
ejpam-5252	137	15	,	,	PUNCT
ejpam-5252	137	16	⟨v1	⟨v1	PROPN
ejpam-5252	137	17	∪	∪	ADP
ejpam-5252	137	18	v2⟩	v2⟩	PROPN
ejpam-5252	137	19	must	must	AUX
ejpam-5252	137	20	be	be	AUX
ejpam-5252	137	21	connected	connect	VERB
ejpam-5252	137	22	and	and	CCONJ
ejpam-5252	137	23	v2	v2	NOUN
ejpam-5252	137	24	is	be	AUX
ejpam-5252	137	25	a	a	DET
ejpam-5252	137	26	dominating	dominating	NOUN
ejpam-5252	137	27	set	set	NOUN
ejpam-5252	137	28	in	in	ADP
ejpam-5252	137	29	g	g	PROPN
ejpam-5252	137	30	,	,	PUNCT
ejpam-5252	137	31	it	it	PRON
ejpam-5252	137	32	follows	follow	VERB
ejpam-5252	137	33	that	that	SCONJ
ejpam-5252	137	34	v2	v2	PROPN
ejpam-5252	137	35	is	be	AUX
ejpam-5252	137	36	a	a	DET
ejpam-5252	137	37	γ	γ	NOUN
ejpam-5252	137	38	-	-	PUNCT
ejpam-5252	137	39	set	set	VERB
ejpam-5252	137	40	in	in	ADP
ejpam-5252	137	41	g.	g.	PROPN
ejpam-5252	137	42	therefore	therefore	ADV
ejpam-5252	137	43	,	,	PUNCT
ejpam-5252	137	44	|v	|v	PROPN
ejpam-5252	137	45	(	(	PUNCT
ejpam-5252	137	46	g)|	g)|	NOUN
ejpam-5252	137	47	=	=	SYM
ejpam-5252	137	48	4	4	NUM
ejpam-5252	137	49	and	and	CCONJ
ejpam-5252	137	50	γ(g	γ(g	PROPN
ejpam-5252	137	51	)	)	PUNCT
ejpam-5252	138	1	=	=	PUNCT
ejpam-5252	138	2	1	1	X
ejpam-5252	138	3	.	.	PUNCT
ejpam-5252	138	4	now	now	ADV
ejpam-5252	138	5	,	,	PUNCT
ejpam-5252	138	6	suppose	suppose	VERB
ejpam-5252	138	7	that	that	SCONJ
ejpam-5252	138	8	|v3|	|v3|	NOUN
ejpam-5252	138	9	=	=	NOUN
ejpam-5252	138	10	1	1	X
ejpam-5252	138	11	.	.	PUNCT
ejpam-5252	139	1	if	if	SCONJ
ejpam-5252	139	2	|v2|	|v2|	NOUN
ejpam-5252	139	3	=	=	SYM
ejpam-5252	139	4	0	0	NUM
ejpam-5252	139	5	,	,	PUNCT
ejpam-5252	139	6	then	then	ADV
ejpam-5252	139	7	|v1|	|v1|	NOUN
ejpam-5252	139	8	=	=	SYM
ejpam-5252	139	9	2	2	X
ejpam-5252	139	10	.	.	PUNCT
ejpam-5252	139	11	consequently	consequently	ADV
ejpam-5252	139	12	,	,	PUNCT
ejpam-5252	139	13	|v	|v	PROPN
ejpam-5252	139	14	(	(	PUNCT
ejpam-5252	139	15	g)|	g)|	NOUN
ejpam-5252	139	16	=	=	SYM
ejpam-5252	139	17	3	3	NUM
ejpam-5252	139	18	.	.	PUNCT
ejpam-5252	140	1	thus	thus	ADV
ejpam-5252	140	2	,	,	PUNCT
ejpam-5252	140	3	g	g	PROPN
ejpam-5252	140	4	∈	∈	PROPN
ejpam-5252	140	5	{	{	PUNCT
ejpam-5252	140	6	k3	k3	PROPN
ejpam-5252	140	7	,	,	PUNCT
ejpam-5252	140	8	p3	p3	PROPN
ejpam-5252	140	9	}	}	PUNCT
ejpam-5252	140	10	,	,	PUNCT
ejpam-5252	140	11	a	a	DET
ejpam-5252	140	12	contradiction	contradiction	NOUN
ejpam-5252	140	13	by	by	ADP
ejpam-5252	140	14	(	(	PUNCT
ejpam-5252	140	15	iii	iii	NOUN
ejpam-5252	140	16	)	)	PUNCT
ejpam-5252	140	17	.	.	PUNCT
ejpam-5252	141	1	if	if	SCONJ
ejpam-5252	141	2	|v2|	|v2|	NOUN
ejpam-5252	141	3	=	=	SYM
ejpam-5252	141	4	1	1	NUM
ejpam-5252	141	5	,	,	PUNCT
ejpam-5252	141	6	then	then	ADV
ejpam-5252	141	7	|v1|	|v1|	VERB
ejpam-5252	141	8	=	=	PROPN
ejpam-5252	141	9	0	0	X
ejpam-5252	141	10	.	.	PUNCT
ejpam-5252	142	1	since	since	SCONJ
ejpam-5252	142	2	v2	v2	PROPN
ejpam-5252	142	3	∪	∪	X
ejpam-5252	142	4	v3	v3	PROPN
ejpam-5252	142	5	is	be	AUX
ejpam-5252	142	6	a	a	DET
ejpam-5252	142	7	2	2	NUM
ejpam-5252	142	8	-	-	PUNCT
ejpam-5252	142	9	dominating	dominating	NOUN
ejpam-5252	142	10	set	set	NOUN
ejpam-5252	142	11	in	in	ADP
ejpam-5252	142	12	g	g	PROPN
ejpam-5252	142	13	and	and	CCONJ
ejpam-5252	142	14	|v2	|v2	NOUN
ejpam-5252	142	15	∪	∪	VERB
ejpam-5252	142	16	v3|	v3|	NOUN
ejpam-5252	142	17	=	=	SYM
ejpam-5252	142	18	2	2	NUM
ejpam-5252	142	19	,	,	PUNCT
ejpam-5252	142	20	it	it	PRON
ejpam-5252	142	21	follows	follow	VERB
ejpam-5252	142	22	that	that	SCONJ
ejpam-5252	142	23	v2	v2	PROPN
ejpam-5252	142	24	∪	∪	X
ejpam-5252	142	25	v3	v3	PROPN
ejpam-5252	142	26	is	be	AUX
ejpam-5252	142	27	a	a	DET
ejpam-5252	142	28	γ2	γ2	NOUN
ejpam-5252	142	29	-	-	PUNCT
ejpam-5252	142	30	set	set	NOUN
ejpam-5252	142	31	in	in	ADP
ejpam-5252	142	32	g.	g.	PROPN
ejpam-5252	142	33	hence	hence	ADV
ejpam-5252	142	34	,	,	PUNCT
ejpam-5252	142	35	|v	|v	PROPN
ejpam-5252	142	36	(	(	PUNCT
ejpam-5252	142	37	g)|	g)|	X
ejpam-5252	142	38	≥	≥	NUM
ejpam-5252	142	39	4	4	NUM
ejpam-5252	142	40	and	and	CCONJ
ejpam-5252	142	41	γ2(g	γ2(g	NUM
ejpam-5252	142	42	)	)	PUNCT
ejpam-5252	142	43	=	=	SYM
ejpam-5252	142	44	2	2	X
ejpam-5252	142	45	.	.	PUNCT
ejpam-5252	142	46	conversely	conversely	ADV
ejpam-5252	142	47	,	,	PUNCT
ejpam-5252	142	48	suppose	suppose	VERB
ejpam-5252	142	49	|v	|v	PROPN
ejpam-5252	142	50	(	(	PUNCT
ejpam-5252	142	51	g)|	g)|	NOUN
ejpam-5252	142	52	=	=	SYM
ejpam-5252	142	53	4	4	NUM
ejpam-5252	142	54	and	and	CCONJ
ejpam-5252	142	55	γ(g	γ(g	PROPN
ejpam-5252	142	56	)	)	PUNCT
ejpam-5252	143	1	=	=	PUNCT
ejpam-5252	143	2	1	1	X
ejpam-5252	143	3	.	.	PUNCT
ejpam-5252	144	1	by	by	ADP
ejpam-5252	144	2	(	(	PUNCT
ejpam-5252	144	3	iii	iii	NOUN
ejpam-5252	144	4	)	)	PUNCT
ejpam-5252	144	5	,	,	PUNCT
ejpam-5252	144	6	γ(g	γ(g	PROPN
ejpam-5252	144	7	)	)	PUNCT
ejpam-5252	144	8	≥	≥	NOUN
ejpam-5252	144	9	5	5	NUM
ejpam-5252	144	10	.	.	PUNCT
ejpam-5252	145	1	let	let	VERB
ejpam-5252	145	2	v	v	PART
ejpam-5252	145	3	be	be	AUX
ejpam-5252	145	4	a	a	DET
ejpam-5252	145	5	dominating	dominating	NOUN
ejpam-5252	145	6	vertex	vertex	NOUN
ejpam-5252	145	7	of	of	ADP
ejpam-5252	145	8	g	g	NOUN
ejpam-5252	145	9	and	and	CCONJ
ejpam-5252	145	10	define	define	VERB
ejpam-5252	145	11	a	a	DET
ejpam-5252	145	12	function	function	NOUN
ejpam-5252	145	13	f	f	NOUN
ejpam-5252	145	14	=	=	SYM
ejpam-5252	145	15	(	(	PUNCT
ejpam-5252	145	16	v0	v0	PROPN
ejpam-5252	145	17	,	,	PUNCT
ejpam-5252	145	18	v1	v1	NOUN
ejpam-5252	145	19	,	,	PUNCT
ejpam-5252	145	20	v2	v2	PROPN
ejpam-5252	145	21	,	,	PUNCT
ejpam-5252	145	22	v3	v3	PROPN
ejpam-5252	145	23	)	)	PUNCT
ejpam-5252	145	24	on	on	ADP
ejpam-5252	145	25	v	v	ADP
ejpam-5252	145	26	(	(	PUNCT
ejpam-5252	145	27	g	g	NOUN
ejpam-5252	145	28	)	)	PUNCT
ejpam-5252	145	29	such	such	ADJ
ejpam-5252	145	30	that	that	DET
ejpam-5252	145	31	v0	v0	NOUN
ejpam-5252	145	32	=	=	SYM
ejpam-5252	145	33	∅	∅	NOUN
ejpam-5252	145	34	=	=	SYM
ejpam-5252	145	35	v3	v3	PROPN
ejpam-5252	145	36	,	,	PUNCT
ejpam-5252	145	37	v2	v2	PROPN
ejpam-5252	145	38	=	=	SYM
ejpam-5252	145	39	{	{	PUNCT
ejpam-5252	145	40	v	v	NOUN
ejpam-5252	145	41	}	}	PUNCT
ejpam-5252	145	42	,	,	PUNCT
ejpam-5252	145	43	v1	v1	NOUN
ejpam-5252	145	44	=	=	SYM
ejpam-5252	145	45	v	v	NOUN
ejpam-5252	145	46	(	(	PUNCT
ejpam-5252	145	47	g	g	NOUN
ejpam-5252	145	48	)	)	PUNCT
ejpam-5252	145	49	\	\	NOUN
ejpam-5252	145	50	{	{	PUNCT
ejpam-5252	145	51	v	v	NOUN
ejpam-5252	145	52	}	}	PUNCT
ejpam-5252	145	53	.	.	PUNCT
ejpam-5252	146	1	then	then	ADV
ejpam-5252	146	2	f	f	PROPN
ejpam-5252	146	3	∈	∈	PROPN
ejpam-5252	146	4	mrdf	mrdf	NOUN
ejpam-5252	146	5	(	(	PUNCT
ejpam-5252	146	6	g	g	NOUN
ejpam-5252	146	7	)	)	PUNCT
ejpam-5252	146	8	and	and	CCONJ
ejpam-5252	146	9	ωmr	ωmr	NOUN
ejpam-5252	146	10	g	g	PROPN
ejpam-5252	146	11	(	(	PUNCT
ejpam-5252	146	12	f	f	X
ejpam-5252	146	13	)	)	PUNCT
ejpam-5252	146	14	=	=	SYM
ejpam-5252	146	15	5	5	X
ejpam-5252	146	16	.	.	PUNCT
ejpam-5252	147	1	this	this	PRON
ejpam-5252	147	2	implies	imply	VERB
ejpam-5252	147	3	that	that	SCONJ
ejpam-5252	147	4	γmr(g	γmr(g	PROPN
ejpam-5252	147	5	)	)	PUNCT
ejpam-5252	147	6	=	=	SYM
ejpam-5252	148	1	5	5	X
ejpam-5252	148	2	.	.	PUNCT
ejpam-5252	149	1	next	next	ADV
ejpam-5252	149	2	,	,	PUNCT
ejpam-5252	149	3	suppose	suppose	VERB
ejpam-5252	149	4	that	that	SCONJ
ejpam-5252	149	5	γ2(g	γ2(g	VERB
ejpam-5252	149	6	)	)	PUNCT
ejpam-5252	149	7	=	=	SYM
ejpam-5252	149	8	2	2	NUM
ejpam-5252	149	9	and	and	CCONJ
ejpam-5252	149	10	|v	|v	PROPN
ejpam-5252	149	11	(	(	PUNCT
ejpam-5252	149	12	g)|	g)|	X
ejpam-5252	149	13	≥	≥	NOUN
ejpam-5252	149	14	4	4	NUM
ejpam-5252	149	15	.	.	PUNCT
ejpam-5252	150	1	let	let	VERB
ejpam-5252	150	2	d	d	NOUN
ejpam-5252	150	3	=	=	PUNCT
ejpam-5252	150	4	{	{	PUNCT
ejpam-5252	150	5	u	u	NOUN
ejpam-5252	150	6	,	,	PUNCT
ejpam-5252	150	7	v	v	NOUN
ejpam-5252	150	8	}	}	PUNCT
ejpam-5252	150	9	be	be	AUX
ejpam-5252	150	10	the	the	DET
ejpam-5252	150	11	γ2	γ2	NOUN
ejpam-5252	150	12	-	-	PUNCT
ejpam-5252	150	13	set	set	NOUN
ejpam-5252	150	14	of	of	ADP
ejpam-5252	150	15	g.	g.	PROPN
ejpam-5252	150	16	define	define	VERB
ejpam-5252	150	17	a	a	DET
ejpam-5252	150	18	function	function	NOUN
ejpam-5252	150	19	g	g	NOUN
ejpam-5252	150	20	=	=	SYM
ejpam-5252	150	21	(	(	PUNCT
ejpam-5252	150	22	v0	v0	PROPN
ejpam-5252	150	23	,	,	PUNCT
ejpam-5252	150	24	v1	v1	NOUN
ejpam-5252	150	25	,	,	PUNCT
ejpam-5252	150	26	v2	v2	PROPN
ejpam-5252	150	27	,	,	PUNCT
ejpam-5252	150	28	v3	v3	PROPN
ejpam-5252	150	29	)	)	PUNCT
ejpam-5252	150	30	such	such	ADJ
ejpam-5252	150	31	that	that	DET
ejpam-5252	150	32	v1	v1	NOUN
ejpam-5252	150	33	=	=	SYM
ejpam-5252	150	34	∅	∅	NOUN
ejpam-5252	150	35	and	and	CCONJ
ejpam-5252	150	36	g(x	g(x	NOUN
ejpam-5252	150	37	)	)	PUNCT
ejpam-5252	151	1	=	=	SYM
ejpam-5252	151	2			NOUN
ejpam-5252	151	3	3	3	NUM
ejpam-5252	151	4	,	,	PUNCT
ejpam-5252	151	5	if	if	SCONJ
ejpam-5252	151	6	x	x	ADP
ejpam-5252	151	7	=	=	PUNCT
ejpam-5252	151	8	u.	u.	NOUN
ejpam-5252	151	9	2	2	NUM
ejpam-5252	151	10	,	,	PUNCT
ejpam-5252	151	11	if	if	SCONJ
ejpam-5252	151	12	x	x	ADP
ejpam-5252	151	13	=	=	SYM
ejpam-5252	151	14	v	v	ADP
ejpam-5252	151	15	0	0	NUM
ejpam-5252	151	16	,	,	PUNCT
ejpam-5252	151	17	if	if	SCONJ
ejpam-5252	151	18	x	x	PROPN
ejpam-5252	151	19	∈	∈	PROPN
ejpam-5252	151	20	v	v	ADP
ejpam-5252	151	21	(	(	PUNCT
ejpam-5252	151	22	g	g	NOUN
ejpam-5252	151	23	)	)	PUNCT
ejpam-5252	151	24	\d	\d	NOUN
ejpam-5252	151	25	.	.	PUNCT
ejpam-5252	152	1	then	then	ADV
ejpam-5252	152	2	g	g	PROPN
ejpam-5252	152	3	∈	∈	PROPN
ejpam-5252	152	4	mrdf	mrdf	NOUN
ejpam-5252	152	5	(	(	PUNCT
ejpam-5252	152	6	g	g	NOUN
ejpam-5252	152	7	)	)	PUNCT
ejpam-5252	152	8	and	and	CCONJ
ejpam-5252	152	9	ωmr	ωmr	NOUN
ejpam-5252	152	10	g	g	PROPN
ejpam-5252	152	11	(	(	PUNCT
ejpam-5252	152	12	g	g	NOUN
ejpam-5252	152	13	)	)	PUNCT
ejpam-5252	152	14	=	=	SYM
ejpam-5252	152	15	5	5	X
ejpam-5252	152	16	.	.	PUNCT
ejpam-5252	152	17	since	since	SCONJ
ejpam-5252	152	18	g	g	PROPN
ejpam-5252	152	19	̸∈	̸∈	PROPN
ejpam-5252	152	20	{	{	PUNCT
ejpam-5252	152	21	k3	k3	PROPN
ejpam-5252	152	22	,	,	PUNCT
ejpam-5252	152	23	p3	p3	PROPN
ejpam-5252	152	24	}	}	PUNCT
ejpam-5252	152	25	,	,	PUNCT
ejpam-5252	152	26	we	we	PRON
ejpam-5252	152	27	must	must	AUX
ejpam-5252	152	28	have	have	VERB
ejpam-5252	152	29	ωmr	ωmr	NOUN
ejpam-5252	152	30	g	g	PROPN
ejpam-5252	152	31	(	(	PUNCT
ejpam-5252	152	32	g	g	NOUN
ejpam-5252	152	33	)	)	PUNCT
ejpam-5252	152	34	=	=	SYM
ejpam-5252	152	35	5	5	X
ejpam-5252	152	36	.	.	PUNCT
ejpam-5252	153	1	hence	hence	ADV
ejpam-5252	153	2	,	,	PUNCT
ejpam-5252	153	3	γmr(g	γmr(g	PROPN
ejpam-5252	153	4	)	)	PUNCT
ejpam-5252	153	5	=	=	SYM
ejpam-5252	153	6	5	5	X
ejpam-5252	153	7	.	.	PUNCT
ejpam-5252	153	8	corollary	corollary	ADJ
ejpam-5252	153	9	1	1	NUM
ejpam-5252	153	10	.	.	PUNCT
ejpam-5252	154	1	for	for	ADP
ejpam-5252	154	2	a	a	DET
ejpam-5252	154	3	connected	connected	ADJ
ejpam-5252	154	4	graph	graph	NOUN
ejpam-5252	154	5	g	g	NOUN
ejpam-5252	154	6	of	of	ADP
ejpam-5252	154	7	order	order	NOUN
ejpam-5252	154	8	4	4	NUM
ejpam-5252	154	9	,	,	PUNCT
ejpam-5252	154	10	γmr(g	γmr(g	PROPN
ejpam-5252	154	11	)	)	PUNCT
ejpam-5252	154	12	=	=	SYM
ejpam-5252	154	13	5	5	NUM
ejpam-5252	154	14	if	if	SCONJ
ejpam-5252	154	15	and	and	CCONJ
ejpam-5252	154	16	only	only	ADV
ejpam-5252	154	17	if	if	SCONJ
ejpam-5252	154	18	g	g	PROPN
ejpam-5252	154	19	∈	∈	PROPN
ejpam-5252	154	20	{	{	PUNCT
ejpam-5252	154	21	k1	k1	NOUN
ejpam-5252	154	22	+	+	CCONJ
ejpam-5252	154	23	(	(	PUNCT
ejpam-5252	154	24	k1	k1	NOUN
ejpam-5252	154	25	∪k2),k1	∪k2),k1	NOUN
ejpam-5252	154	26	+	+	PROPN
ejpam-5252	154	27	k3,k1	k3,k1	PROPN
ejpam-5252	154	28	+	+	PROPN
ejpam-5252	154	29	k3,k1	k3,k1	PROPN
ejpam-5252	154	30	+	+	NUM
ejpam-5252	154	31	p3	p3	PROPN
ejpam-5252	154	32	}	}	PUNCT
ejpam-5252	154	33	.	.	PUNCT
ejpam-5252	155	1	proof	proof	NOUN
ejpam-5252	155	2	.	.	PUNCT
ejpam-5252	156	1	the	the	DET
ejpam-5252	156	2	proof	proof	NOUN
ejpam-5252	156	3	follows	follow	VERB
ejpam-5252	156	4	directly	directly	ADV
ejpam-5252	156	5	from	from	ADP
ejpam-5252	156	6	proposition	proposition	NOUN
ejpam-5252	156	7	5	5	NUM
ejpam-5252	156	8	(	(	PUNCT
ejpam-5252	156	9	iv	iv	NUM
ejpam-5252	156	10	)	)	PUNCT
ejpam-5252	156	11	.	.	PUNCT
ejpam-5252	157	1	remark	remark	PROPN
ejpam-5252	157	2	2	2	NUM
ejpam-5252	157	3	.	.	PUNCT
ejpam-5252	158	1	let	let	VERB
ejpam-5252	158	2	g	g	PRON
ejpam-5252	158	3	be	be	AUX
ejpam-5252	158	4	a	a	DET
ejpam-5252	158	5	graph	graph	NOUN
ejpam-5252	158	6	,	,	PUNCT
ejpam-5252	158	7	then	then	ADV
ejpam-5252	158	8	every	every	DET
ejpam-5252	158	9	γmr	γmr	ADJ
ejpam-5252	158	10	-	-	PUNCT
ejpam-5252	158	11	function	function	NOUN
ejpam-5252	158	12	of	of	ADP
ejpam-5252	158	13	g	g	PROPN
ejpam-5252	158	14	is	be	AUX
ejpam-5252	158	15	a	a	DET
ejpam-5252	158	16	γdr	γdr	NOUN
ejpam-5252	158	17	-	-	PUNCT
ejpam-5252	158	18	function	function	NOUN
ejpam-5252	158	19	of	of	ADP
ejpam-5252	158	20	g	g	PROPN
ejpam-5252	158	21	if	if	SCONJ
ejpam-5252	158	22	v0	v0	NOUN
ejpam-5252	158	23	=	=	PUNCT
ejpam-5252	158	24	∅.	∅.	NOUN
ejpam-5252	158	25	proposition	proposition	NOUN
ejpam-5252	158	26	6	6	NUM
ejpam-5252	158	27	.	.	PUNCT
ejpam-5252	159	1	for	for	ADP
ejpam-5252	159	2	a	a	DET
ejpam-5252	159	3	complete	complete	ADJ
ejpam-5252	159	4	graph	graph	NOUN
ejpam-5252	159	5	kn	kn	PROPN
ejpam-5252	159	6	,	,	PUNCT
ejpam-5252	159	7	γmr(kn	γmr(kn	NOUN
ejpam-5252	159	8	)	)	PUNCT
ejpam-5252	159	9	=	=	SYM
ejpam-5252	159	10	5	5	NUM
ejpam-5252	159	11	for	for	ADP
ejpam-5252	159	12	all	all	DET
ejpam-5252	159	13	n	n	PRON
ejpam-5252	159	14	≥	≥	NOUN
ejpam-5252	159	15	4	4	NUM
ejpam-5252	159	16	.	.	PUNCT
ejpam-5252	160	1	proof	proof	NOUN
ejpam-5252	160	2	.	.	PUNCT
ejpam-5252	161	1	pick	pick	VERB
ejpam-5252	161	2	any	any	DET
ejpam-5252	161	3	x	x	NOUN
ejpam-5252	161	4	,	,	PUNCT
ejpam-5252	161	5	y	y	PROPN
ejpam-5252	161	6	∈	∈	PROPN
ejpam-5252	161	7	v	v	PROPN
ejpam-5252	161	8	(	(	PUNCT
ejpam-5252	161	9	kn	kn	PROPN
ejpam-5252	161	10	)	)	PUNCT
ejpam-5252	161	11	with	with	ADP
ejpam-5252	161	12	x	x	PUNCT
ejpam-5252	161	13	̸=	̸=	PROPN
ejpam-5252	161	14	y.	y.	PROPN
ejpam-5252	161	15	clearly	clearly	ADV
ejpam-5252	161	16	,	,	PUNCT
ejpam-5252	161	17	g	g	PROPN
ejpam-5252	161	18	=	=	SYM
ejpam-5252	161	19	(	(	PUNCT
ejpam-5252	161	20	v	v	NOUN
ejpam-5252	161	21	(	(	PUNCT
ejpam-5252	161	22	kn	kn	PROPN
ejpam-5252	161	23	)	)	PUNCT
ejpam-5252	161	24	\	\	NOUN
ejpam-5252	162	1	{	{	PUNCT
ejpam-5252	162	2	x	x	NOUN
ejpam-5252	162	3	,	,	PUNCT
ejpam-5252	162	4	y},∅	y},∅	PROPN
ejpam-5252	162	5	,	,	PUNCT
ejpam-5252	162	6	{	{	PUNCT
ejpam-5252	162	7	x	x	X
ejpam-5252	162	8	}	}	PUNCT
ejpam-5252	162	9	,	,	PUNCT
ejpam-5252	162	10	{	{	PUNCT
ejpam-5252	162	11	y	y	NOUN
ejpam-5252	162	12	}	}	PUNCT
ejpam-5252	162	13	)	)	PUNCT
ejpam-5252	162	14	∈	∈	PROPN
ejpam-5252	162	15	mrdf	mrdf	NOUN
ejpam-5252	162	16	(	(	PUNCT
ejpam-5252	162	17	kn	kn	PROPN
ejpam-5252	162	18	)	)	PUNCT
ejpam-5252	162	19	.	.	PUNCT
ejpam-5252	163	1	it	it	PRON
ejpam-5252	163	2	follows	follow	VERB
ejpam-5252	163	3	that	that	DET
ejpam-5252	163	4	γmr(kn	γmr(kn	NOUN
ejpam-5252	163	5	)	)	PUNCT
ejpam-5252	163	6	≤	≤	NUM
ejpam-5252	163	7	5	5	NUM
ejpam-5252	163	8	.	.	PUNCT
ejpam-5252	164	1	on	on	ADP
ejpam-5252	164	2	the	the	DET
ejpam-5252	164	3	other	other	ADJ
ejpam-5252	164	4	hand	hand	NOUN
ejpam-5252	164	5	,	,	PUNCT
ejpam-5252	164	6	suppose	suppose	VERB
ejpam-5252	164	7	that	that	SCONJ
ejpam-5252	164	8	f	f	PROPN
ejpam-5252	164	9	=	=	SYM
ejpam-5252	164	10	(	(	PUNCT
ejpam-5252	164	11	v0	v0	PROPN
ejpam-5252	164	12	,	,	PUNCT
ejpam-5252	164	13	v1	v1	NOUN
ejpam-5252	164	14	,	,	PUNCT
ejpam-5252	164	15	v2	v2	PROPN
ejpam-5252	164	16	,	,	PUNCT
ejpam-5252	164	17	v3	v3	PROPN
ejpam-5252	164	18	)	)	PUNCT
ejpam-5252	164	19	is	be	AUX
ejpam-5252	164	20	a	a	DET
ejpam-5252	164	21	γmr	γmr	ADJ
ejpam-5252	164	22	-	-	PUNCT
ejpam-5252	164	23	function	function	NOUN
ejpam-5252	164	24	of	of	ADP
ejpam-5252	164	25	kn	kn	PROPN
ejpam-5252	164	26	.	.	PUNCT
ejpam-5252	165	1	if	if	SCONJ
ejpam-5252	165	2	v0	v0	NOUN
ejpam-5252	165	3	=	=	SYM
ejpam-5252	165	4	∅	∅	NOUN
ejpam-5252	165	5	,	,	PUNCT
ejpam-5252	165	6	then	then	ADV
ejpam-5252	165	7	v3	v3	PROPN
ejpam-5252	165	8	=	=	PUNCT
ejpam-5252	165	9	∅.	∅.	NOUN
ejpam-5252	165	10	since	since	SCONJ
ejpam-5252	165	11	f	f	PROPN
ejpam-5252	165	12	is	be	AUX
ejpam-5252	165	13	a	a	DET
ejpam-5252	165	14	γmrfunction	γmrfunction	NOUN
ejpam-5252	165	15	of	of	ADP
ejpam-5252	165	16	kn	kn	PROPN
ejpam-5252	165	17	,	,	PUNCT
ejpam-5252	165	18	|v2|	|v2|	NOUN
ejpam-5252	165	19	=	=	SYM
ejpam-5252	165	20	1	1	NUM
ejpam-5252	165	21	and	and	CCONJ
ejpam-5252	165	22	|v1|	|v1|	NOUN
ejpam-5252	165	23	=	=	SYM
ejpam-5252	165	24	n	n	CCONJ
ejpam-5252	165	25	−	−	PROPN
ejpam-5252	165	26	1	1	NUM
ejpam-5252	165	27	.	.	PUNCT
ejpam-5252	166	1	hence	hence	ADV
ejpam-5252	166	2	,	,	PUNCT
ejpam-5252	166	3	γmr(kn	γmr(kn	NOUN
ejpam-5252	166	4	)	)	PUNCT
ejpam-5252	167	1	=	=	PUNCT
ejpam-5252	167	2	ωmr	ωmr	PROPN
ejpam-5252	167	3	kn	kn	PROPN
ejpam-5252	167	4	(	(	PUNCT
ejpam-5252	167	5	f	f	X
ejpam-5252	167	6	)	)	PUNCT
ejpam-5252	167	7	=	=	SYM
ejpam-5252	168	1	n	n	PROPN
ejpam-5252	168	2	+	+	NOUN
ejpam-5252	168	3	1	1	NUM
ejpam-5252	168	4	≥	≥	NOUN
ejpam-5252	168	5	5	5	NUM
ejpam-5252	168	6	.	.	PUNCT
ejpam-5252	169	1	if	if	SCONJ
ejpam-5252	169	2	s.	s.	PROPN
ejpam-5252	169	3	ahamad	ahamad	PROPN
ejpam-5252	169	4	,	,	PUNCT
ejpam-5252	169	5	j.	j.	PROPN
ejpam-5252	169	6	cariaga	cariaga	PROPN
ejpam-5252	169	7	,	,	PUNCT
ejpam-5252	169	8	s.	s.	PROPN
ejpam-5252	169	9	menchavez	menchavez	PROPN
ejpam-5252	169	10	/	/	PUNCT
ejpam-5252	169	11	eur	eur	PROPN
ejpam-5252	169	12	.	.	PUNCT
ejpam-5252	170	1	j.	j.	PROPN
ejpam-5252	170	2	pure	pure	PROPN
ejpam-5252	170	3	appl	appl	PROPN
ejpam-5252	170	4	.	.	PROPN
ejpam-5252	170	5	math	math	PROPN
ejpam-5252	170	6	,	,	PUNCT
ejpam-5252	170	7	18	18	NUM
ejpam-5252	170	8	(	(	PUNCT
ejpam-5252	170	9	1	1	NUM
ejpam-5252	170	10	)	)	PUNCT
ejpam-5252	170	11	(	(	PUNCT
ejpam-5252	170	12	2025	2025	NUM
ejpam-5252	170	13	)	)	PUNCT
ejpam-5252	170	14	,	,	PUNCT
ejpam-5252	170	15	5252	5252	NUM
ejpam-5252	170	16	7	7	NUM
ejpam-5252	170	17	of	of	ADP
ejpam-5252	170	18	18	18	NUM
ejpam-5252	170	19	v0	v0	NOUN
ejpam-5252	170	20	̸=	̸=	PROPN
ejpam-5252	170	21	∅	∅	NOUN
ejpam-5252	170	22	,	,	PUNCT
ejpam-5252	170	23	then	then	ADV
ejpam-5252	170	24	|v2|	|v2|	VERB
ejpam-5252	170	25	≥	≥	NOUN
ejpam-5252	170	26	1	1	NUM
ejpam-5252	170	27	and	and	CCONJ
ejpam-5252	170	28	|v3|	|v3|	PROPN
ejpam-5252	170	29	≥	≥	NUM
ejpam-5252	170	30	1	1	NUM
ejpam-5252	170	31	.	.	PUNCT
ejpam-5252	171	1	it	it	PRON
ejpam-5252	171	2	follows	follow	VERB
ejpam-5252	171	3	that	that	DET
ejpam-5252	171	4	γmr(kn	γmr(kn	NOUN
ejpam-5252	171	5	)	)	PUNCT
ejpam-5252	172	1	=	=	PUNCT
ejpam-5252	172	2	ωmr	ωmr	PROPN
ejpam-5252	172	3	kn	kn	PROPN
ejpam-5252	172	4	(	(	PUNCT
ejpam-5252	172	5	f	f	X
ejpam-5252	172	6	)	)	PUNCT
ejpam-5252	172	7	=	=	SYM
ejpam-5252	173	1	2|v2|+3|v3|	2|v2|+3|v3|	NUM
ejpam-5252	173	2	≥	≥	NOUN
ejpam-5252	173	3	5	5	NUM
ejpam-5252	173	4	.	.	PUNCT
ejpam-5252	174	1	therefore	therefore	ADV
ejpam-5252	174	2	,	,	PUNCT
ejpam-5252	174	3	γmr(kn	γmr(kn	NOUN
ejpam-5252	174	4	)	)	PUNCT
ejpam-5252	174	5	=	=	SYM
ejpam-5252	175	1	5	5	X
ejpam-5252	175	2	.	.	PUNCT
ejpam-5252	176	1	in	in	ADP
ejpam-5252	176	2	what	what	PRON
ejpam-5252	176	3	follows	follow	VERB
ejpam-5252	176	4	,	,	PUNCT
ejpam-5252	176	5	we	we	PRON
ejpam-5252	176	6	denote	denote	VERB
ejpam-5252	176	7	by	by	ADP
ejpam-5252	176	8	f	f	PROPN
ejpam-5252	176	9	|g	|g	VERB
ejpam-5252	176	10	the	the	DET
ejpam-5252	176	11	restriction	restriction	NOUN
ejpam-5252	176	12	of	of	ADP
ejpam-5252	176	13	f	f	PROPN
ejpam-5252	176	14	on	on	ADP
ejpam-5252	176	15	the	the	DET
ejpam-5252	176	16	subgraph	subgraph	NOUN
ejpam-5252	176	17	g	g	PROPN
ejpam-5252	176	18	of	of	ADP
ejpam-5252	176	19	the	the	DET
ejpam-5252	176	20	graph	graph	NOUN
ejpam-5252	176	21	h.	h.	NOUN
ejpam-5252	176	22	proposition	proposition	NOUN
ejpam-5252	176	23	7	7	NUM
ejpam-5252	176	24	.	.	PUNCT
ejpam-5252	177	1	let	let	VERB
ejpam-5252	177	2	g	g	PRON
ejpam-5252	177	3	be	be	AUX
ejpam-5252	177	4	a	a	DET
ejpam-5252	177	5	disconnected	disconnected	ADJ
ejpam-5252	177	6	graph	graph	NOUN
ejpam-5252	177	7	with	with	ADP
ejpam-5252	177	8	nontrivial	nontrivial	ADJ
ejpam-5252	177	9	components	component	NOUN
ejpam-5252	177	10	g1	g1	PROPN
ejpam-5252	177	11	,	,	PUNCT
ejpam-5252	177	12	g2	g2	PROPN
ejpam-5252	177	13	,	,	PUNCT
ejpam-5252	177	14	·	·	PUNCT
ejpam-5252	177	15	·	·	PUNCT
ejpam-5252	177	16	·	·	PUNCT
ejpam-5252	177	17	,	,	PUNCT
ejpam-5252	177	18	gn	gn	PROPN
ejpam-5252	177	19	.	.	PUNCT
ejpam-5252	177	20	then	then	ADV
ejpam-5252	177	21	γmr(g	γmr(g	X
ejpam-5252	177	22	)	)	PUNCT
ejpam-5252	178	1	=	=	SYM
ejpam-5252	179	1	∑n	∑n	PROPN
ejpam-5252	179	2	i=1	i=1	PROPN
ejpam-5252	179	3	γmr(gi	γmr(gi	PROPN
ejpam-5252	179	4	)	)	PUNCT
ejpam-5252	179	5	.	.	PUNCT
ejpam-5252	180	1	proof	proof	NOUN
ejpam-5252	180	2	.	.	PUNCT
ejpam-5252	181	1	let	let	VERB
ejpam-5252	181	2	g1	g1	PROPN
ejpam-5252	181	3	,	,	PUNCT
ejpam-5252	181	4	g2	g2	PROPN
ejpam-5252	181	5	,	,	PUNCT
ejpam-5252	181	6	·	·	PUNCT
ejpam-5252	181	7	·	·	PUNCT
ejpam-5252	181	8	·	·	PUNCT
ejpam-5252	181	9	,	,	PUNCT
ejpam-5252	181	10	gn	gn	X
ejpam-5252	181	11	be	be	AUX
ejpam-5252	181	12	the	the	DET
ejpam-5252	181	13	components	component	NOUN
ejpam-5252	181	14	of	of	ADP
ejpam-5252	181	15	g.	g.	PROPN
ejpam-5252	181	16	let	let	VERB
ejpam-5252	181	17	f1	f1	NOUN
ejpam-5252	181	18	,	,	PUNCT
ejpam-5252	181	19	f2	f2	PROPN
ejpam-5252	181	20	,	,	PUNCT
ejpam-5252	181	21	·	·	PUNCT
ejpam-5252	181	22	·	·	PUNCT
ejpam-5252	181	23	·	·	PUNCT
ejpam-5252	181	24	,	,	PUNCT
ejpam-5252	181	25	fn	fn	VERB
ejpam-5252	181	26	be	be	AUX
ejpam-5252	181	27	γmr	γmr	NOUN
ejpam-5252	181	28	-	-	PUNCT
ejpam-5252	181	29	functions	function	NOUN
ejpam-5252	181	30	of	of	ADP
ejpam-5252	181	31	g1	g1	NOUN
ejpam-5252	181	32	,	,	PUNCT
ejpam-5252	181	33	g2	g2	PROPN
ejpam-5252	181	34	,	,	PUNCT
ejpam-5252	181	35	·	·	PUNCT
ejpam-5252	181	36	·	·	PUNCT
ejpam-5252	181	37	·	·	PUNCT
ejpam-5252	182	1	,	,	PUNCT
ejpam-5252	182	2	gn	gn	PROPN
ejpam-5252	182	3	respectively	respectively	ADV
ejpam-5252	182	4	.	.	PUNCT
ejpam-5252	182	5	define	define	VERB
ejpam-5252	182	6	a	a	DET
ejpam-5252	182	7	function	function	NOUN
ejpam-5252	182	8	f	f	NOUN
ejpam-5252	182	9	:	:	PUNCT
ejpam-5252	182	10	v	v	X
ejpam-5252	182	11	(	(	PUNCT
ejpam-5252	182	12	g	g	NOUN
ejpam-5252	182	13	)	)	PUNCT
ejpam-5252	182	14	−→	−→	NOUN
ejpam-5252	182	15	{	{	PUNCT
ejpam-5252	182	16	0	0	NUM
ejpam-5252	182	17	,	,	PUNCT
ejpam-5252	182	18	1	1	NUM
ejpam-5252	182	19	,	,	PUNCT
ejpam-5252	182	20	2	2	NUM
ejpam-5252	182	21	,	,	PUNCT
ejpam-5252	182	22	3	3	NUM
ejpam-5252	182	23	}	}	PUNCT
ejpam-5252	182	24	given	give	VERB
ejpam-5252	182	25	by	by	ADP
ejpam-5252	182	26	f(x	f(x	PROPN
ejpam-5252	182	27	)	)	PUNCT
ejpam-5252	182	28	=	=	PUNCT
ejpam-5252	182	29			NOUN
ejpam-5252	182	30	f1(x	f1(x	NUM
ejpam-5252	182	31	)	)	PUNCT
ejpam-5252	182	32	,	,	PUNCT
ejpam-5252	182	33	if	if	SCONJ
ejpam-5252	182	34	x	x	SYM
ejpam-5252	182	35	∈	∈	PROPN
ejpam-5252	182	36	v	v	NOUN
ejpam-5252	182	37	(	(	PUNCT
ejpam-5252	182	38	g1	g1	PROPN
ejpam-5252	182	39	)	)	PUNCT
ejpam-5252	182	40	.	.	PUNCT
ejpam-5252	183	1	f2(x	f2(x	X
ejpam-5252	183	2	)	)	PUNCT
ejpam-5252	183	3	,	,	PUNCT
ejpam-5252	183	4	if	if	SCONJ
ejpam-5252	183	5	x	x	SYM
ejpam-5252	183	6	∈	∈	PROPN
ejpam-5252	183	7	v	v	X
ejpam-5252	183	8	(	(	PUNCT
ejpam-5252	183	9	g2	g2	PROPN
ejpam-5252	183	10	)	)	PUNCT
ejpam-5252	183	11	.	.	PUNCT
ejpam-5252	183	12	...	...	PUNCT
ejpam-5252	184	1	fn(x	fn(x	X
ejpam-5252	184	2	)	)	PUNCT
ejpam-5252	184	3	,	,	PUNCT
ejpam-5252	184	4	if	if	SCONJ
ejpam-5252	184	5	x	x	SYM
ejpam-5252	184	6	∈	∈	PROPN
ejpam-5252	184	7	v	v	X
ejpam-5252	184	8	(	(	PUNCT
ejpam-5252	184	9	gn	gn	PROPN
ejpam-5252	184	10	)	)	PUNCT
ejpam-5252	184	11	.	.	PUNCT
ejpam-5252	185	1	then	then	ADV
ejpam-5252	185	2	f	f	PROPN
ejpam-5252	185	3	is	be	AUX
ejpam-5252	185	4	a	a	DET
ejpam-5252	185	5	γmr	γmr	ADJ
ejpam-5252	185	6	-	-	PUNCT
ejpam-5252	185	7	function	function	NOUN
ejpam-5252	185	8	ofg	ofg	NOUN
ejpam-5252	185	9	.	.	PUNCT
ejpam-5252	186	1	thus	thus	ADV
ejpam-5252	186	2	γmr(g	γmr(g	X
ejpam-5252	186	3	)	)	PUNCT
ejpam-5252	186	4	≤	≤	NOUN
ejpam-5252	187	1	∑n	∑n	PROPN
ejpam-5252	187	2	i=1	i=1	PROPN
ejpam-5252	187	3	γmr(gi	γmr(gi	PROPN
ejpam-5252	187	4	)	)	PUNCT
ejpam-5252	187	5	.	.	PUNCT
ejpam-5252	188	1	conversely	conversely	ADV
ejpam-5252	188	2	,	,	PUNCT
ejpam-5252	188	3	let	let	VERB
ejpam-5252	188	4	f	f	PRON
ejpam-5252	188	5	be	be	AUX
ejpam-5252	188	6	a	a	DET
ejpam-5252	188	7	γmrfunction	γmrfunction	NOUN
ejpam-5252	188	8	of	of	ADP
ejpam-5252	188	9	g.	g.	PROPN
ejpam-5252	188	10	then	then	ADV
ejpam-5252	188	11	the	the	DET
ejpam-5252	188	12	restriction	restriction	NOUN
ejpam-5252	188	13	f	f	PROPN
ejpam-5252	188	14	|gi	|gi	NOUN
ejpam-5252	188	15	of	of	ADP
ejpam-5252	188	16	f	f	PROPN
ejpam-5252	188	17	to	to	PART
ejpam-5252	188	18	gi	gi	INTJ
ejpam-5252	188	19	,	,	PUNCT
ejpam-5252	188	20	where	where	SCONJ
ejpam-5252	188	21	i	i	PRON
ejpam-5252	188	22	=	=	NOUN
ejpam-5252	188	23	1	1	NUM
ejpam-5252	188	24	,	,	PUNCT
ejpam-5252	188	25	2	2	NUM
ejpam-5252	188	26	,	,	PUNCT
ejpam-5252	188	27	·	·	PUNCT
ejpam-5252	188	28	·	·	PUNCT
ejpam-5252	188	29	·	·	PUNCT
ejpam-5252	188	30	,	,	PUNCT
ejpam-5252	188	31	n	n	X
ejpam-5252	188	32	is	be	AUX
ejpam-5252	188	33	a	a	DET
ejpam-5252	188	34	γmr	γmr	ADJ
ejpam-5252	188	35	-	-	PUNCT
ejpam-5252	188	36	function	function	NOUN
ejpam-5252	188	37	of	of	ADP
ejpam-5252	188	38	gi	gi	NOUN
ejpam-5252	188	39	.	.	PUNCT
ejpam-5252	189	1	thus	thus	ADV
ejpam-5252	189	2	,	,	PUNCT
ejpam-5252	189	3	γmr(gi	γmr(gi	NOUN
ejpam-5252	189	4	)	)	PUNCT
ejpam-5252	189	5	≤	≤	NUM
ejpam-5252	189	6	ωmr	ωmr	NOUN
ejpam-5252	189	7	g	g	PROPN
ejpam-5252	189	8	(	(	PUNCT
ejpam-5252	189	9	f	f	PROPN
ejpam-5252	189	10	|gi	|gi	NOUN
ejpam-5252	189	11	)	)	PUNCT
ejpam-5252	189	12	for	for	ADP
ejpam-5252	189	13	all	all	DET
ejpam-5252	189	14	i	i	PRON
ejpam-5252	189	15	=	=	NOUN
ejpam-5252	189	16	1	1	NUM
ejpam-5252	189	17	,	,	PUNCT
ejpam-5252	189	18	2	2	NUM
ejpam-5252	189	19	,	,	PUNCT
ejpam-5252	189	20	·	·	PUNCT
ejpam-5252	189	21	·	·	PUNCT
ejpam-5252	189	22	·	·	PUNCT
ejpam-5252	189	23	,	,	PUNCT
ejpam-5252	189	24	n.	n.	PROPN
ejpam-5252	189	25	hence	hence	ADV
ejpam-5252	189	26	,	,	PUNCT
ejpam-5252	189	27	∑n	∑n	PROPN
ejpam-5252	189	28	i=1	i=1	PROPN
ejpam-5252	189	29	γmr(gi	γmr(gi	NOUN
ejpam-5252	189	30	)	)	PUNCT
ejpam-5252	189	31	≤	≤	NOUN
ejpam-5252	189	32	γmr(g	γmr(g	PROPN
ejpam-5252	189	33	)	)	PUNCT
ejpam-5252	189	34	.	.	PUNCT
ejpam-5252	190	1	hence	hence	ADV
ejpam-5252	190	2	,	,	PUNCT
ejpam-5252	190	3	the	the	DET
ejpam-5252	190	4	assertion	assertion	NOUN
ejpam-5252	190	5	follows	follow	VERB
ejpam-5252	190	6	by	by	ADP
ejpam-5252	190	7	combining	combine	VERB
ejpam-5252	190	8	the	the	DET
ejpam-5252	190	9	results	result	NOUN
ejpam-5252	190	10	.	.	PUNCT
ejpam-5252	191	1	corollary	corollary	ADJ
ejpam-5252	191	2	2	2	NUM
ejpam-5252	191	3	.	.	PUNCT
ejpam-5252	192	1	let	let	VERB
ejpam-5252	192	2	g	g	PRON
ejpam-5252	192	3	be	be	AUX
ejpam-5252	192	4	a	a	DET
ejpam-5252	192	5	graph	graph	NOUN
ejpam-5252	192	6	of	of	ADP
ejpam-5252	192	7	order	order	NOUN
ejpam-5252	192	8	n.	n.	NOUN
ejpam-5252	192	9	then	then	ADV
ejpam-5252	192	10	γmr(g	γmr(g	PROPN
ejpam-5252	192	11	)	)	PUNCT
ejpam-5252	192	12	=	=	SYM
ejpam-5252	192	13	2n	2n	NUM
ejpam-5252	193	1	if	if	SCONJ
ejpam-5252	193	2	and	and	CCONJ
ejpam-5252	193	3	only	only	ADV
ejpam-5252	193	4	if	if	SCONJ
ejpam-5252	193	5	g	g	PROPN
ejpam-5252	193	6	=	=	PROPN
ejpam-5252	193	7	kn	kn	PROPN
ejpam-5252	193	8	.	.	PUNCT
ejpam-5252	194	1	the	the	DET
ejpam-5252	194	2	n	n	NUM
ejpam-5252	194	3	-	-	PUNCT
ejpam-5252	194	4	barbell	barbell	NOUN
ejpam-5252	194	5	graph	graph	NOUN
ejpam-5252	194	6	is	be	AUX
ejpam-5252	194	7	the	the	DET
ejpam-5252	194	8	simple	simple	ADJ
ejpam-5252	194	9	graph	graph	NOUN
ejpam-5252	194	10	obtained	obtain	VERB
ejpam-5252	194	11	by	by	ADP
ejpam-5252	194	12	joining	join	VERB
ejpam-5252	194	13	two	two	NUM
ejpam-5252	194	14	copies	copy	NOUN
ejpam-5252	194	15	of	of	ADP
ejpam-5252	194	16	complete	complete	ADJ
ejpam-5252	194	17	graph	graph	NOUN
ejpam-5252	194	18	kn≥3	kn≥3	NOUN
ejpam-5252	194	19	by	by	ADP
ejpam-5252	194	20	a	a	DET
ejpam-5252	194	21	bridge	bridge	NOUN
ejpam-5252	194	22	and	and	CCONJ
ejpam-5252	194	23	is	be	AUX
ejpam-5252	194	24	denoted	denote	VERB
ejpam-5252	194	25	by	by	ADP
ejpam-5252	194	26	bn	bn	PROPN
ejpam-5252	194	27	.	.	PUNCT
ejpam-5252	194	28	figure	figure	NOUN
ejpam-5252	194	29	2	2	NUM
ejpam-5252	194	30	shows	show	VERB
ejpam-5252	194	31	the	the	DET
ejpam-5252	194	32	n	n	CCONJ
ejpam-5252	194	33	-	-	PUNCT
ejpam-5252	194	34	barbell	barbell	NOUN
ejpam-5252	194	35	graphs	graph	NOUN
ejpam-5252	194	36	b3	b3	PROPN
ejpam-5252	194	37	and	and	CCONJ
ejpam-5252	194	38	b5	b5	PROPN
ejpam-5252	194	39	,	,	PUNCT
ejpam-5252	194	40	respectively	respectively	ADV
ejpam-5252	194	41	.	.	PUNCT
ejpam-5252	195	1	1	1	NUM
ejpam-5252	195	2	1	1	NUM
ejpam-5252	195	3	1	1	NUM
ejpam-5252	195	4	1	1	NUM
ejpam-5252	195	5	0	0	NUM
ejpam-5252	195	6	0	0	NUM
ejpam-5252	195	7	0	0	NUM
ejpam-5252	195	8	0	0	NUM
ejpam-5252	195	9	0	0	NUM
ejpam-5252	195	10	0	0	NUM
ejpam-5252	195	11	b3	b3	PROPN
ejpam-5252	195	12	:	:	PUNCT
ejpam-5252	195	13	b5	b5	NOUN
ejpam-5252	195	14	:	:	PUNCT
ejpam-5252	195	15	2	2	NUM
ejpam-5252	195	16	2	2	NUM
ejpam-5252	195	17	2	2	NUM
ejpam-5252	195	18	3	3	NUM
ejpam-5252	195	19	3	3	NUM
ejpam-5252	195	20	2	2	NUM
ejpam-5252	195	21	figure	figure	NOUN
ejpam-5252	195	22	2	2	NUM
ejpam-5252	195	23	:	:	PUNCT
ejpam-5252	195	24	the	the	DET
ejpam-5252	195	25	graphs	graph	NOUN
ejpam-5252	195	26	b3	b3	PROPN
ejpam-5252	195	27	and	and	CCONJ
ejpam-5252	195	28	b5	b5	PROPN
ejpam-5252	195	29	with	with	ADP
ejpam-5252	195	30	γmr(b3	γmr(b3	NOUN
ejpam-5252	195	31	)	)	PUNCT
ejpam-5252	195	32	=	=	SYM
ejpam-5252	195	33	8	8	NUM
ejpam-5252	195	34	and	and	CCONJ
ejpam-5252	195	35	γmr(b5	γmr(b5	PROPN
ejpam-5252	195	36	)	)	PUNCT
ejpam-5252	195	37	=	=	SYM
ejpam-5252	195	38	10	10	NUM
ejpam-5252	195	39	,	,	PUNCT
ejpam-5252	195	40	respectively	respectively	ADV
ejpam-5252	195	41	.	.	PUNCT
ejpam-5252	196	1	proposition	proposition	NOUN
ejpam-5252	196	2	8	8	NUM
ejpam-5252	196	3	.	.	PUNCT
ejpam-5252	197	1	for	for	ADP
ejpam-5252	197	2	any	any	DET
ejpam-5252	197	3	n	n	CCONJ
ejpam-5252	197	4	-	-	PUNCT
ejpam-5252	197	5	barbell	barbell	NOUN
ejpam-5252	197	6	graph	graph	NOUN
ejpam-5252	197	7	bn	bn	ADP
ejpam-5252	197	8	where	where	SCONJ
ejpam-5252	197	9	n	n	PRON
ejpam-5252	197	10	≥	≥	X
ejpam-5252	197	11	3	3	NUM
ejpam-5252	197	12	,	,	PUNCT
ejpam-5252	197	13	γmr(bn	γmr(bn	PROPN
ejpam-5252	197	14	)	)	PUNCT
ejpam-5252	197	15	=	=	PRON
ejpam-5252	197	16	{	{	PUNCT
ejpam-5252	197	17	8	8	NUM
ejpam-5252	197	18	,	,	PUNCT
ejpam-5252	197	19	if	if	SCONJ
ejpam-5252	197	20	n	n	NOUN
ejpam-5252	197	21	=	=	SYM
ejpam-5252	197	22	3	3	NUM
ejpam-5252	197	23	.	.	NOUN
ejpam-5252	197	24	10	10	NUM
ejpam-5252	197	25	,	,	PUNCT
ejpam-5252	197	26	if	if	SCONJ
ejpam-5252	197	27	n	n	PRON
ejpam-5252	197	28	≥	≥	NOUN
ejpam-5252	197	29	4	4	NUM
ejpam-5252	197	30	.	.	PUNCT
ejpam-5252	197	31	s.	s.	PROPN
ejpam-5252	197	32	ahamad	ahamad	PROPN
ejpam-5252	197	33	,	,	PUNCT
ejpam-5252	197	34	j.	j.	PROPN
ejpam-5252	197	35	cariaga	cariaga	PROPN
ejpam-5252	197	36	,	,	PUNCT
ejpam-5252	197	37	s.	s.	PROPN
ejpam-5252	197	38	menchavez	menchavez	PROPN
ejpam-5252	197	39	/	/	PUNCT
ejpam-5252	197	40	eur	eur	PROPN
ejpam-5252	197	41	.	.	PUNCT
ejpam-5252	198	1	j.	j.	PROPN
ejpam-5252	198	2	pure	pure	PROPN
ejpam-5252	198	3	appl	appl	PROPN
ejpam-5252	198	4	.	.	PROPN
ejpam-5252	198	5	math	math	PROPN
ejpam-5252	198	6	,	,	PUNCT
ejpam-5252	198	7	18	18	NUM
ejpam-5252	198	8	(	(	PUNCT
ejpam-5252	198	9	1	1	NUM
ejpam-5252	198	10	)	)	PUNCT
ejpam-5252	198	11	(	(	PUNCT
ejpam-5252	198	12	2025	2025	NUM
ejpam-5252	198	13	)	)	PUNCT
ejpam-5252	198	14	,	,	PUNCT
ejpam-5252	198	15	5252	5252	NUM
ejpam-5252	198	16	8	8	NUM
ejpam-5252	198	17	of	of	ADP
ejpam-5252	198	18	18	18	NUM
ejpam-5252	198	19	proof	proof	NOUN
ejpam-5252	198	20	.	.	PUNCT
ejpam-5252	199	1	let	let	VERB
ejpam-5252	199	2	bn	bn	PART
ejpam-5252	199	3	be	be	AUX
ejpam-5252	199	4	an	an	DET
ejpam-5252	199	5	n	n	CCONJ
ejpam-5252	199	6	-	-	PUNCT
ejpam-5252	199	7	barbell	barbell	NOUN
ejpam-5252	199	8	graph	graph	NOUN
ejpam-5252	199	9	and	and	CCONJ
ejpam-5252	199	10	uv	uv	NOUN
ejpam-5252	199	11	∈	∈	PROPN
ejpam-5252	199	12	e(bn	e(bn	PROPN
ejpam-5252	199	13	)	)	PUNCT
ejpam-5252	199	14	be	be	VERB
ejpam-5252	199	15	the	the	DET
ejpam-5252	199	16	bridge	bridge	NOUN
ejpam-5252	199	17	that	that	PRON
ejpam-5252	199	18	joins	join	VERB
ejpam-5252	199	19	the	the	DET
ejpam-5252	199	20	two	two	NUM
ejpam-5252	199	21	copies	copy	NOUN
ejpam-5252	199	22	of	of	ADP
ejpam-5252	199	23	kn	kn	PROPN
ejpam-5252	199	24	.	.	PUNCT
ejpam-5252	200	1	if	if	SCONJ
ejpam-5252	200	2	n	n	NUM
ejpam-5252	200	3	=	=	SYM
ejpam-5252	200	4	3	3	NUM
ejpam-5252	200	5	,	,	PUNCT
ejpam-5252	200	6	define	define	VERB
ejpam-5252	200	7	a	a	DET
ejpam-5252	200	8	function	function	NOUN
ejpam-5252	200	9	f	f	NOUN
ejpam-5252	200	10	=	=	SYM
ejpam-5252	200	11	(	(	PUNCT
ejpam-5252	200	12	v0	v0	PROPN
ejpam-5252	200	13	,	,	PUNCT
ejpam-5252	200	14	v1	v1	NOUN
ejpam-5252	200	15	,	,	PUNCT
ejpam-5252	200	16	v2	v2	PROPN
ejpam-5252	200	17	,	,	PUNCT
ejpam-5252	200	18	v3	v3	PROPN
ejpam-5252	200	19	)	)	PUNCT
ejpam-5252	200	20	given	give	VERB
ejpam-5252	200	21	by	by	ADP
ejpam-5252	200	22	f(x	f(x	PROPN
ejpam-5252	200	23	)	)	PUNCT
ejpam-5252	201	1	=	=	PRON
ejpam-5252	201	2	{	{	PUNCT
ejpam-5252	201	3	2	2	NUM
ejpam-5252	201	4	,	,	PUNCT
ejpam-5252	201	5	x	x	SYM
ejpam-5252	201	6	∈	∈	NOUN
ejpam-5252	201	7	{	{	PUNCT
ejpam-5252	201	8	u	u	NOUN
ejpam-5252	201	9	,	,	PUNCT
ejpam-5252	201	10	v	v	NOUN
ejpam-5252	201	11	}	}	PUNCT
ejpam-5252	201	12	.	.	PUNCT
ejpam-5252	202	1	1	1	NUM
ejpam-5252	202	2	,	,	PUNCT
ejpam-5252	202	3	otherwise	otherwise	ADV
ejpam-5252	202	4	.	.	PUNCT
ejpam-5252	203	1	then	then	ADV
ejpam-5252	203	2	f	f	PROPN
ejpam-5252	203	3	∈	∈	PROPN
ejpam-5252	203	4	mrdf	mrdf	NOUN
ejpam-5252	203	5	of	of	ADP
ejpam-5252	203	6	b3	b3	PROPN
ejpam-5252	203	7	.	.	PUNCT
ejpam-5252	204	1	it	it	PRON
ejpam-5252	204	2	follows	follow	VERB
ejpam-5252	204	3	that	that	SCONJ
ejpam-5252	204	4	γmr(b3	γmr(b3	NOUN
ejpam-5252	204	5	)	)	PUNCT
ejpam-5252	204	6	≤	≤	NUM
ejpam-5252	204	7	8	8	NUM
ejpam-5252	204	8	.	.	PUNCT
ejpam-5252	205	1	now	now	ADV
ejpam-5252	205	2	,	,	PUNCT
ejpam-5252	205	3	suppose	suppose	VERB
ejpam-5252	205	4	that	that	SCONJ
ejpam-5252	205	5	f	f	PROPN
ejpam-5252	206	1	′	′	NUM
ejpam-5252	206	2	=	=	PUNCT
ejpam-5252	207	1	(	(	PUNCT
ejpam-5252	207	2	v	v	NUM
ejpam-5252	207	3	′	′	NUM
ejpam-5252	207	4	0	0	NUM
ejpam-5252	207	5	,	,	PUNCT
ejpam-5252	207	6	v	v	NOUN
ejpam-5252	207	7	′	′	NUM
ejpam-5252	207	8	1	1	NUM
ejpam-5252	207	9	,	,	PUNCT
ejpam-5252	207	10	v	v	NOUN
ejpam-5252	207	11	′	′	NUM
ejpam-5252	207	12	2	2	NUM
ejpam-5252	207	13	,	,	PUNCT
ejpam-5252	207	14	v	v	ADJ
ejpam-5252	207	15	′	′	NUM
ejpam-5252	207	16	3	3	NUM
ejpam-5252	207	17	)	)	PUNCT
ejpam-5252	207	18	is	be	AUX
ejpam-5252	207	19	a	a	DET
ejpam-5252	207	20	γmr	γmr	ADJ
ejpam-5252	207	21	-	-	PUNCT
ejpam-5252	207	22	function	function	NOUN
ejpam-5252	207	23	of	of	ADP
ejpam-5252	207	24	b3	b3	PROPN
ejpam-5252	207	25	.	.	PUNCT
ejpam-5252	208	1	if	if	SCONJ
ejpam-5252	208	2	v	v	NUM
ejpam-5252	208	3	′	′	NOUN
ejpam-5252	208	4	0	0	NUM
ejpam-5252	209	1	=	=	NOUN
ejpam-5252	209	2	∅	∅	NOUN
ejpam-5252	209	3	,	,	PUNCT
ejpam-5252	209	4	then	then	ADV
ejpam-5252	209	5	v	v	ADJ
ejpam-5252	209	6	′	′	NUM
ejpam-5252	209	7	3	3	NUM
ejpam-5252	209	8	=	=	PUNCT
ejpam-5252	209	9	∅.	∅.	NOUN
ejpam-5252	209	10	since	since	SCONJ
ejpam-5252	209	11	f	f	PROPN
ejpam-5252	209	12	′	′	NUM
ejpam-5252	209	13	is	be	AUX
ejpam-5252	209	14	a	a	DET
ejpam-5252	209	15	γmrfunction	γmrfunction	NOUN
ejpam-5252	209	16	of	of	ADP
ejpam-5252	209	17	b3	b3	PROPN
ejpam-5252	209	18	,	,	PUNCT
ejpam-5252	209	19	|v	|v	ADJ
ejpam-5252	209	20	′	′	NOUN
ejpam-5252	209	21	2	2	NUM
ejpam-5252	210	1	|	|	NOUN
ejpam-5252	210	2	=	=	SYM
ejpam-5252	210	3	2	2	NUM
ejpam-5252	210	4	and	and	CCONJ
ejpam-5252	210	5	|v	|v	ADJ
ejpam-5252	210	6	′	′	NUM
ejpam-5252	210	7	1	1	NUM
ejpam-5252	211	1	|	|	ADV
ejpam-5252	211	2	=	=	SYM
ejpam-5252	211	3	v	v	PROPN
ejpam-5252	211	4	(	(	PUNCT
ejpam-5252	211	5	b3	b3	PROPN
ejpam-5252	211	6	)	)	PUNCT
ejpam-5252	211	7	\	\	PROPN
ejpam-5252	212	1	|v	|v	PROPN
ejpam-5252	212	2	′	′	NOUN
ejpam-5252	212	3	2	2	NUM
ejpam-5252	212	4	|	|	NOUN
ejpam-5252	212	5	.	.	PUNCT
ejpam-5252	213	1	hence	hence	ADV
ejpam-5252	213	2	,	,	PUNCT
ejpam-5252	213	3	γmr(b3	γmr(b3	NOUN
ejpam-5252	213	4	)	)	PUNCT
ejpam-5252	213	5	=	=	SYM
ejpam-5252	213	6	ωmr	ωmr	PROPN
ejpam-5252	213	7	b3	b3	PROPN
ejpam-5252	213	8	(	(	PUNCT
ejpam-5252	213	9	f	f	PROPN
ejpam-5252	213	10	′	′	PROPN
ejpam-5252	213	11	)	)	PUNCT
ejpam-5252	213	12	≥	≥	NOUN
ejpam-5252	213	13	8	8	NUM
ejpam-5252	213	14	.	.	PUNCT
ejpam-5252	214	1	if	if	SCONJ
ejpam-5252	214	2	|v	|v	PROPN
ejpam-5252	214	3	′	′	NOUN
ejpam-5252	214	4	0	0	NUM
ejpam-5252	215	1	|	|	ADV
ejpam-5252	215	2	̸=	̸=	PROPN
ejpam-5252	215	3	0	0	NUM
ejpam-5252	215	4	,	,	PUNCT
ejpam-5252	215	5	then	then	ADV
ejpam-5252	215	6	|v	|v	VERB
ejpam-5252	215	7	′	′	NOUN
ejpam-5252	215	8	2	2	NUM
ejpam-5252	215	9	|	|	CCONJ
ejpam-5252	215	10	≥	≥	NOUN
ejpam-5252	215	11	2	2	NUM
ejpam-5252	215	12	and	and	CCONJ
ejpam-5252	215	13	|v	|v	ADJ
ejpam-5252	215	14	′	′	NUM
ejpam-5252	215	15	3	3	NUM
ejpam-5252	215	16	|	|	CCONJ
ejpam-5252	215	17	≥	≥	NUM
ejpam-5252	215	18	1	1	NUM
ejpam-5252	215	19	.	.	PUNCT
ejpam-5252	216	1	it	it	PRON
ejpam-5252	216	2	follows	follow	VERB
ejpam-5252	216	3	that	that	SCONJ
ejpam-5252	216	4	γmr(b3	γmr(b3	NOUN
ejpam-5252	216	5	)	)	PUNCT
ejpam-5252	216	6	=	=	SYM
ejpam-5252	216	7	ωmr	ωmr	PROPN
ejpam-5252	216	8	b3	b3	PROPN
ejpam-5252	216	9	(	(	PUNCT
ejpam-5252	216	10	g	g	NOUN
ejpam-5252	216	11	)	)	PUNCT
ejpam-5252	216	12	≥	≥	NOUN
ejpam-5252	216	13	8	8	NUM
ejpam-5252	216	14	.	.	PUNCT
ejpam-5252	217	1	therefore	therefore	ADV
ejpam-5252	217	2	,	,	PUNCT
ejpam-5252	217	3	γmr(b3	γmr(b3	NOUN
ejpam-5252	217	4	)	)	PUNCT
ejpam-5252	217	5	=	=	SYM
ejpam-5252	217	6	8	8	X
ejpam-5252	217	7	.	.	PUNCT
ejpam-5252	218	1	if	if	SCONJ
ejpam-5252	218	2	n	n	NUM
ejpam-5252	218	3	≥	≥	NOUN
ejpam-5252	218	4	4	4	NUM
ejpam-5252	218	5	.	.	PUNCT
ejpam-5252	218	6	pick	pick	VERB
ejpam-5252	218	7	any	any	DET
ejpam-5252	218	8	v′	v′	NOUN
ejpam-5252	218	9	,	,	PUNCT
ejpam-5252	218	10	u′	u′	PROPN
ejpam-5252	218	11	∈	∈	PROPN
ejpam-5252	218	12	v	v	NOUN
ejpam-5252	218	13	(	(	PUNCT
ejpam-5252	218	14	bn	bn	NOUN
ejpam-5252	218	15	)	)	PUNCT
ejpam-5252	218	16	such	such	ADJ
ejpam-5252	218	17	that	that	DET
ejpam-5252	218	18	v′	v′	PROPN
ejpam-5252	218	19	̸=	̸=	PROPN
ejpam-5252	218	20	u	u	NOUN
ejpam-5252	218	21	,	,	PUNCT
ejpam-5252	218	22	u′	u′	PROPN
ejpam-5252	218	23	̸=	̸=	PROPN
ejpam-5252	218	24	v	v	NOUN
ejpam-5252	218	25	,	,	PUNCT
ejpam-5252	218	26	and	and	CCONJ
ejpam-5252	218	27	v′v	v′v	NUM
ejpam-5252	218	28	,	,	PUNCT
ejpam-5252	218	29	u′u	u′u	ADV
ejpam-5252	218	30	∈	∈	PROPN
ejpam-5252	218	31	e(bn	e(bn	PROPN
ejpam-5252	218	32	)	)	PUNCT
ejpam-5252	218	33	.	.	PUNCT
ejpam-5252	219	1	now	now	ADV
ejpam-5252	219	2	,	,	PUNCT
ejpam-5252	219	3	define	define	VERB
ejpam-5252	219	4	a	a	DET
ejpam-5252	219	5	function	function	NOUN
ejpam-5252	219	6	f	f	NOUN
ejpam-5252	219	7	=	=	SYM
ejpam-5252	219	8	(	(	PUNCT
ejpam-5252	219	9	v0	v0	PROPN
ejpam-5252	219	10	,	,	PUNCT
ejpam-5252	219	11	v1	v1	NOUN
ejpam-5252	219	12	,	,	PUNCT
ejpam-5252	219	13	v2	v2	PROPN
ejpam-5252	219	14	,	,	PUNCT
ejpam-5252	219	15	v3	v3	PROPN
ejpam-5252	219	16	)	)	PUNCT
ejpam-5252	219	17	given	give	VERB
ejpam-5252	219	18	by	by	ADP
ejpam-5252	219	19	f(x	f(x	PROPN
ejpam-5252	219	20	)	)	PUNCT
ejpam-5252	220	1	=	=	PUNCT
ejpam-5252	221	1			NOUN
ejpam-5252	221	2	0	0	NUM
ejpam-5252	221	3	,	,	PUNCT
ejpam-5252	221	4	x	x	SYM
ejpam-5252	221	5	∈	∈	NOUN
ejpam-5252	221	6	v	v	ADP
ejpam-5252	221	7	(	(	PUNCT
ejpam-5252	221	8	bn	bn	NOUN
ejpam-5252	221	9	)	)	PUNCT
ejpam-5252	221	10	\	\	NOUN
ejpam-5252	221	11	{	{	PUNCT
ejpam-5252	221	12	u	u	NOUN
ejpam-5252	221	13	,	,	PUNCT
ejpam-5252	221	14	v	v	NOUN
ejpam-5252	221	15	,	,	PUNCT
ejpam-5252	221	16	u′	u′	PROPN
ejpam-5252	221	17	,	,	PUNCT
ejpam-5252	221	18	v′	v′	PROPN
ejpam-5252	221	19	}	}	PUNCT
ejpam-5252	221	20	.	.	PUNCT
ejpam-5252	222	1	3	3	NUM
ejpam-5252	222	2	,	,	PUNCT
ejpam-5252	222	3	x	x	X
ejpam-5252	222	4	∈	∈	PROPN
ejpam-5252	222	5	{	{	PUNCT
ejpam-5252	222	6	u	u	NOUN
ejpam-5252	222	7	,	,	PUNCT
ejpam-5252	222	8	v	v	NOUN
ejpam-5252	222	9	}	}	PUNCT
ejpam-5252	222	10	.	.	PUNCT
ejpam-5252	223	1	2	2	NUM
ejpam-5252	223	2	,	,	PUNCT
ejpam-5252	223	3	x	x	X
ejpam-5252	223	4	∈	∈	NOUN
ejpam-5252	223	5	{	{	PUNCT
ejpam-5252	223	6	u′	u′	PROPN
ejpam-5252	223	7	,	,	PUNCT
ejpam-5252	223	8	v′	v′	PROPN
ejpam-5252	223	9	}	}	PUNCT
ejpam-5252	223	10	.	.	PUNCT
ejpam-5252	224	1	then	then	ADV
ejpam-5252	224	2	f	f	PROPN
ejpam-5252	224	3	∈	∈	PROPN
ejpam-5252	224	4	mrdf	mrdf	NOUN
ejpam-5252	224	5	of	of	ADP
ejpam-5252	224	6	bn	bn	NOUN
ejpam-5252	224	7	,	,	PUNCT
ejpam-5252	224	8	n	n	PRON
ejpam-5252	224	9	≥	≥	NOUN
ejpam-5252	224	10	4	4	NUM
ejpam-5252	224	11	.	.	PUNCT
ejpam-5252	225	1	it	it	PRON
ejpam-5252	225	2	follows	follow	VERB
ejpam-5252	225	3	that	that	SCONJ
ejpam-5252	225	4	γmr(bn	γmr(bn	NOUN
ejpam-5252	225	5	)	)	PUNCT
ejpam-5252	225	6	≤	≤	NOUN
ejpam-5252	225	7	10	10	NUM
ejpam-5252	225	8	.	.	PUNCT
ejpam-5252	226	1	now	now	ADV
ejpam-5252	226	2	,	,	PUNCT
ejpam-5252	226	3	suppose	suppose	VERB
ejpam-5252	226	4	that	that	SCONJ
ejpam-5252	226	5	g	g	PROPN
ejpam-5252	226	6	=	=	SYM
ejpam-5252	226	7	(	(	PUNCT
ejpam-5252	226	8	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	226	9	)	)	PUNCT
ejpam-5252	226	10	is	be	AUX
ejpam-5252	226	11	a	a	DET
ejpam-5252	226	12	γmr	γmr	ADJ
ejpam-5252	226	13	-	-	PUNCT
ejpam-5252	226	14	function	function	NOUN
ejpam-5252	226	15	of	of	ADP
ejpam-5252	226	16	bn	bn	NOUN
ejpam-5252	226	17	.	.	PUNCT
ejpam-5252	227	1	if	if	SCONJ
ejpam-5252	227	2	w0	w0	PROPN
ejpam-5252	227	3	=	=	SYM
ejpam-5252	227	4	∅	∅	NOUN
ejpam-5252	227	5	,	,	PUNCT
ejpam-5252	227	6	then	then	ADV
ejpam-5252	227	7	w3	w3	PROPN
ejpam-5252	227	8	=	=	PUNCT
ejpam-5252	227	9	∅.	∅.	NOUN
ejpam-5252	227	10	since	since	SCONJ
ejpam-5252	227	11	g	g	PROPN
ejpam-5252	227	12	is	be	AUX
ejpam-5252	227	13	a	a	DET
ejpam-5252	227	14	γmr	γmr	ADJ
ejpam-5252	227	15	-	-	PUNCT
ejpam-5252	227	16	function	function	NOUN
ejpam-5252	227	17	of	of	ADP
ejpam-5252	227	18	bn	bn	NOUN
ejpam-5252	227	19	,	,	PUNCT
ejpam-5252	227	20	|w2|	|w2|	NOUN
ejpam-5252	227	21	=	=	SYM
ejpam-5252	227	22	2	2	NUM
ejpam-5252	227	23	and	and	CCONJ
ejpam-5252	227	24	|w1|	|w1|	NOUN
ejpam-5252	227	25	=	=	SYM
ejpam-5252	227	26	v	v	NOUN
ejpam-5252	227	27	(	(	PUNCT
ejpam-5252	227	28	bn	bn	NOUN
ejpam-5252	227	29	)	)	PUNCT
ejpam-5252	227	30	\	\	NOUN
ejpam-5252	227	31	|w2|	|w2|	NOUN
ejpam-5252	227	32	.	.	PUNCT
ejpam-5252	228	1	hence	hence	ADV
ejpam-5252	228	2	,	,	PUNCT
ejpam-5252	228	3	γmr(bn	γmr(bn	PROPN
ejpam-5252	228	4	)	)	PUNCT
ejpam-5252	228	5	=	=	SYM
ejpam-5252	228	6	ωmr	ωmr	NOUN
ejpam-5252	228	7	bn	bn	ADJ
ejpam-5252	228	8	(	(	PUNCT
ejpam-5252	228	9	g	g	NOUN
ejpam-5252	228	10	)	)	PUNCT
ejpam-5252	228	11	≥	≥	NOUN
ejpam-5252	228	12	10	10	NUM
ejpam-5252	228	13	.	.	PUNCT
ejpam-5252	229	1	if	if	SCONJ
ejpam-5252	229	2	|w0|	|w0|	VERB
ejpam-5252	229	3	̸=	̸=	PROPN
ejpam-5252	229	4	0	0	NUM
ejpam-5252	229	5	,	,	PUNCT
ejpam-5252	229	6	then	then	ADV
ejpam-5252	229	7	|w2|	|w2|	NOUN
ejpam-5252	229	8	≥	≥	NUM
ejpam-5252	229	9	2	2	NUM
ejpam-5252	229	10	and	and	CCONJ
ejpam-5252	229	11	|w3|	|w3|	PRON
ejpam-5252	229	12	≥	≥	NOUN
ejpam-5252	229	13	2	2	NUM
ejpam-5252	229	14	.	.	PUNCT
ejpam-5252	230	1	it	it	PRON
ejpam-5252	230	2	follows	follow	VERB
ejpam-5252	230	3	that	that	PRON
ejpam-5252	230	4	γmr(bn	γmr(bn	PROPN
ejpam-5252	230	5	)	)	PUNCT
ejpam-5252	230	6	=	=	SYM
ejpam-5252	230	7	ωmr	ωmr	NOUN
ejpam-5252	230	8	bn	bn	ADJ
ejpam-5252	230	9	(	(	PUNCT
ejpam-5252	230	10	g	g	NOUN
ejpam-5252	230	11	)	)	PUNCT
ejpam-5252	230	12	=	=	SYM
ejpam-5252	230	13	2|w2|	2|w2|	NUM
ejpam-5252	230	14	+	+	SYM
ejpam-5252	230	15	3|w3|	3|w3|	NUM
ejpam-5252	230	16	≥	≥	NUM
ejpam-5252	230	17	10	10	NUM
ejpam-5252	230	18	.	.	PUNCT
ejpam-5252	231	1	therefore	therefore	ADV
ejpam-5252	231	2	,	,	PUNCT
ejpam-5252	231	3	γmr(b3	γmr(b3	NOUN
ejpam-5252	231	4	)	)	PUNCT
ejpam-5252	231	5	=	=	SYM
ejpam-5252	231	6	10	10	NUM
ejpam-5252	231	7	.	.	PUNCT
ejpam-5252	232	1	the	the	DET
ejpam-5252	232	2	windmill	windmill	NOUN
ejpam-5252	232	3	graph	graph	NOUN
ejpam-5252	232	4	wd(k	wd(k	PROPN
ejpam-5252	232	5	,	,	PUNCT
ejpam-5252	232	6	n)=	n)=	NOUN
ejpam-5252	232	7	g	g	NOUN
ejpam-5252	232	8	=	=	SYM
ejpam-5252	232	9	k1	k1	PROPN
ejpam-5252	232	10	+	+	CCONJ
ejpam-5252	232	11	nkk−1	nkk−1	PROPN
ejpam-5252	232	12	is	be	AUX
ejpam-5252	232	13	constructed	construct	VERB
ejpam-5252	232	14	for	for	ADP
ejpam-5252	232	15	k	k	PROPN
ejpam-5252	232	16	≥	≥	NUM
ejpam-5252	232	17	2	2	NUM
ejpam-5252	232	18	and	and	CCONJ
ejpam-5252	232	19	n	n	PRON
ejpam-5252	232	20	≥	≥	NOUN
ejpam-5252	232	21	2	2	NUM
ejpam-5252	232	22	by	by	ADP
ejpam-5252	232	23	joining	join	VERB
ejpam-5252	232	24	n	n	PRON
ejpam-5252	232	25	copies	copy	NOUN
ejpam-5252	232	26	of	of	ADP
ejpam-5252	232	27	the	the	DET
ejpam-5252	232	28	complete	complete	ADJ
ejpam-5252	232	29	graph	graph	NOUN
ejpam-5252	232	30	kk	kk	X
ejpam-5252	232	31	at	at	ADP
ejpam-5252	232	32	a	a	DET
ejpam-5252	232	33	shared	share	VERB
ejpam-5252	232	34	vertex	vertex	NOUN
ejpam-5252	232	35	.	.	PUNCT
ejpam-5252	233	1	it	it	PRON
ejpam-5252	233	2	has	have	VERB
ejpam-5252	233	3	n(k	n(k	PROPN
ejpam-5252	233	4	−	−	PROPN
ejpam-5252	233	5	1	1	NUM
ejpam-5252	233	6	)	)	PUNCT
ejpam-5252	233	7	+	+	CCONJ
ejpam-5252	233	8	1	1	NUM
ejpam-5252	233	9	vertices	vertex	NOUN
ejpam-5252	233	10	and	and	CCONJ
ejpam-5252	233	11	1	1	NUM
ejpam-5252	233	12	2nk(k	2nk(k	NUM
ejpam-5252	233	13	−	−	NUM
ejpam-5252	233	14	1	1	NUM
ejpam-5252	233	15	)	)	PUNCT
ejpam-5252	233	16	edges	edge	NOUN
ejpam-5252	233	17	.	.	PUNCT
ejpam-5252	234	1	the	the	DET
ejpam-5252	234	2	case	case	NOUN
ejpam-5252	234	3	k	k	NOUN
ejpam-5252	234	4	=	=	SYM
ejpam-5252	234	5	3	3	NUM
ejpam-5252	234	6	corresponds	correspond	VERB
ejpam-5252	234	7	to	to	ADP
ejpam-5252	234	8	the	the	DET
ejpam-5252	234	9	dutch	dutch	ADJ
ejpam-5252	234	10	windmill	windmill	NOUN
ejpam-5252	234	11	graph	graph	NOUN
ejpam-5252	234	12	(	(	PUNCT
ejpam-5252	234	13	also	also	ADV
ejpam-5252	234	14	called	call	VERB
ejpam-5252	234	15	friendship	friendship	NOUN
ejpam-5252	234	16	graph	graph	NOUN
ejpam-5252	234	17	)	)	PUNCT
ejpam-5252	234	18	gn	gn	PROPN
ejpam-5252	234	19	3	3	NUM
ejpam-5252	234	20	=	=	SYM
ejpam-5252	234	21	k1	k1	NOUN
ejpam-5252	234	22	+	+	CCONJ
ejpam-5252	234	23	nk2	nk2	NOUN
ejpam-5252	234	24	and	and	CCONJ
ejpam-5252	234	25	the	the	DET
ejpam-5252	234	26	case	case	NOUN
ejpam-5252	234	27	n	n	NOUN
ejpam-5252	234	28	=	=	SYM
ejpam-5252	234	29	2	2	NUM
ejpam-5252	234	30	corresponds	correspond	NOUN
ejpam-5252	234	31	to	to	ADP
ejpam-5252	234	32	the	the	DET
ejpam-5252	234	33	butterfly	butterfly	NOUN
ejpam-5252	234	34	graph	graph	NOUN
ejpam-5252	234	35	g2	g2	PROPN
ejpam-5252	234	36	3	3	NUM
ejpam-5252	234	37	=	=	SYM
ejpam-5252	234	38	k1	k1	NOUN
ejpam-5252	234	39	+	+	NOUN
ejpam-5252	234	40	2k2	2k2	NUM
ejpam-5252	234	41	.	.	PUNCT
ejpam-5252	235	1	the	the	DET
ejpam-5252	235	2	graphs	graph	NOUN
ejpam-5252	235	3	in	in	ADP
ejpam-5252	235	4	figures	figure	NOUN
ejpam-5252	235	5	3	3	NUM
ejpam-5252	235	6	,	,	PUNCT
ejpam-5252	235	7	4	4	NUM
ejpam-5252	235	8	,	,	PUNCT
ejpam-5252	235	9	and	and	CCONJ
ejpam-5252	235	10	5	5	NUM
ejpam-5252	235	11	are	be	AUX
ejpam-5252	235	12	the	the	DET
ejpam-5252	235	13	windmill	windmill	NOUN
ejpam-5252	235	14	graph	graph	NOUN
ejpam-5252	235	15	wd(4	wd(4	PROPN
ejpam-5252	235	16	,	,	PUNCT
ejpam-5252	235	17	2	2	NUM
ejpam-5252	235	18	)	)	PUNCT
ejpam-5252	235	19	,	,	PUNCT
ejpam-5252	235	20	friendship	friendship	NOUN
ejpam-5252	235	21	graph	graph	NOUN
ejpam-5252	235	22	g4	g4	NOUN
ejpam-5252	235	23	3	3	NUM
ejpam-5252	235	24	and	and	CCONJ
ejpam-5252	235	25	butterfly	butterfly	NOUN
ejpam-5252	235	26	graphs	graph	NOUN
ejpam-5252	235	27	g2	g2	PROPN
ejpam-5252	235	28	3	3	NUM
ejpam-5252	235	29	,	,	PUNCT
ejpam-5252	235	30	respectively	respectively	ADV
ejpam-5252	235	31	.	.	PUNCT
ejpam-5252	236	1	3	3	NUM
ejpam-5252	236	2	22	22	NUM
ejpam-5252	236	3	0	0	NUM
ejpam-5252	236	4	00	00	NUM
ejpam-5252	236	5	0	0	NUM
ejpam-5252	236	6	figure	figure	NOUN
ejpam-5252	236	7	3	3	NUM
ejpam-5252	236	8	:	:	PUNCT
ejpam-5252	236	9	a	a	DET
ejpam-5252	236	10	windmill	windmill	NOUN
ejpam-5252	236	11	graph	graph	NOUN
ejpam-5252	236	12	wd(4	wd(4	PROPN
ejpam-5252	236	13	,	,	PUNCT
ejpam-5252	236	14	2	2	NUM
ejpam-5252	236	15	)	)	PUNCT
ejpam-5252	236	16	with	with	ADP
ejpam-5252	236	17	γmr(wd(4	γmr(wd(4	NOUN
ejpam-5252	236	18	,	,	PUNCT
ejpam-5252	236	19	2	2	NUM
ejpam-5252	236	20	)	)	PUNCT
ejpam-5252	236	21	)	)	PUNCT
ejpam-5252	237	1	=	=	SYM
ejpam-5252	237	2	7	7	NUM
ejpam-5252	237	3	s.	s.	PROPN
ejpam-5252	237	4	ahamad	ahamad	PROPN
ejpam-5252	237	5	,	,	PUNCT
ejpam-5252	237	6	j.	j.	PROPN
ejpam-5252	237	7	cariaga	cariaga	PROPN
ejpam-5252	237	8	,	,	PUNCT
ejpam-5252	237	9	s.	s.	PROPN
ejpam-5252	237	10	menchavez	menchavez	PROPN
ejpam-5252	237	11	/	/	PUNCT
ejpam-5252	237	12	eur	eur	PROPN
ejpam-5252	237	13	.	.	PUNCT
ejpam-5252	238	1	j.	j.	PROPN
ejpam-5252	238	2	pure	pure	PROPN
ejpam-5252	238	3	appl	appl	PROPN
ejpam-5252	238	4	.	.	PROPN
ejpam-5252	238	5	math	math	PROPN
ejpam-5252	238	6	,	,	PUNCT
ejpam-5252	238	7	18	18	NUM
ejpam-5252	238	8	(	(	PUNCT
ejpam-5252	238	9	1	1	NUM
ejpam-5252	238	10	)	)	PUNCT
ejpam-5252	238	11	(	(	PUNCT
ejpam-5252	238	12	2025	2025	NUM
ejpam-5252	238	13	)	)	PUNCT
ejpam-5252	238	14	,	,	PUNCT
ejpam-5252	238	15	5252	5252	NUM
ejpam-5252	238	16	9	9	NUM
ejpam-5252	238	17	of	of	ADP
ejpam-5252	238	18	18	18	NUM
ejpam-5252	238	19	2	2	NUM
ejpam-5252	238	20	1	1	NUM
ejpam-5252	238	21	1	1	NUM
ejpam-5252	238	22	1	1	NUM
ejpam-5252	238	23	1	1	NUM
ejpam-5252	238	24	11	11	NUM
ejpam-5252	238	25	1	1	NUM
ejpam-5252	238	26	1	1	NUM
ejpam-5252	238	27	figure	figure	NOUN
ejpam-5252	238	28	4	4	NUM
ejpam-5252	238	29	:	:	PUNCT
ejpam-5252	238	30	a	a	DET
ejpam-5252	238	31	friendship	friendship	NOUN
ejpam-5252	238	32	graph	graph	NOUN
ejpam-5252	238	33	g4	g4	NOUN
ejpam-5252	238	34	3	3	NUM
ejpam-5252	238	35	with	with	ADP
ejpam-5252	238	36	γmr(g	γmr(g	PROPN
ejpam-5252	238	37	4	4	NUM
ejpam-5252	238	38	3	3	NUM
ejpam-5252	238	39	)	)	PUNCT
ejpam-5252	238	40	=	=	SYM
ejpam-5252	238	41	10	10	NUM
ejpam-5252	238	42	2	2	NUM
ejpam-5252	238	43	1	1	NUM
ejpam-5252	238	44	11	11	NUM
ejpam-5252	238	45	1	1	NUM
ejpam-5252	238	46	figure	figure	NOUN
ejpam-5252	238	47	5	5	NUM
ejpam-5252	238	48	:	:	PUNCT
ejpam-5252	238	49	a	a	DET
ejpam-5252	238	50	butterfly	butterfly	NOUN
ejpam-5252	238	51	graph	graph	NOUN
ejpam-5252	238	52	g2	g2	PROPN
ejpam-5252	238	53	3	3	NUM
ejpam-5252	238	54	with	with	ADP
ejpam-5252	238	55	γmr(g	γmr(g	PROPN
ejpam-5252	238	56	2	2	NUM
ejpam-5252	238	57	3	3	NUM
ejpam-5252	238	58	)	)	PUNCT
ejpam-5252	238	59	=	=	SYM
ejpam-5252	238	60	6	6	NUM
ejpam-5252	238	61	proposition	proposition	NOUN
ejpam-5252	238	62	9	9	NUM
ejpam-5252	238	63	.	.	PUNCT
ejpam-5252	239	1	for	for	ADP
ejpam-5252	239	2	any	any	DET
ejpam-5252	239	3	windmill	windmill	NOUN
ejpam-5252	239	4	graph	graph	NOUN
ejpam-5252	239	5	g	g	PROPN
ejpam-5252	239	6	=	=	PROPN
ejpam-5252	239	7	k1	k1	PROPN
ejpam-5252	239	8	+	+	CCONJ
ejpam-5252	239	9	nkk−1	nkk−1	PROPN
ejpam-5252	239	10	,	,	PUNCT
ejpam-5252	239	11	where	where	SCONJ
ejpam-5252	239	12	k	k	PROPN
ejpam-5252	239	13	≥	≥	VERB
ejpam-5252	239	14	4	4	NUM
ejpam-5252	239	15	and	and	CCONJ
ejpam-5252	239	16	n	n	PRON
ejpam-5252	239	17	≥	≥	NOUN
ejpam-5252	239	18	2	2	NUM
ejpam-5252	239	19	,	,	PUNCT
ejpam-5252	239	20	γmr(g	γmr(g	PROPN
ejpam-5252	239	21	)	)	PUNCT
ejpam-5252	240	1	=	=	SYM
ejpam-5252	241	1	2n+	2n+	NUM
ejpam-5252	241	2	3	3	NUM
ejpam-5252	241	3	.	.	PUNCT
ejpam-5252	241	4	proof	proof	NOUN
ejpam-5252	241	5	.	.	PUNCT
ejpam-5252	242	1	let	let	VERB
ejpam-5252	242	2	g	g	NOUN
ejpam-5252	242	3	=	=	VERB
ejpam-5252	242	4	k1	k1	PROPN
ejpam-5252	242	5	+	+	CCONJ
ejpam-5252	242	6	nkk−1	nkk−1	PROPN
ejpam-5252	242	7	,	,	PUNCT
ejpam-5252	242	8	where	where	SCONJ
ejpam-5252	242	9	k	k	PROPN
ejpam-5252	242	10	≥	≥	VERB
ejpam-5252	242	11	4	4	NUM
ejpam-5252	242	12	and	and	CCONJ
ejpam-5252	242	13	n	n	PRON
ejpam-5252	242	14	≥	≥	NOUN
ejpam-5252	242	15	2	2	NUM
ejpam-5252	242	16	.	.	PUNCT
ejpam-5252	242	17	suppose	suppose	VERB
ejpam-5252	242	18	v	v	X
ejpam-5252	242	19	(	(	PUNCT
ejpam-5252	242	20	k1	k1	NOUN
ejpam-5252	242	21	)	)	PUNCT
ejpam-5252	242	22	=	=	SYM
ejpam-5252	242	23	{	{	PUNCT
ejpam-5252	242	24	u	u	NOUN
ejpam-5252	242	25	}	}	PUNCT
ejpam-5252	242	26	be	be	AUX
ejpam-5252	242	27	the	the	DET
ejpam-5252	242	28	central	central	ADJ
ejpam-5252	242	29	vertex	vertex	NOUN
ejpam-5252	242	30	in	in	ADP
ejpam-5252	242	31	g	g	NOUN
ejpam-5252	242	32	,	,	PUNCT
ejpam-5252	242	33	then	then	ADV
ejpam-5252	242	34	pick	pick	VERB
ejpam-5252	242	35	a	a	DET
ejpam-5252	242	36	vertex	vertex	NOUN
ejpam-5252	242	37	v	v	NOUN
ejpam-5252	242	38	in	in	ADP
ejpam-5252	242	39	each	each	PRON
ejpam-5252	242	40	n	n	NOUN
ejpam-5252	242	41	copies	copy	NOUN
ejpam-5252	242	42	of	of	ADP
ejpam-5252	242	43	the	the	DET
ejpam-5252	242	44	complete	complete	ADJ
ejpam-5252	242	45	graph	graph	NOUN
ejpam-5252	242	46	kk−1	kk−1	PROPN
ejpam-5252	242	47	and	and	CCONJ
ejpam-5252	242	48	define	define	VERB
ejpam-5252	242	49	a	a	DET
ejpam-5252	242	50	function	function	NOUN
ejpam-5252	242	51	f	f	NOUN
ejpam-5252	242	52	=	=	SYM
ejpam-5252	242	53	(	(	PUNCT
ejpam-5252	242	54	v0	v0	PROPN
ejpam-5252	242	55	,	,	PUNCT
ejpam-5252	242	56	v1	v1	NOUN
ejpam-5252	242	57	,	,	PUNCT
ejpam-5252	242	58	v2	v2	PROPN
ejpam-5252	242	59	,	,	PUNCT
ejpam-5252	242	60	v3	v3	PROPN
ejpam-5252	242	61	)	)	PUNCT
ejpam-5252	242	62	given	give	VERB
ejpam-5252	242	63	by	by	ADP
ejpam-5252	242	64	f(x	f(x	PROPN
ejpam-5252	242	65	)	)	PUNCT
ejpam-5252	243	1	=	=	PUNCT
ejpam-5252	244	1			NOUN
ejpam-5252	244	2	3	3	NUM
ejpam-5252	244	3	,	,	PUNCT
ejpam-5252	244	4	x	x	PUNCT
ejpam-5252	244	5	=	=	PUNCT
ejpam-5252	244	6	u.	u.	NOUN
ejpam-5252	244	7	2	2	NUM
ejpam-5252	244	8	,	,	PUNCT
ejpam-5252	244	9	x	x	PUNCT
ejpam-5252	244	10	=	=	PUNCT
ejpam-5252	245	1	v.	v.	ADP
ejpam-5252	245	2	0	0	NUM
ejpam-5252	245	3	,	,	PUNCT
ejpam-5252	245	4	otherwise	otherwise	ADV
ejpam-5252	245	5	.	.	PUNCT
ejpam-5252	246	1	then	then	ADV
ejpam-5252	246	2	f	f	PROPN
ejpam-5252	246	3	∈	∈	PROPN
ejpam-5252	246	4	mrdf	mrdf	NOUN
ejpam-5252	246	5	(	(	PUNCT
ejpam-5252	246	6	g	g	NOUN
ejpam-5252	246	7	)	)	PUNCT
ejpam-5252	246	8	.	.	PUNCT
ejpam-5252	247	1	it	it	PRON
ejpam-5252	247	2	follows	follow	VERB
ejpam-5252	247	3	that	that	PRON
ejpam-5252	247	4	γmr(g	γmr(g	PROPN
ejpam-5252	247	5	)	)	PUNCT
ejpam-5252	247	6	≤	≤	NOUN
ejpam-5252	247	7	2n	2n	NUM
ejpam-5252	248	1	+	+	CCONJ
ejpam-5252	248	2	3	3	X
ejpam-5252	248	3	.	.	PUNCT
ejpam-5252	248	4	now	now	ADV
ejpam-5252	248	5	,	,	PUNCT
ejpam-5252	248	6	suppose	suppose	VERB
ejpam-5252	248	7	that	that	SCONJ
ejpam-5252	248	8	g	g	PROPN
ejpam-5252	248	9	=	=	SYM
ejpam-5252	248	10	(	(	PUNCT
ejpam-5252	248	11	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	248	12	)	)	PUNCT
ejpam-5252	248	13	is	be	AUX
ejpam-5252	248	14	a	a	DET
ejpam-5252	248	15	γmr	γmr	ADJ
ejpam-5252	248	16	-	-	PUNCT
ejpam-5252	248	17	function	function	NOUN
ejpam-5252	248	18	of	of	ADP
ejpam-5252	248	19	g.	g.	PROPN
ejpam-5252	248	20	if	if	SCONJ
ejpam-5252	248	21	w0	w0	PROPN
ejpam-5252	248	22	=	=	SYM
ejpam-5252	248	23	∅	∅	NOUN
ejpam-5252	248	24	,	,	PUNCT
ejpam-5252	248	25	then	then	ADV
ejpam-5252	248	26	w3	w3	PROPN
ejpam-5252	248	27	=	=	PUNCT
ejpam-5252	248	28	∅.	∅.	NOUN
ejpam-5252	248	29	since	since	SCONJ
ejpam-5252	248	30	g	g	PROPN
ejpam-5252	248	31	is	be	AUX
ejpam-5252	248	32	a	a	DET
ejpam-5252	248	33	γmrfunction	γmrfunction	NOUN
ejpam-5252	248	34	of	of	ADP
ejpam-5252	248	35	g	g	NOUN
ejpam-5252	248	36	,	,	PUNCT
ejpam-5252	248	37	|w2|	|w2|	NOUN
ejpam-5252	248	38	=	=	SYM
ejpam-5252	248	39	{	{	PUNCT
ejpam-5252	248	40	u	u	NOUN
ejpam-5252	248	41	}	}	PUNCT
ejpam-5252	248	42	=	=	SYM
ejpam-5252	248	43	1	1	NUM
ejpam-5252	248	44	and	and	CCONJ
ejpam-5252	248	45	|w1|	|w1|	NOUN
ejpam-5252	248	46	=	=	SYM
ejpam-5252	248	47	v	v	NOUN
ejpam-5252	248	48	(	(	PUNCT
ejpam-5252	248	49	g)\{u	g)\{u	PROPN
ejpam-5252	248	50	}	}	PUNCT
ejpam-5252	248	51	.	.	PUNCT
ejpam-5252	249	1	hence	hence	ADV
ejpam-5252	249	2	,	,	PUNCT
ejpam-5252	249	3	γmr(g	γmr(g	PROPN
ejpam-5252	249	4	)	)	PUNCT
ejpam-5252	249	5	=	=	SYM
ejpam-5252	249	6	ωmr	ωmr	NOUN
ejpam-5252	249	7	g	g	PROPN
ejpam-5252	249	8	(	(	PUNCT
ejpam-5252	249	9	g	g	NOUN
ejpam-5252	249	10	)	)	PUNCT
ejpam-5252	249	11	≥	≥	NOUN
ejpam-5252	249	12	2n+3	2n+3	NUM
ejpam-5252	249	13	.	.	PUNCT
ejpam-5252	250	1	if	if	SCONJ
ejpam-5252	250	2	|w0|	|w0|	VERB
ejpam-5252	250	3	̸=	̸=	PROPN
ejpam-5252	250	4	0	0	NUM
ejpam-5252	250	5	,	,	PUNCT
ejpam-5252	250	6	then	then	ADV
ejpam-5252	250	7	|w2|	|w2|	NOUN
ejpam-5252	250	8	≥	≥	NUM
ejpam-5252	250	9	2	2	NUM
ejpam-5252	250	10	and	and	CCONJ
ejpam-5252	250	11	|w3|	|w3|	PRON
ejpam-5252	250	12	≥	≥	NOUN
ejpam-5252	250	13	1	1	NUM
ejpam-5252	250	14	.	.	PUNCT
ejpam-5252	251	1	it	it	PRON
ejpam-5252	251	2	follows	follow	VERB
ejpam-5252	251	3	that	that	PRON
ejpam-5252	251	4	γmr(g	γmr(g	PUNCT
ejpam-5252	251	5	)	)	PUNCT
ejpam-5252	252	1	=	=	SYM
ejpam-5252	252	2	ωmr	ωmr	NOUN
ejpam-5252	252	3	g	g	PROPN
ejpam-5252	252	4	(	(	PUNCT
ejpam-5252	252	5	g	g	NOUN
ejpam-5252	252	6	)	)	PUNCT
ejpam-5252	252	7	=	=	SYM
ejpam-5252	253	1	2|w2|+	2|w2|+	NUM
ejpam-5252	253	2	3|w3|	3|w3|	NUM
ejpam-5252	253	3	≥	≥	NOUN
ejpam-5252	253	4	2n+	2n+	NUM
ejpam-5252	253	5	3	3	NUM
ejpam-5252	253	6	.	.	PUNCT
ejpam-5252	254	1	therefore	therefore	ADV
ejpam-5252	254	2	,	,	PUNCT
ejpam-5252	254	3	γmr(g	γmr(g	PROPN
ejpam-5252	254	4	)	)	PUNCT
ejpam-5252	255	1	=	=	SYM
ejpam-5252	256	1	2n+	2n+	NUM
ejpam-5252	256	2	3	3	X
ejpam-5252	256	3	.	.	PUNCT
ejpam-5252	256	4	proposition	proposition	NOUN
ejpam-5252	256	5	10	10	NUM
ejpam-5252	256	6	.	.	PUNCT
ejpam-5252	257	1	for	for	ADP
ejpam-5252	257	2	any	any	DET
ejpam-5252	257	3	friendship	friendship	NOUN
ejpam-5252	257	4	graph	graph	NOUN
ejpam-5252	257	5	g	g	PROPN
ejpam-5252	257	6	,	,	PUNCT
ejpam-5252	257	7	γmr(g	γmr(g	PROPN
ejpam-5252	257	8	)	)	PUNCT
ejpam-5252	257	9	=	=	SYM
ejpam-5252	257	10	2n+	2n+	NUM
ejpam-5252	257	11	2	2	NUM
ejpam-5252	257	12	.	.	PUNCT
ejpam-5252	257	13	proof	proof	NOUN
ejpam-5252	257	14	.	.	PUNCT
ejpam-5252	258	1	let	let	VERB
ejpam-5252	258	2	g	g	NOUN
ejpam-5252	258	3	=	=	PROPN
ejpam-5252	258	4	k1	k1	PROPN
ejpam-5252	258	5	+	+	CCONJ
ejpam-5252	258	6	nk2	nk2	NOUN
ejpam-5252	258	7	,	,	PUNCT
ejpam-5252	258	8	n	n	PRON
ejpam-5252	258	9	≥	≥	NOUN
ejpam-5252	258	10	2	2	NUM
ejpam-5252	258	11	.	.	PUNCT
ejpam-5252	259	1	let	let	VERB
ejpam-5252	259	2	v	v	NOUN
ejpam-5252	259	3	(	(	PUNCT
ejpam-5252	259	4	k1	k1	NOUN
ejpam-5252	259	5	)	)	PUNCT
ejpam-5252	259	6	=	=	SYM
ejpam-5252	259	7	{	{	PUNCT
ejpam-5252	259	8	u	u	NOUN
ejpam-5252	259	9	}	}	PUNCT
ejpam-5252	259	10	be	be	AUX
ejpam-5252	259	11	the	the	DET
ejpam-5252	259	12	central	central	ADJ
ejpam-5252	259	13	vertex	vertex	NOUN
ejpam-5252	259	14	in	in	ADP
ejpam-5252	259	15	g.	g.	PROPN
ejpam-5252	259	16	define	define	VERB
ejpam-5252	259	17	a	a	DET
ejpam-5252	259	18	function	function	NOUN
ejpam-5252	259	19	f	f	NOUN
ejpam-5252	259	20	=	=	SYM
ejpam-5252	259	21	(	(	PUNCT
ejpam-5252	259	22	∅	∅	NOUN
ejpam-5252	259	23	,	,	PUNCT
ejpam-5252	259	24	v	v	NOUN
ejpam-5252	259	25	(	(	PUNCT
ejpam-5252	259	26	g)\{u	g)\{u	PROPN
ejpam-5252	259	27	}	}	PUNCT
ejpam-5252	259	28	,	,	PUNCT
ejpam-5252	259	29	{	{	PUNCT
ejpam-5252	259	30	u},∅	u},∅	PROPN
ejpam-5252	259	31	)	)	PUNCT
ejpam-5252	259	32	.	.	PUNCT
ejpam-5252	260	1	then	then	ADV
ejpam-5252	260	2	for	for	ADP
ejpam-5252	260	3	all	all	DET
ejpam-5252	260	4	vi	vi	PROPN
ejpam-5252	260	5	∈	∈	PROPN
ejpam-5252	260	6	v1	v1	NOUN
ejpam-5252	260	7	,	,	PUNCT
ejpam-5252	260	8	1	1	NUM
ejpam-5252	260	9	≤	≤	NUM
ejpam-5252	260	10	i	i	PRON
ejpam-5252	260	11	≤	≤	PROPN
ejpam-5252	260	12	n	n	CCONJ
ejpam-5252	260	13	,	,	PUNCT
ejpam-5252	260	14	f(ng[vi	f(ng[vi	PROPN
ejpam-5252	260	15	]	]	X
ejpam-5252	260	16	)	)	PUNCT
ejpam-5252	260	17	=	=	SYM
ejpam-5252	260	18	2n	2n	NUM
ejpam-5252	260	19	+	+	CCONJ
ejpam-5252	260	20	2	2	X
ejpam-5252	260	21	.	.	PUNCT
ejpam-5252	260	22	thus	thus	ADV
ejpam-5252	260	23	,	,	PUNCT
ejpam-5252	260	24	f	f	PROPN
ejpam-5252	260	25	∈	∈	PROPN
ejpam-5252	260	26	mrdf	mrdf	NOUN
ejpam-5252	260	27	(	(	PUNCT
ejpam-5252	260	28	g	g	NOUN
ejpam-5252	260	29	)	)	PUNCT
ejpam-5252	260	30	.	.	PUNCT
ejpam-5252	261	1	it	it	PRON
ejpam-5252	261	2	follows	follow	VERB
ejpam-5252	261	3	that	that	PRON
ejpam-5252	261	4	γmr(g	γmr(g	PROPN
ejpam-5252	261	5	)	)	PUNCT
ejpam-5252	261	6	≤	≤	NOUN
ejpam-5252	261	7	2n	2n	NUM
ejpam-5252	262	1	+	+	CCONJ
ejpam-5252	262	2	2	2	X
ejpam-5252	262	3	.	.	PUNCT
ejpam-5252	262	4	now	now	ADV
ejpam-5252	262	5	,	,	PUNCT
ejpam-5252	262	6	suppose	suppose	VERB
ejpam-5252	262	7	that	that	SCONJ
ejpam-5252	262	8	f	f	PROPN
ejpam-5252	263	1	′	′	NUM
ejpam-5252	263	2	=	=	SYM
ejpam-5252	263	3	(	(	PUNCT
ejpam-5252	263	4	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	263	5	)	)	PUNCT
ejpam-5252	263	6	is	be	AUX
ejpam-5252	263	7	a	a	DET
ejpam-5252	263	8	γmr	γmr	ADJ
ejpam-5252	263	9	-	-	PUNCT
ejpam-5252	263	10	function	function	NOUN
ejpam-5252	263	11	of	of	ADP
ejpam-5252	263	12	g.	g.	PROPN
ejpam-5252	264	1	if	if	SCONJ
ejpam-5252	264	2	w0	w0	PROPN
ejpam-5252	264	3	=	=	SYM
ejpam-5252	264	4	∅	∅	NOUN
ejpam-5252	264	5	,	,	PUNCT
ejpam-5252	264	6	then	then	ADV
ejpam-5252	264	7	w3	w3	PROPN
ejpam-5252	264	8	=	=	PUNCT
ejpam-5252	264	9	∅.	∅.	NOUN
ejpam-5252	264	10	since	since	SCONJ
ejpam-5252	264	11	f	f	PROPN
ejpam-5252	264	12	′	′	NUM
ejpam-5252	264	13	is	be	AUX
ejpam-5252	264	14	a	a	DET
ejpam-5252	264	15	γmr	γmr	ADJ
ejpam-5252	264	16	-	-	PUNCT
ejpam-5252	264	17	function	function	NOUN
ejpam-5252	264	18	of	of	ADP
ejpam-5252	264	19	g	g	NOUN
ejpam-5252	264	20	,	,	PUNCT
ejpam-5252	264	21	by	by	ADP
ejpam-5252	264	22	proposition	proposition	NOUN
ejpam-5252	264	23	4	4	NUM
ejpam-5252	264	24	(	(	PUNCT
ejpam-5252	264	25	i	i	NOUN
ejpam-5252	264	26	)	)	PUNCT
ejpam-5252	264	27	,	,	PUNCT
ejpam-5252	264	28	γmr(g	γmr(g	PROPN
ejpam-5252	264	29	)	)	PUNCT
ejpam-5252	265	1	=	=	SYM
ejpam-5252	266	1	2n+	2n+	NUM
ejpam-5252	266	2	2	2	NUM
ejpam-5252	266	3	.	.	PUNCT
ejpam-5252	266	4	corollary	corollary	ADJ
ejpam-5252	266	5	3	3	NUM
ejpam-5252	266	6	.	.	PUNCT
ejpam-5252	267	1	for	for	ADP
ejpam-5252	267	2	a	a	DET
ejpam-5252	267	3	butterfly	butterfly	NOUN
ejpam-5252	267	4	graph	graph	NOUN
ejpam-5252	267	5	g	g	PROPN
ejpam-5252	267	6	,	,	PUNCT
ejpam-5252	267	7	γmr(g	γmr(g	PROPN
ejpam-5252	267	8	)	)	PUNCT
ejpam-5252	267	9	=	=	SYM
ejpam-5252	267	10	6	6	X
ejpam-5252	267	11	.	.	PUNCT
ejpam-5252	267	12	s.	s.	PROPN
ejpam-5252	267	13	ahamad	ahamad	PROPN
ejpam-5252	267	14	,	,	PUNCT
ejpam-5252	267	15	j.	j.	PROPN
ejpam-5252	267	16	cariaga	cariaga	PROPN
ejpam-5252	267	17	,	,	PUNCT
ejpam-5252	267	18	s.	s.	PROPN
ejpam-5252	267	19	menchavez	menchavez	PROPN
ejpam-5252	267	20	/	/	PUNCT
ejpam-5252	267	21	eur	eur	PROPN
ejpam-5252	267	22	.	.	PUNCT
ejpam-5252	268	1	j.	j.	PROPN
ejpam-5252	268	2	pure	pure	PROPN
ejpam-5252	268	3	appl	appl	PROPN
ejpam-5252	268	4	.	.	PROPN
ejpam-5252	268	5	math	math	PROPN
ejpam-5252	268	6	,	,	PUNCT
ejpam-5252	268	7	18	18	NUM
ejpam-5252	268	8	(	(	PUNCT
ejpam-5252	268	9	1	1	NUM
ejpam-5252	268	10	)	)	PUNCT
ejpam-5252	268	11	(	(	PUNCT
ejpam-5252	268	12	2025	2025	NUM
ejpam-5252	268	13	)	)	PUNCT
ejpam-5252	268	14	,	,	PUNCT
ejpam-5252	268	15	5252	5252	NUM
ejpam-5252	268	16	10	10	NUM
ejpam-5252	268	17	of	of	ADP
ejpam-5252	268	18	18	18	NUM
ejpam-5252	268	19	proof	proof	NOUN
ejpam-5252	268	20	.	.	PUNCT
ejpam-5252	269	1	the	the	DET
ejpam-5252	269	2	result	result	NOUN
ejpam-5252	269	3	follows	follow	VERB
ejpam-5252	269	4	from	from	ADP
ejpam-5252	269	5	proposition	proposition	NOUN
ejpam-5252	269	6	10	10	NUM
ejpam-5252	269	7	.	.	PUNCT
ejpam-5252	270	1	a	a	DET
ejpam-5252	270	2	graph	graph	NOUN
ejpam-5252	270	3	g	g	NOUN
ejpam-5252	270	4	is	be	AUX
ejpam-5252	270	5	called	call	VERB
ejpam-5252	270	6	bipartite	bipartite	ADJ
ejpam-5252	270	7	if	if	SCONJ
ejpam-5252	270	8	the	the	DET
ejpam-5252	270	9	vertex	vertex	NOUN
ejpam-5252	270	10	set	set	VERB
ejpam-5252	270	11	v	v	NOUN
ejpam-5252	270	12	(	(	PUNCT
ejpam-5252	270	13	g	g	NOUN
ejpam-5252	270	14	)	)	PUNCT
ejpam-5252	270	15	of	of	ADP
ejpam-5252	270	16	g	g	NOUN
ejpam-5252	270	17	can	can	AUX
ejpam-5252	270	18	be	be	AUX
ejpam-5252	270	19	partitioned	partition	VERB
ejpam-5252	270	20	into	into	ADP
ejpam-5252	270	21	two	two	NUM
ejpam-5252	270	22	subsets	subset	NOUN
ejpam-5252	270	23	v1	v1	NOUN
ejpam-5252	270	24	and	and	CCONJ
ejpam-5252	270	25	v2	v2	VERB
ejpam-5252	270	26	such	such	ADJ
ejpam-5252	270	27	that	that	SCONJ
ejpam-5252	270	28	every	every	DET
ejpam-5252	270	29	edge	edge	NOUN
ejpam-5252	270	30	in	in	ADP
ejpam-5252	270	31	g	g	PROPN
ejpam-5252	270	32	joins	join	VERB
ejpam-5252	270	33	a	a	DET
ejpam-5252	270	34	vertex	vertex	NOUN
ejpam-5252	270	35	in	in	ADP
ejpam-5252	270	36	v1	v1	NOUN
ejpam-5252	270	37	with	with	ADP
ejpam-5252	270	38	a	a	DET
ejpam-5252	270	39	vertex	vertex	NOUN
ejpam-5252	270	40	in	in	ADP
ejpam-5252	270	41	v2	v2	NOUN
ejpam-5252	270	42	.	.	PUNCT
ejpam-5252	271	1	if	if	SCONJ
ejpam-5252	271	2	g	g	PROPN
ejpam-5252	271	3	is	be	AUX
ejpam-5252	271	4	bipartite	bipartite	ADJ
ejpam-5252	271	5	such	such	ADJ
ejpam-5252	271	6	that	that	SCONJ
ejpam-5252	271	7	g	g	PROPN
ejpam-5252	271	8	contains	contain	VERB
ejpam-5252	271	9	every	every	DET
ejpam-5252	271	10	edge	edge	NOUN
ejpam-5252	271	11	incident	incident	NOUN
ejpam-5252	271	12	with	with	ADP
ejpam-5252	271	13	any	any	DET
ejpam-5252	271	14	pair	pair	NOUN
ejpam-5252	271	15	of	of	ADP
ejpam-5252	271	16	vertices	vertex	NOUN
ejpam-5252	271	17	in	in	ADP
ejpam-5252	271	18	v1	v1	NOUN
ejpam-5252	271	19	and	and	CCONJ
ejpam-5252	271	20	v2	v2	NOUN
ejpam-5252	271	21	,	,	PUNCT
ejpam-5252	271	22	then	then	ADV
ejpam-5252	271	23	g	g	PROPN
ejpam-5252	271	24	is	be	AUX
ejpam-5252	271	25	a	a	DET
ejpam-5252	271	26	complete	complete	ADJ
ejpam-5252	271	27	bipartite	bipartite	NOUN
ejpam-5252	271	28	graph	graph	NOUN
ejpam-5252	271	29	;	;	PUNCT
ejpam-5252	271	30	in	in	ADP
ejpam-5252	271	31	this	this	DET
ejpam-5252	271	32	case	case	NOUN
ejpam-5252	271	33	,	,	PUNCT
ejpam-5252	271	34	g	g	PROPN
ejpam-5252	271	35	=	=	SYM
ejpam-5252	271	36	km	km	PROPN
ejpam-5252	271	37	,	,	PUNCT
ejpam-5252	271	38	n	n	CCONJ
ejpam-5252	271	39	if	if	SCONJ
ejpam-5252	271	40	|v1|	|v1|	NUM
ejpam-5252	271	41	=	=	SYM
ejpam-5252	271	42	m	m	NOUN
ejpam-5252	271	43	and	and	CCONJ
ejpam-5252	271	44	|v2|	|v2|	ADV
ejpam-5252	271	45	=	=	PUNCT
ejpam-5252	271	46	n.	n.	NOUN
ejpam-5252	271	47	figure	figure	NOUN
ejpam-5252	271	48	6	6	NUM
ejpam-5252	271	49	shows	show	VERB
ejpam-5252	271	50	the	the	DET
ejpam-5252	271	51	complete	complete	ADJ
ejpam-5252	271	52	bipartite	bipartite	PROPN
ejpam-5252	271	53	graph	graph	NOUN
ejpam-5252	271	54	k7,5	k7,5	PROPN
ejpam-5252	271	55	.	.	PUNCT
ejpam-5252	271	56	....................................	....................................	PUNCT
ejpam-5252	272	1	....................................	....................................	PUNCT
ejpam-5252	272	2	....................................	....................................	PUNCT
ejpam-5252	273	1	....................................	....................................	PUNCT
ejpam-5252	273	2	....................................	....................................	PUNCT
ejpam-5252	274	1	....................................	....................................	PUNCT
ejpam-5252	274	2	....................................	....................................	PUNCT
ejpam-5252	275	1	....................................	....................................	PUNCT
ejpam-5252	275	2	....................................	....................................	PUNCT
ejpam-5252	276	1	....................................	....................................	PUNCT
ejpam-5252	276	2	....................................	....................................	PUNCT
ejpam-5252	277	1	....................................	....................................	PUNCT
ejpam-5252	277	2	.........................................................................................................................................................................................................................	.........................................................................................................................................................................................................................	PUNCT
ejpam-5252	278	1	...............................................................................................................................................................................................................................................	...............................................................................................................................................................................................................................................	PUNCT
ejpam-5252	278	2	..............................................................................................................................................................................................................................................................................	..............................................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	278	3	....................................................................................................................................................................................................................................................................................................................	....................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	279	1	...........................................................................................................................................................................................................................................................................................................................................................................	...........................................................................................................................................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	279	2	........	........	PUNCT
ejpam-5252	279	3	........	........	PUNCT
ejpam-5252	279	4	........	........	PUNCT
ejpam-5252	279	5	........	........	PUNCT
ejpam-5252	279	6	........	........	PUNCT
ejpam-5252	279	7	........	........	PUNCT
ejpam-5252	279	8	........	........	PUNCT
ejpam-5252	279	9	........	........	PUNCT
ejpam-5252	279	10	........	........	PUNCT
ejpam-5252	279	11	........	........	PUNCT
ejpam-5252	279	12	........	........	PUNCT
ejpam-5252	279	13	........	........	PUNCT
ejpam-5252	279	14	........	........	PUNCT
ejpam-5252	279	15	........	........	PUNCT
ejpam-5252	279	16	........	........	PUNCT
ejpam-5252	279	17	........	........	PUNCT
ejpam-5252	279	18	........	........	PUNCT
ejpam-5252	279	19	........	........	PUNCT
ejpam-5252	279	20	........	........	PUNCT
ejpam-5252	279	21	........	........	PUNCT
ejpam-5252	279	22	........	........	PUNCT
ejpam-5252	279	23	........	........	PUNCT
ejpam-5252	279	24	........	........	PUNCT
ejpam-5252	279	25	........	........	PUNCT
ejpam-5252	280	1	........	........	PUNCT
ejpam-5252	280	2	..........................................................................................................................................................................................................................	..........................................................................................................................................................................................................................	PUNCT
ejpam-5252	281	1	...............................................................................................................................................................................................................................................	...............................................................................................................................................................................................................................................	PUNCT
ejpam-5252	281	2	..............................................................................................................................................................................................................................................................................	..............................................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	282	1	.............................................................................................................................................................................................................................................................................................................................	.............................................................................................................................................................................................................................................................................................................................	PROPN
ejpam-5252	282	2	........	........	PUNCT
ejpam-5252	282	3	........	........	PUNCT
ejpam-5252	282	4	........	........	PUNCT
ejpam-5252	282	5	........	........	PUNCT
ejpam-5252	282	6	........	........	PUNCT
ejpam-5252	282	7	........	........	PUNCT
ejpam-5252	282	8	........	........	PUNCT
ejpam-5252	282	9	........	........	PUNCT
ejpam-5252	282	10	........	........	PUNCT
ejpam-5252	282	11	........	........	PUNCT
ejpam-5252	282	12	........	........	PUNCT
ejpam-5252	282	13	........	........	PUNCT
ejpam-5252	282	14	........	........	PUNCT
ejpam-5252	282	15	........	........	PUNCT
ejpam-5252	282	16	........	........	PUNCT
ejpam-5252	282	17	........	........	PUNCT
ejpam-5252	282	18	........	........	PUNCT
ejpam-5252	282	19	........	........	PUNCT
ejpam-5252	282	20	........	........	PUNCT
ejpam-5252	282	21	........	........	PUNCT
ejpam-5252	282	22	........	........	PUNCT
ejpam-5252	282	23	........	........	PUNCT
ejpam-5252	282	24	........	........	PUNCT
ejpam-5252	282	25	........	........	PUNCT
ejpam-5252	282	26	........	........	PUNCT
ejpam-5252	282	27	........	........	PUNCT
ejpam-5252	282	28	.........	.........	PUNCT
ejpam-5252	283	1	........	........	PUNCT
ejpam-5252	283	2	........	........	PUNCT
ejpam-5252	283	3	........	........	PUNCT
ejpam-5252	283	4	........	........	PUNCT
ejpam-5252	283	5	........	........	PUNCT
ejpam-5252	283	6	........	........	PUNCT
ejpam-5252	283	7	........	........	PUNCT
ejpam-5252	283	8	........	........	PUNCT
ejpam-5252	283	9	........	........	PUNCT
ejpam-5252	283	10	........	........	PUNCT
ejpam-5252	283	11	........	........	PUNCT
ejpam-5252	283	12	........	........	PUNCT
ejpam-5252	283	13	........	........	PUNCT
ejpam-5252	283	14	........	........	PUNCT
ejpam-5252	283	15	........	........	PUNCT
ejpam-5252	283	16	........	........	PUNCT
ejpam-5252	283	17	........	........	PUNCT
ejpam-5252	283	18	........	........	PUNCT
ejpam-5252	283	19	........	........	PUNCT
ejpam-5252	283	20	........	........	PUNCT
ejpam-5252	283	21	........	........	PUNCT
ejpam-5252	283	22	........	........	PUNCT
ejpam-5252	283	23	........	........	PUNCT
ejpam-5252	283	24	........	........	PUNCT
ejpam-5252	284	1	........	........	PUNCT
ejpam-5252	284	2	..........................................................................................................................................................................................................................	..........................................................................................................................................................................................................................	PUNCT
ejpam-5252	285	1	...............................................................................................................................................................................................................................................	...............................................................................................................................................................................................................................................	PUNCT
ejpam-5252	285	2	........................................................................................................................................................................................................................................................................................	........................................................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	286	1	.........	.........	PUNCT
ejpam-5252	287	1	.........	.........	PUNCT
ejpam-5252	287	2	.........	.........	PUNCT
ejpam-5252	288	1	.........	.........	PUNCT
ejpam-5252	288	2	.........	.........	PUNCT
ejpam-5252	289	1	.........	.........	PUNCT
ejpam-5252	289	2	.........	.........	PUNCT
ejpam-5252	290	1	.........	.........	PUNCT
ejpam-5252	290	2	.........	.........	PUNCT
ejpam-5252	291	1	.........	.........	PUNCT
ejpam-5252	291	2	.........	.........	PUNCT
ejpam-5252	292	1	.........	.........	PUNCT
ejpam-5252	292	2	.........	.........	PUNCT
ejpam-5252	293	1	.........	.........	PUNCT
ejpam-5252	293	2	.........	.........	PUNCT
ejpam-5252	294	1	.........	.........	PUNCT
ejpam-5252	294	2	.........	.........	PUNCT
ejpam-5252	295	1	.........	.........	PUNCT
ejpam-5252	295	2	.........	.........	PUNCT
ejpam-5252	296	1	.........	.........	PUNCT
ejpam-5252	296	2	.........	.........	PUNCT
ejpam-5252	297	1	.........	.........	PUNCT
ejpam-5252	297	2	.........	.........	PUNCT
ejpam-5252	298	1	.........	.........	PUNCT
ejpam-5252	298	2	.........	.........	PUNCT
ejpam-5252	299	1	....	....	PUNCT
ejpam-5252	299	2	.........	.........	PUNCT
ejpam-5252	299	3	........	........	PUNCT
ejpam-5252	299	4	........	........	PUNCT
ejpam-5252	299	5	........	........	PUNCT
ejpam-5252	299	6	........	........	PUNCT
ejpam-5252	299	7	........	........	PUNCT
ejpam-5252	299	8	........	........	PUNCT
ejpam-5252	299	9	........	........	PUNCT
ejpam-5252	299	10	........	........	PUNCT
ejpam-5252	299	11	........	........	PUNCT
ejpam-5252	299	12	........	........	PUNCT
ejpam-5252	299	13	........	........	PUNCT
ejpam-5252	299	14	........	........	PUNCT
ejpam-5252	299	15	........	........	PUNCT
ejpam-5252	299	16	........	........	PUNCT
ejpam-5252	299	17	........	........	PUNCT
ejpam-5252	299	18	........	........	PUNCT
ejpam-5252	299	19	........	........	PUNCT
ejpam-5252	299	20	........	........	PUNCT
ejpam-5252	299	21	........	........	PUNCT
ejpam-5252	299	22	........	........	PUNCT
ejpam-5252	299	23	........	........	PUNCT
ejpam-5252	299	24	........	........	PUNCT
ejpam-5252	299	25	........	........	PUNCT
ejpam-5252	299	26	........	........	PUNCT
ejpam-5252	299	27	........	........	PUNCT
ejpam-5252	299	28	........	........	PUNCT
ejpam-5252	299	29	.........	.........	PUNCT
ejpam-5252	299	30	........	........	PUNCT
ejpam-5252	299	31	........	........	PUNCT
ejpam-5252	299	32	........	........	PUNCT
ejpam-5252	299	33	........	........	PUNCT
ejpam-5252	299	34	........	........	PUNCT
ejpam-5252	299	35	........	........	PUNCT
ejpam-5252	299	36	........	........	PUNCT
ejpam-5252	299	37	........	........	PUNCT
ejpam-5252	299	38	........	........	PUNCT
ejpam-5252	299	39	........	........	PUNCT
ejpam-5252	299	40	........	........	PUNCT
ejpam-5252	299	41	........	........	PUNCT
ejpam-5252	299	42	........	........	PUNCT
ejpam-5252	299	43	........	........	PUNCT
ejpam-5252	299	44	........	........	PUNCT
ejpam-5252	299	45	........	........	PUNCT
ejpam-5252	299	46	........	........	PUNCT
ejpam-5252	299	47	........	........	PUNCT
ejpam-5252	299	48	........	........	PUNCT
ejpam-5252	299	49	........	........	PUNCT
ejpam-5252	299	50	........	........	PUNCT
ejpam-5252	299	51	........	........	PUNCT
ejpam-5252	299	52	........	........	PUNCT
ejpam-5252	299	53	........	........	PUNCT
ejpam-5252	299	54	........	........	PUNCT
ejpam-5252	300	1	..........................................................................................................................................................................................................................	..........................................................................................................................................................................................................................	PUNCT
ejpam-5252	300	2	..........................................................................................................................................................................................................................................................	..........................................................................................................................................................................................................................................................	PUNCT
ejpam-5252	301	1	..........	..........	PUNCT
ejpam-5252	301	2	..........	..........	PUNCT
ejpam-5252	302	1	..........	..........	PUNCT
ejpam-5252	302	2	..........	..........	PUNCT
ejpam-5252	303	1	..........	..........	PUNCT
ejpam-5252	303	2	..........	..........	PUNCT
ejpam-5252	304	1	..........	..........	PUNCT
ejpam-5252	304	2	..........	..........	PUNCT
ejpam-5252	305	1	..........	..........	PUNCT
ejpam-5252	305	2	..........	..........	PUNCT
ejpam-5252	306	1	..........	..........	PUNCT
ejpam-5252	306	2	..........	..........	PUNCT
ejpam-5252	307	1	..........	..........	PUNCT
ejpam-5252	307	2	..........	..........	PUNCT
ejpam-5252	308	1	..........	..........	PUNCT
ejpam-5252	308	2	..........	..........	PUNCT
ejpam-5252	309	1	..........	..........	PUNCT
ejpam-5252	309	2	..........	..........	PUNCT
ejpam-5252	310	1	..........	..........	PUNCT
ejpam-5252	310	2	..........	..........	PUNCT
ejpam-5252	311	1	..........	..........	PUNCT
ejpam-5252	311	2	..........	..........	PUNCT
ejpam-5252	312	1	..........	..........	PUNCT
ejpam-5252	312	2	..........	..........	PUNCT
ejpam-5252	313	1	..........	..........	PUNCT
ejpam-5252	313	2	.........	.........	PUNCT
ejpam-5252	314	1	..........	..........	PUNCT
ejpam-5252	314	2	.........	.........	PUNCT
ejpam-5252	315	1	.........	.........	PUNCT
ejpam-5252	315	2	.........	.........	PUNCT
ejpam-5252	316	1	.........	.........	PUNCT
ejpam-5252	316	2	.........	.........	PUNCT
ejpam-5252	317	1	.........	.........	PUNCT
ejpam-5252	317	2	.........	.........	PUNCT
ejpam-5252	318	1	.........	.........	PUNCT
ejpam-5252	318	2	.........	.........	PUNCT
ejpam-5252	319	1	.........	.........	PUNCT
ejpam-5252	319	2	.........	.........	PUNCT
ejpam-5252	320	1	.........	.........	PUNCT
ejpam-5252	320	2	.........	.........	PUNCT
ejpam-5252	321	1	.........	.........	PUNCT
ejpam-5252	321	2	.........	.........	PUNCT
ejpam-5252	322	1	.........	.........	PUNCT
ejpam-5252	322	2	.........	.........	PUNCT
ejpam-5252	323	1	.........	.........	PUNCT
ejpam-5252	323	2	.........	.........	PUNCT
ejpam-5252	324	1	.........	.........	PUNCT
ejpam-5252	324	2	.........	.........	PUNCT
ejpam-5252	325	1	.........	.........	PUNCT
ejpam-5252	325	2	.........	.........	PUNCT
ejpam-5252	326	1	.........	.........	PUNCT
ejpam-5252	326	2	.........	.........	PUNCT
ejpam-5252	327	1	....	....	PUNCT
ejpam-5252	327	2	.........	.........	PUNCT
ejpam-5252	327	3	........	........	PUNCT
ejpam-5252	327	4	........	........	PUNCT
ejpam-5252	327	5	........	........	PUNCT
ejpam-5252	327	6	........	........	PUNCT
ejpam-5252	327	7	........	........	PUNCT
ejpam-5252	327	8	........	........	PUNCT
ejpam-5252	327	9	........	........	PUNCT
ejpam-5252	327	10	........	........	PUNCT
ejpam-5252	327	11	........	........	PUNCT
ejpam-5252	327	12	........	........	PUNCT
ejpam-5252	327	13	........	........	PUNCT
ejpam-5252	327	14	........	........	PUNCT
ejpam-5252	327	15	........	........	PUNCT
ejpam-5252	327	16	........	........	PUNCT
ejpam-5252	327	17	........	........	PUNCT
ejpam-5252	327	18	........	........	PUNCT
ejpam-5252	327	19	........	........	PUNCT
ejpam-5252	327	20	........	........	PUNCT
ejpam-5252	327	21	........	........	PUNCT
ejpam-5252	327	22	........	........	PUNCT
ejpam-5252	327	23	........	........	PUNCT
ejpam-5252	327	24	........	........	PUNCT
ejpam-5252	327	25	........	........	PUNCT
ejpam-5252	327	26	........	........	PUNCT
ejpam-5252	327	27	........	........	PUNCT
ejpam-5252	327	28	........	........	PUNCT
ejpam-5252	327	29	.........	.........	PUNCT
ejpam-5252	327	30	........	........	PUNCT
ejpam-5252	327	31	........	........	PUNCT
ejpam-5252	327	32	........	........	PUNCT
ejpam-5252	327	33	........	........	PUNCT
ejpam-5252	327	34	........	........	PUNCT
ejpam-5252	327	35	........	........	PUNCT
ejpam-5252	327	36	........	........	PUNCT
ejpam-5252	327	37	........	........	PUNCT
ejpam-5252	327	38	........	........	PUNCT
ejpam-5252	327	39	........	........	PUNCT
ejpam-5252	327	40	........	........	PUNCT
ejpam-5252	327	41	........	........	PUNCT
ejpam-5252	327	42	........	........	PUNCT
ejpam-5252	327	43	........	........	PUNCT
ejpam-5252	327	44	........	........	PUNCT
ejpam-5252	327	45	........	........	PUNCT
ejpam-5252	327	46	........	........	PUNCT
ejpam-5252	327	47	........	........	PUNCT
ejpam-5252	327	48	........	........	PUNCT
ejpam-5252	327	49	........	........	PUNCT
ejpam-5252	327	50	........	........	PUNCT
ejpam-5252	327	51	........	........	PUNCT
ejpam-5252	327	52	........	........	PUNCT
ejpam-5252	327	53	........	........	PUNCT
ejpam-5252	327	54	........	........	PUNCT
ejpam-5252	328	1	.......................................................................................................................................................................................................................................	.......................................................................................................................................................................................................................................	PUNCT
ejpam-5252	328	2	............	............	PUNCT
ejpam-5252	328	3	............	............	PUNCT
ejpam-5252	328	4	............	............	PUNCT
ejpam-5252	328	5	............	............	PUNCT
ejpam-5252	328	6	............	............	PUNCT
ejpam-5252	328	7	............	............	PUNCT
ejpam-5252	328	8	............	............	PUNCT
ejpam-5252	328	9	............	............	PUNCT
ejpam-5252	328	10	............	............	PUNCT
ejpam-5252	328	11	............	............	PUNCT
ejpam-5252	328	12	............	............	PUNCT
ejpam-5252	328	13	............	............	PUNCT
ejpam-5252	328	14	............	............	PUNCT
ejpam-5252	328	15	............	............	PUNCT
ejpam-5252	328	16	............	............	PUNCT
ejpam-5252	328	17	............	............	PUNCT
ejpam-5252	328	18	............	............	PUNCT
ejpam-5252	328	19	............	............	PUNCT
ejpam-5252	328	20	............	............	PUNCT
ejpam-5252	328	21	............	............	PUNCT
ejpam-5252	328	22	............	............	PUNCT
ejpam-5252	328	23	............	............	PUNCT
ejpam-5252	328	24	............	............	PUNCT
ejpam-5252	328	25	............	............	PUNCT
ejpam-5252	328	26	.......	.......	PUNCT
ejpam-5252	328	27	...........	...........	PUNCT
ejpam-5252	329	1	..........	..........	PUNCT
ejpam-5252	329	2	..........	..........	PUNCT
ejpam-5252	330	1	..........	..........	PUNCT
ejpam-5252	330	2	..........	..........	PUNCT
ejpam-5252	331	1	..........	..........	PUNCT
ejpam-5252	331	2	..........	..........	PUNCT
ejpam-5252	332	1	..........	..........	PUNCT
ejpam-5252	332	2	..........	..........	PUNCT
ejpam-5252	333	1	..........	..........	PUNCT
ejpam-5252	333	2	..........	..........	PUNCT
ejpam-5252	334	1	..........	..........	PUNCT
ejpam-5252	334	2	..........	..........	PUNCT
ejpam-5252	335	1	..........	..........	PUNCT
ejpam-5252	335	2	..........	..........	PUNCT
ejpam-5252	336	1	..........	..........	PUNCT
ejpam-5252	336	2	..........	..........	PUNCT
ejpam-5252	337	1	..........	..........	PUNCT
ejpam-5252	337	2	..........	..........	PUNCT
ejpam-5252	338	1	..........	..........	PUNCT
ejpam-5252	338	2	..........	..........	PUNCT
ejpam-5252	339	1	..........	..........	PUNCT
ejpam-5252	339	2	..........	..........	PUNCT
ejpam-5252	340	1	..........	..........	PUNCT
ejpam-5252	340	2	..........	..........	PUNCT
ejpam-5252	341	1	..........	..........	PUNCT
ejpam-5252	341	2	.........	.........	PUNCT
ejpam-5252	342	1	..........	..........	PUNCT
ejpam-5252	342	2	.........	.........	PUNCT
ejpam-5252	343	1	.........	.........	PUNCT
ejpam-5252	343	2	.........	.........	PUNCT
ejpam-5252	344	1	.........	.........	PUNCT
ejpam-5252	344	2	.........	.........	PUNCT
ejpam-5252	345	1	.........	.........	PUNCT
ejpam-5252	345	2	.........	.........	PUNCT
ejpam-5252	346	1	.........	.........	PUNCT
ejpam-5252	346	2	.........	.........	PUNCT
ejpam-5252	347	1	.........	.........	PUNCT
ejpam-5252	347	2	.........	.........	PUNCT
ejpam-5252	348	1	.........	.........	PUNCT
ejpam-5252	348	2	.........	.........	PUNCT
ejpam-5252	349	1	.........	.........	PUNCT
ejpam-5252	349	2	.........	.........	PUNCT
ejpam-5252	350	1	.........	.........	PUNCT
ejpam-5252	350	2	.........	.........	PUNCT
ejpam-5252	351	1	.........	.........	PUNCT
ejpam-5252	351	2	.........	.........	PUNCT
ejpam-5252	352	1	.........	.........	PUNCT
ejpam-5252	352	2	.........	.........	PUNCT
ejpam-5252	353	1	.........	.........	PUNCT
ejpam-5252	353	2	.........	.........	PUNCT
ejpam-5252	354	1	.........	.........	PUNCT
ejpam-5252	354	2	.........	.........	PUNCT
ejpam-5252	355	1	....	....	PUNCT
ejpam-5252	355	2	.........	.........	PUNCT
ejpam-5252	355	3	........	........	PUNCT
ejpam-5252	355	4	........	........	PUNCT
ejpam-5252	355	5	........	........	PUNCT
ejpam-5252	355	6	........	........	PUNCT
ejpam-5252	355	7	........	........	PUNCT
ejpam-5252	355	8	........	........	PUNCT
ejpam-5252	355	9	........	........	PUNCT
ejpam-5252	355	10	........	........	PUNCT
ejpam-5252	355	11	........	........	PUNCT
ejpam-5252	355	12	........	........	PUNCT
ejpam-5252	355	13	........	........	PUNCT
ejpam-5252	355	14	........	........	PUNCT
ejpam-5252	355	15	........	........	PUNCT
ejpam-5252	355	16	........	........	PUNCT
ejpam-5252	355	17	........	........	PUNCT
ejpam-5252	355	18	........	........	PUNCT
ejpam-5252	355	19	........	........	PUNCT
ejpam-5252	355	20	........	........	PUNCT
ejpam-5252	355	21	........	........	PUNCT
ejpam-5252	355	22	........	........	PUNCT
ejpam-5252	355	23	........	........	PUNCT
ejpam-5252	355	24	........	........	PUNCT
ejpam-5252	355	25	........	........	PUNCT
ejpam-5252	355	26	........	........	PUNCT
ejpam-5252	355	27	........	........	PUNCT
ejpam-5252	355	28	........	........	PUNCT
ejpam-5252	355	29	.........	.........	PUNCT
ejpam-5252	355	30	........	........	PUNCT
ejpam-5252	355	31	........	........	PUNCT
ejpam-5252	355	32	........	........	PUNCT
ejpam-5252	355	33	........	........	PUNCT
ejpam-5252	355	34	........	........	PUNCT
ejpam-5252	355	35	........	........	PUNCT
ejpam-5252	355	36	........	........	PUNCT
ejpam-5252	355	37	........	........	PUNCT
ejpam-5252	355	38	........	........	PUNCT
ejpam-5252	355	39	........	........	PUNCT
ejpam-5252	355	40	........	........	PUNCT
ejpam-5252	355	41	........	........	PUNCT
ejpam-5252	355	42	........	........	PUNCT
ejpam-5252	355	43	........	........	PUNCT
ejpam-5252	355	44	........	........	PUNCT
ejpam-5252	355	45	........	........	PUNCT
ejpam-5252	355	46	........	........	PUNCT
ejpam-5252	355	47	........	........	PUNCT
ejpam-5252	355	48	........	........	PUNCT
ejpam-5252	355	49	........	........	PUNCT
ejpam-5252	355	50	........	........	PUNCT
ejpam-5252	355	51	........	........	PUNCT
ejpam-5252	355	52	........	........	PUNCT
ejpam-5252	355	53	........	........	PUNCT
ejpam-5252	355	54	........	........	PUNCT
ejpam-5252	355	55	.	.	PUNCT
ejpam-5252	356	1	..............	..............	PUNCT
ejpam-5252	356	2	.............	.............	PUNCT
ejpam-5252	356	3	.............	.............	PUNCT
ejpam-5252	356	4	.............	.............	PUNCT
ejpam-5252	356	5	.............	.............	PUNCT
ejpam-5252	356	6	.............	.............	PUNCT
ejpam-5252	356	7	.............	.............	PUNCT
ejpam-5252	356	8	.............	.............	PUNCT
ejpam-5252	356	9	.............	.............	PUNCT
ejpam-5252	356	10	.............	.............	PUNCT
ejpam-5252	356	11	.............	.............	PUNCT
ejpam-5252	356	12	.............	.............	PUNCT
ejpam-5252	356	13	.............	.............	PUNCT
ejpam-5252	356	14	.............	.............	PUNCT
ejpam-5252	356	15	.............	.............	PUNCT
ejpam-5252	356	16	.............	.............	PUNCT
ejpam-5252	356	17	.............	.............	PUNCT
ejpam-5252	356	18	.............	.............	PUNCT
ejpam-5252	356	19	.............	.............	PUNCT
ejpam-5252	356	20	.............	.............	PUNCT
ejpam-5252	356	21	.............	.............	PUNCT
ejpam-5252	356	22	.............	.............	PUNCT
ejpam-5252	356	23	.............	.............	PUNCT
ejpam-5252	356	24	.............	.............	PUNCT
ejpam-5252	356	25	.............	.............	PUNCT
ejpam-5252	356	26	.............	.............	PUNCT
ejpam-5252	356	27	.............	.............	PUNCT
ejpam-5252	357	1	..	..	PUNCT
ejpam-5252	357	2	.............	.............	PUNCT
ejpam-5252	357	3	............	............	PUNCT
ejpam-5252	357	4	............	............	PUNCT
ejpam-5252	357	5	............	............	PUNCT
ejpam-5252	357	6	............	............	PUNCT
ejpam-5252	357	7	............	............	PUNCT
ejpam-5252	357	8	............	............	PUNCT
ejpam-5252	357	9	............	............	PUNCT
ejpam-5252	357	10	............	............	PUNCT
ejpam-5252	357	11	............	............	PUNCT
ejpam-5252	357	12	............	............	PUNCT
ejpam-5252	357	13	............	............	PUNCT
ejpam-5252	357	14	............	............	PUNCT
ejpam-5252	357	15	............	............	PUNCT
ejpam-5252	357	16	............	............	PUNCT
ejpam-5252	357	17	............	............	PUNCT
ejpam-5252	357	18	............	............	PUNCT
ejpam-5252	357	19	............	............	PUNCT
ejpam-5252	357	20	............	............	PUNCT
ejpam-5252	357	21	............	............	PUNCT
ejpam-5252	357	22	............	............	PUNCT
ejpam-5252	357	23	............	............	PUNCT
ejpam-5252	357	24	............	............	PUNCT
ejpam-5252	357	25	............	............	PUNCT
ejpam-5252	357	26	............	............	PUNCT
ejpam-5252	357	27	.......	.......	PUNCT
ejpam-5252	357	28	...........	...........	PUNCT
ejpam-5252	357	29	..........	..........	PUNCT
ejpam-5252	358	1	..........	..........	PUNCT
ejpam-5252	358	2	..........	..........	PUNCT
ejpam-5252	359	1	..........	..........	PUNCT
ejpam-5252	359	2	..........	..........	PUNCT
ejpam-5252	360	1	..........	..........	PUNCT
ejpam-5252	360	2	..........	..........	PUNCT
ejpam-5252	361	1	..........	..........	PUNCT
ejpam-5252	361	2	..........	..........	PUNCT
ejpam-5252	362	1	..........	..........	PUNCT
ejpam-5252	362	2	..........	..........	PUNCT
ejpam-5252	363	1	..........	..........	PUNCT
ejpam-5252	363	2	..........	..........	PUNCT
ejpam-5252	364	1	..........	..........	PUNCT
ejpam-5252	364	2	..........	..........	PUNCT
ejpam-5252	365	1	..........	..........	PUNCT
ejpam-5252	365	2	..........	..........	PUNCT
ejpam-5252	366	1	..........	..........	PUNCT
ejpam-5252	366	2	..........	..........	PUNCT
ejpam-5252	367	1	..........	..........	PUNCT
ejpam-5252	367	2	..........	..........	PUNCT
ejpam-5252	368	1	..........	..........	PUNCT
ejpam-5252	368	2	..........	..........	PUNCT
ejpam-5252	369	1	..........	..........	PUNCT
ejpam-5252	369	2	..........	..........	PUNCT
ejpam-5252	370	1	.........	.........	PUNCT
ejpam-5252	370	2	..........	..........	PUNCT
ejpam-5252	371	1	.........	.........	PUNCT
ejpam-5252	371	2	.........	.........	PUNCT
ejpam-5252	372	1	.........	.........	PUNCT
ejpam-5252	372	2	.........	.........	PUNCT
ejpam-5252	373	1	.........	.........	PUNCT
ejpam-5252	373	2	.........	.........	PUNCT
ejpam-5252	374	1	.........	.........	PUNCT
ejpam-5252	374	2	.........	.........	PUNCT
ejpam-5252	375	1	.........	.........	PUNCT
ejpam-5252	375	2	.........	.........	PUNCT
ejpam-5252	376	1	.........	.........	PUNCT
ejpam-5252	376	2	.........	.........	PUNCT
ejpam-5252	377	1	.........	.........	PUNCT
ejpam-5252	377	2	.........	.........	PUNCT
ejpam-5252	378	1	.........	.........	PUNCT
ejpam-5252	378	2	.........	.........	PUNCT
ejpam-5252	379	1	.........	.........	PUNCT
ejpam-5252	379	2	.........	.........	PUNCT
ejpam-5252	380	1	.........	.........	PUNCT
ejpam-5252	380	2	.........	.........	PUNCT
ejpam-5252	381	1	.........	.........	PUNCT
ejpam-5252	381	2	.........	.........	PUNCT
ejpam-5252	382	1	.........	.........	PUNCT
ejpam-5252	382	2	.........	.........	PUNCT
ejpam-5252	383	1	.........	.........	PUNCT
ejpam-5252	383	2	....	....	PUNCT
ejpam-5252	383	3	.........	.........	PUNCT
ejpam-5252	383	4	........	........	PUNCT
ejpam-5252	383	5	........	........	PUNCT
ejpam-5252	383	6	........	........	PUNCT
ejpam-5252	383	7	........	........	PUNCT
ejpam-5252	383	8	........	........	PUNCT
ejpam-5252	383	9	........	........	PUNCT
ejpam-5252	383	10	........	........	PUNCT
ejpam-5252	383	11	........	........	PUNCT
ejpam-5252	383	12	........	........	PUNCT
ejpam-5252	383	13	........	........	PUNCT
ejpam-5252	383	14	........	........	PUNCT
ejpam-5252	383	15	........	........	PUNCT
ejpam-5252	383	16	........	........	PUNCT
ejpam-5252	383	17	........	........	PUNCT
ejpam-5252	383	18	........	........	PUNCT
ejpam-5252	383	19	........	........	PUNCT
ejpam-5252	383	20	........	........	PUNCT
ejpam-5252	383	21	........	........	PUNCT
ejpam-5252	383	22	........	........	PUNCT
ejpam-5252	383	23	........	........	PUNCT
ejpam-5252	383	24	........	........	PUNCT
ejpam-5252	383	25	........	........	PUNCT
ejpam-5252	383	26	........	........	PUNCT
ejpam-5252	383	27	........	........	PUNCT
ejpam-5252	383	28	........	........	PUNCT
ejpam-5252	383	29	........	........	PUNCT
ejpam-5252	384	1	k7,5	k7,5	PROPN
ejpam-5252	384	2	:	:	PUNCT
ejpam-5252	384	3	figure	figure	NOUN
ejpam-5252	384	4	6	6	NUM
ejpam-5252	384	5	:	:	PUNCT
ejpam-5252	384	6	a	a	DET
ejpam-5252	384	7	complete	complete	ADJ
ejpam-5252	384	8	bipartite	bipartite	PROPN
ejpam-5252	384	9	g	g	PROPN
ejpam-5252	384	10	=	=	SYM
ejpam-5252	384	11	k7,5	k7,5	PROPN
ejpam-5252	384	12	proposition	proposition	NOUN
ejpam-5252	384	13	11	11	NUM
ejpam-5252	384	14	.	.	PUNCT
ejpam-5252	385	1	for	for	ADP
ejpam-5252	385	2	a	a	DET
ejpam-5252	385	3	complete	complete	ADJ
ejpam-5252	385	4	bipartite	bipartite	NOUN
ejpam-5252	385	5	graph	graph	NOUN
ejpam-5252	385	6	km	km	PROPN
ejpam-5252	385	7	,	,	PUNCT
ejpam-5252	385	8	n	n	CCONJ
ejpam-5252	385	9	,	,	PUNCT
ejpam-5252	385	10	let	let	VERB
ejpam-5252	385	11	p	p	NOUN
ejpam-5252	385	12	=	=	NOUN
ejpam-5252	385	13	min{m	min{m	PROPN
ejpam-5252	385	14	,	,	PUNCT
ejpam-5252	385	15	n},m	n},m	PROPN
ejpam-5252	385	16	,	,	PUNCT
ejpam-5252	385	17	n	n	PRON
ejpam-5252	385	18	≥	≥	NOUN
ejpam-5252	385	19	2	2	NUM
ejpam-5252	385	20	.	.	PUNCT
ejpam-5252	386	1	then	then	ADV
ejpam-5252	386	2	γmr(km	γmr(km	NUM
ejpam-5252	386	3	,	,	PUNCT
ejpam-5252	386	4	n	n	CCONJ
ejpam-5252	386	5	)	)	PUNCT
ejpam-5252	386	6	=	=	SYM
ejpam-5252	386	7			NOUN
ejpam-5252	386	8	5	5	NUM
ejpam-5252	386	9	,	,	PUNCT
ejpam-5252	386	10	if	if	SCONJ
ejpam-5252	386	11	p	p	NOUN
ejpam-5252	386	12	=	=	NOUN
ejpam-5252	386	13	2	2	NUM
ejpam-5252	386	14	.	.	NOUN
ejpam-5252	386	15	7	7	NUM
ejpam-5252	386	16	,	,	PUNCT
ejpam-5252	386	17	if	if	SCONJ
ejpam-5252	386	18	p	p	NOUN
ejpam-5252	386	19	=	=	NOUN
ejpam-5252	386	20	3	3	NUM
ejpam-5252	386	21	9	9	NUM
ejpam-5252	386	22	,	,	PUNCT
ejpam-5252	386	23	if	if	SCONJ
ejpam-5252	386	24	p	p	NOUN
ejpam-5252	386	25	=	=	NOUN
ejpam-5252	386	26	4	4	NUM
ejpam-5252	386	27	10	10	NUM
ejpam-5252	386	28	,	,	PUNCT
ejpam-5252	386	29	if	if	SCONJ
ejpam-5252	386	30	p	p	PRON
ejpam-5252	386	31	≥	≥	NOUN
ejpam-5252	386	32	5	5	NUM
ejpam-5252	386	33	.	.	PUNCT
ejpam-5252	387	1	proof	proof	NOUN
ejpam-5252	387	2	.	.	PUNCT
ejpam-5252	388	1	let	let	VERB
ejpam-5252	388	2	g	g	PRON
ejpam-5252	388	3	be	be	AUX
ejpam-5252	388	4	a	a	DET
ejpam-5252	388	5	complete	complete	ADJ
ejpam-5252	388	6	bipartite	bipartite	NOUN
ejpam-5252	388	7	graph	graph	NOUN
ejpam-5252	388	8	km	km	PROPN
ejpam-5252	388	9	,	,	PUNCT
ejpam-5252	388	10	n	n	PROPN
ejpam-5252	388	11	and	and	CCONJ
ejpam-5252	388	12	x	x	SYM
ejpam-5252	388	13	and	and	CCONJ
ejpam-5252	388	14	y	y	PROPN
ejpam-5252	388	15	be	be	AUX
ejpam-5252	388	16	partite	partite	ADJ
ejpam-5252	388	17	sets	set	NOUN
ejpam-5252	388	18	of	of	ADP
ejpam-5252	388	19	km	km	PROPN
ejpam-5252	388	20	,	,	PUNCT
ejpam-5252	388	21	n	n	CCONJ
ejpam-5252	388	22	,	,	PUNCT
ejpam-5252	388	23	where	where	SCONJ
ejpam-5252	388	24	|x|	|x|	PROPN
ejpam-5252	388	25	=	=	SYM
ejpam-5252	388	26	m	m	PROPN
ejpam-5252	388	27	and	and	CCONJ
ejpam-5252	388	28	|y	|y	X
ejpam-5252	388	29	|	|	NOUN
ejpam-5252	388	30	=	=	PUNCT
ejpam-5252	388	31	n.	n.	NOUN
ejpam-5252	388	32	let	let	VERB
ejpam-5252	388	33	p	p	NOUN
ejpam-5252	388	34	=	=	NOUN
ejpam-5252	388	35	min{m	min{m	PROPN
ejpam-5252	388	36	,	,	PUNCT
ejpam-5252	388	37	n	n	CCONJ
ejpam-5252	388	38	}	}	PUNCT
ejpam-5252	388	39	.	.	PUNCT
ejpam-5252	389	1	for	for	ADP
ejpam-5252	389	2	p	p	NOUN
ejpam-5252	389	3	=	=	PROPN
ejpam-5252	389	4	2	2	NUM
ejpam-5252	389	5	,	,	PUNCT
ejpam-5252	389	6	since	since	SCONJ
ejpam-5252	389	7	|v5|	|v5|	NOUN
ejpam-5252	389	8	=	=	SYM
ejpam-5252	389	9	5	5	NUM
ejpam-5252	389	10	and	and	CCONJ
ejpam-5252	389	11	γ2(g	γ2(g	NUM
ejpam-5252	389	12	)	)	PUNCT
ejpam-5252	389	13	=	=	SYM
ejpam-5252	389	14	2	2	NUM
ejpam-5252	389	15	,	,	PUNCT
ejpam-5252	389	16	it	it	PRON
ejpam-5252	389	17	follows	follow	VERB
ejpam-5252	389	18	from	from	ADP
ejpam-5252	389	19	5(iv	5(iv	NUM
ejpam-5252	389	20	)	)	PUNCT
ejpam-5252	389	21	that	that	SCONJ
ejpam-5252	389	22	γmr(g	γmr(g	PROPN
ejpam-5252	389	23	)	)	PUNCT
ejpam-5252	389	24	=	=	SYM
ejpam-5252	390	1	5	5	X
ejpam-5252	390	2	.	.	X
ejpam-5252	390	3	for	for	ADP
ejpam-5252	390	4	p	p	NOUN
ejpam-5252	390	5	=	=	SYM
ejpam-5252	390	6	3	3	NUM
ejpam-5252	390	7	,	,	PUNCT
ejpam-5252	390	8	let	let	VERB
ejpam-5252	390	9	x	x	PRON
ejpam-5252	390	10	and	and	CCONJ
ejpam-5252	390	11	y	y	PROPN
ejpam-5252	390	12	be	be	AUX
ejpam-5252	390	13	partite	partite	ADJ
ejpam-5252	390	14	sets	set	NOUN
ejpam-5252	390	15	of	of	ADP
ejpam-5252	390	16	g	g	NOUN
ejpam-5252	390	17	and	and	CCONJ
ejpam-5252	390	18	assume	assume	VERB
ejpam-5252	390	19	that	that	SCONJ
ejpam-5252	390	20	|x|	|x|	PROPN
ejpam-5252	390	21	=	=	SYM
ejpam-5252	390	22	3	3	NUM
ejpam-5252	390	23	,	,	PUNCT
ejpam-5252	390	24	say	say	VERB
ejpam-5252	390	25	x	x	X
ejpam-5252	390	26	=	=	PRON
ejpam-5252	390	27	{	{	PUNCT
ejpam-5252	390	28	x1	x1	PROPN
ejpam-5252	390	29	,	,	PUNCT
ejpam-5252	390	30	x2	x2	PROPN
ejpam-5252	390	31	,	,	PUNCT
ejpam-5252	390	32	x3	x3	ADJ
ejpam-5252	390	33	}	}	PUNCT
ejpam-5252	390	34	.	.	PUNCT
ejpam-5252	391	1	let	let	VERB
ejpam-5252	391	2	v0	v0	NOUN
ejpam-5252	391	3	=	=	SYM
ejpam-5252	391	4	y	y	PROPN
ejpam-5252	391	5	,	,	PUNCT
ejpam-5252	391	6	v1	v1	NOUN
ejpam-5252	391	7	=	=	SYM
ejpam-5252	391	8	∅	∅	NOUN
ejpam-5252	391	9	,	,	PUNCT
ejpam-5252	391	10	v2	v2	NOUN
ejpam-5252	391	11	=	=	SYM
ejpam-5252	391	12	{	{	PUNCT
ejpam-5252	391	13	x1	x1	PROPN
ejpam-5252	391	14	,	,	PUNCT
ejpam-5252	391	15	x2	x2	PROPN
ejpam-5252	391	16	}	}	PUNCT
ejpam-5252	391	17	and	and	CCONJ
ejpam-5252	391	18	v3	v3	PROPN
ejpam-5252	391	19	=	=	PUNCT
ejpam-5252	391	20	{	{	PUNCT
ejpam-5252	391	21	x3	x3	ADJ
ejpam-5252	391	22	}	}	PUNCT
ejpam-5252	391	23	.	.	PUNCT
ejpam-5252	392	1	then	then	ADV
ejpam-5252	392	2	f	f	X
ejpam-5252	392	3	=	=	SYM
ejpam-5252	392	4	(	(	PUNCT
ejpam-5252	392	5	v0	v0	PROPN
ejpam-5252	392	6	,	,	PUNCT
ejpam-5252	392	7	v1	v1	NOUN
ejpam-5252	392	8	,	,	PUNCT
ejpam-5252	392	9	v2	v2	PROPN
ejpam-5252	392	10	,	,	PUNCT
ejpam-5252	392	11	v3	v3	PROPN
ejpam-5252	392	12	)	)	PUNCT
ejpam-5252	392	13	∈	∈	PROPN
ejpam-5252	392	14	mrdf	mrdf	NOUN
ejpam-5252	392	15	(	(	PUNCT
ejpam-5252	392	16	g	g	NOUN
ejpam-5252	392	17	)	)	PUNCT
ejpam-5252	392	18	and	and	CCONJ
ejpam-5252	392	19	its	its	PRON
ejpam-5252	392	20	weight	weight	NOUN
ejpam-5252	392	21	is	be	AUX
ejpam-5252	392	22	7	7	NUM
ejpam-5252	392	23	.	.	PUNCT
ejpam-5252	393	1	hence	hence	ADV
ejpam-5252	393	2	,	,	PUNCT
ejpam-5252	393	3	γmr(g	γmr(g	PROPN
ejpam-5252	393	4	)	)	PUNCT
ejpam-5252	393	5	≤	≤	NUM
ejpam-5252	393	6	7	7	NUM
ejpam-5252	393	7	.	.	PUNCT
ejpam-5252	394	1	now	now	ADV
ejpam-5252	394	2	,	,	PUNCT
ejpam-5252	394	3	suppose	suppose	VERB
ejpam-5252	394	4	that	that	SCONJ
ejpam-5252	394	5	g	g	PROPN
ejpam-5252	394	6	=	=	SYM
ejpam-5252	394	7	(	(	PUNCT
ejpam-5252	394	8	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	394	9	)	)	PUNCT
ejpam-5252	394	10	is	be	AUX
ejpam-5252	394	11	a	a	DET
ejpam-5252	394	12	γmr	γmr	ADJ
ejpam-5252	394	13	-	-	PUNCT
ejpam-5252	394	14	function	function	NOUN
ejpam-5252	394	15	.	.	PUNCT
ejpam-5252	395	1	if	if	SCONJ
ejpam-5252	395	2	w0	w0	PROPN
ejpam-5252	395	3	=	=	NOUN
ejpam-5252	395	4	∅	∅	NOUN
ejpam-5252	395	5	then	then	ADV
ejpam-5252	395	6	w3	w3	PROPN
ejpam-5252	395	7	=	=	PUNCT
ejpam-5252	395	8	∅.	∅.	NOUN
ejpam-5252	395	9	since	since	SCONJ
ejpam-5252	395	10	g	g	PROPN
ejpam-5252	395	11	is	be	AUX
ejpam-5252	395	12	a	a	DET
ejpam-5252	395	13	γmr	γmr	ADJ
ejpam-5252	395	14	-	-	PUNCT
ejpam-5252	395	15	function	function	NOUN
ejpam-5252	395	16	,	,	PUNCT
ejpam-5252	395	17	then	then	ADV
ejpam-5252	395	18	γmr(g	γmr(g	NUM
ejpam-5252	395	19	)	)	PUNCT
ejpam-5252	395	20	≥	≥	NOUN
ejpam-5252	395	21	7	7	NUM
ejpam-5252	395	22	.	.	PUNCT
ejpam-5252	396	1	if	if	SCONJ
ejpam-5252	396	2	w0	w0	PROPN
ejpam-5252	396	3	̸=	̸=	PROPN
ejpam-5252	396	4	∅	∅	NOUN
ejpam-5252	396	5	,	,	PUNCT
ejpam-5252	396	6	let	let	VERB
ejpam-5252	396	7	w0	w0	PROPN
ejpam-5252	396	8	=	=	PROPN
ejpam-5252	396	9	y	y	PROPN
ejpam-5252	396	10	,	,	PUNCT
ejpam-5252	396	11	w1	w1	NOUN
ejpam-5252	396	12	=	=	PUNCT
ejpam-5252	396	13	∅,w2	∅,w2	PUNCT
ejpam-5252	397	1	=	=	PRON
ejpam-5252	397	2	{	{	PUNCT
ejpam-5252	397	3	x1	x1	PROPN
ejpam-5252	397	4	,	,	PUNCT
ejpam-5252	397	5	x2	x2	PROPN
ejpam-5252	397	6	}	}	PUNCT
ejpam-5252	397	7	and	and	CCONJ
ejpam-5252	397	8	w3	w3	PROPN
ejpam-5252	397	9	=	=	PUNCT
ejpam-5252	397	10	{	{	PUNCT
ejpam-5252	397	11	x3	x3	ADJ
ejpam-5252	397	12	}	}	PUNCT
ejpam-5252	397	13	.	.	PUNCT
ejpam-5252	398	1	since	since	SCONJ
ejpam-5252	398	2	g	g	PROPN
ejpam-5252	398	3	is	be	AUX
ejpam-5252	398	4	a	a	DET
ejpam-5252	398	5	γmr	γmr	ADJ
ejpam-5252	398	6	-	-	PUNCT
ejpam-5252	398	7	function	function	NOUN
ejpam-5252	398	8	,	,	PUNCT
ejpam-5252	398	9	then	then	ADV
ejpam-5252	398	10	γmr(g	γmr(g	NUM
ejpam-5252	398	11	)	)	PUNCT
ejpam-5252	398	12	≥	≥	NOUN
ejpam-5252	398	13	7	7	NUM
ejpam-5252	398	14	.	.	PUNCT
ejpam-5252	399	1	thus	thus	ADV
ejpam-5252	399	2	,	,	PUNCT
ejpam-5252	399	3	γmr(g	γmr(g	PROPN
ejpam-5252	399	4	)	)	PUNCT
ejpam-5252	399	5	=	=	SYM
ejpam-5252	400	1	7	7	X
ejpam-5252	400	2	.	.	X
ejpam-5252	401	1	if	if	SCONJ
ejpam-5252	401	2	p	p	NOUN
ejpam-5252	401	3	=	=	NOUN
ejpam-5252	401	4	4	4	NUM
ejpam-5252	401	5	and	and	CCONJ
ejpam-5252	401	6	wlog	wlog	NOUN
ejpam-5252	401	7	,	,	PUNCT
ejpam-5252	401	8	let	let	VERB
ejpam-5252	401	9	x	x	PUNCT
ejpam-5252	401	10	=	=	PRON
ejpam-5252	401	11	{	{	PUNCT
ejpam-5252	401	12	x1	x1	PROPN
ejpam-5252	401	13	,	,	PUNCT
ejpam-5252	401	14	x2	x2	PROPN
ejpam-5252	401	15	,	,	PUNCT
ejpam-5252	401	16	x3	x3	ADJ
ejpam-5252	401	17	,	,	PUNCT
ejpam-5252	401	18	x4	x4	PROPN
ejpam-5252	401	19	}	}	PUNCT
ejpam-5252	401	20	such	such	ADJ
ejpam-5252	401	21	that	that	SCONJ
ejpam-5252	401	22	f(x1	f(x1	NOUN
ejpam-5252	401	23	)	)	PUNCT
ejpam-5252	401	24	=	=	SYM
ejpam-5252	401	25	2	2	NUM
ejpam-5252	401	26	,	,	PUNCT
ejpam-5252	401	27	f(x2	f(x2	NOUN
ejpam-5252	401	28	)	)	PUNCT
ejpam-5252	401	29	=	=	SYM
ejpam-5252	401	30	3	3	NUM
ejpam-5252	401	31	and	and	CCONJ
ejpam-5252	401	32	f(x3	f(x3	NUM
ejpam-5252	401	33	)	)	PUNCT
ejpam-5252	402	1	=	=	SYM
ejpam-5252	402	2	2	2	NUM
ejpam-5252	402	3	=	=	SYM
ejpam-5252	402	4	f(x4	f(x4	NOUN
ejpam-5252	402	5	)	)	PUNCT
ejpam-5252	402	6	.	.	PUNCT
ejpam-5252	403	1	then	then	ADV
ejpam-5252	403	2	f(yi	f(yi	NUM
ejpam-5252	403	3	)	)	PUNCT
ejpam-5252	403	4	=	=	SYM
ejpam-5252	404	1	0	0	NUM
ejpam-5252	404	2	,	,	PUNCT
ejpam-5252	404	3	i	i	PRON
ejpam-5252	404	4	=	=	NOUN
ejpam-5252	404	5	1	1	NUM
ejpam-5252	404	6	,	,	PUNCT
ejpam-5252	404	7	2	2	NUM
ejpam-5252	404	8	,	,	PUNCT
ejpam-5252	404	9	·	·	PUNCT
ejpam-5252	404	10	·	·	PUNCT
ejpam-5252	404	11	·	·	PUNCT
ejpam-5252	404	12	,	,	PUNCT
ejpam-5252	404	13	n	n	CCONJ
ejpam-5252	404	14	,	,	PUNCT
ejpam-5252	404	15	and	and	CCONJ
ejpam-5252	404	16	for	for	ADP
ejpam-5252	404	17	every	every	DET
ejpam-5252	404	18	yi	yi	PROPN
ejpam-5252	404	19	∈	∈	PROPN
ejpam-5252	404	20	y	y	PROPN
ejpam-5252	404	21	,	,	PUNCT
ejpam-5252	404	22	yi	yi	PROPN
ejpam-5252	404	23	∈	∈	PROPN
ejpam-5252	404	24	ng(x	ng(x	NUM
ejpam-5252	404	25	)	)	PUNCT
ejpam-5252	404	26	by	by	ADP
ejpam-5252	404	27	(	(	PUNCT
ejpam-5252	404	28	p2	p2	PROPN
ejpam-5252	404	29	)	)	PUNCT
ejpam-5252	404	30	.	.	PUNCT
ejpam-5252	405	1	define	define	VERB
ejpam-5252	405	2	f	f	PROPN
ejpam-5252	405	3	=	=	SYM
ejpam-5252	405	4	(	(	PUNCT
ejpam-5252	405	5	v0	v0	PROPN
ejpam-5252	405	6	,	,	PUNCT
ejpam-5252	405	7	v1	v1	NOUN
ejpam-5252	405	8	,	,	PUNCT
ejpam-5252	405	9	v2	v2	PROPN
ejpam-5252	405	10	,	,	PUNCT
ejpam-5252	405	11	v3	v3	PROPN
ejpam-5252	405	12	)	)	PUNCT
ejpam-5252	405	13	such	such	ADJ
ejpam-5252	405	14	that	that	DET
ejpam-5252	405	15	v0	v0	NOUN
ejpam-5252	405	16	̸=	̸=	PROPN
ejpam-5252	405	17	∅	∅	NOUN
ejpam-5252	405	18	,	,	PUNCT
ejpam-5252	405	19	v1	v1	NOUN
ejpam-5252	405	20	=	=	SYM
ejpam-5252	405	21	∅	∅	NOUN
ejpam-5252	405	22	,	,	PUNCT
ejpam-5252	405	23	v2	v2	NOUN
ejpam-5252	405	24	=	=	SYM
ejpam-5252	405	25	{	{	PUNCT
ejpam-5252	405	26	x1	x1	PROPN
ejpam-5252	405	27	,	,	PUNCT
ejpam-5252	405	28	x3	x3	ADJ
ejpam-5252	405	29	,	,	PUNCT
ejpam-5252	405	30	x4	x4	PROPN
ejpam-5252	405	31	}	}	PUNCT
ejpam-5252	405	32	,	,	PUNCT
ejpam-5252	405	33	v3	v3	PROPN
ejpam-5252	405	34	=	=	SYM
ejpam-5252	405	35	{	{	PUNCT
ejpam-5252	405	36	x2	x2	PROPN
ejpam-5252	405	37	}	}	PUNCT
ejpam-5252	405	38	.	.	PUNCT
ejpam-5252	406	1	then	then	ADV
ejpam-5252	406	2	f	f	PROPN
ejpam-5252	406	3	∈	∈	PROPN
ejpam-5252	406	4	mrdf	mrdf	NOUN
ejpam-5252	406	5	(	(	PUNCT
ejpam-5252	406	6	g	g	NOUN
ejpam-5252	406	7	)	)	PUNCT
ejpam-5252	406	8	.	.	PUNCT
ejpam-5252	407	1	thus	thus	ADV
ejpam-5252	407	2	,	,	PUNCT
ejpam-5252	407	3	γmr(g	γmr(g	PROPN
ejpam-5252	407	4	)	)	PUNCT
ejpam-5252	407	5	≤	≤	NOUN
ejpam-5252	407	6	9	9	NUM
ejpam-5252	407	7	.	.	PUNCT
ejpam-5252	407	8	suppose	suppose	VERB
ejpam-5252	407	9	to	to	ADP
ejpam-5252	407	10	the	the	DET
ejpam-5252	407	11	contrary	contrary	NOUN
ejpam-5252	407	12	that	that	SCONJ
ejpam-5252	407	13	γmr(g	γmr(g	X
ejpam-5252	407	14	)	)	PUNCT
ejpam-5252	407	15	<	<	X
ejpam-5252	407	16	9	9	NUM
ejpam-5252	407	17	and	and	CCONJ
ejpam-5252	407	18	p	p	NOUN
ejpam-5252	407	19	=	=	ADJ
ejpam-5252	407	20	4	4	X
ejpam-5252	407	21	.	.	PUNCT
ejpam-5252	408	1	so	so	ADV
ejpam-5252	408	2	,	,	PUNCT
ejpam-5252	408	3	γmr(g	γmr(g	PROPN
ejpam-5252	408	4	)	)	PUNCT
ejpam-5252	409	1	=	=	SYM
ejpam-5252	409	2	8	8	X
ejpam-5252	409	3	.	.	PUNCT
ejpam-5252	410	1	now	now	ADV
ejpam-5252	410	2	,	,	PUNCT
ejpam-5252	410	3	if	if	SCONJ
ejpam-5252	410	4	x	x	ADP
ejpam-5252	410	5	=	=	PRON
ejpam-5252	410	6	{	{	PUNCT
ejpam-5252	410	7	x1	x1	PROPN
ejpam-5252	410	8	,	,	PUNCT
ejpam-5252	410	9	x2	x2	PROPN
ejpam-5252	410	10	,	,	PUNCT
ejpam-5252	410	11	x3	x3	ADJ
ejpam-5252	410	12	,	,	PUNCT
ejpam-5252	410	13	x4	x4	PROPN
ejpam-5252	410	14	}	}	PUNCT
ejpam-5252	410	15	then	then	ADV
ejpam-5252	410	16	{	{	PUNCT
ejpam-5252	410	17	x1	x1	PROPN
ejpam-5252	410	18	,	,	PUNCT
ejpam-5252	410	19	x2	x2	PROPN
ejpam-5252	410	20	,	,	PUNCT
ejpam-5252	410	21	x3	x3	ADJ
ejpam-5252	410	22	}	}	PUNCT
ejpam-5252	410	23	is	be	AUX
ejpam-5252	410	24	a	a	DET
ejpam-5252	410	25	γ3	γ3	NOUN
ejpam-5252	410	26	-	-	PUNCT
ejpam-5252	410	27	set	set	NOUN
ejpam-5252	410	28	of	of	ADP
ejpam-5252	410	29	g	g	PROPN
ejpam-5252	410	30	\	\	PROPN
ejpam-5252	410	31	{	{	PUNCT
ejpam-5252	410	32	x4	x4	PROPN
ejpam-5252	410	33	}	}	PUNCT
ejpam-5252	410	34	.	.	PUNCT
ejpam-5252	411	1	clearly	clearly	ADV
ejpam-5252	411	2	,	,	PUNCT
ejpam-5252	411	3	7	7	NUM
ejpam-5252	411	4	=	=	SYM
ejpam-5252	411	5	γmr(g	γmr(g	PROPN
ejpam-5252	411	6	\	\	X
ejpam-5252	411	7	{	{	PUNCT
ejpam-5252	411	8	x4	x4	PROPN
ejpam-5252	411	9	}	}	PUNCT
ejpam-5252	411	10	)	)	PUNCT
ejpam-5252	411	11	≤	≤	NUM
ejpam-5252	411	12	γmr(g	γmr(g	X
ejpam-5252	411	13	)	)	PUNCT
ejpam-5252	411	14	=	=	SYM
ejpam-5252	411	15	8	8	X
ejpam-5252	411	16	.	.	PUNCT
ejpam-5252	412	1	this	this	PRON
ejpam-5252	412	2	means	mean	VERB
ejpam-5252	412	3	that	that	SCONJ
ejpam-5252	412	4	f(x4	f(x4	NOUN
ejpam-5252	412	5	)	)	PUNCT
ejpam-5252	412	6	=	=	SYM
ejpam-5252	412	7	1	1	NUM
ejpam-5252	412	8	,	,	PUNCT
ejpam-5252	412	9	a	a	DET
ejpam-5252	412	10	contradiction	contradiction	NOUN
ejpam-5252	412	11	.	.	PUNCT
ejpam-5252	413	1	therefore	therefore	ADV
ejpam-5252	413	2	,	,	PUNCT
ejpam-5252	413	3	γmr(g	γmr(g	PROPN
ejpam-5252	413	4	)	)	PUNCT
ejpam-5252	413	5	=	=	SYM
ejpam-5252	414	1	9	9	X
ejpam-5252	414	2	.	.	X
ejpam-5252	415	1	if	if	SCONJ
ejpam-5252	415	2	p	p	PROPN
ejpam-5252	415	3	≥	≥	NOUN
ejpam-5252	415	4	5	5	NUM
ejpam-5252	415	5	,	,	PUNCT
ejpam-5252	415	6	let	let	VERB
ejpam-5252	415	7	x	x	SYM
ejpam-5252	415	8	=	=	SYM
ejpam-5252	415	9	km	km	NOUN
ejpam-5252	415	10	and	and	CCONJ
ejpam-5252	415	11	y	y	PROPN
ejpam-5252	415	12	=	=	SYM
ejpam-5252	415	13	kn	kn	PROPN
ejpam-5252	415	14	.	.	PROPN
ejpam-5252	415	15	wlog	wlog	PROPN
ejpam-5252	415	16	,	,	PUNCT
ejpam-5252	415	17	let	let	VERB
ejpam-5252	415	18	s.	s.	PROPN
ejpam-5252	415	19	ahamad	ahamad	PROPN
ejpam-5252	415	20	,	,	PUNCT
ejpam-5252	415	21	j.	j.	PROPN
ejpam-5252	415	22	cariaga	cariaga	PROPN
ejpam-5252	415	23	,	,	PUNCT
ejpam-5252	415	24	s.	s.	PROPN
ejpam-5252	415	25	menchavez	menchavez	PROPN
ejpam-5252	415	26	/	/	PUNCT
ejpam-5252	415	27	eur	eur	PROPN
ejpam-5252	415	28	.	.	PUNCT
ejpam-5252	416	1	j.	j.	PROPN
ejpam-5252	416	2	pure	pure	PROPN
ejpam-5252	416	3	appl	appl	PROPN
ejpam-5252	416	4	.	.	PROPN
ejpam-5252	416	5	math	math	PROPN
ejpam-5252	416	6	,	,	PUNCT
ejpam-5252	416	7	18	18	NUM
ejpam-5252	416	8	(	(	PUNCT
ejpam-5252	416	9	1	1	NUM
ejpam-5252	416	10	)	)	PUNCT
ejpam-5252	416	11	(	(	PUNCT
ejpam-5252	416	12	2025	2025	NUM
ejpam-5252	416	13	)	)	PUNCT
ejpam-5252	416	14	,	,	PUNCT
ejpam-5252	416	15	5252	5252	NUM
ejpam-5252	416	16	11	11	NUM
ejpam-5252	416	17	of	of	ADP
ejpam-5252	416	18	18	18	NUM
ejpam-5252	416	19	{	{	PUNCT
ejpam-5252	416	20	u1	u1	NOUN
ejpam-5252	416	21	,	,	PUNCT
ejpam-5252	416	22	u2	u2	PROPN
ejpam-5252	416	23	}	}	PUNCT
ejpam-5252	416	24	∈	∈	PROPN
ejpam-5252	416	25	v	v	NOUN
ejpam-5252	416	26	(	(	PUNCT
ejpam-5252	416	27	km	km	PROPN
ejpam-5252	416	28	)	)	PUNCT
ejpam-5252	416	29	,	,	PUNCT
ejpam-5252	416	30	{	{	PUNCT
ejpam-5252	416	31	v1	v1	NOUN
ejpam-5252	416	32	,	,	PUNCT
ejpam-5252	416	33	v2	v2	NOUN
ejpam-5252	416	34	}	}	PUNCT
ejpam-5252	416	35	∈	∈	PROPN
ejpam-5252	416	36	v	v	NOUN
ejpam-5252	416	37	(	(	PUNCT
ejpam-5252	416	38	kn	kn	PROPN
ejpam-5252	416	39	)	)	PUNCT
ejpam-5252	416	40	and	and	CCONJ
ejpam-5252	416	41	define	define	VERB
ejpam-5252	416	42	a	a	DET
ejpam-5252	416	43	function	function	NOUN
ejpam-5252	416	44	f	f	NOUN
ejpam-5252	416	45	=	=	SYM
ejpam-5252	416	46	(	(	PUNCT
ejpam-5252	416	47	v0	v0	PROPN
ejpam-5252	416	48	,	,	PUNCT
ejpam-5252	416	49	v1	v1	NOUN
ejpam-5252	416	50	,	,	PUNCT
ejpam-5252	416	51	v2	v2	PROPN
ejpam-5252	416	52	,	,	PUNCT
ejpam-5252	416	53	v3	v3	PROPN
ejpam-5252	416	54	)	)	PUNCT
ejpam-5252	416	55	given	give	VERB
ejpam-5252	416	56	by	by	ADP
ejpam-5252	416	57	f(z	f(z	PROPN
ejpam-5252	416	58	)	)	PUNCT
ejpam-5252	416	59	=	=	PUNCT
ejpam-5252	417	1			NOUN
ejpam-5252	417	2	3	3	NUM
ejpam-5252	417	3	,	,	PUNCT
ejpam-5252	417	4	z	z	PROPN
ejpam-5252	417	5	∈	∈	PROPN
ejpam-5252	417	6	{	{	PUNCT
ejpam-5252	417	7	u1	u1	NOUN
ejpam-5252	417	8	,	,	PUNCT
ejpam-5252	417	9	v1	v1	NOUN
ejpam-5252	417	10	}	}	PUNCT
ejpam-5252	417	11	.	.	PUNCT
ejpam-5252	418	1	2	2	NUM
ejpam-5252	418	2	,	,	PUNCT
ejpam-5252	418	3	z	z	NOUN
ejpam-5252	418	4	∈	∈	PROPN
ejpam-5252	418	5	{	{	PUNCT
ejpam-5252	418	6	u2	u2	NOUN
ejpam-5252	418	7	,	,	PUNCT
ejpam-5252	418	8	v2	v2	PROPN
ejpam-5252	418	9	}	}	PUNCT
ejpam-5252	418	10	.	.	PUNCT
ejpam-5252	419	1	0	0	NUM
ejpam-5252	419	2	,	,	PUNCT
ejpam-5252	419	3	otherwise	otherwise	ADV
ejpam-5252	419	4	.	.	PUNCT
ejpam-5252	420	1	then	then	ADV
ejpam-5252	420	2	f	f	PROPN
ejpam-5252	420	3	∈	∈	PROPN
ejpam-5252	420	4	mrdf	mrdf	NOUN
ejpam-5252	420	5	(	(	PUNCT
ejpam-5252	420	6	g	g	NOUN
ejpam-5252	420	7	)	)	PUNCT
ejpam-5252	420	8	.	.	PUNCT
ejpam-5252	421	1	since	since	SCONJ
ejpam-5252	421	2	|v1|	|v1|	NOUN
ejpam-5252	421	3	=	=	SYM
ejpam-5252	421	4	0	0	NUM
ejpam-5252	421	5	and	and	CCONJ
ejpam-5252	421	6	uivi	uivi	PROPN
ejpam-5252	421	7	∈	∈	PROPN
ejpam-5252	421	8	e(g	e(g	PROPN
ejpam-5252	421	9	)	)	PUNCT
ejpam-5252	421	10	,	,	PUNCT
ejpam-5252	421	11	for	for	ADP
ejpam-5252	421	12	all	all	DET
ejpam-5252	421	13	i	i	PRON
ejpam-5252	421	14	=	=	NOUN
ejpam-5252	421	15	1	1	NUM
ejpam-5252	421	16	,	,	PUNCT
ejpam-5252	421	17	2	2	NUM
ejpam-5252	421	18	,	,	PUNCT
ejpam-5252	421	19	{	{	PUNCT
ejpam-5252	421	20	u1	u1	NOUN
ejpam-5252	421	21	,	,	PUNCT
ejpam-5252	421	22	v1	v1	NOUN
ejpam-5252	421	23	}	}	PUNCT
ejpam-5252	421	24	=	=	SYM
ejpam-5252	421	25	v3	v3	PROPN
ejpam-5252	421	26	is	be	AUX
ejpam-5252	421	27	a	a	DET
ejpam-5252	421	28	dominating	dominating	NOUN
ejpam-5252	421	29	set	set	NOUN
ejpam-5252	421	30	of	of	ADP
ejpam-5252	421	31	g.	g.	PROPN
ejpam-5252	421	32	thus	thus	ADV
ejpam-5252	421	33	,	,	PUNCT
ejpam-5252	421	34	γ(g	γ(g	PROPN
ejpam-5252	421	35	)	)	PUNCT
ejpam-5252	421	36	=	=	SYM
ejpam-5252	422	1	2	2	X
ejpam-5252	422	2	.	.	PUNCT
ejpam-5252	422	3	since	since	SCONJ
ejpam-5252	422	4	⟨v2	⟨v2	ADJ
ejpam-5252	422	5	∪	∪	NOUN
ejpam-5252	422	6	v3⟩	v3⟩	PRON
ejpam-5252	422	7	is	be	AUX
ejpam-5252	422	8	connected	connect	VERB
ejpam-5252	422	9	,	,	PUNCT
ejpam-5252	422	10	by	by	ADP
ejpam-5252	422	11	proposition	proposition	NOUN
ejpam-5252	422	12	4	4	NUM
ejpam-5252	422	13	(	(	PUNCT
ejpam-5252	422	14	ii	ii	NOUN
ejpam-5252	422	15	)	)	PUNCT
ejpam-5252	422	16	,	,	PUNCT
ejpam-5252	422	17	γmr(g	γmr(g	PROPN
ejpam-5252	422	18	)	)	PUNCT
ejpam-5252	422	19	=	=	SYM
ejpam-5252	423	1	10	10	NUM
ejpam-5252	423	2	.	.	PUNCT
ejpam-5252	424	1	the	the	DET
ejpam-5252	424	2	fan	fan	NOUN
ejpam-5252	424	3	fn	fn	PROPN
ejpam-5252	424	4	of	of	ADP
ejpam-5252	424	5	order	order	NOUN
ejpam-5252	424	6	n+	n+	ADP
ejpam-5252	424	7	1	1	NUM
ejpam-5252	424	8	is	be	AUX
ejpam-5252	424	9	the	the	DET
ejpam-5252	424	10	graph	graph	NOUN
ejpam-5252	424	11	pn	pn	PROPN
ejpam-5252	424	12	+	+	NOUN
ejpam-5252	424	13	k1	k1	NOUN
ejpam-5252	424	14	and	and	CCONJ
ejpam-5252	424	15	the	the	DET
ejpam-5252	424	16	star	star	NOUN
ejpam-5252	424	17	sn	sn	PROPN
ejpam-5252	424	18	of	of	ADP
ejpam-5252	424	19	order	order	NOUN
ejpam-5252	424	20	n+	n+	ADP
ejpam-5252	424	21	1	1	NUM
ejpam-5252	424	22	is	be	AUX
ejpam-5252	424	23	the	the	DET
ejpam-5252	424	24	graph	graph	NOUN
ejpam-5252	424	25	kn	kn	PROPN
ejpam-5252	424	26	+	+	NOUN
ejpam-5252	424	27	k1	k1	NOUN
ejpam-5252	424	28	.	.	PUNCT
ejpam-5252	425	1	the	the	DET
ejpam-5252	425	2	graphs	graph	NOUN
ejpam-5252	425	3	in	in	ADP
ejpam-5252	425	4	figures	figure	NOUN
ejpam-5252	425	5	7	7	NUM
ejpam-5252	425	6	and	and	CCONJ
ejpam-5252	425	7	8	8	NUM
ejpam-5252	425	8	are	be	AUX
ejpam-5252	425	9	the	the	DET
ejpam-5252	425	10	star	star	NOUN
ejpam-5252	425	11	graph	graph	NOUN
ejpam-5252	425	12	s6	s6	PROPN
ejpam-5252	425	13	and	and	CCONJ
ejpam-5252	425	14	fan	fan	NOUN
ejpam-5252	425	15	graph	graph	NOUN
ejpam-5252	425	16	f6	f6	PROPN
ejpam-5252	425	17	,	,	PUNCT
ejpam-5252	425	18	respectively	respectively	ADV
ejpam-5252	425	19	.	.	PUNCT
ejpam-5252	426	1	2	2	NUM
ejpam-5252	426	2	1	1	NUM
ejpam-5252	426	3	1	1	NUM
ejpam-5252	426	4	1	1	NUM
ejpam-5252	426	5	1	1	NUM
ejpam-5252	426	6	1	1	NUM
ejpam-5252	426	7	2	2	NUM
ejpam-5252	426	8	figure	figure	NOUN
ejpam-5252	426	9	7	7	NUM
ejpam-5252	426	10	:	:	PUNCT
ejpam-5252	426	11	a	a	DET
ejpam-5252	426	12	star	star	NOUN
ejpam-5252	426	13	graph	graph	NOUN
ejpam-5252	426	14	s6	s6	PROPN
ejpam-5252	426	15	with	with	ADP
ejpam-5252	426	16	γmr(s6	γmr(s6	NOUN
ejpam-5252	426	17	)	)	PUNCT
ejpam-5252	426	18	=	=	SYM
ejpam-5252	426	19	7	7	NUM
ejpam-5252	426	20	1	1	NUM
ejpam-5252	426	21	1	1	NUM
ejpam-5252	426	22	1	1	NUM
ejpam-5252	426	23	1	1	NUM
ejpam-5252	426	24	1	1	NUM
ejpam-5252	426	25	2	2	NUM
ejpam-5252	426	26	figure	figure	NOUN
ejpam-5252	426	27	8	8	NUM
ejpam-5252	426	28	:	:	PUNCT
ejpam-5252	426	29	a	a	DET
ejpam-5252	426	30	fan	fan	NOUN
ejpam-5252	426	31	graph	graph	NOUN
ejpam-5252	426	32	f6	f6	PROPN
ejpam-5252	426	33	with	with	ADP
ejpam-5252	426	34	γmr(f6	γmr(f6	PROPN
ejpam-5252	426	35	)	)	PUNCT
ejpam-5252	426	36	=	=	SYM
ejpam-5252	426	37	7	7	NUM
ejpam-5252	426	38	proposition	proposition	NOUN
ejpam-5252	426	39	12	12	NUM
ejpam-5252	426	40	.	.	PUNCT
ejpam-5252	427	1	if	if	SCONJ
ejpam-5252	427	2	g	g	PROPN
ejpam-5252	427	3	∈	∈	PROPN
ejpam-5252	427	4	{	{	PUNCT
ejpam-5252	427	5	fn	fn	NOUN
ejpam-5252	427	6	,	,	PUNCT
ejpam-5252	427	7	sn	sn	PROPN
ejpam-5252	427	8	}	}	PUNCT
ejpam-5252	427	9	,	,	PUNCT
ejpam-5252	427	10	n	n	X
ejpam-5252	427	11	≥	≥	NOUN
ejpam-5252	427	12	1	1	NUM
ejpam-5252	427	13	,	,	PUNCT
ejpam-5252	427	14	then	then	ADV
ejpam-5252	427	15	γmr(g	γmr(g	NUM
ejpam-5252	427	16	)	)	PUNCT
ejpam-5252	427	17	=	=	SYM
ejpam-5252	427	18	n+	n+	PUNCT
ejpam-5252	427	19	2	2	X
ejpam-5252	427	20	.	.	X
ejpam-5252	427	21	proof	proof	NOUN
ejpam-5252	427	22	.	.	PUNCT
ejpam-5252	428	1	wlog	wlog	PROPN
ejpam-5252	428	2	,	,	PUNCT
ejpam-5252	428	3	let	let	VERB
ejpam-5252	428	4	g	g	NOUN
ejpam-5252	428	5	=	=	PUNCT
ejpam-5252	428	6	fn	fn	PROPN
ejpam-5252	428	7	where	where	SCONJ
ejpam-5252	428	8	v	v	NOUN
ejpam-5252	428	9	(	(	PUNCT
ejpam-5252	428	10	g	g	NOUN
ejpam-5252	428	11	)	)	PUNCT
ejpam-5252	429	1	=	=	SYM
ejpam-5252	429	2	v	v	X
ejpam-5252	429	3	(	(	PUNCT
ejpam-5252	429	4	k1	k1	NOUN
ejpam-5252	429	5	+	+	CCONJ
ejpam-5252	429	6	pn	pn	NOUN
ejpam-5252	429	7	)	)	PUNCT
ejpam-5252	429	8	and	and	CCONJ
ejpam-5252	429	9	v	v	NOUN
ejpam-5252	429	10	(	(	PUNCT
ejpam-5252	429	11	k1	k1	NOUN
ejpam-5252	429	12	)	)	PUNCT
ejpam-5252	429	13	=	=	SYM
ejpam-5252	429	14	{	{	PUNCT
ejpam-5252	429	15	u	u	NOUN
ejpam-5252	429	16	}	}	PUNCT
ejpam-5252	429	17	is	be	AUX
ejpam-5252	429	18	a	a	DET
ejpam-5252	429	19	central	central	ADJ
ejpam-5252	429	20	vertex	vertex	NOUN
ejpam-5252	429	21	of	of	ADP
ejpam-5252	429	22	g.	g.	PROPN
ejpam-5252	429	23	now	now	ADV
ejpam-5252	429	24	,	,	PUNCT
ejpam-5252	429	25	define	define	VERB
ejpam-5252	429	26	a	a	DET
ejpam-5252	429	27	function	function	NOUN
ejpam-5252	429	28	f	f	NOUN
ejpam-5252	429	29	=	=	SYM
ejpam-5252	429	30	(	(	PUNCT
ejpam-5252	429	31	v0	v0	PROPN
ejpam-5252	429	32	,	,	PUNCT
ejpam-5252	429	33	v1	v1	NOUN
ejpam-5252	429	34	,	,	PUNCT
ejpam-5252	429	35	v2	v2	PROPN
ejpam-5252	429	36	,	,	PUNCT
ejpam-5252	429	37	v3	v3	PROPN
ejpam-5252	429	38	)	)	PUNCT
ejpam-5252	429	39	given	give	VERB
ejpam-5252	429	40	by	by	ADP
ejpam-5252	429	41	f(x	f(x	PROPN
ejpam-5252	429	42	)	)	PUNCT
ejpam-5252	429	43	=	=	PRON
ejpam-5252	430	1	{	{	PUNCT
ejpam-5252	430	2	2	2	NUM
ejpam-5252	430	3	,	,	PUNCT
ejpam-5252	430	4	x	x	PUNCT
ejpam-5252	430	5	=	=	PRON
ejpam-5252	430	6	{	{	PUNCT
ejpam-5252	430	7	u	u	NOUN
ejpam-5252	430	8	}	}	PUNCT
ejpam-5252	430	9	.	.	PUNCT
ejpam-5252	431	1	1	1	NUM
ejpam-5252	431	2	,	,	PUNCT
ejpam-5252	431	3	otherwise	otherwise	ADV
ejpam-5252	431	4	.	.	PUNCT
ejpam-5252	432	1	then	then	ADV
ejpam-5252	432	2	f	f	PROPN
ejpam-5252	432	3	∈	∈	PROPN
ejpam-5252	432	4	mrdf	mrdf	NOUN
ejpam-5252	432	5	(	(	PUNCT
ejpam-5252	432	6	g	g	NOUN
ejpam-5252	432	7	)	)	PUNCT
ejpam-5252	432	8	.	.	PUNCT
ejpam-5252	433	1	it	it	PRON
ejpam-5252	433	2	follows	follow	VERB
ejpam-5252	433	3	that	that	PRON
ejpam-5252	433	4	γmr(g	γmr(g	PROPN
ejpam-5252	433	5	)	)	PUNCT
ejpam-5252	434	1	≤	≤	NOUN
ejpam-5252	434	2	n	n	PRON
ejpam-5252	434	3	+	+	NOUN
ejpam-5252	434	4	2	2	X
ejpam-5252	434	5	.	.	PUNCT
ejpam-5252	434	6	now	now	ADV
ejpam-5252	434	7	,	,	PUNCT
ejpam-5252	434	8	suppose	suppose	VERB
ejpam-5252	434	9	that	that	SCONJ
ejpam-5252	434	10	g	g	PROPN
ejpam-5252	434	11	=	=	SYM
ejpam-5252	434	12	(	(	PUNCT
ejpam-5252	434	13	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	434	14	)	)	PUNCT
ejpam-5252	434	15	is	be	AUX
ejpam-5252	434	16	a	a	DET
ejpam-5252	434	17	γmr	γmr	ADJ
ejpam-5252	434	18	-	-	PUNCT
ejpam-5252	434	19	function	function	NOUN
ejpam-5252	434	20	of	of	ADP
ejpam-5252	434	21	g.	g.	PROPN
ejpam-5252	434	22	if	if	SCONJ
ejpam-5252	434	23	w0	w0	PROPN
ejpam-5252	434	24	=	=	SYM
ejpam-5252	434	25	∅	∅	NOUN
ejpam-5252	434	26	,	,	PUNCT
ejpam-5252	434	27	then	then	ADV
ejpam-5252	434	28	w3	w3	PROPN
ejpam-5252	434	29	=	=	PUNCT
ejpam-5252	434	30	∅.	∅.	NOUN
ejpam-5252	434	31	since	since	SCONJ
ejpam-5252	434	32	g	g	PROPN
ejpam-5252	434	33	is	be	AUX
ejpam-5252	434	34	a	a	DET
ejpam-5252	434	35	γmr	γmr	ADJ
ejpam-5252	434	36	-	-	PUNCT
ejpam-5252	434	37	function	function	NOUN
ejpam-5252	434	38	of	of	ADP
ejpam-5252	434	39	g	g	NOUN
ejpam-5252	434	40	,	,	PUNCT
ejpam-5252	434	41	|w2|	|w2|	NOUN
ejpam-5252	434	42	=	=	SYM
ejpam-5252	434	43	|v	|v	PROPN
ejpam-5252	434	44	(	(	PUNCT
ejpam-5252	434	45	k1)|	k1)|	NOUN
ejpam-5252	434	46	=	=	SYM
ejpam-5252	434	47	1	1	NUM
ejpam-5252	434	48	and	and	CCONJ
ejpam-5252	434	49	|w1|	|w1|	NOUN
ejpam-5252	434	50	=	=	SYM
ejpam-5252	434	51	|v	|v	X
ejpam-5252	434	52	(	(	PUNCT
ejpam-5252	434	53	pn)|	pn)|	PROPN
ejpam-5252	434	54	=	=	PROPN
ejpam-5252	434	55	n.	n.	PROPN
ejpam-5252	434	56	hence	hence	ADV
ejpam-5252	434	57	,	,	PUNCT
ejpam-5252	434	58	γmr(g	γmr(g	PROPN
ejpam-5252	434	59	)	)	PUNCT
ejpam-5252	434	60	=	=	SYM
ejpam-5252	435	1	ωmr	ωmr	NOUN
ejpam-5252	435	2	g	g	PROPN
ejpam-5252	435	3	(	(	PUNCT
ejpam-5252	435	4	g	g	NOUN
ejpam-5252	435	5	)	)	PUNCT
ejpam-5252	435	6	≥	≥	NOUN
ejpam-5252	435	7	n	n	NOUN
ejpam-5252	435	8	+	+	NUM
ejpam-5252	435	9	2	2	NUM
ejpam-5252	435	10	.	.	X
ejpam-5252	436	1	if	if	SCONJ
ejpam-5252	436	2	|w0|	|w0|	NOUN
ejpam-5252	436	3	=	=	NOUN
ejpam-5252	436	4	̸	̸	NUM
ejpam-5252	436	5	0	0	NUM
ejpam-5252	436	6	,	,	PUNCT
ejpam-5252	436	7	then	then	ADV
ejpam-5252	436	8	|w2|	|w2|	NOUN
ejpam-5252	436	9	≥	≥	NOUN
ejpam-5252	436	10	1	1	NUM
ejpam-5252	436	11	and	and	CCONJ
ejpam-5252	436	12	|w3|	|w3|	PRON
ejpam-5252	436	13	≥	≥	NOUN
ejpam-5252	436	14	1	1	NUM
ejpam-5252	436	15	.	.	PUNCT
ejpam-5252	437	1	it	it	PRON
ejpam-5252	437	2	follows	follow	VERB
ejpam-5252	437	3	that	that	PRON
ejpam-5252	437	4	γmr(g	γmr(g	PUNCT
ejpam-5252	437	5	)	)	PUNCT
ejpam-5252	438	1	=	=	SYM
ejpam-5252	438	2	ωmr	ωmr	NOUN
ejpam-5252	438	3	g	g	PROPN
ejpam-5252	438	4	(	(	PUNCT
ejpam-5252	438	5	g	g	NOUN
ejpam-5252	438	6	)	)	PUNCT
ejpam-5252	438	7	=	=	SYM
ejpam-5252	439	1	2|w2|+	2|w2|+	NUM
ejpam-5252	439	2	3|w3|	3|w3|	NUM
ejpam-5252	439	3	≥	≥	NOUN
ejpam-5252	439	4	n+	n+	PUNCT
ejpam-5252	439	5	2	2	X
ejpam-5252	439	6	.	.	X
ejpam-5252	439	7	therefore	therefore	ADV
ejpam-5252	439	8	,	,	PUNCT
ejpam-5252	439	9	γmr(g	γmr(g	PROPN
ejpam-5252	439	10	)	)	PUNCT
ejpam-5252	439	11	=	=	SYM
ejpam-5252	439	12	n+	n+	PUNCT
ejpam-5252	439	13	2	2	X
ejpam-5252	439	14	.	.	PUNCT
ejpam-5252	439	15	s.	s.	PROPN
ejpam-5252	439	16	ahamad	ahamad	PROPN
ejpam-5252	439	17	,	,	PUNCT
ejpam-5252	439	18	j.	j.	PROPN
ejpam-5252	439	19	cariaga	cariaga	PROPN
ejpam-5252	439	20	,	,	PUNCT
ejpam-5252	439	21	s.	s.	PROPN
ejpam-5252	439	22	menchavez	menchavez	PROPN
ejpam-5252	439	23	/	/	PUNCT
ejpam-5252	439	24	eur	eur	PROPN
ejpam-5252	439	25	.	.	PUNCT
ejpam-5252	440	1	j.	j.	PROPN
ejpam-5252	440	2	pure	pure	PROPN
ejpam-5252	440	3	appl	appl	PROPN
ejpam-5252	440	4	.	.	PROPN
ejpam-5252	440	5	math	math	PROPN
ejpam-5252	440	6	,	,	PUNCT
ejpam-5252	440	7	18	18	NUM
ejpam-5252	440	8	(	(	PUNCT
ejpam-5252	440	9	1	1	NUM
ejpam-5252	440	10	)	)	PUNCT
ejpam-5252	440	11	(	(	PUNCT
ejpam-5252	440	12	2025	2025	NUM
ejpam-5252	440	13	)	)	PUNCT
ejpam-5252	440	14	,	,	PUNCT
ejpam-5252	440	15	5252	5252	NUM
ejpam-5252	440	16	12	12	NUM
ejpam-5252	440	17	of	of	ADP
ejpam-5252	440	18	18	18	NUM
ejpam-5252	440	19	5	5	NUM
ejpam-5252	440	20	.	.	PUNCT
ejpam-5252	441	1	on	on	ADP
ejpam-5252	441	2	the	the	DET
ejpam-5252	441	3	join	join	NOUN
ejpam-5252	441	4	of	of	ADP
ejpam-5252	441	5	graphs	graph	NOUN
ejpam-5252	441	6	given	give	VERB
ejpam-5252	441	7	two	two	NUM
ejpam-5252	441	8	graphs	graph	NOUN
ejpam-5252	441	9	g	g	NOUN
ejpam-5252	441	10	and	and	CCONJ
ejpam-5252	441	11	h	h	NOUN
ejpam-5252	441	12	with	with	ADP
ejpam-5252	441	13	disjoint	disjoint	ADJ
ejpam-5252	441	14	vertex	vertex	NOUN
ejpam-5252	441	15	sets	set	NOUN
ejpam-5252	441	16	,	,	PUNCT
ejpam-5252	441	17	the	the	DET
ejpam-5252	441	18	join	join	NOUN
ejpam-5252	441	19	g+h	g+h	PROPN
ejpam-5252	441	20	of	of	ADP
ejpam-5252	441	21	graphs	graph	NOUN
ejpam-5252	441	22	g	g	PROPN
ejpam-5252	441	23	and	and	CCONJ
ejpam-5252	441	24	h	h	NOUN
ejpam-5252	441	25	,	,	PUNCT
ejpam-5252	441	26	is	be	AUX
ejpam-5252	441	27	the	the	DET
ejpam-5252	441	28	graph	graph	NOUN
ejpam-5252	441	29	with	with	ADP
ejpam-5252	441	30	vertex	vertex	NOUN
ejpam-5252	441	31	-	-	PUNCT
ejpam-5252	441	32	set	set	VERB
ejpam-5252	441	33	v	v	NOUN
ejpam-5252	441	34	(	(	PUNCT
ejpam-5252	441	35	g+h	g+h	NOUN
ejpam-5252	441	36	)	)	PUNCT
ejpam-5252	441	37	=	=	SYM
ejpam-5252	441	38	v	v	X
ejpam-5252	441	39	(	(	PUNCT
ejpam-5252	441	40	g)∪v	g)∪v	NOUN
ejpam-5252	441	41	(	(	PUNCT
ejpam-5252	441	42	h	h	NOUN
ejpam-5252	441	43	)	)	PUNCT
ejpam-5252	441	44	and	and	CCONJ
ejpam-5252	441	45	edge	edge	NOUN
ejpam-5252	441	46	-	-	PUNCT
ejpam-5252	441	47	set	set	VERB
ejpam-5252	441	48	e(g	e(g	NOUN
ejpam-5252	441	49	+	+	CCONJ
ejpam-5252	441	50	h	h	NOUN
ejpam-5252	441	51	)	)	PUNCT
ejpam-5252	441	52	=	=	SYM
ejpam-5252	441	53	e(g	e(g	PROPN
ejpam-5252	441	54	)	)	PUNCT
ejpam-5252	441	55	∪	∪	ADP
ejpam-5252	441	56	e(h	e(h	PROPN
ejpam-5252	441	57	)	)	PUNCT
ejpam-5252	441	58	∪	∪	NOUN
ejpam-5252	441	59	{	{	PUNCT
ejpam-5252	441	60	uv	uv	NOUN
ejpam-5252	441	61	:	:	PUNCT
ejpam-5252	441	62	u	u	PROPN
ejpam-5252	441	63	∈	∈	PROPN
ejpam-5252	441	64	v	v	ADP
ejpam-5252	441	65	(	(	PUNCT
ejpam-5252	441	66	g	g	NOUN
ejpam-5252	441	67	)	)	PUNCT
ejpam-5252	441	68	and	and	CCONJ
ejpam-5252	441	69	v	v	ADP
ejpam-5252	441	70	∈	∈	PROPN
ejpam-5252	441	71	v	v	NOUN
ejpam-5252	441	72	(	(	PUNCT
ejpam-5252	441	73	h	h	NOUN
ejpam-5252	441	74	)	)	PUNCT
ejpam-5252	441	75	}	}	PUNCT
ejpam-5252	442	1	[	[	X
ejpam-5252	442	2	3	3	NUM
ejpam-5252	442	3	]	]	PUNCT
ejpam-5252	442	4	.	.	PUNCT
ejpam-5252	443	1	in	in	ADP
ejpam-5252	443	2	this	this	DET
ejpam-5252	443	3	section	section	NOUN
ejpam-5252	443	4	,	,	PUNCT
ejpam-5252	443	5	the	the	DET
ejpam-5252	443	6	following	follow	VERB
ejpam-5252	443	7	proposition	proposition	NOUN
ejpam-5252	443	8	characterizes	characterize	VERB
ejpam-5252	443	9	allmrdf	allmrdf	NOUN
ejpam-5252	443	10	on	on	ADP
ejpam-5252	443	11	the	the	DET
ejpam-5252	443	12	join	join	NOUN
ejpam-5252	443	13	of	of	ADP
ejpam-5252	443	14	graphs	graph	NOUN
ejpam-5252	443	15	.	.	PUNCT
ejpam-5252	444	1	proposition	proposition	NOUN
ejpam-5252	444	2	13	13	NUM
ejpam-5252	444	3	.	.	PUNCT
ejpam-5252	445	1	let	let	VERB
ejpam-5252	445	2	g	g	NOUN
ejpam-5252	445	3	and	and	CCONJ
ejpam-5252	445	4	h	h	NOUN
ejpam-5252	445	5	be	be	VERB
ejpam-5252	445	6	any	any	DET
ejpam-5252	445	7	graphs	graph	NOUN
ejpam-5252	445	8	and	and	CCONJ
ejpam-5252	445	9	let	let	VERB
ejpam-5252	445	10	f	f	PROPN
ejpam-5252	445	11	∈	∈	PROPN
ejpam-5252	445	12	(	(	PUNCT
ejpam-5252	445	13	v0	v0	NOUN
ejpam-5252	445	14	,	,	PUNCT
ejpam-5252	445	15	v1	v1	NOUN
ejpam-5252	445	16	,	,	PUNCT
ejpam-5252	445	17	v2	v2	PROPN
ejpam-5252	445	18	,	,	PUNCT
ejpam-5252	445	19	v3	v3	PROPN
ejpam-5252	445	20	)	)	PUNCT
ejpam-5252	445	21	be	be	VERB
ejpam-5252	445	22	a	a	DET
ejpam-5252	445	23	function	function	NOUN
ejpam-5252	445	24	on	on	ADP
ejpam-5252	445	25	v	v	NOUN
ejpam-5252	445	26	(	(	PUNCT
ejpam-5252	445	27	g+h	g+h	PROPN
ejpam-5252	445	28	)	)	PUNCT
ejpam-5252	445	29	with	with	ADP
ejpam-5252	445	30	v2	v2	PROPN
ejpam-5252	445	31	̸=	̸=	PROPN
ejpam-5252	445	32	∅	∅	NOUN
ejpam-5252	445	33	and	and	CCONJ
ejpam-5252	445	34	v3	v3	PROPN
ejpam-5252	445	35	̸=	̸=	PROPN
ejpam-5252	445	36	∅.	∅.	ADV
ejpam-5252	445	37	then	then	ADV
ejpam-5252	445	38	f	f	PROPN
ejpam-5252	445	39	∈	∈	PROPN
ejpam-5252	445	40	mrdf	mrdf	NOUN
ejpam-5252	445	41	(	(	PUNCT
ejpam-5252	445	42	g+h	g+h	NOUN
ejpam-5252	445	43	)	)	PUNCT
ejpam-5252	446	1	if	if	SCONJ
ejpam-5252	446	2	and	and	CCONJ
ejpam-5252	446	3	only	only	ADV
ejpam-5252	446	4	if	if	SCONJ
ejpam-5252	446	5	one	one	NUM
ejpam-5252	446	6	of	of	ADP
ejpam-5252	446	7	the	the	DET
ejpam-5252	446	8	following	follow	VERB
ejpam-5252	446	9	holds	hold	VERB
ejpam-5252	446	10	:	:	PUNCT
ejpam-5252	446	11	(	(	PUNCT
ejpam-5252	446	12	i	i	NOUN
ejpam-5252	446	13	)	)	PUNCT
ejpam-5252	446	14	f	f	PROPN
ejpam-5252	446	15	|g	|g	PROPN
ejpam-5252	446	16	∈	∈	PROPN
ejpam-5252	446	17	mrdf	mrdf	NOUN
ejpam-5252	446	18	(	(	PUNCT
ejpam-5252	446	19	g	g	NOUN
ejpam-5252	446	20	)	)	PUNCT
ejpam-5252	446	21	and	and	CCONJ
ejpam-5252	446	22	one	one	NUM
ejpam-5252	446	23	of	of	ADP
ejpam-5252	446	24	the	the	DET
ejpam-5252	446	25	following	following	NOUN
ejpam-5252	446	26	holds	hold	VERB
ejpam-5252	446	27	:	:	PUNCT
ejpam-5252	446	28	(	(	PUNCT
ejpam-5252	446	29	a	a	X
ejpam-5252	446	30	)	)	PUNCT
ejpam-5252	446	31	|v2	|v2	NOUN
ejpam-5252	446	32	∩	∩	ADJ
ejpam-5252	446	33	v	v	X
ejpam-5252	446	34	(	(	PUNCT
ejpam-5252	446	35	g)|	g)|	VERB
ejpam-5252	446	36	≥	≥	NOUN
ejpam-5252	446	37	1	1	NUM
ejpam-5252	446	38	and	and	CCONJ
ejpam-5252	446	39	|v3	|v3	NOUN
ejpam-5252	446	40	∩	∩	ADJ
ejpam-5252	446	41	v	v	X
ejpam-5252	446	42	(	(	PUNCT
ejpam-5252	446	43	g)|	g)|	X
ejpam-5252	446	44	≥	≥	NOUN
ejpam-5252	446	45	1	1	NUM
ejpam-5252	446	46	(	(	PUNCT
ejpam-5252	446	47	b	b	NOUN
ejpam-5252	446	48	)	)	PUNCT
ejpam-5252	446	49	v2	v2	NOUN
ejpam-5252	446	50	∩	∩	ADJ
ejpam-5252	446	51	v	v	NOUN
ejpam-5252	446	52	(	(	PUNCT
ejpam-5252	446	53	g	g	NOUN
ejpam-5252	446	54	)	)	PUNCT
ejpam-5252	446	55	=	=	NOUN
ejpam-5252	446	56	∅	∅	NOUN
ejpam-5252	446	57	and	and	CCONJ
ejpam-5252	446	58	each	each	PRON
ejpam-5252	446	59	of	of	ADP
ejpam-5252	446	60	the	the	DET
ejpam-5252	446	61	following	follow	VERB
ejpam-5252	446	62	holds	hold	VERB
ejpam-5252	446	63	:	:	PUNCT
ejpam-5252	446	64	(	(	PUNCT
ejpam-5252	446	65	b1	b1	NOUN
ejpam-5252	446	66	)	)	PUNCT
ejpam-5252	446	67	v3	v3	PROPN
ejpam-5252	446	68	is	be	AUX
ejpam-5252	446	69	a	a	DET
ejpam-5252	446	70	dominating	dominating	NOUN
ejpam-5252	446	71	set	set	NOUN
ejpam-5252	446	72	of	of	ADP
ejpam-5252	446	73	g.	g.	PROPN
ejpam-5252	446	74	(	(	PUNCT
ejpam-5252	446	75	b2	b2	NOUN
ejpam-5252	446	76	)	)	PUNCT
ejpam-5252	446	77	v2	v2	NOUN
ejpam-5252	446	78	∩	∩	ADJ
ejpam-5252	446	79	v	v	NOUN
ejpam-5252	446	80	(	(	PUNCT
ejpam-5252	446	81	h	h	NOUN
ejpam-5252	446	82	)	)	PUNCT
ejpam-5252	446	83	is	be	AUX
ejpam-5252	446	84	a	a	DET
ejpam-5252	446	85	dominating	dominating	NOUN
ejpam-5252	446	86	set	set	NOUN
ejpam-5252	446	87	of	of	ADP
ejpam-5252	446	88	h[v0	h[v0	NOUN
ejpam-5252	446	89	]	]	PUNCT
ejpam-5252	446	90	.	.	PUNCT
ejpam-5252	447	1	(	(	PUNCT
ejpam-5252	447	2	c	c	X
ejpam-5252	447	3	)	)	PUNCT
ejpam-5252	447	4	v3	v3	PROPN
ejpam-5252	447	5	∩	∩	ADJ
ejpam-5252	447	6	v	v	X
ejpam-5252	447	7	(	(	PUNCT
ejpam-5252	447	8	g	g	NOUN
ejpam-5252	447	9	)	)	PUNCT
ejpam-5252	447	10	=	=	NOUN
ejpam-5252	447	11	∅	∅	NOUN
ejpam-5252	447	12	and	and	CCONJ
ejpam-5252	447	13	each	each	PRON
ejpam-5252	447	14	of	of	ADP
ejpam-5252	447	15	the	the	DET
ejpam-5252	447	16	following	follow	VERB
ejpam-5252	447	17	holds	hold	VERB
ejpam-5252	447	18	:	:	PUNCT
ejpam-5252	447	19	(	(	PUNCT
ejpam-5252	447	20	c1	c1	NOUN
ejpam-5252	447	21	)	)	PUNCT
ejpam-5252	447	22	v2	v2	PROPN
ejpam-5252	447	23	is	be	AUX
ejpam-5252	447	24	a	a	DET
ejpam-5252	447	25	dominating	dominating	NOUN
ejpam-5252	447	26	set	set	NOUN
ejpam-5252	447	27	of	of	ADP
ejpam-5252	447	28	g.	g.	PROPN
ejpam-5252	447	29	(	(	PUNCT
ejpam-5252	447	30	c2	c2	PROPN
ejpam-5252	447	31	)	)	PUNCT
ejpam-5252	447	32	v3	v3	PROPN
ejpam-5252	447	33	∩	∩	ADJ
ejpam-5252	447	34	v	v	X
ejpam-5252	447	35	(	(	PUNCT
ejpam-5252	447	36	h	h	NOUN
ejpam-5252	447	37	)	)	PUNCT
ejpam-5252	447	38	is	be	AUX
ejpam-5252	447	39	a	a	DET
ejpam-5252	447	40	dominating	dominating	NOUN
ejpam-5252	447	41	set	set	NOUN
ejpam-5252	447	42	of	of	ADP
ejpam-5252	447	43	h[v0	h[v0	NOUN
ejpam-5252	447	44	]	]	PUNCT
ejpam-5252	447	45	.	.	PUNCT
ejpam-5252	448	1	(	(	PUNCT
ejpam-5252	448	2	ii	ii	X
ejpam-5252	448	3	)	)	PUNCT
ejpam-5252	448	4	f	f	PROPN
ejpam-5252	448	5	|h	|h	X
ejpam-5252	448	6	∈	∈	PROPN
ejpam-5252	448	7	mrdf	mrdf	NOUN
ejpam-5252	448	8	(	(	PUNCT
ejpam-5252	448	9	h	h	NOUN
ejpam-5252	448	10	)	)	PUNCT
ejpam-5252	448	11	and	and	CCONJ
ejpam-5252	448	12	one	one	NUM
ejpam-5252	448	13	of	of	ADP
ejpam-5252	448	14	the	the	DET
ejpam-5252	448	15	following	following	NOUN
ejpam-5252	448	16	holds	hold	VERB
ejpam-5252	448	17	:	:	PUNCT
ejpam-5252	448	18	(	(	PUNCT
ejpam-5252	448	19	a	a	X
ejpam-5252	448	20	)	)	PUNCT
ejpam-5252	448	21	|v2	|v2	NOUN
ejpam-5252	448	22	∩	∩	ADJ
ejpam-5252	448	23	v	v	X
ejpam-5252	448	24	(	(	PUNCT
ejpam-5252	448	25	h)|	h)|	PROPN
ejpam-5252	448	26	≥	≥	NUM
ejpam-5252	448	27	1	1	NUM
ejpam-5252	448	28	and	and	CCONJ
ejpam-5252	448	29	|v3	|v3	NOUN
ejpam-5252	448	30	∩	∩	ADJ
ejpam-5252	448	31	v	v	X
ejpam-5252	448	32	(	(	PUNCT
ejpam-5252	448	33	h)|	h)|	PROPN
ejpam-5252	448	34	≥	≥	NUM
ejpam-5252	448	35	1	1	NUM
ejpam-5252	448	36	(	(	PUNCT
ejpam-5252	448	37	b	b	NOUN
ejpam-5252	448	38	)	)	PUNCT
ejpam-5252	448	39	v2	v2	NOUN
ejpam-5252	448	40	∩	∩	ADJ
ejpam-5252	448	41	v	v	NOUN
ejpam-5252	448	42	(	(	PUNCT
ejpam-5252	448	43	h	h	NOUN
ejpam-5252	448	44	)	)	PUNCT
ejpam-5252	448	45	=	=	NOUN
ejpam-5252	448	46	∅	∅	NOUN
ejpam-5252	448	47	and	and	CCONJ
ejpam-5252	448	48	each	each	PRON
ejpam-5252	448	49	of	of	ADP
ejpam-5252	448	50	the	the	DET
ejpam-5252	448	51	following	follow	VERB
ejpam-5252	448	52	holds	hold	VERB
ejpam-5252	448	53	:	:	PUNCT
ejpam-5252	448	54	(	(	PUNCT
ejpam-5252	448	55	b1	b1	NOUN
ejpam-5252	448	56	)	)	PUNCT
ejpam-5252	448	57	v3	v3	PROPN
ejpam-5252	448	58	is	be	AUX
ejpam-5252	448	59	a	a	DET
ejpam-5252	448	60	dominating	dominating	NOUN
ejpam-5252	448	61	set	set	NOUN
ejpam-5252	448	62	of	of	ADP
ejpam-5252	448	63	h.	h.	PROPN
ejpam-5252	448	64	(	(	PUNCT
ejpam-5252	448	65	b2	b2	NOUN
ejpam-5252	448	66	)	)	PUNCT
ejpam-5252	448	67	v2	v2	PROPN
ejpam-5252	448	68	∩	∩	ADJ
ejpam-5252	448	69	v	v	NOUN
ejpam-5252	448	70	(	(	PUNCT
ejpam-5252	448	71	g	g	NOUN
ejpam-5252	448	72	)	)	PUNCT
ejpam-5252	448	73	is	be	AUX
ejpam-5252	448	74	a	a	DET
ejpam-5252	448	75	dominating	dominating	NOUN
ejpam-5252	448	76	set	set	NOUN
ejpam-5252	448	77	of	of	ADP
ejpam-5252	448	78	g[v0	g[v0	NOUN
ejpam-5252	448	79	]	]	PUNCT
ejpam-5252	448	80	.	.	PUNCT
ejpam-5252	449	1	(	(	PUNCT
ejpam-5252	449	2	c	c	X
ejpam-5252	449	3	)	)	PUNCT
ejpam-5252	449	4	v3	v3	PROPN
ejpam-5252	449	5	∩	∩	ADJ
ejpam-5252	449	6	v	v	X
ejpam-5252	449	7	(	(	PUNCT
ejpam-5252	449	8	h	h	NOUN
ejpam-5252	449	9	)	)	PUNCT
ejpam-5252	449	10	=	=	NOUN
ejpam-5252	449	11	∅	∅	NOUN
ejpam-5252	449	12	and	and	CCONJ
ejpam-5252	449	13	each	each	PRON
ejpam-5252	449	14	of	of	ADP
ejpam-5252	449	15	the	the	DET
ejpam-5252	449	16	following	follow	VERB
ejpam-5252	449	17	holds	hold	VERB
ejpam-5252	449	18	:	:	PUNCT
ejpam-5252	449	19	(	(	PUNCT
ejpam-5252	449	20	c1	c1	NOUN
ejpam-5252	449	21	)	)	PUNCT
ejpam-5252	449	22	v2	v2	PROPN
ejpam-5252	449	23	is	be	AUX
ejpam-5252	449	24	a	a	DET
ejpam-5252	449	25	dominating	dominating	NOUN
ejpam-5252	449	26	set	set	NOUN
ejpam-5252	449	27	of	of	ADP
ejpam-5252	449	28	h.	h.	PROPN
ejpam-5252	449	29	(	(	PUNCT
ejpam-5252	449	30	c2	c2	PROPN
ejpam-5252	449	31	)	)	PUNCT
ejpam-5252	449	32	v3	v3	PROPN
ejpam-5252	449	33	∩	∩	ADJ
ejpam-5252	449	34	v	v	X
ejpam-5252	449	35	(	(	PUNCT
ejpam-5252	449	36	g	g	NOUN
ejpam-5252	449	37	)	)	PUNCT
ejpam-5252	449	38	is	be	AUX
ejpam-5252	449	39	a	a	DET
ejpam-5252	449	40	dominating	dominating	NOUN
ejpam-5252	449	41	set	set	NOUN
ejpam-5252	449	42	of	of	ADP
ejpam-5252	449	43	g[v0	g[v0	NOUN
ejpam-5252	449	44	]	]	PUNCT
ejpam-5252	449	45	.	.	PUNCT
ejpam-5252	450	1	(	(	PUNCT
ejpam-5252	450	2	iii	iii	X
ejpam-5252	450	3	)	)	PUNCT
ejpam-5252	450	4	f	f	PROPN
ejpam-5252	450	5	|g	|g	VERB
ejpam-5252	450	6	̸∈	̸∈	PROPN
ejpam-5252	450	7	mrdf	mrdf	PROPN
ejpam-5252	450	8	(	(	PUNCT
ejpam-5252	450	9	g	g	NOUN
ejpam-5252	450	10	)	)	PUNCT
ejpam-5252	450	11	,	,	PUNCT
ejpam-5252	450	12	f	f	PROPN
ejpam-5252	450	13	|h	|h	X
ejpam-5252	450	14	̸∈	̸∈	PROPN
ejpam-5252	450	15	mrdf	mrdf	PROPN
ejpam-5252	450	16	(	(	PUNCT
ejpam-5252	450	17	h	h	NOUN
ejpam-5252	450	18	)	)	PUNCT
ejpam-5252	450	19	and	and	CCONJ
ejpam-5252	450	20	each	each	PRON
ejpam-5252	450	21	of	of	ADP
ejpam-5252	450	22	the	the	DET
ejpam-5252	450	23	following	following	NOUN
ejpam-5252	450	24	holds	hold	VERB
ejpam-5252	450	25	:	:	PUNCT
ejpam-5252	450	26	(	(	PUNCT
ejpam-5252	450	27	a	a	X
ejpam-5252	450	28	)	)	PUNCT
ejpam-5252	450	29	v2	v2	PROPN
ejpam-5252	450	30	∩	∩	ADJ
ejpam-5252	450	31	v	v	NOUN
ejpam-5252	450	32	(	(	PUNCT
ejpam-5252	450	33	h	h	NOUN
ejpam-5252	450	34	)	)	PUNCT
ejpam-5252	450	35	̸=	̸=	PROPN
ejpam-5252	450	36	∅	∅	NOUN
ejpam-5252	450	37	whenever	whenever	SCONJ
ejpam-5252	450	38	ng(x	ng(x	NUM
ejpam-5252	450	39	)	)	PUNCT
ejpam-5252	450	40	∩	∩	NOUN
ejpam-5252	450	41	v2	v2	NOUN
ejpam-5252	450	42	=	=	PUNCT
ejpam-5252	450	43	∅	∅	NOUN
ejpam-5252	450	44	for	for	ADP
ejpam-5252	450	45	some	some	DET
ejpam-5252	450	46	x	x	SYM
ejpam-5252	450	47	∈	∈	PROPN
ejpam-5252	450	48	v0	v0	NOUN
ejpam-5252	450	49	.	.	PUNCT
ejpam-5252	451	1	(	(	PUNCT
ejpam-5252	451	2	b	b	X
ejpam-5252	451	3	)	)	PUNCT
ejpam-5252	451	4	v3	v3	PROPN
ejpam-5252	451	5	∩	∩	ADJ
ejpam-5252	451	6	v	v	X
ejpam-5252	451	7	(	(	PUNCT
ejpam-5252	451	8	h	h	NOUN
ejpam-5252	451	9	)	)	PUNCT
ejpam-5252	451	10	̸=	̸=	PROPN
ejpam-5252	451	11	∅	∅	NOUN
ejpam-5252	451	12	whenever	whenever	SCONJ
ejpam-5252	451	13	ng(x	ng(x	NUM
ejpam-5252	451	14	)	)	PUNCT
ejpam-5252	451	15	∩	∩	PROPN
ejpam-5252	451	16	v3	v3	NOUN
ejpam-5252	451	17	=	=	PUNCT
ejpam-5252	451	18	∅	∅	NOUN
ejpam-5252	451	19	for	for	ADP
ejpam-5252	451	20	some	some	DET
ejpam-5252	451	21	x	x	SYM
ejpam-5252	451	22	∈	∈	PROPN
ejpam-5252	451	23	v0	v0	NOUN
ejpam-5252	451	24	.	.	PUNCT
ejpam-5252	452	1	(	(	PUNCT
ejpam-5252	452	2	c	c	X
ejpam-5252	452	3	)	)	PUNCT
ejpam-5252	452	4	v2	v2	NOUN
ejpam-5252	452	5	∩v	∩v	NOUN
ejpam-5252	452	6	(	(	PUNCT
ejpam-5252	452	7	h	h	NOUN
ejpam-5252	452	8	)	)	PUNCT
ejpam-5252	452	9	̸=	̸=	PROPN
ejpam-5252	452	10	∅	∅	NOUN
ejpam-5252	452	11	or	or	CCONJ
ejpam-5252	452	12	v3	v3	PROPN
ejpam-5252	452	13	∩v	∩v	PROPN
ejpam-5252	452	14	(	(	PUNCT
ejpam-5252	452	15	h	h	NOUN
ejpam-5252	452	16	)	)	PUNCT
ejpam-5252	452	17	̸=	̸=	NOUN
ejpam-5252	452	18	∅	∅	NOUN
ejpam-5252	452	19	whenever	whenever	SCONJ
ejpam-5252	452	20	∃x	∃x	PROPN
ejpam-5252	452	21	∈	∈	PROPN
ejpam-5252	452	22	v1	v1	NOUN
ejpam-5252	452	23	with	with	ADP
ejpam-5252	452	24	ng(x)∩v2	ng(x)∩v2	ADJ
ejpam-5252	452	25	=	=	SYM
ejpam-5252	452	26	∅	∅	NOUN
ejpam-5252	452	27	and	and	CCONJ
ejpam-5252	452	28	ng(x	ng(x	NUM
ejpam-5252	452	29	)	)	PUNCT
ejpam-5252	452	30	∩	∩	NOUN
ejpam-5252	452	31	v3	v3	PROPN
ejpam-5252	452	32	=	=	SYM
ejpam-5252	452	33	∅	∅	NOUN
ejpam-5252	452	34	(	(	PUNCT
ejpam-5252	452	35	d	d	X
ejpam-5252	452	36	)	)	PUNCT
ejpam-5252	452	37	v2	v2	NOUN
ejpam-5252	452	38	∩	∩	ADJ
ejpam-5252	452	39	v	v	NOUN
ejpam-5252	452	40	(	(	PUNCT
ejpam-5252	452	41	g	g	NOUN
ejpam-5252	452	42	)	)	PUNCT
ejpam-5252	452	43	̸=	̸=	PROPN
ejpam-5252	452	44	∅	∅	NOUN
ejpam-5252	452	45	whenever	whenever	SCONJ
ejpam-5252	452	46	nh(x	nh(x	NUM
ejpam-5252	452	47	)	)	PUNCT
ejpam-5252	452	48	∩	∩	ADJ
ejpam-5252	452	49	v2	v2	NOUN
ejpam-5252	452	50	=	=	PUNCT
ejpam-5252	452	51	∅	∅	NOUN
ejpam-5252	452	52	for	for	ADP
ejpam-5252	452	53	some	some	DET
ejpam-5252	452	54	x	x	SYM
ejpam-5252	452	55	∈	∈	PROPN
ejpam-5252	452	56	v0	v0	NOUN
ejpam-5252	452	57	.	.	PUNCT
ejpam-5252	453	1	(	(	PUNCT
ejpam-5252	453	2	e	e	X
ejpam-5252	453	3	)	)	PUNCT
ejpam-5252	453	4	v3	v3	PROPN
ejpam-5252	453	5	∩	∩	ADJ
ejpam-5252	453	6	v	v	X
ejpam-5252	453	7	(	(	PUNCT
ejpam-5252	453	8	g	g	NOUN
ejpam-5252	453	9	)	)	PUNCT
ejpam-5252	453	10	̸=	̸=	PROPN
ejpam-5252	453	11	∅	∅	NOUN
ejpam-5252	453	12	whenever	whenever	SCONJ
ejpam-5252	453	13	nh(x	nh(x	NUM
ejpam-5252	453	14	)	)	PUNCT
ejpam-5252	453	15	∩	∩	ADJ
ejpam-5252	453	16	v3	v3	NOUN
ejpam-5252	453	17	=	=	PUNCT
ejpam-5252	453	18	∅	∅	NOUN
ejpam-5252	453	19	for	for	ADP
ejpam-5252	453	20	some	some	DET
ejpam-5252	453	21	x	x	SYM
ejpam-5252	453	22	∈	∈	PROPN
ejpam-5252	453	23	v0	v0	NOUN
ejpam-5252	453	24	.	.	PUNCT
ejpam-5252	454	1	s.	s.	PROPN
ejpam-5252	454	2	ahamad	ahamad	PROPN
ejpam-5252	454	3	,	,	PUNCT
ejpam-5252	454	4	j.	j.	PROPN
ejpam-5252	454	5	cariaga	cariaga	PROPN
ejpam-5252	454	6	,	,	PUNCT
ejpam-5252	454	7	s.	s.	PROPN
ejpam-5252	454	8	menchavez	menchavez	PROPN
ejpam-5252	454	9	/	/	PUNCT
ejpam-5252	454	10	eur	eur	PROPN
ejpam-5252	454	11	.	.	PUNCT
ejpam-5252	455	1	j.	j.	PROPN
ejpam-5252	455	2	pure	pure	PROPN
ejpam-5252	455	3	appl	appl	PROPN
ejpam-5252	455	4	.	.	PROPN
ejpam-5252	455	5	math	math	PROPN
ejpam-5252	455	6	,	,	PUNCT
ejpam-5252	455	7	18	18	NUM
ejpam-5252	455	8	(	(	PUNCT
ejpam-5252	455	9	1	1	NUM
ejpam-5252	455	10	)	)	PUNCT
ejpam-5252	455	11	(	(	PUNCT
ejpam-5252	455	12	2025	2025	NUM
ejpam-5252	455	13	)	)	PUNCT
ejpam-5252	455	14	,	,	PUNCT
ejpam-5252	455	15	5252	5252	NUM
ejpam-5252	455	16	13	13	NUM
ejpam-5252	455	17	of	of	ADP
ejpam-5252	455	18	18	18	NUM
ejpam-5252	455	19	(	(	PUNCT
ejpam-5252	455	20	f	f	X
ejpam-5252	455	21	)	)	PUNCT
ejpam-5252	455	22	v2	v2	PROPN
ejpam-5252	455	23	∩	∩	ADJ
ejpam-5252	455	24	v	v	NOUN
ejpam-5252	455	25	(	(	PUNCT
ejpam-5252	455	26	g	g	NOUN
ejpam-5252	455	27	)	)	PUNCT
ejpam-5252	455	28	̸=	̸=	PROPN
ejpam-5252	455	29	∅	∅	NOUN
ejpam-5252	455	30	or	or	CCONJ
ejpam-5252	455	31	v3	v3	PROPN
ejpam-5252	455	32	∩	∩	ADJ
ejpam-5252	455	33	v	v	X
ejpam-5252	455	34	(	(	PUNCT
ejpam-5252	455	35	g	g	NOUN
ejpam-5252	455	36	)	)	PUNCT
ejpam-5252	455	37	̸=	̸=	PROPN
ejpam-5252	455	38	∅	∅	NOUN
ejpam-5252	455	39	whenever	whenever	SCONJ
ejpam-5252	455	40	∃x	∃x	PROPN
ejpam-5252	455	41	∈	∈	PROPN
ejpam-5252	455	42	v1	v1	NOUN
ejpam-5252	455	43	with	with	ADP
ejpam-5252	455	44	nh(x)∩	nh(x)∩	NOUN
ejpam-5252	455	45	v2	v2	NOUN
ejpam-5252	455	46	=	=	NOUN
ejpam-5252	455	47	∅	∅	NOUN
ejpam-5252	455	48	and	and	CCONJ
ejpam-5252	455	49	nh(x	nh(x	NUM
ejpam-5252	455	50	)	)	PUNCT
ejpam-5252	455	51	∩	∩	ADJ
ejpam-5252	455	52	v3	v3	PROPN
ejpam-5252	455	53	=	=	SYM
ejpam-5252	455	54	∅	∅	NOUN
ejpam-5252	455	55	proof	proof	NOUN
ejpam-5252	455	56	.	.	PUNCT
ejpam-5252	456	1	suppose	suppose	VERB
ejpam-5252	456	2	f	f	PROPN
ejpam-5252	456	3	|g	|g	PROPN
ejpam-5252	456	4	∈	∈	PROPN
ejpam-5252	456	5	mrdf	mrdf	NOUN
ejpam-5252	456	6	(	(	PUNCT
ejpam-5252	456	7	g	g	NOUN
ejpam-5252	456	8	)	)	PUNCT
ejpam-5252	456	9	.	.	PUNCT
ejpam-5252	457	1	assume	assume	VERB
ejpam-5252	457	2	that	that	SCONJ
ejpam-5252	457	3	(	(	PUNCT
ejpam-5252	457	4	i)(a	i)(a	NOUN
ejpam-5252	457	5	)	)	PUNCT
ejpam-5252	457	6	holds	hold	VERB
ejpam-5252	457	7	.	.	PUNCT
ejpam-5252	458	1	let	let	VERB
ejpam-5252	458	2	v	v	NUM
ejpam-5252	458	3	∈	∈	PROPN
ejpam-5252	458	4	v0	v0	NOUN
ejpam-5252	458	5	.	.	PUNCT
ejpam-5252	459	1	if	if	SCONJ
ejpam-5252	459	2	v	v	NUM
ejpam-5252	459	3	∈	∈	PROPN
ejpam-5252	459	4	v	v	NOUN
ejpam-5252	459	5	(	(	PUNCT
ejpam-5252	459	6	g	g	NOUN
ejpam-5252	459	7	)	)	PUNCT
ejpam-5252	459	8	,	,	PUNCT
ejpam-5252	459	9	then	then	ADV
ejpam-5252	459	10	there	there	PRON
ejpam-5252	459	11	exist	exist	VERB
ejpam-5252	459	12	u	u	NOUN
ejpam-5252	459	13	,	,	PUNCT
ejpam-5252	459	14	w	w	PROPN
ejpam-5252	459	15	∈	∈	PROPN
ejpam-5252	459	16	v	v	ADP
ejpam-5252	459	17	(	(	PUNCT
ejpam-5252	459	18	g	g	NOUN
ejpam-5252	459	19	)	)	PUNCT
ejpam-5252	459	20	such	such	ADJ
ejpam-5252	459	21	that	that	SCONJ
ejpam-5252	459	22	{	{	PUNCT
ejpam-5252	459	23	u	u	NOUN
ejpam-5252	459	24	,	,	PUNCT
ejpam-5252	459	25	w	w	NOUN
ejpam-5252	459	26	}	}	PUNCT
ejpam-5252	459	27	⊆	⊆	NUM
ejpam-5252	459	28	ng(v	ng(v	PUNCT
ejpam-5252	459	29	)	)	PUNCT
ejpam-5252	459	30	and	and	CCONJ
ejpam-5252	459	31	f(u	f(u	PROPN
ejpam-5252	459	32	)	)	PUNCT
ejpam-5252	459	33	=	=	SYM
ejpam-5252	459	34	2	2	NUM
ejpam-5252	459	35	and	and	CCONJ
ejpam-5252	459	36	f(w	f(w	NUM
ejpam-5252	459	37	)	)	PUNCT
ejpam-5252	459	38	=	=	SYM
ejpam-5252	459	39	3	3	X
ejpam-5252	459	40	,	,	PUNCT
ejpam-5252	459	41	by	by	ADP
ejpam-5252	459	42	(	(	PUNCT
ejpam-5252	459	43	p1	p1	PROPN
ejpam-5252	459	44	)	)	PUNCT
ejpam-5252	459	45	.	.	PUNCT
ejpam-5252	460	1	this	this	PRON
ejpam-5252	460	2	implies	imply	VERB
ejpam-5252	460	3	that	that	SCONJ
ejpam-5252	460	4	{	{	PUNCT
ejpam-5252	460	5	u	u	NOUN
ejpam-5252	460	6	,	,	PUNCT
ejpam-5252	460	7	w	w	NOUN
ejpam-5252	460	8	}	}	PUNCT
ejpam-5252	460	9	⊆	⊆	NUM
ejpam-5252	460	10	ng+h(v	ng+h(v	NOUN
ejpam-5252	460	11	)	)	PUNCT
ejpam-5252	460	12	.	.	PUNCT
ejpam-5252	461	1	now	now	ADV
ejpam-5252	461	2	,	,	PUNCT
ejpam-5252	461	3	assume	assume	VERB
ejpam-5252	461	4	that	that	SCONJ
ejpam-5252	461	5	v	v	X
ejpam-5252	461	6	∈	∈	PROPN
ejpam-5252	461	7	v	v	NOUN
ejpam-5252	461	8	(	(	PUNCT
ejpam-5252	461	9	h	h	NOUN
ejpam-5252	461	10	)	)	PUNCT
ejpam-5252	461	11	.	.	PUNCT
ejpam-5252	462	1	note	note	VERB
ejpam-5252	462	2	that	that	SCONJ
ejpam-5252	462	3	|v2	|v2	NOUN
ejpam-5252	462	4	∩	∩	PROPN
ejpam-5252	462	5	v	v	X
ejpam-5252	462	6	(	(	PUNCT
ejpam-5252	462	7	g)|	g)|	VERB
ejpam-5252	462	8	≥	≥	NOUN
ejpam-5252	462	9	1	1	NUM
ejpam-5252	462	10	and	and	CCONJ
ejpam-5252	462	11	|v3	|v3	NOUN
ejpam-5252	462	12	∩	∩	ADJ
ejpam-5252	462	13	v	v	X
ejpam-5252	462	14	(	(	PUNCT
ejpam-5252	462	15	g)|	g)|	VERB
ejpam-5252	462	16	≥	≥	NOUN
ejpam-5252	462	17	1	1	NUM
ejpam-5252	462	18	.	.	PUNCT
ejpam-5252	463	1	now	now	ADV
ejpam-5252	463	2	,	,	PUNCT
ejpam-5252	463	3	take	take	VERB
ejpam-5252	463	4	u	u	PRON
ejpam-5252	463	5	∈	∈	PROPN
ejpam-5252	463	6	v2	v2	PROPN
ejpam-5252	463	7	∩	∩	ADJ
ejpam-5252	463	8	v	v	NOUN
ejpam-5252	463	9	(	(	PUNCT
ejpam-5252	463	10	g	g	NOUN
ejpam-5252	463	11	)	)	PUNCT
ejpam-5252	463	12	and	and	CCONJ
ejpam-5252	463	13	w	w	PROPN
ejpam-5252	463	14	∈	∈	PROPN
ejpam-5252	463	15	v3	v3	PROPN
ejpam-5252	463	16	∩	∩	PROPN
ejpam-5252	463	17	v	v	X
ejpam-5252	463	18	(	(	PUNCT
ejpam-5252	463	19	g	g	NOUN
ejpam-5252	463	20	)	)	PUNCT
ejpam-5252	463	21	such	such	ADJ
ejpam-5252	463	22	that	that	SCONJ
ejpam-5252	463	23	vu	vu	PROPN
ejpam-5252	463	24	,	,	PUNCT
ejpam-5252	463	25	vw	vw	PROPN
ejpam-5252	463	26	∈	∈	PROPN
ejpam-5252	463	27	e(g	e(g	PROPN
ejpam-5252	464	1	+	+	NOUN
ejpam-5252	464	2	h	h	NOUN
ejpam-5252	464	3	)	)	PUNCT
ejpam-5252	464	4	.	.	PUNCT
ejpam-5252	465	1	thus	thus	ADV
ejpam-5252	465	2	,	,	PUNCT
ejpam-5252	465	3	{	{	PUNCT
ejpam-5252	465	4	u	u	NOUN
ejpam-5252	465	5	,	,	PUNCT
ejpam-5252	465	6	v	v	NOUN
ejpam-5252	465	7	}	}	PUNCT
ejpam-5252	465	8	⊆	⊆	NUM
ejpam-5252	465	9	ng+h(v	ng+h(v	NOUN
ejpam-5252	465	10	)	)	PUNCT
ejpam-5252	465	11	.	.	PUNCT
ejpam-5252	466	1	moreover	moreover	ADV
ejpam-5252	466	2	,	,	PUNCT
ejpam-5252	466	3	let	let	VERB
ejpam-5252	466	4	v	v	NUM
ejpam-5252	466	5	∈	∈	PROPN
ejpam-5252	466	6	v1	v1	NOUN
ejpam-5252	466	7	.	.	PUNCT
ejpam-5252	467	1	assume	assume	VERB
ejpam-5252	467	2	v	v	ADP
ejpam-5252	467	3	∈	∈	PROPN
ejpam-5252	467	4	v	v	NOUN
ejpam-5252	467	5	(	(	PUNCT
ejpam-5252	467	6	g	g	NOUN
ejpam-5252	467	7	)	)	PUNCT
ejpam-5252	467	8	.	.	PUNCT
ejpam-5252	468	1	then	then	ADV
ejpam-5252	468	2	there	there	PRON
ejpam-5252	468	3	exists	exist	VERB
ejpam-5252	468	4	z	z	NOUN
ejpam-5252	468	5	∈	∈	PROPN
ejpam-5252	468	6	v2	v2	PROPN
ejpam-5252	468	7	∩	∩	ADJ
ejpam-5252	468	8	v	v	NOUN
ejpam-5252	468	9	(	(	PUNCT
ejpam-5252	468	10	g	g	NOUN
ejpam-5252	468	11	)	)	PUNCT
ejpam-5252	468	12	or	or	CCONJ
ejpam-5252	468	13	z	z	NOUN
ejpam-5252	468	14	∈	∈	PROPN
ejpam-5252	468	15	v3	v3	PROPN
ejpam-5252	468	16	∩	∩	PROPN
ejpam-5252	468	17	v	v	X
ejpam-5252	468	18	(	(	PUNCT
ejpam-5252	468	19	g	g	NOUN
ejpam-5252	468	20	)	)	PUNCT
ejpam-5252	468	21	such	such	ADJ
ejpam-5252	468	22	that	that	SCONJ
ejpam-5252	468	23	z	z	PROPN
ejpam-5252	468	24	∈	∈	PROPN
ejpam-5252	468	25	ng(v	ng(v	PUNCT
ejpam-5252	468	26	)	)	PUNCT
ejpam-5252	468	27	by	by	ADP
ejpam-5252	468	28	(	(	PUNCT
ejpam-5252	468	29	p2	p2	PROPN
ejpam-5252	468	30	)	)	PUNCT
ejpam-5252	468	31	.	.	PUNCT
ejpam-5252	469	1	this	this	PRON
ejpam-5252	469	2	means	mean	VERB
ejpam-5252	469	3	that	that	SCONJ
ejpam-5252	469	4	z	z	PROPN
ejpam-5252	469	5	∈	∈	PROPN
ejpam-5252	469	6	ng+h(v	ng+h(v	NOUN
ejpam-5252	469	7	)	)	PUNCT
ejpam-5252	469	8	.	.	PUNCT
ejpam-5252	470	1	now	now	ADV
ejpam-5252	470	2	,	,	PUNCT
ejpam-5252	470	3	assume	assume	VERB
ejpam-5252	470	4	v	v	ADP
ejpam-5252	470	5	∈	∈	PROPN
ejpam-5252	470	6	v	v	NOUN
ejpam-5252	470	7	(	(	PUNCT
ejpam-5252	470	8	h	h	NOUN
ejpam-5252	470	9	)	)	PUNCT
ejpam-5252	470	10	.	.	PUNCT
ejpam-5252	471	1	since	since	SCONJ
ejpam-5252	471	2	|v2	|v2	NOUN
ejpam-5252	471	3	∩	∩	PROPN
ejpam-5252	471	4	v	v	X
ejpam-5252	471	5	(	(	PUNCT
ejpam-5252	471	6	g)|	g)|	VERB
ejpam-5252	471	7	≥	≥	NOUN
ejpam-5252	471	8	1	1	NUM
ejpam-5252	471	9	and	and	CCONJ
ejpam-5252	471	10	|v3	|v3	NOUN
ejpam-5252	471	11	∩	∩	ADJ
ejpam-5252	471	12	v	v	X
ejpam-5252	471	13	(	(	PUNCT
ejpam-5252	471	14	g)|	g)|	VERB
ejpam-5252	471	15	≥	≥	NOUN
ejpam-5252	471	16	1	1	NUM
ejpam-5252	471	17	,	,	PUNCT
ejpam-5252	471	18	there	there	PRON
ejpam-5252	471	19	exists	exist	VERB
ejpam-5252	471	20	z	z	NOUN
ejpam-5252	471	21	∈	∈	PROPN
ejpam-5252	471	22	v2	v2	PROPN
ejpam-5252	471	23	∩	∩	ADJ
ejpam-5252	471	24	v	v	NOUN
ejpam-5252	471	25	(	(	PUNCT
ejpam-5252	471	26	g	g	NOUN
ejpam-5252	471	27	)	)	PUNCT
ejpam-5252	471	28	or	or	CCONJ
ejpam-5252	471	29	z	z	NOUN
ejpam-5252	471	30	∈	∈	PROPN
ejpam-5252	471	31	v3	v3	PROPN
ejpam-5252	471	32	∩	∩	PROPN
ejpam-5252	471	33	v	v	X
ejpam-5252	471	34	(	(	PUNCT
ejpam-5252	471	35	g	g	NOUN
ejpam-5252	471	36	)	)	PUNCT
ejpam-5252	471	37	such	such	ADJ
ejpam-5252	471	38	that	that	SCONJ
ejpam-5252	471	39	z	z	PROPN
ejpam-5252	471	40	∈	∈	PROPN
ejpam-5252	471	41	ng+h(v	ng+h(v	NOUN
ejpam-5252	471	42	)	)	PUNCT
ejpam-5252	471	43	.	.	PUNCT
ejpam-5252	472	1	thus	thus	ADV
ejpam-5252	472	2	,	,	PUNCT
ejpam-5252	472	3	f	f	PROPN
ejpam-5252	472	4	∈	∈	PROPN
ejpam-5252	472	5	mrdf	mrdf	NOUN
ejpam-5252	472	6	(	(	PUNCT
ejpam-5252	472	7	g	g	NOUN
ejpam-5252	472	8	+	+	NOUN
ejpam-5252	472	9	h	h	NOUN
ejpam-5252	472	10	)	)	PUNCT
ejpam-5252	472	11	.	.	PUNCT
ejpam-5252	473	1	similarly	similarly	ADV
ejpam-5252	473	2	,	,	PUNCT
ejpam-5252	473	3	if	if	SCONJ
ejpam-5252	473	4	f	f	PROPN
ejpam-5252	473	5	|h	|h	X
ejpam-5252	473	6	∈	∈	PROPN
ejpam-5252	473	7	mrdf	mrdf	NOUN
ejpam-5252	473	8	(	(	PUNCT
ejpam-5252	473	9	h	h	NOUN
ejpam-5252	473	10	)	)	PUNCT
ejpam-5252	473	11	with	with	ADP
ejpam-5252	473	12	|v2	|v2	NOUN
ejpam-5252	473	13	∩	∩	ADJ
ejpam-5252	473	14	v	v	X
ejpam-5252	473	15	(	(	PUNCT
ejpam-5252	473	16	h)|	h)|	PROPN
ejpam-5252	473	17	≥	≥	NUM
ejpam-5252	473	18	1	1	NUM
ejpam-5252	473	19	and	and	CCONJ
ejpam-5252	473	20	|v3	|v3	NOUN
ejpam-5252	473	21	∩	∩	ADJ
ejpam-5252	473	22	v	v	X
ejpam-5252	473	23	(	(	PUNCT
ejpam-5252	473	24	h)|	h)|	PROPN
ejpam-5252	473	25	≥	≥	NUM
ejpam-5252	473	26	1	1	NUM
ejpam-5252	473	27	,	,	PUNCT
ejpam-5252	473	28	then	then	ADV
ejpam-5252	473	29	f	f	PROPN
ejpam-5252	473	30	∈	∈	PROPN
ejpam-5252	473	31	mrdf	mrdf	NOUN
ejpam-5252	473	32	(	(	PUNCT
ejpam-5252	473	33	g+h	g+h	PROPN
ejpam-5252	473	34	)	)	PUNCT
ejpam-5252	473	35	.	.	PUNCT
ejpam-5252	474	1	assume	assume	VERB
ejpam-5252	474	2	(	(	PUNCT
ejpam-5252	474	3	i)(b	i)(b	NUM
ejpam-5252	474	4	)	)	PUNCT
ejpam-5252	474	5	holds	hold	NOUN
ejpam-5252	474	6	.	.	PUNCT
ejpam-5252	475	1	since	since	SCONJ
ejpam-5252	475	2	v0	v0	NOUN
ejpam-5252	475	3	∩	∩	NOUN
ejpam-5252	475	4	v	v	X
ejpam-5252	475	5	(	(	PUNCT
ejpam-5252	475	6	g	g	NOUN
ejpam-5252	475	7	)	)	PUNCT
ejpam-5252	475	8	=	=	NOUN
ejpam-5252	475	9	∅	∅	NOUN
ejpam-5252	475	10	,	,	PUNCT
ejpam-5252	475	11	v0	v0	PROPN
ejpam-5252	475	12	⊆	⊆	NUM
ejpam-5252	475	13	v	v	NOUN
ejpam-5252	475	14	(	(	PUNCT
ejpam-5252	475	15	h	h	NOUN
ejpam-5252	475	16	)	)	PUNCT
ejpam-5252	475	17	.	.	PUNCT
ejpam-5252	476	1	let	let	VERB
ejpam-5252	476	2	v	v	NUM
ejpam-5252	476	3	∈	∈	PROPN
ejpam-5252	476	4	v0	v0	NOUN
ejpam-5252	476	5	.	.	PUNCT
ejpam-5252	477	1	by	by	ADP
ejpam-5252	477	2	(	(	PUNCT
ejpam-5252	477	3	b2	b2	NOUN
ejpam-5252	477	4	)	)	PUNCT
ejpam-5252	477	5	,	,	PUNCT
ejpam-5252	477	6	there	there	PRON
ejpam-5252	477	7	exists	exist	VERB
ejpam-5252	477	8	u	u	PROPN
ejpam-5252	477	9	∈	∈	PROPN
ejpam-5252	477	10	v2	v2	PROPN
ejpam-5252	477	11	∩	∩	ADJ
ejpam-5252	477	12	v	v	NOUN
ejpam-5252	477	13	(	(	PUNCT
ejpam-5252	477	14	h	h	NOUN
ejpam-5252	477	15	)	)	PUNCT
ejpam-5252	477	16	such	such	ADJ
ejpam-5252	477	17	that	that	SCONJ
ejpam-5252	477	18	uv	uv	PROPN
ejpam-5252	477	19	∈	∈	PROPN
ejpam-5252	477	20	e(h	e(h	PROPN
ejpam-5252	477	21	)	)	PUNCT
ejpam-5252	477	22	⊆	⊆	NUM
ejpam-5252	477	23	e(g+h	e(g+h	NUM
ejpam-5252	477	24	)	)	PUNCT
ejpam-5252	477	25	.	.	PUNCT
ejpam-5252	478	1	also	also	ADV
ejpam-5252	478	2	,	,	PUNCT
ejpam-5252	478	3	since	since	SCONJ
ejpam-5252	478	4	v3	v3	PROPN
ejpam-5252	478	5	is	be	AUX
ejpam-5252	478	6	a	a	DET
ejpam-5252	478	7	dominating	dominating	NOUN
ejpam-5252	478	8	set	set	NOUN
ejpam-5252	478	9	of	of	ADP
ejpam-5252	478	10	g	g	PROPN
ejpam-5252	478	11	,	,	PUNCT
ejpam-5252	478	12	v3	v3	PROPN
ejpam-5252	478	13	∩	∩	PROPN
ejpam-5252	478	14	v	v	X
ejpam-5252	478	15	(	(	PUNCT
ejpam-5252	478	16	g	g	NOUN
ejpam-5252	478	17	)	)	PUNCT
ejpam-5252	478	18	̸=	̸=	PROPN
ejpam-5252	478	19	∅.	∅.	AUX
ejpam-5252	478	20	pick	pick	VERB
ejpam-5252	478	21	u	u	PROPN
ejpam-5252	478	22	∈	∈	PROPN
ejpam-5252	478	23	v3	v3	PROPN
ejpam-5252	478	24	∩	∩	PROPN
ejpam-5252	478	25	v	v	X
ejpam-5252	478	26	(	(	PUNCT
ejpam-5252	478	27	g	g	NOUN
ejpam-5252	478	28	)	)	PUNCT
ejpam-5252	478	29	.	.	PUNCT
ejpam-5252	479	1	then	then	ADV
ejpam-5252	479	2	uv	uv	PROPN
ejpam-5252	479	3	∈	∈	PROPN
ejpam-5252	479	4	e(g+h	e(g+h	NUM
ejpam-5252	479	5	)	)	PUNCT
ejpam-5252	479	6	.	.	PUNCT
ejpam-5252	480	1	let	let	VERB
ejpam-5252	480	2	v	v	NUM
ejpam-5252	480	3	∈	∈	NOUN
ejpam-5252	480	4	v1	v1	NOUN
ejpam-5252	480	5	∩	∩	ADJ
ejpam-5252	480	6	v	v	NOUN
ejpam-5252	480	7	(	(	PUNCT
ejpam-5252	480	8	g	g	NOUN
ejpam-5252	480	9	)	)	PUNCT
ejpam-5252	480	10	.	.	PUNCT
ejpam-5252	481	1	by	by	ADP
ejpam-5252	481	2	(	(	PUNCT
ejpam-5252	481	3	b1	b1	NOUN
ejpam-5252	481	4	)	)	PUNCT
ejpam-5252	481	5	,	,	PUNCT
ejpam-5252	481	6	there	there	PRON
ejpam-5252	481	7	exists	exist	VERB
ejpam-5252	481	8	u	u	PROPN
ejpam-5252	481	9	∈	∈	PROPN
ejpam-5252	481	10	v3	v3	PROPN
ejpam-5252	481	11	∩	∩	PROPN
ejpam-5252	481	12	v	v	X
ejpam-5252	481	13	(	(	PUNCT
ejpam-5252	481	14	g	g	NOUN
ejpam-5252	481	15	)	)	PUNCT
ejpam-5252	481	16	such	such	ADJ
ejpam-5252	481	17	that	that	SCONJ
ejpam-5252	481	18	uv	uv	PROPN
ejpam-5252	481	19	∈	∈	PROPN
ejpam-5252	481	20	e(g	e(g	PROPN
ejpam-5252	481	21	)	)	PUNCT
ejpam-5252	482	1	⊆	⊆	NUM
ejpam-5252	482	2	e(g	e(g	NOUN
ejpam-5252	482	3	+	+	PROPN
ejpam-5252	482	4	h	h	NOUN
ejpam-5252	482	5	)	)	PUNCT
ejpam-5252	482	6	.	.	PUNCT
ejpam-5252	483	1	now	now	ADV
ejpam-5252	483	2	,	,	PUNCT
ejpam-5252	483	3	let	let	VERB
ejpam-5252	483	4	v	v	NUM
ejpam-5252	483	5	∈	∈	NOUN
ejpam-5252	483	6	v1	v1	NOUN
ejpam-5252	483	7	∩	∩	ADJ
ejpam-5252	483	8	v	v	NOUN
ejpam-5252	483	9	(	(	PUNCT
ejpam-5252	483	10	h	h	NOUN
ejpam-5252	483	11	)	)	PUNCT
ejpam-5252	483	12	.	.	PUNCT
ejpam-5252	484	1	by	by	ADP
ejpam-5252	484	2	(	(	PUNCT
ejpam-5252	484	3	b1	b1	NOUN
ejpam-5252	484	4	)	)	PUNCT
ejpam-5252	484	5	,	,	PUNCT
ejpam-5252	484	6	there	there	PRON
ejpam-5252	484	7	exists	exist	VERB
ejpam-5252	484	8	u	u	PROPN
ejpam-5252	484	9	∈	∈	PROPN
ejpam-5252	484	10	v3	v3	PROPN
ejpam-5252	484	11	∩	∩	PROPN
ejpam-5252	484	12	v	v	X
ejpam-5252	484	13	(	(	PUNCT
ejpam-5252	484	14	g	g	NOUN
ejpam-5252	484	15	)	)	PUNCT
ejpam-5252	484	16	.	.	PUNCT
ejpam-5252	485	1	then	then	ADV
ejpam-5252	485	2	uv	uv	PROPN
ejpam-5252	485	3	∈	∈	PROPN
ejpam-5252	485	4	e(g	e(g	NOUN
ejpam-5252	486	1	+	+	CCONJ
ejpam-5252	486	2	h	h	NOUN
ejpam-5252	486	3	)	)	PUNCT
ejpam-5252	486	4	.	.	PUNCT
ejpam-5252	487	1	therefore	therefore	ADV
ejpam-5252	487	2	,	,	PUNCT
ejpam-5252	487	3	f	f	PROPN
ejpam-5252	487	4	∈	∈	PROPN
ejpam-5252	487	5	mdrf	mdrf	NOUN
ejpam-5252	487	6	(	(	PUNCT
ejpam-5252	487	7	g	g	NOUN
ejpam-5252	487	8	+	+	NOUN
ejpam-5252	487	9	h	h	NOUN
ejpam-5252	487	10	)	)	PUNCT
ejpam-5252	487	11	.	.	PUNCT
ejpam-5252	488	1	similarly	similarly	ADV
ejpam-5252	488	2	,	,	PUNCT
ejpam-5252	488	3	if	if	SCONJ
ejpam-5252	488	4	(	(	PUNCT
ejpam-5252	488	5	ii)(b	ii)(b	ADJ
ejpam-5252	488	6	)	)	PUNCT
ejpam-5252	488	7	holds	hold	VERB
ejpam-5252	488	8	,	,	PUNCT
ejpam-5252	488	9	then	then	ADV
ejpam-5252	488	10	f	f	PROPN
ejpam-5252	488	11	∈	∈	PROPN
ejpam-5252	488	12	mrdf	mrdf	NOUN
ejpam-5252	488	13	(	(	PUNCT
ejpam-5252	488	14	g	g	PROPN
ejpam-5252	488	15	+	+	PROPN
ejpam-5252	488	16	h	h	NOUN
ejpam-5252	488	17	)	)	PUNCT
ejpam-5252	488	18	.	.	PUNCT
ejpam-5252	489	1	assume	assume	VERB
ejpam-5252	489	2	(	(	PUNCT
ejpam-5252	489	3	i)(c	i)(c	NOUN
ejpam-5252	489	4	)	)	PUNCT
ejpam-5252	489	5	holds	hold	VERB
ejpam-5252	489	6	.	.	PUNCT
ejpam-5252	490	1	since	since	SCONJ
ejpam-5252	490	2	v0	v0	NOUN
ejpam-5252	490	3	∩	∩	NOUN
ejpam-5252	490	4	v	v	X
ejpam-5252	490	5	(	(	PUNCT
ejpam-5252	490	6	g	g	NOUN
ejpam-5252	490	7	)	)	PUNCT
ejpam-5252	490	8	=	=	NOUN
ejpam-5252	490	9	∅	∅	NOUN
ejpam-5252	490	10	,	,	PUNCT
ejpam-5252	490	11	then	then	ADV
ejpam-5252	490	12	v0	v0	VERB
ejpam-5252	490	13	⊆	⊆	NUM
ejpam-5252	490	14	v	v	NOUN
ejpam-5252	490	15	(	(	PUNCT
ejpam-5252	490	16	h	h	NOUN
ejpam-5252	490	17	)	)	PUNCT
ejpam-5252	490	18	.	.	PUNCT
ejpam-5252	491	1	let	let	VERB
ejpam-5252	491	2	v	v	NUM
ejpam-5252	491	3	∈	∈	PROPN
ejpam-5252	491	4	v0	v0	NOUN
ejpam-5252	491	5	.	.	PUNCT
ejpam-5252	492	1	by	by	ADP
ejpam-5252	492	2	(	(	PUNCT
ejpam-5252	492	3	c2	c2	PROPN
ejpam-5252	492	4	)	)	PUNCT
ejpam-5252	492	5	,	,	PUNCT
ejpam-5252	492	6	there	there	PRON
ejpam-5252	492	7	exists	exist	VERB
ejpam-5252	492	8	u	u	PROPN
ejpam-5252	492	9	∈	∈	PROPN
ejpam-5252	492	10	v3∩v	v3∩v	X
ejpam-5252	492	11	(	(	PUNCT
ejpam-5252	492	12	h	h	NOUN
ejpam-5252	492	13	)	)	PUNCT
ejpam-5252	492	14	such	such	ADJ
ejpam-5252	492	15	that	that	SCONJ
ejpam-5252	492	16	uv	uv	PROPN
ejpam-5252	492	17	∈	∈	PROPN
ejpam-5252	492	18	e(h	e(h	PROPN
ejpam-5252	492	19	)	)	PUNCT
ejpam-5252	492	20	⊆	⊆	NUM
ejpam-5252	492	21	e(g+h	e(g+h	NUM
ejpam-5252	492	22	)	)	PUNCT
ejpam-5252	492	23	.	.	PUNCT
ejpam-5252	493	1	also	also	ADV
ejpam-5252	493	2	,	,	PUNCT
ejpam-5252	493	3	since	since	SCONJ
ejpam-5252	493	4	v2	v2	PROPN
ejpam-5252	493	5	is	be	AUX
ejpam-5252	493	6	a	a	DET
ejpam-5252	493	7	dominating	dominating	NOUN
ejpam-5252	493	8	set	set	NOUN
ejpam-5252	493	9	of	of	ADP
ejpam-5252	493	10	g	g	NOUN
ejpam-5252	493	11	,	,	PUNCT
ejpam-5252	493	12	v2	v2	PROPN
ejpam-5252	493	13	∩	∩	ADJ
ejpam-5252	493	14	v	v	NOUN
ejpam-5252	493	15	(	(	PUNCT
ejpam-5252	493	16	g	g	NOUN
ejpam-5252	493	17	)	)	PUNCT
ejpam-5252	493	18	̸=	̸=	PROPN
ejpam-5252	493	19	∅.	∅.	AUX
ejpam-5252	493	20	pick	pick	VERB
ejpam-5252	493	21	u	u	PRON
ejpam-5252	493	22	∈	∈	PROPN
ejpam-5252	493	23	v2	v2	PROPN
ejpam-5252	493	24	∩	∩	ADJ
ejpam-5252	493	25	v	v	NOUN
ejpam-5252	493	26	(	(	PUNCT
ejpam-5252	493	27	g	g	NOUN
ejpam-5252	493	28	)	)	PUNCT
ejpam-5252	493	29	.	.	PUNCT
ejpam-5252	494	1	then	then	ADV
ejpam-5252	494	2	uv	uv	PROPN
ejpam-5252	494	3	∈	∈	PROPN
ejpam-5252	494	4	e(g+h	e(g+h	NUM
ejpam-5252	494	5	)	)	PUNCT
ejpam-5252	494	6	.	.	PUNCT
ejpam-5252	495	1	let	let	VERB
ejpam-5252	495	2	v	v	NUM
ejpam-5252	495	3	∈	∈	PROPN
ejpam-5252	495	4	v1∩v	v1∩v	NOUN
ejpam-5252	495	5	(	(	PUNCT
ejpam-5252	495	6	g	g	NOUN
ejpam-5252	495	7	)	)	PUNCT
ejpam-5252	495	8	.	.	PUNCT
ejpam-5252	496	1	by	by	ADP
ejpam-5252	496	2	(	(	PUNCT
ejpam-5252	496	3	c1	c1	PROPN
ejpam-5252	496	4	)	)	PUNCT
ejpam-5252	496	5	,	,	PUNCT
ejpam-5252	496	6	there	there	PRON
ejpam-5252	496	7	exists	exist	VERB
ejpam-5252	496	8	u	u	PROPN
ejpam-5252	496	9	∈	∈	PROPN
ejpam-5252	496	10	v2∩v	v2∩v	NOUN
ejpam-5252	496	11	(	(	PUNCT
ejpam-5252	496	12	g	g	NOUN
ejpam-5252	496	13	)	)	PUNCT
ejpam-5252	496	14	such	such	ADJ
ejpam-5252	496	15	that	that	SCONJ
ejpam-5252	496	16	uv	uv	PROPN
ejpam-5252	496	17	∈	∈	PROPN
ejpam-5252	496	18	e(g	e(g	PROPN
ejpam-5252	496	19	)	)	PUNCT
ejpam-5252	497	1	⊆	⊆	NUM
ejpam-5252	497	2	e(g+h	e(g+h	NUM
ejpam-5252	497	3	)	)	PUNCT
ejpam-5252	497	4	.	.	PUNCT
ejpam-5252	498	1	now	now	ADV
ejpam-5252	498	2	,	,	PUNCT
ejpam-5252	498	3	let	let	VERB
ejpam-5252	498	4	v	v	NUM
ejpam-5252	498	5	∈	∈	NOUN
ejpam-5252	498	6	v1	v1	NOUN
ejpam-5252	498	7	∩	∩	ADJ
ejpam-5252	498	8	v	v	NOUN
ejpam-5252	498	9	(	(	PUNCT
ejpam-5252	498	10	h	h	NOUN
ejpam-5252	498	11	)	)	PUNCT
ejpam-5252	498	12	.	.	PUNCT
ejpam-5252	499	1	by	by	ADP
ejpam-5252	499	2	(	(	PUNCT
ejpam-5252	499	3	c1	c1	PROPN
ejpam-5252	499	4	)	)	PUNCT
ejpam-5252	499	5	,	,	PUNCT
ejpam-5252	499	6	there	there	PRON
ejpam-5252	499	7	exists	exist	VERB
ejpam-5252	499	8	u	u	PROPN
ejpam-5252	499	9	∈	∈	PROPN
ejpam-5252	499	10	v2	v2	PROPN
ejpam-5252	499	11	∩	∩	ADJ
ejpam-5252	499	12	v	v	NOUN
ejpam-5252	499	13	(	(	PUNCT
ejpam-5252	499	14	g	g	NOUN
ejpam-5252	499	15	)	)	PUNCT
ejpam-5252	499	16	.	.	PUNCT
ejpam-5252	500	1	then	then	ADV
ejpam-5252	500	2	uv	uv	PROPN
ejpam-5252	500	3	∈	∈	PROPN
ejpam-5252	500	4	e(g	e(g	NOUN
ejpam-5252	501	1	+	+	CCONJ
ejpam-5252	501	2	h	h	NOUN
ejpam-5252	501	3	)	)	PUNCT
ejpam-5252	501	4	.	.	PUNCT
ejpam-5252	502	1	therefore	therefore	ADV
ejpam-5252	502	2	,	,	PUNCT
ejpam-5252	502	3	f	f	PROPN
ejpam-5252	502	4	∈	∈	PROPN
ejpam-5252	502	5	mdrf	mdrf	NOUN
ejpam-5252	502	6	(	(	PUNCT
ejpam-5252	502	7	g+h	g+h	PROPN
ejpam-5252	502	8	)	)	PUNCT
ejpam-5252	502	9	.	.	PUNCT
ejpam-5252	503	1	similarly	similarly	ADV
ejpam-5252	503	2	,	,	PUNCT
ejpam-5252	503	3	if	if	SCONJ
ejpam-5252	503	4	(	(	PUNCT
ejpam-5252	503	5	ii)(c	ii)(c	PROPN
ejpam-5252	503	6	)	)	PUNCT
ejpam-5252	503	7	holds	hold	VERB
ejpam-5252	503	8	,	,	PUNCT
ejpam-5252	503	9	then	then	ADV
ejpam-5252	503	10	f	f	PROPN
ejpam-5252	503	11	∈	∈	PROPN
ejpam-5252	503	12	mrdf	mrdf	NOUN
ejpam-5252	503	13	(	(	PUNCT
ejpam-5252	503	14	g+h	g+h	PROPN
ejpam-5252	503	15	)	)	PUNCT
ejpam-5252	503	16	.	.	PUNCT
ejpam-5252	504	1	suppose	suppose	VERB
ejpam-5252	504	2	(	(	PUNCT
ejpam-5252	504	3	iii	iii	NOUN
ejpam-5252	504	4	)	)	PUNCT
ejpam-5252	504	5	holds	hold	VERB
ejpam-5252	504	6	,	,	PUNCT
ejpam-5252	504	7	that	that	PRON
ejpam-5252	504	8	is	be	AUX
ejpam-5252	504	9	f	f	PROPN
ejpam-5252	504	10	|g	|g	PROPN
ejpam-5252	504	11	̸∈	̸∈	PROPN
ejpam-5252	504	12	mrdf	mrdf	PROPN
ejpam-5252	504	13	(	(	PUNCT
ejpam-5252	504	14	g	g	NOUN
ejpam-5252	504	15	)	)	PUNCT
ejpam-5252	504	16	and	and	CCONJ
ejpam-5252	504	17	f	f	PROPN
ejpam-5252	504	18	|h	|h	X
ejpam-5252	504	19	̸∈	̸∈	PROPN
ejpam-5252	504	20	mrdf	mrdf	PROPN
ejpam-5252	504	21	(	(	PUNCT
ejpam-5252	504	22	g	g	NOUN
ejpam-5252	504	23	)	)	PUNCT
ejpam-5252	504	24	.	.	PUNCT
ejpam-5252	505	1	let	let	VERB
ejpam-5252	505	2	v	v	NUM
ejpam-5252	505	3	∈	∈	PROPN
ejpam-5252	505	4	v0	v0	NOUN
ejpam-5252	505	5	∩	∩	X
ejpam-5252	505	6	v	v	X
ejpam-5252	505	7	(	(	PUNCT
ejpam-5252	505	8	g	g	NOUN
ejpam-5252	505	9	)	)	PUNCT
ejpam-5252	505	10	.	.	PUNCT
ejpam-5252	506	1	if	if	SCONJ
ejpam-5252	506	2	ng(v	ng(v	NOUN
ejpam-5252	506	3	)	)	PUNCT
ejpam-5252	506	4	∩	∩	ADJ
ejpam-5252	506	5	v2	v2	NOUN
ejpam-5252	506	6	=	=	SYM
ejpam-5252	506	7	∅	∅	NOUN
ejpam-5252	506	8	and	and	CCONJ
ejpam-5252	506	9	ng(v	ng(v	NUM
ejpam-5252	506	10	)	)	PUNCT
ejpam-5252	506	11	∩	∩	PROPN
ejpam-5252	506	12	v3	v3	PROPN
ejpam-5252	506	13	̸=	̸=	PROPN
ejpam-5252	506	14	∅.	∅.	ADV
ejpam-5252	506	15	take	take	VERB
ejpam-5252	506	16	u	u	PROPN
ejpam-5252	506	17	∈	∈	PROPN
ejpam-5252	506	18	v3	v3	PROPN
ejpam-5252	506	19	∩	∩	PROPN
ejpam-5252	506	20	v	v	X
ejpam-5252	506	21	(	(	PUNCT
ejpam-5252	506	22	g	g	NOUN
ejpam-5252	506	23	)	)	PUNCT
ejpam-5252	506	24	such	such	ADJ
ejpam-5252	506	25	that	that	SCONJ
ejpam-5252	506	26	uv	uv	PROPN
ejpam-5252	506	27	∈	∈	PROPN
ejpam-5252	506	28	e(g	e(g	PROPN
ejpam-5252	506	29	)	)	PUNCT
ejpam-5252	506	30	⊆	⊆	NUM
ejpam-5252	506	31	e(g	e(g	NOUN
ejpam-5252	506	32	+	+	CCONJ
ejpam-5252	506	33	h	h	NOUN
ejpam-5252	506	34	)	)	PUNCT
ejpam-5252	506	35	.	.	PUNCT
ejpam-5252	507	1	since	since	SCONJ
ejpam-5252	507	2	ng(v	ng(v	NOUN
ejpam-5252	507	3	)	)	PUNCT
ejpam-5252	507	4	∩	∩	ADJ
ejpam-5252	507	5	v2	v2	NOUN
ejpam-5252	507	6	=	=	NOUN
ejpam-5252	507	7	∅	∅	NOUN
ejpam-5252	507	8	,	,	PUNCT
ejpam-5252	507	9	by	by	ADP
ejpam-5252	507	10	assumption	assumption	NOUN
ejpam-5252	507	11	there	there	PRON
ejpam-5252	507	12	exists	exist	VERB
ejpam-5252	507	13	w	w	PROPN
ejpam-5252	507	14	∈	∈	PROPN
ejpam-5252	507	15	v2	v2	PROPN
ejpam-5252	507	16	∩	∩	ADJ
ejpam-5252	507	17	v	v	NOUN
ejpam-5252	507	18	(	(	PUNCT
ejpam-5252	507	19	h	h	NOUN
ejpam-5252	507	20	)	)	PUNCT
ejpam-5252	507	21	such	such	ADJ
ejpam-5252	507	22	that	that	SCONJ
ejpam-5252	507	23	vw	vw	PROPN
ejpam-5252	507	24	∈	∈	PROPN
ejpam-5252	507	25	e(g	e(g	PROPN
ejpam-5252	508	1	+	+	CCONJ
ejpam-5252	508	2	h	h	NOUN
ejpam-5252	508	3	)	)	PUNCT
ejpam-5252	508	4	.	.	PUNCT
ejpam-5252	509	1	if	if	SCONJ
ejpam-5252	509	2	ng(v	ng(v	NOUN
ejpam-5252	509	3	)	)	PUNCT
ejpam-5252	509	4	∩	∩	NOUN
ejpam-5252	509	5	v2	v2	PROPN
ejpam-5252	509	6	̸=	̸=	PROPN
ejpam-5252	509	7	∅	∅	NOUN
ejpam-5252	509	8	and	and	CCONJ
ejpam-5252	509	9	ng(v	ng(v	NUM
ejpam-5252	509	10	)	)	PUNCT
ejpam-5252	509	11	∩	∩	NOUN
ejpam-5252	509	12	v3	v3	NOUN
ejpam-5252	509	13	=	=	PUNCT
ejpam-5252	509	14	∅.	∅.	PART
ejpam-5252	509	15	pick	pick	VERB
ejpam-5252	509	16	u	u	PRON
ejpam-5252	509	17	∈	∈	PROPN
ejpam-5252	509	18	v2	v2	PROPN
ejpam-5252	509	19	∩	∩	ADJ
ejpam-5252	509	20	v	v	NOUN
ejpam-5252	509	21	(	(	PUNCT
ejpam-5252	509	22	g	g	NOUN
ejpam-5252	509	23	)	)	PUNCT
ejpam-5252	509	24	such	such	ADJ
ejpam-5252	509	25	that	that	SCONJ
ejpam-5252	509	26	uv	uv	PROPN
ejpam-5252	509	27	∈	∈	PROPN
ejpam-5252	509	28	e(g	e(g	PROPN
ejpam-5252	509	29	)	)	PUNCT
ejpam-5252	509	30	⊆	⊆	NUM
ejpam-5252	509	31	e(g	e(g	NOUN
ejpam-5252	509	32	+	+	CCONJ
ejpam-5252	509	33	h	h	NOUN
ejpam-5252	509	34	)	)	PUNCT
ejpam-5252	509	35	.	.	PUNCT
ejpam-5252	510	1	since	since	SCONJ
ejpam-5252	510	2	ng(v	ng(v	NOUN
ejpam-5252	510	3	)	)	PUNCT
ejpam-5252	510	4	∩	∩	ADJ
ejpam-5252	510	5	v3	v3	NOUN
ejpam-5252	510	6	=	=	SYM
ejpam-5252	510	7	∅	∅	NOUN
ejpam-5252	510	8	,	,	PUNCT
ejpam-5252	510	9	by	by	ADP
ejpam-5252	510	10	assumption	assumption	NOUN
ejpam-5252	510	11	there	there	PRON
ejpam-5252	510	12	exists	exist	VERB
ejpam-5252	510	13	w	w	PROPN
ejpam-5252	510	14	∈	∈	PROPN
ejpam-5252	510	15	v3	v3	PROPN
ejpam-5252	510	16	∩	∩	PROPN
ejpam-5252	510	17	v	v	X
ejpam-5252	510	18	(	(	PUNCT
ejpam-5252	510	19	h	h	NOUN
ejpam-5252	510	20	)	)	PUNCT
ejpam-5252	510	21	such	such	ADJ
ejpam-5252	510	22	that	that	SCONJ
ejpam-5252	510	23	vw	vw	PROPN
ejpam-5252	510	24	∈	∈	PROPN
ejpam-5252	510	25	e(g+h	e(g+h	NUM
ejpam-5252	510	26	)	)	PUNCT
ejpam-5252	510	27	.	.	PUNCT
ejpam-5252	511	1	if	if	SCONJ
ejpam-5252	511	2	ng(v)∩	ng(v)∩	PRON
ejpam-5252	511	3	v2	v2	VERB
ejpam-5252	511	4	=	=	SYM
ejpam-5252	511	5	∅	∅	NOUN
ejpam-5252	511	6	and	and	CCONJ
ejpam-5252	511	7	ng(v)∩	ng(v)∩	VERB
ejpam-5252	511	8	v3	v3	PROPN
ejpam-5252	511	9	=	=	PUNCT
ejpam-5252	511	10	∅.	∅.	NOUN
ejpam-5252	511	11	then	then	ADV
ejpam-5252	511	12	by	by	ADP
ejpam-5252	511	13	assumption	assumption	NOUN
ejpam-5252	511	14	,	,	PUNCT
ejpam-5252	511	15	v2∩v	v2∩v	PROPN
ejpam-5252	511	16	(	(	PUNCT
ejpam-5252	511	17	h	h	NOUN
ejpam-5252	511	18	)	)	PUNCT
ejpam-5252	511	19	̸=	̸=	PROPN
ejpam-5252	511	20	∅	∅	NOUN
ejpam-5252	511	21	and	and	CCONJ
ejpam-5252	511	22	v3∩v	v3∩v	PROPN
ejpam-5252	511	23	(	(	PUNCT
ejpam-5252	511	24	h	h	NOUN
ejpam-5252	511	25	)	)	PUNCT
ejpam-5252	511	26	̸=	̸=	PROPN
ejpam-5252	511	27	∅	∅	NOUN
ejpam-5252	511	28	and	and	CCONJ
ejpam-5252	511	29	so	so	ADV
ejpam-5252	511	30	,	,	PUNCT
ejpam-5252	511	31	there	there	PRON
ejpam-5252	511	32	exist	exist	VERB
ejpam-5252	511	33	u	u	PROPN
ejpam-5252	511	34	∈	∈	NOUN
ejpam-5252	511	35	v2∩v	v2∩v	NOUN
ejpam-5252	511	36	(	(	PUNCT
ejpam-5252	511	37	h	h	NOUN
ejpam-5252	511	38	)	)	PUNCT
ejpam-5252	511	39	and	and	CCONJ
ejpam-5252	511	40	w	w	PROPN
ejpam-5252	511	41	∈	∈	PROPN
ejpam-5252	511	42	v3	v3	PROPN
ejpam-5252	511	43	∩	∩	PROPN
ejpam-5252	511	44	v	v	X
ejpam-5252	511	45	(	(	PUNCT
ejpam-5252	511	46	h	h	NOUN
ejpam-5252	511	47	)	)	PUNCT
ejpam-5252	511	48	such	such	ADJ
ejpam-5252	511	49	that	that	SCONJ
ejpam-5252	511	50	vu	vu	PROPN
ejpam-5252	511	51	,	,	PUNCT
ejpam-5252	511	52	vw	vw	PROPN
ejpam-5252	511	53	∈	∈	PROPN
ejpam-5252	511	54	e(g+h	e(g+h	NUM
ejpam-5252	511	55	)	)	PUNCT
ejpam-5252	511	56	.	.	PUNCT
ejpam-5252	512	1	now	now	ADV
ejpam-5252	512	2	,	,	PUNCT
ejpam-5252	512	3	suppose	suppose	VERB
ejpam-5252	512	4	f(v	f(v	NOUN
ejpam-5252	512	5	)	)	PUNCT
ejpam-5252	512	6	=	=	SYM
ejpam-5252	513	1	1	1	X
ejpam-5252	513	2	.	.	PUNCT
ejpam-5252	514	1	if	if	SCONJ
ejpam-5252	514	2	ng(v)∩	ng(v)∩	PRON
ejpam-5252	514	3	v2	v2	VERB
ejpam-5252	514	4	=	=	SYM
ejpam-5252	514	5	∅	∅	NOUN
ejpam-5252	514	6	and	and	CCONJ
ejpam-5252	514	7	ng(v	ng(v	NUM
ejpam-5252	514	8	)	)	PUNCT
ejpam-5252	514	9	∩	∩	NOUN
ejpam-5252	514	10	v3	v3	NOUN
ejpam-5252	514	11	=	=	PUNCT
ejpam-5252	514	12	∅.	∅.	NOUN
ejpam-5252	514	13	then	then	ADV
ejpam-5252	514	14	by	by	ADP
ejpam-5252	514	15	assumption	assumption	NOUN
ejpam-5252	514	16	,	,	PUNCT
ejpam-5252	514	17	there	there	PRON
ejpam-5252	514	18	exist	exist	VERB
ejpam-5252	514	19	z	z	NOUN
ejpam-5252	514	20	∈	∈	PROPN
ejpam-5252	514	21	v2	v2	PROPN
ejpam-5252	514	22	∩	∩	ADJ
ejpam-5252	514	23	v	v	NOUN
ejpam-5252	514	24	(	(	PUNCT
ejpam-5252	514	25	h	h	NOUN
ejpam-5252	514	26	)	)	PUNCT
ejpam-5252	514	27	or	or	CCONJ
ejpam-5252	514	28	z	z	NOUN
ejpam-5252	514	29	∈	∈	PROPN
ejpam-5252	514	30	v3	v3	PROPN
ejpam-5252	514	31	∩	∩	PROPN
ejpam-5252	514	32	v	v	X
ejpam-5252	514	33	(	(	PUNCT
ejpam-5252	514	34	h	h	NOUN
ejpam-5252	514	35	)	)	PUNCT
ejpam-5252	514	36	such	such	ADJ
ejpam-5252	514	37	that	that	SCONJ
ejpam-5252	514	38	vz	vz	PROPN
ejpam-5252	514	39	∈	∈	PROPN
ejpam-5252	514	40	e(g+h	e(g+h	PROPN
ejpam-5252	514	41	)	)	PUNCT
ejpam-5252	514	42	satisfying	satisfying	NOUN
ejpam-5252	514	43	(	(	PUNCT
ejpam-5252	514	44	p2	p2	PROPN
ejpam-5252	514	45	)	)	PUNCT
ejpam-5252	514	46	.	.	PUNCT
ejpam-5252	515	1	therefore	therefore	ADV
ejpam-5252	515	2	,	,	PUNCT
ejpam-5252	515	3	f	f	PROPN
ejpam-5252	515	4	∈	∈	PROPN
ejpam-5252	515	5	mrdf	mrdf	NOUN
ejpam-5252	515	6	(	(	PUNCT
ejpam-5252	515	7	g+h	g+h	PROPN
ejpam-5252	515	8	)	)	PUNCT
ejpam-5252	515	9	.	.	PUNCT
ejpam-5252	516	1	similarly	similarly	ADV
ejpam-5252	516	2	,	,	PUNCT
ejpam-5252	516	3	for	for	ADP
ejpam-5252	516	4	v	v	ADP
ejpam-5252	516	5	∈	∈	PROPN
ejpam-5252	516	6	v	v	NOUN
ejpam-5252	516	7	(	(	PUNCT
ejpam-5252	516	8	h	h	NOUN
ejpam-5252	516	9	)	)	PUNCT
ejpam-5252	516	10	such	such	ADJ
ejpam-5252	516	11	that	that	SCONJ
ejpam-5252	516	12	f(v	f(v	NOUN
ejpam-5252	516	13	)	)	PUNCT
ejpam-5252	516	14	∈	∈	PROPN
ejpam-5252	516	15	{	{	PUNCT
ejpam-5252	516	16	0	0	NUM
ejpam-5252	516	17	,	,	PUNCT
ejpam-5252	516	18	1	1	NUM
ejpam-5252	516	19	}	}	PUNCT
ejpam-5252	516	20	,	,	PUNCT
ejpam-5252	516	21	f	f	PROPN
ejpam-5252	516	22	∈	∈	PROPN
ejpam-5252	516	23	mrdf	mrdf	NOUN
ejpam-5252	516	24	(	(	PUNCT
ejpam-5252	516	25	g+h	g+h	NOUN
ejpam-5252	516	26	)	)	PUNCT
ejpam-5252	516	27	.	.	PUNCT
ejpam-5252	517	1	conversely	conversely	ADV
ejpam-5252	517	2	,	,	PUNCT
ejpam-5252	517	3	suppose	suppose	VERB
ejpam-5252	517	4	f	f	PROPN
ejpam-5252	517	5	∈	∈	PROPN
ejpam-5252	517	6	mrdf	mrdf	NOUN
ejpam-5252	517	7	(	(	PUNCT
ejpam-5252	517	8	g+h	g+h	PROPN
ejpam-5252	517	9	)	)	PUNCT
ejpam-5252	517	10	.	.	PUNCT
ejpam-5252	518	1	consider	consider	VERB
ejpam-5252	518	2	the	the	DET
ejpam-5252	518	3	following	follow	VERB
ejpam-5252	518	4	cases	case	NOUN
ejpam-5252	518	5	:	:	PUNCT
ejpam-5252	518	6	case	case	NOUN
ejpam-5252	518	7	1	1	NUM
ejpam-5252	518	8	:	:	PUNCT
ejpam-5252	518	9	suppose	suppose	VERB
ejpam-5252	518	10	f	f	PROPN
ejpam-5252	518	11	|g	|g	PROPN
ejpam-5252	518	12	∈	∈	PROPN
ejpam-5252	518	13	mrdf	mrdf	NOUN
ejpam-5252	518	14	(	(	PUNCT
ejpam-5252	518	15	g	g	NOUN
ejpam-5252	518	16	)	)	PUNCT
ejpam-5252	518	17	.	.	PUNCT
ejpam-5252	519	1	if	if	SCONJ
ejpam-5252	519	2	(	(	PUNCT
ejpam-5252	519	3	i)(a	i)(a	NOUN
ejpam-5252	519	4	)	)	PUNCT
ejpam-5252	519	5	holds	hold	VERB
ejpam-5252	519	6	,	,	PUNCT
ejpam-5252	519	7	we	we	PRON
ejpam-5252	519	8	are	be	AUX
ejpam-5252	519	9	done	do	VERB
ejpam-5252	519	10	.	.	PUNCT
ejpam-5252	520	1	suppose	suppose	VERB
ejpam-5252	520	2	(	(	PUNCT
ejpam-5252	520	3	i)(a	i)(a	NOUN
ejpam-5252	520	4	)	)	PUNCT
ejpam-5252	520	5	does	do	AUX
ejpam-5252	520	6	not	not	PART
ejpam-5252	520	7	hold	hold	VERB
ejpam-5252	520	8	.	.	PUNCT
ejpam-5252	521	1	thus	thus	ADV
ejpam-5252	521	2	,	,	PUNCT
ejpam-5252	521	3	either	either	CCONJ
ejpam-5252	521	4	v2∩v	v2∩v	PROPN
ejpam-5252	521	5	(	(	PUNCT
ejpam-5252	521	6	g	g	NOUN
ejpam-5252	521	7	)	)	PUNCT
ejpam-5252	521	8	=	=	NOUN
ejpam-5252	521	9	∅	∅	NOUN
ejpam-5252	521	10	or	or	CCONJ
ejpam-5252	521	11	v3∩v	v3∩v	NUM
ejpam-5252	521	12	(	(	PUNCT
ejpam-5252	521	13	g	g	NOUN
ejpam-5252	521	14	)	)	PUNCT
ejpam-5252	521	15	=	=	VERB
ejpam-5252	521	16	∅.	∅.	AUX
ejpam-5252	521	17	suppose	suppose	VERB
ejpam-5252	521	18	v2∩v	v2∩v	PROPN
ejpam-5252	521	19	(	(	PUNCT
ejpam-5252	521	20	g	g	NOUN
ejpam-5252	521	21	)	)	PUNCT
ejpam-5252	521	22	=	=	PUNCT
ejpam-5252	521	23	∅.	∅.	ADP
ejpam-5252	521	24	necessarily	necessarily	ADV
ejpam-5252	521	25	,	,	PUNCT
ejpam-5252	521	26	v0	v0	PROPN
ejpam-5252	521	27	∩	∩	ADJ
ejpam-5252	521	28	v	v	X
ejpam-5252	521	29	(	(	PUNCT
ejpam-5252	521	30	g	g	NOUN
ejpam-5252	521	31	)	)	PUNCT
ejpam-5252	521	32	=	=	VERB
ejpam-5252	521	33	∅.	∅.	AUX
ejpam-5252	521	34	let	let	VERB
ejpam-5252	521	35	v	v	NUM
ejpam-5252	521	36	∈	∈	PROPN
ejpam-5252	521	37	v1	v1	NOUN
ejpam-5252	521	38	∩	∩	ADJ
ejpam-5252	521	39	v	v	NOUN
ejpam-5252	521	40	(	(	PUNCT
ejpam-5252	521	41	g	g	NOUN
ejpam-5252	521	42	)	)	PUNCT
ejpam-5252	521	43	.	.	PUNCT
ejpam-5252	522	1	since	since	SCONJ
ejpam-5252	522	2	f	f	PROPN
ejpam-5252	522	3	|g	|g	PROPN
ejpam-5252	522	4	∈	∈	PROPN
ejpam-5252	522	5	mrdf	mrdf	NOUN
ejpam-5252	522	6	(	(	PUNCT
ejpam-5252	522	7	g	g	NOUN
ejpam-5252	522	8	)	)	PUNCT
ejpam-5252	522	9	,	,	PUNCT
ejpam-5252	522	10	there	there	PRON
ejpam-5252	522	11	exists	exist	VERB
ejpam-5252	522	12	u	u	PROPN
ejpam-5252	522	13	∈	∈	PROPN
ejpam-5252	522	14	v3	v3	PROPN
ejpam-5252	522	15	such	such	ADJ
ejpam-5252	522	16	that	that	SCONJ
ejpam-5252	522	17	uv	uv	PROPN
ejpam-5252	522	18	∈	∈	PROPN
ejpam-5252	522	19	e(g	e(g	PROPN
ejpam-5252	522	20	)	)	PUNCT
ejpam-5252	522	21	.	.	PUNCT
ejpam-5252	523	1	thus	thus	ADV
ejpam-5252	523	2	,	,	PUNCT
ejpam-5252	523	3	v3	v3	PROPN
ejpam-5252	523	4	is	be	AUX
ejpam-5252	523	5	a	a	DET
ejpam-5252	523	6	dominating	dominating	NOUN
ejpam-5252	523	7	set	set	NOUN
ejpam-5252	523	8	of	of	ADP
ejpam-5252	523	9	g	g	NOUN
ejpam-5252	523	10	,	,	PUNCT
ejpam-5252	523	11	and	and	CCONJ
ejpam-5252	523	12	so	so	ADV
ejpam-5252	523	13	,	,	PUNCT
ejpam-5252	523	14	(	(	PUNCT
ejpam-5252	523	15	b1	b1	NOUN
ejpam-5252	523	16	)	)	PUNCT
ejpam-5252	523	17	holds	hold	VERB
ejpam-5252	523	18	.	.	PUNCT
ejpam-5252	524	1	also	also	ADV
ejpam-5252	524	2	,	,	PUNCT
ejpam-5252	524	3	since	since	SCONJ
ejpam-5252	524	4	v2∩v	v2∩v	PROPN
ejpam-5252	524	5	(	(	PUNCT
ejpam-5252	524	6	g	g	NOUN
ejpam-5252	524	7	)	)	PUNCT
ejpam-5252	524	8	=	=	NOUN
ejpam-5252	524	9	∅	∅	NOUN
ejpam-5252	524	10	,	,	PUNCT
ejpam-5252	524	11	we	we	PRON
ejpam-5252	524	12	have	have	VERB
ejpam-5252	524	13	v2	v2	PROPN
ejpam-5252	524	14	⊆	⊆	NUM
ejpam-5252	524	15	v	v	NOUN
ejpam-5252	524	16	(	(	PUNCT
ejpam-5252	524	17	h	h	NOUN
ejpam-5252	524	18	)	)	PUNCT
ejpam-5252	524	19	.	.	PUNCT
ejpam-5252	525	1	this	this	PRON
ejpam-5252	525	2	means	mean	VERB
ejpam-5252	525	3	that	that	SCONJ
ejpam-5252	525	4	v2∩v	v2∩v	PROPN
ejpam-5252	525	5	(	(	PUNCT
ejpam-5252	525	6	h	h	NOUN
ejpam-5252	525	7	)	)	PUNCT
ejpam-5252	525	8	̸=	̸=	NOUN
ejpam-5252	525	9	∅	∅	NOUN
ejpam-5252	525	10	,	,	PUNCT
ejpam-5252	525	11	say	say	VERB
ejpam-5252	525	12	w	w	PROPN
ejpam-5252	525	13	∈	∈	PROPN
ejpam-5252	525	14	v2∩v	v2∩v	NOUN
ejpam-5252	525	15	(	(	PUNCT
ejpam-5252	525	16	h	h	NOUN
ejpam-5252	525	17	)	)	PUNCT
ejpam-5252	525	18	.	.	PUNCT
ejpam-5252	526	1	suppose	suppose	VERB
ejpam-5252	526	2	v	v	NUM
ejpam-5252	526	3	∈	∈	PROPN
ejpam-5252	526	4	v0	v0	NOUN
ejpam-5252	526	5	∩	∩	X
ejpam-5252	526	6	v	v	X
ejpam-5252	526	7	(	(	PUNCT
ejpam-5252	526	8	h	h	NOUN
ejpam-5252	526	9	)	)	PUNCT
ejpam-5252	526	10	.	.	PUNCT
ejpam-5252	527	1	then	then	ADV
ejpam-5252	527	2	since	since	SCONJ
ejpam-5252	527	3	f	f	PROPN
ejpam-5252	527	4	∈	∈	PROPN
ejpam-5252	527	5	mrdf	mrdf	NOUN
ejpam-5252	527	6	(	(	PUNCT
ejpam-5252	527	7	g+h	g+h	PROPN
ejpam-5252	527	8	)	)	PUNCT
ejpam-5252	527	9	,	,	PUNCT
ejpam-5252	527	10	vw	vw	PROPN
ejpam-5252	527	11	∈	∈	PROPN
ejpam-5252	527	12	e(h	e(h	PROPN
ejpam-5252	527	13	)	)	PUNCT
ejpam-5252	527	14	⊆	⊆	NUM
ejpam-5252	527	15	e(g+h	e(g+h	NUM
ejpam-5252	527	16	)	)	PUNCT
ejpam-5252	527	17	.	.	PUNCT
ejpam-5252	528	1	and	and	CCONJ
ejpam-5252	528	2	so	so	ADV
ejpam-5252	528	3	,	,	PUNCT
ejpam-5252	528	4	(	(	PUNCT
ejpam-5252	528	5	b2	b2	NOUN
ejpam-5252	528	6	)	)	PUNCT
ejpam-5252	528	7	holds	hold	VERB
ejpam-5252	528	8	.	.	PUNCT
ejpam-5252	529	1	also	also	ADV
ejpam-5252	529	2	,	,	PUNCT
ejpam-5252	529	3	since	since	SCONJ
ejpam-5252	529	4	v3	v3	PROPN
ejpam-5252	529	5	is	be	AUX
ejpam-5252	529	6	a	a	DET
ejpam-5252	529	7	dominating	dominating	NOUN
ejpam-5252	529	8	set	set	NOUN
ejpam-5252	529	9	of	of	ADP
ejpam-5252	529	10	g	g	NOUN
ejpam-5252	529	11	,	,	PUNCT
ejpam-5252	529	12	there	there	PRON
ejpam-5252	529	13	exists	exist	VERB
ejpam-5252	529	14	u	u	PROPN
ejpam-5252	529	15	∈	∈	PROPN
ejpam-5252	529	16	v3	v3	PROPN
ejpam-5252	529	17	∩	∩	PROPN
ejpam-5252	529	18	v	v	X
ejpam-5252	529	19	(	(	PUNCT
ejpam-5252	529	20	g	g	NOUN
ejpam-5252	529	21	)	)	PUNCT
ejpam-5252	529	22	where	where	SCONJ
ejpam-5252	529	23	s.	s.	PROPN
ejpam-5252	529	24	ahamad	ahamad	PROPN
ejpam-5252	529	25	,	,	PUNCT
ejpam-5252	529	26	j.	j.	PROPN
ejpam-5252	529	27	cariaga	cariaga	PROPN
ejpam-5252	529	28	,	,	PUNCT
ejpam-5252	529	29	s.	s.	PROPN
ejpam-5252	529	30	menchavez	menchavez	PROPN
ejpam-5252	529	31	/	/	PUNCT
ejpam-5252	529	32	eur	eur	PROPN
ejpam-5252	529	33	.	.	PUNCT
ejpam-5252	530	1	j.	j.	PROPN
ejpam-5252	530	2	pure	pure	PROPN
ejpam-5252	530	3	appl	appl	PROPN
ejpam-5252	530	4	.	.	PROPN
ejpam-5252	530	5	math	math	PROPN
ejpam-5252	530	6	,	,	PUNCT
ejpam-5252	530	7	18	18	NUM
ejpam-5252	530	8	(	(	PUNCT
ejpam-5252	530	9	1	1	NUM
ejpam-5252	530	10	)	)	PUNCT
ejpam-5252	530	11	(	(	PUNCT
ejpam-5252	530	12	2025	2025	NUM
ejpam-5252	530	13	)	)	PUNCT
ejpam-5252	530	14	,	,	PUNCT
ejpam-5252	530	15	5252	5252	NUM
ejpam-5252	530	16	14	14	NUM
ejpam-5252	530	17	of	of	ADP
ejpam-5252	530	18	18	18	NUM
ejpam-5252	530	19	uv	uv	NOUN
ejpam-5252	530	20	∈	∈	PROPN
ejpam-5252	530	21	e(g+h	e(g+h	NUM
ejpam-5252	530	22	)	)	PUNCT
ejpam-5252	530	23	.	.	PUNCT
ejpam-5252	531	1	suppose	suppose	VERB
ejpam-5252	531	2	v3	v3	PROPN
ejpam-5252	531	3	∩	∩	PROPN
ejpam-5252	531	4	v	v	X
ejpam-5252	531	5	(	(	PUNCT
ejpam-5252	531	6	g	g	NOUN
ejpam-5252	531	7	)	)	PUNCT
ejpam-5252	531	8	=	=	NOUN
ejpam-5252	531	9	∅	∅	NOUN
ejpam-5252	531	10	,	,	PUNCT
ejpam-5252	531	11	then	then	ADV
ejpam-5252	531	12	similarly	similarly	ADV
ejpam-5252	531	13	,	,	PUNCT
ejpam-5252	531	14	(	(	PUNCT
ejpam-5252	531	15	i)(c1	i)(c1	NOUN
ejpam-5252	531	16	)	)	PUNCT
ejpam-5252	531	17	and	and	CCONJ
ejpam-5252	531	18	(	(	PUNCT
ejpam-5252	531	19	i)(c2	i)(c2	NOUN
ejpam-5252	531	20	)	)	PUNCT
ejpam-5252	531	21	hold	hold	NOUN
ejpam-5252	531	22	.	.	PUNCT
ejpam-5252	532	1	case	case	NOUN
ejpam-5252	532	2	2	2	NUM
ejpam-5252	532	3	:	:	PUNCT
ejpam-5252	532	4	suppose	suppose	VERB
ejpam-5252	532	5	f	f	PROPN
ejpam-5252	532	6	|h	|h	PROPN
ejpam-5252	532	7	∈	∈	PROPN
ejpam-5252	532	8	mrdf	mrdf	NOUN
ejpam-5252	532	9	(	(	PUNCT
ejpam-5252	532	10	h	h	NOUN
ejpam-5252	532	11	)	)	PUNCT
ejpam-5252	532	12	.	.	PUNCT
ejpam-5252	533	1	this	this	DET
ejpam-5252	533	2	case	case	NOUN
ejpam-5252	533	3	can	can	AUX
ejpam-5252	533	4	be	be	AUX
ejpam-5252	533	5	proven	prove	VERB
ejpam-5252	533	6	similarly	similarly	ADV
ejpam-5252	533	7	with	with	ADP
ejpam-5252	533	8	case	case	NOUN
ejpam-5252	533	9	1	1	NUM
ejpam-5252	533	10	.	.	PUNCT
ejpam-5252	533	11	case	case	NOUN
ejpam-5252	533	12	3	3	X
ejpam-5252	533	13	:	:	PUNCT
ejpam-5252	533	14	suppose	suppose	VERB
ejpam-5252	533	15	f	f	PROPN
ejpam-5252	533	16	|g	|g	PROPN
ejpam-5252	533	17	/∈	/∈	PUNCT
ejpam-5252	534	1	mrdf	mrdf	NOUN
ejpam-5252	534	2	(	(	PUNCT
ejpam-5252	534	3	g	g	NOUN
ejpam-5252	534	4	)	)	PUNCT
ejpam-5252	534	5	and	and	CCONJ
ejpam-5252	534	6	f	f	PROPN
ejpam-5252	534	7	|h	|h	PROPN
ejpam-5252	534	8	/∈	/∈	PUNCT
ejpam-5252	535	1	mrdf	mrdf	NOUN
ejpam-5252	535	2	(	(	PUNCT
ejpam-5252	535	3	h	h	NOUN
ejpam-5252	535	4	)	)	PUNCT
ejpam-5252	535	5	.	.	PUNCT
ejpam-5252	536	1	if	if	SCONJ
ejpam-5252	536	2	f	f	PROPN
ejpam-5252	536	3	|g	|g	PROPN
ejpam-5252	536	4	/∈	/∈	PUNCT
ejpam-5252	537	1	mrdf	mrdf	NOUN
ejpam-5252	537	2	(	(	PUNCT
ejpam-5252	537	3	g	g	NOUN
ejpam-5252	537	4	)	)	PUNCT
ejpam-5252	537	5	,	,	PUNCT
ejpam-5252	537	6	then	then	ADV
ejpam-5252	537	7	there	there	PRON
ejpam-5252	537	8	exists	exist	VERB
ejpam-5252	537	9	x	x	X
ejpam-5252	537	10	∈	∈	PROPN
ejpam-5252	537	11	v0	v0	NOUN
ejpam-5252	537	12	∩v	∩v	NOUN
ejpam-5252	538	1	(	(	PUNCT
ejpam-5252	538	2	g	g	NOUN
ejpam-5252	538	3	)	)	PUNCT
ejpam-5252	539	1	such	such	ADJ
ejpam-5252	539	2	that	that	SCONJ
ejpam-5252	539	3	ng(x)∩v2	ng(x)∩v2	ADJ
ejpam-5252	539	4	=	=	NOUN
ejpam-5252	539	5	∅	∅	NOUN
ejpam-5252	539	6	or	or	CCONJ
ejpam-5252	539	7	ng(x)∩v3	ng(x)∩v3	ADJ
ejpam-5252	539	8	=	=	PUNCT
ejpam-5252	539	9	∅.	∅.	NOUN
ejpam-5252	539	10	moreover	moreover	ADV
ejpam-5252	539	11	,	,	PUNCT
ejpam-5252	539	12	there	there	PRON
ejpam-5252	539	13	exists	exist	VERB
ejpam-5252	539	14	y	y	PROPN
ejpam-5252	539	15	∈	∈	PROPN
ejpam-5252	539	16	v1∩v	v1∩v	NOUN
ejpam-5252	539	17	(	(	PUNCT
ejpam-5252	539	18	g	g	NOUN
ejpam-5252	539	19	)	)	PUNCT
ejpam-5252	539	20	such	such	ADJ
ejpam-5252	539	21	that	that	DET
ejpam-5252	539	22	ng(y)∩v2	ng(y)∩v2	NOUN
ejpam-5252	539	23	=	=	SYM
ejpam-5252	539	24	∅	∅	NOUN
ejpam-5252	539	25	and	and	CCONJ
ejpam-5252	539	26	ng(y)∩v3	ng(y)∩v3	VERB
ejpam-5252	539	27	=	=	PUNCT
ejpam-5252	539	28	∅.	∅.	NOUN
ejpam-5252	539	29	if	if	SCONJ
ejpam-5252	539	30	ng(x)∩v2	ng(x)∩v2	NOUN
ejpam-5252	539	31	=	=	SYM
ejpam-5252	539	32	∅	∅	NOUN
ejpam-5252	539	33	and	and	CCONJ
ejpam-5252	539	34	ng(x)∩v3	ng(x)∩v3	VERB
ejpam-5252	539	35	̸=	̸=	PROPN
ejpam-5252	539	36	∅.	∅.	NOUN
ejpam-5252	539	37	note	note	VERB
ejpam-5252	539	38	that	that	SCONJ
ejpam-5252	539	39	f	f	PROPN
ejpam-5252	539	40	∈	∈	PROPN
ejpam-5252	539	41	mrdf	mrdf	NOUN
ejpam-5252	539	42	(	(	PUNCT
ejpam-5252	539	43	g+h	g+h	PROPN
ejpam-5252	539	44	)	)	PUNCT
ejpam-5252	539	45	.	.	PUNCT
ejpam-5252	540	1	then	then	ADV
ejpam-5252	540	2	,	,	PUNCT
ejpam-5252	540	3	there	there	PRON
ejpam-5252	540	4	exists	exist	VERB
ejpam-5252	540	5	u	u	PROPN
ejpam-5252	540	6	∈	∈	PROPN
ejpam-5252	540	7	(	(	PUNCT
ejpam-5252	540	8	v2	v2	PROPN
ejpam-5252	540	9	∩v	∩v	NOUN
ejpam-5252	540	10	(	(	PUNCT
ejpam-5252	540	11	g+h	g+h	PROPN
ejpam-5252	540	12	)	)	PUNCT
ejpam-5252	540	13	)	)	PUNCT
ejpam-5252	540	14	such	such	ADJ
ejpam-5252	540	15	that	that	SCONJ
ejpam-5252	540	16	u	u	PROPN
ejpam-5252	540	17	∈	∈	PROPN
ejpam-5252	540	18	ng+h(x	ng+h(x	PROPN
ejpam-5252	540	19	)	)	PUNCT
ejpam-5252	540	20	for	for	ADP
ejpam-5252	540	21	some	some	DET
ejpam-5252	540	22	x	x	SYM
ejpam-5252	540	23	∈	∈	PROPN
ejpam-5252	540	24	v0	v0	NOUN
ejpam-5252	540	25	∩	∩	X
ejpam-5252	540	26	v	v	X
ejpam-5252	540	27	(	(	PUNCT
ejpam-5252	540	28	g	g	NOUN
ejpam-5252	540	29	)	)	PUNCT
ejpam-5252	540	30	.	.	PUNCT
ejpam-5252	541	1	since	since	SCONJ
ejpam-5252	541	2	ng(x	ng(x	NUM
ejpam-5252	541	3	)	)	PUNCT
ejpam-5252	541	4	∩	∩	NOUN
ejpam-5252	541	5	v2	v2	NOUN
ejpam-5252	541	6	=	=	SYM
ejpam-5252	541	7	∅	∅	NOUN
ejpam-5252	541	8	,	,	PUNCT
ejpam-5252	541	9	u	u	NOUN
ejpam-5252	541	10	∈	∈	PROPN
ejpam-5252	541	11	nh(x	nh(x	NUM
ejpam-5252	541	12	)	)	PUNCT
ejpam-5252	541	13	.	.	PUNCT
ejpam-5252	542	1	consequently	consequently	ADV
ejpam-5252	542	2	,	,	PUNCT
ejpam-5252	542	3	u	u	PROPN
ejpam-5252	542	4	∈	∈	PROPN
ejpam-5252	542	5	v2	v2	PROPN
ejpam-5252	542	6	∩	∩	ADJ
ejpam-5252	542	7	v	v	NOUN
ejpam-5252	542	8	(	(	PUNCT
ejpam-5252	542	9	h	h	NOUN
ejpam-5252	542	10	)	)	PUNCT
ejpam-5252	542	11	for	for	ADP
ejpam-5252	542	12	some	some	DET
ejpam-5252	542	13	x	x	SYM
ejpam-5252	542	14	∈	∈	PROPN
ejpam-5252	542	15	v0	v0	NOUN
ejpam-5252	542	16	.	.	PUNCT
ejpam-5252	543	1	thus	thus	ADV
ejpam-5252	543	2	,	,	PUNCT
ejpam-5252	543	3	(	(	PUNCT
ejpam-5252	543	4	iii)(a	iii)(a	NOUN
ejpam-5252	543	5	)	)	PUNCT
ejpam-5252	543	6	holds	hold	VERB
ejpam-5252	543	7	.	.	PUNCT
ejpam-5252	544	1	if	if	SCONJ
ejpam-5252	544	2	ng(x	ng(x	NUM
ejpam-5252	544	3	)	)	PUNCT
ejpam-5252	544	4	∩	∩	PROPN
ejpam-5252	544	5	v3	v3	NOUN
ejpam-5252	544	6	=	=	SYM
ejpam-5252	544	7	∅	∅	NOUN
ejpam-5252	544	8	and	and	CCONJ
ejpam-5252	544	9	ng(x	ng(x	NUM
ejpam-5252	544	10	)	)	PUNCT
ejpam-5252	544	11	∩	∩	NOUN
ejpam-5252	544	12	v2	v2	PROPN
ejpam-5252	544	13	̸=	̸=	PROPN
ejpam-5252	544	14	∅.	∅.	NOUN
ejpam-5252	544	15	since	since	SCONJ
ejpam-5252	544	16	f	f	PROPN
ejpam-5252	544	17	∈	∈	PROPN
ejpam-5252	544	18	mrdf	mrdf	NOUN
ejpam-5252	544	19	(	(	PUNCT
ejpam-5252	544	20	g+h	g+h	PROPN
ejpam-5252	544	21	)	)	PUNCT
ejpam-5252	544	22	,	,	PUNCT
ejpam-5252	544	23	then	then	ADV
ejpam-5252	544	24	there	there	PRON
ejpam-5252	544	25	exists	exist	VERB
ejpam-5252	544	26	w	w	PROPN
ejpam-5252	544	27	∈	∈	PROPN
ejpam-5252	544	28	(	(	PUNCT
ejpam-5252	544	29	v3	v3	PROPN
ejpam-5252	544	30	∩	∩	PROPN
ejpam-5252	544	31	v	v	PROPN
ejpam-5252	544	32	(	(	PUNCT
ejpam-5252	544	33	g+h	g+h	NOUN
ejpam-5252	544	34	)	)	PUNCT
ejpam-5252	544	35	)	)	PUNCT
ejpam-5252	544	36	such	such	ADJ
ejpam-5252	544	37	that	that	SCONJ
ejpam-5252	544	38	w	w	PROPN
ejpam-5252	544	39	∈	∈	PROPN
ejpam-5252	544	40	ng+h(x	ng+h(x	PROPN
ejpam-5252	544	41	)	)	PUNCT
ejpam-5252	544	42	for	for	ADP
ejpam-5252	544	43	some	some	DET
ejpam-5252	544	44	x	x	SYM
ejpam-5252	544	45	∈	∈	PROPN
ejpam-5252	544	46	v0	v0	NOUN
ejpam-5252	544	47	∩	∩	X
ejpam-5252	544	48	v	v	X
ejpam-5252	544	49	(	(	PUNCT
ejpam-5252	544	50	g	g	NOUN
ejpam-5252	544	51	)	)	PUNCT
ejpam-5252	544	52	.	.	PUNCT
ejpam-5252	545	1	consequently	consequently	ADV
ejpam-5252	545	2	,	,	PUNCT
ejpam-5252	545	3	by	by	ADP
ejpam-5252	545	4	assumption	assumption	NOUN
ejpam-5252	545	5	,	,	PUNCT
ejpam-5252	545	6	(	(	PUNCT
ejpam-5252	545	7	iii)(b	iii)(b	ADJ
ejpam-5252	545	8	)	)	PUNCT
ejpam-5252	545	9	holds	hold	VERB
ejpam-5252	545	10	.	.	PUNCT
ejpam-5252	546	1	if	if	SCONJ
ejpam-5252	546	2	ng(x	ng(x	NUM
ejpam-5252	546	3	)	)	PUNCT
ejpam-5252	546	4	∩	∩	NOUN
ejpam-5252	546	5	v2	v2	NOUN
ejpam-5252	546	6	=	=	SYM
ejpam-5252	546	7	∅	∅	NOUN
ejpam-5252	546	8	and	and	CCONJ
ejpam-5252	546	9	ng(x	ng(x	NUM
ejpam-5252	546	10	)	)	PUNCT
ejpam-5252	546	11	∩	∩	NOUN
ejpam-5252	546	12	v3	v3	PROPN
ejpam-5252	546	13	=	=	PUNCT
ejpam-5252	546	14	∅.	∅.	NOUN
ejpam-5252	546	15	since	since	SCONJ
ejpam-5252	546	16	f	f	PROPN
ejpam-5252	546	17	∈	∈	PROPN
ejpam-5252	546	18	mrdf	mrdf	NOUN
ejpam-5252	546	19	(	(	PUNCT
ejpam-5252	546	20	g	g	NOUN
ejpam-5252	546	21	+	+	NOUN
ejpam-5252	546	22	h	h	NOUN
ejpam-5252	546	23	)	)	PUNCT
ejpam-5252	546	24	,	,	PUNCT
ejpam-5252	546	25	then	then	ADV
ejpam-5252	546	26	there	there	PRON
ejpam-5252	546	27	exist	exist	VERB
ejpam-5252	546	28	u	u	PROPN
ejpam-5252	546	29	∈	∈	PROPN
ejpam-5252	546	30	(	(	PUNCT
ejpam-5252	546	31	v2	v2	PROPN
ejpam-5252	546	32	∩	∩	ADJ
ejpam-5252	546	33	v	v	NOUN
ejpam-5252	546	34	(	(	PUNCT
ejpam-5252	546	35	g+h	g+h	NOUN
ejpam-5252	546	36	)	)	PUNCT
ejpam-5252	546	37	)	)	PUNCT
ejpam-5252	546	38	and	and	CCONJ
ejpam-5252	546	39	w	w	PROPN
ejpam-5252	546	40	∈	∈	PROPN
ejpam-5252	546	41	(	(	PUNCT
ejpam-5252	546	42	v3	v3	PROPN
ejpam-5252	546	43	∩	∩	PROPN
ejpam-5252	546	44	v	v	PROPN
ejpam-5252	546	45	(	(	PUNCT
ejpam-5252	546	46	g+h	g+h	NOUN
ejpam-5252	546	47	)	)	PUNCT
ejpam-5252	546	48	)	)	PUNCT
ejpam-5252	546	49	such	such	ADJ
ejpam-5252	546	50	that	that	SCONJ
ejpam-5252	546	51	u	u	NOUN
ejpam-5252	546	52	,	,	PUNCT
ejpam-5252	546	53	w	w	PROPN
ejpam-5252	546	54	∈	∈	PROPN
ejpam-5252	546	55	ng+h(x	ng+h(x	PROPN
ejpam-5252	546	56	)	)	PUNCT
ejpam-5252	546	57	for	for	ADP
ejpam-5252	546	58	some	some	DET
ejpam-5252	546	59	x	x	SYM
ejpam-5252	546	60	∈	∈	PROPN
ejpam-5252	546	61	v0∩v	v0∩v	X
ejpam-5252	546	62	(	(	PUNCT
ejpam-5252	546	63	g	g	NOUN
ejpam-5252	546	64	)	)	PUNCT
ejpam-5252	546	65	.	.	PUNCT
ejpam-5252	547	1	thus	thus	ADV
ejpam-5252	547	2	,	,	PUNCT
ejpam-5252	547	3	by	by	ADP
ejpam-5252	547	4	assumption	assumption	NOUN
ejpam-5252	547	5	,	,	PUNCT
ejpam-5252	547	6	u	u	NOUN
ejpam-5252	547	7	,	,	PUNCT
ejpam-5252	547	8	w	w	PROPN
ejpam-5252	547	9	∈	∈	PROPN
ejpam-5252	547	10	nh(x	nh(x	NUM
ejpam-5252	547	11	)	)	PUNCT
ejpam-5252	547	12	and	and	CCONJ
ejpam-5252	547	13	consequently	consequently	ADV
ejpam-5252	547	14	,	,	PUNCT
ejpam-5252	547	15	v2∩v	v2∩v	PROPN
ejpam-5252	547	16	(	(	PUNCT
ejpam-5252	547	17	h	h	NOUN
ejpam-5252	547	18	)	)	PUNCT
ejpam-5252	547	19	̸=	̸=	PROPN
ejpam-5252	547	20	∅	∅	NOUN
ejpam-5252	547	21	and	and	CCONJ
ejpam-5252	547	22	v3	v3	PROPN
ejpam-5252	547	23	∩	∩	PROPN
ejpam-5252	547	24	v	v	X
ejpam-5252	547	25	(	(	PUNCT
ejpam-5252	547	26	h	h	NOUN
ejpam-5252	547	27	)	)	PUNCT
ejpam-5252	547	28	̸=	̸=	PROPN
ejpam-5252	547	29	∅.	∅.	PRON
ejpam-5252	547	30	hence	hence	ADV
ejpam-5252	547	31	,	,	PUNCT
ejpam-5252	547	32	(	(	PUNCT
ejpam-5252	547	33	iii)(a	iii)(a	PROPN
ejpam-5252	547	34	)	)	PUNCT
ejpam-5252	547	35	and	and	CCONJ
ejpam-5252	547	36	(	(	PUNCT
ejpam-5252	547	37	iii)(b	iii)(b	ADJ
ejpam-5252	547	38	)	)	PUNCT
ejpam-5252	547	39	hold	hold	VERB
ejpam-5252	547	40	.	.	PUNCT
ejpam-5252	548	1	furthermore	furthermore	ADV
ejpam-5252	548	2	,	,	PUNCT
ejpam-5252	548	3	suppose	suppose	VERB
ejpam-5252	548	4	ng(y	ng(y	NOUN
ejpam-5252	548	5	)	)	PUNCT
ejpam-5252	548	6	∩	∩	ADJ
ejpam-5252	548	7	v2	v2	NOUN
ejpam-5252	548	8	=	=	PUNCT
ejpam-5252	548	9	∅	∅	NOUN
ejpam-5252	548	10	and	and	CCONJ
ejpam-5252	548	11	ng(y	ng(y	NOUN
ejpam-5252	548	12	)	)	PUNCT
ejpam-5252	548	13	∩	∩	ADJ
ejpam-5252	548	14	v3	v3	NOUN
ejpam-5252	548	15	=	=	PUNCT
ejpam-5252	548	16	∅	∅	NOUN
ejpam-5252	548	17	for	for	ADP
ejpam-5252	548	18	some	some	DET
ejpam-5252	548	19	y	y	PROPN
ejpam-5252	548	20	∈	∈	PROPN
ejpam-5252	548	21	v1	v1	NOUN
ejpam-5252	548	22	∩	∩	ADJ
ejpam-5252	548	23	v	v	NOUN
ejpam-5252	548	24	(	(	PUNCT
ejpam-5252	548	25	g	g	NOUN
ejpam-5252	548	26	)	)	PUNCT
ejpam-5252	548	27	.	.	PUNCT
ejpam-5252	549	1	since	since	SCONJ
ejpam-5252	549	2	f	f	PROPN
ejpam-5252	549	3	∈	∈	PROPN
ejpam-5252	549	4	mrdf	mrdf	NOUN
ejpam-5252	549	5	(	(	PUNCT
ejpam-5252	549	6	g	g	NOUN
ejpam-5252	549	7	+	+	NOUN
ejpam-5252	549	8	h	h	NOUN
ejpam-5252	549	9	)	)	PUNCT
ejpam-5252	549	10	,	,	PUNCT
ejpam-5252	549	11	there	there	PRON
ejpam-5252	549	12	exist	exist	VERB
ejpam-5252	549	13	u	u	PROPN
ejpam-5252	549	14	∈	∈	PROPN
ejpam-5252	549	15	v2	v2	PROPN
ejpam-5252	549	16	∩	∩	ADJ
ejpam-5252	549	17	v	v	NOUN
ejpam-5252	549	18	(	(	PUNCT
ejpam-5252	549	19	g	g	PROPN
ejpam-5252	549	20	+	+	NOUN
ejpam-5252	549	21	h	h	NOUN
ejpam-5252	549	22	)	)	PUNCT
ejpam-5252	549	23	or	or	CCONJ
ejpam-5252	549	24	w	w	PROPN
ejpam-5252	549	25	∈	∈	PROPN
ejpam-5252	549	26	v3	v3	PROPN
ejpam-5252	549	27	∩	∩	PROPN
ejpam-5252	549	28	v	v	X
ejpam-5252	549	29	(	(	PUNCT
ejpam-5252	549	30	g	g	PROPN
ejpam-5252	549	31	+	+	NOUN
ejpam-5252	549	32	h	h	NOUN
ejpam-5252	549	33	)	)	PUNCT
ejpam-5252	549	34	such	such	ADJ
ejpam-5252	549	35	that	that	SCONJ
ejpam-5252	549	36	u	u	NOUN
ejpam-5252	549	37	,	,	PUNCT
ejpam-5252	549	38	w	w	PROPN
ejpam-5252	549	39	∈	∈	NOUN
ejpam-5252	549	40	ng+h(y	ng+h(y	NUM
ejpam-5252	549	41	)	)	PUNCT
ejpam-5252	549	42	.	.	PUNCT
ejpam-5252	550	1	by	by	ADP
ejpam-5252	550	2	assumption	assumption	NOUN
ejpam-5252	550	3	,	,	PUNCT
ejpam-5252	550	4	u	u	NOUN
ejpam-5252	550	5	,	,	PUNCT
ejpam-5252	550	6	w	w	PROPN
ejpam-5252	550	7	∈	∈	PROPN
ejpam-5252	550	8	nh(y	nh(y	ADV
ejpam-5252	550	9	)	)	PUNCT
ejpam-5252	550	10	and	and	CCONJ
ejpam-5252	550	11	consequently	consequently	ADV
ejpam-5252	550	12	,	,	PUNCT
ejpam-5252	550	13	(	(	PUNCT
ejpam-5252	550	14	iii)(c	iii)(c	NOUN
ejpam-5252	550	15	)	)	PUNCT
ejpam-5252	550	16	holds	hold	VERB
ejpam-5252	550	17	.	.	PUNCT
ejpam-5252	551	1	similarly	similarly	ADV
ejpam-5252	551	2	,	,	PUNCT
ejpam-5252	551	3	if	if	SCONJ
ejpam-5252	551	4	f	f	PROPN
ejpam-5252	551	5	|h	|h	X
ejpam-5252	551	6	/∈	/∈	PUNCT
ejpam-5252	552	1	mrdf	mrdf	NOUN
ejpam-5252	552	2	(	(	PUNCT
ejpam-5252	552	3	h	h	NOUN
ejpam-5252	552	4	)	)	PUNCT
ejpam-5252	552	5	,	,	PUNCT
ejpam-5252	552	6	then	then	ADV
ejpam-5252	552	7	(	(	PUNCT
ejpam-5252	552	8	iii)(d	iii)(d	PROPN
ejpam-5252	552	9	)	)	PUNCT
ejpam-5252	552	10	,	,	PUNCT
ejpam-5252	552	11	(	(	PUNCT
ejpam-5252	552	12	iii)(e	iii)(e	NOUN
ejpam-5252	552	13	)	)	PUNCT
ejpam-5252	552	14	,	,	PUNCT
ejpam-5252	552	15	and	and	CCONJ
ejpam-5252	552	16	(	(	PUNCT
ejpam-5252	552	17	iii)(f	iii)(f	ADJ
ejpam-5252	552	18	)	)	PUNCT
ejpam-5252	552	19	hold	hold	VERB
ejpam-5252	552	20	.	.	PUNCT
ejpam-5252	553	1	proposition	proposition	NOUN
ejpam-5252	553	2	14	14	NUM
ejpam-5252	553	3	.	.	PUNCT
ejpam-5252	554	1	let	let	VERB
ejpam-5252	554	2	g	g	NOUN
ejpam-5252	554	3	and	and	CCONJ
ejpam-5252	554	4	h	h	NOUN
ejpam-5252	554	5	be	be	VERB
ejpam-5252	554	6	any	any	DET
ejpam-5252	554	7	graphs	graph	NOUN
ejpam-5252	554	8	.	.	PUNCT
ejpam-5252	555	1	then	then	ADV
ejpam-5252	555	2	3	3	NUM
ejpam-5252	555	3	≤	≤	NUM
ejpam-5252	555	4	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	555	5	)	)	PUNCT
ejpam-5252	555	6	≤	≤	ADV
ejpam-5252	555	7	10	10	NUM
ejpam-5252	555	8	proof	proof	NOUN
ejpam-5252	555	9	.	.	PUNCT
ejpam-5252	556	1	suppose	suppose	VERB
ejpam-5252	556	2	g	g	PROPN
ejpam-5252	556	3	and	and	CCONJ
ejpam-5252	556	4	h	h	NOUN
ejpam-5252	556	5	are	be	AUX
ejpam-5252	556	6	trivial	trivial	ADJ
ejpam-5252	556	7	graphs	graph	NOUN
ejpam-5252	556	8	.	.	PUNCT
ejpam-5252	557	1	then	then	ADV
ejpam-5252	557	2	by	by	ADP
ejpam-5252	557	3	proposition	proposition	NOUN
ejpam-5252	557	4	5	5	NUM
ejpam-5252	557	5	(	(	PUNCT
ejpam-5252	557	6	ii	ii	NOUN
ejpam-5252	557	7	)	)	PUNCT
ejpam-5252	557	8	,	,	PUNCT
ejpam-5252	557	9	γmr(g+h	γmr(g+h	PROPN
ejpam-5252	557	10	)	)	PUNCT
ejpam-5252	558	1	=	=	SYM
ejpam-5252	558	2	γmr(k2	γmr(k2	NOUN
ejpam-5252	558	3	)	)	PUNCT
ejpam-5252	558	4	=	=	SYM
ejpam-5252	558	5	3	3	X
ejpam-5252	558	6	.	.	PUNCT
ejpam-5252	558	7	suppose	suppose	VERB
ejpam-5252	558	8	g	g	PROPN
ejpam-5252	558	9	and	and	CCONJ
ejpam-5252	558	10	h	h	NOUN
ejpam-5252	558	11	are	be	AUX
ejpam-5252	558	12	not	not	PART
ejpam-5252	558	13	trivial	trivial	ADJ
ejpam-5252	558	14	graphs	graph	NOUN
ejpam-5252	558	15	,	,	PUNCT
ejpam-5252	558	16	then	then	ADV
ejpam-5252	558	17	γmr(g	γmr(g	PROPN
ejpam-5252	559	1	+	+	NUM
ejpam-5252	559	2	h	h	X
ejpam-5252	559	3	)	)	PUNCT
ejpam-5252	559	4	>	>	X
ejpam-5252	560	1	2	2	X
ejpam-5252	560	2	.	.	X
ejpam-5252	560	3	that	that	PRON
ejpam-5252	560	4	is	be	AUX
ejpam-5252	560	5	,	,	PUNCT
ejpam-5252	560	6	γmr(g	γmr(g	PROPN
ejpam-5252	561	1	+	+	NUM
ejpam-5252	561	2	h	h	X
ejpam-5252	561	3	)	)	PUNCT
ejpam-5252	561	4	≥	≥	NOUN
ejpam-5252	561	5	3	3	NUM
ejpam-5252	561	6	.	.	PUNCT
ejpam-5252	562	1	on	on	ADP
ejpam-5252	562	2	the	the	DET
ejpam-5252	562	3	other	other	ADJ
ejpam-5252	562	4	hand	hand	NOUN
ejpam-5252	562	5	,	,	PUNCT
ejpam-5252	562	6	let	let	VERB
ejpam-5252	562	7	v	v	NOUN
ejpam-5252	562	8	(	(	PUNCT
ejpam-5252	562	9	g	g	NOUN
ejpam-5252	562	10	)	)	PUNCT
ejpam-5252	562	11	=	=	SYM
ejpam-5252	562	12	{	{	PUNCT
ejpam-5252	562	13	v1	v1	PROPN
ejpam-5252	562	14	,	,	PUNCT
ejpam-5252	562	15	v2	v2	PROPN
ejpam-5252	562	16	,	,	PUNCT
ejpam-5252	562	17	·	·	PUNCT
ejpam-5252	562	18	·	·	PUNCT
ejpam-5252	562	19	·	·	PUNCT
ejpam-5252	562	20	,	,	PUNCT
ejpam-5252	562	21	vn	vn	INTJ
ejpam-5252	562	22	}	}	PUNCT
ejpam-5252	562	23	and	and	CCONJ
ejpam-5252	562	24	v	v	NOUN
ejpam-5252	562	25	(	(	PUNCT
ejpam-5252	562	26	h	h	NOUN
ejpam-5252	562	27	)	)	PUNCT
ejpam-5252	562	28	=	=	PRON
ejpam-5252	562	29	{	{	PUNCT
ejpam-5252	562	30	u1	u1	NOUN
ejpam-5252	562	31	,	,	PUNCT
ejpam-5252	562	32	u2	u2	PROPN
ejpam-5252	562	33	,	,	PUNCT
ejpam-5252	562	34	·	·	PUNCT
ejpam-5252	562	35	·	·	PUNCT
ejpam-5252	562	36	·	·	PUNCT
ejpam-5252	562	37	,	,	PUNCT
ejpam-5252	562	38	un	un	PROPN
ejpam-5252	562	39	}	}	PUNCT
ejpam-5252	562	40	.	.	PUNCT
ejpam-5252	563	1	now	now	ADV
ejpam-5252	563	2	,	,	PUNCT
ejpam-5252	563	3	define	define	VERB
ejpam-5252	563	4	a	a	DET
ejpam-5252	563	5	function	function	NOUN
ejpam-5252	563	6	f	f	NOUN
ejpam-5252	563	7	=	=	SYM
ejpam-5252	563	8	(	(	PUNCT
ejpam-5252	563	9	v0	v0	PROPN
ejpam-5252	563	10	,	,	PUNCT
ejpam-5252	563	11	v1	v1	NOUN
ejpam-5252	563	12	,	,	PUNCT
ejpam-5252	563	13	v2	v2	PROPN
ejpam-5252	563	14	,	,	PUNCT
ejpam-5252	563	15	v3	v3	PROPN
ejpam-5252	563	16	)	)	PUNCT
ejpam-5252	563	17	on	on	ADP
ejpam-5252	563	18	v	v	ADP
ejpam-5252	563	19	(	(	PUNCT
ejpam-5252	563	20	g+h	g+h	NOUN
ejpam-5252	563	21	)	)	PUNCT
ejpam-5252	563	22	given	give	VERB
ejpam-5252	563	23	by	by	ADP
ejpam-5252	563	24	f(x	f(x	PROPN
ejpam-5252	563	25	)	)	PUNCT
ejpam-5252	564	1	=	=	PUNCT
ejpam-5252	565	1			NOUN
ejpam-5252	565	2	2	2	NUM
ejpam-5252	565	3	,	,	PUNCT
ejpam-5252	565	4	if	if	SCONJ
ejpam-5252	565	5	x	x	SYM
ejpam-5252	565	6	∈	∈	PROPN
ejpam-5252	565	7	{	{	PUNCT
ejpam-5252	565	8	v1	v1	NOUN
ejpam-5252	565	9	,	,	PUNCT
ejpam-5252	565	10	u1	u1	NOUN
ejpam-5252	565	11	}	}	PUNCT
ejpam-5252	565	12	.	.	PUNCT
ejpam-5252	566	1	3	3	NUM
ejpam-5252	566	2	,	,	PUNCT
ejpam-5252	566	3	if	if	SCONJ
ejpam-5252	566	4	x	x	SYM
ejpam-5252	566	5	∈	∈	PROPN
ejpam-5252	566	6	{	{	PUNCT
ejpam-5252	566	7	v2	v2	NOUN
ejpam-5252	566	8	,	,	PUNCT
ejpam-5252	566	9	u2	u2	PROPN
ejpam-5252	566	10	}	}	PUNCT
ejpam-5252	566	11	.	.	PUNCT
ejpam-5252	567	1	0	0	NUM
ejpam-5252	567	2	,	,	PUNCT
ejpam-5252	567	3	if	if	SCONJ
ejpam-5252	567	4	x	x	PROPN
ejpam-5252	567	5	∈	∈	PROPN
ejpam-5252	567	6	v	v	NOUN
ejpam-5252	567	7	(	(	PUNCT
ejpam-5252	567	8	g+h	g+h	NOUN
ejpam-5252	567	9	)	)	PUNCT
ejpam-5252	567	10	\	\	NOUN
ejpam-5252	567	11	{	{	PUNCT
ejpam-5252	567	12	v1	v1	NOUN
ejpam-5252	567	13	,	,	PUNCT
ejpam-5252	567	14	v2	v2	PROPN
ejpam-5252	567	15	,	,	PUNCT
ejpam-5252	567	16	u1	u1	NOUN
ejpam-5252	567	17	,	,	PUNCT
ejpam-5252	567	18	u2	u2	PROPN
ejpam-5252	567	19	}	}	PUNCT
ejpam-5252	567	20	.	.	PUNCT
ejpam-5252	568	1	for	for	ADP
ejpam-5252	568	2	every	every	DET
ejpam-5252	568	3	x	x	SYM
ejpam-5252	568	4	∈	∈	PROPN
ejpam-5252	568	5	v	v	NOUN
ejpam-5252	568	6	(	(	PUNCT
ejpam-5252	568	7	g+h	g+h	PROPN
ejpam-5252	568	8	)	)	PUNCT
ejpam-5252	568	9	.	.	PUNCT
ejpam-5252	569	1	then	then	ADV
ejpam-5252	569	2	f	f	PROPN
ejpam-5252	569	3	∈	∈	PROPN
ejpam-5252	569	4	mrdf	mrdf	NOUN
ejpam-5252	569	5	(	(	PUNCT
ejpam-5252	569	6	g+h	g+h	NOUN
ejpam-5252	569	7	)	)	PUNCT
ejpam-5252	569	8	.	.	PUNCT
ejpam-5252	570	1	thus	thus	ADV
ejpam-5252	570	2	,	,	PUNCT
ejpam-5252	570	3	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	570	4	)	)	PUNCT
ejpam-5252	570	5	≤	≤	NUM
ejpam-5252	570	6	ωmr	ωmr	NOUN
ejpam-5252	570	7	g+h(f	g+h(f	NOUN
ejpam-5252	570	8	)	)	PUNCT
ejpam-5252	570	9	=	=	SYM
ejpam-5252	570	10	10	10	NUM
ejpam-5252	570	11	.	.	PUNCT
ejpam-5252	571	1	hence	hence	ADV
ejpam-5252	571	2	,	,	PUNCT
ejpam-5252	571	3	3	3	NUM
ejpam-5252	571	4	≤	≤	NUM
ejpam-5252	571	5	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	571	6	)	)	PUNCT
ejpam-5252	571	7	≤	≤	NUM
ejpam-5252	571	8	10	10	NUM
ejpam-5252	571	9	.	.	PUNCT
ejpam-5252	572	1	proposition	proposition	NOUN
ejpam-5252	572	2	15	15	NUM
ejpam-5252	572	3	.	.	PUNCT
ejpam-5252	573	1	let	let	VERB
ejpam-5252	573	2	g	g	NOUN
ejpam-5252	574	1	and	and	CCONJ
ejpam-5252	574	2	h	h	NOUN
ejpam-5252	574	3	be	be	VERB
ejpam-5252	574	4	any	any	DET
ejpam-5252	574	5	graphs	graph	NOUN
ejpam-5252	574	6	.	.	PUNCT
ejpam-5252	575	1	then	then	ADV
ejpam-5252	575	2	(	(	PUNCT
ejpam-5252	575	3	i	i	NOUN
ejpam-5252	575	4	)	)	PUNCT
ejpam-5252	575	5	γmr(g+h	γmr(g+h	PROPN
ejpam-5252	575	6	)	)	PUNCT
ejpam-5252	575	7	=	=	SYM
ejpam-5252	575	8	3	3	NUM
ejpam-5252	575	9	if	if	SCONJ
ejpam-5252	575	10	and	and	CCONJ
ejpam-5252	575	11	only	only	ADV
ejpam-5252	575	12	if	if	SCONJ
ejpam-5252	575	13	g	g	PROPN
ejpam-5252	575	14	=	=	SYM
ejpam-5252	575	15	k1	k1	PROPN
ejpam-5252	575	16	and	and	CCONJ
ejpam-5252	575	17	h	h	NOUN
ejpam-5252	575	18	=	=	PROPN
ejpam-5252	575	19	k1	k1	PROPN
ejpam-5252	575	20	(	(	PUNCT
ejpam-5252	575	21	ii	ii	NOUN
ejpam-5252	575	22	)	)	PUNCT
ejpam-5252	575	23	γmr(g+h	γmr(g+h	PROPN
ejpam-5252	575	24	)	)	PUNCT
ejpam-5252	576	1	=	=	PUNCT
ejpam-5252	576	2	4	4	NUM
ejpam-5252	576	3	if	if	SCONJ
ejpam-5252	576	4	and	and	CCONJ
ejpam-5252	576	5	only	only	ADV
ejpam-5252	576	6	if	if	SCONJ
ejpam-5252	576	7	g	g	PROPN
ejpam-5252	576	8	=	=	SYM
ejpam-5252	576	9	k1	k1	PROPN
ejpam-5252	576	10	and	and	CCONJ
ejpam-5252	576	11	h	h	NOUN
ejpam-5252	576	12	∈	∈	PROPN
ejpam-5252	576	13	{	{	PUNCT
ejpam-5252	576	14	k2,k2	k2,k2	PROPN
ejpam-5252	576	15	}	}	PUNCT
ejpam-5252	576	16	(	(	PUNCT
ejpam-5252	576	17	iii	iii	X
ejpam-5252	576	18	)	)	PUNCT
ejpam-5252	576	19	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	576	20	)	)	PUNCT
ejpam-5252	577	1	=	=	PUNCT
ejpam-5252	577	2	5	5	NUM
ejpam-5252	577	3	if	if	SCONJ
ejpam-5252	577	4	and	and	CCONJ
ejpam-5252	577	5	only	only	ADV
ejpam-5252	577	6	if	if	SCONJ
ejpam-5252	577	7	one	one	NUM
ejpam-5252	577	8	of	of	ADP
ejpam-5252	577	9	the	the	DET
ejpam-5252	577	10	following	following	NOUN
ejpam-5252	577	11	holds	hold	VERB
ejpam-5252	577	12	:	:	PUNCT
ejpam-5252	577	13	(	(	PUNCT
ejpam-5252	577	14	a	a	X
ejpam-5252	577	15	)	)	PUNCT
ejpam-5252	577	16	g	g	NOUN
ejpam-5252	577	17	=	=	SYM
ejpam-5252	577	18	k1	k1	PROPN
ejpam-5252	577	19	and	and	CCONJ
ejpam-5252	577	20	h	h	NOUN
ejpam-5252	577	21	∈	∈	PROPN
ejpam-5252	577	22	{	{	PUNCT
ejpam-5252	577	23	p3,k3,k3,k1	p3,k3,k3,k1	NOUN
ejpam-5252	577	24	∪k2	∪k2	X
ejpam-5252	577	25	}	}	PUNCT
ejpam-5252	577	26	or	or	CCONJ
ejpam-5252	577	27	h	h	NOUN
ejpam-5252	577	28	=	=	SYM
ejpam-5252	577	29	k1	k1	PROPN
ejpam-5252	577	30	and	and	CCONJ
ejpam-5252	577	31	g	g	PROPN
ejpam-5252	577	32	∈	∈	PROPN
ejpam-5252	577	33	{	{	PUNCT
ejpam-5252	577	34	p3,k3,k3,k1	p3,k3,k3,k1	NOUN
ejpam-5252	577	35	∪	∪	X
ejpam-5252	577	36	k2	k2	NOUN
ejpam-5252	577	37	}	}	PUNCT
ejpam-5252	577	38	.	.	PUNCT
ejpam-5252	578	1	s.	s.	PROPN
ejpam-5252	578	2	ahamad	ahamad	PROPN
ejpam-5252	578	3	,	,	PUNCT
ejpam-5252	578	4	j.	j.	PROPN
ejpam-5252	578	5	cariaga	cariaga	PROPN
ejpam-5252	578	6	,	,	PUNCT
ejpam-5252	578	7	s.	s.	PROPN
ejpam-5252	578	8	menchavez	menchavez	PROPN
ejpam-5252	578	9	/	/	PUNCT
ejpam-5252	578	10	eur	eur	PROPN
ejpam-5252	578	11	.	.	PUNCT
ejpam-5252	579	1	j.	j.	PROPN
ejpam-5252	579	2	pure	pure	PROPN
ejpam-5252	579	3	appl	appl	PROPN
ejpam-5252	579	4	.	.	PROPN
ejpam-5252	579	5	math	math	PROPN
ejpam-5252	579	6	,	,	PUNCT
ejpam-5252	579	7	18	18	NUM
ejpam-5252	579	8	(	(	PUNCT
ejpam-5252	579	9	1	1	NUM
ejpam-5252	579	10	)	)	PUNCT
ejpam-5252	579	11	(	(	PUNCT
ejpam-5252	579	12	2025	2025	NUM
ejpam-5252	579	13	)	)	PUNCT
ejpam-5252	579	14	,	,	PUNCT
ejpam-5252	579	15	5252	5252	NUM
ejpam-5252	579	16	15	15	NUM
ejpam-5252	579	17	of	of	ADP
ejpam-5252	579	18	18	18	NUM
ejpam-5252	579	19	(	(	PUNCT
ejpam-5252	579	20	b	b	NOUN
ejpam-5252	579	21	)	)	PUNCT
ejpam-5252	579	22	if	if	SCONJ
ejpam-5252	579	23	|v	|v	PROPN
ejpam-5252	579	24	(	(	PUNCT
ejpam-5252	579	25	g+h)|	g+h)|	PROPN
ejpam-5252	579	26	≥	≥	NOUN
ejpam-5252	579	27	4	4	NUM
ejpam-5252	579	28	,	,	PUNCT
ejpam-5252	579	29	then	then	ADV
ejpam-5252	579	30	γ2(g	γ2(g	VERB
ejpam-5252	579	31	)	)	PUNCT
ejpam-5252	579	32	=	=	SYM
ejpam-5252	579	33	2	2	NUM
ejpam-5252	579	34	or	or	CCONJ
ejpam-5252	579	35	γ2(h	γ2(h	NUM
ejpam-5252	579	36	)	)	PUNCT
ejpam-5252	579	37	=	=	SYM
ejpam-5252	580	1	2	2	X
ejpam-5252	580	2	.	.	PUNCT
ejpam-5252	580	3	(	(	PUNCT
ejpam-5252	580	4	c	c	X
ejpam-5252	580	5	)	)	PUNCT
ejpam-5252	580	6	if	if	SCONJ
ejpam-5252	580	7	|v	|v	PROPN
ejpam-5252	580	8	(	(	PUNCT
ejpam-5252	580	9	g+h)|	g+h)|	PROPN
ejpam-5252	580	10	≥	≥	NUM
ejpam-5252	580	11	4	4	NUM
ejpam-5252	580	12	,	,	PUNCT
ejpam-5252	580	13	then	then	ADV
ejpam-5252	580	14	γ(g	γ(g	PROPN
ejpam-5252	580	15	)	)	PUNCT
ejpam-5252	580	16	=	=	SYM
ejpam-5252	580	17	1	1	NUM
ejpam-5252	580	18	and	and	CCONJ
ejpam-5252	580	19	γ(h	γ(h	NOUN
ejpam-5252	580	20	)	)	PUNCT
ejpam-5252	580	21	=	=	SYM
ejpam-5252	581	1	1	1	X
ejpam-5252	581	2	.	.	PUNCT
ejpam-5252	581	3	proof	proof	NOUN
ejpam-5252	581	4	.	.	PUNCT
ejpam-5252	582	1	the	the	DET
ejpam-5252	582	2	proof	proof	NOUN
ejpam-5252	582	3	follows	follow	VERB
ejpam-5252	582	4	immediately	immediately	ADV
ejpam-5252	582	5	from	from	ADP
ejpam-5252	582	6	proposition	proposition	NOUN
ejpam-5252	582	7	5	5	NUM
ejpam-5252	582	8	.	.	PUNCT
ejpam-5252	582	9	corollary	corollary	ADJ
ejpam-5252	582	10	4	4	NUM
ejpam-5252	582	11	.	.	PUNCT
ejpam-5252	583	1	let	let	VERB
ejpam-5252	583	2	m	m	PRON
ejpam-5252	583	3	and	and	CCONJ
ejpam-5252	583	4	n	n	ADV
ejpam-5252	583	5	be	be	AUX
ejpam-5252	583	6	positive	positive	ADJ
ejpam-5252	583	7	integers	integer	NOUN
ejpam-5252	583	8	.	.	PUNCT
ejpam-5252	584	1	(	(	PUNCT
ejpam-5252	584	2	i	i	NOUN
ejpam-5252	584	3	)	)	PUNCT
ejpam-5252	584	4	if	if	SCONJ
ejpam-5252	584	5	g	g	PROPN
ejpam-5252	584	6	=	=	PROPN
ejpam-5252	584	7	kn	kn	PROPN
ejpam-5252	584	8	and	and	CCONJ
ejpam-5252	584	9	h	h	NOUN
ejpam-5252	584	10	=	=	NOUN
ejpam-5252	584	11	km	km	NOUN
ejpam-5252	584	12	with	with	ADP
ejpam-5252	584	13	n	n	CCONJ
ejpam-5252	584	14	,	,	PUNCT
ejpam-5252	584	15	m	m	VERB
ejpam-5252	584	16	≥	≥	NOUN
ejpam-5252	584	17	2	2	NUM
ejpam-5252	584	18	,	,	PUNCT
ejpam-5252	584	19	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	584	20	)	)	PUNCT
ejpam-5252	584	21	=	=	SYM
ejpam-5252	585	1	5	5	X
ejpam-5252	585	2	.	.	PUNCT
ejpam-5252	585	3	(	(	PUNCT
ejpam-5252	585	4	ii	ii	NOUN
ejpam-5252	585	5	)	)	PUNCT
ejpam-5252	585	6	if	if	SCONJ
ejpam-5252	585	7	g	g	PROPN
ejpam-5252	585	8	=	=	PROPN
ejpam-5252	585	9	kn	kn	PROPN
ejpam-5252	585	10	and	and	CCONJ
ejpam-5252	585	11	h	h	NOUN
ejpam-5252	585	12	=	=	NOUN
ejpam-5252	585	13	km	km	NOUN
ejpam-5252	585	14	with	with	ADP
ejpam-5252	585	15	n	n	CCONJ
ejpam-5252	585	16	,	,	PUNCT
ejpam-5252	585	17	m	m	VERB
ejpam-5252	585	18	≥	≥	NOUN
ejpam-5252	585	19	5	5	NUM
ejpam-5252	585	20	,	,	PUNCT
ejpam-5252	585	21	γmr(g+h	γmr(g+h	NOUN
ejpam-5252	585	22	)	)	PUNCT
ejpam-5252	586	1	=	=	PUNCT
ejpam-5252	587	1	10	10	NUM
ejpam-5252	587	2	.	.	NOUN
ejpam-5252	588	1	6	6	NUM
ejpam-5252	588	2	.	.	X
ejpam-5252	588	3	on	on	ADP
ejpam-5252	588	4	the	the	DET
ejpam-5252	588	5	corona	corona	NOUN
ejpam-5252	588	6	of	of	ADP
ejpam-5252	588	7	graphs	graph	NOUN
ejpam-5252	588	8	let	let	VERB
ejpam-5252	588	9	g	g	NOUN
ejpam-5252	588	10	and	and	CCONJ
ejpam-5252	588	11	h	h	NOUN
ejpam-5252	588	12	be	be	AUX
ejpam-5252	588	13	graphs	graph	NOUN
ejpam-5252	588	14	with	with	ADP
ejpam-5252	588	15	disjoint	disjoint	ADJ
ejpam-5252	588	16	vertex	vertex	NOUN
ejpam-5252	588	17	sets	set	NOUN
ejpam-5252	588	18	.	.	PUNCT
ejpam-5252	589	1	the	the	DET
ejpam-5252	589	2	corona	corona	NOUN
ejpam-5252	589	3	of	of	ADP
ejpam-5252	589	4	g	g	PROPN
ejpam-5252	589	5	and	and	CCONJ
ejpam-5252	589	6	h	h	NOUN
ejpam-5252	589	7	is	be	AUX
ejpam-5252	589	8	the	the	DET
ejpam-5252	589	9	graph	graph	NOUN
ejpam-5252	589	10	g	g	PROPN
ejpam-5252	589	11	◦	◦	NOUN
ejpam-5252	589	12	h	h	NOUN
ejpam-5252	589	13	obtained	obtain	VERB
ejpam-5252	589	14	by	by	ADP
ejpam-5252	589	15	taking	take	VERB
ejpam-5252	589	16	one	one	NUM
ejpam-5252	589	17	copy	copy	NOUN
ejpam-5252	589	18	of	of	ADP
ejpam-5252	589	19	g	g	PROPN
ejpam-5252	589	20	and	and	CCONJ
ejpam-5252	589	21	|v	|v	PROPN
ejpam-5252	589	22	(	(	PUNCT
ejpam-5252	589	23	g)|	g)|	NOUN
ejpam-5252	589	24	copies	copy	NOUN
ejpam-5252	589	25	of	of	ADP
ejpam-5252	589	26	h	h	NOUN
ejpam-5252	589	27	,	,	PUNCT
ejpam-5252	589	28	and	and	CCONJ
ejpam-5252	589	29	then	then	ADV
ejpam-5252	589	30	joining	join	VERB
ejpam-5252	589	31	the	the	DET
ejpam-5252	589	32	ith	ith	PROPN
ejpam-5252	589	33	vertex	vertex	NOUN
ejpam-5252	589	34	of	of	ADP
ejpam-5252	589	35	g	g	NOUN
ejpam-5252	589	36	to	to	ADP
ejpam-5252	589	37	every	every	DET
ejpam-5252	589	38	vertex	vertex	NOUN
ejpam-5252	589	39	of	of	ADP
ejpam-5252	589	40	the	the	DET
ejpam-5252	589	41	ith	ith	PROPN
ejpam-5252	589	42	copy	copy	NOUN
ejpam-5252	589	43	of	of	ADP
ejpam-5252	589	44	h.	h.	PROPN
ejpam-5252	589	45	for	for	ADP
ejpam-5252	589	46	convenience	convenience	NOUN
ejpam-5252	589	47	,	,	PUNCT
ejpam-5252	589	48	we	we	PRON
ejpam-5252	589	49	adapt	adapt	VERB
ejpam-5252	589	50	the	the	DET
ejpam-5252	589	51	notation	notation	NOUN
ejpam-5252	589	52	hv	hv	PROPN
ejpam-5252	589	53	+	+	PROPN
ejpam-5252	589	54	v	v	NOUN
ejpam-5252	589	55	used	use	VERB
ejpam-5252	589	56	in	in	ADP
ejpam-5252	589	57	[	[	X
ejpam-5252	589	58	3	3	X
ejpam-5252	589	59	]	]	PUNCT
ejpam-5252	589	60	to	to	PART
ejpam-5252	589	61	denote	denote	VERB
ejpam-5252	589	62	the	the	DET
ejpam-5252	589	63	subgraph	subgraph	NOUN
ejpam-5252	589	64	of	of	ADP
ejpam-5252	589	65	g	g	PROPN
ejpam-5252	589	66	◦	◦	NOUN
ejpam-5252	589	67	h	h	NOUN
ejpam-5252	589	68	corresponding	correspond	VERB
ejpam-5252	589	69	to	to	ADP
ejpam-5252	589	70	the	the	DET
ejpam-5252	589	71	join	join	NOUN
ejpam-5252	589	72	hv+⟨{v}⟩	hv+⟨{v}⟩	NOUN
ejpam-5252	589	73	,	,	PUNCT
ejpam-5252	589	74	v	v	NOUN
ejpam-5252	589	75	∈	∈	PROPN
ejpam-5252	589	76	v	v	NOUN
ejpam-5252	589	77	(	(	PUNCT
ejpam-5252	589	78	g	g	NOUN
ejpam-5252	589	79	)	)	PUNCT
ejpam-5252	589	80	.	.	PUNCT
ejpam-5252	590	1	moreover	moreover	ADV
ejpam-5252	590	2	,	,	PUNCT
ejpam-5252	590	3	for	for	ADP
ejpam-5252	590	4	convenience	convenience	NOUN
ejpam-5252	590	5	,	,	PUNCT
ejpam-5252	590	6	we	we	PRON
ejpam-5252	590	7	define	define	VERB
ejpam-5252	590	8	for	for	ADP
ejpam-5252	590	9	i	i	PROPN
ejpam-5252	590	10	=	=	SYM
ejpam-5252	590	11	0	0	NUM
ejpam-5252	590	12	,	,	PUNCT
ejpam-5252	590	13	1	1	NUM
ejpam-5252	590	14	,	,	PUNCT
ejpam-5252	590	15	2	2	NUM
ejpam-5252	590	16	,	,	PUNCT
ejpam-5252	590	17	3	3	NUM
ejpam-5252	590	18	and	and	CCONJ
ejpam-5252	590	19	v	v	ADP
ejpam-5252	590	20	∈	∈	PROPN
ejpam-5252	590	21	v	v	NOUN
ejpam-5252	590	22	(	(	PUNCT
ejpam-5252	590	23	g	g	NOUN
ejpam-5252	590	24	)	)	PUNCT
ejpam-5252	590	25	,	,	PUNCT
ejpam-5252	590	26	v	v	NOUN
ejpam-5252	590	27	v	v	X
ejpam-5252	590	28	i	i	NOUN
ejpam-5252	590	29	=	=	PUNCT
ejpam-5252	590	30	{	{	PUNCT
ejpam-5252	590	31	u	u	NOUN
ejpam-5252	590	32	∈	∈	PROPN
ejpam-5252	590	33	v	v	ADP
ejpam-5252	590	34	(	(	PUNCT
ejpam-5252	590	35	hv)|f(u	hv)|f(u	NOUN
ejpam-5252	590	36	)	)	PUNCT
ejpam-5252	590	37	=	=	PUNCT
ejpam-5252	590	38	i	i	PROPN
ejpam-5252	590	39	}	}	PUNCT
ejpam-5252	590	40	.	.	PUNCT
ejpam-5252	591	1	proposition	proposition	NOUN
ejpam-5252	591	2	16	16	NUM
ejpam-5252	591	3	.	.	PUNCT
ejpam-5252	592	1	let	let	VERB
ejpam-5252	592	2	g	g	NOUN
ejpam-5252	592	3	be	be	AUX
ejpam-5252	592	4	any	any	DET
ejpam-5252	592	5	nontrivial	nontrivial	ADJ
ejpam-5252	592	6	connected	connect	VERB
ejpam-5252	592	7	graph	graph	NOUN
ejpam-5252	592	8	and	and	CCONJ
ejpam-5252	592	9	h	h	NOUN
ejpam-5252	592	10	be	be	AUX
ejpam-5252	592	11	any	any	DET
ejpam-5252	592	12	graph	graph	NOUN
ejpam-5252	592	13	.	.	PUNCT
ejpam-5252	593	1	let	let	VERB
ejpam-5252	593	2	f	f	PROPN
ejpam-5252	593	3	=	=	SYM
ejpam-5252	593	4	(	(	PUNCT
ejpam-5252	593	5	v0	v0	PROPN
ejpam-5252	593	6	,	,	PUNCT
ejpam-5252	593	7	v1	v1	NOUN
ejpam-5252	593	8	,	,	PUNCT
ejpam-5252	593	9	v2	v2	PROPN
ejpam-5252	593	10	,	,	PUNCT
ejpam-5252	593	11	v3	v3	PROPN
ejpam-5252	593	12	)	)	PUNCT
ejpam-5252	593	13	be	be	VERB
ejpam-5252	593	14	any	any	DET
ejpam-5252	593	15	function	function	NOUN
ejpam-5252	593	16	on	on	ADP
ejpam-5252	593	17	v	v	NOUN
ejpam-5252	593	18	(	(	PUNCT
ejpam-5252	593	19	g	g	PROPN
ejpam-5252	593	20	◦	◦	NOUN
ejpam-5252	593	21	h	h	NOUN
ejpam-5252	593	22	)	)	PUNCT
ejpam-5252	593	23	.	.	PUNCT
ejpam-5252	594	1	then	then	ADV
ejpam-5252	594	2	f	f	PROPN
ejpam-5252	594	3	∈	∈	PROPN
ejpam-5252	594	4	mrdf	mrdf	NOUN
ejpam-5252	594	5	(	(	PUNCT
ejpam-5252	594	6	g	g	NOUN
ejpam-5252	594	7	)	)	PUNCT
ejpam-5252	594	8	if	if	SCONJ
ejpam-5252	595	1	and	and	CCONJ
ejpam-5252	595	2	only	only	ADV
ejpam-5252	595	3	if	if	SCONJ
ejpam-5252	595	4	each	each	PRON
ejpam-5252	595	5	of	of	ADP
ejpam-5252	595	6	the	the	DET
ejpam-5252	595	7	following	follow	VERB
ejpam-5252	595	8	holds	hold	VERB
ejpam-5252	595	9	:	:	PUNCT
ejpam-5252	595	10	(	(	PUNCT
ejpam-5252	595	11	i	i	NOUN
ejpam-5252	595	12	)	)	PUNCT
ejpam-5252	595	13	for	for	ADP
ejpam-5252	595	14	every	every	PRON
ejpam-5252	595	15	v	v	NOUN
ejpam-5252	595	16	∈	∈	NOUN
ejpam-5252	595	17	(	(	PUNCT
ejpam-5252	595	18	v0	v0	NOUN
ejpam-5252	595	19	∪	∪	X
ejpam-5252	595	20	v1	v1	NOUN
ejpam-5252	595	21	)	)	PUNCT
ejpam-5252	595	22	∩	∩	ADJ
ejpam-5252	595	23	v	v	X
ejpam-5252	595	24	(	(	PUNCT
ejpam-5252	595	25	g	g	NOUN
ejpam-5252	595	26	)	)	PUNCT
ejpam-5252	595	27	,	,	PUNCT
ejpam-5252	595	28	f	f	PROPN
ejpam-5252	595	29	|hv	|hv	PROPN
ejpam-5252	595	30	∈	∈	PROPN
ejpam-5252	595	31	mrdf	mrdf	NOUN
ejpam-5252	595	32	(	(	PUNCT
ejpam-5252	595	33	hv	hv	NOUN
ejpam-5252	595	34	)	)	PUNCT
ejpam-5252	595	35	.	.	PUNCT
ejpam-5252	596	1	moreover	moreover	ADV
ejpam-5252	596	2	,	,	PUNCT
ejpam-5252	596	3	if	if	SCONJ
ejpam-5252	596	4	v	v	NUM
ejpam-5252	596	5	∈	∈	PROPN
ejpam-5252	596	6	v0	v0	NOUN
ejpam-5252	596	7	∩	∩	X
ejpam-5252	596	8	v	v	X
ejpam-5252	596	9	(	(	PUNCT
ejpam-5252	596	10	g	g	NOUN
ejpam-5252	596	11	)	)	PUNCT
ejpam-5252	596	12	,	,	PUNCT
ejpam-5252	596	13	then	then	ADV
ejpam-5252	596	14	the	the	DET
ejpam-5252	596	15	following	follow	VERB
ejpam-5252	596	16	holds	hold	VERB
ejpam-5252	596	17	:	:	PUNCT
ejpam-5252	596	18	(	(	PUNCT
ejpam-5252	596	19	a	a	X
ejpam-5252	596	20	)	)	PUNCT
ejpam-5252	596	21	if	if	SCONJ
ejpam-5252	596	22	|v	|v	PROPN
ejpam-5252	596	23	v	v	ADP
ejpam-5252	596	24	2	2	NUM
ejpam-5252	596	25	|	|	ADV
ejpam-5252	596	26	=	=	NOUN
ejpam-5252	596	27	̸	̸	NUM
ejpam-5252	596	28	0	0	PUNCT
ejpam-5252	596	29	and	and	CCONJ
ejpam-5252	596	30	|v	|v	X
ejpam-5252	596	31	v	v	ADP
ejpam-5252	596	32	3	3	NUM
ejpam-5252	596	33	|	|	ADV
ejpam-5252	596	34	=	=	SYM
ejpam-5252	596	35	0	0	NUM
ejpam-5252	596	36	,	,	PUNCT
ejpam-5252	596	37	then	then	ADV
ejpam-5252	596	38	|ng(v	|ng(v	NOUN
ejpam-5252	596	39	)	)	PUNCT
ejpam-5252	596	40	∩	∩	NOUN
ejpam-5252	596	41	v3|	v3|	NOUN
ejpam-5252	596	42	≥	≥	NUM
ejpam-5252	596	43	1	1	NUM
ejpam-5252	596	44	;	;	PUNCT
ejpam-5252	596	45	and	and	CCONJ
ejpam-5252	596	46	(	(	PUNCT
ejpam-5252	596	47	b	b	X
ejpam-5252	596	48	)	)	PUNCT
ejpam-5252	596	49	if	if	SCONJ
ejpam-5252	596	50	|v	|v	PROPN
ejpam-5252	596	51	v	v	ADP
ejpam-5252	596	52	2	2	NUM
ejpam-5252	596	53	|	|	NOUN
ejpam-5252	596	54	=	=	SYM
ejpam-5252	596	55	0	0	NUM
ejpam-5252	596	56	and	and	CCONJ
ejpam-5252	596	57	|v	|v	X
ejpam-5252	596	58	v	v	ADP
ejpam-5252	596	59	3	3	NUM
ejpam-5252	596	60	|	|	ADV
ejpam-5252	596	61	=	=	NOUN
ejpam-5252	596	62	̸	̸	NUM
ejpam-5252	596	63	0	0	NUM
ejpam-5252	596	64	,	,	PUNCT
ejpam-5252	596	65	then	then	ADV
ejpam-5252	596	66	|ng(v	|ng(v	PROPN
ejpam-5252	596	67	)	)	PUNCT
ejpam-5252	596	68	∩	∩	ADJ
ejpam-5252	596	69	v2|	v2|	X
ejpam-5252	596	70	≥	≥	NUM
ejpam-5252	596	71	1	1	NUM
ejpam-5252	596	72	.	.	PUNCT
ejpam-5252	596	73	(	(	PUNCT
ejpam-5252	596	74	ii	ii	NOUN
ejpam-5252	596	75	)	)	PUNCT
ejpam-5252	596	76	for	for	ADP
ejpam-5252	596	77	every	every	DET
ejpam-5252	596	78	v	v	PROPN
ejpam-5252	596	79	∈	∈	PROPN
ejpam-5252	596	80	v2	v2	NOUN
ejpam-5252	596	81	∩	∩	ADJ
ejpam-5252	596	82	v	v	NOUN
ejpam-5252	596	83	(	(	PUNCT
ejpam-5252	596	84	g	g	NOUN
ejpam-5252	596	85	)	)	PUNCT
ejpam-5252	596	86	,	,	PUNCT
ejpam-5252	596	87	v	v	X
ejpam-5252	596	88	v	v	NOUN
ejpam-5252	596	89	3	3	NUM
ejpam-5252	596	90	dominates	dominate	VERB
ejpam-5252	596	91	v	v	ADP
ejpam-5252	596	92	v	v	NOUN
ejpam-5252	596	93	0	0	NUM
ejpam-5252	596	94	.	.	PUNCT
ejpam-5252	597	1	(	(	PUNCT
ejpam-5252	597	2	iii	iii	NOUN
ejpam-5252	597	3	)	)	PUNCT
ejpam-5252	597	4	for	for	ADP
ejpam-5252	597	5	every	every	PRON
ejpam-5252	597	6	v	v	PROPN
ejpam-5252	597	7	∈	∈	PROPN
ejpam-5252	597	8	v3	v3	PROPN
ejpam-5252	597	9	∩	∩	PROPN
ejpam-5252	597	10	v	v	X
ejpam-5252	597	11	(	(	PUNCT
ejpam-5252	597	12	g	g	NOUN
ejpam-5252	597	13	)	)	PUNCT
ejpam-5252	597	14	,	,	PUNCT
ejpam-5252	597	15	v	v	NOUN
ejpam-5252	597	16	v	v	PRON
ejpam-5252	597	17	2	2	NUM
ejpam-5252	597	18	dominates	dominate	VERB
ejpam-5252	597	19	v	v	ADP
ejpam-5252	597	20	v	v	NOUN
ejpam-5252	597	21	0	0	NUM
ejpam-5252	597	22	.	.	PUNCT
ejpam-5252	598	1	proof	proof	NOUN
ejpam-5252	598	2	.	.	PUNCT
ejpam-5252	599	1	suppose	suppose	VERB
ejpam-5252	599	2	f	f	PROPN
ejpam-5252	599	3	∈	∈	PROPN
ejpam-5252	599	4	mrdf	mrdf	NOUN
ejpam-5252	599	5	(	(	PUNCT
ejpam-5252	599	6	g	g	NOUN
ejpam-5252	599	7	◦	◦	NOUN
ejpam-5252	599	8	h	h	NOUN
ejpam-5252	599	9	)	)	PUNCT
ejpam-5252	599	10	and	and	CCONJ
ejpam-5252	599	11	let	let	VERB
ejpam-5252	599	12	v	v	X
ejpam-5252	599	13	∈	∈	PROPN
ejpam-5252	599	14	(	(	PUNCT
ejpam-5252	599	15	v0	v0	NOUN
ejpam-5252	599	16	∪	∪	X
ejpam-5252	599	17	v1	v1	NOUN
ejpam-5252	599	18	)	)	PUNCT
ejpam-5252	599	19	∩	∩	ADJ
ejpam-5252	599	20	v	v	X
ejpam-5252	599	21	(	(	PUNCT
ejpam-5252	599	22	g	g	NOUN
ejpam-5252	599	23	)	)	PUNCT
ejpam-5252	599	24	.	.	PUNCT
ejpam-5252	600	1	let	let	VERB
ejpam-5252	600	2	u	u	PRON
ejpam-5252	600	3	∈	∈	PROPN
ejpam-5252	600	4	v	v	ADP
ejpam-5252	600	5	v	v	NOUN
ejpam-5252	600	6	0	0	NUM
ejpam-5252	600	7	.	.	PUNCT
ejpam-5252	601	1	then	then	ADV
ejpam-5252	601	2	u	u	PROPN
ejpam-5252	601	3	∈	∈	PROPN
ejpam-5252	601	4	v0	v0	NOUN
ejpam-5252	601	5	and	and	CCONJ
ejpam-5252	601	6	by	by	ADP
ejpam-5252	601	7	definition	definition	NOUN
ejpam-5252	601	8	,	,	PUNCT
ejpam-5252	601	9	there	there	PRON
ejpam-5252	601	10	exist	exist	VERB
ejpam-5252	601	11	w	w	NOUN
ejpam-5252	601	12	,	,	PUNCT
ejpam-5252	601	13	z	z	PROPN
ejpam-5252	601	14	∈	∈	PROPN
ejpam-5252	601	15	ng	ng	PROPN
ejpam-5252	601	16	◦	◦	PROPN
ejpam-5252	601	17	h(u	h(u	PROPN
ejpam-5252	601	18	)	)	PUNCT
ejpam-5252	601	19	such	such	ADJ
ejpam-5252	601	20	that	that	SCONJ
ejpam-5252	601	21	w	w	PROPN
ejpam-5252	601	22	∈	∈	PROPN
ejpam-5252	601	23	v2	v2	PROPN
ejpam-5252	601	24	and	and	CCONJ
ejpam-5252	601	25	z	z	NOUN
ejpam-5252	601	26	∈	∈	PROPN
ejpam-5252	601	27	v3	v3	PROPN
ejpam-5252	601	28	.	.	PUNCT
ejpam-5252	602	1	but	but	CCONJ
ejpam-5252	602	2	ng	ng	PROPN
ejpam-5252	602	3	◦	◦	PROPN
ejpam-5252	602	4	h(u	h(u	PROPN
ejpam-5252	602	5	)	)	PUNCT
ejpam-5252	603	1	=	=	PRON
ejpam-5252	603	2	{	{	PUNCT
ejpam-5252	603	3	v	v	NOUN
ejpam-5252	603	4	}	}	PUNCT
ejpam-5252	603	5	∪	∪	ADJ
ejpam-5252	603	6	nhv(u	nhv(u	PROPN
ejpam-5252	603	7	)	)	PUNCT
ejpam-5252	603	8	and	and	CCONJ
ejpam-5252	603	9	v	v	ADP
ejpam-5252	603	10	∈	∈	PROPN
ejpam-5252	603	11	v0	v0	NOUN
ejpam-5252	603	12	∪	∪	X
ejpam-5252	603	13	v1	v1	NOUN
ejpam-5252	603	14	,	,	PUNCT
ejpam-5252	603	15	and	and	CCONJ
ejpam-5252	603	16	thus	thus	ADV
ejpam-5252	603	17	,	,	PUNCT
ejpam-5252	603	18	w	w	PROPN
ejpam-5252	603	19	,	,	PUNCT
ejpam-5252	603	20	z	z	PROPN
ejpam-5252	603	21	∈	∈	PROPN
ejpam-5252	603	22	nhv(u	nhv(u	PROPN
ejpam-5252	603	23	)	)	PUNCT
ejpam-5252	603	24	.	.	PUNCT
ejpam-5252	604	1	moreover	moreover	ADV
ejpam-5252	604	2	,	,	PUNCT
ejpam-5252	604	3	let	let	VERB
ejpam-5252	604	4	u	u	PRON
ejpam-5252	604	5	∈	∈	PROPN
ejpam-5252	604	6	v	v	ADP
ejpam-5252	604	7	v	v	ADP
ejpam-5252	604	8	1	1	NUM
ejpam-5252	604	9	.	.	PUNCT
ejpam-5252	605	1	then	then	ADV
ejpam-5252	605	2	u	u	PROPN
ejpam-5252	605	3	∈	∈	PROPN
ejpam-5252	605	4	v1	v1	NOUN
ejpam-5252	605	5	and	and	CCONJ
ejpam-5252	605	6	by	by	ADP
ejpam-5252	605	7	definition	definition	NOUN
ejpam-5252	605	8	,	,	PUNCT
ejpam-5252	605	9	there	there	PRON
ejpam-5252	605	10	exists	exist	VERB
ejpam-5252	605	11	x	x	X
ejpam-5252	605	12	∈	∈	PROPN
ejpam-5252	605	13	v2	v2	PROPN
ejpam-5252	605	14	∪	∪	X
ejpam-5252	605	15	v3	v3	PROPN
ejpam-5252	605	16	such	such	ADJ
ejpam-5252	605	17	that	that	SCONJ
ejpam-5252	605	18	x	x	SYM
ejpam-5252	605	19	∈	∈	PROPN
ejpam-5252	605	20	ng	ng	PROPN
ejpam-5252	605	21	◦	◦	NOUN
ejpam-5252	605	22	h(u	h(u	PROPN
ejpam-5252	605	23	)	)	PUNCT
ejpam-5252	605	24	,	,	PUNCT
ejpam-5252	605	25	so	so	SCONJ
ejpam-5252	605	26	that	that	SCONJ
ejpam-5252	605	27	x	x	X
ejpam-5252	605	28	∈	∈	NOUN
ejpam-5252	605	29	(	(	PUNCT
ejpam-5252	605	30	v	v	NOUN
ejpam-5252	605	31	v	v	ADP
ejpam-5252	605	32	2	2	NUM
ejpam-5252	605	33	∪	∪	NOUN
ejpam-5252	605	34	v	v	NUM
ejpam-5252	605	35	v	v	ADP
ejpam-5252	605	36	3	3	NUM
ejpam-5252	605	37	)	)	PUNCT
ejpam-5252	605	38	and	and	CCONJ
ejpam-5252	605	39	so	so	ADV
ejpam-5252	605	40	,	,	PUNCT
ejpam-5252	605	41	x	x	PROPN
ejpam-5252	605	42	∈	∈	PROPN
ejpam-5252	605	43	nhv(u	nhv(u	PROPN
ejpam-5252	605	44	)	)	PUNCT
ejpam-5252	605	45	.	.	PUNCT
ejpam-5252	606	1	hence	hence	ADV
ejpam-5252	606	2	,	,	PUNCT
ejpam-5252	606	3	f	f	PROPN
ejpam-5252	606	4	|hv	|hv	PROPN
ejpam-5252	606	5	∈	∈	PROPN
ejpam-5252	606	6	mrdf	mrdf	NOUN
ejpam-5252	606	7	(	(	PUNCT
ejpam-5252	606	8	g	g	NOUN
ejpam-5252	606	9	◦	◦	NOUN
ejpam-5252	606	10	h	h	NOUN
ejpam-5252	606	11	)	)	PUNCT
ejpam-5252	606	12	.	.	PUNCT
ejpam-5252	607	1	now	now	ADV
ejpam-5252	607	2	,	,	PUNCT
ejpam-5252	607	3	let	let	VERB
ejpam-5252	607	4	v	v	PRON
ejpam-5252	607	5	∈	∈	PROPN
ejpam-5252	607	6	v0	v0	NOUN
ejpam-5252	607	7	.	.	PUNCT
ejpam-5252	607	8	suppose	suppose	VERB
ejpam-5252	607	9	that	that	SCONJ
ejpam-5252	607	10	|v	|v	PROPN
ejpam-5252	607	11	v	v	ADP
ejpam-5252	607	12	2	2	NUM
ejpam-5252	607	13	|	|	ADV
ejpam-5252	607	14	=	=	NOUN
ejpam-5252	607	15	̸	̸	NUM
ejpam-5252	607	16	0	0	PUNCT
ejpam-5252	608	1	and	and	CCONJ
ejpam-5252	608	2	|v	|v	X
ejpam-5252	608	3	v	v	ADP
ejpam-5252	608	4	3	3	NUM
ejpam-5252	608	5	|	|	ADV
ejpam-5252	608	6	=	=	NOUN
ejpam-5252	608	7	0	0	X
ejpam-5252	608	8	.	.	PUNCT
ejpam-5252	609	1	then	then	ADV
ejpam-5252	609	2	if	if	SCONJ
ejpam-5252	609	3	u	u	PROPN
ejpam-5252	609	4	∈	∈	PROPN
ejpam-5252	609	5	ng	ng	PROPN
ejpam-5252	609	6	◦	◦	NOUN
ejpam-5252	609	7	h(v	h(v	NOUN
ejpam-5252	609	8	)	)	PUNCT
ejpam-5252	609	9	and	and	CCONJ
ejpam-5252	609	10	u	u	PROPN
ejpam-5252	609	11	∈	∈	PROPN
ejpam-5252	609	12	v3	v3	PROPN
ejpam-5252	609	13	,	,	PUNCT
ejpam-5252	609	14	we	we	PRON
ejpam-5252	609	15	have	have	VERB
ejpam-5252	609	16	u	u	NOUN
ejpam-5252	609	17	∈	∈	NOUN
ejpam-5252	609	18	ng(v	ng(v	NOUN
ejpam-5252	609	19	)	)	PUNCT
ejpam-5252	609	20	∩	∩	PROPN
ejpam-5252	609	21	v3	v3	PROPN
ejpam-5252	609	22	.	.	PUNCT
ejpam-5252	610	1	thus	thus	ADV
ejpam-5252	610	2	,	,	PUNCT
ejpam-5252	610	3	|ng(v	|ng(v	X
ejpam-5252	610	4	)	)	PUNCT
ejpam-5252	610	5	∩	∩	NOUN
ejpam-5252	610	6	v3|	v3|	NOUN
ejpam-5252	610	7	≥	≥	NUM
ejpam-5252	610	8	1	1	NUM
ejpam-5252	610	9	.	.	PUNCT
ejpam-5252	611	1	moreover	moreover	ADV
ejpam-5252	611	2	,	,	PUNCT
ejpam-5252	611	3	suppose	suppose	VERB
ejpam-5252	611	4	that	that	SCONJ
ejpam-5252	611	5	|v	|v	PROPN
ejpam-5252	611	6	v	v	ADP
ejpam-5252	611	7	3	3	NUM
ejpam-5252	611	8	|	|	ADV
ejpam-5252	611	9	̸=	̸=	PROPN
ejpam-5252	611	10	0	0	NUM
ejpam-5252	611	11	and	and	CCONJ
ejpam-5252	611	12	|v	|v	X
ejpam-5252	611	13	v	v	ADP
ejpam-5252	611	14	2	2	NUM
ejpam-5252	612	1	|	|	NOUN
ejpam-5252	612	2	=	=	NOUN
ejpam-5252	612	3	0	0	X
ejpam-5252	612	4	.	.	PUNCT
ejpam-5252	613	1	similarly	similarly	ADV
ejpam-5252	613	2	,	,	PUNCT
ejpam-5252	613	3	if	if	SCONJ
ejpam-5252	613	4	w	w	PROPN
ejpam-5252	613	5	∈	∈	PROPN
ejpam-5252	613	6	ng	ng	PROPN
ejpam-5252	613	7	◦	◦	NOUN
ejpam-5252	613	8	h(v	h(v	NOUN
ejpam-5252	613	9	)	)	PUNCT
ejpam-5252	613	10	and	and	CCONJ
ejpam-5252	613	11	w	w	PROPN
ejpam-5252	613	12	∈	∈	PROPN
ejpam-5252	613	13	v2	v2	PROPN
ejpam-5252	613	14	,	,	PUNCT
ejpam-5252	613	15	then	then	ADV
ejpam-5252	613	16	u	u	PROPN
ejpam-5252	613	17	∈	∈	PROPN
ejpam-5252	613	18	ng(v)∩v2	ng(v)∩v2	NOUN
ejpam-5252	613	19	.	.	PUNCT
ejpam-5252	614	1	thus	thus	ADV
ejpam-5252	614	2	,	,	PUNCT
ejpam-5252	614	3	|ng(v)∩v2|	|ng(v)∩v2|	PRON
ejpam-5252	614	4	≥	≥	NOUN
ejpam-5252	614	5	1	1	NUM
ejpam-5252	614	6	.	.	PUNCT
ejpam-5252	615	1	this	this	PRON
ejpam-5252	615	2	proves	prove	VERB
ejpam-5252	615	3	(	(	PUNCT
ejpam-5252	615	4	i	i	NOUN
ejpam-5252	615	5	)	)	PUNCT
ejpam-5252	615	6	.	.	PUNCT
ejpam-5252	616	1	suppose	suppose	VERB
ejpam-5252	616	2	that	that	SCONJ
ejpam-5252	616	3	v	v	NUM
ejpam-5252	616	4	∈	∈	PROPN
ejpam-5252	616	5	v2	v2	PROPN
ejpam-5252	616	6	∩	∩	ADJ
ejpam-5252	616	7	v	v	NOUN
ejpam-5252	616	8	(	(	PUNCT
ejpam-5252	616	9	g	g	NOUN
ejpam-5252	616	10	)	)	PUNCT
ejpam-5252	616	11	and	and	CCONJ
ejpam-5252	616	12	let	let	VERB
ejpam-5252	616	13	u	u	PRON
ejpam-5252	616	14	∈	∈	PROPN
ejpam-5252	616	15	v	v	ADP
ejpam-5252	616	16	v	v	NOUN
ejpam-5252	616	17	0	0	NUM
ejpam-5252	616	18	.	.	PUNCT
ejpam-5252	617	1	by	by	ADP
ejpam-5252	617	2	definition	definition	NOUN
ejpam-5252	617	3	,	,	PUNCT
ejpam-5252	617	4	there	there	PRON
ejpam-5252	617	5	exists	exist	VERB
ejpam-5252	617	6	{	{	PUNCT
ejpam-5252	617	7	x	x	NOUN
ejpam-5252	617	8	,	,	PUNCT
ejpam-5252	617	9	y	y	PROPN
ejpam-5252	617	10	}	}	PUNCT
ejpam-5252	617	11	⊆	⊆	NUM
ejpam-5252	617	12	ng	ng	PROPN
ejpam-5252	617	13	◦	◦	NOUN
ejpam-5252	617	14	h(u	h(u	PROPN
ejpam-5252	617	15	)	)	PUNCT
ejpam-5252	618	1	=	=	PRON
ejpam-5252	618	2	{	{	PUNCT
ejpam-5252	618	3	v	v	NOUN
ejpam-5252	618	4	}	}	PUNCT
ejpam-5252	618	5	+	+	CCONJ
ejpam-5252	618	6	nhv(u	nhv(u	ADJ
ejpam-5252	618	7	)	)	PUNCT
ejpam-5252	618	8	such	such	ADJ
ejpam-5252	618	9	that	that	SCONJ
ejpam-5252	618	10	x	x	SYM
ejpam-5252	618	11	∈	∈	PROPN
ejpam-5252	618	12	v2	v2	PROPN
ejpam-5252	618	13	and	and	CCONJ
ejpam-5252	618	14	y	y	PROPN
ejpam-5252	618	15	∈	∈	PROPN
ejpam-5252	618	16	v3	v3	PROPN
ejpam-5252	618	17	.	.	PUNCT
ejpam-5252	619	1	if	if	SCONJ
ejpam-5252	619	2	v	v	NUM
ejpam-5252	619	3	∈	∈	PROPN
ejpam-5252	619	4	v2	v2	NOUN
ejpam-5252	619	5	and	and	CCONJ
ejpam-5252	619	6	take	take	VERB
ejpam-5252	619	7	x	x	NOUN
ejpam-5252	619	8	=	=	SYM
ejpam-5252	619	9	v	v	NOUN
ejpam-5252	619	10	,	,	PUNCT
ejpam-5252	619	11	then	then	ADV
ejpam-5252	619	12	s.	s.	PROPN
ejpam-5252	619	13	ahamad	ahamad	PROPN
ejpam-5252	619	14	,	,	PUNCT
ejpam-5252	619	15	j.	j.	PROPN
ejpam-5252	619	16	cariaga	cariaga	PROPN
ejpam-5252	619	17	,	,	PUNCT
ejpam-5252	619	18	s.	s.	PROPN
ejpam-5252	619	19	menchavez	menchavez	PROPN
ejpam-5252	619	20	/	/	PUNCT
ejpam-5252	619	21	eur	eur	PROPN
ejpam-5252	619	22	.	.	PUNCT
ejpam-5252	620	1	j.	j.	PROPN
ejpam-5252	620	2	pure	pure	PROPN
ejpam-5252	620	3	appl	appl	PROPN
ejpam-5252	620	4	.	.	PROPN
ejpam-5252	620	5	math	math	PROPN
ejpam-5252	620	6	,	,	PUNCT
ejpam-5252	620	7	18	18	NUM
ejpam-5252	620	8	(	(	PUNCT
ejpam-5252	620	9	1	1	NUM
ejpam-5252	620	10	)	)	PUNCT
ejpam-5252	620	11	(	(	PUNCT
ejpam-5252	620	12	2025	2025	NUM
ejpam-5252	620	13	)	)	PUNCT
ejpam-5252	620	14	,	,	PUNCT
ejpam-5252	620	15	5252	5252	NUM
ejpam-5252	620	16	16	16	NUM
ejpam-5252	620	17	of	of	ADP
ejpam-5252	620	18	18	18	NUM
ejpam-5252	620	19	y	y	PROPN
ejpam-5252	620	20	∈	∈	PROPN
ejpam-5252	620	21	v	v	ADP
ejpam-5252	620	22	v	v	NOUN
ejpam-5252	620	23	3	3	NUM
ejpam-5252	620	24	and	and	CCONJ
ejpam-5252	620	25	y	y	PROPN
ejpam-5252	620	26	∈	∈	PROPN
ejpam-5252	620	27	nhv(u	nhv(u	PROPN
ejpam-5252	620	28	)	)	PUNCT
ejpam-5252	620	29	.	.	PUNCT
ejpam-5252	621	1	thus	thus	ADV
ejpam-5252	621	2	,	,	PUNCT
ejpam-5252	621	3	v	v	X
ejpam-5252	621	4	v	v	NOUN
ejpam-5252	621	5	3	3	NUM
ejpam-5252	621	6	dominates	dominate	VERB
ejpam-5252	621	7	v	v	ADP
ejpam-5252	621	8	v	v	NOUN
ejpam-5252	621	9	0	0	NUM
ejpam-5252	621	10	.	.	PUNCT
ejpam-5252	622	1	this	this	PRON
ejpam-5252	622	2	proves	prove	VERB
ejpam-5252	622	3	(	(	PUNCT
ejpam-5252	622	4	ii	ii	NOUN
ejpam-5252	622	5	)	)	PUNCT
ejpam-5252	622	6	.	.	PUNCT
ejpam-5252	623	1	similarly	similarly	ADV
ejpam-5252	623	2	,	,	PUNCT
ejpam-5252	623	3	if	if	SCONJ
ejpam-5252	623	4	v	v	PROPN
ejpam-5252	623	5	∈	∈	PROPN
ejpam-5252	623	6	v3	v3	PROPN
ejpam-5252	623	7	and	and	CCONJ
ejpam-5252	623	8	taking	take	VERB
ejpam-5252	623	9	y	y	PROPN
ejpam-5252	623	10	=	=	PUNCT
ejpam-5252	623	11	v	v	NOUN
ejpam-5252	623	12	,	,	PUNCT
ejpam-5252	623	13	then	then	ADV
ejpam-5252	623	14	x	x	SYM
ejpam-5252	623	15	∈	∈	PROPN
ejpam-5252	623	16	v	v	ADP
ejpam-5252	623	17	v	v	NOUN
ejpam-5252	623	18	2	2	NUM
ejpam-5252	623	19	and	and	CCONJ
ejpam-5252	623	20	x	x	ADP
ejpam-5252	623	21	∈	∈	PROPN
ejpam-5252	623	22	nhv(u	nhv(u	PROPN
ejpam-5252	623	23	)	)	PUNCT
ejpam-5252	623	24	.	.	PUNCT
ejpam-5252	624	1	this	this	PRON
ejpam-5252	624	2	means	mean	VERB
ejpam-5252	624	3	that	that	SCONJ
ejpam-5252	624	4	v	v	ADP
ejpam-5252	624	5	v	v	SYM
ejpam-5252	624	6	2	2	NUM
ejpam-5252	624	7	dominates	dominate	VERB
ejpam-5252	624	8	v	v	ADP
ejpam-5252	624	9	v	v	NOUN
ejpam-5252	624	10	0	0	NUM
ejpam-5252	624	11	.	.	PUNCT
ejpam-5252	625	1	this	this	PRON
ejpam-5252	625	2	proves	prove	VERB
ejpam-5252	625	3	(	(	PUNCT
ejpam-5252	625	4	iii	iii	NOUN
ejpam-5252	625	5	)	)	PUNCT
ejpam-5252	625	6	.	.	PUNCT
ejpam-5252	626	1	conversely	conversely	ADV
ejpam-5252	626	2	,	,	PUNCT
ejpam-5252	626	3	let	let	VERB
ejpam-5252	626	4	u	u	PRON
ejpam-5252	626	5	∈	∈	PROPN
ejpam-5252	626	6	v0	v0	NOUN
ejpam-5252	626	7	and	and	CCONJ
ejpam-5252	626	8	let	let	VERB
ejpam-5252	626	9	v	v	NUM
ejpam-5252	626	10	∈	∈	PROPN
ejpam-5252	626	11	v	v	NOUN
ejpam-5252	626	12	(	(	PUNCT
ejpam-5252	626	13	g	g	NOUN
ejpam-5252	626	14	)	)	PUNCT
ejpam-5252	626	15	for	for	ADP
ejpam-5252	626	16	which	which	PRON
ejpam-5252	626	17	u	u	PROPN
ejpam-5252	626	18	∈	∈	PROPN
ejpam-5252	626	19	v	v	NOUN
ejpam-5252	626	20	(	(	PUNCT
ejpam-5252	626	21	hv	hv	PROPN
ejpam-5252	626	22	+	+	PROPN
ejpam-5252	626	23	v	v	NOUN
ejpam-5252	626	24	)	)	PUNCT
ejpam-5252	626	25	.	.	PUNCT
ejpam-5252	627	1	if	if	SCONJ
ejpam-5252	627	2	u	u	PROPN
ejpam-5252	627	3	=	=	PROPN
ejpam-5252	627	4	v	v	NOUN
ejpam-5252	627	5	,	,	PUNCT
ejpam-5252	627	6	by	by	ADP
ejpam-5252	627	7	(	(	PUNCT
ejpam-5252	627	8	i	i	NOUN
ejpam-5252	627	9	)	)	PUNCT
ejpam-5252	627	10	,	,	PUNCT
ejpam-5252	627	11	f	f	PROPN
ejpam-5252	627	12	|hv	|hv	PROPN
ejpam-5252	627	13	∈	∈	PROPN
ejpam-5252	627	14	mrdf	mrdf	NOUN
ejpam-5252	627	15	(	(	PUNCT
ejpam-5252	627	16	g	g	PROPN
ejpam-5252	627	17	◦	◦	NOUN
ejpam-5252	627	18	h	h	NOUN
ejpam-5252	627	19	)	)	PUNCT
ejpam-5252	627	20	.	.	PUNCT
ejpam-5252	628	1	thus	thus	ADV
ejpam-5252	628	2	,	,	PUNCT
ejpam-5252	628	3	there	there	PRON
ejpam-5252	628	4	exist	exist	VERB
ejpam-5252	628	5	w	w	PRON
ejpam-5252	628	6	∈	∈	PROPN
ejpam-5252	628	7	v	v	ADP
ejpam-5252	628	8	v	v	NOUN
ejpam-5252	628	9	2	2	NUM
ejpam-5252	628	10	and	and	CCONJ
ejpam-5252	628	11	z	z	NOUN
ejpam-5252	628	12	∈	∈	PROPN
ejpam-5252	628	13	v	v	ADP
ejpam-5252	628	14	v	v	NUM
ejpam-5252	628	15	3	3	NUM
ejpam-5252	628	16	such	such	ADJ
ejpam-5252	628	17	that	that	PRON
ejpam-5252	628	18	w	w	NOUN
ejpam-5252	628	19	,	,	PUNCT
ejpam-5252	628	20	z	z	PROPN
ejpam-5252	628	21	∈	∈	PROPN
ejpam-5252	628	22	nhv(u	nhv(u	PROPN
ejpam-5252	628	23	)	)	PUNCT
ejpam-5252	628	24	and	and	CCONJ
ejpam-5252	628	25	so	so	ADV
ejpam-5252	628	26	,	,	PUNCT
ejpam-5252	628	27	w	w	PROPN
ejpam-5252	628	28	,	,	PUNCT
ejpam-5252	628	29	z	z	PROPN
ejpam-5252	628	30	∈	∈	PROPN
ejpam-5252	628	31	ng	ng	PROPN
ejpam-5252	628	32	◦	◦	NOUN
ejpam-5252	628	33	h(u	h(u	PROPN
ejpam-5252	628	34	)	)	PUNCT
ejpam-5252	628	35	.	.	PUNCT
ejpam-5252	629	1	now	now	ADV
ejpam-5252	629	2	,	,	PUNCT
ejpam-5252	629	3	if	if	SCONJ
ejpam-5252	629	4	|v	|v	PROPN
ejpam-5252	629	5	v	v	ADP
ejpam-5252	629	6	3	3	NUM
ejpam-5252	629	7	|	|	ADV
ejpam-5252	629	8	=	=	SYM
ejpam-5252	629	9	0	0	NUM
ejpam-5252	629	10	,	,	PUNCT
ejpam-5252	629	11	by	by	ADP
ejpam-5252	629	12	(	(	PUNCT
ejpam-5252	629	13	i	i	NOUN
ejpam-5252	629	14	)	)	PUNCT
ejpam-5252	629	15	|ng(v	|ng(v	PROPN
ejpam-5252	629	16	)	)	PUNCT
ejpam-5252	629	17	∩	∩	NOUN
ejpam-5252	629	18	v3|	v3|	X
ejpam-5252	629	19	≥	≥	NUM
ejpam-5252	629	20	1	1	NUM
ejpam-5252	629	21	.	.	PUNCT
ejpam-5252	629	22	thus	thus	ADV
ejpam-5252	629	23	,	,	PUNCT
ejpam-5252	629	24	there	there	PRON
ejpam-5252	629	25	exists	exist	VERB
ejpam-5252	629	26	w	w	PROPN
ejpam-5252	629	27	∈	∈	PROPN
ejpam-5252	629	28	v	v	ADP
ejpam-5252	629	29	v	v	NOUN
ejpam-5252	629	30	2	2	NUM
ejpam-5252	629	31	and	and	CCONJ
ejpam-5252	629	32	z	z	NOUN
ejpam-5252	629	33	∈	∈	PROPN
ejpam-5252	629	34	ng(v	ng(v	PUNCT
ejpam-5252	629	35	)	)	PUNCT
ejpam-5252	630	1	∩	∩	PROPN
ejpam-5252	630	2	v3	v3	VERB
ejpam-5252	630	3	such	such	ADJ
ejpam-5252	630	4	that	that	SCONJ
ejpam-5252	630	5	w	w	NOUN
ejpam-5252	630	6	,	,	PUNCT
ejpam-5252	630	7	z	z	PROPN
ejpam-5252	630	8	∈	∈	PROPN
ejpam-5252	630	9	ng	ng	PROPN
ejpam-5252	630	10	◦	◦	NOUN
ejpam-5252	630	11	h(u	h(u	PROPN
ejpam-5252	630	12	)	)	PUNCT
ejpam-5252	630	13	.	.	PUNCT
ejpam-5252	631	1	similarly	similarly	ADV
ejpam-5252	631	2	,	,	PUNCT
ejpam-5252	631	3	if	if	SCONJ
ejpam-5252	631	4	|v	|v	PROPN
ejpam-5252	631	5	v	v	ADP
ejpam-5252	631	6	2	2	NUM
ejpam-5252	631	7	|	|	NOUN
ejpam-5252	631	8	=	=	SYM
ejpam-5252	631	9	0	0	NUM
ejpam-5252	631	10	,	,	PUNCT
ejpam-5252	631	11	by	by	ADP
ejpam-5252	631	12	(	(	PUNCT
ejpam-5252	631	13	i	i	NOUN
ejpam-5252	631	14	)	)	PUNCT
ejpam-5252	631	15	,	,	PUNCT
ejpam-5252	631	16	|ng(v)∩v2|	|ng(v)∩v2|	X
ejpam-5252	631	17	≥	≥	NOUN
ejpam-5252	631	18	1	1	NUM
ejpam-5252	631	19	.	.	PUNCT
ejpam-5252	632	1	thus	thus	ADV
ejpam-5252	632	2	,	,	PUNCT
ejpam-5252	632	3	there	there	PRON
ejpam-5252	632	4	exist	exist	VERB
ejpam-5252	632	5	w	w	PROPN
ejpam-5252	632	6	∈	∈	PROPN
ejpam-5252	632	7	ng(v)∩v2	ng(v)∩v2	NOUN
ejpam-5252	632	8	and	and	CCONJ
ejpam-5252	632	9	z	z	NOUN
ejpam-5252	632	10	∈	∈	PROPN
ejpam-5252	632	11	v	v	ADP
ejpam-5252	632	12	v	v	NUM
ejpam-5252	632	13	3	3	NUM
ejpam-5252	632	14	such	such	ADJ
ejpam-5252	633	1	that	that	PRON
ejpam-5252	633	2	w	w	NOUN
ejpam-5252	633	3	,	,	PUNCT
ejpam-5252	633	4	z	z	PROPN
ejpam-5252	633	5	∈	∈	PROPN
ejpam-5252	633	6	ng	ng	PROPN
ejpam-5252	633	7	◦	◦	NOUN
ejpam-5252	633	8	h(u	h(u	PROPN
ejpam-5252	633	9	)	)	PUNCT
ejpam-5252	633	10	.	.	PUNCT
ejpam-5252	634	1	if	if	SCONJ
ejpam-5252	634	2	u	u	PROPN
ejpam-5252	634	3	̸=	̸=	PROPN
ejpam-5252	634	4	v	v	NOUN
ejpam-5252	634	5	,	,	PUNCT
ejpam-5252	634	6	then	then	ADV
ejpam-5252	634	7	u	u	PROPN
ejpam-5252	634	8	∈	∈	PROPN
ejpam-5252	634	9	v	v	ADP
ejpam-5252	634	10	v	v	NOUN
ejpam-5252	634	11	0	0	NUM
ejpam-5252	634	12	.	.	PUNCT
ejpam-5252	635	1	suppose	suppose	VERB
ejpam-5252	635	2	v	v	ADP
ejpam-5252	635	3	∈	∈	PROPN
ejpam-5252	635	4	v1∩v	v1∩v	NOUN
ejpam-5252	635	5	(	(	PUNCT
ejpam-5252	635	6	g	g	NOUN
ejpam-5252	635	7	)	)	PUNCT
ejpam-5252	635	8	.	.	PUNCT
ejpam-5252	636	1	by	by	ADP
ejpam-5252	636	2	(	(	PUNCT
ejpam-5252	636	3	i	i	NOUN
ejpam-5252	636	4	)	)	PUNCT
ejpam-5252	636	5	,	,	PUNCT
ejpam-5252	636	6	f	f	PROPN
ejpam-5252	636	7	|hv	|hv	PROPN
ejpam-5252	636	8	∈	∈	PROPN
ejpam-5252	636	9	mrdf	mrdf	NOUN
ejpam-5252	636	10	(	(	PUNCT
ejpam-5252	636	11	g	g	NOUN
ejpam-5252	636	12	◦	◦	NOUN
ejpam-5252	636	13	h	h	NOUN
ejpam-5252	636	14	)	)	PUNCT
ejpam-5252	636	15	.	.	PUNCT
ejpam-5252	637	1	thus	thus	ADV
ejpam-5252	637	2	,	,	PUNCT
ejpam-5252	637	3	there	there	PRON
ejpam-5252	637	4	exist	exist	VERB
ejpam-5252	637	5	x	x	NOUN
ejpam-5252	637	6	,	,	PUNCT
ejpam-5252	637	7	y	y	PROPN
ejpam-5252	637	8	∈	∈	PROPN
ejpam-5252	637	9	nhv(u	nhv(u	PROPN
ejpam-5252	637	10	)	)	PUNCT
ejpam-5252	637	11	such	such	ADJ
ejpam-5252	637	12	that	that	SCONJ
ejpam-5252	637	13	x	x	SYM
ejpam-5252	637	14	∈	∈	PROPN
ejpam-5252	637	15	v	v	ADP
ejpam-5252	637	16	v	v	ADP
ejpam-5252	637	17	2	2	NUM
ejpam-5252	637	18	and	and	CCONJ
ejpam-5252	637	19	y	y	PROPN
ejpam-5252	637	20	∈	∈	PROPN
ejpam-5252	637	21	v	v	ADP
ejpam-5252	637	22	v	v	NOUN
ejpam-5252	637	23	3	3	NUM
ejpam-5252	637	24	.	.	PUNCT
ejpam-5252	638	1	if	if	SCONJ
ejpam-5252	638	2	v	v	NUM
ejpam-5252	638	3	∈	∈	NOUN
ejpam-5252	638	4	v2∩v	v2∩v	NOUN
ejpam-5252	638	5	(	(	PUNCT
ejpam-5252	638	6	g	g	NOUN
ejpam-5252	638	7	)	)	PUNCT
ejpam-5252	638	8	.	.	PUNCT
ejpam-5252	639	1	by	by	ADP
ejpam-5252	639	2	(	(	PUNCT
ejpam-5252	639	3	ii	ii	NOUN
ejpam-5252	639	4	)	)	PUNCT
ejpam-5252	639	5	,	,	PUNCT
ejpam-5252	639	6	v	v	X
ejpam-5252	639	7	v	v	NOUN
ejpam-5252	639	8	3	3	NUM
ejpam-5252	639	9	dominates	dominate	VERB
ejpam-5252	639	10	v	v	ADP
ejpam-5252	639	11	v	v	NOUN
ejpam-5252	639	12	0	0	NUM
ejpam-5252	639	13	.	.	PUNCT
ejpam-5252	640	1	thus	thus	ADV
ejpam-5252	640	2	,	,	PUNCT
ejpam-5252	640	3	there	there	PRON
ejpam-5252	640	4	exist	exist	VERB
ejpam-5252	640	5	w	w	PROPN
ejpam-5252	640	6	,	,	PUNCT
ejpam-5252	640	7	v	v	NOUN
ejpam-5252	640	8	∈	∈	PROPN
ejpam-5252	640	9	nhv(u	nhv(u	PROPN
ejpam-5252	640	10	)	)	PUNCT
ejpam-5252	640	11	such	such	ADJ
ejpam-5252	640	12	that	that	SCONJ
ejpam-5252	640	13	w	w	PROPN
ejpam-5252	640	14	∈	∈	PROPN
ejpam-5252	640	15	v	v	ADP
ejpam-5252	640	16	v	v	NOUN
ejpam-5252	640	17	3	3	NUM
ejpam-5252	640	18	.	.	PUNCT
ejpam-5252	641	1	also	also	ADV
ejpam-5252	641	2	,	,	PUNCT
ejpam-5252	641	3	if	if	SCONJ
ejpam-5252	641	4	v	v	PROPN
ejpam-5252	641	5	∈	∈	PROPN
ejpam-5252	641	6	v3	v3	PROPN
ejpam-5252	641	7	∩	∩	PROPN
ejpam-5252	641	8	v	v	X
ejpam-5252	641	9	(	(	PUNCT
ejpam-5252	641	10	g	g	NOUN
ejpam-5252	641	11	)	)	PUNCT
ejpam-5252	641	12	.	.	PUNCT
ejpam-5252	642	1	by	by	ADP
ejpam-5252	642	2	(	(	PUNCT
ejpam-5252	642	3	iii	iii	NOUN
ejpam-5252	642	4	)	)	PUNCT
ejpam-5252	642	5	,	,	PUNCT
ejpam-5252	642	6	v	v	NOUN
ejpam-5252	642	7	v	v	PRON
ejpam-5252	642	8	2	2	NUM
ejpam-5252	642	9	dominates	dominate	VERB
ejpam-5252	642	10	v	v	ADP
ejpam-5252	642	11	v	v	NOUN
ejpam-5252	642	12	0	0	NUM
ejpam-5252	642	13	.	.	PUNCT
ejpam-5252	643	1	thus	thus	ADV
ejpam-5252	643	2	,	,	PUNCT
ejpam-5252	643	3	there	there	PRON
ejpam-5252	643	4	exist	exist	VERB
ejpam-5252	643	5	w	w	PROPN
ejpam-5252	643	6	,	,	PUNCT
ejpam-5252	643	7	v	v	NOUN
ejpam-5252	643	8	∈	∈	PROPN
ejpam-5252	643	9	nhv(u	nhv(u	PROPN
ejpam-5252	643	10	)	)	PUNCT
ejpam-5252	643	11	such	such	ADJ
ejpam-5252	643	12	that	that	SCONJ
ejpam-5252	643	13	w	w	PROPN
ejpam-5252	643	14	∈	∈	PROPN
ejpam-5252	643	15	v	v	ADP
ejpam-5252	643	16	v	v	NOUN
ejpam-5252	643	17	2	2	NUM
ejpam-5252	643	18	.	.	PUNCT
ejpam-5252	644	1	now	now	ADV
ejpam-5252	644	2	,	,	PUNCT
ejpam-5252	644	3	let	let	VERB
ejpam-5252	644	4	u	u	PRON
ejpam-5252	644	5	∈	∈	PROPN
ejpam-5252	644	6	v1	v1	NOUN
ejpam-5252	644	7	.	.	PUNCT
ejpam-5252	645	1	if	if	SCONJ
ejpam-5252	645	2	u	u	PROPN
ejpam-5252	645	3	∈	∈	PROPN
ejpam-5252	645	4	v	v	X
ejpam-5252	645	5	(	(	PUNCT
ejpam-5252	645	6	g	g	NOUN
ejpam-5252	645	7	)	)	PUNCT
ejpam-5252	645	8	,	,	PUNCT
ejpam-5252	645	9	then	then	ADV
ejpam-5252	645	10	there	there	PRON
ejpam-5252	645	11	exist	exist	VERB
ejpam-5252	645	12	x	x	PUNCT
ejpam-5252	645	13	∈	∈	PROPN
ejpam-5252	645	14	v	v	NUM
ejpam-5252	645	15	u	u	NOUN
ejpam-5252	645	16	2	2	NUM
ejpam-5252	645	17	∪v	∪v	NOUN
ejpam-5252	645	18	u	u	NOUN
ejpam-5252	645	19	3	3	NUM
ejpam-5252	645	20	such	such	ADJ
ejpam-5252	645	21	that	that	SCONJ
ejpam-5252	645	22	x	x	SYM
ejpam-5252	645	23	∈	∈	PROPN
ejpam-5252	645	24	nhu(u	nhu(u	NOUN
ejpam-5252	645	25	)	)	PUNCT
ejpam-5252	645	26	since	since	SCONJ
ejpam-5252	645	27	f	f	PROPN
ejpam-5252	645	28	|hu	|hu	NUM
ejpam-5252	645	29	∈	∈	PROPN
ejpam-5252	645	30	mrdf	mrdf	NOUN
ejpam-5252	645	31	(	(	PUNCT
ejpam-5252	645	32	hu	hu	PROPN
ejpam-5252	645	33	)	)	PUNCT
ejpam-5252	645	34	.	.	PUNCT
ejpam-5252	646	1	this	this	PRON
ejpam-5252	646	2	implies	imply	VERB
ejpam-5252	646	3	that	that	SCONJ
ejpam-5252	646	4	x	x	PUNCT
ejpam-5252	646	5	∈	∈	PROPN
ejpam-5252	646	6	v2	v2	NOUN
ejpam-5252	646	7	∪	∪	X
ejpam-5252	646	8	v3	v3	PROPN
ejpam-5252	646	9	and	and	CCONJ
ejpam-5252	646	10	x	x	PUNCT
ejpam-5252	646	11	∈	∈	PROPN
ejpam-5252	646	12	ng	ng	PROPN
ejpam-5252	646	13	◦	◦	NOUN
ejpam-5252	646	14	h(u	h(u	PROPN
ejpam-5252	646	15	)	)	PUNCT
ejpam-5252	646	16	.	.	PUNCT
ejpam-5252	647	1	suppose	suppose	VERB
ejpam-5252	647	2	u	u	PRON
ejpam-5252	647	3	∈	∈	PROPN
ejpam-5252	647	4	v	v	ADP
ejpam-5252	647	5	(	(	PUNCT
ejpam-5252	647	6	hv	hv	PROPN
ejpam-5252	647	7	)	)	PUNCT
ejpam-5252	647	8	for	for	ADP
ejpam-5252	647	9	some	some	DET
ejpam-5252	647	10	v	v	ADP
ejpam-5252	647	11	∈	∈	PROPN
ejpam-5252	647	12	v	v	NOUN
ejpam-5252	647	13	(	(	PUNCT
ejpam-5252	647	14	g	g	NOUN
ejpam-5252	647	15	)	)	PUNCT
ejpam-5252	647	16	.	.	PUNCT
ejpam-5252	648	1	if	if	SCONJ
ejpam-5252	648	2	v	v	NUM
ejpam-5252	648	3	∈	∈	PROPN
ejpam-5252	648	4	(	(	PUNCT
ejpam-5252	648	5	v2	v2	PROPN
ejpam-5252	648	6	∪	∪	X
ejpam-5252	648	7	v3	v3	PROPN
ejpam-5252	648	8	)	)	PUNCT
ejpam-5252	648	9	∩	∩	PROPN
ejpam-5252	648	10	v	v	X
ejpam-5252	648	11	(	(	PUNCT
ejpam-5252	648	12	g	g	NOUN
ejpam-5252	648	13	)	)	PUNCT
ejpam-5252	648	14	,	,	PUNCT
ejpam-5252	648	15	then	then	ADV
ejpam-5252	648	16	v	v	X
ejpam-5252	648	17	∈	∈	PROPN
ejpam-5252	648	18	(	(	PUNCT
ejpam-5252	648	19	v2	v2	PROPN
ejpam-5252	648	20	∪	∪	X
ejpam-5252	648	21	v3	v3	PROPN
ejpam-5252	648	22	)	)	PUNCT
ejpam-5252	648	23	∩	∩	PROPN
ejpam-5252	648	24	ng	ng	PROPN
ejpam-5252	648	25	◦	◦	PROPN
ejpam-5252	648	26	h(u	h(u	PROPN
ejpam-5252	648	27	)	)	PUNCT
ejpam-5252	648	28	.	.	PUNCT
ejpam-5252	649	1	if	if	SCONJ
ejpam-5252	649	2	v	v	NUM
ejpam-5252	649	3	∈	∈	PROPN
ejpam-5252	649	4	v0	v0	NOUN
ejpam-5252	649	5	∪	∪	X
ejpam-5252	649	6	v1	v1	PROPN
ejpam-5252	649	7	,	,	PUNCT
ejpam-5252	649	8	then	then	ADV
ejpam-5252	649	9	there	there	PRON
ejpam-5252	649	10	exist	exist	VERB
ejpam-5252	649	11	w	w	PRON
ejpam-5252	649	12	∈	∈	PROPN
ejpam-5252	649	13	v	v	ADP
ejpam-5252	649	14	v	v	ADP
ejpam-5252	649	15	2	2	NUM
ejpam-5252	649	16	∪	∪	NOUN
ejpam-5252	649	17	v	v	NUM
ejpam-5252	649	18	v	v	ADP
ejpam-5252	649	19	3	3	NUM
ejpam-5252	649	20	such	such	ADJ
ejpam-5252	649	21	that	that	DET
ejpam-5252	649	22	w	w	PROPN
ejpam-5252	649	23	∈	∈	PROPN
ejpam-5252	649	24	nhv(u	nhv(u	PROPN
ejpam-5252	649	25	)	)	PUNCT
ejpam-5252	649	26	since	since	SCONJ
ejpam-5252	649	27	f	f	PROPN
ejpam-5252	649	28	|hv	|hv	PROPN
ejpam-5252	649	29	∈	∈	PROPN
ejpam-5252	649	30	mrdf	mrdf	NOUN
ejpam-5252	649	31	(	(	PUNCT
ejpam-5252	649	32	hv	hv	NOUN
ejpam-5252	649	33	)	)	PUNCT
ejpam-5252	649	34	by	by	ADP
ejpam-5252	649	35	(	(	PUNCT
ejpam-5252	649	36	i	i	NOUN
ejpam-5252	649	37	)	)	PUNCT
ejpam-5252	649	38	.	.	PUNCT
ejpam-5252	650	1	it	it	PRON
ejpam-5252	650	2	implies	imply	VERB
ejpam-5252	650	3	that	that	SCONJ
ejpam-5252	650	4	w	w	PROPN
ejpam-5252	650	5	∈	∈	PROPN
ejpam-5252	650	6	v2	v2	NOUN
ejpam-5252	650	7	∪	∪	X
ejpam-5252	650	8	v3	v3	PROPN
ejpam-5252	650	9	and	and	CCONJ
ejpam-5252	650	10	w	w	PROPN
ejpam-5252	650	11	∈	∈	PROPN
ejpam-5252	650	12	ng	ng	PROPN
ejpam-5252	650	13	◦	◦	NOUN
ejpam-5252	650	14	h(u	h(u	PROPN
ejpam-5252	650	15	)	)	PUNCT
ejpam-5252	650	16	.	.	PUNCT
ejpam-5252	651	1	therefore	therefore	ADV
ejpam-5252	651	2	,	,	PUNCT
ejpam-5252	651	3	f	f	PROPN
ejpam-5252	651	4	∈	∈	PROPN
ejpam-5252	651	5	mrdf	mrdf	NOUN
ejpam-5252	651	6	(	(	PUNCT
ejpam-5252	651	7	g	g	PROPN
ejpam-5252	651	8	◦	◦	NOUN
ejpam-5252	651	9	h	h	NOUN
ejpam-5252	651	10	)	)	PUNCT
ejpam-5252	651	11	.	.	PUNCT
ejpam-5252	652	1	proposition	proposition	NOUN
ejpam-5252	652	2	17	17	NUM
ejpam-5252	652	3	.	.	PUNCT
ejpam-5252	653	1	let	let	VERB
ejpam-5252	653	2	g	g	NOUN
ejpam-5252	654	1	and	and	CCONJ
ejpam-5252	654	2	h	h	NOUN
ejpam-5252	654	3	be	be	VERB
ejpam-5252	654	4	any	any	DET
ejpam-5252	654	5	graph	graph	NOUN
ejpam-5252	654	6	with	with	ADP
ejpam-5252	654	7	|v	|v	PROPN
ejpam-5252	654	8	(	(	PUNCT
ejpam-5252	654	9	g)|	g)|	NOUN
ejpam-5252	654	10	=	=	PUNCT
ejpam-5252	654	11	n	n	NOUN
ejpam-5252	654	12	and	and	CCONJ
ejpam-5252	654	13	|v	|v	PROPN
ejpam-5252	654	14	(	(	PUNCT
ejpam-5252	654	15	h)|	h)|	NOUN
ejpam-5252	654	16	=	=	PUNCT
ejpam-5252	654	17	m	m	NOUN
ejpam-5252	654	18	and	and	CCONJ
ejpam-5252	654	19	let	let	VERB
ejpam-5252	654	20	f	f	PROPN
ejpam-5252	654	21	=	=	SYM
ejpam-5252	654	22	(	(	PUNCT
ejpam-5252	654	23	v0	v0	PROPN
ejpam-5252	654	24	,	,	PUNCT
ejpam-5252	654	25	v1	v1	NOUN
ejpam-5252	654	26	,	,	PUNCT
ejpam-5252	654	27	v2	v2	PROPN
ejpam-5252	654	28	,	,	PUNCT
ejpam-5252	654	29	v3	v3	PROPN
ejpam-5252	654	30	)	)	PUNCT
ejpam-5252	654	31	be	be	VERB
ejpam-5252	654	32	a	a	DET
ejpam-5252	654	33	γmr	γmr	ADJ
ejpam-5252	654	34	-	-	PUNCT
ejpam-5252	654	35	function	function	NOUN
ejpam-5252	654	36	of	of	ADP
ejpam-5252	654	37	g	g	PROPN
ejpam-5252	654	38	◦	◦	NOUN
ejpam-5252	654	39	h.	h.	NOUN
ejpam-5252	654	40	then	then	ADV
ejpam-5252	654	41	3n	3n	NUM
ejpam-5252	654	42	≤	≤	NUM
ejpam-5252	654	43	∑	∑	PUNCT
ejpam-5252	654	44	a∈v	a∈v	PROPN
ejpam-5252	654	45	(	(	PUNCT
ejpam-5252	654	46	v+hv	v+hv	NOUN
ejpam-5252	654	47	)	)	PUNCT
ejpam-5252	654	48	f(a	f(a	NOUN
ejpam-5252	654	49	)	)	PUNCT
ejpam-5252	654	50	≤	≤	NOUN
ejpam-5252	654	51	2n+	2n+	NUM
ejpam-5252	654	52	nm	nm	NOUN
ejpam-5252	654	53	,	,	PUNCT
ejpam-5252	654	54	for	for	ADP
ejpam-5252	654	55	each	each	DET
ejpam-5252	654	56	v	v	NUM
ejpam-5252	654	57	∈	∈	PROPN
ejpam-5252	654	58	v	v	NOUN
ejpam-5252	654	59	(	(	PUNCT
ejpam-5252	654	60	g	g	NOUN
ejpam-5252	654	61	)	)	PUNCT
ejpam-5252	654	62	.	.	PUNCT
ejpam-5252	655	1	proof	proof	NOUN
ejpam-5252	655	2	.	.	PUNCT
ejpam-5252	656	1	let	let	VERB
ejpam-5252	656	2	v	v	NUM
ejpam-5252	656	3	∈	∈	PROPN
ejpam-5252	656	4	v	v	NOUN
ejpam-5252	656	5	(	(	PUNCT
ejpam-5252	656	6	g	g	NOUN
ejpam-5252	656	7	)	)	PUNCT
ejpam-5252	656	8	.	.	PUNCT
ejpam-5252	657	1	if	if	SCONJ
ejpam-5252	657	2	v	v	NUM
ejpam-5252	657	3	∈	∈	PROPN
ejpam-5252	657	4	v2	v2	PROPN
ejpam-5252	657	5	∪	∪	X
ejpam-5252	657	6	v3	v3	PROPN
ejpam-5252	657	7	,	,	PUNCT
ejpam-5252	657	8	then	then	ADV
ejpam-5252	657	9	3n	3n	NUM
ejpam-5252	657	10	≤	≤	NUM
ejpam-5252	657	11	∑	∑	PUNCT
ejpam-5252	657	12	p∈v	p∈v	NOUN
ejpam-5252	657	13	(	(	PUNCT
ejpam-5252	657	14	hv	hv	NOUN
ejpam-5252	657	15	)	)	PUNCT
ejpam-5252	657	16	f(p	f(p	PROPN
ejpam-5252	657	17	)	)	PUNCT
ejpam-5252	657	18	≤	≤	NUM
ejpam-5252	657	19	∑	∑	PUNCT
ejpam-5252	657	20	a∈v	a∈v	PROPN
ejpam-5252	657	21	(	(	PUNCT
ejpam-5252	657	22	v+hv	v+hv	NOUN
ejpam-5252	657	23	)	)	PUNCT
ejpam-5252	657	24	f(a	f(a	NOUN
ejpam-5252	657	25	)	)	PUNCT
ejpam-5252	657	26	≤	≤	NUM
ejpam-5252	657	27	2n	2n	NUM
ejpam-5252	657	28	+	+	CCONJ
ejpam-5252	657	29	nm	nm	X
ejpam-5252	657	30	.	.	PUNCT
ejpam-5252	657	31	suppose	suppose	VERB
ejpam-5252	657	32	that	that	SCONJ
ejpam-5252	657	33	v	v	PROPN
ejpam-5252	657	34	∈	∈	PROPN
ejpam-5252	657	35	v0	v0	NOUN
ejpam-5252	657	36	.	.	PUNCT
ejpam-5252	658	1	by	by	ADP
ejpam-5252	658	2	proposition	proposition	NOUN
ejpam-5252	658	3	16	16	NUM
ejpam-5252	658	4	,	,	PUNCT
ejpam-5252	658	5	f	f	PROPN
ejpam-5252	658	6	|hv	|hv	PROPN
ejpam-5252	658	7	∈	∈	PROPN
ejpam-5252	658	8	mrdf	mrdf	NOUN
ejpam-5252	658	9	(	(	PUNCT
ejpam-5252	658	10	hv	hv	NOUN
ejpam-5252	658	11	)	)	PUNCT
ejpam-5252	658	12	.	.	PUNCT
ejpam-5252	659	1	thus	thus	ADV
ejpam-5252	659	2	,	,	PUNCT
ejpam-5252	659	3	3n	3n	NUM
ejpam-5252	659	4	≤∑	≤∑	PROPN
ejpam-5252	659	5	a∈v	a∈v	PROPN
ejpam-5252	659	6	(	(	PUNCT
ejpam-5252	659	7	v+hv	v+hv	NOUN
ejpam-5252	659	8	)	)	PUNCT
ejpam-5252	659	9	f(a	f(a	NOUN
ejpam-5252	659	10	)	)	PUNCT
ejpam-5252	659	11	≤	≤	NUM
ejpam-5252	659	12	2n	2n	NUM
ejpam-5252	659	13	+	+	CCONJ
ejpam-5252	659	14	nm	nm	NOUN
ejpam-5252	659	15	.	.	PUNCT
ejpam-5252	660	1	if	if	SCONJ
ejpam-5252	660	2	v	v	NUM
ejpam-5252	660	3	∈	∈	PROPN
ejpam-5252	660	4	v1	v1	NOUN
ejpam-5252	660	5	,	,	PUNCT
ejpam-5252	660	6	then	then	ADV
ejpam-5252	660	7	by	by	ADP
ejpam-5252	660	8	proposition	proposition	NOUN
ejpam-5252	660	9	16	16	NUM
ejpam-5252	660	10	,	,	PUNCT
ejpam-5252	660	11	f	f	PROPN
ejpam-5252	660	12	|hv	|hv	PROPN
ejpam-5252	660	13	∈	∈	PROPN
ejpam-5252	660	14	mrdf	mrdf	NOUN
ejpam-5252	660	15	(	(	PUNCT
ejpam-5252	660	16	hv	hv	NOUN
ejpam-5252	660	17	)	)	PUNCT
ejpam-5252	660	18	.	.	PUNCT
ejpam-5252	661	1	thus	thus	ADV
ejpam-5252	661	2	,	,	PUNCT
ejpam-5252	661	3	3n	3n	NUM
ejpam-5252	661	4	≤	≤	NUM
ejpam-5252	661	5	∑	∑	PUNCT
ejpam-5252	661	6	p∈v	p∈v	NOUN
ejpam-5252	661	7	(	(	PUNCT
ejpam-5252	661	8	hv	hv	NOUN
ejpam-5252	661	9	)	)	PUNCT
ejpam-5252	661	10	f(p	f(p	PROPN
ejpam-5252	661	11	)	)	PUNCT
ejpam-5252	661	12	≤	≤	NUM
ejpam-5252	661	13	∑	∑	PUNCT
ejpam-5252	661	14	a∈v	a∈v	PROPN
ejpam-5252	661	15	(	(	PUNCT
ejpam-5252	661	16	v+hv	v+hv	NOUN
ejpam-5252	661	17	)	)	PUNCT
ejpam-5252	661	18	f(a	f(a	NOUN
ejpam-5252	661	19	)	)	PUNCT
ejpam-5252	661	20	≤	≤	NUM
ejpam-5252	661	21	2n	2n	NUM
ejpam-5252	661	22	+	+	CCONJ
ejpam-5252	661	23	nm	nm	X
ejpam-5252	661	24	.	.	PUNCT
ejpam-5252	662	1	moreover	moreover	ADV
ejpam-5252	662	2	,	,	PUNCT
ejpam-5252	662	3	the	the	DET
ejpam-5252	662	4	bounds	bound	NOUN
ejpam-5252	662	5	are	be	AUX
ejpam-5252	662	6	sharp	sharp	ADJ
ejpam-5252	662	7	if	if	SCONJ
ejpam-5252	662	8	h	h	NOUN
ejpam-5252	662	9	=	=	SYM
ejpam-5252	662	10	k1	k1	PROPN
ejpam-5252	662	11	and	and	CCONJ
ejpam-5252	662	12	g	g	PROPN
ejpam-5252	662	13	∈	∈	PROPN
ejpam-5252	662	14	{	{	PUNCT
ejpam-5252	662	15	pn	pn	PROPN
ejpam-5252	662	16	,	,	PUNCT
ejpam-5252	662	17	cn	cn	PROPN
ejpam-5252	662	18	,	,	PUNCT
ejpam-5252	662	19	kn	kn	PROPN
ejpam-5252	662	20	}	}	PUNCT
ejpam-5252	662	21	.	.	PUNCT
ejpam-5252	663	1	proposition	proposition	NOUN
ejpam-5252	663	2	18	18	NUM
ejpam-5252	663	3	.	.	PUNCT
ejpam-5252	664	1	let	let	VERB
ejpam-5252	664	2	g	g	PRON
ejpam-5252	664	3	be	be	AUX
ejpam-5252	664	4	a	a	DET
ejpam-5252	664	5	connected	connected	ADJ
ejpam-5252	664	6	graph	graph	NOUN
ejpam-5252	664	7	of	of	ADP
ejpam-5252	664	8	order	order	NOUN
ejpam-5252	664	9	n	n	PRON
ejpam-5252	664	10	≥	≥	NOUN
ejpam-5252	664	11	1	1	NUM
ejpam-5252	664	12	and	and	CCONJ
ejpam-5252	664	13	km	km	PROPN
ejpam-5252	664	14	be	be	AUX
ejpam-5252	664	15	the	the	DET
ejpam-5252	664	16	complete	complete	ADJ
ejpam-5252	664	17	graph	graph	NOUN
ejpam-5252	664	18	of	of	ADP
ejpam-5252	664	19	order	order	NOUN
ejpam-5252	664	20	m	m	VERB
ejpam-5252	664	21	≥	≥	NOUN
ejpam-5252	664	22	2	2	NUM
ejpam-5252	664	23	,	,	PUNCT
ejpam-5252	664	24	then	then	ADV
ejpam-5252	664	25	γmr(g	γmr(g	PROPN
ejpam-5252	664	26	◦	◦	NOUN
ejpam-5252	664	27	km	km	NOUN
ejpam-5252	664	28	)	)	PUNCT
ejpam-5252	665	1	=	=	PRON
ejpam-5252	665	2	{	{	PUNCT
ejpam-5252	665	3	4n	4n	NOUN
ejpam-5252	665	4	,	,	PUNCT
ejpam-5252	665	5	if	if	SCONJ
ejpam-5252	665	6	m	m	VERB
ejpam-5252	665	7	=	=	NOUN
ejpam-5252	665	8	2	2	NUM
ejpam-5252	665	9	.	.	NOUN
ejpam-5252	665	10	5n	5n	NUM
ejpam-5252	665	11	,	,	PUNCT
ejpam-5252	665	12	if	if	SCONJ
ejpam-5252	665	13	m	m	PROPN
ejpam-5252	665	14	≥	≥	NOUN
ejpam-5252	665	15	3	3	NUM
ejpam-5252	665	16	.	.	PUNCT
ejpam-5252	666	1	proof	proof	NOUN
ejpam-5252	666	2	.	.	PUNCT
ejpam-5252	667	1	if	if	SCONJ
ejpam-5252	667	2	n	n	NOUN
ejpam-5252	667	3	=	=	SYM
ejpam-5252	667	4	1	1	NUM
ejpam-5252	667	5	,	,	PUNCT
ejpam-5252	667	6	then	then	ADV
ejpam-5252	667	7	g	g	PROPN
ejpam-5252	667	8	◦	◦	NOUN
ejpam-5252	667	9	km	km	NOUN
ejpam-5252	667	10	=	=	SYM
ejpam-5252	667	11	km+1	km+1	PROPN
ejpam-5252	667	12	.	.	NOUN
ejpam-5252	668	1	hence	hence	ADV
ejpam-5252	668	2	,	,	PUNCT
ejpam-5252	668	3	if	if	SCONJ
ejpam-5252	668	4	m	m	ADV
ejpam-5252	668	5	=	=	NOUN
ejpam-5252	668	6	2	2	NUM
ejpam-5252	668	7	,	,	PUNCT
ejpam-5252	668	8	γmr(g	γmr(g	PROPN
ejpam-5252	668	9	◦	◦	NOUN
ejpam-5252	668	10	k2	k2	NOUN
ejpam-5252	668	11	)	)	PUNCT
ejpam-5252	668	12	=	=	SYM
ejpam-5252	668	13	γmr(k3	γmr(k3	PROPN
ejpam-5252	668	14	)	)	PUNCT
ejpam-5252	668	15	=	=	SYM
ejpam-5252	668	16	4	4	NUM
ejpam-5252	668	17	by	by	ADP
ejpam-5252	668	18	proposition	proposition	NOUN
ejpam-5252	668	19	5	5	NUM
ejpam-5252	668	20	(	(	PUNCT
ejpam-5252	668	21	iii	iii	NOUN
ejpam-5252	668	22	)	)	PUNCT
ejpam-5252	668	23	.	.	PUNCT
ejpam-5252	669	1	if	if	SCONJ
ejpam-5252	669	2	m	m	PROPN
ejpam-5252	669	3	≥	≥	VERB
ejpam-5252	669	4	4	4	NUM
ejpam-5252	669	5	,	,	PUNCT
ejpam-5252	669	6	then	then	ADV
ejpam-5252	669	7	γmr(km+1	γmr(km+1	NOUN
ejpam-5252	669	8	)	)	PUNCT
ejpam-5252	670	1	=	=	SYM
ejpam-5252	670	2	5	5	NUM
ejpam-5252	670	3	by	by	ADP
ejpam-5252	670	4	proposition	proposition	NOUN
ejpam-5252	670	5	6	6	NUM
ejpam-5252	670	6	.	.	PUNCT
ejpam-5252	671	1	now	now	ADV
ejpam-5252	671	2	,	,	PUNCT
ejpam-5252	671	3	if	if	SCONJ
ejpam-5252	671	4	n	n	PROPN
ejpam-5252	671	5	>	>	X
ejpam-5252	671	6	1	1	NUM
ejpam-5252	671	7	,	,	PUNCT
ejpam-5252	671	8	then	then	ADV
ejpam-5252	671	9	for	for	ADP
ejpam-5252	671	10	m	m	PROPN
ejpam-5252	671	11	=	=	SYM
ejpam-5252	671	12	2	2	NUM
ejpam-5252	671	13	,	,	PUNCT
ejpam-5252	671	14	let	let	VERB
ejpam-5252	671	15	v	v	NOUN
ejpam-5252	671	16	(	(	PUNCT
ejpam-5252	671	17	k2	k2	NOUN
ejpam-5252	671	18	)	)	PUNCT
ejpam-5252	671	19	=	=	SYM
ejpam-5252	671	20	{	{	PUNCT
ejpam-5252	671	21	x	x	NOUN
ejpam-5252	671	22	,	,	PUNCT
ejpam-5252	671	23	y	y	NOUN
ejpam-5252	671	24	}	}	PUNCT
ejpam-5252	671	25	and	and	CCONJ
ejpam-5252	671	26	v	v	X
ejpam-5252	671	27	(	(	PUNCT
ejpam-5252	671	28	g	g	NOUN
ejpam-5252	671	29	)	)	PUNCT
ejpam-5252	671	30	=	=	SYM
ejpam-5252	671	31	{	{	PUNCT
ejpam-5252	671	32	v1	v1	PROPN
ejpam-5252	671	33	,	,	PUNCT
ejpam-5252	671	34	v2	v2	PROPN
ejpam-5252	671	35	,	,	PUNCT
ejpam-5252	671	36	·	·	PUNCT
ejpam-5252	671	37	·	·	PUNCT
ejpam-5252	671	38	·	·	PUNCT
ejpam-5252	671	39	,	,	PUNCT
ejpam-5252	671	40	vn	vn	PROPN
ejpam-5252	671	41	}	}	PUNCT
ejpam-5252	671	42	.	.	PUNCT
ejpam-5252	672	1	define	define	VERB
ejpam-5252	672	2	a	a	DET
ejpam-5252	672	3	function	function	NOUN
ejpam-5252	672	4	f	f	NOUN
ejpam-5252	672	5	=	=	SYM
ejpam-5252	672	6	(	(	PUNCT
ejpam-5252	672	7	v0	v0	PROPN
ejpam-5252	672	8	,	,	PUNCT
ejpam-5252	672	9	v1	v1	NOUN
ejpam-5252	672	10	,	,	PUNCT
ejpam-5252	672	11	v2	v2	PROPN
ejpam-5252	672	12	,	,	PUNCT
ejpam-5252	672	13	v3	v3	PROPN
ejpam-5252	672	14	)	)	PUNCT
ejpam-5252	672	15	on	on	ADP
ejpam-5252	672	16	v	v	NUM
ejpam-5252	672	17	(	(	PUNCT
ejpam-5252	672	18	g	g	PROPN
ejpam-5252	672	19	◦	◦	NOUN
ejpam-5252	672	20	k2	k2	NOUN
ejpam-5252	672	21	)	)	PUNCT
ejpam-5252	672	22	where	where	SCONJ
ejpam-5252	672	23	v0	v0	NOUN
ejpam-5252	672	24	=	=	SYM
ejpam-5252	672	25	∅	∅	NOUN
ejpam-5252	672	26	=	=	SYM
ejpam-5252	672	27	v3	v3	PROPN
ejpam-5252	672	28	,	,	PUNCT
ejpam-5252	672	29	v1	v1	NOUN
ejpam-5252	672	30	=	=	SYM
ejpam-5252	672	31	⋃	⋃	NOUN
ejpam-5252	672	32	v∈v	v∈v	NOUN
ejpam-5252	672	33	(	(	PUNCT
ejpam-5252	672	34	g	g	NOUN
ejpam-5252	672	35	)	)	PUNCT
ejpam-5252	672	36	v	v	NOUN
ejpam-5252	672	37	(	(	PUNCT
ejpam-5252	672	38	hv	hv	PROPN
ejpam-5252	672	39	)	)	PUNCT
ejpam-5252	672	40	,	,	PUNCT
ejpam-5252	672	41	v2	v2	PROPN
ejpam-5252	672	42	=	=	SYM
ejpam-5252	672	43	v	v	NOUN
ejpam-5252	672	44	(	(	PUNCT
ejpam-5252	672	45	g	g	NOUN
ejpam-5252	672	46	)	)	PUNCT
ejpam-5252	672	47	.	.	PUNCT
ejpam-5252	673	1	then	then	ADV
ejpam-5252	673	2	f	f	PROPN
ejpam-5252	673	3	∈	∈	PROPN
ejpam-5252	673	4	mrdf	mrdf	NOUN
ejpam-5252	673	5	(	(	PUNCT
ejpam-5252	673	6	g	g	NOUN
ejpam-5252	673	7	◦	◦	NOUN
ejpam-5252	673	8	k2	k2	NOUN
ejpam-5252	673	9	)	)	PUNCT
ejpam-5252	673	10	.	.	PUNCT
ejpam-5252	674	1	it	it	PRON
ejpam-5252	674	2	follows	follow	VERB
ejpam-5252	674	3	that	that	SCONJ
ejpam-5252	674	4	γmr(g	γmr(g	PROPN
ejpam-5252	674	5	◦	◦	NOUN
ejpam-5252	674	6	k2	k2	ADJ
ejpam-5252	674	7	)	)	PUNCT
ejpam-5252	674	8	≤	≤	NOUN
ejpam-5252	674	9	4n	4n	NOUN
ejpam-5252	674	10	.	.	PUNCT
ejpam-5252	675	1	now	now	ADV
ejpam-5252	675	2	,	,	PUNCT
ejpam-5252	675	3	suppose	suppose	VERB
ejpam-5252	675	4	that	that	SCONJ
ejpam-5252	675	5	g	g	PROPN
ejpam-5252	675	6	=	=	SYM
ejpam-5252	675	7	(	(	PUNCT
ejpam-5252	675	8	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	675	9	)	)	PUNCT
ejpam-5252	675	10	is	be	AUX
ejpam-5252	675	11	a	a	DET
ejpam-5252	675	12	γmr	γmr	ADJ
ejpam-5252	675	13	-	-	PUNCT
ejpam-5252	675	14	function	function	NOUN
ejpam-5252	675	15	of	of	ADP
ejpam-5252	675	16	g	g	NOUN
ejpam-5252	675	17	◦	◦	NOUN
ejpam-5252	675	18	k2	k2	ADJ
ejpam-5252	675	19	.	.	PUNCT
ejpam-5252	676	1	if	if	SCONJ
ejpam-5252	676	2	w0	w0	PROPN
ejpam-5252	676	3	=	=	SYM
ejpam-5252	676	4	∅	∅	NOUN
ejpam-5252	676	5	,	,	PUNCT
ejpam-5252	676	6	then	then	ADV
ejpam-5252	676	7	w3	w3	PROPN
ejpam-5252	676	8	=	=	PUNCT
ejpam-5252	676	9	∅.	∅.	NOUN
ejpam-5252	676	10	since	since	SCONJ
ejpam-5252	676	11	g	g	PROPN
ejpam-5252	676	12	is	be	AUX
ejpam-5252	676	13	a	a	DET
ejpam-5252	676	14	γmr	γmr	ADJ
ejpam-5252	676	15	-	-	PUNCT
ejpam-5252	676	16	function	function	NOUN
ejpam-5252	676	17	of	of	ADP
ejpam-5252	676	18	g	g	NOUN
ejpam-5252	676	19	◦	◦	NOUN
ejpam-5252	676	20	k2	k2	NOUN
ejpam-5252	676	21	,	,	PUNCT
ejpam-5252	676	22	by	by	ADP
ejpam-5252	676	23	proposition	proposition	NOUN
ejpam-5252	676	24	4	4	NUM
ejpam-5252	676	25	,	,	PUNCT
ejpam-5252	676	26	γmr(g	γmr(g	PROPN
ejpam-5252	676	27	◦	◦	NOUN
ejpam-5252	676	28	k2	k2	NOUN
ejpam-5252	676	29	)	)	PUNCT
ejpam-5252	676	30	=	=	SYM
ejpam-5252	676	31	4n	4n	X
ejpam-5252	676	32	.	.	PUNCT
ejpam-5252	677	1	s.	s.	PROPN
ejpam-5252	677	2	ahamad	ahamad	PROPN
ejpam-5252	677	3	,	,	PUNCT
ejpam-5252	677	4	j.	j.	PROPN
ejpam-5252	677	5	cariaga	cariaga	PROPN
ejpam-5252	677	6	,	,	PUNCT
ejpam-5252	677	7	s.	s.	PROPN
ejpam-5252	677	8	menchavez	menchavez	PROPN
ejpam-5252	677	9	/	/	PUNCT
ejpam-5252	677	10	eur	eur	PROPN
ejpam-5252	677	11	.	.	PUNCT
ejpam-5252	678	1	j.	j.	PROPN
ejpam-5252	678	2	pure	pure	PROPN
ejpam-5252	678	3	appl	appl	PROPN
ejpam-5252	678	4	.	.	PROPN
ejpam-5252	678	5	math	math	PROPN
ejpam-5252	678	6	,	,	PUNCT
ejpam-5252	678	7	18	18	NUM
ejpam-5252	678	8	(	(	PUNCT
ejpam-5252	678	9	1	1	NUM
ejpam-5252	678	10	)	)	PUNCT
ejpam-5252	678	11	(	(	PUNCT
ejpam-5252	678	12	2025	2025	NUM
ejpam-5252	678	13	)	)	PUNCT
ejpam-5252	678	14	,	,	PUNCT
ejpam-5252	678	15	5252	5252	NUM
ejpam-5252	678	16	17	17	NUM
ejpam-5252	678	17	of	of	ADP
ejpam-5252	678	18	18	18	NUM
ejpam-5252	678	19	for	for	ADP
ejpam-5252	678	20	m	m	PROPN
ejpam-5252	678	21	≥	≥	NOUN
ejpam-5252	678	22	3	3	NUM
ejpam-5252	678	23	,	,	PUNCT
ejpam-5252	678	24	let	let	VERB
ejpam-5252	678	25	v	v	X
ejpam-5252	678	26	(	(	PUNCT
ejpam-5252	678	27	g	g	NOUN
ejpam-5252	678	28	)	)	PUNCT
ejpam-5252	678	29	=	=	SYM
ejpam-5252	678	30	{	{	PUNCT
ejpam-5252	678	31	v1	v1	PROPN
ejpam-5252	678	32	,	,	PUNCT
ejpam-5252	678	33	v2	v2	PROPN
ejpam-5252	678	34	,	,	PUNCT
ejpam-5252	678	35	·	·	PUNCT
ejpam-5252	678	36	·	·	PUNCT
ejpam-5252	678	37	·	·	PUNCT
ejpam-5252	678	38	,	,	PUNCT
ejpam-5252	678	39	vn	vn	INTJ
ejpam-5252	678	40	}	}	PUNCT
ejpam-5252	678	41	and	and	CCONJ
ejpam-5252	678	42	wlog	wlog	NOUN
ejpam-5252	678	43	,	,	PUNCT
ejpam-5252	678	44	pick	pick	VERB
ejpam-5252	678	45	a	a	DET
ejpam-5252	678	46	vertex	vertex	NOUN
ejpam-5252	678	47	u	u	NOUN
ejpam-5252	678	48	∈	∈	PROPN
ejpam-5252	678	49	v	v	NOUN
ejpam-5252	678	50	(	(	PUNCT
ejpam-5252	678	51	km	km	PROPN
ejpam-5252	678	52	)	)	PUNCT
ejpam-5252	678	53	.	.	PUNCT
ejpam-5252	679	1	define	define	VERB
ejpam-5252	679	2	a	a	DET
ejpam-5252	679	3	function	function	NOUN
ejpam-5252	679	4	f	f	NOUN
ejpam-5252	679	5	=	=	SYM
ejpam-5252	679	6	(	(	PUNCT
ejpam-5252	679	7	v0	v0	PROPN
ejpam-5252	679	8	,	,	PUNCT
ejpam-5252	679	9	v1	v1	NOUN
ejpam-5252	679	10	,	,	PUNCT
ejpam-5252	679	11	v2	v2	PROPN
ejpam-5252	679	12	,	,	PUNCT
ejpam-5252	679	13	v3	v3	PROPN
ejpam-5252	679	14	)	)	PUNCT
ejpam-5252	679	15	on	on	ADP
ejpam-5252	679	16	v	v	NUM
ejpam-5252	679	17	(	(	PUNCT
ejpam-5252	679	18	g	g	PROPN
ejpam-5252	679	19	◦	◦	NOUN
ejpam-5252	679	20	km	km	NOUN
ejpam-5252	679	21	)	)	PUNCT
ejpam-5252	679	22	by	by	ADP
ejpam-5252	679	23	f(x	f(x	PROPN
ejpam-5252	679	24	)	)	PUNCT
ejpam-5252	680	1	=	=	PUNCT
ejpam-5252	681	1			NOUN
ejpam-5252	681	2	3	3	NUM
ejpam-5252	681	3	,	,	PUNCT
ejpam-5252	681	4	if	if	SCONJ
ejpam-5252	681	5	x	x	PROPN
ejpam-5252	681	6	∈	∈	PROPN
ejpam-5252	681	7	v	v	X
ejpam-5252	681	8	(	(	PUNCT
ejpam-5252	681	9	g	g	NOUN
ejpam-5252	681	10	)	)	PUNCT
ejpam-5252	681	11	.	.	PUNCT
ejpam-5252	682	1	2	2	NUM
ejpam-5252	682	2	,	,	PUNCT
ejpam-5252	682	3	if	if	SCONJ
ejpam-5252	682	4	x	x	PROPN
ejpam-5252	682	5	∈	∈	PROPN
ejpam-5252	682	6	⋃	⋃	NOUN
ejpam-5252	682	7	v∈v	v∈v	NOUN
ejpam-5252	682	8	(	(	PUNCT
ejpam-5252	682	9	g	g	NOUN
ejpam-5252	682	10	)	)	PUNCT
ejpam-5252	682	11	v	v	NOUN
ejpam-5252	682	12	(	(	PUNCT
ejpam-5252	682	13	uv	uv	NOUN
ejpam-5252	682	14	)	)	PUNCT
ejpam-5252	682	15	.	.	PUNCT
ejpam-5252	683	1	0	0	PUNCT
ejpam-5252	683	2	,	,	PUNCT
ejpam-5252	683	3	if	if	SCONJ
ejpam-5252	683	4	x	x	SYM
ejpam-5252	683	5	∈	∈	PROPN
ejpam-5252	683	6	⋃	⋃	NOUN
ejpam-5252	683	7	v∈v	v∈v	NOUN
ejpam-5252	683	8	(	(	PUNCT
ejpam-5252	683	9	g	g	NOUN
ejpam-5252	683	10	)	)	PUNCT
ejpam-5252	683	11	v	v	NOUN
ejpam-5252	683	12	(	(	PUNCT
ejpam-5252	683	13	(	(	PUNCT
ejpam-5252	683	14	h	h	NOUN
ejpam-5252	683	15	\	\	NOUN
ejpam-5252	683	16	u)v	u)v	PUNCT
ejpam-5252	683	17	)	)	PUNCT
ejpam-5252	683	18	.	.	PUNCT
ejpam-5252	684	1	then	then	ADV
ejpam-5252	684	2	f	f	PROPN
ejpam-5252	684	3	∈	∈	PROPN
ejpam-5252	684	4	mrdf	mrdf	NOUN
ejpam-5252	684	5	(	(	PUNCT
ejpam-5252	684	6	g	g	PROPN
ejpam-5252	684	7	◦	◦	NOUN
ejpam-5252	684	8	km	km	PROPN
ejpam-5252	684	9	)	)	PUNCT
ejpam-5252	684	10	.	.	PUNCT
ejpam-5252	685	1	it	it	PRON
ejpam-5252	685	2	follows	follow	VERB
ejpam-5252	685	3	that	that	SCONJ
ejpam-5252	685	4	γmr(g	γmr(g	PROPN
ejpam-5252	685	5	◦	◦	NOUN
ejpam-5252	685	6	km	km	NOUN
ejpam-5252	685	7	)	)	PUNCT
ejpam-5252	685	8	≤	≤	NUM
ejpam-5252	685	9	5n	5n	NOUN
ejpam-5252	685	10	.	.	PUNCT
ejpam-5252	686	1	now	now	ADV
ejpam-5252	686	2	,	,	PUNCT
ejpam-5252	686	3	suppose	suppose	VERB
ejpam-5252	686	4	that	that	SCONJ
ejpam-5252	686	5	g	g	PROPN
ejpam-5252	686	6	=	=	SYM
ejpam-5252	686	7	(	(	PUNCT
ejpam-5252	686	8	w0,w1,w2,w3	w0,w1,w2,w3	PROPN
ejpam-5252	686	9	)	)	PUNCT
ejpam-5252	686	10	is	be	AUX
ejpam-5252	686	11	a	a	DET
ejpam-5252	686	12	γmr	γmr	ADJ
ejpam-5252	686	13	-	-	PUNCT
ejpam-5252	686	14	function	function	NOUN
ejpam-5252	686	15	of	of	ADP
ejpam-5252	686	16	g	g	PROPN
ejpam-5252	686	17	◦	◦	NOUN
ejpam-5252	686	18	km	km	NOUN
ejpam-5252	686	19	.	.	PUNCT
ejpam-5252	687	1	if	if	SCONJ
ejpam-5252	687	2	w0	w0	PROPN
ejpam-5252	687	3	=	=	SYM
ejpam-5252	687	4	∅	∅	NOUN
ejpam-5252	687	5	,	,	PUNCT
ejpam-5252	687	6	then	then	ADV
ejpam-5252	687	7	w3	w3	PROPN
ejpam-5252	687	8	=	=	PUNCT
ejpam-5252	687	9	∅.	∅.	NOUN
ejpam-5252	687	10	since	since	SCONJ
ejpam-5252	687	11	g	g	PROPN
ejpam-5252	687	12	is	be	AUX
ejpam-5252	687	13	a	a	DET
ejpam-5252	687	14	γmr	γmr	ADJ
ejpam-5252	687	15	-	-	PUNCT
ejpam-5252	687	16	function	function	NOUN
ejpam-5252	687	17	of	of	ADP
ejpam-5252	687	18	g	g	PROPN
ejpam-5252	687	19	◦	◦	NOUN
ejpam-5252	687	20	km	km	PROPN
ejpam-5252	687	21	,	,	PUNCT
ejpam-5252	687	22	|w2|	|w2|	NOUN
ejpam-5252	687	23	=	=	SYM
ejpam-5252	687	24	v	v	PROPN
ejpam-5252	687	25	(	(	PUNCT
ejpam-5252	687	26	g	g	NOUN
ejpam-5252	687	27	)	)	PUNCT
ejpam-5252	687	28	and	and	CCONJ
ejpam-5252	687	29	|w1|	|w1|	NOUN
ejpam-5252	687	30	=	=	SYM
ejpam-5252	687	31	v	v	NOUN
ejpam-5252	687	32	(	(	PUNCT
ejpam-5252	687	33	hv	hv	PROPN
ejpam-5252	687	34	)	)	PUNCT
ejpam-5252	687	35	.	.	PUNCT
ejpam-5252	688	1	hence	hence	ADV
ejpam-5252	688	2	,	,	PUNCT
ejpam-5252	688	3	γmr(g	γmr(g	PROPN
ejpam-5252	688	4	◦	◦	NOUN
ejpam-5252	688	5	km	km	NOUN
ejpam-5252	688	6	)	)	PUNCT
ejpam-5252	688	7	=	=	PUNCT
ejpam-5252	689	1	ωmr	ωmr	NOUN
ejpam-5252	689	2	g	g	PROPN
ejpam-5252	689	3	◦	◦	NOUN
ejpam-5252	689	4	km	km	NOUN
ejpam-5252	689	5	(	(	PUNCT
ejpam-5252	689	6	g	g	NOUN
ejpam-5252	689	7	)	)	PUNCT
ejpam-5252	689	8	≥	≥	NUM
ejpam-5252	689	9	5n	5n	NOUN
ejpam-5252	689	10	.	.	PUNCT
ejpam-5252	690	1	if	if	SCONJ
ejpam-5252	690	2	|w0|	|w0|	NOUN
ejpam-5252	690	3	=	=	NOUN
ejpam-5252	690	4	̸	̸	NUM
ejpam-5252	690	5	0	0	NUM
ejpam-5252	690	6	,	,	PUNCT
ejpam-5252	690	7	then	then	ADV
ejpam-5252	690	8	|w2|	|w2|	NOUN
ejpam-5252	690	9	≥	≥	NOUN
ejpam-5252	690	10	1	1	NUM
ejpam-5252	690	11	and	and	CCONJ
ejpam-5252	690	12	|w3|	|w3|	PRON
ejpam-5252	690	13	≥	≥	NOUN
ejpam-5252	690	14	1	1	NUM
ejpam-5252	690	15	.	.	PUNCT
ejpam-5252	691	1	it	it	PRON
ejpam-5252	691	2	follows	follow	VERB
ejpam-5252	691	3	that	that	SCONJ
ejpam-5252	691	4	γmr(g	γmr(g	PROPN
ejpam-5252	691	5	◦	◦	NOUN
ejpam-5252	691	6	km	km	NOUN
ejpam-5252	691	7	)	)	PUNCT
ejpam-5252	691	8	=	=	PUNCT
ejpam-5252	691	9	ωmr	ωmr	NOUN
ejpam-5252	691	10	g	g	PROPN
ejpam-5252	691	11	◦	◦	NOUN
ejpam-5252	691	12	km	km	NOUN
ejpam-5252	691	13	(	(	PUNCT
ejpam-5252	691	14	g	g	NOUN
ejpam-5252	691	15	)	)	PUNCT
ejpam-5252	691	16	=	=	SYM
ejpam-5252	691	17	2|w2|+	2|w2|+	NUM
ejpam-5252	691	18	3|w3|	3|w3|	NUM
ejpam-5252	691	19	≥	≥	NUM
ejpam-5252	691	20	5n	5n	NUM
ejpam-5252	691	21	.	.	PUNCT
ejpam-5252	692	1	therefore	therefore	ADV
ejpam-5252	692	2	,	,	PUNCT
ejpam-5252	692	3	γmr(g	γmr(g	PROPN
ejpam-5252	692	4	◦	◦	NOUN
ejpam-5252	692	5	km	km	NOUN
ejpam-5252	692	6	)	)	PUNCT
ejpam-5252	693	1	=	=	SYM
ejpam-5252	693	2	5n	5n	X
ejpam-5252	693	3	.	.	PUNCT
ejpam-5252	694	1	corollary	corollary	ADJ
ejpam-5252	694	2	5	5	NUM
ejpam-5252	694	3	.	.	PUNCT
ejpam-5252	695	1	if	if	SCONJ
ejpam-5252	695	2	kn	kn	PROPN
ejpam-5252	695	3	is	be	AUX
ejpam-5252	695	4	a	a	DET
ejpam-5252	695	5	complete	complete	ADJ
ejpam-5252	695	6	graph	graph	NOUN
ejpam-5252	695	7	of	of	ADP
ejpam-5252	695	8	order	order	NOUN
ejpam-5252	695	9	n	n	PRON
ejpam-5252	695	10	≥	≥	NOUN
ejpam-5252	695	11	1	1	NUM
ejpam-5252	695	12	,	,	PUNCT
ejpam-5252	695	13	then	then	ADV
ejpam-5252	695	14	(	(	PUNCT
ejpam-5252	695	15	i	i	NOUN
ejpam-5252	695	16	)	)	PUNCT
ejpam-5252	695	17	γmr(k1	γmr(k1	PROPN
ejpam-5252	695	18	◦	◦	PROPN
ejpam-5252	695	19	kn	kn	PROPN
ejpam-5252	695	20	)	)	PUNCT
ejpam-5252	695	21	=	=	PUNCT
ejpam-5252	695	22	n+	n+	PUNCT
ejpam-5252	695	23	2	2	X
ejpam-5252	695	24	.	.	PUNCT
ejpam-5252	695	25	(	(	PUNCT
ejpam-5252	695	26	ii	ii	NOUN
ejpam-5252	695	27	)	)	PUNCT
ejpam-5252	695	28	γmr(kn	γmr(kn	NOUN
ejpam-5252	695	29	◦	◦	NOUN
ejpam-5252	695	30	k1	k1	NOUN
ejpam-5252	695	31	)	)	PUNCT
ejpam-5252	695	32	=	=	SYM
ejpam-5252	695	33	3n	3n	NOUN
ejpam-5252	695	34	.	.	PUNCT
ejpam-5252	696	1	proof	proof	NOUN
ejpam-5252	696	2	.	.	PUNCT
ejpam-5252	697	1	statement	statement	NOUN
ejpam-5252	697	2	(	(	PUNCT
ejpam-5252	697	3	i	i	NOUN
ejpam-5252	697	4	)	)	PUNCT
ejpam-5252	697	5	follows	follow	VERB
ejpam-5252	697	6	from	from	ADP
ejpam-5252	697	7	the	the	DET
ejpam-5252	697	8	fact	fact	NOUN
ejpam-5252	697	9	that	that	SCONJ
ejpam-5252	697	10	k1	k1	PROPN
ejpam-5252	697	11	◦	◦	NOUN
ejpam-5252	697	12	kn	kn	NOUN
ejpam-5252	697	13	=	=	PUNCT
ejpam-5252	697	14	sn	sn	PROPN
ejpam-5252	697	15	and	and	CCONJ
ejpam-5252	697	16	by	by	ADP
ejpam-5252	697	17	proposition	proposition	NOUN
ejpam-5252	697	18	12	12	NUM
ejpam-5252	697	19	(	(	PUNCT
ejpam-5252	697	20	ii	ii	NOUN
ejpam-5252	697	21	)	)	PUNCT
ejpam-5252	697	22	,	,	PUNCT
ejpam-5252	697	23	γmr(k1	γmr(k1	PROPN
ejpam-5252	697	24	◦	◦	PROPN
ejpam-5252	697	25	kn	kn	PROPN
ejpam-5252	697	26	)	)	PUNCT
ejpam-5252	697	27	=	=	PUNCT
ejpam-5252	697	28	γmr(sn	γmr(sn	X
ejpam-5252	697	29	)	)	PUNCT
ejpam-5252	697	30	=	=	SYM
ejpam-5252	698	1	n+2	n+2	X
ejpam-5252	698	2	.	.	PUNCT
ejpam-5252	699	1	for	for	ADP
ejpam-5252	699	2	(	(	PUNCT
ejpam-5252	699	3	ii	ii	NOUN
ejpam-5252	699	4	)	)	PUNCT
ejpam-5252	699	5	,	,	PUNCT
ejpam-5252	699	6	note	note	VERB
ejpam-5252	699	7	that	that	SCONJ
ejpam-5252	699	8	kn	kn	PROPN
ejpam-5252	699	9	◦	◦	PROPN
ejpam-5252	699	10	k1	k1	PROPN
ejpam-5252	699	11	is	be	AUX
ejpam-5252	699	12	the	the	DET
ejpam-5252	699	13	disjoint	disjoint	PROPN
ejpam-5252	699	14	union	union	NOUN
ejpam-5252	699	15	n	n	PROPN
ejpam-5252	699	16	copies	copy	NOUN
ejpam-5252	699	17	of	of	ADP
ejpam-5252	699	18	k2	k2	NOUN
ejpam-5252	699	19	.	.	PUNCT
ejpam-5252	700	1	using	use	VERB
ejpam-5252	700	2	proposition	proposition	NOUN
ejpam-5252	700	3	5	5	NUM
ejpam-5252	700	4	(	(	PUNCT
ejpam-5252	700	5	ii	ii	NOUN
ejpam-5252	700	6	)	)	PUNCT
ejpam-5252	700	7	and	and	CCONJ
ejpam-5252	700	8	proposition	proposition	NOUN
ejpam-5252	700	9	7	7	NUM
ejpam-5252	700	10	,	,	PUNCT
ejpam-5252	700	11	we	we	PRON
ejpam-5252	700	12	have	have	VERB
ejpam-5252	700	13	γmr(kn	γmr(kn	NOUN
ejpam-5252	700	14	◦	◦	NOUN
ejpam-5252	700	15	k1	k1	NOUN
ejpam-5252	700	16	)	)	PUNCT
ejpam-5252	701	1	=	=	SYM
ejpam-5252	701	2	3n	3n	NOUN
ejpam-5252	701	3	.	.	PUNCT
ejpam-5252	702	1	acknowledgements	acknowledgement	NOUN
ejpam-5252	702	2	the	the	DET
ejpam-5252	702	3	authors	author	NOUN
ejpam-5252	702	4	are	be	AUX
ejpam-5252	702	5	grateful	grateful	ADJ
ejpam-5252	702	6	to	to	ADP
ejpam-5252	702	7	the	the	DET
ejpam-5252	702	8	reviewers	reviewer	NOUN
ejpam-5252	702	9	for	for	ADP
ejpam-5252	702	10	their	their	PRON
ejpam-5252	702	11	invaluable	invaluable	ADJ
ejpam-5252	702	12	assistance	assistance	NOUN
ejpam-5252	702	13	through	through	ADP
ejpam-5252	702	14	their	their	PRON
ejpam-5252	702	15	comments	comment	NOUN
ejpam-5252	702	16	and	and	CCONJ
ejpam-5252	702	17	suggestions	suggestion	NOUN
ejpam-5252	702	18	,	,	PUNCT
ejpam-5252	702	19	which	which	PRON
ejpam-5252	702	20	led	lead	VERB
ejpam-5252	702	21	to	to	ADP
ejpam-5252	702	22	the	the	DET
ejpam-5252	702	23	improvement	improvement	NOUN
ejpam-5252	702	24	of	of	ADP
ejpam-5252	702	25	the	the	DET
ejpam-5252	702	26	paper	paper	NOUN
ejpam-5252	702	27	.	.	PUNCT
ejpam-5252	703	1	also	also	ADV
ejpam-5252	703	2	,	,	PUNCT
ejpam-5252	703	3	one	one	NUM
ejpam-5252	703	4	of	of	ADP
ejpam-5252	703	5	the	the	DET
ejpam-5252	703	6	authors	author	NOUN
ejpam-5252	703	7	,	,	PUNCT
ejpam-5252	703	8	s.	s.	PROPN
ejpam-5252	703	9	ahamad	ahamad	PROPN
ejpam-5252	703	10	,	,	PUNCT
ejpam-5252	703	11	would	would	AUX
ejpam-5252	703	12	like	like	VERB
ejpam-5252	703	13	to	to	PART
ejpam-5252	703	14	recognize	recognize	VERB
ejpam-5252	703	15	the	the	DET
ejpam-5252	703	16	financial	financial	ADJ
ejpam-5252	703	17	support	support	NOUN
ejpam-5252	703	18	of	of	ADP
ejpam-5252	703	19	the	the	DET
ejpam-5252	703	20	department	department	NOUN
ejpam-5252	703	21	of	of	ADP
ejpam-5252	703	22	science	science	NOUN
ejpam-5252	703	23	and	and	CCONJ
ejpam-5252	703	24	technology	technology	NOUN
ejpam-5252	703	25	accelerated	accelerate	VERB
ejpam-5252	703	26	science	science	NOUN
ejpam-5252	703	27	and	and	CCONJ
ejpam-5252	703	28	technology	technology	NOUN
ejpam-5252	703	29	human	human	ADJ
ejpam-5252	703	30	resource	resource	NOUN
ejpam-5252	703	31	development	development	NOUN
ejpam-5252	703	32	program	program	NOUN
ejpam-5252	703	33	(	(	PUNCT
ejpam-5252	703	34	dost	dost	NOUN
ejpam-5252	703	35	-	-	PUNCT
ejpam-5252	703	36	asthrdp)-philippines	asthrdp)-philippine	NOUN
ejpam-5252	703	37	.	.	PUNCT
ejpam-5252	704	1	s.	s.	PROPN
ejpam-5252	704	2	ahamad	ahamad	PROPN
ejpam-5252	704	3	,	,	PUNCT
ejpam-5252	704	4	j.	j.	PROPN
ejpam-5252	704	5	cariaga	cariaga	PROPN
ejpam-5252	704	6	,	,	PUNCT
ejpam-5252	704	7	s.	s.	PROPN
ejpam-5252	704	8	menchavez	menchavez	PROPN
ejpam-5252	704	9	/	/	PUNCT
ejpam-5252	704	10	eur	eur	PROPN
ejpam-5252	704	11	.	.	PUNCT
ejpam-5252	705	1	j.	j.	PROPN
ejpam-5252	705	2	pure	pure	PROPN
ejpam-5252	705	3	appl	appl	PROPN
ejpam-5252	705	4	.	.	PROPN
ejpam-5252	705	5	math	math	PROPN
ejpam-5252	705	6	,	,	PUNCT
ejpam-5252	705	7	18	18	NUM
ejpam-5252	705	8	(	(	PUNCT
ejpam-5252	705	9	1	1	NUM
ejpam-5252	705	10	)	)	PUNCT
ejpam-5252	705	11	(	(	PUNCT
ejpam-5252	705	12	2025	2025	NUM
ejpam-5252	705	13	)	)	PUNCT
ejpam-5252	705	14	,	,	PUNCT
ejpam-5252	705	15	5252	5252	NUM
ejpam-5252	705	16	18	18	NUM
ejpam-5252	705	17	of	of	ADP
ejpam-5252	705	18	18	18	NUM
ejpam-5252	705	19	references	reference	NOUN
ejpam-5252	705	20	[	[	X
ejpam-5252	705	21	1	1	NUM
ejpam-5252	705	22	]	]	X
ejpam-5252	705	23	a.o	a.o	PROPN
ejpam-5252	705	24	.	.	PROPN
ejpam-5252	705	25	ahmed	ahmed	PROPN
ejpam-5252	705	26	and	and	CCONJ
ejpam-5252	705	27	n.a	n.a	PROPN
ejpam-5252	705	28	.	.	PROPN
ejpam-5252	705	29	manal	manal	PROPN
ejpam-5252	705	30	.	.	PUNCT
ejpam-5252	706	1	calculating	calculate	VERB
ejpam-5252	706	2	modern	modern	ADJ
ejpam-5252	706	3	roman	roman	ADJ
ejpam-5252	706	4	domination	domination	NOUN
ejpam-5252	706	5	of	of	ADP
ejpam-5252	706	6	fan	fan	NOUN
ejpam-5252	706	7	graph	graph	NOUN
ejpam-5252	706	8	and	and	CCONJ
ejpam-5252	706	9	double	double	ADJ
ejpam-5252	706	10	fan	fan	NOUN
ejpam-5252	706	11	graph	graph	NOUN
ejpam-5252	706	12	.	.	PUNCT
ejpam-5252	707	1	journal	journal	PROPN
ejpam-5252	707	2	of	of	ADP
ejpam-5252	707	3	applied	apply	VERB
ejpam-5252	707	4	sciences	science	NOUN
ejpam-5252	707	5	and	and	CCONJ
ejpam-5252	707	6	nanotechnology	nanotechnology	NOUN
ejpam-5252	707	7	,	,	PUNCT
ejpam-5252	707	8	2:47–54	2:47–54	NUM
ejpam-5252	707	9	,	,	PUNCT
ejpam-5252	707	10	2022	2022	NUM
ejpam-5252	707	11	.	.	PUNCT
ejpam-5252	708	1	[	[	X
ejpam-5252	708	2	2	2	NUM
ejpam-5252	708	3	]	]	X
ejpam-5252	708	4	f.	f.	PROPN
ejpam-5252	708	5	buckley	buckley	PROPN
ejpam-5252	708	6	and	and	CCONJ
ejpam-5252	708	7	f.	f.	PROPN
ejpam-5252	708	8	harary	harary	PROPN
ejpam-5252	708	9	.	.	PUNCT
ejpam-5252	709	1	distance	distance	NOUN
ejpam-5252	709	2	in	in	ADP
ejpam-5252	709	3	graphs	graph	NOUN
ejpam-5252	709	4	.	.	PUNCT
ejpam-5252	710	1	addison	addison	PROPN
ejpam-5252	710	2	-	-	PUNCT
ejpam-5252	710	3	wesley	wesley	PROPN
ejpam-5252	710	4	,	,	PUNCT
ejpam-5252	710	5	redwood	redwood	NOUN
ejpam-5252	710	6	city	city	NOUN
ejpam-5252	710	7	,	,	PUNCT
ejpam-5252	710	8	ca	ca	NOUN
ejpam-5252	710	9	,	,	PUNCT
ejpam-5252	710	10	1990	1990	NUM
ejpam-5252	710	11	.	.	PUNCT
ejpam-5252	711	1	[	[	X
ejpam-5252	711	2	3	3	X
ejpam-5252	711	3	]	]	X
ejpam-5252	711	4	j.b	j.b	PROPN
ejpam-5252	711	5	.	.	PROPN
ejpam-5252	711	6	cariaga	cariaga	PROPN
ejpam-5252	711	7	and	and	CCONJ
ejpam-5252	711	8	f.p	f.p	PROPN
ejpam-5252	711	9	.	.	PROPN
ejpam-5252	711	10	jamil	jamil	PROPN
ejpam-5252	711	11	.	.	PUNCT
ejpam-5252	712	1	on	on	ADP
ejpam-5252	712	2	double	double	ADJ
ejpam-5252	712	3	roman	roman	ADJ
ejpam-5252	712	4	dominating	dominating	NOUN
ejpam-5252	712	5	functions	function	NOUN
ejpam-5252	712	6	in	in	ADP
ejpam-5252	712	7	graphs	graph	NOUN
ejpam-5252	712	8	.	.	PUNCT
ejpam-5252	713	1	european	european	ADJ
ejpam-5252	713	2	journal	journal	PROPN
ejpam-5252	713	3	of	of	ADP
ejpam-5252	713	4	pure	pure	ADJ
ejpam-5252	713	5	and	and	CCONJ
ejpam-5252	713	6	applied	applied	ADJ
ejpam-5252	713	7	mathematics	mathematic	NOUN
ejpam-5252	713	8	,	,	PUNCT
ejpam-5252	713	9	16:847–863	16:847–863	NUM
ejpam-5252	713	10	,	,	PUNCT
ejpam-5252	713	11	2023	2023	NUM
ejpam-5252	713	12	.	.	PUNCT
ejpam-5252	714	1	[	[	X
ejpam-5252	714	2	4	4	NUM
ejpam-5252	714	3	]	]	X
ejpam-5252	714	4	e.w	e.w	PROPN
ejpam-5252	714	5	.	.	PROPN
ejpam-5252	714	6	chambers	chamber	NOUN
ejpam-5252	714	7	,	,	PUNCT
ejpam-5252	714	8	p.	p.	NOUN
ejpam-5252	714	9	erdos	erdo	NOUN
ejpam-5252	714	10	,	,	PUNCT
ejpam-5252	714	11	and	and	CCONJ
ejpam-5252	714	12	j.	j.	PROPN
ejpam-5252	714	13	chvatal	chvatal	PROPN
ejpam-5252	714	14	.	.	PUNCT
ejpam-5252	715	1	extremal	extremal	ADJ
ejpam-5252	715	2	problems	problem	NOUN
ejpam-5252	715	3	for	for	ADP
ejpam-5252	715	4	roman	roman	ADJ
ejpam-5252	715	5	domination	domination	NOUN
ejpam-5252	715	6	.	.	PUNCT
ejpam-5252	716	1	society	society	NOUN
ejpam-5252	716	2	for	for	ADP
ejpam-5252	716	3	industrial	industrial	ADJ
ejpam-5252	716	4	and	and	CCONJ
ejpam-5252	716	5	applied	apply	VERB
ejpam-5252	716	6	mathematics	mathematic	NOUN
ejpam-5252	716	7	journal	journal	NOUN
ejpam-5252	716	8	of	of	ADP
ejpam-5252	716	9	discrete	discrete	ADJ
ejpam-5252	716	10	mathematics	mathematic	NOUN
ejpam-5252	716	11	,	,	PUNCT
ejpam-5252	716	12	23:1575–1586	23:1575–1586	NUM
ejpam-5252	716	13	,	,	PUNCT
ejpam-5252	716	14	2004	2004	NUM
ejpam-5252	716	15	.	.	PUNCT
ejpam-5252	717	1	[	[	X
ejpam-5252	717	2	5	5	NUM
ejpam-5252	717	3	]	]	SYM
ejpam-5252	717	4	e.j	e.j	PROPN
ejpam-5252	717	5	.	.	PROPN
ejpam-5252	717	6	cockayne	cockayne	PROPN
ejpam-5252	717	7	and	and	CCONJ
ejpam-5252	717	8	s.t	s.t	PROPN
ejpam-5252	717	9	.	.	PROPN
ejpam-5252	717	10	hedetniemi	hedetniemi	PROPN
ejpam-5252	717	11	.	.	PUNCT
ejpam-5252	718	1	towards	towards	ADP
ejpam-5252	718	2	a	a	DET
ejpam-5252	718	3	theory	theory	NOUN
ejpam-5252	718	4	of	of	ADP
ejpam-5252	718	5	domination	domination	NOUN
ejpam-5252	718	6	in	in	ADP
ejpam-5252	718	7	graphs	graph	NOUN
ejpam-5252	718	8	.	.	PUNCT
ejpam-5252	719	1	networks	network	NOUN
ejpam-5252	719	2	:	:	PUNCT
ejpam-5252	719	3	an	an	DET
ejpam-5252	719	4	international	international	ADJ
ejpam-5252	719	5	journal	journal	NOUN
ejpam-5252	719	6	,	,	PUNCT
ejpam-5252	719	7	7:247–261	7:247–261	NUM
ejpam-5252	719	8	,	,	PUNCT
ejpam-5252	719	9	1977	1977	NUM
ejpam-5252	719	10	.	.	PUNCT
ejpam-5252	720	1	[	[	X
ejpam-5252	720	2	6	6	NUM
ejpam-5252	720	3	]	]	SYM
ejpam-5252	720	4	e.j	e.j	PROPN
ejpam-5252	720	5	.	.	PROPN
ejpam-5252	720	6	cockayne	cockayne	PROPN
ejpam-5252	720	7	,	,	PUNCT
ejpam-5252	720	8	p.m.	p.m.	NOUN
ejpam-5252	721	1	dreyer	dreyer	PROPN
ejpam-5252	721	2	sr	sr	PROPN
ejpam-5252	721	3	.	.	PROPN
ejpam-5252	721	4	,	,	PUNCT
ejpam-5252	721	5	s.m	s.m	PROPN
ejpam-5252	721	6	.	.	PROPN
ejpam-5252	721	7	hedetniemi	hedetniemi	PROPN
ejpam-5252	721	8	,	,	PUNCT
ejpam-5252	721	9	and	and	CCONJ
ejpam-5252	721	10	s.t	s.t	PROPN
ejpam-5252	721	11	.	.	PROPN
ejpam-5252	721	12	hedetniemi	hedetniemi	PROPN
ejpam-5252	721	13	.	.	PUNCT
ejpam-5252	722	1	roman	roman	ADJ
ejpam-5252	722	2	domination	domination	NOUN
ejpam-5252	722	3	in	in	ADP
ejpam-5252	722	4	graphs	graph	NOUN
ejpam-5252	722	5	.	.	PUNCT
ejpam-5252	723	1	discrete	discrete	ADJ
ejpam-5252	723	2	mathematics	mathematic	NOUN
ejpam-5252	723	3	,	,	PUNCT
ejpam-5252	723	4	278:1–3	278:1–3	NUM
ejpam-5252	723	5	,	,	PUNCT
ejpam-5252	723	6	2004	2004	NUM
ejpam-5252	723	7	.	.	PUNCT
ejpam-5252	724	1	[	[	X
ejpam-5252	724	2	7	7	NUM
ejpam-5252	724	3	]	]	X
ejpam-5252	724	4	r.j	r.j	PROPN
ejpam-5252	724	5	.	.	PROPN
ejpam-5252	724	6	fortosa	fortosa	PROPN
ejpam-5252	724	7	,	,	PUNCT
ejpam-5252	724	8	f.p	f.p	PROPN
ejpam-5252	724	9	.	.	PROPN
ejpam-5252	724	10	jamil	jamil	PROPN
ejpam-5252	724	11	,	,	PUNCT
ejpam-5252	724	12	and	and	CCONJ
ejpam-5252	724	13	s.r	s.r	PROPN
ejpam-5252	724	14	.	.	PROPN
ejpam-5252	724	15	canoy	canoy	PROPN
ejpam-5252	724	16	.	.	PUNCT
ejpam-5252	725	1	convex	convex	VERB
ejpam-5252	725	2	roman	roman	ADJ
ejpam-5252	725	3	dominating	dominating	NOUN
ejpam-5252	725	4	functions	function	NOUN
ejpam-5252	725	5	on	on	ADP
ejpam-5252	725	6	graphs	graph	NOUN
ejpam-5252	725	7	under	under	ADP
ejpam-5252	725	8	some	some	DET
ejpam-5252	725	9	binary	binary	ADJ
ejpam-5252	725	10	operations	operation	NOUN
ejpam-5252	725	11	.	.	PUNCT
ejpam-5252	726	1	european	european	ADJ
ejpam-5252	726	2	journal	journal	PROPN
ejpam-5252	726	3	of	of	ADP
ejpam-5252	726	4	pure	pure	ADJ
ejpam-5252	726	5	and	and	CCONJ
ejpam-5252	726	6	applied	applied	ADJ
ejpam-5252	726	7	mathematics	mathematic	NOUN
ejpam-5252	726	8	,	,	PUNCT
ejpam-5252	726	9	17:1335–1351	17:1335–1351	NUM
ejpam-5252	726	10	,	,	PUNCT
ejpam-5252	726	11	2024	2024	NUM
ejpam-5252	726	12	.	.	PUNCT
ejpam-5252	727	1	[	[	X
ejpam-5252	727	2	8	8	NUM
ejpam-5252	727	3	]	]	X
ejpam-5252	727	4	a.h	a.h	PROPN
ejpam-5252	727	5	.	.	PROPN
ejpam-5252	727	6	hassan	hassan	PROPN
ejpam-5252	727	7	and	and	CCONJ
ejpam-5252	727	8	a.o	a.o	PROPN
ejpam-5252	727	9	.	.	PROPN
ejpam-5252	727	10	ahmed	ahmed	PROPN
ejpam-5252	727	11	.	.	PUNCT
ejpam-5252	728	1	modern	modern	ADJ
ejpam-5252	728	2	roman	roman	ADJ
ejpam-5252	728	3	domination	domination	NOUN
ejpam-5252	728	4	in	in	ADP
ejpam-5252	728	5	graphs	graph	NOUN
ejpam-5252	728	6	.	.	PUNCT
ejpam-5252	729	1	basrah	basrah	PROPN
ejpam-5252	729	2	journal	journal	PROPN
ejpam-5252	729	3	of	of	ADP
ejpam-5252	729	4	agricultural	agricultural	ADJ
ejpam-5252	729	5	sciences	science	NOUN
ejpam-5252	729	6	,	,	PUNCT
ejpam-5252	729	7	36:45–54	36:45–54	NUM
ejpam-5252	729	8	,	,	PUNCT
ejpam-5252	729	9	2018	2018	NUM
ejpam-5252	729	10	.	.	PUNCT
ejpam-5252	730	1	[	[	X
ejpam-5252	730	2	9	9	NUM
ejpam-5252	730	3	]	]	X
ejpam-5252	730	4	m.a	m.a	PROPN
ejpam-5252	730	5	.	.	PROPN
ejpam-5252	730	6	henning	henning	PROPN
ejpam-5252	730	7	and	and	CCONJ
ejpam-5252	730	8	s.t	s.t	PROPN
ejpam-5252	730	9	.	.	PROPN
ejpam-5252	730	10	hedetniemi	hedetniemi	PROPN
ejpam-5252	730	11	.	.	PUNCT
ejpam-5252	731	1	defending	defend	VERB
ejpam-5252	731	2	the	the	DET
ejpam-5252	731	3	roman	roman	ADJ
ejpam-5252	731	4	empire	empire	NOUN
ejpam-5252	731	5	—	—	PUNCT
ejpam-5252	731	6	a	a	DET
ejpam-5252	731	7	new	new	ADJ
ejpam-5252	731	8	strategy	strategy	NOUN
ejpam-5252	731	9	.	.	PUNCT
ejpam-5252	732	1	discrete	discrete	ADJ
ejpam-5252	732	2	mathematics	mathematic	NOUN
ejpam-5252	732	3	,	,	PUNCT
ejpam-5252	732	4	266:1–3	266:1–3	NUM
ejpam-5252	732	5	,	,	PUNCT
ejpam-5252	732	6	2003	2003	NUM
ejpam-5252	732	7	.	.	PUNCT
ejpam-5252	733	1	[	[	X
ejpam-5252	733	2	10	10	NUM
ejpam-5252	733	3	]	]	X
ejpam-5252	733	4	a.a	a.a	PROPN
ejpam-5252	733	5	.	.	PROPN
ejpam-5252	733	6	hossein	hossein	PROPN
ejpam-5252	733	7	,	,	PUNCT
ejpam-5252	733	8	a.h	a.h	PROPN
ejpam-5252	733	9	.	.	PROPN
ejpam-5252	733	10	michael	michael	PROPN
ejpam-5252	733	11	,	,	PUNCT
ejpam-5252	733	12	s.	s.	PROPN
ejpam-5252	733	13	vladimir	vladimir	PROPN
ejpam-5252	733	14	,	,	PUNCT
ejpam-5252	733	15	and	and	CCONJ
ejpam-5252	733	16	g.y	g.y	PROPN
ejpam-5252	733	17	.	.	PUNCT
ejpam-5252	733	18	ismael	ismael	PROPN
ejpam-5252	733	19	.	.	PUNCT
ejpam-5252	734	1	total	total	ADJ
ejpam-5252	734	2	roman	roman	ADJ
ejpam-5252	734	3	domination	domination	NOUN
ejpam-5252	734	4	in	in	ADP
ejpam-5252	734	5	graphs	graph	NOUN
ejpam-5252	734	6	.	.	PUNCT
ejpam-5252	735	1	applicable	applicable	ADJ
ejpam-5252	735	2	analysis	analysis	NOUN
ejpam-5252	735	3	and	and	CCONJ
ejpam-5252	735	4	discrete	discrete	ADJ
ejpam-5252	735	5	mathematics	mathematic	NOUN
ejpam-5252	735	6	,	,	PUNCT
ejpam-5252	735	7	10:501–517	10:501–517	PROPN
ejpam-5252	735	8	,	,	PUNCT
ejpam-5252	735	9	2016	2016	NUM
ejpam-5252	735	10	.	.	PUNCT
ejpam-5252	736	1	[	[	X
ejpam-5252	736	2	11	11	NUM
ejpam-5252	736	3	]	]	X
ejpam-5252	736	4	s.s	s.s	PROPN
ejpam-5252	736	5	.	.	PROPN
ejpam-5252	736	6	majeed	majeed	PROPN
ejpam-5252	736	7	,	,	PUNCT
ejpam-5252	736	8	a.a	a.a	PROPN
ejpam-5252	736	9	.	.	PROPN
ejpam-5252	736	10	omran	omran	PROPN
ejpam-5252	736	11	,	,	PUNCT
ejpam-5252	736	12	and	and	CCONJ
ejpam-5252	736	13	m.n	m.n	PROPN
ejpam-5252	736	14	.	.	PROPN
ejpam-5252	736	15	yaqoob	yaqoob	PROPN
ejpam-5252	736	16	.	.	PUNCT
ejpam-5252	737	1	modern	modern	ADJ
ejpam-5252	737	2	roman	roman	ADJ
ejpam-5252	737	3	domination	domination	NOUN
ejpam-5252	737	4	of	of	ADP
ejpam-5252	737	5	corona	corona	NOUN
ejpam-5252	737	6	of	of	ADP
ejpam-5252	737	7	cycle	cycle	NOUN
ejpam-5252	737	8	graph	graph	NOUN
ejpam-5252	737	9	with	with	ADP
ejpam-5252	737	10	some	some	DET
ejpam-5252	737	11	certain	certain	ADJ
ejpam-5252	737	12	graphs	graph	NOUN
ejpam-5252	737	13	.	.	PUNCT
ejpam-5252	738	1	international	international	ADJ
ejpam-5252	738	2	journal	journal	NOUN
ejpam-5252	738	3	of	of	ADP
ejpam-5252	738	4	mathematics	mathematic	NOUN
ejpam-5252	738	5	and	and	CCONJ
ejpam-5252	738	6	computer	computer	NOUN
ejpam-5252	738	7	science	science	NOUN
ejpam-5252	738	8	,	,	PUNCT
ejpam-5252	738	9	17	17	NUM
ejpam-5252	738	10	,	,	PUNCT
ejpam-5252	738	11	2022	2022	NUM
ejpam-5252	738	12	.	.	PUNCT
ejpam-5252	739	1	[	[	X
ejpam-5252	739	2	12	12	NUM
ejpam-5252	739	3	]	]	X
ejpam-5252	739	4	l.m	l.m	PROPN
ejpam-5252	739	5	.	.	PROPN
ejpam-5252	739	6	paleta	paleta	PROPN
ejpam-5252	739	7	and	and	CCONJ
ejpam-5252	739	8	f.p	f.p	PROPN
ejpam-5252	739	9	.	.	PROPN
ejpam-5252	739	10	jamil	jamil	PROPN
ejpam-5252	739	11	.	.	PUNCT
ejpam-5252	740	1	more	more	ADJ
ejpam-5252	740	2	on	on	ADP
ejpam-5252	740	3	perfect	perfect	ADJ
ejpam-5252	740	4	roman	roman	ADJ
ejpam-5252	740	5	domination	domination	NOUN
ejpam-5252	740	6	in	in	ADP
ejpam-5252	740	7	graphs	graph	NOUN
ejpam-5252	740	8	.	.	PUNCT
ejpam-5252	741	1	european	european	ADJ
ejpam-5252	741	2	journal	journal	PROPN
ejpam-5252	741	3	of	of	ADP
ejpam-5252	741	4	pure	pure	ADJ
ejpam-5252	741	5	and	and	CCONJ
ejpam-5252	741	6	applied	applied	ADJ
ejpam-5252	741	7	mathematics	mathematic	NOUN
ejpam-5252	741	8	,	,	PUNCT
ejpam-5252	741	9	13:529–548	13:529–548	NUM
ejpam-5252	741	10	,	,	PUNCT
ejpam-5252	741	11	2020	2020	NUM
ejpam-5252	741	12	.	.	PUNCT
ejpam-5252	742	1	[	[	X
ejpam-5252	742	2	13	13	NUM
ejpam-5252	742	3	]	]	X
ejpam-5252	742	4	c.s	c.s	PROPN
ejpam-5252	742	5	.	.	PROPN
ejpam-5252	742	6	revelle	revelle	PROPN
ejpam-5252	742	7	and	and	CCONJ
ejpam-5252	742	8	k.e	k.e	PROPN
ejpam-5252	742	9	.	.	PUNCT
ejpam-5252	743	1	rosing	rosing	PROPN
ejpam-5252	743	2	.	.	PUNCT
ejpam-5252	744	1	defendens	defenden	VERB
ejpam-5252	744	2	imperium	imperium	NOUN
ejpam-5252	744	3	romanum	romanum	NOUN
ejpam-5252	744	4	:	:	PUNCT
ejpam-5252	744	5	a	a	DET
ejpam-5252	744	6	classical	classical	ADJ
ejpam-5252	744	7	problem	problem	NOUN
ejpam-5252	744	8	in	in	ADP
ejpam-5252	744	9	military	military	ADJ
ejpam-5252	744	10	strategy	strategy	NOUN
ejpam-5252	744	11	.	.	PUNCT
ejpam-5252	745	1	american	american	PROPN
ejpam-5252	745	2	mathematical	mathematical	PROPN
ejpam-5252	745	3	monthly	monthly	ADJ
ejpam-5252	745	4	,	,	PUNCT
ejpam-5252	745	5	107:585–594	107:585–594	NUM
ejpam-5252	745	6	,	,	PUNCT
ejpam-5252	745	7	2000	2000	NUM
ejpam-5252	745	8	.	.	PUNCT
ejpam-5252	746	1	[	[	X
ejpam-5252	746	2	14	14	NUM
ejpam-5252	746	3	]	]	X
ejpam-5252	746	4	s.	s.	PROPN
ejpam-5252	746	5	salah	salah	PROPN
ejpam-5252	746	6	,	,	PUNCT
ejpam-5252	746	7	a.a	a.a	PROPN
ejpam-5252	746	8	.	.	PROPN
ejpam-5252	746	9	omran	omran	PROPN
ejpam-5252	746	10	,	,	PUNCT
ejpam-5252	746	11	and	and	CCONJ
ejpam-5252	746	12	m.n	m.n	PROPN
ejpam-5252	746	13	.	.	PROPN
ejpam-5252	746	14	al	al	PROPN
ejpam-5252	746	15	-	-	PUNCT
ejpam-5252	746	16	harere	harere	PROPN
ejpam-5252	746	17	.	.	PUNCT
ejpam-5252	747	1	modern	modern	ADJ
ejpam-5252	747	2	roman	roman	ADJ
ejpam-5252	747	3	domination	domination	NOUN
ejpam-5252	747	4	on	on	ADP
ejpam-5252	747	5	two	two	NUM
ejpam-5252	747	6	operations	operation	NOUN
ejpam-5252	747	7	in	in	ADP
ejpam-5252	747	8	certain	certain	ADJ
ejpam-5252	747	9	graphs	graph	NOUN
ejpam-5252	747	10	.	.	PUNCT
ejpam-5252	748	1	aip	aip	PROPN
ejpam-5252	748	2	conference	conference	NOUN
ejpam-5252	748	3	proceedings	proceeding	NOUN
ejpam-5252	748	4	,	,	PUNCT
ejpam-5252	748	5	2386	2386	NUM
ejpam-5252	748	6	,	,	PUNCT
ejpam-5252	748	7	2022	2022	NUM
ejpam-5252	748	8	.	.	PUNCT
ejpam-5252	749	1	[	[	X
ejpam-5252	749	2	15	15	NUM
ejpam-5252	749	3	]	]	X
ejpam-5252	749	4	i.	i.	PROPN
ejpam-5252	749	5	stewart	stewart	PROPN
ejpam-5252	749	6	.	.	PUNCT
ejpam-5252	750	1	defend	defend	VERB
ejpam-5252	750	2	the	the	DET
ejpam-5252	750	3	roman	roman	ADJ
ejpam-5252	750	4	empire	empire	NOUN
ejpam-5252	750	5	!	!	PUNCT
ejpam-5252	751	1	scientific	scientific	ADJ
ejpam-5252	751	2	american	american	PROPN
ejpam-5252	751	3	,	,	PUNCT
ejpam-5252	751	4	281:136–139	281:136–139	NUM
ejpam-5252	751	5	,	,	PUNCT
ejpam-5252	751	6	1999	1999	NUM
ejpam-5252	751	7	.	.	PUNCT