id	sid	tid	token	lemma	pos
ejpam-4962	1	1	european	european	PROPN
ejpam-4962	1	2	journal	journal	PROPN
ejpam-4962	1	3	of	of	ADP
ejpam-4962	1	4	pure	pure	ADJ
ejpam-4962	1	5	and	and	CCONJ
ejpam-4962	1	6	applied	apply	VERB
ejpam-4962	1	7	mathematics	mathematic	NOUN
ejpam-4962	1	8	vol	vol	NOUN
ejpam-4962	1	9	.	.	PUNCT
ejpam-4962	2	1	16	16	NUM
ejpam-4962	2	2	,	,	PUNCT
ejpam-4962	2	3	no	no	INTJ
ejpam-4962	2	4	.	.	NOUN
ejpam-4962	2	5	4	4	NUM
ejpam-4962	2	6	,	,	PUNCT
ejpam-4962	2	7	2023	2023	NUM
ejpam-4962	2	8	,	,	PUNCT
ejpam-4962	2	9	2368	2368	NUM
ejpam-4962	2	10	-	-	SYM
ejpam-4962	2	11	2383	2383	NUM
ejpam-4962	2	12	issn	issn	VERB
ejpam-4962	2	13	1307	1307	NUM
ejpam-4962	2	14	-	-	SYM
ejpam-4962	2	15	5543	5543	NUM
ejpam-4962	2	16	–	–	PUNCT
ejpam-4962	2	17	ejpam.com	ejpam.com	X
ejpam-4962	2	18	published	publish	VERB
ejpam-4962	2	19	by	by	ADP
ejpam-4962	2	20	new	new	PROPN
ejpam-4962	2	21	york	york	PROPN
ejpam-4962	2	22	business	business	PROPN
ejpam-4962	2	23	global	global	ADJ
ejpam-4962	2	24	geodetic	geodetic	ADJ
ejpam-4962	2	25	roman	roman	ADJ
ejpam-4962	2	26	dominating	dominating	NOUN
ejpam-4962	2	27	functions	function	NOUN
ejpam-4962	2	28	in	in	ADP
ejpam-4962	2	29	a	a	DET
ejpam-4962	2	30	graph	graph	NOUN
ejpam-4962	2	31	rona	rona	PROPN
ejpam-4962	2	32	jane	jane	PROPN
ejpam-4962	2	33	g.	g.	PROPN
ejpam-4962	2	34	fortosa1,∗	fortosa1,∗	PROPN
ejpam-4962	2	35	,	,	PUNCT
ejpam-4962	2	36	sergio	sergio	PROPN
ejpam-4962	2	37	r.	r.	PROPN
ejpam-4962	2	38	canoy	canoy	PROPN
ejpam-4962	2	39	,	,	PUNCT
ejpam-4962	2	40	jr.1,2	jr.1,2	ADJ
ejpam-4962	2	41	1	1	NUM
ejpam-4962	2	42	department	department	NOUN
ejpam-4962	2	43	of	of	ADP
ejpam-4962	2	44	mathematics	mathematic	NOUN
ejpam-4962	2	45	and	and	CCONJ
ejpam-4962	2	46	statistics	statistic	NOUN
ejpam-4962	2	47	,	,	PUNCT
ejpam-4962	2	48	college	college	NOUN
ejpam-4962	2	49	of	of	ADP
ejpam-4962	2	50	science	science	NOUN
ejpam-4962	2	51	and	and	CCONJ
ejpam-4962	2	52	mathematics	mathematic	NOUN
ejpam-4962	2	53	,	,	PUNCT
ejpam-4962	2	54	2	2	NUM
ejpam-4962	2	55	center	center	NOUN
ejpam-4962	2	56	for	for	ADP
ejpam-4962	2	57	graph	graph	NOUN
ejpam-4962	2	58	theory	theory	NOUN
ejpam-4962	2	59	,	,	PUNCT
ejpam-4962	2	60	algebra	algebra	NOUN
ejpam-4962	2	61	,	,	PUNCT
ejpam-4962	2	62	and	and	CCONJ
ejpam-4962	2	63	analysisprism	analysisprism	NOUN
ejpam-4962	2	64	,	,	PUNCT
ejpam-4962	2	65	msu	msu	PROPN
ejpam-4962	2	66	-	-	PUNCT
ejpam-4962	2	67	iligan	iligan	PROPN
ejpam-4962	2	68	institute	institute	PROPN
ejpam-4962	2	69	of	of	ADP
ejpam-4962	2	70	technology	technology	PROPN
ejpam-4962	2	71	,	,	PUNCT
ejpam-4962	2	72	9200	9200	NUM
ejpam-4962	2	73	iligan	iligan	ADJ
ejpam-4962	2	74	city	city	NOUN
ejpam-4962	2	75	,	,	PUNCT
ejpam-4962	3	1	philippines	philippine	NOUN
ejpam-4962	3	2	abstract	abstract	ADJ
ejpam-4962	3	3	.	.	PUNCT
ejpam-4962	4	1	let	let	VERB
ejpam-4962	4	2	g	g	PRON
ejpam-4962	4	3	be	be	AUX
ejpam-4962	4	4	a	a	DET
ejpam-4962	4	5	connected	connected	ADJ
ejpam-4962	4	6	graph	graph	NOUN
ejpam-4962	4	7	.	.	PUNCT
ejpam-4962	5	1	a	a	DET
ejpam-4962	5	2	function	function	NOUN
ejpam-4962	5	3	f	f	NOUN
ejpam-4962	5	4	:	:	PUNCT
ejpam-4962	5	5	v	v	X
ejpam-4962	5	6	(	(	PUNCT
ejpam-4962	5	7	g	g	NOUN
ejpam-4962	5	8	)	)	PUNCT
ejpam-4962	5	9	→	→	SYM
ejpam-4962	5	10	{	{	PUNCT
ejpam-4962	5	11	0	0	NUM
ejpam-4962	5	12	,	,	PUNCT
ejpam-4962	5	13	1	1	NUM
ejpam-4962	5	14	,	,	PUNCT
ejpam-4962	5	15	2	2	NUM
ejpam-4962	5	16	}	}	PUNCT
ejpam-4962	5	17	is	be	AUX
ejpam-4962	5	18	a	a	DET
ejpam-4962	5	19	geodetic	geodetic	ADJ
ejpam-4962	5	20	roman	roman	ADJ
ejpam-4962	5	21	dominating	dominating	NOUN
ejpam-4962	5	22	function	function	NOUN
ejpam-4962	5	23	(	(	PUNCT
ejpam-4962	5	24	or	or	CCONJ
ejpam-4962	5	25	grdf	grdf	NOUN
ejpam-4962	5	26	)	)	PUNCT
ejpam-4962	5	27	if	if	SCONJ
ejpam-4962	5	28	every	every	DET
ejpam-4962	5	29	vertex	vertex	NOUN
ejpam-4962	5	30	u	u	NOUN
ejpam-4962	5	31	for	for	ADP
ejpam-4962	5	32	which	which	PRON
ejpam-4962	5	33	f(u	f(u	PROPN
ejpam-4962	5	34	)	)	PUNCT
ejpam-4962	6	1	=	=	SYM
ejpam-4962	6	2	0	0	NUM
ejpam-4962	6	3	is	be	AUX
ejpam-4962	6	4	adjacent	adjacent	ADJ
ejpam-4962	6	5	to	to	ADP
ejpam-4962	6	6	at	at	ADV
ejpam-4962	6	7	least	least	ADV
ejpam-4962	6	8	one	one	NUM
ejpam-4962	6	9	vertex	vertex	NOUN
ejpam-4962	6	10	v	v	NOUN
ejpam-4962	6	11	for	for	ADP
ejpam-4962	6	12	which	which	PRON
ejpam-4962	6	13	f(v	f(v	NOUN
ejpam-4962	6	14	)	)	PUNCT
ejpam-4962	6	15	=	=	SYM
ejpam-4962	6	16	2	2	NUM
ejpam-4962	6	17	and	and	CCONJ
ejpam-4962	6	18	v1	v1	VERB
ejpam-4962	6	19	∪	∪	X
ejpam-4962	6	20	v2	v2	NOUN
ejpam-4962	6	21	is	be	AUX
ejpam-4962	6	22	a	a	DET
ejpam-4962	6	23	geodetic	geodetic	ADJ
ejpam-4962	6	24	set	set	NOUN
ejpam-4962	6	25	in	in	ADP
ejpam-4962	6	26	g.	g.	PROPN
ejpam-4962	6	27	the	the	DET
ejpam-4962	6	28	weight	weight	NOUN
ejpam-4962	6	29	of	of	ADP
ejpam-4962	6	30	a	a	DET
ejpam-4962	6	31	geodetic	geodetic	ADJ
ejpam-4962	6	32	roman	roman	ADJ
ejpam-4962	6	33	dominating	dominating	NOUN
ejpam-4962	6	34	function	function	NOUN
ejpam-4962	6	35	f	f	PROPN
ejpam-4962	6	36	,	,	PUNCT
ejpam-4962	6	37	denoted	denote	VERB
ejpam-4962	6	38	by	by	ADP
ejpam-4962	6	39	ωgr	ωgr	PROPN
ejpam-4962	6	40	g	g	PROPN
ejpam-4962	6	41	(	(	PUNCT
ejpam-4962	6	42	f	f	PROPN
ejpam-4962	6	43	)	)	PUNCT
ejpam-4962	6	44	,	,	PUNCT
ejpam-4962	6	45	is	be	AUX
ejpam-4962	6	46	given	give	VERB
ejpam-4962	6	47	by	by	ADP
ejpam-4962	6	48	ωgr	ωgr	NOUN
ejpam-4962	6	49	g	g	PROPN
ejpam-4962	6	50	(	(	PUNCT
ejpam-4962	6	51	f	f	X
ejpam-4962	6	52	)	)	PUNCT
ejpam-4962	7	1	=	=	SYM
ejpam-4962	7	2	∑	∑	PUNCT
ejpam-4962	7	3	v∈v	v∈v	PROPN
ejpam-4962	7	4	(	(	PUNCT
ejpam-4962	7	5	g	g	NOUN
ejpam-4962	7	6	)	)	PUNCT
ejpam-4962	7	7	f(v	f(v	NOUN
ejpam-4962	7	8	)	)	PUNCT
ejpam-4962	7	9	.	.	PUNCT
ejpam-4962	8	1	the	the	DET
ejpam-4962	8	2	minimum	minimum	ADJ
ejpam-4962	8	3	weight	weight	NOUN
ejpam-4962	8	4	of	of	ADP
ejpam-4962	8	5	a	a	DET
ejpam-4962	8	6	grdf	grdf	NOUN
ejpam-4962	8	7	on	on	ADP
ejpam-4962	8	8	g	g	NOUN
ejpam-4962	8	9	,	,	PUNCT
ejpam-4962	8	10	denoted	denote	VERB
ejpam-4962	8	11	by	by	ADP
ejpam-4962	8	12	γgr(g	γgr(g	PROPN
ejpam-4962	8	13	)	)	PUNCT
ejpam-4962	8	14	,	,	PUNCT
ejpam-4962	8	15	is	be	AUX
ejpam-4962	8	16	called	call	VERB
ejpam-4962	8	17	the	the	DET
ejpam-4962	8	18	geodetic	geodetic	ADJ
ejpam-4962	8	19	roman	roman	ADJ
ejpam-4962	8	20	domination	domination	NOUN
ejpam-4962	8	21	number	number	NOUN
ejpam-4962	8	22	of	of	ADP
ejpam-4962	8	23	g.	g.	PROPN
ejpam-4962	8	24	in	in	ADP
ejpam-4962	8	25	this	this	DET
ejpam-4962	8	26	paper	paper	NOUN
ejpam-4962	8	27	,	,	PUNCT
ejpam-4962	8	28	we	we	PRON
ejpam-4962	8	29	give	give	VERB
ejpam-4962	8	30	some	some	DET
ejpam-4962	8	31	properties	property	NOUN
ejpam-4962	8	32	of	of	ADP
ejpam-4962	8	33	geodetic	geodetic	ADJ
ejpam-4962	8	34	roman	roman	ADJ
ejpam-4962	8	35	domination	domination	NOUN
ejpam-4962	8	36	and	and	CCONJ
ejpam-4962	8	37	determine	determine	VERB
ejpam-4962	8	38	the	the	DET
ejpam-4962	8	39	geodetic	geodetic	ADJ
ejpam-4962	8	40	roman	roman	ADJ
ejpam-4962	8	41	domination	domination	NOUN
ejpam-4962	8	42	number	number	NOUN
ejpam-4962	8	43	of	of	ADP
ejpam-4962	8	44	some	some	DET
ejpam-4962	8	45	graphs	graph	NOUN
ejpam-4962	8	46	.	.	PUNCT
ejpam-4962	9	1	2020	2020	NUM
ejpam-4962	9	2	mathematics	mathematic	NOUN
ejpam-4962	9	3	subject	subject	NOUN
ejpam-4962	9	4	classifications	classification	NOUN
ejpam-4962	9	5	:	:	PUNCT
ejpam-4962	9	6	05c69	05c69	X
ejpam-4962	9	7	key	key	ADJ
ejpam-4962	9	8	words	word	NOUN
ejpam-4962	9	9	and	and	CCONJ
ejpam-4962	9	10	phrases	phrase	NOUN
ejpam-4962	9	11	:	:	PUNCT
ejpam-4962	9	12	geodetic	geodetic	ADJ
ejpam-4962	9	13	set	set	NOUN
ejpam-4962	9	14	,	,	PUNCT
ejpam-4962	9	15	geodetic	geodetic	ADJ
ejpam-4962	9	16	dominating	dominating	NOUN
ejpam-4962	9	17	set	set	NOUN
ejpam-4962	9	18	,	,	PUNCT
ejpam-4962	9	19	roman	roman	ADJ
ejpam-4962	9	20	dominating	dominating	NOUN
ejpam-4962	9	21	function	function	NOUN
ejpam-4962	9	22	,	,	PUNCT
ejpam-4962	9	23	roman	roman	ADJ
ejpam-4962	9	24	domination	domination	NOUN
ejpam-4962	9	25	number	number	NOUN
ejpam-4962	9	26	,	,	PUNCT
ejpam-4962	9	27	geodetic	geodetic	ADJ
ejpam-4962	9	28	roman	roman	ADJ
ejpam-4962	9	29	dominating	dominating	NOUN
ejpam-4962	9	30	function	function	NOUN
ejpam-4962	9	31	,	,	PUNCT
ejpam-4962	9	32	geodetic	geodetic	ADJ
ejpam-4962	9	33	roman	roman	ADJ
ejpam-4962	9	34	domination	domination	NOUN
ejpam-4962	9	35	number	number	NOUN
ejpam-4962	9	36	1	1	NUM
ejpam-4962	9	37	.	.	PUNCT
ejpam-4962	10	1	introduction	introduction	NOUN
ejpam-4962	10	2	roman	roman	ADJ
ejpam-4962	10	3	domination	domination	NOUN
ejpam-4962	10	4	was	be	AUX
ejpam-4962	10	5	inspired	inspire	VERB
ejpam-4962	10	6	by	by	ADP
ejpam-4962	10	7	the	the	DET
ejpam-4962	10	8	strategies	strategy	NOUN
ejpam-4962	10	9	for	for	ADP
ejpam-4962	10	10	defending	defend	VERB
ejpam-4962	10	11	the	the	DET
ejpam-4962	10	12	roman	roman	ADJ
ejpam-4962	10	13	empire	empire	NOUN
ejpam-4962	10	14	against	against	ADP
ejpam-4962	10	15	invaders	invader	NOUN
ejpam-4962	10	16	,	,	PUNCT
ejpam-4962	10	17	as	as	SCONJ
ejpam-4962	10	18	presented	present	VERB
ejpam-4962	10	19	by	by	ADP
ejpam-4962	10	20	stewart	stewart	PROPN
ejpam-4962	11	1	[	[	X
ejpam-4962	11	2	22	22	NUM
ejpam-4962	11	3	]	]	PUNCT
ejpam-4962	11	4	and	and	CCONJ
ejpam-4962	11	5	revelle	revelle	NOUN
ejpam-4962	11	6	and	and	CCONJ
ejpam-4962	11	7	rosing	rose	VERB
ejpam-4962	11	8	[	[	X
ejpam-4962	11	9	20	20	NUM
ejpam-4962	11	10	]	]	PUNCT
ejpam-4962	11	11	.	.	PUNCT
ejpam-4962	12	1	motivated	motivate	VERB
ejpam-4962	12	2	by	by	ADP
ejpam-4962	12	3	this	this	DET
ejpam-4962	12	4	strategy	strategy	NOUN
ejpam-4962	12	5	,	,	PUNCT
ejpam-4962	12	6	cockayne	cockayne	NOUN
ejpam-4962	12	7	,	,	PUNCT
ejpam-4962	12	8	dreyer	dreyer	PROPN
ejpam-4962	12	9	and	and	CCONJ
ejpam-4962	12	10	hedetniemi	hedetniemi	PROPN
ejpam-4962	12	11	introduced	introduce	VERB
ejpam-4962	12	12	the	the	DET
ejpam-4962	12	13	concept	concept	NOUN
ejpam-4962	12	14	of	of	ADP
ejpam-4962	12	15	roman	roman	ADJ
ejpam-4962	12	16	domination	domination	NOUN
ejpam-4962	12	17	in	in	ADP
ejpam-4962	12	18	2004	2004	NUM
ejpam-4962	12	19	[	[	X
ejpam-4962	12	20	12	12	NUM
ejpam-4962	12	21	]	]	PUNCT
ejpam-4962	12	22	.	.	PUNCT
ejpam-4962	13	1	roman	roman	ADJ
ejpam-4962	13	2	domination	domination	NOUN
ejpam-4962	13	3	in	in	ADP
ejpam-4962	13	4	a	a	DET
ejpam-4962	13	5	graph	graph	NOUN
ejpam-4962	13	6	is	be	AUX
ejpam-4962	13	7	a	a	DET
ejpam-4962	13	8	well	well	ADV
ejpam-4962	13	9	studied	study	VERB
ejpam-4962	13	10	concept	concept	NOUN
ejpam-4962	13	11	under	under	ADP
ejpam-4962	13	12	the	the	DET
ejpam-4962	13	13	topic	topic	NOUN
ejpam-4962	13	14	of	of	ADP
ejpam-4962	13	15	domination	domination	NOUN
ejpam-4962	13	16	.	.	PUNCT
ejpam-4962	14	1	as	as	ADP
ejpam-4962	14	2	a	a	DET
ejpam-4962	14	3	protection	protection	NOUN
ejpam-4962	14	4	strategy	strategy	NOUN
ejpam-4962	14	5	involving	involve	VERB
ejpam-4962	14	6	field	field	NOUN
ejpam-4962	14	7	armies	army	NOUN
ejpam-4962	14	8	,	,	PUNCT
ejpam-4962	14	9	the	the	DET
ejpam-4962	14	10	roman	roman	ADJ
ejpam-4962	14	11	domination	domination	NOUN
ejpam-4962	14	12	concept	concept	NOUN
ejpam-4962	14	13	ensures	ensure	VERB
ejpam-4962	14	14	that	that	SCONJ
ejpam-4962	14	15	an	an	DET
ejpam-4962	14	16	unsecured	unsecured	ADJ
ejpam-4962	14	17	location	location	NOUN
ejpam-4962	14	18	is	be	AUX
ejpam-4962	14	19	made	make	VERB
ejpam-4962	14	20	secured	secure	VERB
ejpam-4962	14	21	by	by	ADP
ejpam-4962	14	22	sending	send	VERB
ejpam-4962	14	23	an	an	DET
ejpam-4962	14	24	army	army	NOUN
ejpam-4962	14	25	to	to	ADP
ejpam-4962	14	26	the	the	DET
ejpam-4962	14	27	location	location	NOUN
ejpam-4962	14	28	from	from	ADP
ejpam-4962	14	29	an	an	DET
ejpam-4962	14	30	adjacent	adjacent	ADJ
ejpam-4962	14	31	secured	secure	VERB
ejpam-4962	14	32	location	location	NOUN
ejpam-4962	14	33	subject	subject	NOUN
ejpam-4962	14	34	to	to	ADP
ejpam-4962	14	35	the	the	DET
ejpam-4962	14	36	constraint	constraint	NOUN
ejpam-4962	14	37	that	that	PRON
ejpam-4962	14	38	one	one	NUM
ejpam-4962	14	39	army	army	NOUN
ejpam-4962	14	40	must	must	AUX
ejpam-4962	14	41	be	be	AUX
ejpam-4962	14	42	left	leave	VERB
ejpam-4962	14	43	behind	behind	ADV
ejpam-4962	14	44	in	in	ADP
ejpam-4962	14	45	the	the	DET
ejpam-4962	14	46	secured	secure	VERB
ejpam-4962	14	47	location	location	NOUN
ejpam-4962	14	48	.	.	PUNCT
ejpam-4962	15	1	other	other	ADJ
ejpam-4962	15	2	applications	application	NOUN
ejpam-4962	15	3	of	of	ADP
ejpam-4962	15	4	the	the	DET
ejpam-4962	15	5	concept	concept	NOUN
ejpam-4962	15	6	and	and	CCONJ
ejpam-4962	15	7	some	some	PRON
ejpam-4962	15	8	of	of	ADP
ejpam-4962	15	9	its	its	PRON
ejpam-4962	15	10	variations	variation	NOUN
ejpam-4962	15	11	can	can	AUX
ejpam-4962	15	12	be	be	AUX
ejpam-4962	15	13	found	find	VERB
ejpam-4962	15	14	in	in	ADP
ejpam-4962	15	15	[	[	X
ejpam-4962	15	16	1	1	NUM
ejpam-4962	15	17	]	]	PUNCT
ejpam-4962	15	18	,	,	PUNCT
ejpam-4962	15	19	[	[	X
ejpam-4962	15	20	2	2	NUM
ejpam-4962	15	21	]	]	PUNCT
ejpam-4962	15	22	,	,	PUNCT
ejpam-4962	15	23	[	[	X
ejpam-4962	15	24	3	3	NUM
ejpam-4962	15	25	]	]	PUNCT
ejpam-4962	15	26	,	,	PUNCT
ejpam-4962	15	27	[	[	X
ejpam-4962	15	28	4	4	NUM
ejpam-4962	15	29	]	]	PUNCT
ejpam-4962	15	30	,	,	PUNCT
ejpam-4962	15	31	[	[	X
ejpam-4962	15	32	5	5	NUM
ejpam-4962	15	33	]	]	PUNCT
ejpam-4962	15	34	,	,	PUNCT
ejpam-4962	15	35	[	[	X
ejpam-4962	15	36	10	10	NUM
ejpam-4962	15	37	]	]	PUNCT
ejpam-4962	15	38	,	,	PUNCT
ejpam-4962	15	39	[	[	X
ejpam-4962	15	40	12	12	NUM
ejpam-4962	15	41	]	]	PUNCT
ejpam-4962	15	42	,	,	PUNCT
ejpam-4962	15	43	[	[	X
ejpam-4962	15	44	15	15	NUM
ejpam-4962	15	45	]	]	PUNCT
ejpam-4962	15	46	,	,	PUNCT
ejpam-4962	15	47	[	[	X
ejpam-4962	15	48	16	16	NUM
ejpam-4962	15	49	]	]	PUNCT
ejpam-4962	15	50	,	,	PUNCT
ejpam-4962	15	51	[	[	X
ejpam-4962	15	52	17	17	NUM
ejpam-4962	15	53	]	]	PUNCT
ejpam-4962	15	54	,	,	PUNCT
ejpam-4962	15	55	and	and	CCONJ
ejpam-4962	15	56	[	[	X
ejpam-4962	15	57	19	19	NUM
ejpam-4962	15	58	]	]	PUNCT
ejpam-4962	15	59	.	.	PUNCT
ejpam-4962	16	1	another	another	DET
ejpam-4962	16	2	variant	variant	NOUN
ejpam-4962	16	3	of	of	ADP
ejpam-4962	16	4	domination	domination	NOUN
ejpam-4962	16	5	is	be	AUX
ejpam-4962	16	6	the	the	DET
ejpam-4962	16	7	concept	concept	NOUN
ejpam-4962	16	8	geodetic	geodetic	ADJ
ejpam-4962	16	9	domination	domination	NOUN
ejpam-4962	16	10	which	which	PRON
ejpam-4962	16	11	was	be	AUX
ejpam-4962	16	12	introduced	introduce	VERB
ejpam-4962	16	13	by	by	ADP
ejpam-4962	16	14	buckley	buckley	NOUN
ejpam-4962	16	15	,	,	PUNCT
ejpam-4962	16	16	harary	harary	NOUN
ejpam-4962	16	17	and	and	CCONJ
ejpam-4962	16	18	quintas	quinta	NOUN
ejpam-4962	16	19	[	[	X
ejpam-4962	16	20	6	6	NUM
ejpam-4962	16	21	]	]	PUNCT
ejpam-4962	16	22	.	.	PUNCT
ejpam-4962	17	1	geodesics	geodesic	NOUN
ejpam-4962	17	2	refers	refer	VERB
ejpam-4962	17	3	to	to	ADP
ejpam-4962	17	4	the	the	DET
ejpam-4962	17	5	shortest	short	ADJ
ejpam-4962	17	6	paths	path	NOUN
ejpam-4962	17	7	between	between	ADP
ejpam-4962	17	8	two	two	NUM
ejpam-4962	17	9	vertices	vertex	NOUN
ejpam-4962	17	10	in	in	ADP
ejpam-4962	17	11	a	a	DET
ejpam-4962	17	12	graph	graph	NOUN
ejpam-4962	17	13	.	.	PUNCT
ejpam-4962	18	1	the	the	DET
ejpam-4962	18	2	concept	concept	NOUN
ejpam-4962	18	3	of	of	ADP
ejpam-4962	18	4	geodesics	geodesic	NOUN
ejpam-4962	18	5	is	be	AUX
ejpam-4962	18	6	closely	closely	ADV
ejpam-4962	18	7	related	relate	VERB
ejpam-4962	18	8	to	to	ADP
ejpam-4962	18	9	the	the	DET
ejpam-4962	18	10	∗corresponding	∗corresponde	VERB
ejpam-4962	18	11	author	author	NOUN
ejpam-4962	18	12	.	.	PUNCT
ejpam-4962	19	1	doi	doi	NOUN
ejpam-4962	19	2	:	:	PUNCT
ejpam-4962	19	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4962	https://doi.org/10.29020/nybg.ejpam.v16i4.4962	NOUN
ejpam-4962	19	4	email	email	NOUN
ejpam-4962	19	5	addresses	address	NOUN
ejpam-4962	19	6	:	:	PUNCT
ejpam-4962	20	1	ronajane.fortosa@g.msuiit.edu.ph	ronajane.fortosa@g.msuiit.edu.ph	PROPN
ejpam-4962	20	2	(	(	PUNCT
ejpam-4962	20	3	r.j	r.j	PROPN
ejpam-4962	20	4	.	.	PROPN
ejpam-4962	20	5	fortosa	fortosa	PROPN
ejpam-4962	20	6	)	)	PUNCT
ejpam-4962	20	7	,	,	PUNCT
ejpam-4962	20	8	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-4962	20	9	(	(	PUNCT
ejpam-4962	20	10	s.	s.	PROPN
ejpam-4962	20	11	canoy	canoy	PROPN
ejpam-4962	20	12	)	)	PUNCT
ejpam-4962	20	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4962	20	14	2368	2368	NUM
ejpam-4962	21	1	©	©	ADP
ejpam-4962	21	2	2023	2023	NUM
ejpam-4962	21	3	ejpam	ejpam	NOUN
ejpam-4962	21	4	all	all	DET
ejpam-4962	21	5	rights	right	NOUN
ejpam-4962	21	6	reserved	reserve	VERB
ejpam-4962	21	7	.	.	PUNCT
ejpam-4962	22	1	r.	r.	PROPN
ejpam-4962	22	2	fortosa	fortosa	PROPN
ejpam-4962	22	3	,	,	PUNCT
ejpam-4962	22	4	s.	s.	PROPN
ejpam-4962	22	5	canoy	canoy	PROPN
ejpam-4962	22	6	jr	jr	PROPN
ejpam-4962	22	7	.	.	PROPN
ejpam-4962	22	8	/	/	SYM
ejpam-4962	22	9	eur	eur	PROPN
ejpam-4962	22	10	.	.	PUNCT
ejpam-4962	23	1	j.	j.	PROPN
ejpam-4962	23	2	pure	pure	PROPN
ejpam-4962	23	3	appl	appl	PROPN
ejpam-4962	23	4	.	.	PROPN
ejpam-4962	23	5	math	math	PROPN
ejpam-4962	23	6	,	,	PUNCT
ejpam-4962	23	7	16	16	NUM
ejpam-4962	23	8	(	(	PUNCT
ejpam-4962	23	9	4	4	NUM
ejpam-4962	23	10	)	)	PUNCT
ejpam-4962	23	11	(	(	PUNCT
ejpam-4962	23	12	2023	2023	NUM
ejpam-4962	23	13	)	)	PUNCT
ejpam-4962	23	14	,	,	PUNCT
ejpam-4962	23	15	2368	2368	NUM
ejpam-4962	23	16	-	-	SYM
ejpam-4962	23	17	2383	2383	NUM
ejpam-4962	23	18	2369	2369	NUM
ejpam-4962	23	19	notion	notion	NOUN
ejpam-4962	23	20	of	of	ADP
ejpam-4962	23	21	distance	distance	NOUN
ejpam-4962	23	22	in	in	ADP
ejpam-4962	23	23	a	a	DET
ejpam-4962	23	24	graph	graph	NOUN
ejpam-4962	23	25	.	.	PUNCT
ejpam-4962	24	1	as	as	ADP
ejpam-4962	24	2	a	a	DET
ejpam-4962	24	3	matter	matter	NOUN
ejpam-4962	24	4	of	of	ADP
ejpam-4962	24	5	fact	fact	NOUN
ejpam-4962	24	6	,	,	PUNCT
ejpam-4962	24	7	the	the	DET
ejpam-4962	24	8	distance	distance	NOUN
ejpam-4962	24	9	between	between	ADP
ejpam-4962	24	10	two	two	NUM
ejpam-4962	24	11	vertices	vertex	NOUN
ejpam-4962	24	12	is	be	AUX
ejpam-4962	24	13	defined	define	VERB
ejpam-4962	24	14	as	as	ADP
ejpam-4962	24	15	the	the	DET
ejpam-4962	24	16	length	length	NOUN
ejpam-4962	24	17	of	of	ADP
ejpam-4962	24	18	the	the	DET
ejpam-4962	24	19	shortest	short	ADJ
ejpam-4962	24	20	geodesic	geodesic	NOUN
ejpam-4962	24	21	between	between	ADP
ejpam-4962	24	22	them	they	PRON
ejpam-4962	24	23	.	.	PUNCT
ejpam-4962	25	1	in	in	ADP
ejpam-4962	25	2	simple	simple	ADJ
ejpam-4962	25	3	terms	term	NOUN
ejpam-4962	25	4	,	,	PUNCT
ejpam-4962	25	5	the	the	DET
ejpam-4962	25	6	concept	concept	NOUN
ejpam-4962	25	7	represents	represent	VERB
ejpam-4962	25	8	the	the	DET
ejpam-4962	25	9	minimum	minimum	ADJ
ejpam-4962	25	10	number	number	NOUN
ejpam-4962	25	11	of	of	ADP
ejpam-4962	25	12	edges	edge	NOUN
ejpam-4962	25	13	that	that	PRON
ejpam-4962	25	14	must	must	AUX
ejpam-4962	25	15	be	be	AUX
ejpam-4962	25	16	traversed	traverse	VERB
ejpam-4962	25	17	to	to	PART
ejpam-4962	25	18	travel	travel	VERB
ejpam-4962	25	19	from	from	ADP
ejpam-4962	25	20	one	one	NUM
ejpam-4962	25	21	vertex	vertex	NOUN
ejpam-4962	25	22	to	to	ADP
ejpam-4962	25	23	another	another	PRON
ejpam-4962	25	24	.	.	PUNCT
ejpam-4962	26	1	geodetic	geodetic	ADJ
ejpam-4962	26	2	sets	set	NOUN
ejpam-4962	26	3	and	and	CCONJ
ejpam-4962	26	4	geodetic	geodetic	ADJ
ejpam-4962	26	5	domination	domination	NOUN
ejpam-4962	26	6	have	have	VERB
ejpam-4962	26	7	plenty	plenty	NOUN
ejpam-4962	26	8	of	of	ADP
ejpam-4962	26	9	applications	application	NOUN
ejpam-4962	26	10	and	and	CCONJ
ejpam-4962	26	11	researchers	researcher	NOUN
ejpam-4962	26	12	continue	continue	VERB
ejpam-4962	26	13	to	to	PART
ejpam-4962	26	14	investigate	investigate	VERB
ejpam-4962	26	15	various	various	ADJ
ejpam-4962	26	16	concepts	concept	NOUN
ejpam-4962	26	17	involving	involve	VERB
ejpam-4962	26	18	them	they	PRON
ejpam-4962	26	19	.	.	PUNCT
ejpam-4962	27	1	some	some	DET
ejpam-4962	27	2	studies	study	NOUN
ejpam-4962	27	3	on	on	ADP
ejpam-4962	27	4	geodetic	geodetic	ADJ
ejpam-4962	27	5	sets	set	NOUN
ejpam-4962	27	6	and	and	CCONJ
ejpam-4962	27	7	related	related	ADJ
ejpam-4962	27	8	concepts	concept	NOUN
ejpam-4962	27	9	can	can	AUX
ejpam-4962	27	10	be	be	AUX
ejpam-4962	27	11	found	find	VERB
ejpam-4962	27	12	in	in	ADP
ejpam-4962	27	13	[	[	X
ejpam-4962	27	14	7	7	NUM
ejpam-4962	27	15	]	]	PUNCT
ejpam-4962	27	16	,	,	PUNCT
ejpam-4962	27	17	[	[	X
ejpam-4962	27	18	8	8	NUM
ejpam-4962	27	19	]	]	PUNCT
ejpam-4962	27	20	,	,	PUNCT
ejpam-4962	27	21	[	[	X
ejpam-4962	27	22	11	11	NUM
ejpam-4962	27	23	]	]	PUNCT
ejpam-4962	27	24	,	,	PUNCT
ejpam-4962	27	25	[	[	X
ejpam-4962	27	26	13	13	NUM
ejpam-4962	27	27	]	]	PUNCT
ejpam-4962	27	28	,	,	PUNCT
ejpam-4962	27	29	[	[	X
ejpam-4962	27	30	14	14	NUM
ejpam-4962	27	31	]	]	PUNCT
ejpam-4962	27	32	,	,	PUNCT
ejpam-4962	27	33	[	[	X
ejpam-4962	27	34	18	18	NUM
ejpam-4962	27	35	]	]	PUNCT
ejpam-4962	27	36	,	,	PUNCT
ejpam-4962	27	37	[	[	X
ejpam-4962	27	38	21	21	NUM
ejpam-4962	27	39	]	]	PUNCT
ejpam-4962	27	40	,	,	PUNCT
ejpam-4962	27	41	[	[	X
ejpam-4962	27	42	23	23	NUM
ejpam-4962	27	43	]	]	PUNCT
ejpam-4962	27	44	and	and	CCONJ
ejpam-4962	27	45	[	[	X
ejpam-4962	27	46	24	24	NUM
ejpam-4962	27	47	]	]	PUNCT
ejpam-4962	27	48	.	.	PUNCT
ejpam-4962	28	1	in	in	ADP
ejpam-4962	28	2	this	this	DET
ejpam-4962	28	3	study	study	NOUN
ejpam-4962	28	4	,	,	PUNCT
ejpam-4962	28	5	we	we	PRON
ejpam-4962	28	6	introduce	introduce	VERB
ejpam-4962	28	7	the	the	DET
ejpam-4962	28	8	concept	concept	NOUN
ejpam-4962	28	9	of	of	ADP
ejpam-4962	28	10	geodetic	geodetic	ADJ
ejpam-4962	28	11	roman	roman	ADJ
ejpam-4962	28	12	domination	domination	NOUN
ejpam-4962	28	13	,	,	PUNCT
ejpam-4962	28	14	a	a	DET
ejpam-4962	28	15	concept	concept	NOUN
ejpam-4962	28	16	which	which	PRON
ejpam-4962	28	17	combines	combine	VERB
ejpam-4962	28	18	the	the	DET
ejpam-4962	28	19	concepts	concept	NOUN
ejpam-4962	28	20	of	of	ADP
ejpam-4962	28	21	geodetic	geodetic	ADJ
ejpam-4962	28	22	set	set	NOUN
ejpam-4962	28	23	and	and	CCONJ
ejpam-4962	28	24	roman	roman	ADJ
ejpam-4962	28	25	domination	domination	NOUN
ejpam-4962	28	26	.	.	PUNCT
ejpam-4962	29	1	geodetic	geodetic	ADJ
ejpam-4962	29	2	roman	roman	ADJ
ejpam-4962	29	3	domination	domination	NOUN
ejpam-4962	29	4	as	as	ADP
ejpam-4962	29	5	a	a	DET
ejpam-4962	29	6	protection	protection	NOUN
ejpam-4962	29	7	strategy	strategy	NOUN
ejpam-4962	29	8	(	(	PUNCT
ejpam-4962	29	9	involving	involve	VERB
ejpam-4962	29	10	of	of	ADP
ejpam-4962	29	11	field	field	NOUN
ejpam-4962	29	12	armies	army	NOUN
ejpam-4962	29	13	)	)	PUNCT
ejpam-4962	29	14	guarantees	guarantee	NOUN
ejpam-4962	29	15	,	,	PUNCT
ejpam-4962	29	16	in	in	ADP
ejpam-4962	29	17	addition	addition	NOUN
ejpam-4962	29	18	to	to	ADP
ejpam-4962	29	19	what	what	PRON
ejpam-4962	29	20	the	the	DET
ejpam-4962	29	21	roman	roman	ADJ
ejpam-4962	29	22	domination	domination	NOUN
ejpam-4962	29	23	requires	require	VERB
ejpam-4962	29	24	,	,	PUNCT
ejpam-4962	29	25	that	that	SCONJ
ejpam-4962	29	26	every	every	DET
ejpam-4962	29	27	unsecured	unsecured	ADJ
ejpam-4962	29	28	location	location	NOUN
ejpam-4962	29	29	lies	lie	VERB
ejpam-4962	29	30	along	along	ADP
ejpam-4962	29	31	a	a	DET
ejpam-4962	29	32	shortest	short	ADJ
ejpam-4962	29	33	path	path	NOUN
ejpam-4962	29	34	between	between	ADP
ejpam-4962	29	35	two	two	NUM
ejpam-4962	29	36	secured	secured	ADJ
ejpam-4962	29	37	locations	location	NOUN
ejpam-4962	29	38	.	.	PUNCT
ejpam-4962	30	1	2	2	X
ejpam-4962	30	2	.	.	NOUN
ejpam-4962	30	3	terminologies	terminology	NOUN
ejpam-4962	30	4	and	and	CCONJ
ejpam-4962	30	5	notations	notation	NOUN
ejpam-4962	30	6	let	let	VERB
ejpam-4962	30	7	g	g	PRON
ejpam-4962	30	8	be	be	AUX
ejpam-4962	30	9	a	a	DET
ejpam-4962	30	10	connected	connected	ADJ
ejpam-4962	30	11	graph	graph	NOUN
ejpam-4962	30	12	.	.	PUNCT
ejpam-4962	31	1	for	for	ADP
ejpam-4962	31	2	vertices	vertex	NOUN
ejpam-4962	31	3	u	u	NOUN
ejpam-4962	31	4	and	and	CCONJ
ejpam-4962	31	5	v	v	NOUN
ejpam-4962	31	6	in	in	ADP
ejpam-4962	31	7	g	g	PROPN
ejpam-4962	31	8	,	,	PUNCT
ejpam-4962	31	9	a	a	DET
ejpam-4962	31	10	u	u	NOUN
ejpam-4962	31	11	-	-	NOUN
ejpam-4962	31	12	v	v	ADJ
ejpam-4962	31	13	geodesic	geodesic	NOUN
ejpam-4962	31	14	is	be	AUX
ejpam-4962	31	15	any	any	DET
ejpam-4962	31	16	shortest	short	ADJ
ejpam-4962	31	17	path	path	NOUN
ejpam-4962	31	18	in	in	ADP
ejpam-4962	31	19	g	g	NOUN
ejpam-4962	31	20	joining	join	VERB
ejpam-4962	31	21	u	u	NOUN
ejpam-4962	31	22	and	and	CCONJ
ejpam-4962	31	23	v.	v.	ADP
ejpam-4962	31	24	the	the	DET
ejpam-4962	31	25	length	length	NOUN
ejpam-4962	31	26	of	of	ADP
ejpam-4962	31	27	a	a	DET
ejpam-4962	31	28	u	u	NOUN
ejpam-4962	31	29	-	-	NOUN
ejpam-4962	31	30	v	v	ADJ
ejpam-4962	31	31	geodesic	geodesic	NOUN
ejpam-4962	31	32	is	be	AUX
ejpam-4962	31	33	called	call	VERB
ejpam-4962	31	34	the	the	DET
ejpam-4962	31	35	distance	distance	NOUN
ejpam-4962	31	36	dg(u	dg(u	X
ejpam-4962	31	37	,	,	PUNCT
ejpam-4962	31	38	v	v	NOUN
ejpam-4962	31	39	)	)	PUNCT
ejpam-4962	31	40	between	between	ADP
ejpam-4962	31	41	u	u	PROPN
ejpam-4962	31	42	and	and	CCONJ
ejpam-4962	31	43	v.	v.	NOUN
ejpam-4962	31	44	for	for	ADP
ejpam-4962	31	45	every	every	DET
ejpam-4962	31	46	two	two	NUM
ejpam-4962	31	47	vertices	vertex	NOUN
ejpam-4962	31	48	u	u	NOUN
ejpam-4962	31	49	and	and	CCONJ
ejpam-4962	31	50	v	v	NOUN
ejpam-4962	31	51	of	of	ADP
ejpam-4962	31	52	g	g	NOUN
ejpam-4962	31	53	,	,	PUNCT
ejpam-4962	31	54	the	the	DET
ejpam-4962	31	55	symbol	symbol	NOUN
ejpam-4962	31	56	ig[u	ig[u	PROPN
ejpam-4962	31	57	,	,	PUNCT
ejpam-4962	31	58	v	v	NOUN
ejpam-4962	31	59	]	]	PUNCT
ejpam-4962	31	60	is	be	AUX
ejpam-4962	31	61	used	use	VERB
ejpam-4962	31	62	to	to	PART
ejpam-4962	31	63	denote	denote	VERB
ejpam-4962	31	64	the	the	DET
ejpam-4962	31	65	set	set	NOUN
ejpam-4962	31	66	consisting	consisting	NOUN
ejpam-4962	31	67	of	of	ADP
ejpam-4962	31	68	u	u	NOUN
ejpam-4962	31	69	and	and	CCONJ
ejpam-4962	31	70	v	v	NOUN
ejpam-4962	31	71	and	and	CCONJ
ejpam-4962	31	72	the	the	DET
ejpam-4962	31	73	vertices	vertex	NOUN
ejpam-4962	31	74	lying	lie	VERB
ejpam-4962	31	75	on	on	ADP
ejpam-4962	31	76	any	any	PRON
ejpam-4962	31	77	of	of	ADP
ejpam-4962	31	78	the	the	DET
ejpam-4962	31	79	u	u	NOUN
ejpam-4962	31	80	-	-	NOUN
ejpam-4962	31	81	v	v	ADJ
ejpam-4962	31	82	geodesics	geodesic	NOUN
ejpam-4962	31	83	.	.	PUNCT
ejpam-4962	32	1	the	the	DET
ejpam-4962	32	2	set	set	NOUN
ejpam-4962	32	3	ig(u	ig(u	NOUN
ejpam-4962	32	4	,	,	PUNCT
ejpam-4962	32	5	v	v	NOUN
ejpam-4962	32	6	)	)	PUNCT
ejpam-4962	32	7	is	be	AUX
ejpam-4962	32	8	the	the	DET
ejpam-4962	32	9	set	set	NOUN
ejpam-4962	32	10	ig[u	ig[u	PROPN
ejpam-4962	32	11	,	,	PUNCT
ejpam-4962	32	12	v	v	NOUN
ejpam-4962	32	13	]	]	PUNCT
ejpam-4962	32	14	\	\	PUNCT
ejpam-4962	32	15	{	{	PUNCT
ejpam-4962	32	16	u	u	NOUN
ejpam-4962	32	17	,	,	PUNCT
ejpam-4962	32	18	v	v	NOUN
ejpam-4962	32	19	}	}	PUNCT
ejpam-4962	32	20	.	.	PUNCT
ejpam-4962	33	1	the	the	DET
ejpam-4962	33	2	geodetic	geodetic	ADJ
ejpam-4962	33	3	closure	closure	NOUN
ejpam-4962	33	4	of	of	ADP
ejpam-4962	33	5	a	a	DET
ejpam-4962	33	6	subset	subset	NOUN
ejpam-4962	33	7	s	s	NOUN
ejpam-4962	33	8	of	of	ADP
ejpam-4962	33	9	g	g	PROPN
ejpam-4962	33	10	is	be	AUX
ejpam-4962	33	11	the	the	DET
ejpam-4962	33	12	set	set	VERB
ejpam-4962	33	13	ig[s	ig[	NOUN
ejpam-4962	33	14	]	]	X
ejpam-4962	33	15	=	=	PUNCT
ejpam-4962	33	16	∪u	∪u	NUM
ejpam-4962	33	17	,	,	PUNCT
ejpam-4962	33	18	v∈sig[u	v∈sig[u	NUM
ejpam-4962	33	19	,	,	PUNCT
ejpam-4962	33	20	v	v	NOUN
ejpam-4962	33	21	]	]	PUNCT
ejpam-4962	33	22	.	.	PUNCT
ejpam-4962	34	1	also	also	ADV
ejpam-4962	34	2	,	,	PUNCT
ejpam-4962	34	3	ig(s	ig(s	X
ejpam-4962	34	4	)	)	PUNCT
ejpam-4962	34	5	=	=	PUNCT
ejpam-4962	34	6	∪u	∪u	ADP
ejpam-4962	34	7	,	,	PUNCT
ejpam-4962	34	8	v∈sig(u	v∈sig(u	PROPN
ejpam-4962	34	9	,	,	PUNCT
ejpam-4962	34	10	v	v	NOUN
ejpam-4962	34	11	)	)	PUNCT
ejpam-4962	34	12	.	.	PUNCT
ejpam-4962	35	1	the	the	DET
ejpam-4962	35	2	open	open	ADJ
ejpam-4962	35	3	neighborhood	neighborhood	NOUN
ejpam-4962	35	4	of	of	ADP
ejpam-4962	35	5	u	u	PROPN
ejpam-4962	35	6	∈	∈	PROPN
ejpam-4962	35	7	v	v	ADP
ejpam-4962	35	8	(	(	PUNCT
ejpam-4962	35	9	g	g	NOUN
ejpam-4962	35	10	)	)	PUNCT
ejpam-4962	35	11	is	be	AUX
ejpam-4962	35	12	given	give	VERB
ejpam-4962	35	13	by	by	ADP
ejpam-4962	35	14	ng(u	ng(u	NOUN
ejpam-4962	35	15	)	)	PUNCT
ejpam-4962	35	16	=	=	PRON
ejpam-4962	35	17	{	{	PUNCT
ejpam-4962	35	18	v	v	NUM
ejpam-4962	35	19	∈	∈	NOUN
ejpam-4962	35	20	v	v	NOUN
ejpam-4962	35	21	(	(	PUNCT
ejpam-4962	35	22	g	g	NOUN
ejpam-4962	35	23	)	)	PUNCT
ejpam-4962	35	24	:	:	PUNCT
ejpam-4962	35	25	vu	vu	PROPN
ejpam-4962	35	26	∈	∈	PROPN
ejpam-4962	35	27	e(g	e(g	PROPN
ejpam-4962	35	28	)	)	PUNCT
ejpam-4962	35	29	.	.	PUNCT
ejpam-4962	36	1	the	the	DET
ejpam-4962	36	2	closed	closed	ADJ
ejpam-4962	36	3	neighborhood	neighborhood	NOUN
ejpam-4962	36	4	of	of	ADP
ejpam-4962	36	5	u	u	NOUN
ejpam-4962	36	6	is	be	AUX
ejpam-4962	36	7	the	the	DET
ejpam-4962	36	8	set	set	NOUN
ejpam-4962	36	9	ng[u	ng[u	PROPN
ejpam-4962	36	10	]	]	X
ejpam-4962	36	11	=	=	SYM
ejpam-4962	36	12	ng(u	ng(u	PROPN
ejpam-4962	36	13	)	)	PUNCT
ejpam-4962	36	14	∪	∪	NOUN
ejpam-4962	36	15	{	{	PUNCT
ejpam-4962	36	16	u	u	NOUN
ejpam-4962	36	17	}	}	PUNCT
ejpam-4962	36	18	.	.	PUNCT
ejpam-4962	37	1	if	if	SCONJ
ejpam-4962	37	2	x	x	PROPN
ejpam-4962	37	3	⊆	⊆	NUM
ejpam-4962	37	4	v	v	X
ejpam-4962	37	5	(	(	PUNCT
ejpam-4962	37	6	g	g	NOUN
ejpam-4962	37	7	)	)	PUNCT
ejpam-4962	37	8	,	,	PUNCT
ejpam-4962	37	9	the	the	DET
ejpam-4962	37	10	open	open	ADJ
ejpam-4962	37	11	neighborhood	neighborhood	NOUN
ejpam-4962	37	12	of	of	ADP
ejpam-4962	37	13	x	x	SYM
ejpam-4962	37	14	is	be	AUX
ejpam-4962	37	15	the	the	DET
ejpam-4962	37	16	set	set	NOUN
ejpam-4962	37	17	ng(x	ng(x	NUM
ejpam-4962	37	18	)	)	PUNCT
ejpam-4962	37	19	=	=	SYM
ejpam-4962	37	20	∪u∈xng(u	∪u∈xng(u	PROPN
ejpam-4962	37	21	)	)	PUNCT
ejpam-4962	37	22	.	.	PUNCT
ejpam-4962	38	1	the	the	DET
ejpam-4962	38	2	closed	closed	ADJ
ejpam-4962	38	3	neighborhood	neighborhood	NOUN
ejpam-4962	38	4	of	of	ADP
ejpam-4962	38	5	x	x	SYM
ejpam-4962	38	6	is	be	AUX
ejpam-4962	38	7	the	the	DET
ejpam-4962	38	8	set	set	NOUN
ejpam-4962	38	9	ng[x	ng[x	PROPN
ejpam-4962	38	10	]	]	X
ejpam-4962	38	11	=	=	SYM
ejpam-4962	38	12	ng(x)∪x	ng(x)∪x	PROPN
ejpam-4962	38	13	.	.	PUNCT
ejpam-4962	39	1	the	the	DET
ejpam-4962	39	2	degree	degree	NOUN
ejpam-4962	39	3	of	of	ADP
ejpam-4962	39	4	a	a	DET
ejpam-4962	39	5	vertex	vertex	NOUN
ejpam-4962	39	6	v	v	NOUN
ejpam-4962	39	7	in	in	ADP
ejpam-4962	39	8	g	g	PROPN
ejpam-4962	39	9	is	be	AUX
ejpam-4962	39	10	given	give	VERB
ejpam-4962	39	11	by	by	ADP
ejpam-4962	39	12	degg(v	degg(v	PROPN
ejpam-4962	39	13	)	)	PUNCT
ejpam-4962	39	14	=	=	NOUN
ejpam-4962	39	15	|ng(u)|	|ng(u)|	NOUN
ejpam-4962	39	16	.	.	PUNCT
ejpam-4962	40	1	a	a	DET
ejpam-4962	40	2	vertex	vertex	NOUN
ejpam-4962	40	3	of	of	ADP
ejpam-4962	40	4	a	a	DET
ejpam-4962	40	5	connected	connected	ADJ
ejpam-4962	40	6	graph	graph	NOUN
ejpam-4962	40	7	g	g	PROPN
ejpam-4962	40	8	is	be	AUX
ejpam-4962	40	9	an	an	DET
ejpam-4962	40	10	extreme	extreme	ADJ
ejpam-4962	40	11	or	or	CCONJ
ejpam-4962	40	12	simplicial	simplicial	ADJ
ejpam-4962	40	13	vertex	vertex	NOUN
ejpam-4962	40	14	if	if	SCONJ
ejpam-4962	40	15	its	its	PRON
ejpam-4962	40	16	open	open	ADJ
ejpam-4962	40	17	neighborhood	neighborhood	NOUN
ejpam-4962	40	18	induces	induce	VERB
ejpam-4962	40	19	a	a	DET
ejpam-4962	40	20	complete	complete	ADJ
ejpam-4962	40	21	subgraph	subgraph	NOUN
ejpam-4962	40	22	of	of	ADP
ejpam-4962	40	23	g.	g.	PROPN
ejpam-4962	40	24	the	the	DET
ejpam-4962	40	25	set	set	NOUN
ejpam-4962	40	26	of	of	ADP
ejpam-4962	40	27	extreme	extreme	ADJ
ejpam-4962	40	28	vertices	vertex	NOUN
ejpam-4962	40	29	of	of	ADP
ejpam-4962	40	30	g	g	PROPN
ejpam-4962	40	31	is	be	AUX
ejpam-4962	40	32	denoted	denote	VERB
ejpam-4962	40	33	by	by	ADP
ejpam-4962	40	34	ext(g	ext(g	PROPN
ejpam-4962	40	35	)	)	PUNCT
ejpam-4962	40	36	.	.	PUNCT
ejpam-4962	41	1	a	a	DET
ejpam-4962	41	2	set	set	NOUN
ejpam-4962	41	3	s	s	NOUN
ejpam-4962	41	4	⊆	⊆	NUM
ejpam-4962	41	5	v	v	NOUN
ejpam-4962	41	6	(	(	PUNCT
ejpam-4962	41	7	g	g	NOUN
ejpam-4962	41	8	)	)	PUNCT
ejpam-4962	41	9	is	be	AUX
ejpam-4962	41	10	said	say	VERB
ejpam-4962	41	11	to	to	PART
ejpam-4962	41	12	be	be	AUX
ejpam-4962	41	13	a	a	DET
ejpam-4962	41	14	dominating	dominating	NOUN
ejpam-4962	41	15	set	set	NOUN
ejpam-4962	41	16	of	of	ADP
ejpam-4962	41	17	a	a	DET
ejpam-4962	41	18	graph	graph	NOUN
ejpam-4962	41	19	g	g	NOUN
ejpam-4962	41	20	if	if	SCONJ
ejpam-4962	41	21	for	for	ADP
ejpam-4962	41	22	every	every	DET
ejpam-4962	41	23	vertex	vertex	NOUN
ejpam-4962	41	24	v	v	ADP
ejpam-4962	41	25	∈	∈	NOUN
ejpam-4962	41	26	v	v	NOUN
ejpam-4962	41	27	(	(	PUNCT
ejpam-4962	41	28	g	g	NOUN
ejpam-4962	41	29	)	)	PUNCT
ejpam-4962	41	30	\	\	PROPN
ejpam-4962	42	1	s	s	VERB
ejpam-4962	42	2	there	there	PRON
ejpam-4962	42	3	exists	exist	VERB
ejpam-4962	42	4	an	an	DET
ejpam-4962	42	5	element	element	NOUN
ejpam-4962	42	6	of	of	ADP
ejpam-4962	42	7	w	w	PROPN
ejpam-4962	42	8	∈	∈	PROPN
ejpam-4962	42	9	s	s	VERB
ejpam-4962	42	10	such	such	ADJ
ejpam-4962	42	11	that	that	SCONJ
ejpam-4962	42	12	vw	vw	PROPN
ejpam-4962	42	13	∈	∈	PROPN
ejpam-4962	42	14	e(g	e(g	PROPN
ejpam-4962	42	15	)	)	PUNCT
ejpam-4962	42	16	,	,	PUNCT
ejpam-4962	42	17	i.e.	i.e.	X
ejpam-4962	42	18	,	,	PUNCT
ejpam-4962	42	19	n	n	X
ejpam-4962	42	20	[	[	X
ejpam-4962	42	21	s	s	X
ejpam-4962	42	22	]	]	X
ejpam-4962	42	23	=	=	SYM
ejpam-4962	42	24	v	v	NOUN
ejpam-4962	42	25	(	(	PUNCT
ejpam-4962	42	26	g	g	NOUN
ejpam-4962	42	27	)	)	PUNCT
ejpam-4962	42	28	.	.	PUNCT
ejpam-4962	43	1	the	the	DET
ejpam-4962	43	2	smallest	small	ADJ
ejpam-4962	43	3	cardinality	cardinality	NOUN
ejpam-4962	43	4	of	of	ADP
ejpam-4962	43	5	a	a	DET
ejpam-4962	43	6	dominating	dominating	NOUN
ejpam-4962	43	7	set	set	NOUN
ejpam-4962	43	8	in	in	ADP
ejpam-4962	43	9	g	g	PROPN
ejpam-4962	43	10	is	be	AUX
ejpam-4962	43	11	called	call	VERB
ejpam-4962	43	12	the	the	DET
ejpam-4962	43	13	domination	domination	NOUN
ejpam-4962	43	14	number	number	NOUN
ejpam-4962	43	15	of	of	ADP
ejpam-4962	43	16	g	g	NOUN
ejpam-4962	43	17	and	and	CCONJ
ejpam-4962	43	18	is	be	AUX
ejpam-4962	43	19	denoted	denote	VERB
ejpam-4962	43	20	by	by	ADP
ejpam-4962	43	21	γ(g	γ(g	PROPN
ejpam-4962	43	22	)	)	PUNCT
ejpam-4962	43	23	.	.	PUNCT
ejpam-4962	44	1	any	any	DET
ejpam-4962	44	2	dominating	dominating	NOUN
ejpam-4962	44	3	set	set	VERB
ejpam-4962	44	4	in	in	ADP
ejpam-4962	44	5	g	g	PROPN
ejpam-4962	44	6	with	with	ADP
ejpam-4962	44	7	cardinality	cardinality	PROPN
ejpam-4962	44	8	γ(g	γ(g	PROPN
ejpam-4962	44	9	)	)	PUNCT
ejpam-4962	44	10	is	be	AUX
ejpam-4962	44	11	called	call	VERB
ejpam-4962	44	12	a	a	DET
ejpam-4962	44	13	γ	γ	NOUN
ejpam-4962	44	14	-	-	PUNCT
ejpam-4962	44	15	set	set	NOUN
ejpam-4962	44	16	in	in	ADP
ejpam-4962	44	17	g.	g.	PROPN
ejpam-4962	44	18	a	a	DET
ejpam-4962	44	19	set	set	NOUN
ejpam-4962	44	20	s	s	NOUN
ejpam-4962	44	21	of	of	ADP
ejpam-4962	44	22	vertices	vertex	NOUN
ejpam-4962	44	23	in	in	ADP
ejpam-4962	44	24	a	a	DET
ejpam-4962	44	25	graph	graph	NOUN
ejpam-4962	44	26	g	g	NOUN
ejpam-4962	44	27	is	be	AUX
ejpam-4962	44	28	a	a	DET
ejpam-4962	44	29	geodetic	geodetic	ADJ
ejpam-4962	44	30	set	set	NOUN
ejpam-4962	44	31	if	if	SCONJ
ejpam-4962	44	32	ig[s	ig[	NOUN
ejpam-4962	44	33	]	]	X
ejpam-4962	44	34	=	=	SYM
ejpam-4962	44	35	v	v	X
ejpam-4962	44	36	(	(	PUNCT
ejpam-4962	44	37	g	g	NOUN
ejpam-4962	44	38	)	)	PUNCT
ejpam-4962	44	39	.	.	PUNCT
ejpam-4962	45	1	the	the	DET
ejpam-4962	45	2	minimum	minimum	ADJ
ejpam-4962	45	3	cardinality	cardinality	NOUN
ejpam-4962	45	4	of	of	ADP
ejpam-4962	45	5	a	a	DET
ejpam-4962	45	6	geodetic	geodetic	ADJ
ejpam-4962	45	7	set	set	NOUN
ejpam-4962	45	8	in	in	ADP
ejpam-4962	45	9	g	g	NOUN
ejpam-4962	45	10	,	,	PUNCT
ejpam-4962	45	11	denoted	denote	VERB
ejpam-4962	45	12	by	by	ADP
ejpam-4962	45	13	g(g	g(g	PROPN
ejpam-4962	45	14	)	)	PUNCT
ejpam-4962	45	15	,	,	PUNCT
ejpam-4962	45	16	is	be	AUX
ejpam-4962	45	17	the	the	DET
ejpam-4962	45	18	geodetic	geodetic	ADJ
ejpam-4962	45	19	number	number	NOUN
ejpam-4962	45	20	of	of	ADP
ejpam-4962	45	21	g.	g.	PROPN
ejpam-4962	45	22	a	a	DET
ejpam-4962	45	23	set	set	NOUN
ejpam-4962	45	24	s	s	PROPN
ejpam-4962	45	25	⊆	⊆	NUM
ejpam-4962	45	26	v	v	NOUN
ejpam-4962	45	27	(	(	PUNCT
ejpam-4962	45	28	g	g	NOUN
ejpam-4962	45	29	)	)	PUNCT
ejpam-4962	45	30	is	be	AUX
ejpam-4962	45	31	called	call	VERB
ejpam-4962	45	32	a	a	DET
ejpam-4962	45	33	geodetic	geodetic	ADJ
ejpam-4962	45	34	dominating	dominating	NOUN
ejpam-4962	45	35	set	set	NOUN
ejpam-4962	45	36	if	if	SCONJ
ejpam-4962	45	37	s	s	VERB
ejpam-4962	45	38	is	be	AUX
ejpam-4962	45	39	both	both	PRON
ejpam-4962	45	40	a	a	DET
ejpam-4962	45	41	geodetic	geodetic	ADJ
ejpam-4962	45	42	and	and	CCONJ
ejpam-4962	45	43	a	a	DET
ejpam-4962	45	44	dominating	dominating	NOUN
ejpam-4962	45	45	set	set	NOUN
ejpam-4962	45	46	.	.	PUNCT
ejpam-4962	46	1	the	the	DET
ejpam-4962	46	2	minimum	minimum	ADJ
ejpam-4962	46	3	cardinality	cardinality	NOUN
ejpam-4962	46	4	of	of	ADP
ejpam-4962	46	5	a	a	DET
ejpam-4962	46	6	geodetic	geodetic	ADJ
ejpam-4962	46	7	dominating	dominating	NOUN
ejpam-4962	46	8	set	set	VERB
ejpam-4962	46	9	in	in	ADP
ejpam-4962	46	10	g	g	NOUN
ejpam-4962	46	11	,	,	PUNCT
ejpam-4962	46	12	denoted	denote	VERB
ejpam-4962	46	13	by	by	ADP
ejpam-4962	46	14	γg(g	γg(g	NOUN
ejpam-4962	46	15	)	)	PUNCT
ejpam-4962	46	16	,	,	PUNCT
ejpam-4962	46	17	is	be	AUX
ejpam-4962	46	18	the	the	DET
ejpam-4962	46	19	geodetic	geodetic	ADJ
ejpam-4962	46	20	domination	domination	NOUN
ejpam-4962	46	21	number	number	NOUN
ejpam-4962	46	22	of	of	ADP
ejpam-4962	46	23	g.	g.	PROPN
ejpam-4962	46	24	any	any	DET
ejpam-4962	46	25	geodetic	geodetic	ADJ
ejpam-4962	46	26	dominating	dominating	NOUN
ejpam-4962	46	27	set	set	VERB
ejpam-4962	46	28	in	in	ADP
ejpam-4962	46	29	g	g	PROPN
ejpam-4962	46	30	with	with	ADP
ejpam-4962	46	31	cardinality	cardinality	NOUN
ejpam-4962	46	32	γg(g	γg(g	CCONJ
ejpam-4962	46	33	)	)	PUNCT
ejpam-4962	46	34	is	be	AUX
ejpam-4962	46	35	called	call	VERB
ejpam-4962	46	36	a	a	DET
ejpam-4962	46	37	γg	γg	ADV
ejpam-4962	46	38	-	-	PUNCT
ejpam-4962	46	39	set	set	NOUN
ejpam-4962	46	40	in	in	ADP
ejpam-4962	46	41	g.	g.	PROPN
ejpam-4962	46	42	a	a	DET
ejpam-4962	46	43	set	set	NOUN
ejpam-4962	46	44	s	s	PROPN
ejpam-4962	46	45	⊆	⊆	NUM
ejpam-4962	46	46	v	v	NOUN
ejpam-4962	46	47	(	(	PUNCT
ejpam-4962	46	48	g	g	NOUN
ejpam-4962	46	49	)	)	PUNCT
ejpam-4962	46	50	of	of	ADP
ejpam-4962	46	51	a	a	DET
ejpam-4962	46	52	graph	graph	NOUN
ejpam-4962	46	53	g	g	NOUN
ejpam-4962	46	54	is	be	AUX
ejpam-4962	46	55	2	2	NUM
ejpam-4962	46	56	-	-	PUNCT
ejpam-4962	46	57	path	path	NOUN
ejpam-4962	46	58	closure	closure	NOUN
ejpam-4962	46	59	absorbing	absorb	VERB
ejpam-4962	46	60	if	if	SCONJ
ejpam-4962	46	61	for	for	ADP
ejpam-4962	46	62	each	each	DET
ejpam-4962	46	63	x	x	SYM
ejpam-4962	46	64	∈	∈	PROPN
ejpam-4962	46	65	v	v	ADP
ejpam-4962	46	66	(	(	PUNCT
ejpam-4962	46	67	g	g	NOUN
ejpam-4962	46	68	)	)	PUNCT
ejpam-4962	46	69	\	\	PROPN
ejpam-4962	46	70	s	s	VERB
ejpam-4962	46	71	there	there	PRON
ejpam-4962	46	72	exist	exist	VERB
ejpam-4962	46	73	u	u	NOUN
ejpam-4962	46	74	,	,	PUNCT
ejpam-4962	46	75	v	v	PROPN
ejpam-4962	46	76	∈	∈	NOUN
ejpam-4962	46	77	s	s	VERB
ejpam-4962	46	78	such	such	ADJ
ejpam-4962	46	79	that	that	DET
ejpam-4962	46	80	dg(u	dg(u	ADJ
ejpam-4962	46	81	,	,	PUNCT
ejpam-4962	46	82	v	v	NOUN
ejpam-4962	46	83	)	)	PUNCT
ejpam-4962	46	84	=	=	SYM
ejpam-4962	46	85	2	2	NUM
ejpam-4962	46	86	and	and	CCONJ
ejpam-4962	46	87	x	x	PROPN
ejpam-4962	46	88	∈	∈	PROPN
ejpam-4962	46	89	ig(u	ig(u	NOUN
ejpam-4962	46	90	,	,	PUNCT
ejpam-4962	46	91	v	v	NOUN
ejpam-4962	46	92	)	)	PUNCT
ejpam-4962	46	93	.	.	PUNCT
ejpam-4962	47	1	the	the	DET
ejpam-4962	47	2	minimum	minimum	ADJ
ejpam-4962	47	3	cardinality	cardinality	NOUN
ejpam-4962	47	4	of	of	ADP
ejpam-4962	47	5	a	a	DET
ejpam-4962	47	6	2	2	NUM
ejpam-4962	47	7	-	-	PUNCT
ejpam-4962	47	8	path	path	NOUN
ejpam-4962	47	9	closure	closure	NOUN
ejpam-4962	47	10	absorbing	absorb	VERB
ejpam-4962	47	11	set	set	NOUN
ejpam-4962	47	12	in	in	ADP
ejpam-4962	47	13	g	g	PROPN
ejpam-4962	47	14	is	be	AUX
ejpam-4962	47	15	denoted	denote	VERB
ejpam-4962	47	16	by	by	ADP
ejpam-4962	47	17	ρ2(g	ρ2(g	NOUN
ejpam-4962	47	18	)	)	PUNCT
ejpam-4962	47	19	.	.	PUNCT
ejpam-4962	48	1	any	any	DET
ejpam-4962	48	2	2	2	NUM
ejpam-4962	48	3	-	-	PUNCT
ejpam-4962	48	4	path	path	NOUN
ejpam-4962	48	5	closure	closure	NOUN
ejpam-4962	48	6	absorbing	absorb	VERB
ejpam-4962	48	7	set	set	NOUN
ejpam-4962	48	8	in	in	ADP
ejpam-4962	48	9	g	g	NOUN
ejpam-4962	48	10	with	with	ADP
ejpam-4962	48	11	cardinality	cardinality	PROPN
ejpam-4962	48	12	ρ2(g	ρ2(g	NUM
ejpam-4962	48	13	)	)	PUNCT
ejpam-4962	48	14	is	be	AUX
ejpam-4962	48	15	called	call	VERB
ejpam-4962	48	16	a	a	DET
ejpam-4962	48	17	ρ2	ρ2	NOUN
ejpam-4962	48	18	-	-	PUNCT
ejpam-4962	48	19	set	set	NOUN
ejpam-4962	48	20	.	.	PUNCT
ejpam-4962	49	1	a	a	DET
ejpam-4962	49	2	function	function	NOUN
ejpam-4962	49	3	f	f	NOUN
ejpam-4962	49	4	:	:	PUNCT
ejpam-4962	49	5	v	v	X
ejpam-4962	49	6	(	(	PUNCT
ejpam-4962	49	7	g	g	NOUN
ejpam-4962	49	8	)	)	PUNCT
ejpam-4962	49	9	→	→	SYM
ejpam-4962	49	10	{	{	PUNCT
ejpam-4962	49	11	0	0	NUM
ejpam-4962	49	12	,	,	PUNCT
ejpam-4962	49	13	1	1	NUM
ejpam-4962	49	14	,	,	PUNCT
ejpam-4962	49	15	2	2	NUM
ejpam-4962	49	16	}	}	PUNCT
ejpam-4962	49	17	is	be	AUX
ejpam-4962	49	18	a	a	DET
ejpam-4962	49	19	roman	roman	ADJ
ejpam-4962	49	20	dominating	dominating	NOUN
ejpam-4962	49	21	function	function	NOUN
ejpam-4962	49	22	(	(	PUNCT
ejpam-4962	49	23	or	or	CCONJ
ejpam-4962	49	24	just	just	ADV
ejpam-4962	49	25	rdf	rdf	VERB
ejpam-4962	49	26	)	)	PUNCT
ejpam-4962	49	27	if	if	SCONJ
ejpam-4962	49	28	r.	r.	PROPN
ejpam-4962	49	29	fortosa	fortosa	PROPN
ejpam-4962	49	30	,	,	PUNCT
ejpam-4962	49	31	s.	s.	PROPN
ejpam-4962	49	32	canoy	canoy	PROPN
ejpam-4962	49	33	jr	jr	PROPN
ejpam-4962	49	34	.	.	PROPN
ejpam-4962	49	35	/	/	SYM
ejpam-4962	49	36	eur	eur	PROPN
ejpam-4962	49	37	.	.	PUNCT
ejpam-4962	50	1	j.	j.	PROPN
ejpam-4962	50	2	pure	pure	PROPN
ejpam-4962	50	3	appl	appl	PROPN
ejpam-4962	50	4	.	.	PROPN
ejpam-4962	50	5	math	math	PROPN
ejpam-4962	50	6	,	,	PUNCT
ejpam-4962	50	7	16	16	NUM
ejpam-4962	50	8	(	(	PUNCT
ejpam-4962	50	9	4	4	NUM
ejpam-4962	50	10	)	)	PUNCT
ejpam-4962	50	11	(	(	PUNCT
ejpam-4962	50	12	2023	2023	NUM
ejpam-4962	50	13	)	)	PUNCT
ejpam-4962	50	14	,	,	PUNCT
ejpam-4962	50	15	2368	2368	NUM
ejpam-4962	50	16	-	-	SYM
ejpam-4962	50	17	2383	2383	NUM
ejpam-4962	50	18	2370	2370	NUM
ejpam-4962	50	19	every	every	DET
ejpam-4962	50	20	vertex	vertex	NOUN
ejpam-4962	50	21	u	u	NOUN
ejpam-4962	50	22	for	for	ADP
ejpam-4962	50	23	which	which	PRON
ejpam-4962	50	24	f(u	f(u	PROPN
ejpam-4962	50	25	)	)	PUNCT
ejpam-4962	51	1	=	=	SYM
ejpam-4962	51	2	0	0	NUM
ejpam-4962	51	3	is	be	AUX
ejpam-4962	51	4	adjacent	adjacent	ADJ
ejpam-4962	51	5	to	to	ADP
ejpam-4962	51	6	at	at	ADV
ejpam-4962	51	7	least	least	ADV
ejpam-4962	51	8	one	one	NUM
ejpam-4962	51	9	vertex	vertex	NOUN
ejpam-4962	51	10	v	v	NOUN
ejpam-4962	51	11	for	for	ADP
ejpam-4962	51	12	which	which	PRON
ejpam-4962	51	13	f(v	f(v	NOUN
ejpam-4962	51	14	)	)	PUNCT
ejpam-4962	51	15	=	=	SYM
ejpam-4962	52	1	2	2	X
ejpam-4962	52	2	.	.	PUNCT
ejpam-4962	52	3	the	the	DET
ejpam-4962	52	4	weight	weight	NOUN
ejpam-4962	52	5	of	of	ADP
ejpam-4962	52	6	an	an	DET
ejpam-4962	52	7	rdf	rdf	NOUN
ejpam-4962	52	8	f	f	NOUN
ejpam-4962	52	9	is	be	AUX
ejpam-4962	52	10	given	give	VERB
ejpam-4962	52	11	by	by	ADP
ejpam-4962	52	12	ωg(f	ωg(f	NOUN
ejpam-4962	52	13	)	)	PUNCT
ejpam-4962	52	14	=	=	SYM
ejpam-4962	52	15	∑	∑	PUNCT
ejpam-4962	52	16	v∈v	v∈v	PROPN
ejpam-4962	52	17	(	(	PUNCT
ejpam-4962	52	18	g	g	NOUN
ejpam-4962	52	19	)	)	PUNCT
ejpam-4962	52	20	f(v	f(v	NOUN
ejpam-4962	52	21	)	)	PUNCT
ejpam-4962	52	22	.	.	PUNCT
ejpam-4962	53	1	the	the	DET
ejpam-4962	53	2	roman	roman	ADJ
ejpam-4962	53	3	domination	domination	NOUN
ejpam-4962	53	4	number	number	NOUN
ejpam-4962	53	5	of	of	ADP
ejpam-4962	53	6	a	a	DET
ejpam-4962	53	7	graph	graph	NOUN
ejpam-4962	53	8	g	g	NOUN
ejpam-4962	53	9	,	,	PUNCT
ejpam-4962	53	10	denoted	denote	VERB
ejpam-4962	53	11	by	by	ADP
ejpam-4962	53	12	γr(g	γr(g	PROPN
ejpam-4962	53	13	)	)	PUNCT
ejpam-4962	53	14	,	,	PUNCT
ejpam-4962	53	15	is	be	AUX
ejpam-4962	53	16	the	the	DET
ejpam-4962	53	17	minimum	minimum	ADJ
ejpam-4962	53	18	weight	weight	NOUN
ejpam-4962	53	19	of	of	ADP
ejpam-4962	53	20	an	an	DET
ejpam-4962	53	21	rdf	rdf	NOUN
ejpam-4962	53	22	on	on	ADP
ejpam-4962	53	23	g.	g.	PROPN
ejpam-4962	53	24	any	any	DET
ejpam-4962	53	25	rdf	rdf	VERB
ejpam-4962	53	26	f	f	NOUN
ejpam-4962	53	27	on	on	ADP
ejpam-4962	53	28	g	g	NOUN
ejpam-4962	53	29	with	with	ADP
ejpam-4962	53	30	ωg(f	ωg(f	NOUN
ejpam-4962	53	31	)	)	PUNCT
ejpam-4962	53	32	=	=	SYM
ejpam-4962	53	33	γr(g	γr(g	NOUN
ejpam-4962	53	34	)	)	PUNCT
ejpam-4962	53	35	is	be	AUX
ejpam-4962	53	36	called	call	VERB
ejpam-4962	53	37	a	a	DET
ejpam-4962	53	38	γr	γr	PROPN
ejpam-4962	53	39	-	-	NOUN
ejpam-4962	53	40	function	function	NOUN
ejpam-4962	53	41	.	.	PUNCT
ejpam-4962	54	1	if	if	SCONJ
ejpam-4962	54	2	f	f	PROPN
ejpam-4962	54	3	is	be	AUX
ejpam-4962	54	4	an	an	DET
ejpam-4962	54	5	rdf	rdf	NOUN
ejpam-4962	54	6	on	on	ADP
ejpam-4962	54	7	g	g	PROPN
ejpam-4962	54	8	and	and	CCONJ
ejpam-4962	54	9	vi	vi	NOUN
ejpam-4962	54	10	=	=	NOUN
ejpam-4962	54	11	{	{	PUNCT
ejpam-4962	54	12	v	v	NUM
ejpam-4962	54	13	∈	∈	NOUN
ejpam-4962	54	14	v	v	NOUN
ejpam-4962	54	15	(	(	PUNCT
ejpam-4962	54	16	g	g	NOUN
ejpam-4962	54	17	)	)	PUNCT
ejpam-4962	54	18	:	:	PUNCT
ejpam-4962	54	19	f(v	f(v	NOUN
ejpam-4962	54	20	)	)	PUNCT
ejpam-4962	55	1	=	=	PUNCT
ejpam-4962	55	2	i	i	PROPN
ejpam-4962	55	3	}	}	PUNCT
ejpam-4962	55	4	for	for	ADP
ejpam-4962	55	5	i	i	PROPN
ejpam-4962	55	6	∈	∈	PROPN
ejpam-4962	55	7	{	{	PUNCT
ejpam-4962	55	8	0	0	NUM
ejpam-4962	55	9	,	,	PUNCT
ejpam-4962	55	10	1	1	NUM
ejpam-4962	55	11	,	,	PUNCT
ejpam-4962	55	12	2	2	NUM
ejpam-4962	55	13	}	}	PUNCT
ejpam-4962	55	14	,	,	PUNCT
ejpam-4962	55	15	then	then	ADV
ejpam-4962	55	16	we	we	PRON
ejpam-4962	55	17	denote	denote	VERB
ejpam-4962	55	18	f	f	PROPN
ejpam-4962	55	19	by	by	ADP
ejpam-4962	55	20	f	f	PROPN
ejpam-4962	55	21	=	=	SYM
ejpam-4962	55	22	(	(	PUNCT
ejpam-4962	55	23	v0	v0	PROPN
ejpam-4962	55	24	,	,	PUNCT
ejpam-4962	55	25	v1	v1	NOUN
ejpam-4962	55	26	,	,	PUNCT
ejpam-4962	55	27	v2	v2	PROPN
ejpam-4962	55	28	)	)	PUNCT
ejpam-4962	55	29	.	.	PUNCT
ejpam-4962	56	1	in	in	ADP
ejpam-4962	56	2	this	this	DET
ejpam-4962	56	3	case	case	NOUN
ejpam-4962	56	4	,	,	PUNCT
ejpam-4962	56	5	ωg(f	ωg(f	NUM
ejpam-4962	56	6	)	)	PUNCT
ejpam-4962	56	7	=	=	PUNCT
ejpam-4962	56	8	|v1|+	|v1|+	DET
ejpam-4962	56	9	2|v2|	2|v2|	NUM
ejpam-4962	56	10	.	.	PUNCT
ejpam-4962	57	1	a	a	DET
ejpam-4962	57	2	roman	roman	ADJ
ejpam-4962	57	3	dominating	dominating	NOUN
ejpam-4962	57	4	function	function	NOUN
ejpam-4962	57	5	f	f	PROPN
ejpam-4962	57	6	=	=	SYM
ejpam-4962	57	7	(	(	PUNCT
ejpam-4962	57	8	v0	v0	PROPN
ejpam-4962	57	9	,	,	PUNCT
ejpam-4962	57	10	v1	v1	NOUN
ejpam-4962	57	11	,	,	PUNCT
ejpam-4962	57	12	v2	v2	PROPN
ejpam-4962	57	13	)	)	PUNCT
ejpam-4962	57	14	on	on	ADP
ejpam-4962	57	15	g	g	PROPN
ejpam-4962	57	16	is	be	AUX
ejpam-4962	57	17	a	a	DET
ejpam-4962	57	18	geodetic	geodetic	ADJ
ejpam-4962	57	19	roman	roman	ADJ
ejpam-4962	57	20	dominating	dominating	NOUN
ejpam-4962	57	21	function	function	NOUN
ejpam-4962	57	22	(	(	PUNCT
ejpam-4962	57	23	or	or	CCONJ
ejpam-4962	57	24	grdf	grdf	NOUN
ejpam-4962	57	25	)	)	PUNCT
ejpam-4962	57	26	if	if	SCONJ
ejpam-4962	57	27	v1	v1	NOUN
ejpam-4962	57	28	∪	∪	VERB
ejpam-4962	57	29	v2	v2	NOUN
ejpam-4962	57	30	is	be	AUX
ejpam-4962	57	31	a	a	DET
ejpam-4962	57	32	geodetic	geodetic	ADJ
ejpam-4962	57	33	set	set	NOUN
ejpam-4962	57	34	in	in	ADP
ejpam-4962	57	35	g.	g.	PROPN
ejpam-4962	57	36	the	the	DET
ejpam-4962	57	37	weight	weight	NOUN
ejpam-4962	57	38	of	of	ADP
ejpam-4962	57	39	a	a	DET
ejpam-4962	57	40	geodetic	geodetic	ADJ
ejpam-4962	57	41	roman	roman	ADJ
ejpam-4962	57	42	dominating	dominating	NOUN
ejpam-4962	57	43	function	function	NOUN
ejpam-4962	57	44	f	f	PROPN
ejpam-4962	57	45	=	=	SYM
ejpam-4962	57	46	(	(	PUNCT
ejpam-4962	57	47	v0	v0	PROPN
ejpam-4962	57	48	,	,	PUNCT
ejpam-4962	57	49	v1	v1	NOUN
ejpam-4962	57	50	,	,	PUNCT
ejpam-4962	57	51	v2	v2	PROPN
ejpam-4962	57	52	)	)	PUNCT
ejpam-4962	57	53	on	on	ADP
ejpam-4962	57	54	g	g	PROPN
ejpam-4962	57	55	is	be	AUX
ejpam-4962	57	56	given	give	VERB
ejpam-4962	57	57	by	by	ADP
ejpam-4962	57	58	ωgr	ωgr	NOUN
ejpam-4962	57	59	g	g	PROPN
ejpam-4962	57	60	(	(	PUNCT
ejpam-4962	57	61	f	f	X
ejpam-4962	57	62	)	)	PUNCT
ejpam-4962	57	63	=	=	PUNCT
ejpam-4962	57	64	|v1|+2|v2|	|v1|+2|v2|	NOUN
ejpam-4962	57	65	.	.	PUNCT
ejpam-4962	58	1	the	the	DET
ejpam-4962	58	2	minimum	minimum	ADJ
ejpam-4962	58	3	weight	weight	NOUN
ejpam-4962	58	4	of	of	ADP
ejpam-4962	58	5	a	a	DET
ejpam-4962	58	6	grdf	grdf	NOUN
ejpam-4962	58	7	on	on	ADP
ejpam-4962	58	8	g	g	NOUN
ejpam-4962	58	9	,	,	PUNCT
ejpam-4962	58	10	denoted	denote	VERB
ejpam-4962	58	11	by	by	ADP
ejpam-4962	58	12	γgr(g	γgr(g	PROPN
ejpam-4962	58	13	)	)	PUNCT
ejpam-4962	58	14	,	,	PUNCT
ejpam-4962	58	15	is	be	AUX
ejpam-4962	58	16	called	call	VERB
ejpam-4962	58	17	the	the	DET
ejpam-4962	58	18	geodetic	geodetic	ADJ
ejpam-4962	58	19	roman	roman	ADJ
ejpam-4962	58	20	domination	domination	NOUN
ejpam-4962	58	21	number	number	NOUN
ejpam-4962	58	22	of	of	ADP
ejpam-4962	58	23	g.	g.	PROPN
ejpam-4962	58	24	any	any	DET
ejpam-4962	58	25	grdf	grdf	NOUN
ejpam-4962	58	26	f	f	PROPN
ejpam-4962	58	27	on	on	ADP
ejpam-4962	58	28	g	g	PROPN
ejpam-4962	58	29	with	with	ADP
ejpam-4962	58	30	ωgr	ωgr	NOUN
ejpam-4962	58	31	g	g	PROPN
ejpam-4962	58	32	(	(	PUNCT
ejpam-4962	58	33	f	f	X
ejpam-4962	58	34	)	)	PUNCT
ejpam-4962	58	35	=	=	SYM
ejpam-4962	58	36	γgr(g	γgr(g	PROPN
ejpam-4962	58	37	)	)	PUNCT
ejpam-4962	58	38	is	be	AUX
ejpam-4962	58	39	called	call	VERB
ejpam-4962	58	40	a	a	DET
ejpam-4962	58	41	γgr	γgr	NOUN
ejpam-4962	58	42	-	-	PUNCT
ejpam-4962	58	43	function	function	NOUN
ejpam-4962	58	44	.	.	PUNCT
ejpam-4962	59	1	the	the	DET
ejpam-4962	59	2	join	join	NOUN
ejpam-4962	59	3	of	of	ADP
ejpam-4962	59	4	two	two	NUM
ejpam-4962	59	5	graphs	graph	NOUN
ejpam-4962	59	6	g	g	NOUN
ejpam-4962	59	7	and	and	CCONJ
ejpam-4962	59	8	h	h	NOUN
ejpam-4962	59	9	,	,	PUNCT
ejpam-4962	59	10	denoted	denote	VERB
ejpam-4962	59	11	by	by	ADP
ejpam-4962	59	12	g	g	PROPN
ejpam-4962	59	13	+	+	PROPN
ejpam-4962	59	14	h	h	NOUN
ejpam-4962	59	15	,	,	PUNCT
ejpam-4962	59	16	is	be	AUX
ejpam-4962	59	17	the	the	DET
ejpam-4962	59	18	graph	graph	NOUN
ejpam-4962	59	19	with	with	ADP
ejpam-4962	59	20	v	v	NOUN
ejpam-4962	59	21	(	(	PUNCT
ejpam-4962	59	22	g+h	g+h	NOUN
ejpam-4962	59	23	)	)	PUNCT
ejpam-4962	59	24	=	=	SYM
ejpam-4962	59	25	v	v	X
ejpam-4962	59	26	(	(	PUNCT
ejpam-4962	59	27	g)∪v	g)∪v	NOUN
ejpam-4962	59	28	(	(	PUNCT
ejpam-4962	59	29	h	h	NOUN
ejpam-4962	59	30	)	)	PUNCT
ejpam-4962	59	31	and	and	CCONJ
ejpam-4962	59	32	e(g+h	e(g+h	NUM
ejpam-4962	59	33	)	)	PUNCT
ejpam-4962	60	1	=	=	PUNCT
ejpam-4962	60	2	e(g)∪e(h)∪{uv	e(g)∪e(h)∪{uv	X
ejpam-4962	60	3	:	:	PUNCT
ejpam-4962	60	4	u	u	PROPN
ejpam-4962	60	5	∈	∈	PROPN
ejpam-4962	60	6	v	v	ADP
ejpam-4962	60	7	(	(	PUNCT
ejpam-4962	60	8	g	g	NOUN
ejpam-4962	60	9	)	)	PUNCT
ejpam-4962	60	10	and	and	CCONJ
ejpam-4962	60	11	v	v	ADP
ejpam-4962	60	12	∈	∈	NOUN
ejpam-4962	60	13	v	v	NOUN
ejpam-4962	60	14	(	(	PUNCT
ejpam-4962	60	15	g	g	NOUN
ejpam-4962	60	16	)	)	PUNCT
ejpam-4962	60	17	}	}	PUNCT
ejpam-4962	60	18	,	,	PUNCT
ejpam-4962	60	19	where	where	SCONJ
ejpam-4962	60	20	“	"	PUNCT
ejpam-4962	60	21	∪	∪	NOUN
ejpam-4962	60	22	”	"	PUNCT
ejpam-4962	60	23	refers	refer	VERB
ejpam-4962	60	24	to	to	ADP
ejpam-4962	60	25	a	a	DET
ejpam-4962	60	26	disjoint	disjoint	NOUN
ejpam-4962	60	27	union	union	NOUN
ejpam-4962	60	28	of	of	ADP
ejpam-4962	60	29	sets	set	NOUN
ejpam-4962	60	30	.	.	PUNCT
ejpam-4962	61	1	3	3	X
ejpam-4962	61	2	.	.	X
ejpam-4962	61	3	known	know	VERB
ejpam-4962	61	4	results	result	NOUN
ejpam-4962	61	5	we	we	PRON
ejpam-4962	61	6	state	state	VERB
ejpam-4962	61	7	some	some	DET
ejpam-4962	61	8	results	result	NOUN
ejpam-4962	61	9	that	that	PRON
ejpam-4962	61	10	will	will	AUX
ejpam-4962	61	11	be	be	AUX
ejpam-4962	61	12	needed	need	VERB
ejpam-4962	61	13	in	in	ADP
ejpam-4962	61	14	this	this	DET
ejpam-4962	61	15	study	study	NOUN
ejpam-4962	61	16	.	.	PUNCT
ejpam-4962	62	1	remark	remark	NOUN
ejpam-4962	62	2	1	1	NUM
ejpam-4962	62	3	.	.	PUNCT
ejpam-4962	63	1	[	[	X
ejpam-4962	63	2	9	9	NUM
ejpam-4962	63	3	]	]	PUNCT
ejpam-4962	63	4	every	every	DET
ejpam-4962	63	5	geodetic	geodetic	ADJ
ejpam-4962	63	6	set	set	NOUN
ejpam-4962	63	7	in	in	ADP
ejpam-4962	63	8	a	a	DET
ejpam-4962	63	9	graph	graph	NOUN
ejpam-4962	63	10	contains	contain	VERB
ejpam-4962	63	11	the	the	DET
ejpam-4962	63	12	extreme	extreme	ADJ
ejpam-4962	63	13	vertices	vertex	NOUN
ejpam-4962	63	14	.	.	PUNCT
ejpam-4962	64	1	remark	remark	NOUN
ejpam-4962	64	2	2	2	NUM
ejpam-4962	64	3	.	.	PUNCT
ejpam-4962	65	1	[	[	X
ejpam-4962	65	2	11	11	NUM
ejpam-4962	65	3	]	]	PUNCT
ejpam-4962	65	4	let	let	VERB
ejpam-4962	65	5	n	n	PRON
ejpam-4962	65	6	be	be	AUX
ejpam-4962	65	7	a	a	DET
ejpam-4962	65	8	positive	positive	ADJ
ejpam-4962	65	9	integer	integer	NOUN
ejpam-4962	65	10	.	.	PUNCT
ejpam-4962	66	1	then	then	ADV
ejpam-4962	66	2	(	(	PUNCT
ejpam-4962	66	3	i	i	NOUN
ejpam-4962	66	4	)	)	PUNCT
ejpam-4962	66	5	γg(cn	γg(cn	NOUN
ejpam-4962	66	6	)	)	PUNCT
ejpam-4962	67	1	=	=	PUNCT
ejpam-4962	67	2	⌈n3	⌈n3	ADJ
ejpam-4962	67	3	⌉	⌉	X
ejpam-4962	67	4	(	(	PUNCT
ejpam-4962	67	5	ii	ii	NOUN
ejpam-4962	67	6	)	)	PUNCT
ejpam-4962	67	7	γg(pn	γg(pn	NOUN
ejpam-4962	67	8	)	)	PUNCT
ejpam-4962	67	9	=	=	PRON
ejpam-4962	68	1	⌈n+2	⌈n+2	NUM
ejpam-4962	68	2	3	3	NUM
ejpam-4962	68	3	⌉.	⌉.	ADV
ejpam-4962	68	4	theorem	theorem	ADJ
ejpam-4962	68	5	1	1	NUM
ejpam-4962	68	6	.	.	PUNCT
ejpam-4962	69	1	[	[	X
ejpam-4962	69	2	13	13	NUM
ejpam-4962	69	3	]	]	PUNCT
ejpam-4962	69	4	let	let	VERB
ejpam-4962	69	5	g	g	PRON
ejpam-4962	69	6	be	be	AUX
ejpam-4962	69	7	a	a	DET
ejpam-4962	69	8	connected	connected	ADJ
ejpam-4962	69	9	graph	graph	NOUN
ejpam-4962	69	10	of	of	ADP
ejpam-4962	69	11	order	order	NOUN
ejpam-4962	69	12	n	n	PRON
ejpam-4962	69	13	≥	≥	NOUN
ejpam-4962	69	14	2	2	NUM
ejpam-4962	69	15	.	.	PUNCT
ejpam-4962	70	1	then	then	ADV
ejpam-4962	70	2	the	the	DET
ejpam-4962	70	3	following	follow	VERB
ejpam-4962	70	4	hold	hold	NOUN
ejpam-4962	70	5	:	:	PUNCT
ejpam-4962	70	6	(	(	PUNCT
ejpam-4962	70	7	i	i	NOUN
ejpam-4962	70	8	)	)	PUNCT
ejpam-4962	70	9	γg(g	γg(g	ADP
ejpam-4962	70	10	)	)	PUNCT
ejpam-4962	70	11	=	=	SYM
ejpam-4962	70	12	2	2	NUM
ejpam-4962	70	13	if	if	SCONJ
ejpam-4962	70	14	and	and	CCONJ
ejpam-4962	70	15	only	only	ADV
ejpam-4962	70	16	if	if	SCONJ
ejpam-4962	70	17	there	there	PRON
ejpam-4962	70	18	exists	exist	VERB
ejpam-4962	70	19	a	a	DET
ejpam-4962	70	20	geodetic	geodetic	ADJ
ejpam-4962	70	21	set	set	NOUN
ejpam-4962	70	22	s	s	PART
ejpam-4962	70	23	=	=	PUNCT
ejpam-4962	70	24	{	{	PUNCT
ejpam-4962	70	25	u	u	NOUN
ejpam-4962	70	26	,	,	PUNCT
ejpam-4962	70	27	v	v	NOUN
ejpam-4962	70	28	}	}	PUNCT
ejpam-4962	70	29	of	of	ADP
ejpam-4962	70	30	g	g	NOUN
ejpam-4962	70	31	such	such	ADJ
ejpam-4962	70	32	that	that	SCONJ
ejpam-4962	70	33	d(u	d(u	PROPN
ejpam-4962	70	34	,	,	PUNCT
ejpam-4962	70	35	v	v	NOUN
ejpam-4962	70	36	)	)	PUNCT
ejpam-4962	70	37	≤	≤	NOUN
ejpam-4962	70	38	3	3	NUM
ejpam-4962	70	39	.	.	PUNCT
ejpam-4962	70	40	(	(	PUNCT
ejpam-4962	70	41	ii	ii	NOUN
ejpam-4962	70	42	)	)	PUNCT
ejpam-4962	70	43	γg(g	γg(g	ADP
ejpam-4962	70	44	)	)	PUNCT
ejpam-4962	70	45	=	=	SYM
ejpam-4962	71	1	n	n	NOUN
ejpam-4962	71	2	if	if	SCONJ
ejpam-4962	72	1	and	and	CCONJ
ejpam-4962	72	2	only	only	ADV
ejpam-4962	72	3	if	if	SCONJ
ejpam-4962	72	4	g	g	PROPN
ejpam-4962	72	5	is	be	AUX
ejpam-4962	72	6	the	the	DET
ejpam-4962	72	7	complete	complete	ADJ
ejpam-4962	72	8	graph	graph	NOUN
ejpam-4962	72	9	on	on	ADP
ejpam-4962	72	10	n	n	DET
ejpam-4962	72	11	vertices	vertex	NOUN
ejpam-4962	72	12	.	.	PUNCT
ejpam-4962	73	1	(	(	PUNCT
ejpam-4962	73	2	iii	iii	NOUN
ejpam-4962	73	3	)	)	PUNCT
ejpam-4962	73	4	γg(g	γg(g	NOUN
ejpam-4962	73	5	)	)	PUNCT
ejpam-4962	74	1	=	=	SYM
ejpam-4962	74	2	n−1	n−1	PROPN
ejpam-4962	74	3	if	if	SCONJ
ejpam-4962	74	4	and	and	CCONJ
ejpam-4962	74	5	only	only	ADV
ejpam-4962	74	6	if	if	SCONJ
ejpam-4962	74	7	there	there	PRON
ejpam-4962	74	8	exists	exist	VERB
ejpam-4962	74	9	a	a	DET
ejpam-4962	74	10	vertex	vertex	NOUN
ejpam-4962	74	11	v	v	NOUN
ejpam-4962	74	12	in	in	ADP
ejpam-4962	74	13	g	g	PROPN
ejpam-4962	74	14	such	such	DET
ejpam-4962	74	15	that	that	DET
ejpam-4962	74	16	v	v	NOUN
ejpam-4962	74	17	(	(	PUNCT
ejpam-4962	74	18	g)\{v	g)\{v	PROPN
ejpam-4962	74	19	}	}	PUNCT
ejpam-4962	74	20	⊆	⊆	NUM
ejpam-4962	74	21	ng(v	ng(v	PUNCT
ejpam-4962	74	22	)	)	PUNCT
ejpam-4962	74	23	and	and	CCONJ
ejpam-4962	74	24	g	g	PROPN
ejpam-4962	74	25	\	\	PROPN
ejpam-4962	74	26	v	v	NOUN
ejpam-4962	74	27	is	be	AUX
ejpam-4962	74	28	the	the	DET
ejpam-4962	74	29	union	union	NOUN
ejpam-4962	74	30	of	of	ADP
ejpam-4962	74	31	at	at	ADV
ejpam-4962	74	32	least	least	ADV
ejpam-4962	74	33	two	two	NUM
ejpam-4962	74	34	complete	complete	ADJ
ejpam-4962	74	35	graphs	graph	NOUN
ejpam-4962	74	36	.	.	PUNCT
ejpam-4962	75	1	remark	remark	NOUN
ejpam-4962	75	2	3	3	NUM
ejpam-4962	75	3	.	.	PUNCT
ejpam-4962	76	1	[	[	X
ejpam-4962	76	2	24	24	NUM
ejpam-4962	76	3	]	]	PUNCT
ejpam-4962	76	4	every	every	DET
ejpam-4962	76	5	2	2	NUM
ejpam-4962	76	6	-	-	PUNCT
ejpam-4962	76	7	path	path	NOUN
ejpam-4962	76	8	closure	closure	NOUN
ejpam-4962	76	9	absorbing	absorb	VERB
ejpam-4962	76	10	set	set	VERB
ejpam-4962	76	11	in	in	ADP
ejpam-4962	76	12	a	a	DET
ejpam-4962	76	13	connected	connected	ADJ
ejpam-4962	76	14	graph	graph	NOUN
ejpam-4962	76	15	g	g	PROPN
ejpam-4962	76	16	is	be	AUX
ejpam-4962	76	17	a	a	DET
ejpam-4962	76	18	dominating	dominating	NOUN
ejpam-4962	76	19	set	set	VERB
ejpam-4962	76	20	in	in	ADP
ejpam-4962	76	21	g.	g.	PROPN
ejpam-4962	76	22	4	4	NUM
ejpam-4962	76	23	.	.	PUNCT
ejpam-4962	77	1	results	result	NOUN
ejpam-4962	77	2	proposition	proposition	NOUN
ejpam-4962	77	3	1	1	X
ejpam-4962	77	4	.	.	PUNCT
ejpam-4962	78	1	let	let	VERB
ejpam-4962	78	2	g	g	PRON
ejpam-4962	78	3	be	be	AUX
ejpam-4962	78	4	a	a	DET
ejpam-4962	78	5	graph	graph	NOUN
ejpam-4962	78	6	of	of	ADP
ejpam-4962	78	7	order	order	NOUN
ejpam-4962	78	8	n	n	NOUN
ejpam-4962	78	9	and	and	CCONJ
ejpam-4962	78	10	let	let	VERB
ejpam-4962	78	11	f	f	PROPN
ejpam-4962	78	12	=	=	SYM
ejpam-4962	78	13	(	(	PUNCT
ejpam-4962	78	14	v0	v0	PROPN
ejpam-4962	78	15	,	,	PUNCT
ejpam-4962	78	16	v1	v1	NOUN
ejpam-4962	78	17	,	,	PUNCT
ejpam-4962	78	18	v2	v2	PROPN
ejpam-4962	78	19	)	)	PUNCT
ejpam-4962	78	20	be	be	AUX
ejpam-4962	78	21	a	a	DET
ejpam-4962	78	22	γgr	γgr	NOUN
ejpam-4962	78	23	-	-	PUNCT
ejpam-4962	78	24	function	function	NOUN
ejpam-4962	78	25	.	.	PUNCT
ejpam-4962	79	1	then	then	ADV
ejpam-4962	79	2	each	each	PRON
ejpam-4962	79	3	of	of	ADP
ejpam-4962	79	4	the	the	DET
ejpam-4962	79	5	following	following	ADJ
ejpam-4962	79	6	statements	statement	NOUN
ejpam-4962	79	7	holds	hold	VERB
ejpam-4962	79	8	:	:	PUNCT
ejpam-4962	79	9	(	(	PUNCT
ejpam-4962	79	10	i	i	NOUN
ejpam-4962	79	11	)	)	PUNCT
ejpam-4962	79	12	v1	v1	VERB
ejpam-4962	79	13	∪	∪	NOUN
ejpam-4962	79	14	v2	v2	NOUN
ejpam-4962	79	15	contains	contain	VERB
ejpam-4962	79	16	all	all	DET
ejpam-4962	79	17	the	the	DET
ejpam-4962	79	18	extreme	extreme	ADJ
ejpam-4962	79	19	vertices	vertex	NOUN
ejpam-4962	79	20	of	of	ADP
ejpam-4962	79	21	g.	g.	PROPN
ejpam-4962	79	22	r.	r.	PROPN
ejpam-4962	79	23	fortosa	fortosa	PROPN
ejpam-4962	79	24	,	,	PUNCT
ejpam-4962	79	25	s.	s.	PROPN
ejpam-4962	79	26	canoy	canoy	PROPN
ejpam-4962	79	27	jr	jr	PROPN
ejpam-4962	79	28	.	.	PROPN
ejpam-4962	79	29	/	/	SYM
ejpam-4962	79	30	eur	eur	PROPN
ejpam-4962	79	31	.	.	PUNCT
ejpam-4962	80	1	j.	j.	PROPN
ejpam-4962	80	2	pure	pure	PROPN
ejpam-4962	80	3	appl	appl	PROPN
ejpam-4962	80	4	.	.	PROPN
ejpam-4962	80	5	math	math	PROPN
ejpam-4962	80	6	,	,	PUNCT
ejpam-4962	80	7	16	16	NUM
ejpam-4962	80	8	(	(	PUNCT
ejpam-4962	80	9	4	4	NUM
ejpam-4962	80	10	)	)	PUNCT
ejpam-4962	80	11	(	(	PUNCT
ejpam-4962	80	12	2023	2023	NUM
ejpam-4962	80	13	)	)	PUNCT
ejpam-4962	80	14	,	,	PUNCT
ejpam-4962	80	15	2368	2368	NUM
ejpam-4962	80	16	-	-	SYM
ejpam-4962	80	17	2383	2383	NUM
ejpam-4962	80	18	2371	2371	NUM
ejpam-4962	80	19	(	(	PUNCT
ejpam-4962	80	20	ii	ii	NOUN
ejpam-4962	80	21	)	)	PUNCT
ejpam-4962	80	22	|v0|	|v0|	NOUN
ejpam-4962	80	23	=	=	SYM
ejpam-4962	80	24	0	0	PUNCT
ejpam-4962	81	1	if	if	SCONJ
ejpam-4962	81	2	and	and	CCONJ
ejpam-4962	81	3	only	only	ADV
ejpam-4962	81	4	if	if	SCONJ
ejpam-4962	81	5	|v2|	|v2|	ADV
ejpam-4962	81	6	=	=	SYM
ejpam-4962	81	7	0	0	X
ejpam-4962	81	8	.	.	PUNCT
ejpam-4962	82	1	(	(	PUNCT
ejpam-4962	82	2	iii	iii	X
ejpam-4962	82	3	)	)	PUNCT
ejpam-4962	82	4	if	if	SCONJ
ejpam-4962	82	5	|v0|	|v0|	NOUN
ejpam-4962	82	6	=	=	SYM
ejpam-4962	82	7	0	0	NUM
ejpam-4962	82	8	,	,	PUNCT
ejpam-4962	82	9	then	then	ADV
ejpam-4962	82	10	γgr(g	γgr(g	PROPN
ejpam-4962	82	11	)	)	PUNCT
ejpam-4962	83	1	=	=	SYM
ejpam-4962	83	2	n.	n.	NOUN
ejpam-4962	83	3	(	(	PUNCT
ejpam-4962	83	4	iv	iv	X
ejpam-4962	83	5	)	)	PUNCT
ejpam-4962	83	6	if	if	SCONJ
ejpam-4962	83	7	|v1|	|v1|	NUM
ejpam-4962	83	8	=	=	SYM
ejpam-4962	83	9	0	0	NUM
ejpam-4962	83	10	,	,	PUNCT
ejpam-4962	83	11	then	then	ADV
ejpam-4962	83	12	v2	v2	PROPN
ejpam-4962	83	13	is	be	AUX
ejpam-4962	83	14	γg	γg	ADV
ejpam-4962	83	15	-	-	PUNCT
ejpam-4962	83	16	set	set	NOUN
ejpam-4962	83	17	of	of	ADP
ejpam-4962	83	18	g	g	PROPN
ejpam-4962	83	19	and	and	CCONJ
ejpam-4962	83	20	γgr(g	γgr(g	PROPN
ejpam-4962	83	21	)	)	PUNCT
ejpam-4962	83	22	=	=	SYM
ejpam-4962	83	23	2γg(g	2γg(g	NUM
ejpam-4962	83	24	)	)	PUNCT
ejpam-4962	83	25	.	.	PUNCT
ejpam-4962	84	1	proof	proof	NOUN
ejpam-4962	84	2	.	.	PUNCT
ejpam-4962	85	1	(	(	PUNCT
ejpam-4962	85	2	i	i	NOUN
ejpam-4962	85	3	)	)	PUNCT
ejpam-4962	85	4	by	by	ADP
ejpam-4962	85	5	remark	remark	NOUN
ejpam-4962	85	6	1	1	NUM
ejpam-4962	85	7	,	,	PUNCT
ejpam-4962	85	8	v1	v1	VERB
ejpam-4962	85	9	∪	∪	NOUN
ejpam-4962	85	10	v2	v2	NOUN
ejpam-4962	85	11	contains	contain	VERB
ejpam-4962	85	12	all	all	DET
ejpam-4962	85	13	the	the	DET
ejpam-4962	85	14	extreme	extreme	ADJ
ejpam-4962	85	15	vertices	vertex	NOUN
ejpam-4962	85	16	of	of	ADP
ejpam-4962	85	17	g.	g.	PROPN
ejpam-4962	85	18	(	(	PUNCT
ejpam-4962	85	19	ii	ii	PROPN
ejpam-4962	85	20	)	)	PUNCT
ejpam-4962	85	21	suppose	suppose	VERB
ejpam-4962	85	22	|v0|	|v0|	NOUN
ejpam-4962	85	23	=	=	SYM
ejpam-4962	85	24	0	0	X
ejpam-4962	85	25	.	.	PUNCT
ejpam-4962	85	26	suppose	suppose	VERB
ejpam-4962	85	27	further	far	ADV
ejpam-4962	85	28	that	that	DET
ejpam-4962	85	29	|v2|	|v2|	NOUN
ejpam-4962	85	30	=	=	SYM
ejpam-4962	85	31	̸	̸	NUM
ejpam-4962	85	32	0	0	NUM
ejpam-4962	85	33	.	.	PUNCT
ejpam-4962	86	1	define	define	VERB
ejpam-4962	86	2	g	g	PROPN
ejpam-4962	86	3	=	=	SYM
ejpam-4962	86	4	(	(	PUNCT
ejpam-4962	86	5	∅	∅	NOUN
ejpam-4962	86	6	,	,	PUNCT
ejpam-4962	86	7	v	v	NOUN
ejpam-4962	86	8	(	(	PUNCT
ejpam-4962	86	9	g),∅	g),∅	NOUN
ejpam-4962	86	10	)	)	PUNCT
ejpam-4962	86	11	.	.	PUNCT
ejpam-4962	87	1	then	then	ADV
ejpam-4962	87	2	ωgr	ωgr	X
ejpam-4962	87	3	g	g	PROPN
ejpam-4962	87	4	(	(	PUNCT
ejpam-4962	87	5	g	g	NOUN
ejpam-4962	87	6	)	)	PUNCT
ejpam-4962	87	7	=	=	SYM
ejpam-4962	87	8	n	n	NOUN
ejpam-4962	87	9	=	=	NOUN
ejpam-4962	87	10	|v1|	|v1|	NOUN
ejpam-4962	87	11	+	+	CCONJ
ejpam-4962	87	12	|v2|	|v2|	X
ejpam-4962	87	13	<	<	X
ejpam-4962	87	14	|v1|	|v1|	NOUN
ejpam-4962	87	15	+	+	CCONJ
ejpam-4962	87	16	2|v2|	2|v2|	NUM
ejpam-4962	87	17	=	=	SYM
ejpam-4962	87	18	γgr(g	γgr(g	PROPN
ejpam-4962	87	19	)	)	PUNCT
ejpam-4962	87	20	,	,	PUNCT
ejpam-4962	87	21	a	a	DET
ejpam-4962	87	22	contradiction	contradiction	NOUN
ejpam-4962	87	23	.	.	PUNCT
ejpam-4962	88	1	thus	thus	ADV
ejpam-4962	88	2	,	,	PUNCT
ejpam-4962	88	3	|v2|	|v2|	NOUN
ejpam-4962	88	4	=	=	SYM
ejpam-4962	88	5	0	0	X
ejpam-4962	88	6	.	.	PUNCT
ejpam-4962	89	1	the	the	DET
ejpam-4962	89	2	converse	converse	NOUN
ejpam-4962	89	3	is	be	AUX
ejpam-4962	89	4	clear	clear	ADJ
ejpam-4962	89	5	.	.	PUNCT
ejpam-4962	90	1	(	(	PUNCT
ejpam-4962	90	2	iii	iii	X
ejpam-4962	90	3	)	)	PUNCT
ejpam-4962	90	4	suppose	suppose	VERB
ejpam-4962	90	5	|v0|	|v0|	NOUN
ejpam-4962	90	6	=	=	SYM
ejpam-4962	90	7	0	0	X
ejpam-4962	90	8	.	.	PUNCT
ejpam-4962	91	1	then	then	ADV
ejpam-4962	91	2	|v2|	|v2|	ADV
ejpam-4962	91	3	=	=	SYM
ejpam-4962	91	4	0	0	NUM
ejpam-4962	91	5	by	by	ADP
ejpam-4962	91	6	(	(	PUNCT
ejpam-4962	91	7	ii	ii	NOUN
ejpam-4962	91	8	)	)	PUNCT
ejpam-4962	91	9	.	.	PUNCT
ejpam-4962	92	1	hence	hence	ADV
ejpam-4962	92	2	,	,	PUNCT
ejpam-4962	92	3	γgr(g	γgr(g	PROPN
ejpam-4962	92	4	)	)	PUNCT
ejpam-4962	92	5	=	=	SYM
ejpam-4962	92	6	|v1|	|v1|	NOUN
ejpam-4962	92	7	=	=	SYM
ejpam-4962	92	8	n.	n.	NOUN
ejpam-4962	92	9	(	(	PUNCT
ejpam-4962	92	10	iv	iv	X
ejpam-4962	92	11	)	)	PUNCT
ejpam-4962	92	12	suppose	suppose	VERB
ejpam-4962	92	13	|v1|	|v1|	NOUN
ejpam-4962	92	14	=	=	SYM
ejpam-4962	92	15	0	0	X
ejpam-4962	92	16	.	.	PUNCT
ejpam-4962	93	1	then	then	ADV
ejpam-4962	93	2	v2	v2	PROPN
ejpam-4962	93	3	is	be	AUX
ejpam-4962	93	4	a	a	DET
ejpam-4962	93	5	geodetic	geodetic	ADJ
ejpam-4962	93	6	dominating	dominating	NOUN
ejpam-4962	93	7	set	set	NOUN
ejpam-4962	93	8	of	of	ADP
ejpam-4962	93	9	g.	g.	PROPN
ejpam-4962	93	10	suppose	suppose	VERB
ejpam-4962	93	11	v2	v2	NOUN
ejpam-4962	93	12	is	be	AUX
ejpam-4962	93	13	not	not	PART
ejpam-4962	93	14	a	a	DET
ejpam-4962	93	15	γg	γg	ADV
ejpam-4962	93	16	-	-	PUNCT
ejpam-4962	93	17	set	set	NOUN
ejpam-4962	93	18	.	.	PUNCT
ejpam-4962	94	1	let	let	VERB
ejpam-4962	94	2	d	d	PRON
ejpam-4962	94	3	be	be	AUX
ejpam-4962	94	4	a	a	DET
ejpam-4962	94	5	γg	γg	ADV
ejpam-4962	94	6	-	-	PUNCT
ejpam-4962	94	7	set	set	NOUN
ejpam-4962	94	8	of	of	ADP
ejpam-4962	94	9	g.	g.	PROPN
ejpam-4962	94	10	then	then	ADV
ejpam-4962	94	11	|d|	|d|	PROPN
ejpam-4962	94	12	<	<	X
ejpam-4962	94	13	|v2|	|v2|	NOUN
ejpam-4962	94	14	.	.	PUNCT
ejpam-4962	95	1	define	define	VERB
ejpam-4962	95	2	h	h	NOUN
ejpam-4962	95	3	=	=	SYM
ejpam-4962	95	4	(	(	PUNCT
ejpam-4962	95	5	v	v	X
ejpam-4962	95	6	(	(	PUNCT
ejpam-4962	95	7	g)\d,∅	g)\d,∅	PROPN
ejpam-4962	95	8	,	,	PUNCT
ejpam-4962	95	9	d	d	NOUN
ejpam-4962	95	10	)	)	PUNCT
ejpam-4962	95	11	.	.	PUNCT
ejpam-4962	96	1	then	then	ADV
ejpam-4962	96	2	h	h	PROPN
ejpam-4962	96	3	is	be	AUX
ejpam-4962	96	4	a	a	DET
ejpam-4962	96	5	grdf	grdf	NOUN
ejpam-4962	96	6	on	on	ADP
ejpam-4962	96	7	g.	g.	PROPN
ejpam-4962	96	8	hence	hence	ADV
ejpam-4962	96	9	,	,	PUNCT
ejpam-4962	96	10	ωgr	ωgr	PROPN
ejpam-4962	96	11	g	g	PROPN
ejpam-4962	96	12	(	(	PUNCT
ejpam-4962	96	13	h	h	NOUN
ejpam-4962	96	14	)	)	PUNCT
ejpam-4962	96	15	=	=	SYM
ejpam-4962	96	16	2|d|	2|d|	NUM
ejpam-4962	96	17	<	<	X
ejpam-4962	96	18	2|v2|	2|v2|	NUM
ejpam-4962	96	19	=	=	SYM
ejpam-4962	96	20	γgr(g	γgr(g	PROPN
ejpam-4962	96	21	)	)	PUNCT
ejpam-4962	96	22	,	,	PUNCT
ejpam-4962	96	23	a	a	DET
ejpam-4962	96	24	contradiction	contradiction	NOUN
ejpam-4962	96	25	.	.	PUNCT
ejpam-4962	97	1	thus	thus	ADV
ejpam-4962	97	2	v2	v2	NOUN
ejpam-4962	97	3	is	be	AUX
ejpam-4962	97	4	a	a	DET
ejpam-4962	97	5	γg	γg	ADV
ejpam-4962	97	6	-	-	PUNCT
ejpam-4962	97	7	set	set	NOUN
ejpam-4962	97	8	in	in	ADP
ejpam-4962	97	9	g	g	PROPN
ejpam-4962	97	10	and	and	CCONJ
ejpam-4962	97	11	γgr(g	γgr(g	PROPN
ejpam-4962	97	12	)	)	PUNCT
ejpam-4962	97	13	=	=	SYM
ejpam-4962	97	14	2|v2|	2|v2|	NUM
ejpam-4962	97	15	=	=	SYM
ejpam-4962	97	16	2γg(g	2γg(g	NUM
ejpam-4962	97	17	)	)	PUNCT
ejpam-4962	97	18	.	.	PUNCT
ejpam-4962	98	1	proposition	proposition	NOUN
ejpam-4962	98	2	2	2	NUM
ejpam-4962	98	3	.	.	X
ejpam-4962	98	4	for	for	ADP
ejpam-4962	98	5	any	any	DET
ejpam-4962	98	6	graph	graph	NOUN
ejpam-4962	98	7	g	g	NOUN
ejpam-4962	98	8	,	,	PUNCT
ejpam-4962	98	9	1	1	NUM
ejpam-4962	98	10	≤	≤	NOUN
ejpam-4962	98	11	γg(g	γg(g	NOUN
ejpam-4962	98	12	)	)	PUNCT
ejpam-4962	98	13	≤	≤	NUM
ejpam-4962	98	14	γgr(g	γgr(g	PROPN
ejpam-4962	98	15	)	)	PUNCT
ejpam-4962	98	16	≤	≤	NUM
ejpam-4962	98	17	min{n	min{n	NOUN
ejpam-4962	98	18	,	,	PUNCT
ejpam-4962	98	19	2γg(g	2γg(g	NUM
ejpam-4962	98	20	)	)	PUNCT
ejpam-4962	98	21	}	}	PUNCT
ejpam-4962	98	22	.	.	PUNCT
ejpam-4962	99	1	proof	proof	NOUN
ejpam-4962	99	2	.	.	PUNCT
ejpam-4962	100	1	let	let	VERB
ejpam-4962	100	2	f	f	PROPN
ejpam-4962	100	3	=	=	SYM
ejpam-4962	100	4	(	(	PUNCT
ejpam-4962	100	5	v0	v0	PROPN
ejpam-4962	100	6	,	,	PUNCT
ejpam-4962	100	7	v1	v1	NOUN
ejpam-4962	100	8	,	,	PUNCT
ejpam-4962	100	9	v2	v2	PROPN
ejpam-4962	100	10	)	)	PUNCT
ejpam-4962	100	11	be	be	AUX
ejpam-4962	100	12	a	a	DET
ejpam-4962	100	13	γgr	γgr	NOUN
ejpam-4962	100	14	-	-	PUNCT
ejpam-4962	100	15	function	function	NOUN
ejpam-4962	100	16	.	.	PUNCT
ejpam-4962	101	1	then	then	ADV
ejpam-4962	101	2	v1	v1	VERB
ejpam-4962	101	3	∪	∪	ADJ
ejpam-4962	101	4	v2	v2	NOUN
ejpam-4962	101	5	is	be	AUX
ejpam-4962	101	6	a	a	DET
ejpam-4962	101	7	geodetic	geodetic	ADJ
ejpam-4962	101	8	dominating	dominating	NOUN
ejpam-4962	101	9	set	set	NOUN
ejpam-4962	101	10	of	of	ADP
ejpam-4962	101	11	g.	g.	PROPN
ejpam-4962	101	12	hence	hence	ADV
ejpam-4962	101	13	,	,	PUNCT
ejpam-4962	101	14	1	1	NUM
ejpam-4962	101	15	≤	≤	NOUN
ejpam-4962	101	16	γg(g	γg(g	NOUN
ejpam-4962	101	17	)	)	PUNCT
ejpam-4962	101	18	≤	≤	NOUN
ejpam-4962	101	19	|v1|	|v1|	NOUN
ejpam-4962	101	20	+	+	CCONJ
ejpam-4962	101	21	|v2|	|v2|	NOUN
ejpam-4962	101	22	≤	≤	NOUN
ejpam-4962	101	23	|v1|	|v1|	NOUN
ejpam-4962	101	24	+	+	CCONJ
ejpam-4962	101	25	2|v2|	2|v2|	NUM
ejpam-4962	101	26	=	=	SYM
ejpam-4962	101	27	γgr(g	γgr(g	PROPN
ejpam-4962	101	28	)	)	PUNCT
ejpam-4962	101	29	.	.	PUNCT
ejpam-4962	102	1	now	now	ADV
ejpam-4962	102	2	,	,	PUNCT
ejpam-4962	102	3	let	let	VERB
ejpam-4962	102	4	v	v	NOUN
ejpam-4962	102	5	′	′	NOUN
ejpam-4962	102	6	0	0	NUM
ejpam-4962	103	1	=	=	SYM
ejpam-4962	103	2	v	v	NUM
ejpam-4962	103	3	′	′	NUM
ejpam-4962	103	4	2	2	NUM
ejpam-4962	103	5	=	=	NOUN
ejpam-4962	103	6	∅	∅	NOUN
ejpam-4962	103	7	and	and	CCONJ
ejpam-4962	103	8	v	v	NOUN
ejpam-4962	103	9	′	′	NUM
ejpam-4962	103	10	1	1	NUM
ejpam-4962	103	11	=	=	SYM
ejpam-4962	103	12	v	v	NOUN
ejpam-4962	103	13	(	(	PUNCT
ejpam-4962	103	14	g	g	NOUN
ejpam-4962	103	15	)	)	PUNCT
ejpam-4962	103	16	.	.	PUNCT
ejpam-4962	104	1	then	then	ADV
ejpam-4962	104	2	g	g	PROPN
ejpam-4962	104	3	=	=	PUNCT
ejpam-4962	104	4	(	(	PUNCT
ejpam-4962	104	5	v	v	NUM
ejpam-4962	104	6	′	′	NUM
ejpam-4962	104	7	0	0	NUM
ejpam-4962	104	8	,	,	PUNCT
ejpam-4962	104	9	v	v	NOUN
ejpam-4962	104	10	′	′	NUM
ejpam-4962	104	11	1	1	NUM
ejpam-4962	104	12	,	,	PUNCT
ejpam-4962	104	13	v	v	NOUN
ejpam-4962	104	14	′	′	NUM
ejpam-4962	104	15	2	2	NUM
ejpam-4962	104	16	)	)	PUNCT
ejpam-4962	104	17	is	be	AUX
ejpam-4962	104	18	a	a	DET
ejpam-4962	104	19	grdf	grdf	NOUN
ejpam-4962	104	20	on	on	ADP
ejpam-4962	104	21	g	g	PROPN
ejpam-4962	104	22	and	and	CCONJ
ejpam-4962	104	23	γgr(g	γgr(g	PROPN
ejpam-4962	104	24	)	)	PUNCT
ejpam-4962	104	25	≤	≤	NOUN
ejpam-4962	104	26	|v	|v	ADV
ejpam-4962	104	27	′	′	NOUN
ejpam-4962	104	28	1	1	NUM
ejpam-4962	105	1	|	|	ADV
ejpam-4962	105	2	=	=	PUNCT
ejpam-4962	105	3	n.	n.	PROPN
ejpam-4962	105	4	finally	finally	ADV
ejpam-4962	105	5	,	,	PUNCT
ejpam-4962	105	6	let	let	VERB
ejpam-4962	105	7	s	s	PRON
ejpam-4962	105	8	be	be	AUX
ejpam-4962	105	9	a	a	DET
ejpam-4962	105	10	γg	γg	ADV
ejpam-4962	105	11	-	-	PUNCT
ejpam-4962	105	12	set	set	NOUN
ejpam-4962	105	13	of	of	ADP
ejpam-4962	105	14	g.	g.	PROPN
ejpam-4962	105	15	define	define	VERB
ejpam-4962	105	16	h	h	NOUN
ejpam-4962	105	17	=	=	SYM
ejpam-4962	105	18	(	(	PUNCT
ejpam-4962	105	19	v	v	NUM
ejpam-4962	105	20	′′	′′	PROPN
ejpam-4962	105	21	0	0	NUM
ejpam-4962	105	22	,	,	PUNCT
ejpam-4962	105	23	v	v	ADP
ejpam-4962	105	24	′′	′′	PROPN
ejpam-4962	105	25	1	1	NUM
ejpam-4962	105	26	,	,	PUNCT
ejpam-4962	105	27	v	v	ADP
ejpam-4962	105	28	′′	′′	PROPN
ejpam-4962	105	29	2	2	NUM
ejpam-4962	105	30	)	)	PUNCT
ejpam-4962	105	31	by	by	ADP
ejpam-4962	105	32	setting	set	VERB
ejpam-4962	105	33	v	v	ADP
ejpam-4962	105	34	′′	′′	PROPN
ejpam-4962	105	35	2	2	NUM
ejpam-4962	105	36	=	=	SYM
ejpam-4962	105	37	s	s	PROPN
ejpam-4962	105	38	,	,	PUNCT
ejpam-4962	105	39	v	v	ADP
ejpam-4962	105	40	′′	′′	PROPN
ejpam-4962	105	41	0	0	NUM
ejpam-4962	106	1	=	=	SYM
ejpam-4962	106	2	v	v	NOUN
ejpam-4962	106	3	(	(	PUNCT
ejpam-4962	106	4	g	g	NOUN
ejpam-4962	106	5	)	)	PUNCT
ejpam-4962	106	6	\	\	PROPN
ejpam-4962	107	1	s	s	PROPN
ejpam-4962	107	2	,	,	PUNCT
ejpam-4962	107	3	and	and	CCONJ
ejpam-4962	107	4	v	v	ADP
ejpam-4962	107	5	′′	′′	PROPN
ejpam-4962	107	6	1	1	NUM
ejpam-4962	107	7	=	=	SYM
ejpam-4962	107	8	∅.	∅.	NOUN
ejpam-4962	107	9	then	then	ADV
ejpam-4962	107	10	h	h	PROPN
ejpam-4962	107	11	is	be	AUX
ejpam-4962	107	12	a	a	DET
ejpam-4962	107	13	grdf	grdf	NOUN
ejpam-4962	107	14	on	on	ADP
ejpam-4962	107	15	g	g	PROPN
ejpam-4962	107	16	and	and	CCONJ
ejpam-4962	107	17	γgr(g	γgr(g	PROPN
ejpam-4962	107	18	)	)	PUNCT
ejpam-4962	107	19	≤	≤	NUM
ejpam-4962	107	20	ωgr	ωgr	X
ejpam-4962	107	21	g	g	PROPN
ejpam-4962	107	22	(	(	PUNCT
ejpam-4962	107	23	g	g	NOUN
ejpam-4962	107	24	)	)	PUNCT
ejpam-4962	107	25	=	=	SYM
ejpam-4962	108	1	2|v	2|v	X
ejpam-4962	109	1	′′	′′	NOUN
ejpam-4962	109	2	2	2	NUM
ejpam-4962	109	3	|	|	NOUN
ejpam-4962	109	4	=	=	SYM
ejpam-4962	109	5	2γg(g	2γg(g	NUM
ejpam-4962	109	6	)	)	PUNCT
ejpam-4962	109	7	.	.	PUNCT
ejpam-4962	110	1	therefore	therefore	ADV
ejpam-4962	110	2	,	,	PUNCT
ejpam-4962	110	3	1	1	NUM
ejpam-4962	110	4	≤	≤	NOUN
ejpam-4962	110	5	γg(g	γg(g	NOUN
ejpam-4962	110	6	)	)	PUNCT
ejpam-4962	110	7	≤	≤	NUM
ejpam-4962	110	8	γgr(g	γgr(g	PROPN
ejpam-4962	110	9	)	)	PUNCT
ejpam-4962	110	10	≤	≤	NUM
ejpam-4962	110	11	min{n	min{n	NOUN
ejpam-4962	110	12	,	,	PUNCT
ejpam-4962	110	13	2γg(g	2γg(g	NUM
ejpam-4962	110	14	)	)	PUNCT
ejpam-4962	110	15	}	}	PUNCT
ejpam-4962	110	16	.	.	PUNCT
ejpam-4962	111	1	theorem	theorem	NOUN
ejpam-4962	111	2	2	2	NUM
ejpam-4962	111	3	.	.	PUNCT
ejpam-4962	112	1	let	let	VERB
ejpam-4962	112	2	g	g	NOUN
ejpam-4962	112	3	be	be	AUX
ejpam-4962	112	4	any	any	DET
ejpam-4962	112	5	graph	graph	NOUN
ejpam-4962	112	6	of	of	ADP
ejpam-4962	112	7	order	order	NOUN
ejpam-4962	112	8	n.	n.	NOUN
ejpam-4962	112	9	then	then	ADV
ejpam-4962	112	10	each	each	PRON
ejpam-4962	112	11	of	of	ADP
ejpam-4962	112	12	the	the	DET
ejpam-4962	112	13	following	following	ADJ
ejpam-4962	112	14	statements	statement	NOUN
ejpam-4962	112	15	holds	hold	VERB
ejpam-4962	112	16	.	.	PUNCT
ejpam-4962	113	1	(	(	PUNCT
ejpam-4962	113	2	i	i	NOUN
ejpam-4962	113	3	)	)	PUNCT
ejpam-4962	113	4	γgr(g	γgr(g	PROPN
ejpam-4962	113	5	)	)	PUNCT
ejpam-4962	113	6	=	=	PUNCT
ejpam-4962	113	7	1	1	NUM
ejpam-4962	113	8	if	if	SCONJ
ejpam-4962	113	9	and	and	CCONJ
ejpam-4962	113	10	only	only	ADV
ejpam-4962	113	11	if	if	SCONJ
ejpam-4962	113	12	g	g	PROPN
ejpam-4962	113	13	=	=	PROPN
ejpam-4962	113	14	k1	k1	PROPN
ejpam-4962	113	15	.	.	PUNCT
ejpam-4962	113	16	(	(	PUNCT
ejpam-4962	113	17	ii	ii	NOUN
ejpam-4962	113	18	)	)	PUNCT
ejpam-4962	113	19	γgr(g	γgr(g	PROPN
ejpam-4962	113	20	)	)	PUNCT
ejpam-4962	113	21	=	=	SYM
ejpam-4962	113	22	2	2	NUM
ejpam-4962	113	23	if	if	SCONJ
ejpam-4962	113	24	and	and	CCONJ
ejpam-4962	113	25	only	only	ADV
ejpam-4962	113	26	if	if	SCONJ
ejpam-4962	113	27	g	g	PROPN
ejpam-4962	113	28	=	=	SYM
ejpam-4962	113	29	k2	k2	PROPN
ejpam-4962	113	30	or	or	CCONJ
ejpam-4962	113	31	g	g	NOUN
ejpam-4962	113	32	=	=	SYM
ejpam-4962	113	33	k2	k2	PROPN
ejpam-4962	113	34	.	.	PUNCT
ejpam-4962	114	1	(	(	PUNCT
ejpam-4962	114	2	iii	iii	X
ejpam-4962	114	3	)	)	PUNCT
ejpam-4962	114	4	γgr(g	γgr(g	PROPN
ejpam-4962	114	5	)	)	PUNCT
ejpam-4962	114	6	=	=	SYM
ejpam-4962	114	7	3	3	NUM
ejpam-4962	114	8	if	if	SCONJ
ejpam-4962	114	9	and	and	CCONJ
ejpam-4962	114	10	only	only	ADV
ejpam-4962	114	11	if	if	SCONJ
ejpam-4962	114	12	g	g	PROPN
ejpam-4962	114	13	∈	∈	PROPN
ejpam-4962	114	14	{	{	PUNCT
ejpam-4962	114	15	k3,k3,k1	k3,k3,k1	NOUN
ejpam-4962	114	16	∪k2	∪k2	PUNCT
ejpam-4962	114	17	}	}	PUNCT
ejpam-4962	114	18	or	or	CCONJ
ejpam-4962	114	19	g	g	NOUN
ejpam-4962	114	20	=	=	PUNCT
ejpam-4962	114	21	k2+h	k2+h	VERB
ejpam-4962	114	22	for	for	ADP
ejpam-4962	114	23	some	some	DET
ejpam-4962	114	24	graph	graph	NOUN
ejpam-4962	114	25	h	h	NOUN
ejpam-4962	114	26	of	of	ADP
ejpam-4962	114	27	order	order	NOUN
ejpam-4962	114	28	n−	n−	NOUN
ejpam-4962	114	29	2	2	NUM
ejpam-4962	114	30	.	.	PUNCT
ejpam-4962	115	1	proof	proof	NOUN
ejpam-4962	115	2	.	.	PUNCT
ejpam-4962	116	1	let	let	VERB
ejpam-4962	116	2	f	f	PROPN
ejpam-4962	116	3	=	=	SYM
ejpam-4962	116	4	(	(	PUNCT
ejpam-4962	116	5	v0	v0	PROPN
ejpam-4962	116	6	,	,	PUNCT
ejpam-4962	116	7	v1	v1	NOUN
ejpam-4962	116	8	,	,	PUNCT
ejpam-4962	116	9	v2	v2	PROPN
ejpam-4962	116	10	)	)	PUNCT
ejpam-4962	116	11	be	be	AUX
ejpam-4962	116	12	a	a	DET
ejpam-4962	116	13	γgr	γgr	NOUN
ejpam-4962	116	14	-	-	PUNCT
ejpam-4962	116	15	function	function	NOUN
ejpam-4962	116	16	on	on	ADP
ejpam-4962	116	17	g.	g.	PROPN
ejpam-4962	116	18	(	(	PUNCT
ejpam-4962	116	19	i	i	NOUN
ejpam-4962	116	20	)	)	PUNCT
ejpam-4962	116	21	assume	assume	VERB
ejpam-4962	116	22	that	that	SCONJ
ejpam-4962	116	23	γgr(g	γgr(g	PROPN
ejpam-4962	116	24	)	)	PUNCT
ejpam-4962	116	25	=	=	SYM
ejpam-4962	117	1	1	1	X
ejpam-4962	117	2	.	.	PUNCT
ejpam-4962	118	1	then	then	ADV
ejpam-4962	118	2	|v1|	|v1|	NOUN
ejpam-4962	118	3	=	=	SYM
ejpam-4962	118	4	1	1	NUM
ejpam-4962	118	5	and	and	CCONJ
ejpam-4962	118	6	|v0|	|v0|	NOUN
ejpam-4962	118	7	=	=	SYM
ejpam-4962	118	8	0	0	X
ejpam-4962	118	9	.	.	PUNCT
ejpam-4962	119	1	hence	hence	ADV
ejpam-4962	119	2	g	g	PROPN
ejpam-4962	119	3	=	=	SYM
ejpam-4962	119	4	k1	k1	PROPN
ejpam-4962	119	5	.	.	PUNCT
ejpam-4962	120	1	the	the	DET
ejpam-4962	120	2	converse	converse	NOUN
ejpam-4962	120	3	is	be	AUX
ejpam-4962	120	4	clear	clear	ADJ
ejpam-4962	120	5	.	.	PUNCT
ejpam-4962	121	1	(	(	PUNCT
ejpam-4962	121	2	ii	ii	NOUN
ejpam-4962	121	3	)	)	PUNCT
ejpam-4962	121	4	suppose	suppose	VERB
ejpam-4962	121	5	γgr(g	γgr(g	PROPN
ejpam-4962	121	6	)	)	PUNCT
ejpam-4962	121	7	=	=	SYM
ejpam-4962	122	1	2	2	X
ejpam-4962	122	2	.	.	PUNCT
ejpam-4962	123	1	then	then	ADV
ejpam-4962	123	2	ωgr	ωgr	INTJ
ejpam-4962	123	3	g	g	PROPN
ejpam-4962	123	4	(	(	PUNCT
ejpam-4962	123	5	f	f	X
ejpam-4962	123	6	)	)	PUNCT
ejpam-4962	123	7	=	=	NOUN
ejpam-4962	123	8	|v1|	|v1|	NOUN
ejpam-4962	123	9	+	+	CCONJ
ejpam-4962	123	10	2|v2|	2|v2|	NUM
ejpam-4962	123	11	=	=	SYM
ejpam-4962	123	12	2	2	X
ejpam-4962	123	13	.	.	X
ejpam-4962	123	14	suppose	suppose	VERB
ejpam-4962	123	15	|v2|	|v2|	NOUN
ejpam-4962	123	16	=	=	SYM
ejpam-4962	123	17	1	1	X
ejpam-4962	123	18	.	.	PUNCT
ejpam-4962	123	19	then	then	ADV
ejpam-4962	123	20	|v1|	|v1|	VERB
ejpam-4962	123	21	=	=	SYM
ejpam-4962	123	22	0	0	NUM
ejpam-4962	123	23	and	and	CCONJ
ejpam-4962	123	24	∅	∅	NOUN
ejpam-4962	123	25	̸=	̸=	PROPN
ejpam-4962	123	26	v0	v0	NOUN
ejpam-4962	123	27	=	=	SYM
ejpam-4962	123	28	v	v	PROPN
ejpam-4962	123	29	(	(	PUNCT
ejpam-4962	123	30	g	g	NOUN
ejpam-4962	123	31	)	)	PUNCT
ejpam-4962	123	32	\	\	PUNCT
ejpam-4962	124	1	v2	v2	PROPN
ejpam-4962	124	2	⊆	⊆	NUM
ejpam-4962	124	3	ng(v2	ng(v2	NOUN
ejpam-4962	124	4	)	)	PUNCT
ejpam-4962	124	5	.	.	PUNCT
ejpam-4962	125	1	this	this	PRON
ejpam-4962	125	2	implies	imply	VERB
ejpam-4962	125	3	that	that	SCONJ
ejpam-4962	125	4	v2	v2	PROPN
ejpam-4962	125	5	is	be	AUX
ejpam-4962	125	6	not	not	PART
ejpam-4962	125	7	a	a	DET
ejpam-4962	125	8	geodetic	geodetic	ADJ
ejpam-4962	125	9	set	set	NOUN
ejpam-4962	125	10	in	in	ADP
ejpam-4962	125	11	g	g	PROPN
ejpam-4962	125	12	,	,	PUNCT
ejpam-4962	125	13	a	a	DET
ejpam-4962	125	14	contradiction	contradiction	NOUN
ejpam-4962	125	15	.	.	PUNCT
ejpam-4962	126	1	hence	hence	ADV
ejpam-4962	126	2	,	,	PUNCT
ejpam-4962	126	3	|v2|	|v2|	NOUN
ejpam-4962	126	4	=	=	SYM
ejpam-4962	126	5	0	0	NUM
ejpam-4962	126	6	and	and	CCONJ
ejpam-4962	126	7	|v1|	|v1|	NOUN
ejpam-4962	126	8	=	=	SYM
ejpam-4962	126	9	n.	n.	NOUN
ejpam-4962	126	10	it	it	PRON
ejpam-4962	126	11	follows	follow	VERB
ejpam-4962	126	12	that	that	SCONJ
ejpam-4962	126	13	g	g	PROPN
ejpam-4962	126	14	=	=	SYM
ejpam-4962	126	15	k2	k2	PROPN
ejpam-4962	126	16	or	or	CCONJ
ejpam-4962	126	17	g	g	NOUN
ejpam-4962	126	18	=	=	SYM
ejpam-4962	126	19	k2	k2	PROPN
ejpam-4962	126	20	.	.	PUNCT
ejpam-4962	127	1	r.	r.	PROPN
ejpam-4962	127	2	fortosa	fortosa	PROPN
ejpam-4962	127	3	,	,	PUNCT
ejpam-4962	127	4	s.	s.	PROPN
ejpam-4962	127	5	canoy	canoy	PROPN
ejpam-4962	127	6	jr	jr	PROPN
ejpam-4962	127	7	.	.	PROPN
ejpam-4962	127	8	/	/	SYM
ejpam-4962	127	9	eur	eur	PROPN
ejpam-4962	127	10	.	.	PUNCT
ejpam-4962	128	1	j.	j.	PROPN
ejpam-4962	128	2	pure	pure	PROPN
ejpam-4962	128	3	appl	appl	PROPN
ejpam-4962	128	4	.	.	PROPN
ejpam-4962	128	5	math	math	PROPN
ejpam-4962	128	6	,	,	PUNCT
ejpam-4962	128	7	16	16	NUM
ejpam-4962	128	8	(	(	PUNCT
ejpam-4962	128	9	4	4	NUM
ejpam-4962	128	10	)	)	PUNCT
ejpam-4962	128	11	(	(	PUNCT
ejpam-4962	128	12	2023	2023	NUM
ejpam-4962	128	13	)	)	PUNCT
ejpam-4962	128	14	,	,	PUNCT
ejpam-4962	128	15	2368	2368	NUM
ejpam-4962	128	16	-	-	SYM
ejpam-4962	128	17	2383	2383	NUM
ejpam-4962	128	18	2372	2372	NUM
ejpam-4962	128	19	conversely	conversely	ADV
ejpam-4962	128	20	,	,	PUNCT
ejpam-4962	128	21	if	if	SCONJ
ejpam-4962	128	22	g	g	PROPN
ejpam-4962	128	23	=	=	SYM
ejpam-4962	128	24	k2	k2	PROPN
ejpam-4962	128	25	or	or	CCONJ
ejpam-4962	128	26	g	g	NOUN
ejpam-4962	128	27	=	=	SYM
ejpam-4962	128	28	k2	k2	ADJ
ejpam-4962	128	29	,	,	PUNCT
ejpam-4962	128	30	γgr(g)=2	γgr(g)=2	NOUN
ejpam-4962	128	31	.	.	PUNCT
ejpam-4962	129	1	(	(	PUNCT
ejpam-4962	129	2	iii	iii	X
ejpam-4962	129	3	)	)	PUNCT
ejpam-4962	129	4	suppose	suppose	VERB
ejpam-4962	129	5	γgr(g	γgr(g	PROPN
ejpam-4962	129	6	)	)	PUNCT
ejpam-4962	129	7	=	=	SYM
ejpam-4962	130	1	3	3	X
ejpam-4962	130	2	.	.	PUNCT
ejpam-4962	130	3	then	then	ADV
ejpam-4962	130	4	|v1|+	|v1|+	ADV
ejpam-4962	130	5	2|v2|	2|v2|	NUM
ejpam-4962	130	6	=	=	SYM
ejpam-4962	130	7	3	3	X
ejpam-4962	130	8	.	.	X
ejpam-4962	130	9	hence	hence	ADV
ejpam-4962	130	10	|v2|	|v2|	ADV
ejpam-4962	130	11	≤	≤	NUM
ejpam-4962	130	12	1	1	NUM
ejpam-4962	130	13	.	.	PUNCT
ejpam-4962	130	14	consider	consider	VERB
ejpam-4962	130	15	the	the	DET
ejpam-4962	130	16	following	follow	VERB
ejpam-4962	130	17	cases	case	NOUN
ejpam-4962	130	18	:	:	PUNCT
ejpam-4962	130	19	case	case	NOUN
ejpam-4962	130	20	1	1	NUM
ejpam-4962	130	21	:	:	PUNCT
ejpam-4962	130	22	|v2|	|v2|	NOUN
ejpam-4962	130	23	=	=	SYM
ejpam-4962	130	24	0	0	PUNCT
ejpam-4962	130	25	then	then	ADV
ejpam-4962	130	26	|v0|	|v0|	NOUN
ejpam-4962	130	27	=	=	SYM
ejpam-4962	130	28	0	0	NUM
ejpam-4962	130	29	and	and	CCONJ
ejpam-4962	130	30	|v1|	|v1|	NOUN
ejpam-4962	130	31	=	=	SYM
ejpam-4962	130	32	n	n	NOUN
ejpam-4962	130	33	=	=	SYM
ejpam-4962	130	34	3	3	X
ejpam-4962	130	35	.	.	PUNCT
ejpam-4962	131	1	this	this	PRON
ejpam-4962	131	2	implies	imply	VERB
ejpam-4962	131	3	that	that	SCONJ
ejpam-4962	131	4	g	g	PROPN
ejpam-4962	131	5	∈	∈	PROPN
ejpam-4962	131	6	{	{	PUNCT
ejpam-4962	131	7	p3,k3,k3,k1	p3,k3,k3,k1	NOUN
ejpam-4962	131	8	∪k2	∪k2	X
ejpam-4962	131	9	}	}	PUNCT
ejpam-4962	131	10	.	.	PUNCT
ejpam-4962	132	1	case	case	NOUN
ejpam-4962	132	2	2	2	NUM
ejpam-4962	132	3	:	:	PUNCT
ejpam-4962	132	4	|v2|	|v2|	NOUN
ejpam-4962	132	5	=	=	SYM
ejpam-4962	132	6	1	1	NUM
ejpam-4962	132	7	then	then	ADV
ejpam-4962	132	8	|v1|	|v1|	NOUN
ejpam-4962	132	9	=	=	SYM
ejpam-4962	132	10	1	1	X
ejpam-4962	132	11	.	.	PUNCT
ejpam-4962	133	1	let	let	VERB
ejpam-4962	133	2	v1	v1	VERB
ejpam-4962	133	3	=	=	SYM
ejpam-4962	133	4	{	{	PUNCT
ejpam-4962	133	5	x	x	NOUN
ejpam-4962	133	6	}	}	PUNCT
ejpam-4962	133	7	and	and	CCONJ
ejpam-4962	133	8	v2	v2	NOUN
ejpam-4962	133	9	=	=	SYM
ejpam-4962	133	10	{	{	PUNCT
ejpam-4962	133	11	y	y	NOUN
ejpam-4962	133	12	}	}	PUNCT
ejpam-4962	133	13	.	.	PUNCT
ejpam-4962	134	1	then	then	ADV
ejpam-4962	134	2	v0	v0	PROPN
ejpam-4962	134	3	=	=	SYM
ejpam-4962	134	4	v	v	PROPN
ejpam-4962	134	5	(	(	PUNCT
ejpam-4962	134	6	g	g	NOUN
ejpam-4962	134	7	)	)	PUNCT
ejpam-4962	134	8	\	\	NOUN
ejpam-4962	135	1	{	{	PUNCT
ejpam-4962	135	2	x	x	NOUN
ejpam-4962	135	3	,	,	PUNCT
ejpam-4962	135	4	y	y	PROPN
ejpam-4962	135	5	}	}	PUNCT
ejpam-4962	135	6	⊆	⊆	NUM
ejpam-4962	135	7	ng(y	ng(y	NOUN
ejpam-4962	135	8	)	)	PUNCT
ejpam-4962	135	9	.	.	PUNCT
ejpam-4962	136	1	since	since	SCONJ
ejpam-4962	136	2	v1	v1	NOUN
ejpam-4962	136	3	∪	∪	NOUN
ejpam-4962	136	4	v2	v2	NOUN
ejpam-4962	136	5	is	be	AUX
ejpam-4962	136	6	a	a	DET
ejpam-4962	136	7	geodetic	geodetic	ADJ
ejpam-4962	136	8	set	set	NOUN
ejpam-4962	136	9	,	,	PUNCT
ejpam-4962	136	10	xy	xy	PROPN
ejpam-4962	136	11	/∈	/∈	PUNCT
ejpam-4962	136	12	e(g	e(g	PROPN
ejpam-4962	136	13	)	)	PUNCT
ejpam-4962	136	14	.	.	PUNCT
ejpam-4962	137	1	let	let	VERB
ejpam-4962	137	2	w	w	NOUN
ejpam-4962	137	3	∈	∈	PROPN
ejpam-4962	137	4	ng(x	ng(x	NUM
ejpam-4962	137	5	)	)	PUNCT
ejpam-4962	137	6	.	.	PUNCT
ejpam-4962	138	1	then	then	ADV
ejpam-4962	138	2	w	w	PROPN
ejpam-4962	138	3	∈	∈	PROPN
ejpam-4962	138	4	v0	v0	NOUN
ejpam-4962	138	5	.	.	PUNCT
ejpam-4962	139	1	this	this	PRON
ejpam-4962	139	2	implies	imply	VERB
ejpam-4962	139	3	that	that	SCONJ
ejpam-4962	139	4	dg(x	dg(x	PROPN
ejpam-4962	139	5	,	,	PUNCT
ejpam-4962	139	6	y	y	NOUN
ejpam-4962	139	7	)	)	PUNCT
ejpam-4962	139	8	=	=	SYM
ejpam-4962	139	9	2	2	X
ejpam-4962	139	10	.	.	PUNCT
ejpam-4962	139	11	since	since	SCONJ
ejpam-4962	139	12	v0	v0	NOUN
ejpam-4962	139	13	⊆	⊆	NUM
ejpam-4962	139	14	ig(x	ig(x	X
ejpam-4962	139	15	,	,	PUNCT
ejpam-4962	139	16	y	y	NOUN
ejpam-4962	139	17	)	)	PUNCT
ejpam-4962	139	18	,	,	PUNCT
ejpam-4962	139	19	v0	v0	NOUN
ejpam-4962	139	20	=	=	SYM
ejpam-4962	139	21	ng(x	ng(x	NUM
ejpam-4962	139	22	)	)	PUNCT
ejpam-4962	139	23	∩	∩	NOUN
ejpam-4962	139	24	ng(y	ng(y	NOUN
ejpam-4962	139	25	)	)	PUNCT
ejpam-4962	139	26	.	.	PUNCT
ejpam-4962	140	1	let	let	VERB
ejpam-4962	140	2	h	h	NOUN
ejpam-4962	140	3	=	=	PUNCT
ejpam-4962	140	4	⟨v0⟩.	⟨v0⟩.	NOUN
ejpam-4962	140	5	then	then	ADV
ejpam-4962	141	1	g	g	PROPN
ejpam-4962	141	2	=	=	SYM
ejpam-4962	141	3	⟨{x	⟨{x	PROPN
ejpam-4962	141	4	,	,	PUNCT
ejpam-4962	141	5	y}⟩+h	y}⟩+h	PROPN
ejpam-4962	141	6	(	(	PUNCT
ejpam-4962	141	7	isomorphic	isomorphic	ADJ
ejpam-4962	141	8	to	to	ADP
ejpam-4962	141	9	k2	k2	PROPN
ejpam-4962	141	10	+	+	PROPN
ejpam-4962	141	11	h	h	NOUN
ejpam-4962	141	12	)	)	PUNCT
ejpam-4962	141	13	.	.	PUNCT
ejpam-4962	142	1	for	for	ADP
ejpam-4962	142	2	the	the	DET
ejpam-4962	142	3	converse	converse	NOUN
ejpam-4962	142	4	,	,	PUNCT
ejpam-4962	142	5	suppose	suppose	VERB
ejpam-4962	142	6	that	that	SCONJ
ejpam-4962	142	7	g	g	PROPN
ejpam-4962	142	8	∈	∈	PROPN
ejpam-4962	142	9	{	{	PUNCT
ejpam-4962	142	10	k3,k3,k1	k3,k3,k1	NOUN
ejpam-4962	142	11	∪	∪	ADP
ejpam-4962	142	12	k2	k2	NOUN
ejpam-4962	142	13	}	}	PUNCT
ejpam-4962	142	14	.	.	PUNCT
ejpam-4962	143	1	then	then	ADV
ejpam-4962	143	2	γgr(g	γgr(g	PROPN
ejpam-4962	143	3	)	)	PUNCT
ejpam-4962	143	4	=	=	SYM
ejpam-4962	144	1	3	3	X
ejpam-4962	144	2	.	.	PUNCT
ejpam-4962	145	1	next	next	ADV
ejpam-4962	145	2	,	,	PUNCT
ejpam-4962	145	3	suppose	suppose	VERB
ejpam-4962	145	4	that	that	SCONJ
ejpam-4962	145	5	g	g	PROPN
ejpam-4962	145	6	=	=	PROPN
ejpam-4962	145	7	k2	k2	PROPN
ejpam-4962	145	8	+	+	CCONJ
ejpam-4962	145	9	h	h	NOUN
ejpam-4962	145	10	for	for	ADP
ejpam-4962	145	11	some	some	DET
ejpam-4962	145	12	graph	graph	NOUN
ejpam-4962	145	13	h.	h.	PROPN
ejpam-4962	145	14	let	let	VERB
ejpam-4962	145	15	v	v	NOUN
ejpam-4962	145	16	(	(	PUNCT
ejpam-4962	145	17	k2	k2	NOUN
ejpam-4962	145	18	)	)	PUNCT
ejpam-4962	145	19	=	=	PRON
ejpam-4962	145	20	{	{	PUNCT
ejpam-4962	145	21	p	p	X
ejpam-4962	145	22	,	,	PUNCT
ejpam-4962	145	23	q	q	NOUN
ejpam-4962	145	24	}	}	PUNCT
ejpam-4962	145	25	and	and	CCONJ
ejpam-4962	145	26	let	let	VERB
ejpam-4962	145	27	v0	v0	NOUN
ejpam-4962	145	28	=	=	SYM
ejpam-4962	145	29	v	v	PROPN
ejpam-4962	145	30	(	(	PUNCT
ejpam-4962	145	31	h	h	NOUN
ejpam-4962	145	32	)	)	PUNCT
ejpam-4962	145	33	,	,	PUNCT
ejpam-4962	145	34	v1	v1	NOUN
ejpam-4962	145	35	=	=	SYM
ejpam-4962	145	36	{	{	PUNCT
ejpam-4962	145	37	p	p	X
ejpam-4962	145	38	}	}	PUNCT
ejpam-4962	145	39	,	,	PUNCT
ejpam-4962	145	40	and	and	CCONJ
ejpam-4962	145	41	v2	v2	NOUN
ejpam-4962	145	42	=	=	SYM
ejpam-4962	145	43	{	{	PUNCT
ejpam-4962	145	44	q	q	X
ejpam-4962	145	45	}	}	PUNCT
ejpam-4962	145	46	.	.	PUNCT
ejpam-4962	146	1	then	then	ADV
ejpam-4962	146	2	g	g	PROPN
ejpam-4962	146	3	=	=	SYM
ejpam-4962	146	4	(	(	PUNCT
ejpam-4962	146	5	v0	v0	PROPN
ejpam-4962	146	6	,	,	PUNCT
ejpam-4962	146	7	v1	v1	NOUN
ejpam-4962	146	8	,	,	PUNCT
ejpam-4962	146	9	v2	v2	PROPN
ejpam-4962	146	10	)	)	PUNCT
ejpam-4962	146	11	is	be	AUX
ejpam-4962	146	12	a	a	DET
ejpam-4962	146	13	grdf	grdf	NOUN
ejpam-4962	146	14	on	on	ADP
ejpam-4962	146	15	g.	g.	PROPN
ejpam-4962	146	16	it	it	PRON
ejpam-4962	146	17	follows	follow	VERB
ejpam-4962	146	18	that	that	SCONJ
ejpam-4962	146	19	γgr(g	γgr(g	PROPN
ejpam-4962	146	20	)	)	PUNCT
ejpam-4962	146	21	≤	≤	NUM
ejpam-4962	146	22	ωgr	ωgr	X
ejpam-4962	146	23	g	g	PROPN
ejpam-4962	146	24	(	(	PUNCT
ejpam-4962	146	25	g	g	NOUN
ejpam-4962	146	26	)	)	PUNCT
ejpam-4962	146	27	=	=	SYM
ejpam-4962	147	1	3	3	X
ejpam-4962	147	2	.	.	PUNCT
ejpam-4962	147	3	by	by	ADP
ejpam-4962	147	4	(	(	PUNCT
ejpam-4962	147	5	i	i	NOUN
ejpam-4962	147	6	)	)	PUNCT
ejpam-4962	147	7	and	and	CCONJ
ejpam-4962	147	8	(	(	PUNCT
ejpam-4962	147	9	ii	ii	NOUN
ejpam-4962	147	10	)	)	PUNCT
ejpam-4962	147	11	,	,	PUNCT
ejpam-4962	147	12	γgr(g	γgr(g	PROPN
ejpam-4962	147	13	)	)	PUNCT
ejpam-4962	147	14	=	=	SYM
ejpam-4962	148	1	3	3	X
ejpam-4962	148	2	.	.	X
ejpam-4962	148	3	lemma	lemma	PROPN
ejpam-4962	148	4	1	1	X
ejpam-4962	148	5	.	.	PUNCT
ejpam-4962	149	1	let	let	VERB
ejpam-4962	149	2	g	g	PRON
ejpam-4962	149	3	be	be	AUX
ejpam-4962	149	4	a	a	DET
ejpam-4962	149	5	graph	graph	NOUN
ejpam-4962	149	6	of	of	ADP
ejpam-4962	149	7	order	order	NOUN
ejpam-4962	149	8	n.	n.	NOUN
ejpam-4962	149	9	then	then	ADV
ejpam-4962	149	10	γg(g	γg(g	PRON
ejpam-4962	149	11	)	)	PUNCT
ejpam-4962	150	1	=	=	SYM
ejpam-4962	150	2	n	n	NOUN
ejpam-4962	150	3	if	if	SCONJ
ejpam-4962	150	4	and	and	CCONJ
ejpam-4962	150	5	only	only	ADV
ejpam-4962	150	6	if	if	SCONJ
ejpam-4962	150	7	every	every	DET
ejpam-4962	150	8	component	component	NOUN
ejpam-4962	150	9	of	of	ADP
ejpam-4962	150	10	g	g	PROPN
ejpam-4962	150	11	is	be	AUX
ejpam-4962	150	12	complete	complete	ADJ
ejpam-4962	150	13	.	.	PUNCT
ejpam-4962	151	1	proof	proof	NOUN
ejpam-4962	151	2	.	.	PUNCT
ejpam-4962	152	1	suppose	suppose	VERB
ejpam-4962	152	2	γg(g	γg(g	PRON
ejpam-4962	152	3	)	)	PUNCT
ejpam-4962	152	4	=	=	VERB
ejpam-4962	153	1	n.	n.	NOUN
ejpam-4962	153	2	if	if	SCONJ
ejpam-4962	153	3	g	g	PROPN
ejpam-4962	153	4	is	be	AUX
ejpam-4962	153	5	connected	connect	VERB
ejpam-4962	153	6	,	,	PUNCT
ejpam-4962	153	7	then	then	ADV
ejpam-4962	153	8	g	g	PROPN
ejpam-4962	153	9	is	be	AUX
ejpam-4962	153	10	complete	complete	ADJ
ejpam-4962	153	11	by	by	ADP
ejpam-4962	153	12	theorem	theorem	NOUN
ejpam-4962	153	13	1(ii	1(ii	NUM
ejpam-4962	153	14	)	)	PUNCT
ejpam-4962	153	15	.	.	PUNCT
ejpam-4962	154	1	suppose	suppose	VERB
ejpam-4962	154	2	g	g	PROPN
ejpam-4962	154	3	is	be	AUX
ejpam-4962	154	4	disconnected	disconnect	VERB
ejpam-4962	154	5	with	with	ADP
ejpam-4962	154	6	components	component	NOUN
ejpam-4962	154	7	g1	g1	PROPN
ejpam-4962	154	8	,	,	PUNCT
ejpam-4962	154	9	g2	g2	PROPN
ejpam-4962	154	10	,	,	PUNCT
ejpam-4962	154	11	.	.	PUNCT
ejpam-4962	154	12	.	.	PUNCT
ejpam-4962	155	1	.	.	PUNCT
ejpam-4962	156	1	,	,	PUNCT
ejpam-4962	156	2	gk	gk	PROPN
ejpam-4962	156	3	.	.	PROPN
ejpam-4962	156	4	suppose	suppose	VERB
ejpam-4962	156	5	g	g	PROPN
ejpam-4962	156	6	has	have	VERB
ejpam-4962	156	7	a	a	DET
ejpam-4962	156	8	component	component	NOUN
ejpam-4962	156	9	gj	gj	NOUN
ejpam-4962	156	10	that	that	PRON
ejpam-4962	156	11	is	be	AUX
ejpam-4962	156	12	not	not	PART
ejpam-4962	156	13	complete	complete	ADJ
ejpam-4962	156	14	.	.	PUNCT
ejpam-4962	157	1	then	then	ADV
ejpam-4962	157	2	γg(gj	γg(gj	PROPN
ejpam-4962	157	3	)	)	PUNCT
ejpam-4962	158	1	<	<	X
ejpam-4962	158	2	|v	|v	X
ejpam-4962	158	3	(	(	PUNCT
ejpam-4962	158	4	gj)|	gj)|	PROPN
ejpam-4962	158	5	by	by	ADP
ejpam-4962	158	6	theorem	theorem	NOUN
ejpam-4962	158	7	1(ii	1(ii	NUM
ejpam-4962	158	8	)	)	PUNCT
ejpam-4962	158	9	.	.	PUNCT
ejpam-4962	159	1	hence	hence	ADV
ejpam-4962	159	2	,	,	PUNCT
ejpam-4962	159	3	γg(g	γg(g	PUNCT
ejpam-4962	159	4	)	)	PUNCT
ejpam-4962	160	1	=	=	PUNCT
ejpam-4962	161	1	∑k	∑k	PROPN
ejpam-4962	161	2	i=1	i=1	PROPN
ejpam-4962	161	3	γg(gi	γg(gi	PROPN
ejpam-4962	161	4	)	)	PUNCT
ejpam-4962	161	5	<	<	X
ejpam-4962	161	6	n	n	CCONJ
ejpam-4962	161	7	,	,	PUNCT
ejpam-4962	161	8	a	a	DET
ejpam-4962	161	9	contradiction	contradiction	NOUN
ejpam-4962	161	10	.	.	PUNCT
ejpam-4962	162	1	thus	thus	ADV
ejpam-4962	162	2	,	,	PUNCT
ejpam-4962	162	3	every	every	DET
ejpam-4962	162	4	component	component	NOUN
ejpam-4962	162	5	of	of	ADP
ejpam-4962	162	6	g	g	PROPN
ejpam-4962	162	7	is	be	AUX
ejpam-4962	162	8	complete	complete	ADJ
ejpam-4962	162	9	.	.	PUNCT
ejpam-4962	163	1	the	the	DET
ejpam-4962	163	2	converse	converse	NOUN
ejpam-4962	163	3	is	be	AUX
ejpam-4962	163	4	clear	clear	ADJ
ejpam-4962	163	5	.	.	PUNCT
ejpam-4962	164	1	theorem	theorem	NOUN
ejpam-4962	164	2	3	3	X
ejpam-4962	164	3	.	.	PUNCT
ejpam-4962	165	1	let	let	VERB
ejpam-4962	165	2	g	g	PRON
ejpam-4962	165	3	be	be	AUX
ejpam-4962	165	4	a	a	DET
ejpam-4962	165	5	graph	graph	NOUN
ejpam-4962	165	6	of	of	ADP
ejpam-4962	165	7	order	order	NOUN
ejpam-4962	165	8	n.	n.	NOUN
ejpam-4962	165	9	then	then	ADV
ejpam-4962	165	10	γg(g	γg(g	PRON
ejpam-4962	165	11	)	)	PUNCT
ejpam-4962	165	12	=	=	SYM
ejpam-4962	165	13	γgr(g	γgr(g	PROPN
ejpam-4962	165	14	)	)	PUNCT
ejpam-4962	165	15	if	if	SCONJ
ejpam-4962	165	16	and	and	CCONJ
ejpam-4962	165	17	only	only	ADV
ejpam-4962	165	18	if	if	SCONJ
ejpam-4962	165	19	every	every	DET
ejpam-4962	165	20	component	component	NOUN
ejpam-4962	165	21	of	of	ADP
ejpam-4962	165	22	g	g	PROPN
ejpam-4962	165	23	is	be	AUX
ejpam-4962	165	24	complete	complete	ADJ
ejpam-4962	165	25	.	.	PUNCT
ejpam-4962	166	1	proof	proof	NOUN
ejpam-4962	166	2	.	.	PUNCT
ejpam-4962	167	1	suppose	suppose	VERB
ejpam-4962	167	2	γg(g	γg(g	PRON
ejpam-4962	167	3	)	)	PUNCT
ejpam-4962	167	4	=	=	SYM
ejpam-4962	167	5	γgr(g	γgr(g	PROPN
ejpam-4962	167	6	)	)	PUNCT
ejpam-4962	167	7	.	.	PUNCT
ejpam-4962	168	1	let	let	VERB
ejpam-4962	168	2	f	f	PROPN
ejpam-4962	168	3	=	=	SYM
ejpam-4962	168	4	(	(	PUNCT
ejpam-4962	168	5	v0	v0	PROPN
ejpam-4962	168	6	,	,	PUNCT
ejpam-4962	168	7	v1	v1	NOUN
ejpam-4962	168	8	,	,	PUNCT
ejpam-4962	168	9	v2	v2	PROPN
ejpam-4962	168	10	)	)	PUNCT
ejpam-4962	168	11	be	be	AUX
ejpam-4962	168	12	a	a	DET
ejpam-4962	168	13	γgr	γgr	NOUN
ejpam-4962	168	14	-	-	PUNCT
ejpam-4962	168	15	function	function	NOUN
ejpam-4962	168	16	on	on	ADP
ejpam-4962	168	17	g.	g.	PROPN
ejpam-4962	168	18	then	then	ADV
ejpam-4962	168	19	γg(g	γg(g	X
ejpam-4962	168	20	)	)	PUNCT
ejpam-4962	168	21	≤	≤	NUM
ejpam-4962	168	22	|v1|	|v1|	NOUN
ejpam-4962	168	23	+	+	CCONJ
ejpam-4962	168	24	|v2|	|v2|	NOUN
ejpam-4962	168	25	≤	≤	NOUN
ejpam-4962	168	26	|v1|	|v1|	NOUN
ejpam-4962	168	27	+	+	CCONJ
ejpam-4962	168	28	2|v2|	2|v2|	NUM
ejpam-4962	168	29	=	=	SYM
ejpam-4962	168	30	γgr(g	γgr(g	PROPN
ejpam-4962	168	31	)	)	PUNCT
ejpam-4962	168	32	.	.	PUNCT
ejpam-4962	169	1	since	since	SCONJ
ejpam-4962	169	2	γg(g	γg(g	NUM
ejpam-4962	169	3	)	)	PUNCT
ejpam-4962	169	4	=	=	SYM
ejpam-4962	169	5	γgr(g	γgr(g	PROPN
ejpam-4962	169	6	)	)	PUNCT
ejpam-4962	169	7	,	,	PUNCT
ejpam-4962	169	8	it	it	PRON
ejpam-4962	169	9	follows	follow	VERB
ejpam-4962	169	10	that	that	PRON
ejpam-4962	169	11	γg(g	γg(g	PUNCT
ejpam-4962	169	12	)	)	PUNCT
ejpam-4962	170	1	=	=	NOUN
ejpam-4962	170	2	|v1|	|v1|	NOUN
ejpam-4962	170	3	+	+	CCONJ
ejpam-4962	170	4	|v2|	|v2|	NOUN
ejpam-4962	170	5	=	=	SYM
ejpam-4962	170	6	|v1|	|v1|	NOUN
ejpam-4962	170	7	+	+	CCONJ
ejpam-4962	170	8	2|v2|	2|v2|	NUM
ejpam-4962	170	9	.	.	PUNCT
ejpam-4962	171	1	consequently	consequently	ADV
ejpam-4962	171	2	,	,	PUNCT
ejpam-4962	171	3	|v2|	|v2|	NOUN
ejpam-4962	171	4	=	=	SYM
ejpam-4962	171	5	0	0	NUM
ejpam-4962	171	6	,	,	PUNCT
ejpam-4962	171	7	|v0|	|v0|	NOUN
ejpam-4962	171	8	=	=	SYM
ejpam-4962	171	9	0	0	NUM
ejpam-4962	171	10	,	,	PUNCT
ejpam-4962	171	11	and	and	CCONJ
ejpam-4962	171	12	|v1|	|v1|	NOUN
ejpam-4962	171	13	=	=	SYM
ejpam-4962	171	14	|v	|v	PROPN
ejpam-4962	171	15	(	(	PUNCT
ejpam-4962	171	16	g)|	g)|	NOUN
ejpam-4962	171	17	.	.	PUNCT
ejpam-4962	172	1	thus	thus	ADV
ejpam-4962	172	2	,	,	PUNCT
ejpam-4962	172	3	γg(g	γg(g	NOUN
ejpam-4962	172	4	)	)	PUNCT
ejpam-4962	172	5	=	=	VERB
ejpam-4962	173	1	n.	n.	NOUN
ejpam-4962	173	2	by	by	ADP
ejpam-4962	173	3	lemma	lemma	PROPN
ejpam-4962	173	4	1	1	NUM
ejpam-4962	173	5	,	,	PUNCT
ejpam-4962	173	6	every	every	DET
ejpam-4962	173	7	component	component	NOUN
ejpam-4962	173	8	of	of	ADP
ejpam-4962	173	9	g	g	PROPN
ejpam-4962	173	10	is	be	AUX
ejpam-4962	173	11	complete	complete	ADJ
ejpam-4962	173	12	.	.	PUNCT
ejpam-4962	174	1	for	for	ADP
ejpam-4962	174	2	the	the	DET
ejpam-4962	174	3	converse	converse	NOUN
ejpam-4962	174	4	,	,	PUNCT
ejpam-4962	174	5	suppose	suppose	VERB
ejpam-4962	174	6	that	that	SCONJ
ejpam-4962	174	7	every	every	DET
ejpam-4962	174	8	component	component	NOUN
ejpam-4962	174	9	of	of	ADP
ejpam-4962	174	10	g	g	PROPN
ejpam-4962	174	11	is	be	AUX
ejpam-4962	174	12	complete	complete	ADJ
ejpam-4962	174	13	.	.	PUNCT
ejpam-4962	175	1	then	then	ADV
ejpam-4962	175	2	γg(g	γg(g	PRON
ejpam-4962	175	3	)	)	PUNCT
ejpam-4962	175	4	=	=	SYM
ejpam-4962	175	5	n	n	NOUN
ejpam-4962	175	6	by	by	ADP
ejpam-4962	175	7	lemma	lemma	PROPN
ejpam-4962	175	8	1	1	NUM
ejpam-4962	175	9	.	.	PUNCT
ejpam-4962	176	1	thus	thus	ADV
ejpam-4962	176	2	,	,	PUNCT
ejpam-4962	176	3	γgr(g	γgr(g	PROPN
ejpam-4962	176	4	)	)	PUNCT
ejpam-4962	176	5	=	=	SYM
ejpam-4962	176	6	n	n	X
ejpam-4962	176	7	by	by	ADP
ejpam-4962	176	8	proposition	proposition	NOUN
ejpam-4962	176	9	2	2	NUM
ejpam-4962	176	10	.	.	X
ejpam-4962	176	11	proposition	proposition	NOUN
ejpam-4962	176	12	3	3	NUM
ejpam-4962	176	13	.	.	PUNCT
ejpam-4962	177	1	let	let	VERB
ejpam-4962	177	2	n	n	PRON
ejpam-4962	177	3	be	be	AUX
ejpam-4962	177	4	a	a	DET
ejpam-4962	177	5	positive	positive	ADJ
ejpam-4962	177	6	integer	integer	NOUN
ejpam-4962	177	7	.	.	PUNCT
ejpam-4962	178	1	then	then	ADV
ejpam-4962	178	2	(	(	PUNCT
ejpam-4962	178	3	i	i	NOUN
ejpam-4962	178	4	)	)	PUNCT
ejpam-4962	178	5	γgr(cn	γgr(cn	ADV
ejpam-4962	178	6	)	)	PUNCT
ejpam-4962	178	7	=	=	PUNCT
ejpam-4962	178	8			NOUN
ejpam-4962	178	9	3	3	NUM
ejpam-4962	178	10	,	,	PUNCT
ejpam-4962	178	11	if	if	SCONJ
ejpam-4962	178	12	n	n	NOUN
ejpam-4962	178	13	=	=	SYM
ejpam-4962	178	14	3	3	NUM
ejpam-4962	178	15	2n	2n	NUM
ejpam-4962	178	16	3	3	NUM
ejpam-4962	178	17	,	,	PUNCT
ejpam-4962	178	18	if	if	SCONJ
ejpam-4962	178	19	n	n	PRON
ejpam-4962	178	20	≡	≡	PROPN
ejpam-4962	178	21	0(mod	0(mod	NOUN
ejpam-4962	178	22	3	3	X
ejpam-4962	178	23	)	)	PUNCT
ejpam-4962	178	24	2n+1	2n+1	NOUN
ejpam-4962	178	25	3	3	NUM
ejpam-4962	178	26	,	,	PUNCT
ejpam-4962	178	27	if	if	SCONJ
ejpam-4962	178	28	n	n	PRON
ejpam-4962	178	29	≡	≡	PROPN
ejpam-4962	178	30	1(mod	1(mod	NUM
ejpam-4962	178	31	3	3	X
ejpam-4962	178	32	)	)	PUNCT
ejpam-4962	178	33	2n+2	2n+2	NUM
ejpam-4962	178	34	3	3	NUM
ejpam-4962	178	35	,	,	PUNCT
ejpam-4962	178	36	if	if	SCONJ
ejpam-4962	178	37	n	n	PRON
ejpam-4962	178	38	≡	≡	PROPN
ejpam-4962	178	39	2(mod	2(mod	NUM
ejpam-4962	178	40	3	3	X
ejpam-4962	178	41	)	)	PUNCT
ejpam-4962	178	42	r.	r.	PROPN
ejpam-4962	178	43	fortosa	fortosa	PROPN
ejpam-4962	178	44	,	,	PUNCT
ejpam-4962	178	45	s.	s.	PROPN
ejpam-4962	178	46	canoy	canoy	PROPN
ejpam-4962	178	47	jr	jr	PROPN
ejpam-4962	178	48	.	.	PROPN
ejpam-4962	178	49	/	/	SYM
ejpam-4962	178	50	eur	eur	PROPN
ejpam-4962	178	51	.	.	PUNCT
ejpam-4962	179	1	j.	j.	PROPN
ejpam-4962	179	2	pure	pure	PROPN
ejpam-4962	179	3	appl	appl	PROPN
ejpam-4962	179	4	.	.	PROPN
ejpam-4962	179	5	math	math	PROPN
ejpam-4962	179	6	,	,	PUNCT
ejpam-4962	179	7	16	16	NUM
ejpam-4962	179	8	(	(	PUNCT
ejpam-4962	179	9	4	4	NUM
ejpam-4962	179	10	)	)	PUNCT
ejpam-4962	179	11	(	(	PUNCT
ejpam-4962	179	12	2023	2023	NUM
ejpam-4962	179	13	)	)	PUNCT
ejpam-4962	179	14	,	,	PUNCT
ejpam-4962	179	15	2368	2368	NUM
ejpam-4962	179	16	-	-	SYM
ejpam-4962	179	17	2383	2383	NUM
ejpam-4962	179	18	2373	2373	NUM
ejpam-4962	179	19	(	(	PUNCT
ejpam-4962	179	20	ii	ii	NOUN
ejpam-4962	179	21	)	)	PUNCT
ejpam-4962	179	22	γgr(pn	γgr(pn	NOUN
ejpam-4962	179	23	)	)	PUNCT
ejpam-4962	179	24	=	=	PUNCT
ejpam-4962	179	25			NOUN
ejpam-4962	179	26	1	1	NUM
ejpam-4962	179	27	,	,	PUNCT
ejpam-4962	179	28	if	if	SCONJ
ejpam-4962	179	29	n	n	NOUN
ejpam-4962	179	30	=	=	SYM
ejpam-4962	179	31	1	1	NUM
ejpam-4962	179	32	2n+3	2n+3	NOUN
ejpam-4962	179	33	3	3	NUM
ejpam-4962	179	34	,	,	PUNCT
ejpam-4962	179	35	if	if	SCONJ
ejpam-4962	179	36	n	n	PRON
ejpam-4962	179	37	≡	≡	PROPN
ejpam-4962	179	38	0(mod	0(mod	NOUN
ejpam-4962	179	39	3	3	X
ejpam-4962	179	40	)	)	PUNCT
ejpam-4962	179	41	2n+4	2n+4	NOUN
ejpam-4962	179	42	3	3	NUM
ejpam-4962	179	43	,	,	PUNCT
ejpam-4962	179	44	if	if	SCONJ
ejpam-4962	179	45	n	n	PRON
ejpam-4962	179	46	≡	≡	PROPN
ejpam-4962	179	47	1(mod	1(mod	NUM
ejpam-4962	179	48	3	3	X
ejpam-4962	179	49	)	)	PUNCT
ejpam-4962	179	50	2n+2	2n+2	NUM
ejpam-4962	179	51	3	3	NUM
ejpam-4962	179	52	,	,	PUNCT
ejpam-4962	179	53	if	if	SCONJ
ejpam-4962	179	54	n	n	PRON
ejpam-4962	179	55	≡	≡	PROPN
ejpam-4962	179	56	2(mod	2(mod	NUM
ejpam-4962	179	57	3	3	X
ejpam-4962	179	58	)	)	PUNCT
ejpam-4962	179	59	proof	proof	NOUN
ejpam-4962	179	60	.	.	PUNCT
ejpam-4962	180	1	(	(	PUNCT
ejpam-4962	180	2	i	i	NOUN
ejpam-4962	180	3	)	)	PUNCT
ejpam-4962	180	4	clearly	clearly	ADV
ejpam-4962	180	5	,	,	PUNCT
ejpam-4962	180	6	γgr(c3	γgr(c3	NOUN
ejpam-4962	180	7	)	)	PUNCT
ejpam-4962	180	8	=	=	SYM
ejpam-4962	181	1	3	3	X
ejpam-4962	181	2	.	.	X
ejpam-4962	181	3	let	let	VERB
ejpam-4962	181	4	n	n	PRON
ejpam-4962	181	5	≥	≥	X
ejpam-4962	181	6	4	4	NUM
ejpam-4962	181	7	and	and	CCONJ
ejpam-4962	181	8	let	let	VERB
ejpam-4962	181	9	cn	cn	PROPN
ejpam-4962	181	10	=	=	PUNCT
ejpam-4962	182	1	[	[	X
ejpam-4962	182	2	v1	v1	NOUN
ejpam-4962	182	3	,	,	PUNCT
ejpam-4962	182	4	v2	v2	NOUN
ejpam-4962	182	5	,	,	PUNCT
ejpam-4962	182	6	.	.	PUNCT
ejpam-4962	182	7	.	.	PUNCT
ejpam-4962	182	8	.	.	PUNCT
ejpam-4962	183	1	,	,	PUNCT
ejpam-4962	183	2	vn	vn	X
ejpam-4962	183	3	,	,	PUNCT
ejpam-4962	183	4	v1	v1	PROPN
ejpam-4962	183	5	]	]	PUNCT
ejpam-4962	183	6	.	.	PUNCT
ejpam-4962	184	1	consider	consider	VERB
ejpam-4962	184	2	the	the	DET
ejpam-4962	184	3	following	follow	VERB
ejpam-4962	184	4	cases	case	NOUN
ejpam-4962	184	5	:	:	PUNCT
ejpam-4962	184	6	case	case	NOUN
ejpam-4962	184	7	1	1	NUM
ejpam-4962	184	8	:	:	PUNCT
ejpam-4962	184	9	n	n	NUM
ejpam-4962	184	10	≡	≡	PROPN
ejpam-4962	184	11	0(mod	0(mod	NOUN
ejpam-4962	184	12	3	3	X
ejpam-4962	184	13	)	)	PUNCT
ejpam-4962	184	14	let	let	VERB
ejpam-4962	184	15	n	n	NOUN
ejpam-4962	184	16	=	=	NOUN
ejpam-4962	184	17	3r	3r	NUM
ejpam-4962	184	18	for	for	ADP
ejpam-4962	184	19	some	some	DET
ejpam-4962	184	20	positive	positive	ADJ
ejpam-4962	184	21	integer	integer	NOUN
ejpam-4962	184	22	r.	r.	NOUN
ejpam-4962	184	23	let	let	VERB
ejpam-4962	184	24	v1	v1	NOUN
ejpam-4962	184	25	=	=	NOUN
ejpam-4962	184	26	∅	∅	NOUN
ejpam-4962	184	27	,	,	PUNCT
ejpam-4962	184	28	v2	v2	NOUN
ejpam-4962	184	29	=	=	SYM
ejpam-4962	184	30	{	{	PUNCT
ejpam-4962	184	31	v1	v1	PROPN
ejpam-4962	184	32	,	,	PUNCT
ejpam-4962	184	33	v4	v4	NOUN
ejpam-4962	184	34	,	,	PUNCT
ejpam-4962	184	35	v7	v7	NOUN
ejpam-4962	184	36	,	,	PUNCT
ejpam-4962	184	37	.	.	PUNCT
ejpam-4962	184	38	.	.	PUNCT
ejpam-4962	185	1	.	.	PUNCT
ejpam-4962	186	1	,	,	PUNCT
ejpam-4962	186	2	v3r−2	v3r−2	PROPN
ejpam-4962	186	3	}	}	PUNCT
ejpam-4962	186	4	,	,	PUNCT
ejpam-4962	186	5	and	and	CCONJ
ejpam-4962	186	6	v0	v0	PROPN
ejpam-4962	186	7	=	=	SYM
ejpam-4962	186	8	v	v	PROPN
ejpam-4962	186	9	(	(	PUNCT
ejpam-4962	186	10	cn	cn	PROPN
ejpam-4962	186	11	)	)	PUNCT
ejpam-4962	186	12	\	\	PROPN
ejpam-4962	186	13	v2	v2	PROPN
ejpam-4962	186	14	.	.	PUNCT
ejpam-4962	187	1	then	then	ADV
ejpam-4962	187	2	f	f	PROPN
ejpam-4962	187	3	=	=	SYM
ejpam-4962	187	4	(	(	PUNCT
ejpam-4962	187	5	v0	v0	PROPN
ejpam-4962	187	6	,	,	PUNCT
ejpam-4962	187	7	v1	v1	NOUN
ejpam-4962	187	8	,	,	PUNCT
ejpam-4962	187	9	v2	v2	PROPN
ejpam-4962	187	10	)	)	PUNCT
ejpam-4962	187	11	is	be	AUX
ejpam-4962	187	12	a	a	DET
ejpam-4962	187	13	grdf	grdf	NOUN
ejpam-4962	187	14	on	on	ADP
ejpam-4962	187	15	cn	cn	PROPN
ejpam-4962	187	16	.	.	PUNCT
ejpam-4962	188	1	hence	hence	ADV
ejpam-4962	188	2	,	,	PUNCT
ejpam-4962	188	3	γgr(cn	γgr(cn	ADV
ejpam-4962	188	4	)	)	PUNCT
ejpam-4962	188	5	≤	≤	NOUN
ejpam-4962	188	6	ωcn(f	ωcn(f	NUM
ejpam-4962	188	7	)	)	PUNCT
ejpam-4962	188	8	=	=	SYM
ejpam-4962	188	9	2|v2|	2|v2|	NUM
ejpam-4962	188	10	=	=	SYM
ejpam-4962	188	11	2n	2n	NUM
ejpam-4962	188	12	3	3	X
ejpam-4962	188	13	.	.	PUNCT
ejpam-4962	189	1	let	let	VERB
ejpam-4962	189	2	g	g	PROPN
ejpam-4962	189	3	=	=	PUNCT
ejpam-4962	189	4	(	(	PUNCT
ejpam-4962	189	5	v	v	NUM
ejpam-4962	189	6	′	′	NUM
ejpam-4962	189	7	0	0	NUM
ejpam-4962	189	8	,	,	PUNCT
ejpam-4962	189	9	v	v	NOUN
ejpam-4962	189	10	′	′	NUM
ejpam-4962	189	11	1	1	NUM
ejpam-4962	189	12	,	,	PUNCT
ejpam-4962	189	13	v	v	NOUN
ejpam-4962	189	14	′	′	NUM
ejpam-4962	189	15	2	2	NUM
ejpam-4962	189	16	)	)	PUNCT
ejpam-4962	189	17	be	be	AUX
ejpam-4962	189	18	a	a	DET
ejpam-4962	189	19	γgr	γgr	NOUN
ejpam-4962	189	20	-	-	PUNCT
ejpam-4962	189	21	function	function	NOUN
ejpam-4962	189	22	on	on	ADP
ejpam-4962	189	23	cn	cn	PROPN
ejpam-4962	189	24	.	.	PUNCT
ejpam-4962	190	1	since	since	SCONJ
ejpam-4962	190	2	γgr(g	γgr(g	PROPN
ejpam-4962	190	3	)	)	PUNCT
ejpam-4962	190	4	≤	≤	NOUN
ejpam-4962	190	5	2n	2n	NUM
ejpam-4962	190	6	3	3	NUM
ejpam-4962	190	7	,	,	PUNCT
ejpam-4962	190	8	it	it	PRON
ejpam-4962	190	9	follows	follow	VERB
ejpam-4962	190	10	that	that	SCONJ
ejpam-4962	190	11	v	v	X
ejpam-4962	190	12	′	′	NOUN
ejpam-4962	190	13	2	2	NUM
ejpam-4962	190	14	̸=	̸=	PROPN
ejpam-4962	190	15	∅.	∅.	ADV
ejpam-4962	190	16	suppose	suppose	VERB
ejpam-4962	190	17	|v	|v	PROPN
ejpam-4962	190	18	′	′	ADP
ejpam-4962	190	19	1	1	NUM
ejpam-4962	191	1	|	|	ADV
ejpam-4962	191	2	=	=	PUNCT
ejpam-4962	191	3	k.	k.	PROPN
ejpam-4962	192	1	then	then	ADV
ejpam-4962	192	2	k	k	PROPN
ejpam-4962	193	1	+	+	PROPN
ejpam-4962	194	1	2|v	2|v	NUM
ejpam-4962	195	1	′	′	NUM
ejpam-4962	195	2	2	2	NUM
ejpam-4962	196	1	|	|	ADV
ejpam-4962	196	2	≤	≤	NUM
ejpam-4962	196	3	2n	2n	NUM
ejpam-4962	196	4	3	3	NUM
ejpam-4962	196	5	.	.	PUNCT
ejpam-4962	197	1	this	this	PRON
ejpam-4962	197	2	implies	imply	VERB
ejpam-4962	197	3	that	that	SCONJ
ejpam-4962	197	4	|v	|v	PROPN
ejpam-4962	197	5	′	′	NUM
ejpam-4962	197	6	2	2	NUM
ejpam-4962	197	7	|	|	ADV
ejpam-4962	197	8	≤	≤	NUM
ejpam-4962	197	9	r	r	NOUN
ejpam-4962	197	10	−	−	NOUN
ejpam-4962	197	11	k	k	NOUN
ejpam-4962	197	12	2	2	NUM
ejpam-4962	197	13	and	and	CCONJ
ejpam-4962	197	14	|v	|v	ADJ
ejpam-4962	197	15	′	′	NOUN
ejpam-4962	197	16	0	0	NUM
ejpam-4962	198	1	|	|	ADV
ejpam-4962	198	2	=	=	SYM
ejpam-4962	198	3	n	n	PRON
ejpam-4962	198	4	−	−	PROPN
ejpam-4962	198	5	(	(	PUNCT
ejpam-4962	198	6	|v	|v	PROPN
ejpam-4962	198	7	′	′	NUM
ejpam-4962	198	8	1	1	NUM
ejpam-4962	199	1	|	|	ADV
ejpam-4962	199	2	+	+	CCONJ
ejpam-4962	199	3	|v	|v	ADJ
ejpam-4962	199	4	′	′	NOUN
ejpam-4962	199	5	2	2	NUM
ejpam-4962	199	6	|	|	NOUN
ejpam-4962	199	7	)	)	PUNCT
ejpam-4962	199	8	≥	≥	NOUN
ejpam-4962	200	1	2r	2r	NUM
ejpam-4962	201	1	−	−	PUNCT
ejpam-4962	202	1	k	k	NOUN
ejpam-4962	202	2	2	2	PROPN
ejpam-4962	202	3	.	.	PUNCT
ejpam-4962	202	4	suppose	suppose	VERB
ejpam-4962	202	5	that	that	SCONJ
ejpam-4962	202	6	k	k	PROPN
ejpam-4962	202	7	≥	≥	NUM
ejpam-4962	202	8	1	1	NUM
ejpam-4962	202	9	.	.	PUNCT
ejpam-4962	202	10	then	then	ADV
ejpam-4962	202	11	|v	|v	PROPN
ejpam-4962	202	12	′	′	NOUN
ejpam-4962	202	13	2	2	NUM
ejpam-4962	203	1	|	|	ADV
ejpam-4962	203	2	≤	≤	NUM
ejpam-4962	203	3	r	r	NOUN
ejpam-4962	203	4	−	−	NOUN
ejpam-4962	203	5	k	k	SYM
ejpam-4962	203	6	2	2	NUM
ejpam-4962	203	7	implies	imply	VERB
ejpam-4962	203	8	that	that	SCONJ
ejpam-4962	203	9	|v	|v	PROPN
ejpam-4962	203	10	′	′	NOUN
ejpam-4962	203	11	0	0	NUM
ejpam-4962	204	1	|	|	ADV
ejpam-4962	204	2	≤	≤	NUM
ejpam-4962	204	3	2|v	2|v	NUM
ejpam-4962	205	1	′	′	NUM
ejpam-4962	205	2	2	2	NUM
ejpam-4962	206	1	|	|	ADV
ejpam-4962	206	2	≤	≤	NOUN
ejpam-4962	206	3	2r	2r	NUM
ejpam-4962	206	4	−	−	PROPN
ejpam-4962	207	1	k.	k.	PROPN
ejpam-4962	207	2	this	this	PRON
ejpam-4962	207	3	contradicts	contradict	VERB
ejpam-4962	207	4	the	the	DET
ejpam-4962	207	5	fact	fact	NOUN
ejpam-4962	207	6	that	that	SCONJ
ejpam-4962	207	7	|v	|v	PROPN
ejpam-4962	207	8	′	′	NOUN
ejpam-4962	207	9	0	0	NUM
ejpam-4962	208	1	|	|	CCONJ
ejpam-4962	208	2	≥	≥	X
ejpam-4962	209	1	2r	2r	NUM
ejpam-4962	209	2	−	−	PUNCT
ejpam-4962	210	1	k	k	NOUN
ejpam-4962	210	2	2	2	NUM
ejpam-4962	210	3	>	>	SYM
ejpam-4962	210	4	2r	2r	NUM
ejpam-4962	210	5	−	−	PROPN
ejpam-4962	210	6	k.	k.	PROPN
ejpam-4962	211	1	therefore	therefore	ADV
ejpam-4962	211	2	,	,	PUNCT
ejpam-4962	211	3	k	k	PROPN
ejpam-4962	211	4	=	=	PUNCT
ejpam-4962	211	5	0	0	X
ejpam-4962	211	6	.	.	PUNCT
ejpam-4962	211	7	by	by	ADP
ejpam-4962	211	8	proposition	proposition	NOUN
ejpam-4962	211	9	1(iv	1(iv	NUM
ejpam-4962	211	10	)	)	PUNCT
ejpam-4962	211	11	and	and	CCONJ
ejpam-4962	211	12	remark	remark	NOUN
ejpam-4962	211	13	2(i	2(i	NUM
ejpam-4962	211	14	)	)	PUNCT
ejpam-4962	211	15	,	,	PUNCT
ejpam-4962	211	16	γgr(cn	γgr(cn	ADV
ejpam-4962	211	17	)	)	PUNCT
ejpam-4962	211	18	=	=	SYM
ejpam-4962	212	1	ωgr	ωgr	NUM
ejpam-4962	212	2	cn	cn	X
ejpam-4962	212	3	(	(	PUNCT
ejpam-4962	212	4	g	g	NOUN
ejpam-4962	212	5	)	)	PUNCT
ejpam-4962	212	6	=	=	SYM
ejpam-4962	212	7	2n	2n	NUM
ejpam-4962	212	8	3	3	NUM
ejpam-4962	212	9	.	.	PUNCT
ejpam-4962	212	10	case	case	NOUN
ejpam-4962	212	11	2	2	NUM
ejpam-4962	212	12	:	:	PUNCT
ejpam-4962	212	13	n	n	NUM
ejpam-4962	212	14	≡	≡	PROPN
ejpam-4962	212	15	1(mod	1(mod	NUM
ejpam-4962	212	16	3	3	X
ejpam-4962	212	17	)	)	PUNCT
ejpam-4962	212	18	let	let	VERB
ejpam-4962	212	19	n	n	PRON
ejpam-4962	212	20	=	=	SYM
ejpam-4962	212	21	3s+1	3s+1	PROPN
ejpam-4962	212	22	for	for	ADP
ejpam-4962	212	23	some	some	DET
ejpam-4962	212	24	positive	positive	ADJ
ejpam-4962	212	25	integer	integer	NOUN
ejpam-4962	212	26	s.	s.	PROPN
ejpam-4962	212	27	let	let	VERB
ejpam-4962	212	28	v1	v1	VERB
ejpam-4962	212	29	=	=	SYM
ejpam-4962	212	30	{	{	PUNCT
ejpam-4962	212	31	v3s	v3s	PROPN
ejpam-4962	212	32	}	}	PUNCT
ejpam-4962	212	33	,	,	PUNCT
ejpam-4962	212	34	v2	v2	PROPN
ejpam-4962	212	35	=	=	SYM
ejpam-4962	212	36	{	{	PUNCT
ejpam-4962	212	37	v1	v1	PROPN
ejpam-4962	212	38	,	,	PUNCT
ejpam-4962	212	39	v4	v4	NOUN
ejpam-4962	212	40	,	,	PUNCT
ejpam-4962	212	41	v7	v7	NOUN
ejpam-4962	212	42	,	,	PUNCT
ejpam-4962	212	43	.	.	PUNCT
ejpam-4962	212	44	.	.	PUNCT
ejpam-4962	212	45	.	.	PUNCT
ejpam-4962	213	1	,	,	PUNCT
ejpam-4962	213	2	v3s−2	v3s−2	VERB
ejpam-4962	213	3	}	}	PUNCT
ejpam-4962	213	4	,	,	PUNCT
ejpam-4962	213	5	and	and	CCONJ
ejpam-4962	213	6	v0	v0	PROPN
ejpam-4962	213	7	=	=	SYM
ejpam-4962	213	8	v	v	PROPN
ejpam-4962	213	9	(	(	PUNCT
ejpam-4962	213	10	g	g	NOUN
ejpam-4962	213	11	)	)	PUNCT
ejpam-4962	213	12	\	\	PUNCT
ejpam-4962	213	13	(	(	PUNCT
ejpam-4962	213	14	v1	v1	VERB
ejpam-4962	213	15	∪	∪	NOUN
ejpam-4962	213	16	v2	v2	NOUN
ejpam-4962	213	17	)	)	PUNCT
ejpam-4962	213	18	.	.	PUNCT
ejpam-4962	214	1	thus	thus	ADV
ejpam-4962	214	2	f	f	X
ejpam-4962	214	3	=	=	SYM
ejpam-4962	214	4	(	(	PUNCT
ejpam-4962	214	5	v0	v0	PROPN
ejpam-4962	214	6	,	,	PUNCT
ejpam-4962	214	7	v1	v1	NOUN
ejpam-4962	214	8	,	,	PUNCT
ejpam-4962	214	9	v2	v2	PROPN
ejpam-4962	214	10	)	)	PUNCT
ejpam-4962	214	11	is	be	AUX
ejpam-4962	214	12	a	a	DET
ejpam-4962	214	13	grdf	grdf	NOUN
ejpam-4962	214	14	in	in	ADP
ejpam-4962	214	15	cn	cn	PROPN
ejpam-4962	214	16	.	.	PUNCT
ejpam-4962	215	1	hence	hence	ADV
ejpam-4962	215	2	,	,	PUNCT
ejpam-4962	215	3	γgr(cn	γgr(cn	ADV
ejpam-4962	215	4	)	)	PUNCT
ejpam-4962	215	5	≤	≤	NUM
ejpam-4962	215	6	ωgr	ωgr	X
ejpam-4962	215	7	cn	cn	X
ejpam-4962	215	8	(	(	PUNCT
ejpam-4962	215	9	f	f	X
ejpam-4962	215	10	)	)	PUNCT
ejpam-4962	215	11	=	=	SYM
ejpam-4962	215	12	2n+1	2n+1	NOUN
ejpam-4962	215	13	3	3	NUM
ejpam-4962	215	14	.	.	PUNCT
ejpam-4962	216	1	suppose	suppose	VERB
ejpam-4962	216	2	g	g	PROPN
ejpam-4962	216	3	=	=	PUNCT
ejpam-4962	216	4	(	(	PUNCT
ejpam-4962	216	5	v	v	NUM
ejpam-4962	216	6	′	′	NUM
ejpam-4962	216	7	0	0	NUM
ejpam-4962	216	8	,	,	PUNCT
ejpam-4962	216	9	v	v	NOUN
ejpam-4962	216	10	′	′	NUM
ejpam-4962	216	11	1	1	NUM
ejpam-4962	216	12	,	,	PUNCT
ejpam-4962	216	13	v	v	NOUN
ejpam-4962	216	14	′	′	NUM
ejpam-4962	216	15	2	2	NUM
ejpam-4962	216	16	)	)	PUNCT
ejpam-4962	216	17	is	be	AUX
ejpam-4962	216	18	a	a	DET
ejpam-4962	216	19	γgr	γgr	NOUN
ejpam-4962	216	20	-	-	PUNCT
ejpam-4962	216	21	function	function	NOUN
ejpam-4962	216	22	on	on	ADP
ejpam-4962	216	23	cn	cn	PROPN
ejpam-4962	216	24	.	.	PUNCT
ejpam-4962	217	1	since	since	SCONJ
ejpam-4962	217	2	γgr(g	γgr(g	PROPN
ejpam-4962	217	3	)	)	PUNCT
ejpam-4962	217	4	≤	≤	NUM
ejpam-4962	217	5	2n+1	2n+1	PROPN
ejpam-4962	217	6	3	3	NUM
ejpam-4962	217	7	,	,	PUNCT
ejpam-4962	217	8	it	it	PRON
ejpam-4962	217	9	follows	follow	VERB
ejpam-4962	217	10	that	that	SCONJ
ejpam-4962	217	11	v	v	X
ejpam-4962	217	12	′	′	NOUN
ejpam-4962	217	13	2	2	NUM
ejpam-4962	217	14	̸=	̸=	PROPN
ejpam-4962	217	15	∅.	∅.	ADV
ejpam-4962	217	16	suppose	suppose	VERB
ejpam-4962	217	17	|v	|v	PROPN
ejpam-4962	217	18	′	′	ADP
ejpam-4962	217	19	1	1	NUM
ejpam-4962	218	1	|	|	ADV
ejpam-4962	218	2	=	=	PUNCT
ejpam-4962	218	3	k.	k.	PROPN
ejpam-4962	219	1	then	then	ADV
ejpam-4962	219	2	k	k	PROPN
ejpam-4962	220	1	+	+	PROPN
ejpam-4962	221	1	2|v	2|v	NUM
ejpam-4962	222	1	′	′	NUM
ejpam-4962	222	2	2	2	NUM
ejpam-4962	223	1	|	|	ADV
ejpam-4962	223	2	≤	≤	NUM
ejpam-4962	223	3	2n+1	2n+1	PROPN
ejpam-4962	223	4	3	3	NUM
ejpam-4962	223	5	.	.	PUNCT
ejpam-4962	224	1	thus	thus	ADV
ejpam-4962	224	2	|v	|v	ADV
ejpam-4962	224	3	′	′	NUM
ejpam-4962	224	4	2	2	NUM
ejpam-4962	224	5	|	|	ADV
ejpam-4962	224	6	≤	≤	PROPN
ejpam-4962	224	7	s	s	AUX
ejpam-4962	224	8	−	−	PROPN
ejpam-4962	224	9	1	1	NUM
ejpam-4962	224	10	2(k	2(k	NUM
ejpam-4962	224	11	−	−	NOUN
ejpam-4962	224	12	1	1	NUM
ejpam-4962	224	13	)	)	PUNCT
ejpam-4962	224	14	and	and	CCONJ
ejpam-4962	224	15	|v	|v	ADJ
ejpam-4962	224	16	′	′	NOUN
ejpam-4962	224	17	0	0	NUM
ejpam-4962	225	1	|	|	ADV
ejpam-4962	225	2	=	=	SYM
ejpam-4962	225	3	n	n	PRON
ejpam-4962	225	4	−	−	PROPN
ejpam-4962	225	5	(	(	PUNCT
ejpam-4962	225	6	|v	|v	PROPN
ejpam-4962	225	7	′	′	NUM
ejpam-4962	225	8	1	1	NUM
ejpam-4962	226	1	|	|	ADV
ejpam-4962	226	2	+	+	CCONJ
ejpam-4962	226	3	|v	|v	ADJ
ejpam-4962	226	4	′	′	NOUN
ejpam-4962	226	5	2	2	NUM
ejpam-4962	226	6	|	|	NOUN
ejpam-4962	226	7	)	)	PUNCT
ejpam-4962	226	8	≥	≥	NOUN
ejpam-4962	227	1	2s	2s	NUM
ejpam-4962	227	2	−	−	NUM
ejpam-4962	227	3	1	1	NUM
ejpam-4962	227	4	2(k	2(k	NUM
ejpam-4962	227	5	−	−	NOUN
ejpam-4962	227	6	1	1	NUM
ejpam-4962	227	7	)	)	PUNCT
ejpam-4962	227	8	.	.	PUNCT
ejpam-4962	228	1	if	if	SCONJ
ejpam-4962	228	2	k	k	PROPN
ejpam-4962	228	3	=	=	SYM
ejpam-4962	228	4	0	0	PROPN
ejpam-4962	228	5	,	,	PUNCT
ejpam-4962	228	6	then	then	ADV
ejpam-4962	228	7	|v	|v	VERB
ejpam-4962	228	8	′	′	NOUN
ejpam-4962	228	9	2	2	NUM
ejpam-4962	228	10	|	|	ADV
ejpam-4962	228	11	≤	≤	PROPN
ejpam-4962	228	12	s	s	PART
ejpam-4962	228	13	+	+	CCONJ
ejpam-4962	228	14	1	1	NUM
ejpam-4962	228	15	2	2	NUM
ejpam-4962	228	16	and	and	CCONJ
ejpam-4962	228	17	|v	|v	ADJ
ejpam-4962	228	18	′	′	NOUN
ejpam-4962	228	19	0	0	NUM
ejpam-4962	229	1	|	|	ADV
ejpam-4962	229	2	≥	≥	X
ejpam-4962	229	3	2s	2s	NUM
ejpam-4962	229	4	+	+	CCONJ
ejpam-4962	229	5	1	1	NUM
ejpam-4962	229	6	2	2	NUM
ejpam-4962	229	7	.	.	PUNCT
ejpam-4962	230	1	hence	hence	ADV
ejpam-4962	230	2	,	,	PUNCT
ejpam-4962	230	3	|v	|v	ADJ
ejpam-4962	230	4	′	′	NOUN
ejpam-4962	230	5	2	2	NUM
ejpam-4962	231	1	|	|	ADV
ejpam-4962	231	2	≤	≤	NOUN
ejpam-4962	231	3	s	s	PART
ejpam-4962	231	4	and	and	CCONJ
ejpam-4962	231	5	|v	|v	ADJ
ejpam-4962	231	6	′	′	NOUN
ejpam-4962	231	7	0	0	NUM
ejpam-4962	232	1	|	|	ADV
ejpam-4962	232	2	≥	≥	X
ejpam-4962	232	3	2s	2s	X
ejpam-4962	233	1	+	+	NOUN
ejpam-4962	233	2	1	1	X
ejpam-4962	233	3	.	.	X
ejpam-4962	234	1	this	this	PRON
ejpam-4962	234	2	is	be	AUX
ejpam-4962	234	3	not	not	PART
ejpam-4962	234	4	possible	possible	ADJ
ejpam-4962	234	5	.	.	PUNCT
ejpam-4962	235	1	hence	hence	ADV
ejpam-4962	235	2	k	k	PROPN
ejpam-4962	235	3	≥	≥	NUM
ejpam-4962	235	4	1	1	X
ejpam-4962	235	5	.	.	PUNCT
ejpam-4962	235	6	suppose	suppose	VERB
ejpam-4962	235	7	k	k	PROPN
ejpam-4962	235	8	≥	≥	NUM
ejpam-4962	235	9	2	2	NUM
ejpam-4962	235	10	.	.	PUNCT
ejpam-4962	235	11	then	then	ADV
ejpam-4962	235	12	|v	|v	VERB
ejpam-4962	235	13	′	′	NOUN
ejpam-4962	235	14	2	2	NUM
ejpam-4962	236	1	|	|	ADV
ejpam-4962	236	2	≤	≤	PROPN
ejpam-4962	236	3	s	s	VERB
ejpam-4962	236	4	−	−	PROPN
ejpam-4962	236	5	1	1	NUM
ejpam-4962	236	6	2(k	2(k	NUM
ejpam-4962	236	7	−	−	NOUN
ejpam-4962	236	8	1	1	NUM
ejpam-4962	236	9	)	)	PUNCT
ejpam-4962	236	10	implies	imply	VERB
ejpam-4962	236	11	that	that	SCONJ
ejpam-4962	236	12	|v	|v	PROPN
ejpam-4962	236	13	′	′	NOUN
ejpam-4962	236	14	0	0	NUM
ejpam-4962	237	1	|	|	ADV
ejpam-4962	237	2	≤	≤	VERB
ejpam-4962	238	1	2s	2s	NUM
ejpam-4962	238	2	−	−	PROPN
ejpam-4962	238	3	(	(	PUNCT
ejpam-4962	238	4	k	k	NOUN
ejpam-4962	238	5	−	−	PROPN
ejpam-4962	238	6	1	1	NUM
ejpam-4962	238	7	)	)	PUNCT
ejpam-4962	238	8	.	.	PUNCT
ejpam-4962	239	1	however	however	ADV
ejpam-4962	239	2	,	,	PUNCT
ejpam-4962	239	3	|v	|v	ADJ
ejpam-4962	239	4	′	′	NOUN
ejpam-4962	239	5	0	0	NUM
ejpam-4962	240	1	|	|	ADV
ejpam-4962	240	2	≥	≥	X
ejpam-4962	240	3	2s	2s	NUM
ejpam-4962	240	4	−	−	NUM
ejpam-4962	240	5	1	1	NUM
ejpam-4962	240	6	2(k	2(k	NUM
ejpam-4962	240	7	−	−	NOUN
ejpam-4962	240	8	1	1	NUM
ejpam-4962	240	9	)	)	PUNCT
ejpam-4962	240	10	>	>	X
ejpam-4962	241	1	2s	2s	NUM
ejpam-4962	241	2	−	−	PROPN
ejpam-4962	241	3	(	(	PUNCT
ejpam-4962	241	4	k	k	NOUN
ejpam-4962	242	1	−	−	PROPN
ejpam-4962	242	2	1	1	NUM
ejpam-4962	242	3	)	)	PUNCT
ejpam-4962	242	4	,	,	PUNCT
ejpam-4962	242	5	a	a	DET
ejpam-4962	242	6	contradiction	contradiction	NOUN
ejpam-4962	242	7	.	.	PUNCT
ejpam-4962	243	1	therefore	therefore	ADV
ejpam-4962	243	2	,	,	PUNCT
ejpam-4962	243	3	k	k	PROPN
ejpam-4962	243	4	=	=	SYM
ejpam-4962	243	5	1	1	NUM
ejpam-4962	243	6	and	and	CCONJ
ejpam-4962	243	7	γgr(cn	γgr(cn	NUM
ejpam-4962	243	8	)	)	PUNCT
ejpam-4962	243	9	=	=	SYM
ejpam-4962	244	1	ωgr	ωgr	NUM
ejpam-4962	244	2	cn	cn	X
ejpam-4962	244	3	(	(	PUNCT
ejpam-4962	244	4	g	g	NOUN
ejpam-4962	244	5	)	)	PUNCT
ejpam-4962	244	6	=	=	SYM
ejpam-4962	244	7	2n+1	2n+1	NOUN
ejpam-4962	244	8	3	3	NUM
ejpam-4962	244	9	.	.	PUNCT
ejpam-4962	245	1	case	case	NOUN
ejpam-4962	245	2	3	3	NUM
ejpam-4962	245	3	:	:	PUNCT
ejpam-4962	245	4	n	n	NUM
ejpam-4962	245	5	≡	≡	PROPN
ejpam-4962	245	6	2(mod	2(mod	NUM
ejpam-4962	245	7	3	3	X
ejpam-4962	245	8	)	)	PUNCT
ejpam-4962	245	9	let	let	VERB
ejpam-4962	245	10	n	n	NOUN
ejpam-4962	245	11	=	=	SYM
ejpam-4962	245	12	3t+2	3t+2	PROPN
ejpam-4962	245	13	for	for	ADP
ejpam-4962	245	14	some	some	DET
ejpam-4962	245	15	positive	positive	ADJ
ejpam-4962	245	16	integer	integer	NOUN
ejpam-4962	245	17	t.	t.	PROPN
ejpam-4962	245	18	let	let	VERB
ejpam-4962	245	19	v1	v1	VERB
ejpam-4962	245	20	=	=	SYM
ejpam-4962	245	21	{	{	PUNCT
ejpam-4962	245	22	v3	v3	PROPN
ejpam-4962	245	23	t	t	PROPN
ejpam-4962	245	24	,	,	PUNCT
ejpam-4962	245	25	v3t+1	v3t+1	PROPN
ejpam-4962	245	26	}	}	PUNCT
ejpam-4962	245	27	,	,	PUNCT
ejpam-4962	245	28	v2	v2	NOUN
ejpam-4962	245	29	=	=	SYM
ejpam-4962	245	30	{	{	PUNCT
ejpam-4962	245	31	v1	v1	PROPN
ejpam-4962	245	32	,	,	PUNCT
ejpam-4962	245	33	v4	v4	NOUN
ejpam-4962	245	34	,	,	PUNCT
ejpam-4962	245	35	v7	v7	NOUN
ejpam-4962	245	36	,	,	PUNCT
ejpam-4962	245	37	.	.	PUNCT
ejpam-4962	245	38	.	.	PUNCT
ejpam-4962	246	1	.	.	PUNCT
ejpam-4962	247	1	,	,	PUNCT
ejpam-4962	247	2	v3t−2	v3t−2	PROPN
ejpam-4962	247	3	}	}	PUNCT
ejpam-4962	247	4	,	,	PUNCT
ejpam-4962	247	5	and	and	CCONJ
ejpam-4962	247	6	v0	v0	PROPN
ejpam-4962	247	7	=	=	SYM
ejpam-4962	247	8	v	v	PROPN
ejpam-4962	247	9	(	(	PUNCT
ejpam-4962	247	10	g	g	NOUN
ejpam-4962	247	11	)	)	PUNCT
ejpam-4962	247	12	\	\	PUNCT
ejpam-4962	248	1	(	(	PUNCT
ejpam-4962	248	2	v1	v1	VERB
ejpam-4962	248	3	∪	∪	NOUN
ejpam-4962	248	4	v2	v2	NOUN
ejpam-4962	248	5	)	)	PUNCT
ejpam-4962	248	6	.	.	PUNCT
ejpam-4962	249	1	thus	thus	ADV
ejpam-4962	249	2	f	f	X
ejpam-4962	249	3	=	=	SYM
ejpam-4962	249	4	(	(	PUNCT
ejpam-4962	249	5	v0	v0	PROPN
ejpam-4962	249	6	,	,	PUNCT
ejpam-4962	249	7	v1	v1	NOUN
ejpam-4962	249	8	,	,	PUNCT
ejpam-4962	249	9	v2	v2	PROPN
ejpam-4962	249	10	)	)	PUNCT
ejpam-4962	249	11	is	be	AUX
ejpam-4962	249	12	a	a	DET
ejpam-4962	249	13	grdf	grdf	NOUN
ejpam-4962	249	14	in	in	ADP
ejpam-4962	249	15	cn	cn	PROPN
ejpam-4962	249	16	.	.	PUNCT
ejpam-4962	250	1	hence	hence	ADV
ejpam-4962	250	2	,	,	PUNCT
ejpam-4962	250	3	γgr(cn	γgr(cn	ADV
ejpam-4962	250	4	)	)	PUNCT
ejpam-4962	250	5	≤	≤	NUM
ejpam-4962	250	6	ωgr	ωgr	PUNCT
ejpam-4962	250	7	cn	cn	PROPN
ejpam-4962	250	8	=	=	PROPN
ejpam-4962	250	9	2n+2	2n+2	PROPN
ejpam-4962	250	10	3	3	NUM
ejpam-4962	250	11	.	.	PUNCT
ejpam-4962	251	1	let	let	VERB
ejpam-4962	251	2	g	g	PROPN
ejpam-4962	251	3	=	=	PUNCT
ejpam-4962	251	4	(	(	PUNCT
ejpam-4962	251	5	v	v	NUM
ejpam-4962	251	6	′	′	NUM
ejpam-4962	251	7	0	0	NUM
ejpam-4962	251	8	,	,	PUNCT
ejpam-4962	251	9	v	v	NOUN
ejpam-4962	251	10	′	′	NUM
ejpam-4962	251	11	1	1	NUM
ejpam-4962	251	12	,	,	PUNCT
ejpam-4962	251	13	v	v	NOUN
ejpam-4962	251	14	′	′	NUM
ejpam-4962	251	15	2	2	NUM
ejpam-4962	251	16	)	)	PUNCT
ejpam-4962	251	17	be	be	AUX
ejpam-4962	251	18	a	a	DET
ejpam-4962	251	19	γgr	γgr	NOUN
ejpam-4962	251	20	-	-	PUNCT
ejpam-4962	251	21	function	function	NOUN
ejpam-4962	251	22	on	on	ADP
ejpam-4962	251	23	cn	cn	PROPN
ejpam-4962	251	24	.	.	PUNCT
ejpam-4962	252	1	since	since	SCONJ
ejpam-4962	252	2	γgr(g	γgr(g	PROPN
ejpam-4962	252	3	)	)	PUNCT
ejpam-4962	252	4	≤	≤	NOUN
ejpam-4962	253	1	2n+2	2n+2	NUM
ejpam-4962	253	2	3	3	NUM
ejpam-4962	253	3	,	,	PUNCT
ejpam-4962	253	4	it	it	PRON
ejpam-4962	253	5	follows	follow	VERB
ejpam-4962	253	6	that	that	SCONJ
ejpam-4962	253	7	|v	|v	PROPN
ejpam-4962	253	8	′	′	NUM
ejpam-4962	253	9	2	2	NUM
ejpam-4962	253	10	|	|	ADV
ejpam-4962	253	11	̸=	̸=	PROPN
ejpam-4962	253	12	∅.	∅.	ADV
ejpam-4962	253	13	suppose	suppose	VERB
ejpam-4962	253	14	|v	|v	PROPN
ejpam-4962	253	15	′	′	ADP
ejpam-4962	253	16	1	1	NUM
ejpam-4962	254	1	|	|	ADV
ejpam-4962	254	2	=	=	PUNCT
ejpam-4962	254	3	k.	k.	PROPN
ejpam-4962	255	1	then	then	ADV
ejpam-4962	255	2	k	k	PROPN
ejpam-4962	256	1	+	+	PROPN
ejpam-4962	257	1	2|v	2|v	NUM
ejpam-4962	257	2	′	′	NUM
ejpam-4962	257	3	2	2	NUM
ejpam-4962	258	1	|	|	ADV
ejpam-4962	258	2	≤	≤	PUNCT
ejpam-4962	258	3	2n+2	2n+2	PROPN
ejpam-4962	258	4	3	3	NUM
ejpam-4962	258	5	.	.	PUNCT
ejpam-4962	259	1	thus	thus	ADV
ejpam-4962	259	2	|v	|v	ADV
ejpam-4962	259	3	′	′	NUM
ejpam-4962	259	4	2	2	NUM
ejpam-4962	259	5	|	|	ADV
ejpam-4962	259	6	≤	≤	X
ejpam-4962	259	7	t	t	NOUN
ejpam-4962	259	8	+	+	CCONJ
ejpam-4962	259	9	2−k	2−k	NUM
ejpam-4962	259	10	2	2	NUM
ejpam-4962	259	11	and	and	CCONJ
ejpam-4962	259	12	|v	|v	ADJ
ejpam-4962	259	13	′	′	NOUN
ejpam-4962	259	14	0	0	NUM
ejpam-4962	260	1	|	|	ADV
ejpam-4962	260	2	=	=	SYM
ejpam-4962	260	3	n	n	PRON
ejpam-4962	260	4	−	−	PROPN
ejpam-4962	260	5	(	(	PUNCT
ejpam-4962	260	6	|v	|v	PROPN
ejpam-4962	260	7	′	′	NUM
ejpam-4962	260	8	1	1	NUM
ejpam-4962	261	1	|	|	ADV
ejpam-4962	261	2	+	+	CCONJ
ejpam-4962	261	3	|v	|v	ADJ
ejpam-4962	261	4	′	′	NOUN
ejpam-4962	261	5	2	2	NUM
ejpam-4962	261	6	|	|	NOUN
ejpam-4962	261	7	)	)	PUNCT
ejpam-4962	261	8	≥	≥	NOUN
ejpam-4962	261	9	2	2	NUM
ejpam-4962	261	10	t	t	NOUN
ejpam-4962	261	11	+	+	CCONJ
ejpam-4962	261	12	2−k	2−k	NUM
ejpam-4962	261	13	2	2	NUM
ejpam-4962	261	14	.	.	PUNCT
ejpam-4962	262	1	if	if	SCONJ
ejpam-4962	262	2	k	k	PROPN
ejpam-4962	262	3	=	=	SYM
ejpam-4962	262	4	0	0	PROPN
ejpam-4962	262	5	,	,	PUNCT
ejpam-4962	262	6	then	then	ADV
ejpam-4962	262	7	|v	|v	VERB
ejpam-4962	262	8	′	′	NOUN
ejpam-4962	262	9	2	2	NUM
ejpam-4962	262	10	|	|	ADV
ejpam-4962	262	11	≤	≤	NUM
ejpam-4962	262	12	t	t	NOUN
ejpam-4962	262	13	+	+	CCONJ
ejpam-4962	262	14	1	1	NUM
ejpam-4962	262	15	and	and	CCONJ
ejpam-4962	262	16	|v	|v	ADJ
ejpam-4962	262	17	′	′	NOUN
ejpam-4962	262	18	0	0	NUM
ejpam-4962	263	1	|	|	CCONJ
ejpam-4962	263	2	≥	≥	NUM
ejpam-4962	263	3	2	2	NUM
ejpam-4962	263	4	t	t	NOUN
ejpam-4962	263	5	+	+	NOUN
ejpam-4962	263	6	1	1	NUM
ejpam-4962	263	7	.	.	PUNCT
ejpam-4962	263	8	hence	hence	ADV
ejpam-4962	263	9	,	,	PUNCT
ejpam-4962	263	10	|v	|v	ADJ
ejpam-4962	263	11	′	′	NOUN
ejpam-4962	263	12	2	2	NUM
ejpam-4962	263	13	|	|	ADV
ejpam-4962	263	14	≤	≤	X
ejpam-4962	263	15	t	t	NOUN
ejpam-4962	263	16	and	and	CCONJ
ejpam-4962	263	17	|v	|v	PROPN
ejpam-4962	263	18	′	′	NOUN
ejpam-4962	263	19	0	0	NUM
ejpam-4962	264	1	|	|	CCONJ
ejpam-4962	264	2	≥	≥	NUM
ejpam-4962	264	3	2	2	NUM
ejpam-4962	264	4	t	t	NOUN
ejpam-4962	264	5	+	+	NOUN
ejpam-4962	264	6	1	1	X
ejpam-4962	264	7	.	.	PUNCT
ejpam-4962	265	1	if	if	SCONJ
ejpam-4962	265	2	k	k	PROPN
ejpam-4962	265	3	=	=	SYM
ejpam-4962	265	4	1	1	NUM
ejpam-4962	265	5	,	,	PUNCT
ejpam-4962	265	6	then	then	ADV
ejpam-4962	265	7	|v	|v	VERB
ejpam-4962	265	8	′	′	NOUN
ejpam-4962	265	9	2	2	NUM
ejpam-4962	265	10	|	|	ADV
ejpam-4962	265	11	≤	≤	X
ejpam-4962	265	12	t	t	NOUN
ejpam-4962	265	13	+	+	CCONJ
ejpam-4962	265	14	1	1	NUM
ejpam-4962	265	15	2	2	NUM
ejpam-4962	265	16	and	and	CCONJ
ejpam-4962	265	17	|v	|v	ADJ
ejpam-4962	265	18	′	′	NOUN
ejpam-4962	265	19	0	0	NUM
ejpam-4962	266	1	|	|	CCONJ
ejpam-4962	266	2	≥	≥	NUM
ejpam-4962	266	3	2	2	NUM
ejpam-4962	266	4	t	t	NOUN
ejpam-4962	266	5	+	+	NOUN
ejpam-4962	266	6	1	1	NUM
ejpam-4962	266	7	2	2	NUM
ejpam-4962	266	8	.	.	PUNCT
ejpam-4962	267	1	hence	hence	ADV
ejpam-4962	267	2	,	,	PUNCT
ejpam-4962	267	3	|v	|v	ADJ
ejpam-4962	267	4	′	′	NOUN
ejpam-4962	267	5	2	2	NUM
ejpam-4962	267	6	|	|	ADV
ejpam-4962	267	7	≤	≤	X
ejpam-4962	267	8	t	t	NOUN
ejpam-4962	267	9	and	and	CCONJ
ejpam-4962	267	10	|v	|v	PROPN
ejpam-4962	267	11	′	′	NOUN
ejpam-4962	267	12	0	0	NUM
ejpam-4962	268	1	|	|	CCONJ
ejpam-4962	268	2	≥	≥	NUM
ejpam-4962	268	3	2	2	NUM
ejpam-4962	268	4	t	t	NOUN
ejpam-4962	268	5	+	+	NOUN
ejpam-4962	268	6	1	1	NUM
ejpam-4962	268	7	,	,	PUNCT
ejpam-4962	268	8	which	which	PRON
ejpam-4962	268	9	is	be	AUX
ejpam-4962	268	10	not	not	PART
ejpam-4962	268	11	possible	possible	ADJ
ejpam-4962	268	12	.	.	PUNCT
ejpam-4962	269	1	thus	thus	ADV
ejpam-4962	269	2	,	,	PUNCT
ejpam-4962	269	3	k	k	PROPN
ejpam-4962	269	4	̸=	̸=	PROPN
ejpam-4962	269	5	1	1	NUM
ejpam-4962	269	6	.	.	PUNCT
ejpam-4962	269	7	suppose	suppose	VERB
ejpam-4962	269	8	k	k	PROPN
ejpam-4962	269	9	≥	≥	NUM
ejpam-4962	269	10	3	3	NUM
ejpam-4962	269	11	.	.	PUNCT
ejpam-4962	269	12	then	then	ADV
ejpam-4962	269	13	|v	|v	PROPN
ejpam-4962	269	14	′	′	NOUN
ejpam-4962	269	15	2	2	NUM
ejpam-4962	269	16	|	|	ADV
ejpam-4962	269	17	≤	≤	NUM
ejpam-4962	269	18	t+	t+	PUNCT
ejpam-4962	269	19	2−k	2−k	NUM
ejpam-4962	269	20	2	2	NUM
ejpam-4962	269	21	implies	imply	VERB
ejpam-4962	269	22	that	that	SCONJ
ejpam-4962	269	23	|v	|v	PROPN
ejpam-4962	269	24	′	′	NOUN
ejpam-4962	269	25	0	0	NUM
ejpam-4962	270	1	|	|	ADV
ejpam-4962	270	2	≤	≤	NUM
ejpam-4962	270	3	2|v	2|v	NUM
ejpam-4962	271	1	′	′	NUM
ejpam-4962	271	2	2	2	NUM
ejpam-4962	271	3	|	|	ADV
ejpam-4962	271	4	≤	≤	NOUN
ejpam-4962	271	5	2t+2−k	2t+2−k	ADV
ejpam-4962	271	6	.	.	PUNCT
ejpam-4962	272	1	however	however	ADV
ejpam-4962	272	2	,	,	PUNCT
ejpam-4962	272	3	|v	|v	ADJ
ejpam-4962	272	4	′	′	NOUN
ejpam-4962	272	5	0	0	NUM
ejpam-4962	273	1	|	|	CCONJ
ejpam-4962	273	2	≥	≥	NUM
ejpam-4962	273	3	2t+	2t+	NUM
ejpam-4962	273	4	2−k	2−k	NUM
ejpam-4962	273	5	2	2	NUM
ejpam-4962	273	6	>	>	X
ejpam-4962	273	7	2t+2−k	2t+2−k	NUM
ejpam-4962	273	8	,	,	PUNCT
ejpam-4962	273	9	a	a	DET
ejpam-4962	273	10	contradiction	contradiction	NOUN
ejpam-4962	273	11	.	.	PUNCT
ejpam-4962	274	1	therefore	therefore	ADV
ejpam-4962	274	2	,	,	PUNCT
ejpam-4962	274	3	k	k	PROPN
ejpam-4962	274	4	=	=	PUNCT
ejpam-4962	274	5	0	0	PROPN
ejpam-4962	274	6	or	or	CCONJ
ejpam-4962	274	7	k	k	X
ejpam-4962	274	8	=	=	SYM
ejpam-4962	274	9	2	2	X
ejpam-4962	274	10	.	.	PUNCT
ejpam-4962	275	1	if	if	SCONJ
ejpam-4962	275	2	k	k	PROPN
ejpam-4962	275	3	=	=	SYM
ejpam-4962	275	4	0	0	PROPN
ejpam-4962	275	5	,	,	PUNCT
ejpam-4962	275	6	by	by	ADP
ejpam-4962	275	7	proposition	proposition	NOUN
ejpam-4962	275	8	1(iv	1(iv	NUM
ejpam-4962	275	9	)	)	PUNCT
ejpam-4962	275	10	and	and	CCONJ
ejpam-4962	275	11	remark	remark	VERB
ejpam-4962	275	12	r.	r.	PROPN
ejpam-4962	275	13	fortosa	fortosa	PROPN
ejpam-4962	275	14	,	,	PUNCT
ejpam-4962	275	15	s.	s.	PROPN
ejpam-4962	275	16	canoy	canoy	PROPN
ejpam-4962	275	17	jr	jr	PROPN
ejpam-4962	275	18	.	.	PROPN
ejpam-4962	275	19	/	/	SYM
ejpam-4962	275	20	eur	eur	PROPN
ejpam-4962	275	21	.	.	PUNCT
ejpam-4962	276	1	j.	j.	PROPN
ejpam-4962	276	2	pure	pure	PROPN
ejpam-4962	276	3	appl	appl	PROPN
ejpam-4962	276	4	.	.	PROPN
ejpam-4962	276	5	math	math	PROPN
ejpam-4962	276	6	,	,	PUNCT
ejpam-4962	276	7	16	16	NUM
ejpam-4962	276	8	(	(	PUNCT
ejpam-4962	276	9	4	4	NUM
ejpam-4962	276	10	)	)	PUNCT
ejpam-4962	276	11	(	(	PUNCT
ejpam-4962	276	12	2023	2023	NUM
ejpam-4962	276	13	)	)	PUNCT
ejpam-4962	276	14	,	,	PUNCT
ejpam-4962	276	15	2368	2368	NUM
ejpam-4962	276	16	-	-	SYM
ejpam-4962	276	17	2383	2383	NUM
ejpam-4962	276	18	2374	2374	NUM
ejpam-4962	276	19	2(i	2(i	NUM
ejpam-4962	276	20	)	)	PUNCT
ejpam-4962	276	21	,	,	PUNCT
ejpam-4962	276	22	γgr(cn	γgr(cn	ADV
ejpam-4962	276	23	)	)	PUNCT
ejpam-4962	276	24	=	=	SYM
ejpam-4962	277	1	ωgr	ωgr	NUM
ejpam-4962	277	2	cn	cn	X
ejpam-4962	277	3	(	(	PUNCT
ejpam-4962	277	4	g	g	NOUN
ejpam-4962	277	5	)	)	PUNCT
ejpam-4962	277	6	=	=	PUNCT
ejpam-4962	278	1	2n+2	2n+2	NUM
ejpam-4962	278	2	3	3	NUM
ejpam-4962	278	3	.	.	PUNCT
ejpam-4962	279	1	if	if	SCONJ
ejpam-4962	279	2	k	k	PROPN
ejpam-4962	279	3	=	=	SYM
ejpam-4962	279	4	2	2	NUM
ejpam-4962	279	5	,	,	PUNCT
ejpam-4962	279	6	then	then	ADV
ejpam-4962	279	7	g	g	PROPN
ejpam-4962	279	8	is	be	AUX
ejpam-4962	279	9	of	of	ADP
ejpam-4962	279	10	the	the	DET
ejpam-4962	279	11	same	same	ADJ
ejpam-4962	279	12	type	type	NOUN
ejpam-4962	279	13	as	as	ADP
ejpam-4962	279	14	the	the	DET
ejpam-4962	279	15	function	function	NOUN
ejpam-4962	279	16	f	f	PROPN
ejpam-4962	279	17	defined	define	VERB
ejpam-4962	279	18	earlier	early	ADV
ejpam-4962	279	19	.	.	PUNCT
ejpam-4962	280	1	hence	hence	ADV
ejpam-4962	280	2	,	,	PUNCT
ejpam-4962	280	3	γgr(cn	γgr(cn	ADV
ejpam-4962	280	4	)	)	PUNCT
ejpam-4962	281	1	=	=	SYM
ejpam-4962	281	2	ωgr	ωgr	NUM
ejpam-4962	281	3	cn	cn	X
ejpam-4962	281	4	(	(	PUNCT
ejpam-4962	281	5	g	g	NOUN
ejpam-4962	281	6	)	)	PUNCT
ejpam-4962	281	7	=	=	PUNCT
ejpam-4962	282	1	2n+2	2n+2	NUM
ejpam-4962	282	2	3	3	NUM
ejpam-4962	282	3	.	.	PUNCT
ejpam-4962	283	1	(	(	PUNCT
ejpam-4962	283	2	ii	ii	NOUN
ejpam-4962	283	3	)	)	PUNCT
ejpam-4962	283	4	let	let	VERB
ejpam-4962	283	5	pn	pn	NOUN
ejpam-4962	283	6	=	=	PUNCT
ejpam-4962	284	1	[	[	X
ejpam-4962	284	2	v1	v1	NOUN
ejpam-4962	284	3	,	,	PUNCT
ejpam-4962	284	4	v2	v2	NOUN
ejpam-4962	284	5	,	,	PUNCT
ejpam-4962	284	6	.	.	PUNCT
ejpam-4962	284	7	.	.	PUNCT
ejpam-4962	284	8	.	.	PUNCT
ejpam-4962	285	1	,	,	PUNCT
ejpam-4962	285	2	vn	vn	X
ejpam-4962	285	3	]	]	PUNCT
ejpam-4962	285	4	.	.	PUNCT
ejpam-4962	286	1	clearly	clearly	ADV
ejpam-4962	286	2	,	,	PUNCT
ejpam-4962	286	3	γgr(p1	γgr(p1	NOUN
ejpam-4962	286	4	)	)	PUNCT
ejpam-4962	287	1	=	=	SYM
ejpam-4962	287	2	1	1	X
ejpam-4962	287	3	.	.	PUNCT
ejpam-4962	287	4	suppose	suppose	VERB
ejpam-4962	287	5	n	n	PRON
ejpam-4962	287	6	≥	≥	NUM
ejpam-4962	287	7	2	2	NUM
ejpam-4962	287	8	.	.	PUNCT
ejpam-4962	287	9	consider	consider	VERB
ejpam-4962	287	10	the	the	DET
ejpam-4962	287	11	following	follow	VERB
ejpam-4962	287	12	cases	case	NOUN
ejpam-4962	287	13	:	:	PUNCT
ejpam-4962	287	14	case	case	NOUN
ejpam-4962	287	15	1	1	NUM
ejpam-4962	287	16	:	:	PUNCT
ejpam-4962	287	17	n	n	NUM
ejpam-4962	287	18	≡	≡	PROPN
ejpam-4962	287	19	0(mod	0(mod	NOUN
ejpam-4962	287	20	3	3	X
ejpam-4962	287	21	)	)	PUNCT
ejpam-4962	287	22	let	let	VERB
ejpam-4962	287	23	n	n	NOUN
ejpam-4962	287	24	=	=	NOUN
ejpam-4962	287	25	3r	3r	NUM
ejpam-4962	287	26	for	for	SCONJ
ejpam-4962	287	27	some	some	DET
ejpam-4962	287	28	positve	positve	NOUN
ejpam-4962	287	29	integer	integer	NOUN
ejpam-4962	287	30	r.	r.	PROPN
ejpam-4962	287	31	let	let	VERB
ejpam-4962	287	32	v2	v2	PROPN
ejpam-4962	287	33	=	=	SYM
ejpam-4962	287	34	{	{	PUNCT
ejpam-4962	287	35	v1	v1	PROPN
ejpam-4962	287	36	,	,	PUNCT
ejpam-4962	287	37	v3	v3	PROPN
ejpam-4962	287	38	,	,	PUNCT
ejpam-4962	287	39	.	.	PUNCT
ejpam-4962	287	40	.	.	PUNCT
ejpam-4962	288	1	.	.	PUNCT
ejpam-4962	289	1	,	,	PUNCT
ejpam-4962	289	2	v3r−2	v3r−2	PROPN
ejpam-4962	289	3	}	}	PUNCT
ejpam-4962	289	4	,	,	PUNCT
ejpam-4962	289	5	v1	v1	NOUN
ejpam-4962	289	6	=	=	SYM
ejpam-4962	289	7	{	{	PUNCT
ejpam-4962	289	8	v3r	v3r	NOUN
ejpam-4962	289	9	}	}	PUNCT
ejpam-4962	289	10	and	and	CCONJ
ejpam-4962	289	11	v0	v0	PROPN
ejpam-4962	289	12	=	=	SYM
ejpam-4962	289	13	v	v	PROPN
ejpam-4962	289	14	(	(	PUNCT
ejpam-4962	289	15	pn	pn	NOUN
ejpam-4962	289	16	)	)	PUNCT
ejpam-4962	289	17	\	\	PUNCT
ejpam-4962	290	1	(	(	PUNCT
ejpam-4962	290	2	v1	v1	VERB
ejpam-4962	290	3	∪	∪	NOUN
ejpam-4962	290	4	v2	v2	NOUN
ejpam-4962	290	5	)	)	PUNCT
ejpam-4962	290	6	.	.	PUNCT
ejpam-4962	291	1	thus	thus	ADV
ejpam-4962	291	2	f	f	X
ejpam-4962	291	3	=	=	SYM
ejpam-4962	291	4	(	(	PUNCT
ejpam-4962	291	5	v0	v0	PROPN
ejpam-4962	291	6	,	,	PUNCT
ejpam-4962	291	7	v1	v1	NOUN
ejpam-4962	291	8	,	,	PUNCT
ejpam-4962	291	9	v2	v2	PROPN
ejpam-4962	291	10	)	)	PUNCT
ejpam-4962	291	11	is	be	AUX
ejpam-4962	291	12	a	a	DET
ejpam-4962	291	13	grdf	grdf	NOUN
ejpam-4962	291	14	on	on	ADP
ejpam-4962	291	15	pn	pn	PROPN
ejpam-4962	291	16	.	.	PROPN
ejpam-4962	292	1	hence	hence	ADV
ejpam-4962	292	2	,	,	PUNCT
ejpam-4962	292	3	γgr(pn	γgr(pn	NOUN
ejpam-4962	292	4	)	)	PUNCT
ejpam-4962	292	5	≤	≤	NUM
ejpam-4962	292	6	ωgr	ωgr	X
ejpam-4962	292	7	pn	pn	X
ejpam-4962	292	8	(	(	PUNCT
ejpam-4962	292	9	f	f	X
ejpam-4962	292	10	)	)	PUNCT
ejpam-4962	292	11	=	=	PUNCT
ejpam-4962	292	12	|v1|+	|v1|+	PRON
ejpam-4962	292	13	2|v2|	2|v2|	NUM
ejpam-4962	292	14	=	=	SYM
ejpam-4962	292	15	1	1	NUM
ejpam-4962	292	16	+	+	NUM
ejpam-4962	292	17	2(n3	2(n3	NUM
ejpam-4962	292	18	)	)	PUNCT
ejpam-4962	292	19	=	=	PUNCT
ejpam-4962	292	20	2n+3	2n+3	NOUN
ejpam-4962	292	21	3	3	NUM
ejpam-4962	292	22	.	.	PUNCT
ejpam-4962	293	1	let	let	VERB
ejpam-4962	293	2	g	g	PROPN
ejpam-4962	293	3	=	=	PUNCT
ejpam-4962	293	4	(	(	PUNCT
ejpam-4962	293	5	v	v	NUM
ejpam-4962	293	6	′	′	NUM
ejpam-4962	293	7	0	0	NUM
ejpam-4962	293	8	,	,	PUNCT
ejpam-4962	293	9	v	v	NOUN
ejpam-4962	293	10	′	′	NUM
ejpam-4962	293	11	1	1	NUM
ejpam-4962	293	12	,	,	PUNCT
ejpam-4962	293	13	v	v	NOUN
ejpam-4962	293	14	′	′	NUM
ejpam-4962	293	15	2	2	NUM
ejpam-4962	293	16	)	)	PUNCT
ejpam-4962	293	17	be	be	AUX
ejpam-4962	293	18	a	a	DET
ejpam-4962	293	19	γgr	γgr	NOUN
ejpam-4962	293	20	-	-	PUNCT
ejpam-4962	293	21	function	function	NOUN
ejpam-4962	293	22	.	.	PUNCT
ejpam-4962	294	1	since	since	SCONJ
ejpam-4962	294	2	γgr(g	γgr(g	PROPN
ejpam-4962	294	3	)	)	PUNCT
ejpam-4962	294	4	≤	≤	NOUN
ejpam-4962	295	1	2n+3	2n+3	NOUN
ejpam-4962	295	2	3	3	NUM
ejpam-4962	295	3	,	,	PUNCT
ejpam-4962	295	4	it	it	PRON
ejpam-4962	295	5	follows	follow	VERB
ejpam-4962	295	6	that	that	SCONJ
ejpam-4962	295	7	v	v	X
ejpam-4962	295	8	′	′	NOUN
ejpam-4962	295	9	2	2	NUM
ejpam-4962	295	10	̸=	̸=	PROPN
ejpam-4962	295	11	∅.	∅.	ADV
ejpam-4962	295	12	suppose	suppose	VERB
ejpam-4962	295	13	|v	|v	PROPN
ejpam-4962	295	14	′	′	ADP
ejpam-4962	295	15	1	1	NUM
ejpam-4962	296	1	|	|	ADV
ejpam-4962	296	2	=	=	PUNCT
ejpam-4962	296	3	k.	k.	PROPN
ejpam-4962	297	1	then	then	ADV
ejpam-4962	297	2	k	k	PROPN
ejpam-4962	298	1	+	+	PROPN
ejpam-4962	299	1	2|v	2|v	NUM
ejpam-4962	300	1	′	′	NUM
ejpam-4962	300	2	2	2	NUM
ejpam-4962	301	1	|	|	ADV
ejpam-4962	301	2	≤	≤	NUM
ejpam-4962	301	3	2n+3	2n+3	NOUN
ejpam-4962	301	4	3	3	NUM
ejpam-4962	301	5	.	.	PUNCT
ejpam-4962	302	1	thus	thus	ADV
ejpam-4962	302	2	|v	|v	ADV
ejpam-4962	302	3	′	′	NUM
ejpam-4962	302	4	2	2	NUM
ejpam-4962	302	5	|	|	ADV
ejpam-4962	302	6	≤	≤	NUM
ejpam-4962	302	7	r	r	NOUN
ejpam-4962	302	8	−	−	NUM
ejpam-4962	302	9	1	1	NUM
ejpam-4962	302	10	2(k	2(k	NUM
ejpam-4962	302	11	−	−	NOUN
ejpam-4962	302	12	1	1	NUM
ejpam-4962	302	13	)	)	PUNCT
ejpam-4962	302	14	and	and	CCONJ
ejpam-4962	302	15	|v	|v	ADJ
ejpam-4962	302	16	′	′	NOUN
ejpam-4962	302	17	0	0	NUM
ejpam-4962	303	1	|	|	ADV
ejpam-4962	303	2	=	=	SYM
ejpam-4962	303	3	n	n	PRON
ejpam-4962	303	4	−	−	PROPN
ejpam-4962	303	5	(	(	PUNCT
ejpam-4962	303	6	|v	|v	PROPN
ejpam-4962	303	7	′	′	NUM
ejpam-4962	303	8	1	1	NUM
ejpam-4962	304	1	|	|	ADV
ejpam-4962	304	2	+	+	CCONJ
ejpam-4962	304	3	|v	|v	ADJ
ejpam-4962	304	4	′	′	NOUN
ejpam-4962	304	5	2	2	NUM
ejpam-4962	304	6	|	|	NOUN
ejpam-4962	304	7	)	)	PUNCT
ejpam-4962	304	8	≥	≥	NOUN
ejpam-4962	305	1	2r	2r	NUM
ejpam-4962	305	2	−	−	NOUN
ejpam-4962	305	3	1	1	NUM
ejpam-4962	305	4	2(k	2(k	NUM
ejpam-4962	305	5	+	+	CCONJ
ejpam-4962	305	6	1	1	NUM
ejpam-4962	305	7	)	)	PUNCT
ejpam-4962	305	8	.	.	PUNCT
ejpam-4962	306	1	suppose	suppose	VERB
ejpam-4962	306	2	k	k	PROPN
ejpam-4962	306	3	=	=	PUNCT
ejpam-4962	306	4	0	0	PROPN
ejpam-4962	306	5	.	.	PUNCT
ejpam-4962	307	1	then	then	ADV
ejpam-4962	307	2	|v	|v	PROPN
ejpam-4962	307	3	′	′	NOUN
ejpam-4962	307	4	2	2	NUM
ejpam-4962	308	1	|	|	ADV
ejpam-4962	308	2	≤	≤	NUM
ejpam-4962	308	3	r	r	NOUN
ejpam-4962	308	4	+	+	CCONJ
ejpam-4962	308	5	1	1	NUM
ejpam-4962	308	6	2	2	NUM
ejpam-4962	308	7	and	and	CCONJ
ejpam-4962	308	8	|v	|v	ADJ
ejpam-4962	308	9	′	′	NOUN
ejpam-4962	308	10	0	0	NUM
ejpam-4962	309	1	|	|	CCONJ
ejpam-4962	309	2	≥	≥	X
ejpam-4962	309	3	2r	2r	NUM
ejpam-4962	309	4	−	−	NOUN
ejpam-4962	309	5	1	1	NUM
ejpam-4962	309	6	2	2	NUM
ejpam-4962	309	7	.	.	PUNCT
ejpam-4962	310	1	this	this	PRON
ejpam-4962	310	2	implies	imply	VERB
ejpam-4962	310	3	that	that	SCONJ
ejpam-4962	310	4	|v	|v	PROPN
ejpam-4962	310	5	′	′	NUM
ejpam-4962	310	6	2	2	NUM
ejpam-4962	310	7	|	|	ADV
ejpam-4962	310	8	≤	≤	NUM
ejpam-4962	310	9	r	r	NOUN
ejpam-4962	310	10	and	and	CCONJ
ejpam-4962	310	11	|v	|v	ADJ
ejpam-4962	310	12	′	′	NOUN
ejpam-4962	310	13	0	0	NUM
ejpam-4962	311	1	|	|	CCONJ
ejpam-4962	311	2	≥	≥	NOUN
ejpam-4962	311	3	2r	2r	NUM
ejpam-4962	311	4	.	.	PUNCT
ejpam-4962	312	1	since	since	SCONJ
ejpam-4962	312	2	|v	|v	PROPN
ejpam-4962	312	3	′	′	NUM
ejpam-4962	312	4	1	1	NUM
ejpam-4962	313	1	|	|	ADV
ejpam-4962	313	2	=	=	SYM
ejpam-4962	313	3	0	0	NUM
ejpam-4962	313	4	,	,	PUNCT
ejpam-4962	313	5	|v	|v	VERB
ejpam-4962	313	6	′	′	NOUN
ejpam-4962	313	7	0	0	NUM
ejpam-4962	314	1	|	|	ADV
ejpam-4962	314	2	<	<	X
ejpam-4962	314	3	2|v	2|v	NOUN
ejpam-4962	314	4	′	′	NUM
ejpam-4962	314	5	2	2	NUM
ejpam-4962	315	1	|	|	ADV
ejpam-4962	315	2	(	(	PUNCT
ejpam-4962	315	3	as	as	ADP
ejpam-4962	315	4	v1	v1	NOUN
ejpam-4962	315	5	∈	∈	NOUN
ejpam-4962	315	6	v	v	NOUN
ejpam-4962	315	7	′	′	NUM
ejpam-4962	315	8	2	2	NUM
ejpam-4962	315	9	or	or	CCONJ
ejpam-4962	315	10	vn	vn	ADP
ejpam-4962	315	11	∈	∈	PROPN
ejpam-4962	315	12	v	v	ADP
ejpam-4962	315	13	′	′	NUM
ejpam-4962	315	14	2	2	NUM
ejpam-4962	315	15	;	;	PUNCT
ejpam-4962	315	16	hence	hence	ADV
ejpam-4962	315	17	,	,	PUNCT
ejpam-4962	315	18	at	at	ADV
ejpam-4962	315	19	least	least	ADJ
ejpam-4962	315	20	one	one	NUM
ejpam-4962	315	21	of	of	ADP
ejpam-4962	315	22	them	they	PRON
ejpam-4962	315	23	has	have	VERB
ejpam-4962	315	24	only	only	ADV
ejpam-4962	315	25	one	one	NUM
ejpam-4962	315	26	neighbor	neighbor	NOUN
ejpam-4962	315	27	in	in	ADP
ejpam-4962	315	28	v	v	NUM
ejpam-4962	315	29	′	′	NUM
ejpam-4962	315	30	0	0	NUM
ejpam-4962	315	31	)	)	PUNCT
ejpam-4962	315	32	.	.	PUNCT
ejpam-4962	316	1	thus	thus	ADV
ejpam-4962	316	2	,	,	PUNCT
ejpam-4962	316	3	|v	|v	ADJ
ejpam-4962	316	4	′	′	NOUN
ejpam-4962	316	5	2	2	NUM
ejpam-4962	316	6	|	|	ADV
ejpam-4962	316	7	≤	≤	NUM
ejpam-4962	316	8	r	r	NOUN
ejpam-4962	316	9	implies	imply	VERB
ejpam-4962	316	10	that	that	SCONJ
ejpam-4962	316	11	|v	|v	PROPN
ejpam-4962	316	12	′	′	NOUN
ejpam-4962	316	13	0	0	NUM
ejpam-4962	317	1	|	|	ADV
ejpam-4962	317	2	<	<	X
ejpam-4962	317	3	2r	2r	NUM
ejpam-4962	317	4	.	.	PUNCT
ejpam-4962	318	1	this	this	PRON
ejpam-4962	318	2	contradicts	contradict	VERB
ejpam-4962	318	3	the	the	DET
ejpam-4962	318	4	fact	fact	NOUN
ejpam-4962	318	5	that	that	SCONJ
ejpam-4962	318	6	|v	|v	PROPN
ejpam-4962	318	7	′	′	NOUN
ejpam-4962	318	8	0	0	NUM
ejpam-4962	319	1	|	|	CCONJ
ejpam-4962	319	2	≥	≥	X
ejpam-4962	319	3	2r	2r	NUM
ejpam-4962	319	4	.	.	PUNCT
ejpam-4962	319	5	suppose	suppose	VERB
ejpam-4962	319	6	k	k	X
ejpam-4962	319	7	=	=	SYM
ejpam-4962	319	8	2	2	X
ejpam-4962	319	9	.	.	PUNCT
ejpam-4962	319	10	then	then	ADV
ejpam-4962	319	11	|v	|v	VERB
ejpam-4962	319	12	′	′	NOUN
ejpam-4962	319	13	2	2	NUM
ejpam-4962	320	1	|	|	ADV
ejpam-4962	320	2	≤	≤	NUM
ejpam-4962	320	3	r−	r−	PROPN
ejpam-4962	320	4	1	1	NUM
ejpam-4962	320	5	2	2	NUM
ejpam-4962	320	6	and	and	CCONJ
ejpam-4962	320	7	|v	|v	ADJ
ejpam-4962	320	8	′	′	NOUN
ejpam-4962	320	9	0	0	NUM
ejpam-4962	321	1	|	|	CCONJ
ejpam-4962	321	2	≥	≥	X
ejpam-4962	321	3	2r−	2r−	NOUN
ejpam-4962	321	4	3	3	NUM
ejpam-4962	321	5	2	2	NUM
ejpam-4962	321	6	.	.	PUNCT
ejpam-4962	322	1	this	this	PRON
ejpam-4962	322	2	implies	imply	VERB
ejpam-4962	322	3	that	that	SCONJ
ejpam-4962	322	4	|v	|v	PROPN
ejpam-4962	322	5	′	′	NUM
ejpam-4962	322	6	2	2	NUM
ejpam-4962	322	7	|	|	ADV
ejpam-4962	322	8	≤	≤	PUNCT
ejpam-4962	322	9	r−1	r−1	PROPN
ejpam-4962	322	10	and	and	CCONJ
ejpam-4962	322	11	|v	|v	VERB
ejpam-4962	322	12	′	′	NOUN
ejpam-4962	322	13	0	0	NUM
ejpam-4962	323	1	|	|	CCONJ
ejpam-4962	323	2	≥	≥	NUM
ejpam-4962	323	3	2r−1	2r−1	NUM
ejpam-4962	323	4	.	.	PUNCT
ejpam-4962	324	1	this	this	PRON
ejpam-4962	324	2	is	be	AUX
ejpam-4962	324	3	not	not	PART
ejpam-4962	324	4	possible	possible	ADJ
ejpam-4962	324	5	.	.	PUNCT
ejpam-4962	325	1	suppose	suppose	VERB
ejpam-4962	325	2	k	k	PROPN
ejpam-4962	325	3	≥	≥	NUM
ejpam-4962	325	4	4	4	NUM
ejpam-4962	325	5	.	.	PUNCT
ejpam-4962	325	6	then	then	ADV
ejpam-4962	325	7	|v	|v	VERB
ejpam-4962	325	8	′	′	NOUN
ejpam-4962	325	9	2	2	NUM
ejpam-4962	326	1	|	|	ADV
ejpam-4962	326	2	≤	≤	NUM
ejpam-4962	326	3	r−	r−	PROPN
ejpam-4962	326	4	1	1	NUM
ejpam-4962	326	5	2(k−1	2(k−1	NOUN
ejpam-4962	326	6	)	)	PUNCT
ejpam-4962	326	7	implies	imply	VERB
ejpam-4962	326	8	that	that	SCONJ
ejpam-4962	326	9	|v	|v	PROPN
ejpam-4962	326	10	′	′	NOUN
ejpam-4962	326	11	0	0	NUM
ejpam-4962	327	1	|	|	ADV
ejpam-4962	327	2	≤	≤	NUM
ejpam-4962	327	3	2r−(k−1	2r−(k−1	NUM
ejpam-4962	327	4	)	)	PUNCT
ejpam-4962	327	5	.	.	PUNCT
ejpam-4962	328	1	however	however	ADV
ejpam-4962	328	2	,	,	PUNCT
ejpam-4962	328	3	|v	|v	ADJ
ejpam-4962	328	4	′	′	NOUN
ejpam-4962	328	5	0	0	NUM
ejpam-4962	329	1	|	|	CCONJ
ejpam-4962	329	2	≥	≥	X
ejpam-4962	330	1	2r	2r	NUM
ejpam-4962	330	2	−	−	NOUN
ejpam-4962	330	3	1	1	NUM
ejpam-4962	330	4	2(k	2(k	NUM
ejpam-4962	330	5	+	+	CCONJ
ejpam-4962	330	6	1	1	X
ejpam-4962	330	7	)	)	PUNCT
ejpam-4962	330	8	>	>	X
ejpam-4962	331	1	2r	2r	NUM
ejpam-4962	331	2	−	−	PROPN
ejpam-4962	332	1	(	(	PUNCT
ejpam-4962	332	2	k	k	NOUN
ejpam-4962	332	3	−	−	PROPN
ejpam-4962	332	4	1	1	NUM
ejpam-4962	332	5	)	)	PUNCT
ejpam-4962	332	6	,	,	PUNCT
ejpam-4962	332	7	a	a	DET
ejpam-4962	332	8	contradiction	contradiction	NOUN
ejpam-4962	332	9	.	.	PUNCT
ejpam-4962	333	1	thus	thus	ADV
ejpam-4962	333	2	,	,	PUNCT
ejpam-4962	333	3	k	k	PROPN
ejpam-4962	333	4	=	=	SYM
ejpam-4962	333	5	1	1	NUM
ejpam-4962	333	6	or	or	CCONJ
ejpam-4962	333	7	k	k	NOUN
ejpam-4962	333	8	=	=	SYM
ejpam-4962	333	9	3	3	X
ejpam-4962	333	10	.	.	PUNCT
ejpam-4962	334	1	if	if	SCONJ
ejpam-4962	334	2	k	k	PROPN
ejpam-4962	334	3	=	=	SYM
ejpam-4962	334	4	1	1	NUM
ejpam-4962	334	5	,	,	PUNCT
ejpam-4962	334	6	then	then	ADV
ejpam-4962	334	7	g	g	PROPN
ejpam-4962	334	8	is	be	AUX
ejpam-4962	334	9	of	of	ADP
ejpam-4962	334	10	the	the	DET
ejpam-4962	334	11	same	same	ADJ
ejpam-4962	334	12	type	type	NOUN
ejpam-4962	334	13	as	as	ADP
ejpam-4962	334	14	the	the	DET
ejpam-4962	334	15	function	function	NOUN
ejpam-4962	334	16	f	f	PROPN
ejpam-4962	334	17	defined	define	VERB
ejpam-4962	334	18	earlier	early	ADV
ejpam-4962	334	19	.	.	PUNCT
ejpam-4962	335	1	hence	hence	ADV
ejpam-4962	335	2	γgr(pn	γgr(pn	NUM
ejpam-4962	335	3	)	)	PUNCT
ejpam-4962	336	1	=	=	PUNCT
ejpam-4962	336	2	2n+3	2n+3	NOUN
ejpam-4962	336	3	3	3	NUM
ejpam-4962	336	4	.	.	PUNCT
ejpam-4962	337	1	if	if	SCONJ
ejpam-4962	337	2	k	k	PROPN
ejpam-4962	337	3	=	=	SYM
ejpam-4962	337	4	3	3	NUM
ejpam-4962	337	5	,	,	PUNCT
ejpam-4962	337	6	then	then	ADV
ejpam-4962	337	7	we	we	PRON
ejpam-4962	337	8	may	may	AUX
ejpam-4962	337	9	consider	consider	VERB
ejpam-4962	337	10	h	h	NOUN
ejpam-4962	337	11	=	=	SYM
ejpam-4962	337	12	(	(	PUNCT
ejpam-4962	337	13	v	v	NUM
ejpam-4962	337	14	′′	′′	PROPN
ejpam-4962	337	15	0	0	NUM
ejpam-4962	337	16	,	,	PUNCT
ejpam-4962	337	17	v	v	ADP
ejpam-4962	337	18	′′	′′	PROPN
ejpam-4962	337	19	1	1	NUM
ejpam-4962	337	20	,	,	PUNCT
ejpam-4962	337	21	v	v	ADP
ejpam-4962	337	22	′′	′′	PROPN
ejpam-4962	337	23	2	2	NUM
ejpam-4962	337	24	)	)	PUNCT
ejpam-4962	337	25	where	where	SCONJ
ejpam-4962	337	26	v	v	X
ejpam-4962	337	27	′′	′′	PROPN
ejpam-4962	337	28	1	1	NUM
ejpam-4962	337	29	=	=	SYM
ejpam-4962	337	30	{	{	PUNCT
ejpam-4962	337	31	v1	v1	PROPN
ejpam-4962	337	32	,	,	PUNCT
ejpam-4962	337	33	v2	v2	NOUN
ejpam-4962	337	34	,	,	PUNCT
ejpam-4962	337	35	v3r	v3r	NOUN
ejpam-4962	337	36	}	}	PUNCT
ejpam-4962	337	37	,	,	PUNCT
ejpam-4962	337	38	v	v	ADP
ejpam-4962	337	39	′′	′′	PROPN
ejpam-4962	337	40	2	2	NUM
ejpam-4962	337	41	=	=	SYM
ejpam-4962	337	42	{	{	PUNCT
ejpam-4962	337	43	v4	v4	NOUN
ejpam-4962	337	44	,	,	PUNCT
ejpam-4962	337	45	v7	v7	NUM
ejpam-4962	337	46	,	,	PUNCT
ejpam-4962	337	47	.	.	PUNCT
ejpam-4962	337	48	.	.	PUNCT
ejpam-4962	338	1	.	.	PUNCT
ejpam-4962	339	1	,	,	PUNCT
ejpam-4962	339	2	v3r−2	v3r−2	NOUN
ejpam-4962	339	3	}	}	PUNCT
ejpam-4962	339	4	and	and	CCONJ
ejpam-4962	339	5	v	v	ADP
ejpam-4962	339	6	′′	′′	PROPN
ejpam-4962	339	7	0	0	NUM
ejpam-4962	339	8	=	=	SYM
ejpam-4962	339	9	v	v	PROPN
ejpam-4962	339	10	(	(	PUNCT
ejpam-4962	339	11	pn	pn	NOUN
ejpam-4962	339	12	)	)	PUNCT
ejpam-4962	339	13	\	\	PUNCT
ejpam-4962	340	1	(	(	PUNCT
ejpam-4962	340	2	v	v	NUM
ejpam-4962	340	3	′′	′′	PROPN
ejpam-4962	340	4	1	1	NUM
ejpam-4962	340	5	∪	∪	X
ejpam-4962	340	6	v	v	ADP
ejpam-4962	340	7	′′	′′	PROPN
ejpam-4962	340	8	2	2	NUM
ejpam-4962	340	9	)	)	PUNCT
ejpam-4962	340	10	.	.	PUNCT
ejpam-4962	341	1	hence	hence	ADV
ejpam-4962	341	2	,	,	PUNCT
ejpam-4962	341	3	h	h	NOUN
ejpam-4962	341	4	is	be	AUX
ejpam-4962	341	5	a	a	DET
ejpam-4962	341	6	grdf	grdf	NOUN
ejpam-4962	341	7	on	on	ADP
ejpam-4962	341	8	pn	pn	PROPN
ejpam-4962	341	9	and	and	CCONJ
ejpam-4962	341	10	ωgr	ωgr	NUM
ejpam-4962	341	11	pn	pn	PROPN
ejpam-4962	341	12	(	(	PUNCT
ejpam-4962	341	13	h	h	NOUN
ejpam-4962	341	14	)	)	PUNCT
ejpam-4962	341	15	=	=	PUNCT
ejpam-4962	342	1	2n+3	2n+3	NOUN
ejpam-4962	342	2	3	3	NUM
ejpam-4962	342	3	.	.	PUNCT
ejpam-4962	343	1	case	case	NOUN
ejpam-4962	343	2	2	2	NUM
ejpam-4962	343	3	:	:	PUNCT
ejpam-4962	343	4	n	n	NUM
ejpam-4962	343	5	≡	≡	PROPN
ejpam-4962	343	6	1(mod	1(mod	NUM
ejpam-4962	343	7	3	3	X
ejpam-4962	343	8	)	)	PUNCT
ejpam-4962	343	9	let	let	VERB
ejpam-4962	343	10	n	n	NOUN
ejpam-4962	343	11	=	=	SYM
ejpam-4962	343	12	3s	3s	NUM
ejpam-4962	343	13	+	+	CCONJ
ejpam-4962	343	14	1	1	NUM
ejpam-4962	343	15	for	for	ADP
ejpam-4962	343	16	some	some	DET
ejpam-4962	343	17	positive	positive	ADJ
ejpam-4962	343	18	integer	integer	NOUN
ejpam-4962	343	19	s.	s.	PROPN
ejpam-4962	343	20	let	let	VERB
ejpam-4962	343	21	v1	v1	NOUN
ejpam-4962	343	22	=	=	SYM
ejpam-4962	344	1	∅	∅	NOUN
ejpam-4962	344	2	v2	v2	NOUN
ejpam-4962	344	3	=	=	SYM
ejpam-4962	344	4	{	{	PUNCT
ejpam-4962	344	5	v1	v1	PROPN
ejpam-4962	344	6	,	,	PUNCT
ejpam-4962	344	7	v4	v4	NOUN
ejpam-4962	344	8	,	,	PUNCT
ejpam-4962	344	9	v7	v7	NOUN
ejpam-4962	344	10	,	,	PUNCT
ejpam-4962	344	11	.	.	PUNCT
ejpam-4962	344	12	.	.	PUNCT
ejpam-4962	344	13	.	.	PUNCT
ejpam-4962	345	1	,	,	PUNCT
ejpam-4962	345	2	v3s+1	v3s+1	NOUN
ejpam-4962	345	3	}	}	PUNCT
ejpam-4962	345	4	,	,	PUNCT
ejpam-4962	345	5	and	and	CCONJ
ejpam-4962	345	6	v0	v0	PROPN
ejpam-4962	345	7	=	=	SYM
ejpam-4962	345	8	v	v	PROPN
ejpam-4962	345	9	(	(	PUNCT
ejpam-4962	345	10	pn	pn	NOUN
ejpam-4962	345	11	)	)	PUNCT
ejpam-4962	345	12	\	\	PUNCT
ejpam-4962	345	13	(	(	PUNCT
ejpam-4962	345	14	v1	v1	VERB
ejpam-4962	345	15	∪	∪	NOUN
ejpam-4962	345	16	v2	v2	NOUN
ejpam-4962	345	17	)	)	PUNCT
ejpam-4962	345	18	.	.	PUNCT
ejpam-4962	346	1	thus	thus	ADV
ejpam-4962	346	2	f	f	X
ejpam-4962	346	3	=	=	SYM
ejpam-4962	346	4	(	(	PUNCT
ejpam-4962	346	5	v0	v0	PROPN
ejpam-4962	346	6	,	,	PUNCT
ejpam-4962	346	7	v1	v1	NOUN
ejpam-4962	346	8	,	,	PUNCT
ejpam-4962	346	9	v2	v2	PROPN
ejpam-4962	346	10	)	)	PUNCT
ejpam-4962	346	11	is	be	AUX
ejpam-4962	346	12	a	a	DET
ejpam-4962	346	13	grdf	grdf	NOUN
ejpam-4962	346	14	in	in	ADP
ejpam-4962	346	15	pn	pn	PROPN
ejpam-4962	346	16	.	.	PROPN
ejpam-4962	347	1	hence	hence	ADV
ejpam-4962	347	2	,	,	PUNCT
ejpam-4962	347	3	γgr(pn	γgr(pn	NOUN
ejpam-4962	347	4	)	)	PUNCT
ejpam-4962	347	5	≤	≤	NUM
ejpam-4962	347	6	ωgr	ωgr	NOUN
ejpam-4962	347	7	pn	pn	PROPN
ejpam-4962	347	8	=	=	SYM
ejpam-4962	347	9	2(n+2	2(n+2	NUM
ejpam-4962	347	10	3	3	NUM
ejpam-4962	347	11	)	)	PUNCT
ejpam-4962	347	12	=	=	SYM
ejpam-4962	348	1	2n+4	2n+4	NOUN
ejpam-4962	348	2	3	3	X
ejpam-4962	348	3	.	.	PUNCT
ejpam-4962	349	1	let	let	VERB
ejpam-4962	349	2	g	g	PROPN
ejpam-4962	349	3	=	=	PUNCT
ejpam-4962	349	4	(	(	PUNCT
ejpam-4962	349	5	v	v	NUM
ejpam-4962	349	6	′	′	NUM
ejpam-4962	349	7	0	0	NUM
ejpam-4962	349	8	,	,	PUNCT
ejpam-4962	349	9	v	v	NOUN
ejpam-4962	349	10	′	′	NUM
ejpam-4962	349	11	2	2	NUM
ejpam-4962	349	12	,	,	PUNCT
ejpam-4962	349	13	v	v	NOUN
ejpam-4962	349	14	′	′	NUM
ejpam-4962	349	15	2	2	NUM
ejpam-4962	349	16	)	)	PUNCT
ejpam-4962	349	17	be	be	AUX
ejpam-4962	349	18	a	a	DET
ejpam-4962	349	19	γgr	γgr	NOUN
ejpam-4962	349	20	-	-	PUNCT
ejpam-4962	349	21	function	function	NOUN
ejpam-4962	349	22	on	on	ADP
ejpam-4962	349	23	pn	pn	PROPN
ejpam-4962	349	24	.	.	PUNCT
ejpam-4962	350	1	since	since	SCONJ
ejpam-4962	350	2	γgr(pn	γgr(pn	NOUN
ejpam-4962	350	3	)	)	PUNCT
ejpam-4962	350	4	≤	≤	NUM
ejpam-4962	351	1	2n+4	2n+4	NOUN
ejpam-4962	351	2	3	3	NUM
ejpam-4962	351	3	,	,	PUNCT
ejpam-4962	351	4	it	it	PRON
ejpam-4962	351	5	follows	follow	VERB
ejpam-4962	351	6	that	that	SCONJ
ejpam-4962	351	7	v	v	X
ejpam-4962	351	8	′	′	NOUN
ejpam-4962	351	9	2	2	NUM
ejpam-4962	351	10	̸=	̸=	PROPN
ejpam-4962	351	11	∅.	∅.	ADV
ejpam-4962	351	12	suppose	suppose	VERB
ejpam-4962	351	13	that	that	SCONJ
ejpam-4962	351	14	|v	|v	PROPN
ejpam-4962	351	15	′	′	NUM
ejpam-4962	351	16	1	1	NUM
ejpam-4962	352	1	|	|	ADV
ejpam-4962	352	2	=	=	PUNCT
ejpam-4962	352	3	k.	k.	PROPN
ejpam-4962	353	1	then	then	ADV
ejpam-4962	353	2	k	k	PROPN
ejpam-4962	354	1	+	+	PROPN
ejpam-4962	355	1	2|v	2|v	NUM
ejpam-4962	356	1	′	′	NUM
ejpam-4962	356	2	2	2	NUM
ejpam-4962	357	1	|	|	ADV
ejpam-4962	357	2	≤	≤	NUM
ejpam-4962	357	3	2n+4	2n+4	PROPN
ejpam-4962	357	4	3	3	NUM
ejpam-4962	357	5	.	.	PUNCT
ejpam-4962	358	1	thus	thus	ADV
ejpam-4962	358	2	|v	|v	ADV
ejpam-4962	358	3	′	′	NUM
ejpam-4962	358	4	2	2	NUM
ejpam-4962	358	5	|	|	ADV
ejpam-4962	358	6	≤	≤	PROPN
ejpam-4962	358	7	s	s	AUX
ejpam-4962	358	8	−	−	PROPN
ejpam-4962	358	9	1	1	NUM
ejpam-4962	358	10	2(k	2(k	NUM
ejpam-4962	358	11	−	−	NOUN
ejpam-4962	358	12	2	2	NUM
ejpam-4962	358	13	)	)	PUNCT
ejpam-4962	358	14	and	and	CCONJ
ejpam-4962	358	15	|v	|v	ADJ
ejpam-4962	358	16	′	′	NOUN
ejpam-4962	358	17	0	0	NUM
ejpam-4962	359	1	|	|	ADV
ejpam-4962	359	2	=	=	SYM
ejpam-4962	359	3	n−	n−	PROPN
ejpam-4962	359	4	(	(	PUNCT
ejpam-4962	359	5	|v	|v	PROPN
ejpam-4962	359	6	′	′	NUM
ejpam-4962	359	7	1	1	NUM
ejpam-4962	359	8	|+	|+	NOUN
ejpam-4962	359	9	|v	|v	X
ejpam-4962	360	1	′	′	NOUN
ejpam-4962	360	2	2	2	NUM
ejpam-4962	360	3	|	|	NOUN
ejpam-4962	360	4	)	)	PUNCT
ejpam-4962	360	5	≥	≥	NOUN
ejpam-4962	361	1	2s−	2s−	NUM
ejpam-4962	361	2	k	k	SYM
ejpam-4962	361	3	2	2	NUM
ejpam-4962	361	4	.	.	PUNCT
ejpam-4962	361	5	suppose	suppose	VERB
ejpam-4962	361	6	that	that	SCONJ
ejpam-4962	361	7	k	k	PROPN
ejpam-4962	361	8	=	=	PUNCT
ejpam-4962	361	9	1	1	X
ejpam-4962	361	10	.	.	PUNCT
ejpam-4962	361	11	then	then	ADV
ejpam-4962	361	12	|v	|v	PROPN
ejpam-4962	361	13	′	′	NOUN
ejpam-4962	361	14	2	2	NUM
ejpam-4962	361	15	|	|	ADV
ejpam-4962	361	16	≤	≤	NUM
ejpam-4962	361	17	s+	s+	PUNCT
ejpam-4962	361	18	1	1	NUM
ejpam-4962	361	19	2	2	NUM
ejpam-4962	361	20	and	and	CCONJ
ejpam-4962	361	21	|v	|v	ADJ
ejpam-4962	361	22	′	′	NOUN
ejpam-4962	361	23	0	0	NUM
ejpam-4962	362	1	|	|	CCONJ
ejpam-4962	362	2	≥	≥	NUM
ejpam-4962	362	3	2s−	2s−	NUM
ejpam-4962	362	4	1	1	NUM
ejpam-4962	362	5	2	2	NUM
ejpam-4962	362	6	.	.	PUNCT
ejpam-4962	363	1	this	this	PRON
ejpam-4962	363	2	implies	imply	VERB
ejpam-4962	363	3	that	that	SCONJ
ejpam-4962	363	4	|v	|v	PROPN
ejpam-4962	363	5	′	′	NOUN
ejpam-4962	363	6	2	2	NUM
ejpam-4962	363	7	|	|	ADV
ejpam-4962	363	8	≤	≤	NOUN
ejpam-4962	363	9	s	s	PART
ejpam-4962	363	10	and	and	CCONJ
ejpam-4962	363	11	|v	|v	ADJ
ejpam-4962	363	12	′	′	NOUN
ejpam-4962	363	13	0	0	NUM
ejpam-4962	364	1	|	|	ADV
ejpam-4962	364	2	≥	≥	X
ejpam-4962	364	3	2s	2s	X
ejpam-4962	364	4	.	.	PUNCT
ejpam-4962	365	1	since	since	SCONJ
ejpam-4962	365	2	|v	|v	PROPN
ejpam-4962	365	3	′	′	NUM
ejpam-4962	365	4	1	1	NUM
ejpam-4962	366	1	|	|	ADV
ejpam-4962	366	2	=	=	SYM
ejpam-4962	366	3	1	1	NUM
ejpam-4962	366	4	,	,	PUNCT
ejpam-4962	366	5	|v	|v	VERB
ejpam-4962	366	6	′	′	NOUN
ejpam-4962	366	7	0	0	NUM
ejpam-4962	367	1	|	|	ADV
ejpam-4962	367	2	<	<	X
ejpam-4962	367	3	2|v	2|v	NOUN
ejpam-4962	367	4	′	′	NUM
ejpam-4962	367	5	2	2	NUM
ejpam-4962	368	1	|	|	ADV
ejpam-4962	368	2	(	(	PUNCT
ejpam-4962	368	3	as	as	ADP
ejpam-4962	368	4	v1	v1	NOUN
ejpam-4962	368	5	∈	∈	NOUN
ejpam-4962	368	6	v	v	NOUN
ejpam-4962	368	7	′	′	NUM
ejpam-4962	368	8	2	2	NUM
ejpam-4962	368	9	or	or	CCONJ
ejpam-4962	368	10	vn	vn	ADP
ejpam-4962	368	11	∈	∈	PROPN
ejpam-4962	368	12	v	v	ADP
ejpam-4962	368	13	′	′	NUM
ejpam-4962	368	14	2	2	NUM
ejpam-4962	368	15	;	;	PUNCT
ejpam-4962	368	16	hence	hence	ADV
ejpam-4962	368	17	,	,	PUNCT
ejpam-4962	368	18	at	at	ADV
ejpam-4962	368	19	least	least	ADJ
ejpam-4962	368	20	one	one	NUM
ejpam-4962	368	21	of	of	ADP
ejpam-4962	368	22	them	they	PRON
ejpam-4962	368	23	has	have	VERB
ejpam-4962	368	24	only	only	ADV
ejpam-4962	368	25	one	one	NUM
ejpam-4962	368	26	neighbor	neighbor	NOUN
ejpam-4962	368	27	in	in	ADP
ejpam-4962	368	28	v	v	NUM
ejpam-4962	368	29	′	′	NUM
ejpam-4962	368	30	0	0	NUM
ejpam-4962	368	31	)	)	PUNCT
ejpam-4962	368	32	.	.	PUNCT
ejpam-4962	369	1	thus	thus	ADV
ejpam-4962	369	2	,	,	PUNCT
ejpam-4962	369	3	|v	|v	ADJ
ejpam-4962	369	4	′	′	NOUN
ejpam-4962	369	5	2	2	NUM
ejpam-4962	369	6	|	|	ADV
ejpam-4962	369	7	≤	≤	PROPN
ejpam-4962	369	8	s	s	VERB
ejpam-4962	369	9	implies	imply	VERB
ejpam-4962	369	10	that	that	SCONJ
ejpam-4962	369	11	|v	|v	PROPN
ejpam-4962	369	12	′	′	NOUN
ejpam-4962	369	13	0	0	NUM
ejpam-4962	370	1	|	|	ADV
ejpam-4962	370	2	<	<	X
ejpam-4962	370	3	2s	2s	X
ejpam-4962	370	4	.	.	PUNCT
ejpam-4962	371	1	this	this	PRON
ejpam-4962	371	2	contradicts	contradict	VERB
ejpam-4962	371	3	the	the	DET
ejpam-4962	371	4	fact	fact	NOUN
ejpam-4962	371	5	that	that	SCONJ
ejpam-4962	371	6	|v	|v	PROPN
ejpam-4962	371	7	′	′	NOUN
ejpam-4962	371	8	0	0	NUM
ejpam-4962	372	1	|	|	ADV
ejpam-4962	372	2	≥	≥	X
ejpam-4962	372	3	2s	2s	X
ejpam-4962	372	4	.	.	PUNCT
ejpam-4962	373	1	suppose	suppose	VERB
ejpam-4962	373	2	that	that	SCONJ
ejpam-4962	373	3	k	k	PROPN
ejpam-4962	373	4	=	=	PUNCT
ejpam-4962	373	5	3	3	X
ejpam-4962	373	6	.	.	PUNCT
ejpam-4962	373	7	then	then	ADV
ejpam-4962	373	8	|v	|v	PROPN
ejpam-4962	373	9	′	′	NOUN
ejpam-4962	373	10	2	2	NUM
ejpam-4962	373	11	|	|	ADV
ejpam-4962	373	12	≤	≤	NUM
ejpam-4962	373	13	s−	s−	PROPN
ejpam-4962	373	14	1	1	NUM
ejpam-4962	373	15	2	2	NUM
ejpam-4962	373	16	and	and	CCONJ
ejpam-4962	373	17	|v	|v	ADJ
ejpam-4962	373	18	′	′	NOUN
ejpam-4962	373	19	0	0	NUM
ejpam-4962	374	1	|	|	CCONJ
ejpam-4962	374	2	≥	≥	NUM
ejpam-4962	374	3	2s−	2s−	NUM
ejpam-4962	374	4	3	3	NUM
ejpam-4962	374	5	2	2	NUM
ejpam-4962	374	6	.	.	PUNCT
ejpam-4962	375	1	this	this	PRON
ejpam-4962	375	2	implies	imply	VERB
ejpam-4962	375	3	that	that	SCONJ
ejpam-4962	375	4	|v	|v	PROPN
ejpam-4962	375	5	′	′	NUM
ejpam-4962	375	6	2	2	NUM
ejpam-4962	375	7	|	|	ADV
ejpam-4962	375	8	≤	≤	NUM
ejpam-4962	375	9	s−	s−	PROPN
ejpam-4962	375	10	1	1	NUM
ejpam-4962	375	11	and	and	CCONJ
ejpam-4962	375	12	|v	|v	PROPN
ejpam-4962	375	13	′	′	NOUN
ejpam-4962	375	14	0	0	NUM
ejpam-4962	376	1	|	|	CCONJ
ejpam-4962	376	2	≥	≥	NUM
ejpam-4962	376	3	2s−	2s−	NUM
ejpam-4962	376	4	1	1	NUM
ejpam-4962	376	5	.	.	PUNCT
ejpam-4962	377	1	this	this	PRON
ejpam-4962	377	2	is	be	AUX
ejpam-4962	377	3	not	not	PART
ejpam-4962	377	4	possible	possible	ADJ
ejpam-4962	377	5	.	.	PUNCT
ejpam-4962	378	1	suppose	suppose	VERB
ejpam-4962	378	2	k	k	PROPN
ejpam-4962	378	3	≥	≥	NUM
ejpam-4962	378	4	5	5	NUM
ejpam-4962	378	5	.	.	PUNCT
ejpam-4962	378	6	then	then	ADV
ejpam-4962	378	7	|v	|v	PROPN
ejpam-4962	378	8	′	′	NOUN
ejpam-4962	378	9	2	2	NUM
ejpam-4962	379	1	|	|	ADV
ejpam-4962	379	2	≤	≤	PROPN
ejpam-4962	379	3	s	s	VERB
ejpam-4962	379	4	−	−	PROPN
ejpam-4962	379	5	1	1	NUM
ejpam-4962	379	6	2(k	2(k	NUM
ejpam-4962	379	7	−	−	NOUN
ejpam-4962	379	8	2	2	NUM
ejpam-4962	379	9	)	)	PUNCT
ejpam-4962	379	10	implies	imply	VERB
ejpam-4962	379	11	that	that	SCONJ
ejpam-4962	379	12	|v	|v	PROPN
ejpam-4962	379	13	′	′	NOUN
ejpam-4962	379	14	0	0	NUM
ejpam-4962	380	1	|	|	ADV
ejpam-4962	380	2	≤	≤	VERB
ejpam-4962	381	1	2s	2s	NUM
ejpam-4962	381	2	−	−	PROPN
ejpam-4962	381	3	(	(	PUNCT
ejpam-4962	381	4	k	k	NOUN
ejpam-4962	381	5	−	−	PROPN
ejpam-4962	381	6	2	2	NUM
ejpam-4962	381	7	)	)	PUNCT
ejpam-4962	381	8	.	.	PUNCT
ejpam-4962	382	1	however	however	ADV
ejpam-4962	382	2	,	,	PUNCT
ejpam-4962	382	3	|v	|v	ADJ
ejpam-4962	382	4	′	′	NOUN
ejpam-4962	382	5	0	0	NUM
ejpam-4962	383	1	|	|	ADV
ejpam-4962	383	2	≥	≥	X
ejpam-4962	383	3	2s	2s	NUM
ejpam-4962	383	4	−	−	PROPN
ejpam-4962	384	1	k	k	NOUN
ejpam-4962	384	2	2	2	X
ejpam-4962	384	3	>	>	PUNCT
ejpam-4962	384	4	2s	2s	NUM
ejpam-4962	384	5	−	−	PROPN
ejpam-4962	384	6	(	(	PUNCT
ejpam-4962	384	7	k	k	NOUN
ejpam-4962	384	8	−	−	PROPN
ejpam-4962	384	9	2	2	NUM
ejpam-4962	384	10	)	)	PUNCT
ejpam-4962	384	11	,	,	PUNCT
ejpam-4962	384	12	a	a	DET
ejpam-4962	384	13	contradiction	contradiction	NOUN
ejpam-4962	384	14	.	.	PUNCT
ejpam-4962	385	1	thus	thus	ADV
ejpam-4962	385	2	,	,	PUNCT
ejpam-4962	385	3	k	k	PROPN
ejpam-4962	385	4	=	=	PUNCT
ejpam-4962	385	5	0	0	PROPN
ejpam-4962	385	6	or	or	CCONJ
ejpam-4962	385	7	k	k	X
ejpam-4962	385	8	=	=	SYM
ejpam-4962	385	9	2	2	NUM
ejpam-4962	385	10	or	or	CCONJ
ejpam-4962	385	11	k	k	NOUN
ejpam-4962	385	12	=	=	NOUN
ejpam-4962	385	13	4	4	X
ejpam-4962	385	14	.	.	PUNCT
ejpam-4962	386	1	if	if	SCONJ
ejpam-4962	386	2	k	k	PROPN
ejpam-4962	386	3	=	=	SYM
ejpam-4962	386	4	0	0	PROPN
ejpam-4962	386	5	,	,	PUNCT
ejpam-4962	386	6	by	by	ADP
ejpam-4962	386	7	proposition	proposition	NOUN
ejpam-4962	386	8	1(iv	1(iv	NUM
ejpam-4962	386	9	)	)	PUNCT
ejpam-4962	386	10	and	and	CCONJ
ejpam-4962	386	11	remark	remark	NOUN
ejpam-4962	386	12	2(ii	2(ii	NUM
ejpam-4962	386	13	)	)	PUNCT
ejpam-4962	386	14	,	,	PUNCT
ejpam-4962	386	15	γgr(pn	γgr(pn	NUM
ejpam-4962	386	16	)	)	PUNCT
ejpam-4962	386	17	=	=	SYM
ejpam-4962	387	1	2n+4	2n+4	NOUN
ejpam-4962	387	2	3	3	X
ejpam-4962	387	3	.	.	PUNCT
ejpam-4962	388	1	if	if	SCONJ
ejpam-4962	388	2	k	k	PROPN
ejpam-4962	388	3	=	=	SYM
ejpam-4962	388	4	2	2	NUM
ejpam-4962	388	5	,	,	PUNCT
ejpam-4962	388	6	then	then	ADV
ejpam-4962	388	7	we	we	PRON
ejpam-4962	388	8	may	may	AUX
ejpam-4962	388	9	consider	consider	VERB
ejpam-4962	388	10	j	j	NOUN
ejpam-4962	388	11	=	=	SYM
ejpam-4962	388	12	(	(	PUNCT
ejpam-4962	388	13	v	v	NUM
ejpam-4962	388	14	′′	′′	PROPN
ejpam-4962	388	15	0	0	NUM
ejpam-4962	388	16	,	,	PUNCT
ejpam-4962	388	17	v	v	ADP
ejpam-4962	388	18	′′	′′	PROPN
ejpam-4962	388	19	1	1	NUM
ejpam-4962	388	20	,	,	PUNCT
ejpam-4962	388	21	v	v	ADP
ejpam-4962	388	22	′′	′′	PROPN
ejpam-4962	388	23	2	2	NUM
ejpam-4962	388	24	)	)	PUNCT
ejpam-4962	388	25	where	where	SCONJ
ejpam-4962	388	26	v	v	X
ejpam-4962	388	27	′′	′′	PROPN
ejpam-4962	388	28	1	1	NUM
ejpam-4962	388	29	=	=	SYM
ejpam-4962	388	30	{	{	PUNCT
ejpam-4962	388	31	v1	v1	PROPN
ejpam-4962	388	32	,	,	PUNCT
ejpam-4962	388	33	v3s+1	v3s+1	PROPN
ejpam-4962	388	34	}	}	PUNCT
ejpam-4962	388	35	,	,	PUNCT
ejpam-4962	388	36	v	v	ADP
ejpam-4962	388	37	′′	′′	PROPN
ejpam-4962	388	38	2	2	NUM
ejpam-4962	388	39	=	=	SYM
ejpam-4962	388	40	{	{	PUNCT
ejpam-4962	388	41	v3	v3	PROPN
ejpam-4962	388	42	,	,	PUNCT
ejpam-4962	388	43	v6	v6	PROPN
ejpam-4962	388	44	,	,	PUNCT
ejpam-4962	388	45	v9	v9	PROPN
ejpam-4962	388	46	,	,	PUNCT
ejpam-4962	388	47	.	.	PUNCT
ejpam-4962	388	48	.	.	PUNCT
ejpam-4962	389	1	.	.	PUNCT
ejpam-4962	390	1	,	,	PUNCT
ejpam-4962	390	2	v3s	v3s	PROPN
ejpam-4962	390	3	}	}	PUNCT
ejpam-4962	390	4	and	and	CCONJ
ejpam-4962	390	5	v	v	ADP
ejpam-4962	390	6	′′	′′	PROPN
ejpam-4962	390	7	0	0	NUM
ejpam-4962	391	1	=	=	SYM
ejpam-4962	391	2	v	v	PROPN
ejpam-4962	391	3	(	(	PUNCT
ejpam-4962	391	4	pn	pn	NOUN
ejpam-4962	391	5	)	)	PUNCT
ejpam-4962	391	6	\	\	PUNCT
ejpam-4962	392	1	(	(	PUNCT
ejpam-4962	392	2	v	v	NUM
ejpam-4962	392	3	′′	′′	PROPN
ejpam-4962	392	4	1	1	NUM
ejpam-4962	392	5	∪	∪	X
ejpam-4962	392	6	v	v	ADP
ejpam-4962	392	7	′′	′′	PROPN
ejpam-4962	392	8	2	2	NUM
ejpam-4962	392	9	)	)	PUNCT
ejpam-4962	392	10	.	.	PUNCT
ejpam-4962	393	1	hence	hence	ADV
ejpam-4962	393	2	,	,	PUNCT
ejpam-4962	393	3	j	j	PROPN
ejpam-4962	393	4	is	be	AUX
ejpam-4962	393	5	a	a	DET
ejpam-4962	393	6	grdf	grdf	NOUN
ejpam-4962	393	7	on	on	ADP
ejpam-4962	393	8	pn	pn	PROPN
ejpam-4962	393	9	and	and	CCONJ
ejpam-4962	393	10	ωgr	ωgr	PROPN
ejpam-4962	393	11	pn	pn	PROPN
ejpam-4962	393	12	(	(	PUNCT
ejpam-4962	393	13	j	j	PROPN
ejpam-4962	393	14	)	)	PUNCT
ejpam-4962	393	15	=	=	SYM
ejpam-4962	394	1	2n+4	2n+4	NOUN
ejpam-4962	394	2	3	3	X
ejpam-4962	394	3	.	.	PUNCT
ejpam-4962	395	1	if	if	SCONJ
ejpam-4962	395	2	k	k	PROPN
ejpam-4962	395	3	=	=	SYM
ejpam-4962	395	4	4	4	NUM
ejpam-4962	395	5	,	,	PUNCT
ejpam-4962	395	6	then	then	ADV
ejpam-4962	395	7	we	we	PRON
ejpam-4962	395	8	may	may	AUX
ejpam-4962	395	9	consider	consider	VERB
ejpam-4962	395	10	l	l	NOUN
ejpam-4962	395	11	=	=	SYM
ejpam-4962	395	12	(	(	PUNCT
ejpam-4962	395	13	v	v	NOUN
ejpam-4962	395	14	∗	∗	NOUN
ejpam-4962	395	15	0	0	NUM
ejpam-4962	395	16	,	,	PUNCT
ejpam-4962	395	17	v	v	NOUN
ejpam-4962	395	18	∗	∗	NOUN
ejpam-4962	395	19	1	1	NUM
ejpam-4962	395	20	,	,	PUNCT
ejpam-4962	395	21	v	v	NOUN
ejpam-4962	395	22	∗	∗	NOUN
ejpam-4962	395	23	2	2	NUM
ejpam-4962	395	24	)	)	PUNCT
ejpam-4962	395	25	where	where	SCONJ
ejpam-4962	395	26	v	v	NOUN
ejpam-4962	395	27	∗	∗	X
ejpam-4962	395	28	1	1	NUM
ejpam-4962	395	29	=	=	SYM
ejpam-4962	395	30	{	{	PUNCT
ejpam-4962	395	31	v1	v1	PROPN
ejpam-4962	395	32	,	,	PUNCT
ejpam-4962	395	33	v2	v2	PROPN
ejpam-4962	395	34	,	,	PUNCT
ejpam-4962	395	35	v3s	v3s	PROPN
ejpam-4962	395	36	,	,	PUNCT
ejpam-4962	395	37	v3s+1	v3s+1	PROPN
ejpam-4962	395	38	}	}	PUNCT
ejpam-4962	395	39	,	,	PUNCT
ejpam-4962	395	40	v	v	NOUN
ejpam-4962	395	41	∗	∗	NOUN
ejpam-4962	395	42	2	2	NUM
ejpam-4962	395	43	=	=	SYM
ejpam-4962	395	44	{	{	PUNCT
ejpam-4962	395	45	v4	v4	NOUN
ejpam-4962	395	46	,	,	PUNCT
ejpam-4962	395	47	v7	v7	NUM
ejpam-4962	395	48	,	,	PUNCT
ejpam-4962	395	49	.	.	PUNCT
ejpam-4962	395	50	.	.	PUNCT
ejpam-4962	396	1	.	.	PUNCT
ejpam-4962	397	1	,	,	PUNCT
ejpam-4962	397	2	v3s−1	v3s−1	PROPN
ejpam-4962	397	3	}	}	PUNCT
ejpam-4962	397	4	and	and	CCONJ
ejpam-4962	397	5	v	v	ADP
ejpam-4962	397	6	∗	∗	NOUN
ejpam-4962	397	7	0	0	NUM
ejpam-4962	398	1	=	=	SYM
ejpam-4962	398	2	v	v	NOUN
ejpam-4962	398	3	(	(	PUNCT
ejpam-4962	398	4	pn	pn	NOUN
ejpam-4962	398	5	)	)	PUNCT
ejpam-4962	398	6	\	\	PUNCT
ejpam-4962	399	1	(	(	PUNCT
ejpam-4962	399	2	v	v	NOUN
ejpam-4962	399	3	∗	∗	NOUN
ejpam-4962	399	4	1	1	NUM
ejpam-4962	399	5	∪	∪	NOUN
ejpam-4962	399	6	v	v	NOUN
ejpam-4962	399	7	∗	∗	NOUN
ejpam-4962	399	8	2	2	NUM
ejpam-4962	399	9	)	)	PUNCT
ejpam-4962	399	10	.	.	PUNCT
ejpam-4962	400	1	hence	hence	ADV
ejpam-4962	400	2	,	,	PUNCT
ejpam-4962	400	3	l	l	NOUN
ejpam-4962	400	4	is	be	AUX
ejpam-4962	400	5	a	a	DET
ejpam-4962	400	6	grdf	grdf	NOUN
ejpam-4962	400	7	on	on	ADP
ejpam-4962	400	8	pn	pn	PROPN
ejpam-4962	400	9	and	and	CCONJ
ejpam-4962	400	10	ωgr	ωgr	NUM
ejpam-4962	400	11	pn	pn	PROPN
ejpam-4962	400	12	(	(	PUNCT
ejpam-4962	400	13	l	l	NOUN
ejpam-4962	400	14	)	)	PUNCT
ejpam-4962	400	15	=	=	SYM
ejpam-4962	401	1	2n+4	2n+4	NOUN
ejpam-4962	401	2	3	3	NUM
ejpam-4962	401	3	.	.	PUNCT
ejpam-4962	402	1	therefore	therefore	ADV
ejpam-4962	402	2	,	,	PUNCT
ejpam-4962	402	3	γgr(pn	γgr(pn	X
ejpam-4962	402	4	)	)	PUNCT
ejpam-4962	402	5	=	=	SYM
ejpam-4962	403	1	2n+4	2n+4	NOUN
ejpam-4962	403	2	3	3	NUM
ejpam-4962	403	3	.	.	PUNCT
ejpam-4962	404	1	r.	r.	PROPN
ejpam-4962	404	2	fortosa	fortosa	PROPN
ejpam-4962	404	3	,	,	PUNCT
ejpam-4962	404	4	s.	s.	PROPN
ejpam-4962	404	5	canoy	canoy	PROPN
ejpam-4962	404	6	jr	jr	PROPN
ejpam-4962	404	7	.	.	PROPN
ejpam-4962	404	8	/	/	SYM
ejpam-4962	404	9	eur	eur	PROPN
ejpam-4962	404	10	.	.	PUNCT
ejpam-4962	405	1	j.	j.	PROPN
ejpam-4962	405	2	pure	pure	PROPN
ejpam-4962	405	3	appl	appl	PROPN
ejpam-4962	405	4	.	.	PROPN
ejpam-4962	405	5	math	math	PROPN
ejpam-4962	405	6	,	,	PUNCT
ejpam-4962	405	7	16	16	NUM
ejpam-4962	405	8	(	(	PUNCT
ejpam-4962	405	9	4	4	NUM
ejpam-4962	405	10	)	)	PUNCT
ejpam-4962	405	11	(	(	PUNCT
ejpam-4962	405	12	2023	2023	NUM
ejpam-4962	405	13	)	)	PUNCT
ejpam-4962	405	14	,	,	PUNCT
ejpam-4962	405	15	2368	2368	NUM
ejpam-4962	405	16	-	-	SYM
ejpam-4962	405	17	2383	2383	NUM
ejpam-4962	405	18	2375	2375	NUM
ejpam-4962	405	19	case	case	NOUN
ejpam-4962	405	20	3	3	NUM
ejpam-4962	405	21	:	:	PUNCT
ejpam-4962	405	22	n	n	NUM
ejpam-4962	405	23	≡	≡	PROPN
ejpam-4962	405	24	2(mod	2(mod	NUM
ejpam-4962	405	25	3	3	X
ejpam-4962	405	26	)	)	PUNCT
ejpam-4962	405	27	let	let	VERB
ejpam-4962	405	28	n	n	NOUN
ejpam-4962	405	29	=	=	SYM
ejpam-4962	405	30	3t+2	3t+2	PROPN
ejpam-4962	405	31	for	for	ADP
ejpam-4962	405	32	some	some	DET
ejpam-4962	405	33	positive	positive	ADJ
ejpam-4962	405	34	integer	integer	NOUN
ejpam-4962	405	35	t.	t.	NOUN
ejpam-4962	405	36	define	define	VERB
ejpam-4962	405	37	v1	v1	PROPN
ejpam-4962	405	38	=	=	SYM
ejpam-4962	405	39	{	{	PUNCT
ejpam-4962	405	40	v1	v1	NOUN
ejpam-4962	405	41	,	,	PUNCT
ejpam-4962	405	42	v3t+2	v3t+2	PROPN
ejpam-4962	405	43	}	}	PUNCT
ejpam-4962	405	44	,	,	PUNCT
ejpam-4962	405	45	v2	v2	PROPN
ejpam-4962	405	46	=	=	SYM
ejpam-4962	405	47	{	{	PUNCT
ejpam-4962	405	48	v3	v3	PROPN
ejpam-4962	405	49	,	,	PUNCT
ejpam-4962	405	50	v6	v6	PROPN
ejpam-4962	405	51	,	,	PUNCT
ejpam-4962	405	52	v9	v9	PROPN
ejpam-4962	405	53	,	,	PUNCT
ejpam-4962	405	54	.	.	PUNCT
ejpam-4962	405	55	.	.	PUNCT
ejpam-4962	405	56	.	.	PUNCT
ejpam-4962	406	1	,	,	PUNCT
ejpam-4962	406	2	v3	v3	PROPN
ejpam-4962	406	3	t	t	PROPN
ejpam-4962	406	4	}	}	PUNCT
ejpam-4962	406	5	and	and	CCONJ
ejpam-4962	406	6	v0	v0	PROPN
ejpam-4962	406	7	=	=	SYM
ejpam-4962	406	8	v	v	PROPN
ejpam-4962	406	9	(	(	PUNCT
ejpam-4962	406	10	g	g	NOUN
ejpam-4962	406	11	)	)	PUNCT
ejpam-4962	406	12	\	\	PUNCT
ejpam-4962	406	13	(	(	PUNCT
ejpam-4962	406	14	v1	v1	VERB
ejpam-4962	406	15	∪	∪	NOUN
ejpam-4962	406	16	v2	v2	NOUN
ejpam-4962	406	17	)	)	PUNCT
ejpam-4962	406	18	.	.	PUNCT
ejpam-4962	407	1	thus	thus	ADV
ejpam-4962	407	2	,	,	PUNCT
ejpam-4962	407	3	f	f	PROPN
ejpam-4962	407	4	=	=	SYM
ejpam-4962	407	5	(	(	PUNCT
ejpam-4962	407	6	v0	v0	PROPN
ejpam-4962	407	7	,	,	PUNCT
ejpam-4962	407	8	v1	v1	NOUN
ejpam-4962	407	9	,	,	PUNCT
ejpam-4962	407	10	v2	v2	PROPN
ejpam-4962	407	11	)	)	PUNCT
ejpam-4962	407	12	is	be	AUX
ejpam-4962	407	13	a	a	DET
ejpam-4962	407	14	grdf	grdf	NOUN
ejpam-4962	407	15	in	in	ADP
ejpam-4962	407	16	pn	pn	PROPN
ejpam-4962	407	17	.	.	PROPN
ejpam-4962	408	1	hence	hence	ADV
ejpam-4962	408	2	,	,	PUNCT
ejpam-4962	408	3	γgr(pn	γgr(pn	NOUN
ejpam-4962	408	4	)	)	PUNCT
ejpam-4962	408	5	≤	≤	NUM
ejpam-4962	408	6	ωgr	ωgr	X
ejpam-4962	408	7	g	g	PROPN
ejpam-4962	408	8	(	(	PUNCT
ejpam-4962	408	9	f	f	X
ejpam-4962	408	10	)	)	PUNCT
ejpam-4962	408	11	=	=	PUNCT
ejpam-4962	408	12	|v1|+	|v1|+	PRON
ejpam-4962	408	13	2|v2|	2|v2|	NUM
ejpam-4962	408	14	=	=	SYM
ejpam-4962	408	15	3	3	NUM
ejpam-4962	408	16	+	+	NUM
ejpam-4962	408	17	2(n−2	2(n−2	PROPN
ejpam-4962	408	18	3	3	NUM
ejpam-4962	408	19	)	)	PUNCT
ejpam-4962	408	20	=	=	PUNCT
ejpam-4962	408	21	2n+2	2n+2	NUM
ejpam-4962	408	22	3	3	NUM
ejpam-4962	408	23	.	.	PUNCT
ejpam-4962	409	1	let	let	VERB
ejpam-4962	409	2	g	g	PROPN
ejpam-4962	409	3	=	=	SYM
ejpam-4962	409	4	(	(	PUNCT
ejpam-4962	409	5	v0	v0	PROPN
ejpam-4962	409	6	,	,	PUNCT
ejpam-4962	409	7	v1	v1	NOUN
ejpam-4962	409	8	,	,	PUNCT
ejpam-4962	409	9	v2	v2	PROPN
ejpam-4962	409	10	)	)	PUNCT
ejpam-4962	409	11	be	be	AUX
ejpam-4962	409	12	a	a	DET
ejpam-4962	409	13	γgr	γgr	NOUN
ejpam-4962	409	14	-	-	PUNCT
ejpam-4962	409	15	function	function	NOUN
ejpam-4962	409	16	on	on	ADP
ejpam-4962	409	17	pn	pn	PROPN
ejpam-4962	409	18	.	.	PUNCT
ejpam-4962	410	1	since	since	SCONJ
ejpam-4962	410	2	γgr(g	γgr(g	PROPN
ejpam-4962	410	3	)	)	PUNCT
ejpam-4962	410	4	≤	≤	NOUN
ejpam-4962	411	1	2n+2	2n+2	NUM
ejpam-4962	411	2	3	3	NUM
ejpam-4962	411	3	,	,	PUNCT
ejpam-4962	411	4	it	it	PRON
ejpam-4962	411	5	follows	follow	VERB
ejpam-4962	411	6	that	that	SCONJ
ejpam-4962	411	7	v	v	X
ejpam-4962	411	8	′	′	NOUN
ejpam-4962	411	9	2	2	NUM
ejpam-4962	411	10	̸=	̸=	PROPN
ejpam-4962	411	11	∅.	∅.	ADV
ejpam-4962	411	12	suppose	suppose	VERB
ejpam-4962	411	13	that	that	SCONJ
ejpam-4962	411	14	|v	|v	PROPN
ejpam-4962	411	15	′	′	NUM
ejpam-4962	411	16	1	1	NUM
ejpam-4962	412	1	|	|	ADV
ejpam-4962	412	2	=	=	PUNCT
ejpam-4962	412	3	k.	k.	PROPN
ejpam-4962	413	1	then	then	ADV
ejpam-4962	413	2	k	k	PROPN
ejpam-4962	414	1	+	+	PROPN
ejpam-4962	415	1	2|v	2|v	NUM
ejpam-4962	415	2	′	′	NUM
ejpam-4962	415	3	2	2	NUM
ejpam-4962	416	1	|	|	ADV
ejpam-4962	416	2	≤	≤	PUNCT
ejpam-4962	416	3	2n+2	2n+2	PROPN
ejpam-4962	416	4	3	3	NUM
ejpam-4962	416	5	.	.	PUNCT
ejpam-4962	417	1	thus	thus	ADV
ejpam-4962	417	2	,	,	PUNCT
ejpam-4962	417	3	|v	|v	ADJ
ejpam-4962	417	4	′	′	NOUN
ejpam-4962	417	5	2	2	NUM
ejpam-4962	418	1	|	|	ADV
ejpam-4962	418	2	≤	≤	X
ejpam-4962	418	3	t	t	NOUN
ejpam-4962	418	4	−	−	NUM
ejpam-4962	418	5	1	1	NUM
ejpam-4962	418	6	2(k	2(k	NUM
ejpam-4962	418	7	−	−	NOUN
ejpam-4962	418	8	2	2	NUM
ejpam-4962	418	9	)	)	PUNCT
ejpam-4962	418	10	and	and	CCONJ
ejpam-4962	418	11	|v	|v	ADJ
ejpam-4962	418	12	′	′	NOUN
ejpam-4962	418	13	0	0	NUM
ejpam-4962	419	1	|	|	ADV
ejpam-4962	419	2	=	=	SYM
ejpam-4962	419	3	n−	n−	PROPN
ejpam-4962	419	4	(	(	PUNCT
ejpam-4962	419	5	|v	|v	PROPN
ejpam-4962	419	6	′	′	NUM
ejpam-4962	419	7	1	1	NUM
ejpam-4962	419	8	|+	|+	NOUN
ejpam-4962	419	9	|v	|v	X
ejpam-4962	420	1	′	′	NOUN
ejpam-4962	420	2	2	2	NUM
ejpam-4962	420	3	|	|	NOUN
ejpam-4962	420	4	)	)	PUNCT
ejpam-4962	420	5	≥	≥	NOUN
ejpam-4962	421	1	2t−	2t−	NUM
ejpam-4962	421	2	1	1	NUM
ejpam-4962	421	3	2(k−	2(k−	NUM
ejpam-4962	421	4	2	2	NUM
ejpam-4962	421	5	)	)	PUNCT
ejpam-4962	421	6	.	.	PUNCT
ejpam-4962	422	1	if	if	SCONJ
ejpam-4962	422	2	k	k	PROPN
ejpam-4962	422	3	=	=	SYM
ejpam-4962	422	4	0	0	PROPN
ejpam-4962	422	5	,	,	PUNCT
ejpam-4962	422	6	then	then	ADV
ejpam-4962	422	7	|v	|v	VERB
ejpam-4962	422	8	′	′	NOUN
ejpam-4962	422	9	2	2	NUM
ejpam-4962	422	10	|	|	ADV
ejpam-4962	422	11	≤	≤	X
ejpam-4962	422	12	t+1	t+1	PRON
ejpam-4962	422	13	and	and	CCONJ
ejpam-4962	422	14	|v	|v	PROPN
ejpam-4962	422	15	′	′	NOUN
ejpam-4962	422	16	0	0	NUM
ejpam-4962	423	1	|	|	CCONJ
ejpam-4962	423	2	≥	≥	NOUN
ejpam-4962	423	3	2t+1	2t+1	NUM
ejpam-4962	423	4	.	.	PUNCT
ejpam-4962	424	1	this	this	PRON
ejpam-4962	424	2	is	be	AUX
ejpam-4962	424	3	not	not	PART
ejpam-4962	424	4	possible	possible	ADJ
ejpam-4962	424	5	.	.	PUNCT
ejpam-4962	425	1	suppose	suppose	VERB
ejpam-4962	425	2	that	that	SCONJ
ejpam-4962	425	3	k	k	PROPN
ejpam-4962	425	4	=	=	PUNCT
ejpam-4962	425	5	1	1	X
ejpam-4962	425	6	.	.	PUNCT
ejpam-4962	425	7	then	then	ADV
ejpam-4962	425	8	|v	|v	PROPN
ejpam-4962	425	9	′	′	NOUN
ejpam-4962	425	10	2	2	NUM
ejpam-4962	426	1	|	|	ADV
ejpam-4962	426	2	≤	≤	X
ejpam-4962	426	3	t	t	NOUN
ejpam-4962	426	4	+	+	CCONJ
ejpam-4962	426	5	1	1	NUM
ejpam-4962	426	6	2	2	NUM
ejpam-4962	426	7	and	and	CCONJ
ejpam-4962	426	8	|v	|v	ADJ
ejpam-4962	426	9	′	′	NOUN
ejpam-4962	426	10	0	0	NUM
ejpam-4962	427	1	|	|	CCONJ
ejpam-4962	427	2	≥	≥	NUM
ejpam-4962	427	3	2	2	NUM
ejpam-4962	427	4	t	t	NOUN
ejpam-4962	427	5	+	+	NOUN
ejpam-4962	427	6	1	1	NUM
ejpam-4962	427	7	2	2	NUM
ejpam-4962	427	8	.	.	PUNCT
ejpam-4962	428	1	this	this	PRON
ejpam-4962	428	2	implies	imply	VERB
ejpam-4962	428	3	that	that	SCONJ
ejpam-4962	428	4	|v	|v	PROPN
ejpam-4962	428	5	′	′	NUM
ejpam-4962	428	6	2	2	NUM
ejpam-4962	428	7	|	|	ADV
ejpam-4962	428	8	≤	≤	X
ejpam-4962	428	9	t	t	NOUN
ejpam-4962	428	10	and	and	CCONJ
ejpam-4962	428	11	|v	|v	PROPN
ejpam-4962	428	12	′	′	NOUN
ejpam-4962	428	13	0	0	NUM
ejpam-4962	429	1	|	|	CCONJ
ejpam-4962	429	2	≥	≥	NUM
ejpam-4962	429	3	2	2	NUM
ejpam-4962	429	4	t	t	NOUN
ejpam-4962	429	5	+	+	NOUN
ejpam-4962	429	6	1	1	X
ejpam-4962	429	7	.	.	PUNCT
ejpam-4962	430	1	this	this	PRON
ejpam-4962	430	2	is	be	AUX
ejpam-4962	430	3	also	also	ADV
ejpam-4962	430	4	not	not	PART
ejpam-4962	430	5	possible	possible	ADJ
ejpam-4962	430	6	.	.	PUNCT
ejpam-4962	431	1	suppose	suppose	VERB
ejpam-4962	431	2	that	that	SCONJ
ejpam-4962	431	3	k	k	PROPN
ejpam-4962	431	4	≥	≥	NUM
ejpam-4962	431	5	3	3	NUM
ejpam-4962	431	6	.	.	PUNCT
ejpam-4962	431	7	then	then	ADV
ejpam-4962	431	8	|v	|v	PROPN
ejpam-4962	431	9	′	′	NOUN
ejpam-4962	431	10	2	2	NUM
ejpam-4962	431	11	|	|	ADV
ejpam-4962	431	12	≤	≤	NUM
ejpam-4962	431	13	t−	t−	PROPN
ejpam-4962	431	14	1	1	NUM
ejpam-4962	431	15	2(k−2	2(k−2	NOUN
ejpam-4962	431	16	)	)	PUNCT
ejpam-4962	431	17	implies	imply	VERB
ejpam-4962	431	18	that	that	SCONJ
ejpam-4962	431	19	|v	|v	PROPN
ejpam-4962	431	20	′	′	NOUN
ejpam-4962	431	21	0	0	NUM
ejpam-4962	432	1	|	|	ADV
ejpam-4962	432	2	≤	≤	NUM
ejpam-4962	432	3	2t−(k−2	2t−(k−2	NUM
ejpam-4962	432	4	)	)	PUNCT
ejpam-4962	432	5	.	.	PUNCT
ejpam-4962	433	1	however	however	ADV
ejpam-4962	433	2	,	,	PUNCT
ejpam-4962	433	3	|v	|v	ADJ
ejpam-4962	433	4	′	′	NOUN
ejpam-4962	433	5	0	0	NUM
ejpam-4962	434	1	|	|	CCONJ
ejpam-4962	434	2	≥	≥	X
ejpam-4962	434	3	2t−	2t−	NOUN
ejpam-4962	434	4	1	1	NUM
ejpam-4962	434	5	2(k−2	2(k−2	NUM
ejpam-4962	434	6	)	)	PUNCT
ejpam-4962	434	7	>	>	X
ejpam-4962	434	8	2t−(k−2	2t−(k−2	NUM
ejpam-4962	434	9	)	)	PUNCT
ejpam-4962	434	10	,	,	PUNCT
ejpam-4962	434	11	a	a	DET
ejpam-4962	434	12	contradiction	contradiction	NOUN
ejpam-4962	434	13	.	.	PUNCT
ejpam-4962	435	1	thus	thus	ADV
ejpam-4962	435	2	,	,	PUNCT
ejpam-4962	435	3	k	k	PROPN
ejpam-4962	435	4	=	=	SYM
ejpam-4962	435	5	2	2	NUM
ejpam-4962	435	6	and	and	CCONJ
ejpam-4962	435	7	γgr(pn	γgr(pn	NUM
ejpam-4962	435	8	)	)	PUNCT
ejpam-4962	435	9	=	=	PUNCT
ejpam-4962	435	10	2n+2	2n+2	NUM
ejpam-4962	435	11	3	3	NUM
ejpam-4962	435	12	.	.	PUNCT
ejpam-4962	435	13	theorem	theorem	VERB
ejpam-4962	435	14	4	4	NUM
ejpam-4962	435	15	.	.	PUNCT
ejpam-4962	436	1	let	let	VERB
ejpam-4962	436	2	g1	g1	PROPN
ejpam-4962	436	3	,	,	PUNCT
ejpam-4962	436	4	.	.	PUNCT
ejpam-4962	436	5	.	.	PUNCT
ejpam-4962	437	1	.	.	PUNCT
ejpam-4962	438	1	,	,	PUNCT
ejpam-4962	438	2	gk	gk	PROPN
ejpam-4962	438	3	(	(	PUNCT
ejpam-4962	438	4	k	k	X
ejpam-4962	438	5	≥	≥	NUM
ejpam-4962	438	6	2	2	NUM
ejpam-4962	438	7	)	)	PUNCT
ejpam-4962	438	8	be	be	AUX
ejpam-4962	438	9	the	the	DET
ejpam-4962	438	10	components	component	NOUN
ejpam-4962	438	11	of	of	ADP
ejpam-4962	438	12	g.	g.	PROPN
ejpam-4962	438	13	then	then	ADV
ejpam-4962	438	14	γgr(g	γgr(g	PROPN
ejpam-4962	438	15	)	)	PUNCT
ejpam-4962	438	16	=	=	SYM
ejpam-4962	439	1	k∑	k∑	PROPN
ejpam-4962	440	1	j=1	j=1	NOUN
ejpam-4962	440	2	γgr(gj	γgr(gj	NUM
ejpam-4962	440	3	)	)	PUNCT
ejpam-4962	440	4	.	.	PUNCT
ejpam-4962	441	1	proof	proof	NOUN
ejpam-4962	441	2	.	.	PUNCT
ejpam-4962	442	1	let	let	VERB
ejpam-4962	442	2	g1	g1	PROPN
ejpam-4962	442	3	,	,	PUNCT
ejpam-4962	442	4	.	.	PUNCT
ejpam-4962	442	5	.	.	PUNCT
ejpam-4962	443	1	.	.	PUNCT
ejpam-4962	444	1	,	,	PUNCT
ejpam-4962	444	2	gk	gk	PROPN
ejpam-4962	444	3	be	be	AUX
ejpam-4962	444	4	the	the	DET
ejpam-4962	444	5	components	component	NOUN
ejpam-4962	444	6	of	of	ADP
ejpam-4962	444	7	g.	g.	PROPN
ejpam-4962	444	8	for	for	ADP
ejpam-4962	444	9	each	each	DET
ejpam-4962	444	10	j	j	PROPN
ejpam-4962	444	11	∈	∈	PROPN
ejpam-4962	444	12	{	{	PUNCT
ejpam-4962	444	13	1	1	NUM
ejpam-4962	444	14	,	,	PUNCT
ejpam-4962	444	15	2	2	NUM
ejpam-4962	444	16	,	,	PUNCT
ejpam-4962	444	17	.	.	PUNCT
ejpam-4962	444	18	.	.	PUNCT
ejpam-4962	445	1	.	.	PUNCT
ejpam-4962	446	1	,	,	PUNCT
ejpam-4962	446	2	k	k	X
ejpam-4962	446	3	}	}	PUNCT
ejpam-4962	446	4	,	,	PUNCT
ejpam-4962	446	5	let	let	VERB
ejpam-4962	446	6	gj	gj	NOUN
ejpam-4962	446	7	=	=	PUNCT
ejpam-4962	446	8	(	(	PUNCT
ejpam-4962	446	9	v	v	NUM
ejpam-4962	446	10	j	j	PROPN
ejpam-4962	446	11	0	0	NUM
ejpam-4962	446	12	,	,	PUNCT
ejpam-4962	446	13	v	v	PART
ejpam-4962	446	14	j	j	PROPN
ejpam-4962	446	15	1	1	NUM
ejpam-4962	446	16	,	,	PUNCT
ejpam-4962	446	17	v	v	PART
ejpam-4962	446	18	j	j	PROPN
ejpam-4962	446	19	2	2	NUM
ejpam-4962	446	20	)	)	PUNCT
ejpam-4962	446	21	be	be	AUX
ejpam-4962	446	22	a	a	DET
ejpam-4962	446	23	γgr	γgr	NOUN
ejpam-4962	446	24	-	-	PUNCT
ejpam-4962	446	25	functions	function	NOUN
ejpam-4962	446	26	of	of	ADP
ejpam-4962	446	27	gj	gj	NOUN
ejpam-4962	446	28	.	.	PUNCT
ejpam-4962	447	1	let	let	VERB
ejpam-4962	447	2	v0	v0	NOUN
ejpam-4962	447	3	=	=	SYM
ejpam-4962	447	4	∪k	∪k	PROPN
ejpam-4962	447	5	j=1v	j=1v	PROPN
ejpam-4962	447	6	j	j	PROPN
ejpam-4962	447	7	0	0	NUM
ejpam-4962	447	8	,	,	PUNCT
ejpam-4962	447	9	v1	v1	PROPN
ejpam-4962	447	10	=	=	SYM
ejpam-4962	447	11	∪k	∪k	PROPN
ejpam-4962	447	12	j=1v	j=1v	PROPN
ejpam-4962	447	13	j	j	PROPN
ejpam-4962	447	14	1	1	NUM
ejpam-4962	447	15	,	,	PUNCT
ejpam-4962	447	16	and	and	CCONJ
ejpam-4962	447	17	v2	v2	X
ejpam-4962	447	18	=	=	SYM
ejpam-4962	447	19	∪k	∪k	PROPN
ejpam-4962	447	20	j=1v	j=1v	PROPN
ejpam-4962	447	21	j	j	PROPN
ejpam-4962	447	22	2	2	NUM
ejpam-4962	447	23	.	.	PUNCT
ejpam-4962	448	1	then	then	ADV
ejpam-4962	448	2	g	g	PROPN
ejpam-4962	448	3	=	=	SYM
ejpam-4962	448	4	(	(	PUNCT
ejpam-4962	448	5	v0	v0	PROPN
ejpam-4962	448	6	,	,	PUNCT
ejpam-4962	448	7	v1	v1	NOUN
ejpam-4962	448	8	,	,	PUNCT
ejpam-4962	448	9	v2	v2	PROPN
ejpam-4962	448	10	)	)	PUNCT
ejpam-4962	448	11	is	be	AUX
ejpam-4962	448	12	a	a	DET
ejpam-4962	448	13	grdf	grdf	NOUN
ejpam-4962	448	14	on	on	ADP
ejpam-4962	448	15	g.	g.	PROPN
ejpam-4962	448	16	hence	hence	ADV
ejpam-4962	448	17	,	,	PUNCT
ejpam-4962	448	18	γgr(g	γgr(g	PROPN
ejpam-4962	448	19	)	)	PUNCT
ejpam-4962	448	20	≤	≤	NUM
ejpam-4962	448	21	ωgr	ωgr	X
ejpam-4962	448	22	g	g	PROPN
ejpam-4962	448	23	(	(	PUNCT
ejpam-4962	448	24	g	g	NOUN
ejpam-4962	448	25	)	)	PUNCT
ejpam-4962	448	26	=	=	PUNCT
ejpam-4962	448	27	|v1|+	|v1|+	PRON
ejpam-4962	448	28	2|v2|	2|v2|	NUM
ejpam-4962	448	29	=	=	SYM
ejpam-4962	448	30	k∑	k∑	NOUN
ejpam-4962	448	31	j=1	j=1	NOUN
ejpam-4962	448	32	γgr(gj	γgr(gj	NUM
ejpam-4962	448	33	)	)	PUNCT
ejpam-4962	448	34	.	.	PUNCT
ejpam-4962	449	1	next	next	ADV
ejpam-4962	449	2	,	,	PUNCT
ejpam-4962	449	3	suppose	suppose	VERB
ejpam-4962	449	4	that	that	SCONJ
ejpam-4962	449	5	f	f	PROPN
ejpam-4962	449	6	=	=	SYM
ejpam-4962	449	7	(	(	PUNCT
ejpam-4962	449	8	v0	v0	PROPN
ejpam-4962	449	9	,	,	PUNCT
ejpam-4962	449	10	v1	v1	NOUN
ejpam-4962	449	11	,	,	PUNCT
ejpam-4962	449	12	v2	v2	PROPN
ejpam-4962	449	13	)	)	PUNCT
ejpam-4962	449	14	is	be	AUX
ejpam-4962	449	15	a	a	DET
ejpam-4962	449	16	γgr	γgr	NOUN
ejpam-4962	449	17	-	-	PUNCT
ejpam-4962	449	18	function	function	NOUN
ejpam-4962	449	19	on	on	ADP
ejpam-4962	449	20	g.	g.	PROPN
ejpam-4962	449	21	then	then	ADV
ejpam-4962	449	22	fj	fj	PROPN
ejpam-4962	449	23	=	=	PUNCT
ejpam-4962	449	24	(	(	PUNCT
ejpam-4962	449	25	v	v	NUM
ejpam-4962	449	26	j	j	PROPN
ejpam-4962	449	27	0	0	NUM
ejpam-4962	449	28	,	,	PUNCT
ejpam-4962	449	29	v	v	PART
ejpam-4962	449	30	j	j	PROPN
ejpam-4962	449	31	1	1	NUM
ejpam-4962	449	32	,	,	PUNCT
ejpam-4962	449	33	v	v	NOUN
ejpam-4962	449	34	j	j	PROPN
ejpam-4962	449	35	2	2	NUM
ejpam-4962	449	36	)	)	PUNCT
ejpam-4962	449	37	,	,	PUNCT
ejpam-4962	449	38	where	where	SCONJ
ejpam-4962	449	39	v	v	ADP
ejpam-4962	449	40	j	j	PROPN
ejpam-4962	449	41	0	0	PUNCT
ejpam-4962	450	1	=	=	SYM
ejpam-4962	450	2	v0	v0	NOUN
ejpam-4962	450	3	∩	∩	X
ejpam-4962	450	4	v	v	X
ejpam-4962	450	5	(	(	PUNCT
ejpam-4962	450	6	gj	gj	PROPN
ejpam-4962	450	7	)	)	PUNCT
ejpam-4962	450	8	,	,	PUNCT
ejpam-4962	450	9	v	v	X
ejpam-4962	450	10	j	j	PROPN
ejpam-4962	450	11	1	1	NUM
ejpam-4962	450	12	=	=	SYM
ejpam-4962	450	13	v1	v1	PROPN
ejpam-4962	450	14	∩	∩	ADJ
ejpam-4962	450	15	v	v	NOUN
ejpam-4962	450	16	(	(	PUNCT
ejpam-4962	450	17	gj	gj	NOUN
ejpam-4962	450	18	)	)	PUNCT
ejpam-4962	450	19	,	,	PUNCT
ejpam-4962	450	20	and	and	CCONJ
ejpam-4962	450	21	v	v	X
ejpam-4962	450	22	j	j	PROPN
ejpam-4962	450	23	2	2	NUM
ejpam-4962	450	24	=	=	SYM
ejpam-4962	450	25	v2	v2	PROPN
ejpam-4962	450	26	∩	∩	ADJ
ejpam-4962	450	27	v	v	NOUN
ejpam-4962	450	28	(	(	PUNCT
ejpam-4962	450	29	gj	gj	NOUN
ejpam-4962	450	30	)	)	PUNCT
ejpam-4962	450	31	,	,	PUNCT
ejpam-4962	450	32	is	be	AUX
ejpam-4962	450	33	a	a	DET
ejpam-4962	450	34	grdf	grdf	NOUN
ejpam-4962	450	35	on	on	ADP
ejpam-4962	450	36	gj	gj	NOUN
ejpam-4962	450	37	for	for	ADP
ejpam-4962	450	38	each	each	DET
ejpam-4962	450	39	j	j	PROPN
ejpam-4962	450	40	∈	∈	PROPN
ejpam-4962	450	41	{	{	PUNCT
ejpam-4962	450	42	1	1	NUM
ejpam-4962	450	43	,	,	PUNCT
ejpam-4962	450	44	2	2	NUM
ejpam-4962	450	45	,	,	PUNCT
ejpam-4962	450	46	.	.	PUNCT
ejpam-4962	450	47	.	.	PUNCT
ejpam-4962	450	48	.	.	PUNCT
ejpam-4962	451	1	k	k	X
ejpam-4962	451	2	}	}	PUNCT
ejpam-4962	451	3	.	.	PUNCT
ejpam-4962	452	1	thus	thus	ADV
ejpam-4962	452	2	,	,	PUNCT
ejpam-4962	452	3	γgr(gj	γgr(gj	NOUN
ejpam-4962	452	4	)	)	PUNCT
ejpam-4962	452	5	≤	≤	NUM
ejpam-4962	452	6	ωgr	ωgr	NOUN
ejpam-4962	452	7	gj	gj	NOUN
ejpam-4962	452	8	(	(	PUNCT
ejpam-4962	452	9	fj	fj	PROPN
ejpam-4962	452	10	)	)	PUNCT
ejpam-4962	452	11	for	for	ADP
ejpam-4962	452	12	all	all	DET
ejpam-4962	452	13	j	j	PROPN
ejpam-4962	452	14	∈	∈	PROPN
ejpam-4962	452	15	{	{	PUNCT
ejpam-4962	452	16	1	1	NUM
ejpam-4962	452	17	,	,	PUNCT
ejpam-4962	452	18	2	2	NUM
ejpam-4962	452	19	,	,	PUNCT
ejpam-4962	452	20	.	.	PUNCT
ejpam-4962	452	21	.	.	PUNCT
ejpam-4962	453	1	.	.	PUNCT
ejpam-4962	454	1	,	,	PUNCT
ejpam-4962	454	2	k	k	X
ejpam-4962	454	3	}	}	PUNCT
ejpam-4962	454	4	.	.	PUNCT
ejpam-4962	455	1	hence,∑k	hence,∑k	PROPN
ejpam-4962	456	1	j=1	j=1	PROPN
ejpam-4962	456	2	γgr(gj	γgr(gj	NUM
ejpam-4962	456	3	)	)	PUNCT
ejpam-4962	456	4	≤	≤	PUNCT
ejpam-4962	456	5	γgr(g	γgr(g	PROPN
ejpam-4962	456	6	)	)	PUNCT
ejpam-4962	456	7	.	.	PUNCT
ejpam-4962	457	1	this	this	PRON
ejpam-4962	457	2	establishes	establish	VERB
ejpam-4962	457	3	the	the	DET
ejpam-4962	457	4	desired	desire	VERB
ejpam-4962	457	5	equality	equality	NOUN
ejpam-4962	457	6	.	.	PUNCT
ejpam-4962	458	1	proposition	proposition	NOUN
ejpam-4962	458	2	4	4	NUM
ejpam-4962	458	3	.	.	PUNCT
ejpam-4962	459	1	let	let	VERB
ejpam-4962	459	2	g	g	PRON
ejpam-4962	459	3	be	be	AUX
ejpam-4962	459	4	a	a	DET
ejpam-4962	459	5	graph	graph	NOUN
ejpam-4962	459	6	of	of	ADP
ejpam-4962	459	7	order	order	NOUN
ejpam-4962	459	8	n.	n.	VERB
ejpam-4962	459	9	if	if	SCONJ
ejpam-4962	459	10	γg(g	γg(g	PRON
ejpam-4962	459	11	)	)	PUNCT
ejpam-4962	459	12	=	=	PUNCT
ejpam-4962	459	13	n−	n−	NOUN
ejpam-4962	459	14	1	1	NUM
ejpam-4962	459	15	,	,	PUNCT
ejpam-4962	459	16	then	then	ADV
ejpam-4962	459	17	γgr(g	γgr(g	PROPN
ejpam-4962	459	18	)	)	PUNCT
ejpam-4962	459	19	=	=	SYM
ejpam-4962	459	20	n.	n.	NOUN
ejpam-4962	459	21	proof	proof	NOUN
ejpam-4962	459	22	.	.	PUNCT
ejpam-4962	460	1	suppose	suppose	VERB
ejpam-4962	460	2	γg(g	γg(g	PRON
ejpam-4962	460	3	)	)	PUNCT
ejpam-4962	460	4	=	=	SYM
ejpam-4962	461	1	n	n	CCONJ
ejpam-4962	461	2	−	−	NOUN
ejpam-4962	462	1	1	1	X
ejpam-4962	462	2	.	.	PUNCT
ejpam-4962	462	3	let	let	VERB
ejpam-4962	462	4	f	f	PROPN
ejpam-4962	462	5	=	=	SYM
ejpam-4962	462	6	(	(	PUNCT
ejpam-4962	462	7	v0	v0	PROPN
ejpam-4962	462	8	,	,	PUNCT
ejpam-4962	462	9	v1	v1	NOUN
ejpam-4962	462	10	,	,	PUNCT
ejpam-4962	462	11	v2	v2	PROPN
ejpam-4962	462	12	)	)	PUNCT
ejpam-4962	462	13	be	be	AUX
ejpam-4962	462	14	a	a	DET
ejpam-4962	462	15	γgr	γgr	NOUN
ejpam-4962	462	16	-	-	PUNCT
ejpam-4962	462	17	function	function	NOUN
ejpam-4962	462	18	on	on	ADP
ejpam-4962	462	19	g.	g.	PROPN
ejpam-4962	462	20	since	since	SCONJ
ejpam-4962	462	21	v1	v1	PROPN
ejpam-4962	462	22	∪	∪	NOUN
ejpam-4962	462	23	v2	v2	NOUN
ejpam-4962	462	24	is	be	AUX
ejpam-4962	462	25	a	a	DET
ejpam-4962	462	26	geodetic	geodetic	ADJ
ejpam-4962	462	27	set	set	NOUN
ejpam-4962	462	28	,	,	PUNCT
ejpam-4962	462	29	v1	v1	VERB
ejpam-4962	462	30	∪	∪	NOUN
ejpam-4962	462	31	v2	v2	PROPN
ejpam-4962	462	32	=	=	SYM
ejpam-4962	462	33	v	v	NOUN
ejpam-4962	462	34	(	(	PUNCT
ejpam-4962	462	35	g	g	NOUN
ejpam-4962	462	36	)	)	PUNCT
ejpam-4962	462	37	or	or	CCONJ
ejpam-4962	462	38	v1	v1	VERB
ejpam-4962	462	39	∪	∪	ADJ
ejpam-4962	462	40	v2	v2	NOUN
ejpam-4962	462	41	=	=	SYM
ejpam-4962	462	42	v	v	NOUN
ejpam-4962	462	43	(	(	PUNCT
ejpam-4962	462	44	g	g	NOUN
ejpam-4962	462	45	)	)	PUNCT
ejpam-4962	462	46	\	\	NOUN
ejpam-4962	463	1	{	{	PUNCT
ejpam-4962	463	2	x	x	NOUN
ejpam-4962	463	3	}	}	PUNCT
ejpam-4962	463	4	for	for	ADP
ejpam-4962	463	5	some	some	DET
ejpam-4962	463	6	x	x	SYM
ejpam-4962	463	7	∈	∈	PROPN
ejpam-4962	463	8	v	v	NOUN
ejpam-4962	463	9	(	(	PUNCT
ejpam-4962	463	10	g	g	NOUN
ejpam-4962	463	11	)	)	PUNCT
ejpam-4962	463	12	.	.	PUNCT
ejpam-4962	464	1	if	if	SCONJ
ejpam-4962	464	2	v1	v1	VERB
ejpam-4962	464	3	∪	∪	VERB
ejpam-4962	464	4	v2	v2	PROPN
ejpam-4962	464	5	=	=	SYM
ejpam-4962	464	6	v	v	NOUN
ejpam-4962	464	7	(	(	PUNCT
ejpam-4962	464	8	g	g	NOUN
ejpam-4962	464	9	)	)	PUNCT
ejpam-4962	464	10	,	,	PUNCT
ejpam-4962	464	11	then	then	ADV
ejpam-4962	464	12	|v0|	|v0|	NOUN
ejpam-4962	464	13	=	=	SYM
ejpam-4962	464	14	0	0	X
ejpam-4962	464	15	.	.	PUNCT
ejpam-4962	465	1	thus	thus	ADV
ejpam-4962	465	2	,	,	PUNCT
ejpam-4962	465	3	|v2|	|v2|	NOUN
ejpam-4962	465	4	=	=	SYM
ejpam-4962	465	5	0	0	NUM
ejpam-4962	465	6	and	and	CCONJ
ejpam-4962	465	7	|v1|	|v1|	NOUN
ejpam-4962	465	8	=	=	SYM
ejpam-4962	465	9	n.	n.	PROPN
ejpam-4962	465	10	hence	hence	ADV
ejpam-4962	465	11	,	,	PUNCT
ejpam-4962	465	12	γgr(g	γgr(g	PROPN
ejpam-4962	465	13	)	)	PUNCT
ejpam-4962	465	14	=	=	SYM
ejpam-4962	465	15	n.	n.	NOUN
ejpam-4962	465	16	suppose	suppose	VERB
ejpam-4962	465	17	v1	v1	NOUN
ejpam-4962	465	18	∪	∪	NOUN
ejpam-4962	465	19	v2	v2	PROPN
ejpam-4962	465	20	=	=	SYM
ejpam-4962	465	21	v	v	NOUN
ejpam-4962	465	22	(	(	PUNCT
ejpam-4962	465	23	g	g	NOUN
ejpam-4962	465	24	)	)	PUNCT
ejpam-4962	465	25	\	\	NOUN
ejpam-4962	465	26	{	{	PUNCT
ejpam-4962	465	27	x	x	NOUN
ejpam-4962	465	28	}	}	PUNCT
ejpam-4962	465	29	for	for	ADP
ejpam-4962	465	30	some	some	DET
ejpam-4962	465	31	x	x	SYM
ejpam-4962	465	32	∈	∈	PROPN
ejpam-4962	465	33	v	v	NOUN
ejpam-4962	465	34	(	(	PUNCT
ejpam-4962	465	35	g	g	NOUN
ejpam-4962	465	36	)	)	PUNCT
ejpam-4962	465	37	.	.	PUNCT
ejpam-4962	466	1	then	then	ADV
ejpam-4962	466	2	v0	v0	PROPN
ejpam-4962	466	3	=	=	SYM
ejpam-4962	466	4	{	{	PUNCT
ejpam-4962	466	5	x	x	NOUN
ejpam-4962	466	6	}	}	PUNCT
ejpam-4962	466	7	.	.	PUNCT
ejpam-4962	467	1	since	since	SCONJ
ejpam-4962	467	2	f	f	PROPN
ejpam-4962	467	3	is	be	AUX
ejpam-4962	467	4	a	a	DET
ejpam-4962	467	5	γgr	γgr	NOUN
ejpam-4962	467	6	-	-	PUNCT
ejpam-4962	467	7	function	function	NOUN
ejpam-4962	467	8	,	,	PUNCT
ejpam-4962	467	9	|v2|	|v2|	NOUN
ejpam-4962	467	10	=	=	SYM
ejpam-4962	467	11	1	1	X
ejpam-4962	467	12	.	.	PUNCT
ejpam-4962	467	13	therefore	therefore	ADV
ejpam-4962	467	14	,	,	PUNCT
ejpam-4962	467	15	γgr(g	γgr(g	PROPN
ejpam-4962	467	16	)	)	PUNCT
ejpam-4962	467	17	=	=	PUNCT
ejpam-4962	467	18	|v1|+	|v1|+	PRON
ejpam-4962	467	19	2|v2|	2|v2|	NUM
ejpam-4962	467	20	=	=	SYM
ejpam-4962	467	21	(	(	PUNCT
ejpam-4962	467	22	n−	n−	NOUN
ejpam-4962	467	23	2	2	NUM
ejpam-4962	467	24	)	)	PUNCT
ejpam-4962	467	25	+	+	CCONJ
ejpam-4962	467	26	2	2	NUM
ejpam-4962	467	27	=	=	SYM
ejpam-4962	467	28	n.	n.	NOUN
ejpam-4962	467	29	corollary	corollary	NOUN
ejpam-4962	467	30	1	1	NUM
ejpam-4962	467	31	.	.	PUNCT
ejpam-4962	468	1	for	for	ADP
ejpam-4962	468	2	any	any	DET
ejpam-4962	468	3	positive	positive	ADJ
ejpam-4962	468	4	integer	integer	NOUN
ejpam-4962	468	5	n	n	CCONJ
ejpam-4962	468	6	,	,	PUNCT
ejpam-4962	468	7	γgr(k1,n	γgr(k1,n	PROPN
ejpam-4962	468	8	)	)	PUNCT
ejpam-4962	468	9	=	=	VERB
ejpam-4962	468	10	n.	n.	NOUN
ejpam-4962	468	11	the	the	DET
ejpam-4962	468	12	next	next	ADJ
ejpam-4962	468	13	result	result	NOUN
ejpam-4962	468	14	follows	follow	VERB
ejpam-4962	468	15	from	from	ADP
ejpam-4962	468	16	theorem	theorem	ADJ
ejpam-4962	468	17	3	3	NUM
ejpam-4962	468	18	,	,	PUNCT
ejpam-4962	468	19	corollary	corollary	ADJ
ejpam-4962	468	20	1	1	NUM
ejpam-4962	468	21	,	,	PUNCT
ejpam-4962	468	22	and	and	CCONJ
ejpam-4962	468	23	theorem	theorem	VERB
ejpam-4962	468	24	4	4	NUM
ejpam-4962	468	25	.	.	PUNCT
ejpam-4962	468	26	corollary	corollary	ADJ
ejpam-4962	468	27	2	2	NUM
ejpam-4962	468	28	.	.	PUNCT
ejpam-4962	469	1	let	let	VERB
ejpam-4962	469	2	g	g	PRON
ejpam-4962	469	3	be	be	AUX
ejpam-4962	469	4	a	a	DET
ejpam-4962	469	5	graph	graph	NOUN
ejpam-4962	469	6	of	of	ADP
ejpam-4962	469	7	order	order	NOUN
ejpam-4962	469	8	n.	n.	NOUN
ejpam-4962	469	9	if	if	SCONJ
ejpam-4962	469	10	every	every	DET
ejpam-4962	469	11	component	component	NOUN
ejpam-4962	469	12	of	of	ADP
ejpam-4962	469	13	g	g	PROPN
ejpam-4962	469	14	is	be	AUX
ejpam-4962	469	15	either	either	CCONJ
ejpam-4962	469	16	complete	complete	ADJ
ejpam-4962	469	17	or	or	CCONJ
ejpam-4962	469	18	a	a	DET
ejpam-4962	469	19	star	star	NOUN
ejpam-4962	469	20	,	,	PUNCT
ejpam-4962	469	21	then	then	ADV
ejpam-4962	469	22	γgr(g	γgr(g	PROPN
ejpam-4962	469	23	)	)	PUNCT
ejpam-4962	469	24	=	=	SYM
ejpam-4962	469	25	n.	n.	PROPN
ejpam-4962	469	26	r.	r.	PROPN
ejpam-4962	469	27	fortosa	fortosa	PROPN
ejpam-4962	469	28	,	,	PUNCT
ejpam-4962	469	29	s.	s.	PROPN
ejpam-4962	469	30	canoy	canoy	PROPN
ejpam-4962	469	31	jr	jr	PROPN
ejpam-4962	469	32	.	.	PROPN
ejpam-4962	469	33	/	/	SYM
ejpam-4962	469	34	eur	eur	PROPN
ejpam-4962	469	35	.	.	PUNCT
ejpam-4962	470	1	j.	j.	PROPN
ejpam-4962	470	2	pure	pure	PROPN
ejpam-4962	470	3	appl	appl	PROPN
ejpam-4962	470	4	.	.	PROPN
ejpam-4962	470	5	math	math	PROPN
ejpam-4962	470	6	,	,	PUNCT
ejpam-4962	470	7	16	16	NUM
ejpam-4962	470	8	(	(	PUNCT
ejpam-4962	470	9	4	4	NUM
ejpam-4962	470	10	)	)	PUNCT
ejpam-4962	470	11	(	(	PUNCT
ejpam-4962	470	12	2023	2023	NUM
ejpam-4962	470	13	)	)	PUNCT
ejpam-4962	470	14	,	,	PUNCT
ejpam-4962	470	15	2368	2368	NUM
ejpam-4962	470	16	-	-	SYM
ejpam-4962	470	17	2383	2383	NUM
ejpam-4962	470	18	2376	2376	NUM
ejpam-4962	470	19	proposition	proposition	NOUN
ejpam-4962	470	20	5	5	NUM
ejpam-4962	470	21	.	.	PUNCT
ejpam-4962	471	1	if	if	SCONJ
ejpam-4962	471	2	g	g	PROPN
ejpam-4962	471	3	is	be	AUX
ejpam-4962	471	4	a	a	DET
ejpam-4962	471	5	graph	graph	NOUN
ejpam-4962	471	6	of	of	ADP
ejpam-4962	471	7	order	order	NOUN
ejpam-4962	471	8	n	n	PRON
ejpam-4962	471	9	≥	≥	NOUN
ejpam-4962	471	10	5	5	NUM
ejpam-4962	471	11	and	and	CCONJ
ejpam-4962	471	12	γgr(g	γgr(g	PROPN
ejpam-4962	471	13	)	)	PUNCT
ejpam-4962	471	14	=	=	SYM
ejpam-4962	472	1	n	n	CCONJ
ejpam-4962	472	2	,	,	PUNCT
ejpam-4962	472	3	then	then	ADV
ejpam-4962	472	4	g	g	PROPN
ejpam-4962	472	5	has	have	VERB
ejpam-4962	472	6	no	no	DET
ejpam-4962	472	7	induced	induce	VERB
ejpam-4962	472	8	subgraph	subgraph	NOUN
ejpam-4962	472	9	p5	p5	PROPN
ejpam-4962	472	10	.	.	PUNCT
ejpam-4962	473	1	proof	proof	NOUN
ejpam-4962	473	2	.	.	PUNCT
ejpam-4962	474	1	suppose	suppose	VERB
ejpam-4962	474	2	g	g	PROPN
ejpam-4962	474	3	has	have	VERB
ejpam-4962	474	4	an	an	DET
ejpam-4962	474	5	induced	induced	ADJ
ejpam-4962	474	6	subgraph	subgraph	NOUN
ejpam-4962	474	7	p5	p5	NOUN
ejpam-4962	474	8	=	=	PUNCT
ejpam-4962	475	1	[	[	X
ejpam-4962	475	2	v1	v1	NOUN
ejpam-4962	475	3	,	,	PUNCT
ejpam-4962	475	4	v2	v2	PROPN
ejpam-4962	475	5	,	,	PUNCT
ejpam-4962	475	6	v3	v3	PROPN
ejpam-4962	475	7	,	,	PUNCT
ejpam-4962	475	8	v4	v4	PROPN
ejpam-4962	475	9	,	,	PUNCT
ejpam-4962	475	10	v5	v5	PROPN
ejpam-4962	475	11	]	]	PUNCT
ejpam-4962	475	12	.	.	PUNCT
ejpam-4962	476	1	define	define	VERB
ejpam-4962	476	2	v0	v0	NOUN
ejpam-4962	476	3	=	=	SYM
ejpam-4962	476	4	{	{	PUNCT
ejpam-4962	476	5	v2	v2	PROPN
ejpam-4962	476	6	,	,	PUNCT
ejpam-4962	476	7	v4	v4	PROPN
ejpam-4962	476	8	}	}	PUNCT
ejpam-4962	476	9	,	,	PUNCT
ejpam-4962	476	10	v2	v2	PROPN
ejpam-4962	476	11	=	=	SYM
ejpam-4962	476	12	{	{	PUNCT
ejpam-4962	476	13	v3	v3	NOUN
ejpam-4962	476	14	}	}	PUNCT
ejpam-4962	476	15	and	and	CCONJ
ejpam-4962	476	16	v1	v1	PROPN
ejpam-4962	476	17	=	=	SYM
ejpam-4962	476	18	v	v	NOUN
ejpam-4962	476	19	(	(	PUNCT
ejpam-4962	476	20	g	g	NOUN
ejpam-4962	476	21	)	)	PUNCT
ejpam-4962	476	22	\	\	NOUN
ejpam-4962	476	23	{	{	PUNCT
ejpam-4962	476	24	v2	v2	PROPN
ejpam-4962	476	25	,	,	PUNCT
ejpam-4962	476	26	v3	v3	PROPN
ejpam-4962	476	27	,	,	PUNCT
ejpam-4962	476	28	v4	v4	PROPN
ejpam-4962	476	29	}	}	PUNCT
ejpam-4962	476	30	.	.	PUNCT
ejpam-4962	477	1	then	then	ADV
ejpam-4962	477	2	v1	v1	VERB
ejpam-4962	477	3	∪	∪	ADJ
ejpam-4962	477	4	v2	v2	NOUN
ejpam-4962	477	5	is	be	AUX
ejpam-4962	477	6	a	a	DET
ejpam-4962	477	7	geodetic	geodetic	ADJ
ejpam-4962	477	8	dominating	dominating	NOUN
ejpam-4962	477	9	set	set	VERB
ejpam-4962	477	10	in	in	ADP
ejpam-4962	477	11	g	g	PROPN
ejpam-4962	477	12	and	and	CCONJ
ejpam-4962	477	13	v0	v0	PROPN
ejpam-4962	477	14	⊆	⊆	NUM
ejpam-4962	477	15	ng(v3	ng(v3	NOUN
ejpam-4962	477	16	)	)	PUNCT
ejpam-4962	477	17	.	.	PUNCT
ejpam-4962	478	1	this	this	PRON
ejpam-4962	478	2	implies	imply	VERB
ejpam-4962	478	3	that	that	SCONJ
ejpam-4962	478	4	f	f	PROPN
ejpam-4962	478	5	=	=	SYM
ejpam-4962	478	6	(	(	PUNCT
ejpam-4962	478	7	v0	v0	PROPN
ejpam-4962	478	8	,	,	PUNCT
ejpam-4962	478	9	v1	v1	NOUN
ejpam-4962	478	10	,	,	PUNCT
ejpam-4962	478	11	v2	v2	PROPN
ejpam-4962	478	12	)	)	PUNCT
ejpam-4962	478	13	is	be	AUX
ejpam-4962	478	14	a	a	DET
ejpam-4962	478	15	grdf	grdf	NOUN
ejpam-4962	478	16	on	on	ADP
ejpam-4962	478	17	g.	g.	PROPN
ejpam-4962	478	18	thus	thus	ADV
ejpam-4962	478	19	,	,	PUNCT
ejpam-4962	478	20	ωgr	ωgr	PROPN
ejpam-4962	478	21	g	g	PROPN
ejpam-4962	478	22	(	(	PUNCT
ejpam-4962	478	23	f	f	X
ejpam-4962	478	24	)	)	PUNCT
ejpam-4962	478	25	=	=	NOUN
ejpam-4962	478	26	|v1|	|v1|	NOUN
ejpam-4962	478	27	+	+	CCONJ
ejpam-4962	478	28	2|v2|	2|v2|	NUM
ejpam-4962	478	29	=	=	SYM
ejpam-4962	478	30	(	(	PUNCT
ejpam-4962	478	31	n	n	CCONJ
ejpam-4962	478	32	−	−	PROPN
ejpam-4962	478	33	3	3	NUM
ejpam-4962	478	34	)	)	PUNCT
ejpam-4962	478	35	+	+	NOUN
ejpam-4962	478	36	2(1	2(1	NUM
ejpam-4962	478	37	)	)	PUNCT
ejpam-4962	478	38	=	=	SYM
ejpam-4962	479	1	n	n	CCONJ
ejpam-4962	479	2	−	−	PROPN
ejpam-4962	479	3	1	1	NUM
ejpam-4962	479	4	,	,	PUNCT
ejpam-4962	479	5	a	a	DET
ejpam-4962	479	6	contradiction	contradiction	NOUN
ejpam-4962	479	7	.	.	PUNCT
ejpam-4962	480	1	therefore	therefore	ADV
ejpam-4962	480	2	,	,	PUNCT
ejpam-4962	480	3	g	g	PROPN
ejpam-4962	480	4	is	be	AUX
ejpam-4962	480	5	p5	p5	ADJ
ejpam-4962	480	6	-	-	PUNCT
ejpam-4962	480	7	free	free	ADJ
ejpam-4962	480	8	.	.	PUNCT
ejpam-4962	481	1	the	the	DET
ejpam-4962	481	2	converse	converse	NOUN
ejpam-4962	481	3	of	of	ADP
ejpam-4962	481	4	proposition	proposition	NOUN
ejpam-4962	481	5	5	5	NUM
ejpam-4962	481	6	is	be	AUX
ejpam-4962	481	7	not	not	PART
ejpam-4962	481	8	true	true	ADJ
ejpam-4962	481	9	.	.	PUNCT
ejpam-4962	482	1	the	the	DET
ejpam-4962	482	2	cycle	cycle	NOUN
ejpam-4962	482	3	c5	c5	PROPN
ejpam-4962	482	4	has	have	VERB
ejpam-4962	482	5	no	no	DET
ejpam-4962	482	6	induced	induce	VERB
ejpam-4962	482	7	subgraph	subgraph	NOUN
ejpam-4962	482	8	p5	p5	NOUN
ejpam-4962	482	9	but	but	CCONJ
ejpam-4962	482	10	γgr(c5	γgr(c5	NOUN
ejpam-4962	482	11	)	)	PUNCT
ejpam-4962	482	12	=	=	SYM
ejpam-4962	482	13	4	4	NUM
ejpam-4962	482	14	̸=	̸=	PROPN
ejpam-4962	482	15	5	5	NUM
ejpam-4962	482	16	by	by	ADP
ejpam-4962	482	17	proposition	proposition	NOUN
ejpam-4962	482	18	3	3	NUM
ejpam-4962	482	19	.	.	X
ejpam-4962	482	20	proposition	proposition	NOUN
ejpam-4962	482	21	6	6	NUM
ejpam-4962	482	22	.	.	PUNCT
ejpam-4962	483	1	let	let	VERB
ejpam-4962	483	2	g	g	PRON
ejpam-4962	483	3	be	be	AUX
ejpam-4962	483	4	a	a	DET
ejpam-4962	483	5	connected	connected	ADJ
ejpam-4962	483	6	graph	graph	NOUN
ejpam-4962	483	7	such	such	ADJ
ejpam-4962	483	8	that	that	PRON
ejpam-4962	483	9	γg(g	γg(g	NOUN
ejpam-4962	483	10	)	)	PUNCT
ejpam-4962	483	11	̸=	̸=	PROPN
ejpam-4962	483	12	γgr(g	γgr(g	PROPN
ejpam-4962	483	13	)	)	PUNCT
ejpam-4962	483	14	.	.	PUNCT
ejpam-4962	484	1	then	then	ADV
ejpam-4962	484	2	γgr(g	γgr(g	PROPN
ejpam-4962	484	3	)	)	PUNCT
ejpam-4962	484	4	=	=	SYM
ejpam-4962	484	5	γg(g	γg(g	X
ejpam-4962	484	6	)	)	PUNCT
ejpam-4962	484	7	+	+	CCONJ
ejpam-4962	484	8	1	1	NUM
ejpam-4962	484	9	if	if	SCONJ
ejpam-4962	484	10	and	and	CCONJ
ejpam-4962	484	11	only	only	ADV
ejpam-4962	484	12	if	if	SCONJ
ejpam-4962	484	13	one	one	NUM
ejpam-4962	484	14	of	of	ADP
ejpam-4962	484	15	the	the	DET
ejpam-4962	484	16	following	follow	VERB
ejpam-4962	484	17	holds	hold	VERB
ejpam-4962	484	18	:	:	PUNCT
ejpam-4962	484	19	(	(	PUNCT
ejpam-4962	484	20	i	i	NOUN
ejpam-4962	484	21	)	)	PUNCT
ejpam-4962	484	22	there	there	PRON
ejpam-4962	484	23	exists	exist	VERB
ejpam-4962	484	24	a	a	DET
ejpam-4962	484	25	vertex	vertex	NOUN
ejpam-4962	484	26	v	v	NOUN
ejpam-4962	484	27	in	in	ADP
ejpam-4962	484	28	g	g	PROPN
ejpam-4962	485	1	such	such	DET
ejpam-4962	485	2	that	that	PRON
ejpam-4962	485	3	v	v	NOUN
ejpam-4962	485	4	(	(	PUNCT
ejpam-4962	485	5	g	g	NOUN
ejpam-4962	485	6	)	)	PUNCT
ejpam-4962	485	7	\	\	NOUN
ejpam-4962	485	8	{	{	PUNCT
ejpam-4962	485	9	v	v	NOUN
ejpam-4962	485	10	}	}	PUNCT
ejpam-4962	485	11	⊆	⊆	NUM
ejpam-4962	485	12	ng(v	ng(v	PUNCT
ejpam-4962	485	13	)	)	PUNCT
ejpam-4962	485	14	and	and	CCONJ
ejpam-4962	485	15	g	g	PROPN
ejpam-4962	485	16	\	\	PROPN
ejpam-4962	485	17	v	v	NOUN
ejpam-4962	485	18	is	be	AUX
ejpam-4962	485	19	the	the	DET
ejpam-4962	485	20	union	union	NOUN
ejpam-4962	485	21	of	of	ADP
ejpam-4962	485	22	at	at	ADV
ejpam-4962	485	23	least	least	ADV
ejpam-4962	485	24	two	two	NUM
ejpam-4962	485	25	complete	complete	ADJ
ejpam-4962	485	26	graphs	graph	NOUN
ejpam-4962	485	27	.	.	PUNCT
ejpam-4962	486	1	(	(	PUNCT
ejpam-4962	486	2	ii	ii	NOUN
ejpam-4962	486	3	)	)	PUNCT
ejpam-4962	486	4	there	there	PRON
ejpam-4962	486	5	exists	exist	VERB
ejpam-4962	486	6	a	a	DET
ejpam-4962	486	7	vertex	vertex	NOUN
ejpam-4962	486	8	v	v	NOUN
ejpam-4962	486	9	in	in	ADP
ejpam-4962	486	10	g	g	PROPN
ejpam-4962	486	11	and	and	CCONJ
ejpam-4962	486	12	s	s	VERB
ejpam-4962	486	13	⊆	⊆	NUM
ejpam-4962	486	14	v	v	NOUN
ejpam-4962	486	15	(	(	PUNCT
ejpam-4962	486	16	g	g	NOUN
ejpam-4962	486	17	)	)	PUNCT
ejpam-4962	486	18	such	such	ADJ
ejpam-4962	486	19	that	that	PRON
ejpam-4962	486	20	s	s	VERB
ejpam-4962	486	21	⊆	⊆	NUM
ejpam-4962	486	22	ng(v	ng(v	PUNCT
ejpam-4962	486	23	)	)	PUNCT
ejpam-4962	486	24	and	and	CCONJ
ejpam-4962	486	25	v	v	NOUN
ejpam-4962	486	26	(	(	PUNCT
ejpam-4962	486	27	g	g	NOUN
ejpam-4962	486	28	)	)	PUNCT
ejpam-4962	486	29	\	\	PROPN
ejpam-4962	487	1	s	s	PART
ejpam-4962	487	2	is	be	AUX
ejpam-4962	487	3	a	a	DET
ejpam-4962	487	4	γg	γg	ADV
ejpam-4962	487	5	-	-	PUNCT
ejpam-4962	487	6	set	set	VERB
ejpam-4962	487	7	in	in	ADP
ejpam-4962	487	8	g.	g.	PROPN
ejpam-4962	487	9	proof	proof	PROPN
ejpam-4962	487	10	.	.	PUNCT
ejpam-4962	488	1	suppose	suppose	VERB
ejpam-4962	488	2	γg(g	γg(g	ADP
ejpam-4962	488	3	)	)	PUNCT
ejpam-4962	489	1	+	+	CCONJ
ejpam-4962	489	2	1	1	NUM
ejpam-4962	489	3	=	=	SYM
ejpam-4962	489	4	γgr(g	γgr(g	PROPN
ejpam-4962	489	5	)	)	PUNCT
ejpam-4962	489	6	.	.	PUNCT
ejpam-4962	490	1	let	let	VERB
ejpam-4962	490	2	f	f	PROPN
ejpam-4962	490	3	=	=	SYM
ejpam-4962	490	4	(	(	PUNCT
ejpam-4962	490	5	v0	v0	PROPN
ejpam-4962	490	6	,	,	PUNCT
ejpam-4962	490	7	v1	v1	NOUN
ejpam-4962	490	8	,	,	PUNCT
ejpam-4962	490	9	v2	v2	PROPN
ejpam-4962	490	10	)	)	PUNCT
ejpam-4962	490	11	be	be	AUX
ejpam-4962	490	12	a	a	DET
ejpam-4962	490	13	γgr	γgr	NOUN
ejpam-4962	490	14	-	-	PUNCT
ejpam-4962	490	15	function	function	NOUN
ejpam-4962	490	16	.	.	PUNCT
ejpam-4962	491	1	consider	consider	VERB
ejpam-4962	491	2	the	the	DET
ejpam-4962	491	3	following	follow	VERB
ejpam-4962	491	4	cases	case	NOUN
ejpam-4962	491	5	:	:	PUNCT
ejpam-4962	491	6	case	case	NOUN
ejpam-4962	491	7	1	1	NUM
ejpam-4962	491	8	:	:	PUNCT
ejpam-4962	491	9	γg(g	γg(g	NUM
ejpam-4962	491	10	)	)	PUNCT
ejpam-4962	491	11	<	<	X
ejpam-4962	491	12	|v1|+	|v1|+	ADV
ejpam-4962	491	13	|v2|	|v2|	ADV
ejpam-4962	491	14	then	then	ADV
ejpam-4962	491	15	γg(g	γg(g	PUNCT
ejpam-4962	491	16	)	)	PUNCT
ejpam-4962	492	1	+	+	CCONJ
ejpam-4962	492	2	1	1	NUM
ejpam-4962	492	3	≤	≤	NUM
ejpam-4962	492	4	|v1|	|v1|	NOUN
ejpam-4962	492	5	+	+	CCONJ
ejpam-4962	492	6	|v2|	|v2|	NOUN
ejpam-4962	492	7	≤	≤	NOUN
ejpam-4962	492	8	|v1|	|v1|	NOUN
ejpam-4962	492	9	+	+	CCONJ
ejpam-4962	492	10	2|v2|	2|v2|	NUM
ejpam-4962	492	11	≤	≤	ADJ
ejpam-4962	492	12	γgr(g	γgr(g	PROPN
ejpam-4962	492	13	)	)	PUNCT
ejpam-4962	492	14	.	.	PUNCT
ejpam-4962	493	1	the	the	DET
ejpam-4962	493	2	assumption	assumption	NOUN
ejpam-4962	493	3	that	that	SCONJ
ejpam-4962	493	4	γgr(g	γgr(g	PROPN
ejpam-4962	493	5	)	)	PUNCT
ejpam-4962	493	6	=	=	SYM
ejpam-4962	493	7	γg(g	γg(g	X
ejpam-4962	493	8	)	)	PUNCT
ejpam-4962	493	9	+	+	CCONJ
ejpam-4962	493	10	1	1	NUM
ejpam-4962	493	11	implies	imply	VERB
ejpam-4962	493	12	that	that	DET
ejpam-4962	493	13	|v2|	|v2|	NOUN
ejpam-4962	493	14	=	=	SYM
ejpam-4962	493	15	0	0	X
ejpam-4962	493	16	.	.	PUNCT
ejpam-4962	493	17	by	by	ADP
ejpam-4962	493	18	proposition	proposition	NOUN
ejpam-4962	493	19	1(ii	1(ii	NUM
ejpam-4962	493	20	)	)	PUNCT
ejpam-4962	493	21	,	,	PUNCT
ejpam-4962	493	22	|v0|	|v0|	NOUN
ejpam-4962	493	23	=	=	SYM
ejpam-4962	493	24	0	0	NUM
ejpam-4962	493	25	and	and	CCONJ
ejpam-4962	493	26	γgr(g	γgr(g	PROPN
ejpam-4962	493	27	)	)	PUNCT
ejpam-4962	493	28	=	=	VERB
ejpam-4962	494	1	n.	n.	NOUN
ejpam-4962	494	2	it	it	PRON
ejpam-4962	494	3	follows	follow	VERB
ejpam-4962	494	4	that	that	PRON
ejpam-4962	494	5	γg(g	γg(g	PUNCT
ejpam-4962	494	6	)	)	PUNCT
ejpam-4962	494	7	=	=	PUNCT
ejpam-4962	494	8	n−	n−	NOUN
ejpam-4962	494	9	1	1	NUM
ejpam-4962	494	10	.	.	PUNCT
ejpam-4962	495	1	by	by	ADP
ejpam-4962	495	2	theorem	theorem	NOUN
ejpam-4962	495	3	1(iii	1(iii	NUM
ejpam-4962	495	4	)	)	PUNCT
ejpam-4962	495	5	,	,	PUNCT
ejpam-4962	495	6	(	(	PUNCT
ejpam-4962	495	7	i	i	NOUN
ejpam-4962	495	8	)	)	PUNCT
ejpam-4962	495	9	follows	follow	VERB
ejpam-4962	495	10	.	.	PUNCT
ejpam-4962	496	1	case	case	NOUN
ejpam-4962	496	2	2	2	NUM
ejpam-4962	496	3	:	:	PUNCT
ejpam-4962	496	4	γg(g	γg(g	NUM
ejpam-4962	496	5	)	)	PUNCT
ejpam-4962	496	6	=	=	PUNCT
ejpam-4962	497	1	|v1|+	|v1|+	ADV
ejpam-4962	497	2	|v2|	|v2|	ADV
ejpam-4962	497	3	then	then	ADV
ejpam-4962	497	4	γg(g	γg(g	PUNCT
ejpam-4962	497	5	)	)	PUNCT
ejpam-4962	498	1	+	+	CCONJ
ejpam-4962	498	2	1	1	NUM
ejpam-4962	498	3	=	=	NOUN
ejpam-4962	498	4	|v1|	|v1|	NOUN
ejpam-4962	498	5	+	+	CCONJ
ejpam-4962	498	6	|v2|	|v2|	ADV
ejpam-4962	498	7	+	+	CCONJ
ejpam-4962	498	8	1	1	NUM
ejpam-4962	498	9	=	=	NOUN
ejpam-4962	498	10	|v1|	|v1|	NOUN
ejpam-4962	498	11	+	+	CCONJ
ejpam-4962	498	12	2|v2|	2|v2|	NUM
ejpam-4962	498	13	=	=	SYM
ejpam-4962	498	14	γgr(g	γgr(g	PROPN
ejpam-4962	498	15	)	)	PUNCT
ejpam-4962	498	16	.	.	PUNCT
ejpam-4962	499	1	hence	hence	ADV
ejpam-4962	499	2	,	,	PUNCT
ejpam-4962	499	3	|v2|	|v2|	NOUN
ejpam-4962	499	4	=	=	SYM
ejpam-4962	499	5	1	1	NUM
ejpam-4962	499	6	and	and	CCONJ
ejpam-4962	499	7	|v1|	|v1|	NOUN
ejpam-4962	499	8	=	=	SYM
ejpam-4962	499	9	γg(g	γg(g	NOUN
ejpam-4962	499	10	)	)	PUNCT
ejpam-4962	499	11	−	−	PROPN
ejpam-4962	500	1	1	1	X
ejpam-4962	500	2	.	.	PUNCT
ejpam-4962	501	1	this	this	PRON
ejpam-4962	501	2	implies	imply	VERB
ejpam-4962	501	3	that	that	SCONJ
ejpam-4962	501	4	|v0|	|v0|	NOUN
ejpam-4962	501	5	=	=	SYM
ejpam-4962	501	6	|v	|v	X
ejpam-4962	501	7	(	(	PUNCT
ejpam-4962	501	8	g	g	NOUN
ejpam-4962	501	9	)	)	PUNCT
ejpam-4962	501	10	\	\	PUNCT
ejpam-4962	502	1	(	(	PUNCT
ejpam-4962	502	2	v1	v1	VERB
ejpam-4962	502	3	∪	∪	ADJ
ejpam-4962	502	4	v2)|	v2)|	NOUN
ejpam-4962	502	5	=	=	SYM
ejpam-4962	502	6	n	n	CCONJ
ejpam-4962	502	7	−	−	NOUN
ejpam-4962	502	8	γg(g	γg(g	NUM
ejpam-4962	502	9	)	)	PUNCT
ejpam-4962	502	10	.	.	PUNCT
ejpam-4962	503	1	let	let	VERB
ejpam-4962	503	2	v2	v2	VERB
ejpam-4962	503	3	=	=	PUNCT
ejpam-4962	503	4	{	{	PUNCT
ejpam-4962	503	5	v	v	NOUN
ejpam-4962	503	6	}	}	PUNCT
ejpam-4962	503	7	and	and	CCONJ
ejpam-4962	503	8	s	s	NOUN
ejpam-4962	503	9	=	=	PROPN
ejpam-4962	503	10	v0	v0	PROPN
ejpam-4962	503	11	.	.	PUNCT
ejpam-4962	504	1	then	then	ADV
ejpam-4962	504	2	s	s	VERB
ejpam-4962	504	3	⊆	⊆	NUM
ejpam-4962	504	4	ng(v	ng(v	NUM
ejpam-4962	504	5	)	)	PUNCT
ejpam-4962	504	6	.	.	PUNCT
ejpam-4962	505	1	moreover	moreover	ADV
ejpam-4962	505	2	,	,	PUNCT
ejpam-4962	505	3	v	v	INTJ
ejpam-4962	505	4	(	(	PUNCT
ejpam-4962	505	5	g	g	NOUN
ejpam-4962	505	6	)	)	PUNCT
ejpam-4962	505	7	\	\	PROPN
ejpam-4962	505	8	s	s	PART
ejpam-4962	505	9	=	=	SYM
ejpam-4962	505	10	v1	v1	NOUN
ejpam-4962	505	11	∪	∪	NOUN
ejpam-4962	505	12	v2	v2	NOUN
ejpam-4962	505	13	is	be	AUX
ejpam-4962	505	14	a	a	DET
ejpam-4962	505	15	γg	γg	ADV
ejpam-4962	505	16	-	-	PUNCT
ejpam-4962	505	17	set	set	NOUN
ejpam-4962	505	18	because	because	SCONJ
ejpam-4962	505	19	it	it	PRON
ejpam-4962	505	20	is	be	AUX
ejpam-4962	505	21	a	a	DET
ejpam-4962	505	22	geodetic	geodetic	ADJ
ejpam-4962	505	23	set	set	NOUN
ejpam-4962	505	24	and	and	CCONJ
ejpam-4962	505	25	|v1	|v1	X
ejpam-4962	505	26	∪	∪	X
ejpam-4962	505	27	v2|	v2|	X
ejpam-4962	505	28	=	=	PUNCT
ejpam-4962	505	29	γg(g	γg(g	NOUN
ejpam-4962	505	30	)	)	PUNCT
ejpam-4962	505	31	.	.	PUNCT
ejpam-4962	506	1	therefore	therefore	ADV
ejpam-4962	506	2	(	(	PUNCT
ejpam-4962	506	3	ii	ii	NOUN
ejpam-4962	506	4	)	)	PUNCT
ejpam-4962	506	5	holds	hold	VERB
ejpam-4962	506	6	.	.	PUNCT
ejpam-4962	507	1	for	for	ADP
ejpam-4962	507	2	the	the	DET
ejpam-4962	507	3	converse	converse	NOUN
ejpam-4962	507	4	,	,	PUNCT
ejpam-4962	507	5	suppose	suppose	VERB
ejpam-4962	507	6	first	first	ADV
ejpam-4962	507	7	that	that	SCONJ
ejpam-4962	507	8	(	(	PUNCT
ejpam-4962	507	9	i	i	NOUN
ejpam-4962	507	10	)	)	PUNCT
ejpam-4962	507	11	holds	hold	VERB
ejpam-4962	507	12	.	.	PUNCT
ejpam-4962	508	1	let	let	VERB
ejpam-4962	508	2	s	s	NOUN
ejpam-4962	508	3	=	=	X
ejpam-4962	508	4	v	v	ADJ
ejpam-4962	508	5	(	(	PUNCT
ejpam-4962	508	6	g	g	NOUN
ejpam-4962	508	7	)	)	PUNCT
ejpam-4962	508	8	\	\	NOUN
ejpam-4962	508	9	{	{	PUNCT
ejpam-4962	508	10	v	v	NOUN
ejpam-4962	508	11	}	}	PUNCT
ejpam-4962	508	12	.	.	PUNCT
ejpam-4962	509	1	let	let	VERB
ejpam-4962	509	2	w	w	PROPN
ejpam-4962	509	3	∈	∈	VERB
ejpam-4962	509	4	s.	s.	PROPN
ejpam-4962	509	5	since	since	SCONJ
ejpam-4962	509	6	g\v	g\v	PROPN
ejpam-4962	509	7	=	=	PUNCT
ejpam-4962	509	8	⟨s⟩	⟨s⟩	PROPN
ejpam-4962	509	9	is	be	AUX
ejpam-4962	509	10	the	the	DET
ejpam-4962	509	11	union	union	NOUN
ejpam-4962	509	12	of	of	ADP
ejpam-4962	509	13	at	at	ADV
ejpam-4962	509	14	least	least	ADV
ejpam-4962	509	15	two	two	NUM
ejpam-4962	509	16	complete	complete	ADJ
ejpam-4962	509	17	graphs	graph	NOUN
ejpam-4962	509	18	,	,	PUNCT
ejpam-4962	509	19	the	the	DET
ejpam-4962	509	20	component	component	NOUN
ejpam-4962	509	21	c	c	NOUN
ejpam-4962	509	22	ofg\v	ofg\v	PROPN
ejpam-4962	509	23	containing	contain	VERB
ejpam-4962	509	24	w	w	PROPN
ejpam-4962	509	25	as	as	ADP
ejpam-4962	509	26	a	a	DET
ejpam-4962	509	27	vertex	vertex	NOUN
ejpam-4962	509	28	is	be	AUX
ejpam-4962	509	29	complete	complete	ADJ
ejpam-4962	509	30	.	.	PUNCT
ejpam-4962	510	1	this	this	PRON
ejpam-4962	510	2	implies	imply	VERB
ejpam-4962	510	3	that	that	SCONJ
ejpam-4962	510	4	s	s	VERB
ejpam-4962	510	5	=	=	ADJ
ejpam-4962	510	6	ext(g	ext(g	NOUN
ejpam-4962	510	7	)	)	PUNCT
ejpam-4962	510	8	.	.	PUNCT
ejpam-4962	511	1	now	now	ADV
ejpam-4962	511	2	,	,	PUNCT
ejpam-4962	511	3	let	let	VERB
ejpam-4962	511	4	c1	c1	PROPN
ejpam-4962	511	5	and	and	CCONJ
ejpam-4962	511	6	c2	c2	PROPN
ejpam-4962	511	7	be	be	AUX
ejpam-4962	511	8	distinct	distinct	ADJ
ejpam-4962	511	9	components	component	NOUN
ejpam-4962	511	10	of	of	ADP
ejpam-4962	511	11	g	g	PROPN
ejpam-4962	511	12	\	\	PROPN
ejpam-4962	511	13	v	v	NOUN
ejpam-4962	511	14	and	and	CCONJ
ejpam-4962	511	15	let	let	VERB
ejpam-4962	511	16	x	x	SYM
ejpam-4962	511	17	∈	∈	PROPN
ejpam-4962	511	18	v	v	X
ejpam-4962	511	19	(	(	PUNCT
ejpam-4962	511	20	c1	c1	PROPN
ejpam-4962	511	21	)	)	PUNCT
ejpam-4962	511	22	and	and	CCONJ
ejpam-4962	511	23	y	y	PROPN
ejpam-4962	511	24	∈	∈	PROPN
ejpam-4962	511	25	v	v	PROPN
ejpam-4962	511	26	(	(	PUNCT
ejpam-4962	511	27	c2	c2	PROPN
ejpam-4962	511	28	)	)	PUNCT
ejpam-4962	511	29	.	.	PUNCT
ejpam-4962	512	1	then	then	ADV
ejpam-4962	512	2	v	v	ADP
ejpam-4962	512	3	∈	∈	PROPN
ejpam-4962	512	4	ig(x	ig(x	X
ejpam-4962	512	5	,	,	PUNCT
ejpam-4962	512	6	y	y	NOUN
ejpam-4962	512	7	)	)	PUNCT
ejpam-4962	512	8	.	.	PUNCT
ejpam-4962	513	1	hence	hence	ADV
ejpam-4962	513	2	,	,	PUNCT
ejpam-4962	513	3	s	s	PART
ejpam-4962	513	4	=	=	SYM
ejpam-4962	513	5	ext(g	ext(g	NOUN
ejpam-4962	513	6	)	)	PUNCT
ejpam-4962	513	7	is	be	AUX
ejpam-4962	513	8	the	the	DET
ejpam-4962	513	9	unique	unique	ADJ
ejpam-4962	513	10	γg	γg	ADV
ejpam-4962	513	11	-	-	PUNCT
ejpam-4962	513	12	set	set	NOUN
ejpam-4962	513	13	of	of	ADP
ejpam-4962	513	14	g	g	NOUN
ejpam-4962	513	15	and	and	CCONJ
ejpam-4962	513	16	γg(g	γg(g	NOUN
ejpam-4962	513	17	)	)	PUNCT
ejpam-4962	513	18	=	=	SYM
ejpam-4962	514	1	n	n	CCONJ
ejpam-4962	515	1	−	−	NOUN
ejpam-4962	515	2	1	1	NUM
ejpam-4962	515	3	.	.	PUNCT
ejpam-4962	516	1	by	by	ADP
ejpam-4962	516	2	proposition	proposition	NOUN
ejpam-4962	516	3	4	4	NUM
ejpam-4962	516	4	,	,	PUNCT
ejpam-4962	516	5	we	we	PRON
ejpam-4962	516	6	have	have	VERB
ejpam-4962	516	7	γgr(g	γgr(g	PROPN
ejpam-4962	516	8	)	)	PUNCT
ejpam-4962	516	9	=	=	SYM
ejpam-4962	517	1	n	n	NOUN
ejpam-4962	517	2	=	=	PUNCT
ejpam-4962	517	3	γg(g	γg(g	X
ejpam-4962	517	4	)	)	PUNCT
ejpam-4962	517	5	+	+	CCONJ
ejpam-4962	518	1	1	1	X
ejpam-4962	518	2	.	.	PUNCT
ejpam-4962	518	3	next	next	ADV
ejpam-4962	518	4	,	,	PUNCT
ejpam-4962	518	5	suppose	suppose	VERB
ejpam-4962	518	6	that	that	SCONJ
ejpam-4962	518	7	(	(	PUNCT
ejpam-4962	518	8	ii	ii	NOUN
ejpam-4962	518	9	)	)	PUNCT
ejpam-4962	518	10	holds	hold	VERB
ejpam-4962	518	11	.	.	PUNCT
ejpam-4962	519	1	let	let	VERB
ejpam-4962	519	2	v0	v0	NOUN
ejpam-4962	519	3	=	=	SYM
ejpam-4962	519	4	s	s	PROPN
ejpam-4962	519	5	,	,	PUNCT
ejpam-4962	519	6	v2	v2	X
ejpam-4962	519	7	=	=	SYM
ejpam-4962	519	8	{	{	PUNCT
ejpam-4962	519	9	v	v	NOUN
ejpam-4962	519	10	}	}	PUNCT
ejpam-4962	519	11	and	and	CCONJ
ejpam-4962	519	12	v1	v1	PROPN
ejpam-4962	519	13	=	=	SYM
ejpam-4962	519	14	v	v	NOUN
ejpam-4962	519	15	(	(	PUNCT
ejpam-4962	519	16	g	g	NOUN
ejpam-4962	519	17	)	)	PUNCT
ejpam-4962	519	18	\	\	PUNCT
ejpam-4962	520	1	(	(	PUNCT
ejpam-4962	520	2	s	s	NOUN
ejpam-4962	520	3	∪	∪	X
ejpam-4962	520	4	{	{	PUNCT
ejpam-4962	520	5	v	v	NOUN
ejpam-4962	520	6	}	}	PUNCT
ejpam-4962	520	7	)	)	PUNCT
ejpam-4962	520	8	.	.	PUNCT
ejpam-4962	521	1	then	then	ADV
ejpam-4962	521	2	v1	v1	VERB
ejpam-4962	521	3	∪	∪	ADJ
ejpam-4962	521	4	v2	v2	PROPN
ejpam-4962	521	5	=	=	SYM
ejpam-4962	521	6	v	v	NOUN
ejpam-4962	521	7	(	(	PUNCT
ejpam-4962	521	8	g	g	NOUN
ejpam-4962	521	9	)	)	PUNCT
ejpam-4962	521	10	\	\	PROPN
ejpam-4962	522	1	s	s	PART
ejpam-4962	522	2	is	be	AUX
ejpam-4962	522	3	a	a	DET
ejpam-4962	522	4	γg	γg	ADV
ejpam-4962	522	5	-	-	PUNCT
ejpam-4962	522	6	set	set	NOUN
ejpam-4962	522	7	of	of	ADP
ejpam-4962	522	8	g	g	NOUN
ejpam-4962	522	9	and	and	CCONJ
ejpam-4962	522	10	v0	v0	PROPN
ejpam-4962	522	11	⊆	⊆	NUM
ejpam-4962	522	12	ng(v	ng(v	NOUN
ejpam-4962	522	13	)	)	PUNCT
ejpam-4962	522	14	.	.	PUNCT
ejpam-4962	523	1	it	it	PRON
ejpam-4962	523	2	follows	follow	VERB
ejpam-4962	523	3	that	that	SCONJ
ejpam-4962	523	4	g	g	PROPN
ejpam-4962	523	5	=	=	SYM
ejpam-4962	523	6	(	(	PUNCT
ejpam-4962	523	7	v0	v0	PROPN
ejpam-4962	523	8	,	,	PUNCT
ejpam-4962	523	9	v1	v1	NOUN
ejpam-4962	523	10	,	,	PUNCT
ejpam-4962	523	11	v2	v2	PROPN
ejpam-4962	523	12	)	)	PUNCT
ejpam-4962	523	13	is	be	AUX
ejpam-4962	523	14	a	a	DET
ejpam-4962	523	15	grdf	grdf	NOUN
ejpam-4962	523	16	on	on	ADP
ejpam-4962	523	17	g	g	PROPN
ejpam-4962	523	18	and	and	CCONJ
ejpam-4962	523	19	γgr(g	γgr(g	PROPN
ejpam-4962	523	20	)	)	PUNCT
ejpam-4962	523	21	≤	≤	NUM
ejpam-4962	523	22	ωgr	ωgr	X
ejpam-4962	523	23	g	g	PROPN
ejpam-4962	523	24	(	(	PUNCT
ejpam-4962	523	25	g	g	NOUN
ejpam-4962	523	26	)	)	PUNCT
ejpam-4962	523	27	=	=	PUNCT
ejpam-4962	523	28	|v1|+	|v1|+	PRON
ejpam-4962	523	29	2|v2|	2|v2|	NUM
ejpam-4962	523	30	=	=	NOUN
ejpam-4962	523	31	γg(g)−	γg(g)−	NOUN
ejpam-4962	523	32	1	1	NUM
ejpam-4962	523	33	+	+	CCONJ
ejpam-4962	523	34	2	2	NUM
ejpam-4962	523	35	=	=	SYM
ejpam-4962	523	36	γg(g	γg(g	X
ejpam-4962	523	37	)	)	PUNCT
ejpam-4962	524	1	+	+	CCONJ
ejpam-4962	524	2	1	1	X
ejpam-4962	524	3	.	.	PUNCT
ejpam-4962	524	4	since	since	SCONJ
ejpam-4962	524	5	γg(g	γg(g	NUM
ejpam-4962	524	6	)	)	PUNCT
ejpam-4962	524	7	<	<	X
ejpam-4962	524	8	γgr(g	γgr(g	PROPN
ejpam-4962	524	9	)	)	PUNCT
ejpam-4962	524	10	,	,	PUNCT
ejpam-4962	524	11	γg(g	γg(g	PUNCT
ejpam-4962	524	12	)	)	PUNCT
ejpam-4962	524	13	+	+	CCONJ
ejpam-4962	524	14	1	1	NUM
ejpam-4962	524	15	≤	≤	NUM
ejpam-4962	524	16	γgr(g	γgr(g	PROPN
ejpam-4962	524	17	)	)	PUNCT
ejpam-4962	524	18	.	.	PUNCT
ejpam-4962	525	1	thus	thus	ADV
ejpam-4962	525	2	,	,	PUNCT
ejpam-4962	525	3	γgr(g	γgr(g	PROPN
ejpam-4962	525	4	)	)	PUNCT
ejpam-4962	525	5	=	=	SYM
ejpam-4962	525	6	γg(g	γg(g	X
ejpam-4962	525	7	)	)	PUNCT
ejpam-4962	525	8	+	+	NOUN
ejpam-4962	525	9	1	1	X
ejpam-4962	525	10	.	.	X
ejpam-4962	525	11	r.	r.	PROPN
ejpam-4962	525	12	fortosa	fortosa	PROPN
ejpam-4962	525	13	,	,	PUNCT
ejpam-4962	525	14	s.	s.	PROPN
ejpam-4962	525	15	canoy	canoy	PROPN
ejpam-4962	525	16	jr	jr	PROPN
ejpam-4962	525	17	.	.	PROPN
ejpam-4962	525	18	/	/	SYM
ejpam-4962	525	19	eur	eur	PROPN
ejpam-4962	525	20	.	.	PUNCT
ejpam-4962	526	1	j.	j.	PROPN
ejpam-4962	526	2	pure	pure	PROPN
ejpam-4962	526	3	appl	appl	PROPN
ejpam-4962	526	4	.	.	PROPN
ejpam-4962	526	5	math	math	PROPN
ejpam-4962	526	6	,	,	PUNCT
ejpam-4962	526	7	16	16	NUM
ejpam-4962	526	8	(	(	PUNCT
ejpam-4962	526	9	4	4	NUM
ejpam-4962	526	10	)	)	PUNCT
ejpam-4962	526	11	(	(	PUNCT
ejpam-4962	526	12	2023	2023	NUM
ejpam-4962	526	13	)	)	PUNCT
ejpam-4962	526	14	,	,	PUNCT
ejpam-4962	526	15	2368	2368	NUM
ejpam-4962	526	16	-	-	SYM
ejpam-4962	526	17	2383	2383	NUM
ejpam-4962	526	18	2377	2377	NUM
ejpam-4962	526	19	theorem	theorem	NOUN
ejpam-4962	526	20	5	5	NUM
ejpam-4962	526	21	.	.	PUNCT
ejpam-4962	527	1	let	let	VERB
ejpam-4962	527	2	g	g	PROPN
ejpam-4962	527	3	=	=	SYM
ejpam-4962	527	4	kn1,	kn1,	X
ejpam-4962	527	5	...	...	PUNCT
ejpam-4962	527	6	,nk	,nk	PUNCT
ejpam-4962	527	7	be	be	VERB
ejpam-4962	527	8	the	the	DET
ejpam-4962	527	9	complete	complete	ADJ
ejpam-4962	527	10	k	k	ADJ
ejpam-4962	527	11	-	-	ADJ
ejpam-4962	527	12	partite	partite	ADJ
ejpam-4962	527	13	graph	graph	NOUN
ejpam-4962	527	14	with	with	ADP
ejpam-4962	527	15	1	1	NUM
ejpam-4962	527	16	≤	≤	NUM
ejpam-4962	527	17	n1	n1	ADJ
ejpam-4962	527	18	≤	≤	NOUN
ejpam-4962	527	19	n2	n2	NOUN
ejpam-4962	527	20	.	.	PUNCT
ejpam-4962	527	21	.	.	PUNCT
ejpam-4962	527	22	.	.	PUNCT
ejpam-4962	528	1	≤	≤	PROPN
ejpam-4962	528	2	nk	nk	PROPN
ejpam-4962	528	3	and	and	CCONJ
ejpam-4962	528	4	|{nj	|{nj	NUM
ejpam-4962	528	5	:	:	PUNCT
ejpam-4962	528	6	nj	nj	PROPN
ejpam-4962	528	7	̸=	̸=	PROPN
ejpam-4962	528	8	1}|	1}|	NUM
ejpam-4962	528	9	≥	≥	NOUN
ejpam-4962	528	10	2	2	NUM
ejpam-4962	528	11	.	.	PUNCT
ejpam-4962	528	12	then	then	ADV
ejpam-4962	528	13	γgr(g	γgr(g	PROPN
ejpam-4962	528	14	)	)	PUNCT
ejpam-4962	528	15	=	=	SYM
ejpam-4962	528	16	min{n(g	min{n(g	NOUN
ejpam-4962	528	17	)	)	PUNCT
ejpam-4962	529	1	+	+	CCONJ
ejpam-4962	529	2	1	1	NUM
ejpam-4962	529	3	,	,	PUNCT
ejpam-4962	529	4	6	6	NUM
ejpam-4962	529	5	}	}	PUNCT
ejpam-4962	529	6	,	,	PUNCT
ejpam-4962	529	7	where	where	SCONJ
ejpam-4962	529	8	n(g	n(g	NUM
ejpam-4962	529	9	)	)	PUNCT
ejpam-4962	530	1	=	=	PRON
ejpam-4962	530	2	min{nj	min{nj	NOUN
ejpam-4962	530	3	:	:	PUNCT
ejpam-4962	530	4	nj	nj	PROPN
ejpam-4962	530	5	≥	≥	VERB
ejpam-4962	530	6	2	2	NUM
ejpam-4962	530	7	}	}	PUNCT
ejpam-4962	530	8	.	.	PUNCT
ejpam-4962	531	1	proof	proof	NOUN
ejpam-4962	531	2	.	.	PUNCT
ejpam-4962	532	1	let	let	VERB
ejpam-4962	532	2	sn1	sn1	PROPN
ejpam-4962	532	3	,	,	PUNCT
ejpam-4962	532	4	sn2	sn2	PROPN
ejpam-4962	532	5	,	,	PUNCT
ejpam-4962	532	6	.	.	PUNCT
ejpam-4962	532	7	.	.	PUNCT
ejpam-4962	533	1	.	.	PUNCT
ejpam-4962	534	1	,	,	PUNCT
ejpam-4962	534	2	snk	snk	PROPN
ejpam-4962	534	3	be	be	AUX
ejpam-4962	534	4	the	the	DET
ejpam-4962	534	5	partite	partite	ADJ
ejpam-4962	534	6	sets	set	NOUN
ejpam-4962	534	7	in	in	ADP
ejpam-4962	534	8	g	g	NOUN
ejpam-4962	534	9	and	and	CCONJ
ejpam-4962	534	10	let	let	VERB
ejpam-4962	534	11	n(g	n(g	NUM
ejpam-4962	534	12	)	)	PUNCT
ejpam-4962	535	1	=	=	PUNCT
ejpam-4962	535	2	min{nj	min{nj	NOUN
ejpam-4962	535	3	:	:	PUNCT
ejpam-4962	535	4	nj	nj	PROPN
ejpam-4962	535	5	≥	≥	VERB
ejpam-4962	535	6	2	2	NUM
ejpam-4962	535	7	}	}	PUNCT
ejpam-4962	535	8	.	.	PUNCT
ejpam-4962	536	1	suppose	suppose	VERB
ejpam-4962	536	2	n(g	n(g	NUM
ejpam-4962	536	3	)	)	PUNCT
ejpam-4962	536	4	=	=	SYM
ejpam-4962	537	1	2	2	X
ejpam-4962	537	2	.	.	X
ejpam-4962	537	3	then	then	ADV
ejpam-4962	537	4	γgr(g	γgr(g	PROPN
ejpam-4962	537	5	)	)	PUNCT
ejpam-4962	537	6	=	=	SYM
ejpam-4962	537	7	3	3	NUM
ejpam-4962	537	8	=	=	SYM
ejpam-4962	537	9	n(g	n(g	NUM
ejpam-4962	537	10	)	)	PUNCT
ejpam-4962	537	11	+	+	NUM
ejpam-4962	537	12	1	1	NUM
ejpam-4962	537	13	,	,	PUNCT
ejpam-4962	537	14	by	by	ADP
ejpam-4962	537	15	theorem	theorem	ADJ
ejpam-4962	537	16	2(iii	2(iii	NUM
ejpam-4962	537	17	)	)	PUNCT
ejpam-4962	537	18	.	.	PUNCT
ejpam-4962	538	1	next	next	ADV
ejpam-4962	538	2	,	,	PUNCT
ejpam-4962	538	3	suppose	suppose	VERB
ejpam-4962	538	4	that	that	SCONJ
ejpam-4962	538	5	n(g	n(g	NUM
ejpam-4962	538	6	)	)	PUNCT
ejpam-4962	538	7	≥	≥	NOUN
ejpam-4962	538	8	3	3	NUM
ejpam-4962	538	9	.	.	PUNCT
ejpam-4962	538	10	pick	pick	VERB
ejpam-4962	538	11	u	u	PROPN
ejpam-4962	538	12	∈	∈	PROPN
ejpam-4962	538	13	sn	sn	PROPN
ejpam-4962	538	14	.	.	PUNCT
ejpam-4962	539	1	let	let	VERB
ejpam-4962	539	2	v2	v2	VERB
ejpam-4962	539	3	=	=	PUNCT
ejpam-4962	539	4	{	{	PUNCT
ejpam-4962	539	5	u	u	NOUN
ejpam-4962	539	6	}	}	PUNCT
ejpam-4962	539	7	,	,	PUNCT
ejpam-4962	539	8	v0	v0	NOUN
ejpam-4962	539	9	=	=	SYM
ejpam-4962	539	10	v	v	PROPN
ejpam-4962	539	11	(	(	PUNCT
ejpam-4962	539	12	g	g	NOUN
ejpam-4962	539	13	)	)	PUNCT
ejpam-4962	539	14	\	\	NOUN
ejpam-4962	539	15	sn(g	sn(g	NUM
ejpam-4962	539	16	)	)	PUNCT
ejpam-4962	539	17	,	,	PUNCT
ejpam-4962	539	18	and	and	CCONJ
ejpam-4962	539	19	v1	v1	NOUN
ejpam-4962	539	20	=	=	SYM
ejpam-4962	539	21	sn(g	sn(g	X
ejpam-4962	539	22	)	)	PUNCT
ejpam-4962	539	23	\	\	NOUN
ejpam-4962	539	24	{	{	PUNCT
ejpam-4962	539	25	u	u	NOUN
ejpam-4962	539	26	}	}	PUNCT
ejpam-4962	539	27	.	.	PUNCT
ejpam-4962	540	1	then	then	ADV
ejpam-4962	540	2	f	f	X
ejpam-4962	540	3	=	=	SYM
ejpam-4962	540	4	(	(	PUNCT
ejpam-4962	540	5	v0	v0	PROPN
ejpam-4962	540	6	,	,	PUNCT
ejpam-4962	540	7	v1	v1	NOUN
ejpam-4962	540	8	,	,	PUNCT
ejpam-4962	540	9	v2	v2	PROPN
ejpam-4962	540	10	)	)	PUNCT
ejpam-4962	540	11	is	be	AUX
ejpam-4962	540	12	a	a	DET
ejpam-4962	540	13	grdf	grdf	NOUN
ejpam-4962	540	14	on	on	ADP
ejpam-4962	540	15	g.	g.	PROPN
ejpam-4962	540	16	this	this	PRON
ejpam-4962	540	17	implies	imply	VERB
ejpam-4962	540	18	that	that	SCONJ
ejpam-4962	540	19	γgr(g	γgr(g	PROPN
ejpam-4962	540	20	)	)	PUNCT
ejpam-4962	540	21	≤	≤	NUM
ejpam-4962	540	22	ωgr	ωgr	X
ejpam-4962	540	23	g	g	PROPN
ejpam-4962	540	24	(	(	PUNCT
ejpam-4962	540	25	f	f	X
ejpam-4962	540	26	)	)	PUNCT
ejpam-4962	540	27	=	=	SYM
ejpam-4962	541	1	(	(	PUNCT
ejpam-4962	541	2	n(g)−	n(g)−	INTJ
ejpam-4962	541	3	1	1	NUM
ejpam-4962	541	4	)	)	PUNCT
ejpam-4962	541	5	+	+	CCONJ
ejpam-4962	541	6	2	2	NUM
ejpam-4962	541	7	=	=	SYM
ejpam-4962	541	8	n(g	n(g	NUM
ejpam-4962	541	9	)	)	PUNCT
ejpam-4962	541	10	+	+	CCONJ
ejpam-4962	541	11	1	1	X
ejpam-4962	541	12	.	.	PUNCT
ejpam-4962	541	13	next	next	ADV
ejpam-4962	541	14	,	,	PUNCT
ejpam-4962	541	15	let	let	VERB
ejpam-4962	541	16	v	v	NUM
ejpam-4962	541	17	∗	∗	NOUN
ejpam-4962	541	18	2	2	NUM
ejpam-4962	541	19	=	=	SYM
ejpam-4962	541	20	{	{	PUNCT
ejpam-4962	541	21	x	x	NOUN
ejpam-4962	541	22	,	,	PUNCT
ejpam-4962	541	23	y	y	PROPN
ejpam-4962	541	24	}	}	PUNCT
ejpam-4962	541	25	,	,	PUNCT
ejpam-4962	541	26	v	v	X
ejpam-4962	541	27	∗	∗	NOUN
ejpam-4962	541	28	1	1	NUM
ejpam-4962	541	29	=	=	SYM
ejpam-4962	541	30	{	{	PUNCT
ejpam-4962	541	31	w	w	PROPN
ejpam-4962	541	32	,	,	PUNCT
ejpam-4962	541	33	z	z	NOUN
ejpam-4962	541	34	}	}	PUNCT
ejpam-4962	541	35	,	,	PUNCT
ejpam-4962	541	36	and	and	CCONJ
ejpam-4962	541	37	v	v	ADP
ejpam-4962	541	38	∗	∗	NOUN
ejpam-4962	541	39	0	0	NUM
ejpam-4962	541	40	=	=	SYM
ejpam-4962	541	41	v	v	NOUN
ejpam-4962	541	42	(	(	PUNCT
ejpam-4962	541	43	g	g	NOUN
ejpam-4962	541	44	)	)	PUNCT
ejpam-4962	541	45	\	\	PUNCT
ejpam-4962	542	1	(	(	PUNCT
ejpam-4962	542	2	v	v	NOUN
ejpam-4962	542	3	∗	∗	NOUN
ejpam-4962	542	4	1	1	NUM
ejpam-4962	542	5	∪	∪	NOUN
ejpam-4962	542	6	v	v	NOUN
ejpam-4962	542	7	∗	∗	NOUN
ejpam-4962	542	8	2	2	NUM
ejpam-4962	542	9	)	)	PUNCT
ejpam-4962	542	10	where	where	SCONJ
ejpam-4962	542	11	x	x	X
ejpam-4962	542	12	,	,	PUNCT
ejpam-4962	542	13	w	w	PROPN
ejpam-4962	542	14	∈	∈	PROPN
ejpam-4962	542	15	snr	snr	PROPN
ejpam-4962	542	16	and	and	CCONJ
ejpam-4962	542	17	y	y	PROPN
ejpam-4962	542	18	,	,	PUNCT
ejpam-4962	542	19	z	z	PROPN
ejpam-4962	542	20	∈	∈	PROPN
ejpam-4962	542	21	snt	snt	NOUN
ejpam-4962	542	22	where	where	SCONJ
ejpam-4962	542	23	nr	nr	PRON
ejpam-4962	542	24	̸=	̸=	PROPN
ejpam-4962	542	25	1	1	NUM
ejpam-4962	542	26	and	and	CCONJ
ejpam-4962	542	27	nt	not	PART
ejpam-4962	542	28	̸=	̸=	PROPN
ejpam-4962	542	29	1	1	NUM
ejpam-4962	542	30	.	.	PUNCT
ejpam-4962	543	1	then	then	ADV
ejpam-4962	543	2	f	f	NOUN
ejpam-4962	544	1	′	′	NUM
ejpam-4962	545	1	=	=	SYM
ejpam-4962	546	1	(	(	PUNCT
ejpam-4962	546	2	v	v	NOUN
ejpam-4962	546	3	∗	∗	NOUN
ejpam-4962	546	4	0	0	NUM
ejpam-4962	546	5	,	,	PUNCT
ejpam-4962	546	6	v	v	NOUN
ejpam-4962	546	7	∗	∗	NOUN
ejpam-4962	546	8	1	1	NUM
ejpam-4962	546	9	,	,	PUNCT
ejpam-4962	546	10	v	v	NOUN
ejpam-4962	546	11	∗	∗	NOUN
ejpam-4962	546	12	2	2	NUM
ejpam-4962	546	13	)	)	PUNCT
ejpam-4962	546	14	is	be	AUX
ejpam-4962	546	15	a	a	DET
ejpam-4962	546	16	grdf	grdf	NOUN
ejpam-4962	546	17	on	on	ADP
ejpam-4962	546	18	g	g	PROPN
ejpam-4962	546	19	and	and	CCONJ
ejpam-4962	546	20	γgr(g	γgr(g	PROPN
ejpam-4962	546	21	)	)	PUNCT
ejpam-4962	546	22	≤	≤	NUM
ejpam-4962	546	23	ωgr	ωgr	X
ejpam-4962	546	24	g	g	PROPN
ejpam-4962	546	25	(	(	PUNCT
ejpam-4962	546	26	f	f	PROPN
ejpam-4962	546	27	′	′	NUM
ejpam-4962	546	28	)	)	PUNCT
ejpam-4962	547	1	=	=	PUNCT
ejpam-4962	547	2	|v	|v	PROPN
ejpam-4962	547	3	∗	∗	NOUN
ejpam-4962	547	4	1	1	NUM
ejpam-4962	547	5	|+	|+	NOUN
ejpam-4962	547	6	2|v	2|v	PROPN
ejpam-4962	547	7	∗	∗	NOUN
ejpam-4962	547	8	2	2	NUM
ejpam-4962	547	9	|	|	NOUN
ejpam-4962	547	10	=	=	SYM
ejpam-4962	547	11	2	2	NUM
ejpam-4962	547	12	+	+	NUM
ejpam-4962	547	13	2(2	2(2	NUM
ejpam-4962	547	14	)	)	PUNCT
ejpam-4962	547	15	=	=	SYM
ejpam-4962	548	1	6	6	X
ejpam-4962	548	2	.	.	PUNCT
ejpam-4962	548	3	therefore	therefore	ADV
ejpam-4962	548	4	,	,	PUNCT
ejpam-4962	548	5	γgr(g	γgr(g	PROPN
ejpam-4962	548	6	)	)	PUNCT
ejpam-4962	548	7	≤	≤	NOUN
ejpam-4962	548	8	min{n(g	min{n(g	PUNCT
ejpam-4962	548	9	)	)	PUNCT
ejpam-4962	549	1	+	+	CCONJ
ejpam-4962	549	2	1	1	NUM
ejpam-4962	549	3	,	,	PUNCT
ejpam-4962	549	4	6	6	NUM
ejpam-4962	549	5	}	}	PUNCT
ejpam-4962	549	6	.	.	PUNCT
ejpam-4962	550	1	now	now	ADV
ejpam-4962	550	2	,	,	PUNCT
ejpam-4962	550	3	let	let	VERB
ejpam-4962	550	4	g	g	NOUN
ejpam-4962	550	5	=	=	SYM
ejpam-4962	550	6	(	(	PUNCT
ejpam-4962	550	7	v	v	NUM
ejpam-4962	550	8	′′	′′	PROPN
ejpam-4962	550	9	0	0	NUM
ejpam-4962	550	10	,	,	PUNCT
ejpam-4962	550	11	v	v	ADP
ejpam-4962	550	12	′′	′′	PROPN
ejpam-4962	550	13	1	1	NUM
ejpam-4962	550	14	,	,	PUNCT
ejpam-4962	550	15	v	v	ADP
ejpam-4962	550	16	′′	′′	PROPN
ejpam-4962	550	17	2	2	NUM
ejpam-4962	550	18	)	)	PUNCT
ejpam-4962	550	19	be	be	AUX
ejpam-4962	550	20	a	a	DET
ejpam-4962	550	21	γgr	γgr	NOUN
ejpam-4962	550	22	-	-	PUNCT
ejpam-4962	550	23	function	function	NOUN
ejpam-4962	550	24	on	on	ADP
ejpam-4962	550	25	g.	g.	PROPN
ejpam-4962	550	26	suppose	suppose	VERB
ejpam-4962	550	27	that	that	SCONJ
ejpam-4962	550	28	γgr(g	γgr(g	PROPN
ejpam-4962	550	29	)	)	PUNCT
ejpam-4962	550	30	<	<	X
ejpam-4962	550	31	n(g	n(g	NUM
ejpam-4962	550	32	)	)	PUNCT
ejpam-4962	551	1	+	+	CCONJ
ejpam-4962	551	2	1	1	NUM
ejpam-4962	551	3	≤	≤	NUM
ejpam-4962	551	4	6	6	NUM
ejpam-4962	551	5	.	.	PUNCT
ejpam-4962	551	6	then	then	ADV
ejpam-4962	551	7	γgr(g	γgr(g	PROPN
ejpam-4962	551	8	)	)	PUNCT
ejpam-4962	552	1	=	=	SYM
ejpam-4962	552	2	ωgr	ωgr	NUM
ejpam-4962	552	3	g	g	PROPN
ejpam-4962	552	4	(	(	PUNCT
ejpam-4962	552	5	g	g	NOUN
ejpam-4962	552	6	)	)	PUNCT
ejpam-4962	552	7	=	=	PUNCT
ejpam-4962	552	8	|v	|v	PROPN
ejpam-4962	552	9	′′	′′	PROPN
ejpam-4962	552	10	1	1	NUM
ejpam-4962	553	1	|	|	ADV
ejpam-4962	553	2	+	+	CCONJ
ejpam-4962	553	3	2|v	2|v	NUM
ejpam-4962	554	1	′′	′′	NOUN
ejpam-4962	554	2	2	2	NUM
ejpam-4962	554	3	|	|	ADV
ejpam-4962	554	4	<	<	X
ejpam-4962	554	5	n(g	n(g	NUM
ejpam-4962	554	6	)	)	PUNCT
ejpam-4962	554	7	+	+	CCONJ
ejpam-4962	554	8	1	1	X
ejpam-4962	554	9	.	.	PUNCT
ejpam-4962	554	10	this	this	PRON
ejpam-4962	554	11	implies	imply	VERB
ejpam-4962	554	12	that	that	SCONJ
ejpam-4962	554	13	|v	|v	PROPN
ejpam-4962	554	14	′′	′′	PROPN
ejpam-4962	554	15	2	2	NUM
ejpam-4962	554	16	|	|	ADV
ejpam-4962	554	17	≤	≤	NUM
ejpam-4962	554	18	2	2	NUM
ejpam-4962	554	19	.	.	PUNCT
ejpam-4962	555	1	if	if	SCONJ
ejpam-4962	555	2	|v	|v	PROPN
ejpam-4962	555	3	′′	′′	PROPN
ejpam-4962	555	4	2	2	NUM
ejpam-4962	555	5	|	|	NOUN
ejpam-4962	555	6	=	=	SYM
ejpam-4962	555	7	0	0	NUM
ejpam-4962	555	8	,	,	PUNCT
ejpam-4962	555	9	then	then	ADV
ejpam-4962	555	10	|v	|v	VERB
ejpam-4962	555	11	′′	′′	PROPN
ejpam-4962	555	12	0	0	PUNCT
ejpam-4962	556	1	|	|	ADV
ejpam-4962	556	2	=	=	SYM
ejpam-4962	556	3	0	0	NUM
ejpam-4962	556	4	and	and	CCONJ
ejpam-4962	556	5	|v	|v	PROPN
ejpam-4962	556	6	′′	′′	PROPN
ejpam-4962	556	7	1	1	NUM
ejpam-4962	557	1	|	|	ADV
ejpam-4962	558	1	=	=	PRON
ejpam-4962	559	1	∑k	∑k	PROPN
ejpam-4962	560	1	i=1	i=1	PROPN
ejpam-4962	561	1	ni	ni	PROPN
ejpam-4962	561	2	≥	≥	PROPN
ejpam-4962	561	3	6	6	NUM
ejpam-4962	561	4	,	,	PUNCT
ejpam-4962	561	5	a	a	DET
ejpam-4962	561	6	contradiction	contradiction	NOUN
ejpam-4962	561	7	.	.	PUNCT
ejpam-4962	562	1	suppose	suppose	VERB
ejpam-4962	562	2	that	that	SCONJ
ejpam-4962	562	3	|v	|v	PROPN
ejpam-4962	562	4	′′	′′	PROPN
ejpam-4962	562	5	2	2	NUM
ejpam-4962	562	6	|	|	NOUN
ejpam-4962	562	7	=	=	SYM
ejpam-4962	562	8	1	1	NUM
ejpam-4962	562	9	,	,	PUNCT
ejpam-4962	562	10	say	say	VERB
ejpam-4962	562	11	v	v	ADP
ejpam-4962	562	12	′′	′′	PROPN
ejpam-4962	562	13	2	2	NUM
ejpam-4962	562	14	=	=	NOUN
ejpam-4962	562	15	{	{	PUNCT
ejpam-4962	562	16	v′′	v′′	NOUN
ejpam-4962	562	17	}	}	PUNCT
ejpam-4962	562	18	.	.	PUNCT
ejpam-4962	563	1	we	we	PRON
ejpam-4962	563	2	may	may	AUX
ejpam-4962	563	3	assume	assume	VERB
ejpam-4962	563	4	that	that	SCONJ
ejpam-4962	563	5	v′′	v′′	VERB
ejpam-4962	563	6	∈	∈	PROPN
ejpam-4962	563	7	sn(g	sn(g	NUM
ejpam-4962	563	8	)	)	PUNCT
ejpam-4962	563	9	.	.	PUNCT
ejpam-4962	564	1	then	then	ADV
ejpam-4962	564	2	sn(g	sn(g	PUNCT
ejpam-4962	564	3	)	)	PUNCT
ejpam-4962	564	4	\	\	NOUN
ejpam-4962	564	5	{	{	PUNCT
ejpam-4962	564	6	v′′	v′′	NOUN
ejpam-4962	564	7	}	}	PUNCT
ejpam-4962	564	8	⊆	⊆	NUM
ejpam-4962	564	9	v	v	ADP
ejpam-4962	564	10	′′	′′	PROPN
ejpam-4962	564	11	1	1	NUM
ejpam-4962	564	12	.	.	PUNCT
ejpam-4962	565	1	this	this	PRON
ejpam-4962	565	2	implies	imply	VERB
ejpam-4962	565	3	that	that	SCONJ
ejpam-4962	565	4	n(g	n(g	NUM
ejpam-4962	565	5	)	)	PUNCT
ejpam-4962	566	1	+	+	CCONJ
ejpam-4962	566	2	1	1	NUM
ejpam-4962	566	3	=	=	SYM
ejpam-4962	566	4	|sn(g	|sn(g	NOUN
ejpam-4962	566	5	)	)	PUNCT
ejpam-4962	566	6	\	\	NOUN
ejpam-4962	566	7	{	{	PUNCT
ejpam-4962	566	8	v′′}|+	v′′}|+	PROPN
ejpam-4962	566	9	2|v	2|v	NUM
ejpam-4962	567	1	′′	′′	NOUN
ejpam-4962	567	2	2	2	NUM
ejpam-4962	567	3	|	|	ADV
ejpam-4962	567	4	≤	≤	NUM
ejpam-4962	567	5	|v	|v	VERB
ejpam-4962	567	6	′′	′′	NOUN
ejpam-4962	567	7	1	1	NUM
ejpam-4962	567	8	|+	|+	NOUN
ejpam-4962	567	9	2|v	2|v	NUM
ejpam-4962	568	1	′′	′′	NOUN
ejpam-4962	568	2	2	2	NUM
ejpam-4962	568	3	|	|	ADV
ejpam-4962	568	4	<	<	X
ejpam-4962	568	5	n(g	n(g	NUM
ejpam-4962	568	6	)	)	PUNCT
ejpam-4962	568	7	+	+	CCONJ
ejpam-4962	568	8	1	1	NUM
ejpam-4962	568	9	,	,	PUNCT
ejpam-4962	568	10	a	a	DET
ejpam-4962	568	11	contradiction	contradiction	NOUN
ejpam-4962	568	12	.	.	PUNCT
ejpam-4962	569	1	suppose	suppose	VERB
ejpam-4962	569	2	now	now	ADV
ejpam-4962	569	3	that	that	SCONJ
ejpam-4962	569	4	|v	|v	PROPN
ejpam-4962	569	5	′′	′′	PROPN
ejpam-4962	569	6	2	2	NUM
ejpam-4962	569	7	|	|	NOUN
ejpam-4962	569	8	=	=	NOUN
ejpam-4962	569	9	2	2	X
ejpam-4962	569	10	.	.	PUNCT
ejpam-4962	569	11	suppose	suppose	VERB
ejpam-4962	569	12	|v	|v	PROPN
ejpam-4962	569	13	′′	′′	PROPN
ejpam-4962	569	14	1	1	NUM
ejpam-4962	569	15	|	|	ADV
ejpam-4962	569	16	=	=	NOUN
ejpam-4962	569	17	1	1	X
ejpam-4962	569	18	.	.	PUNCT
ejpam-4962	569	19	then	then	ADV
ejpam-4962	569	20	n(g	n(g	NUM
ejpam-4962	569	21	)	)	PUNCT
ejpam-4962	569	22	=	=	SYM
ejpam-4962	570	1	5	5	X
ejpam-4962	570	2	.	.	PUNCT
ejpam-4962	570	3	let	let	VERB
ejpam-4962	570	4	v	v	VERB
ejpam-4962	570	5	′′	′′	PROPN
ejpam-4962	570	6	2	2	NUM
ejpam-4962	570	7	=	=	SYM
ejpam-4962	570	8	{	{	PUNCT
ejpam-4962	570	9	p	p	X
ejpam-4962	570	10	,	,	PUNCT
ejpam-4962	570	11	q	q	ADJ
ejpam-4962	570	12	}	}	PUNCT
ejpam-4962	570	13	and	and	CCONJ
ejpam-4962	570	14	v	v	ADP
ejpam-4962	570	15	′′	′′	PROPN
ejpam-4962	570	16	1	1	NUM
ejpam-4962	570	17	=	=	SYM
ejpam-4962	570	18	{	{	PUNCT
ejpam-4962	570	19	s	s	NOUN
ejpam-4962	570	20	}	}	PUNCT
ejpam-4962	570	21	.	.	PUNCT
ejpam-4962	571	1	since	since	SCONJ
ejpam-4962	571	2	v	v	NUM
ejpam-4962	571	3	′′	′′	PROPN
ejpam-4962	571	4	1	1	NUM
ejpam-4962	571	5	∪	∪	X
ejpam-4962	571	6	v	v	ADP
ejpam-4962	571	7	′′	′′	PROPN
ejpam-4962	571	8	2	2	NUM
ejpam-4962	571	9	is	be	AUX
ejpam-4962	571	10	a	a	DET
ejpam-4962	571	11	geodetic	geodetic	ADJ
ejpam-4962	571	12	set	set	NOUN
ejpam-4962	571	13	,	,	PUNCT
ejpam-4962	571	14	at	at	ADP
ejpam-4962	571	15	least	least	ADJ
ejpam-4962	571	16	two	two	NUM
ejpam-4962	571	17	of	of	ADP
ejpam-4962	571	18	the	the	DET
ejpam-4962	571	19	vertices	vertex	NOUN
ejpam-4962	571	20	p	p	X
ejpam-4962	571	21	,	,	PUNCT
ejpam-4962	571	22	q	q	X
ejpam-4962	571	23	,	,	PUNCT
ejpam-4962	571	24	and	and	CCONJ
ejpam-4962	571	25	s	s	AUX
ejpam-4962	571	26	belong	belong	VERB
ejpam-4962	571	27	to	to	ADP
ejpam-4962	571	28	the	the	DET
ejpam-4962	571	29	same	same	ADJ
ejpam-4962	571	30	partite	partite	ADJ
ejpam-4962	571	31	set	set	NOUN
ejpam-4962	571	32	,	,	PUNCT
ejpam-4962	571	33	say	say	VERB
ejpam-4962	571	34	sni	sni	PROPN
ejpam-4962	571	35	where	where	SCONJ
ejpam-4962	571	36	i	i	PRON
ejpam-4962	571	37	∈	∈	PROPN
ejpam-4962	571	38	{	{	PUNCT
ejpam-4962	571	39	1	1	NUM
ejpam-4962	571	40	,	,	PUNCT
ejpam-4962	571	41	2	2	NUM
ejpam-4962	571	42	,	,	PUNCT
ejpam-4962	571	43	.	.	PUNCT
ejpam-4962	571	44	.	.	PUNCT
ejpam-4962	571	45	.	.	PUNCT
ejpam-4962	572	1	,	,	PUNCT
ejpam-4962	572	2	k	k	X
ejpam-4962	572	3	}	}	PUNCT
ejpam-4962	572	4	.	.	PUNCT
ejpam-4962	573	1	choose	choose	VERB
ejpam-4962	573	2	any	any	DET
ejpam-4962	573	3	z	z	NOUN
ejpam-4962	573	4	∈	∈	PROPN
ejpam-4962	573	5	sni	sni	PROPN
ejpam-4962	573	6	\	\	PROPN
ejpam-4962	573	7	{	{	PUNCT
ejpam-4962	573	8	p	p	X
ejpam-4962	573	9	,	,	PUNCT
ejpam-4962	573	10	q	q	ADJ
ejpam-4962	573	11	,	,	PUNCT
ejpam-4962	573	12	s}(such	s}(such	NUM
ejpam-4962	573	13	z	z	NOUN
ejpam-4962	573	14	exists	exist	VERB
ejpam-4962	573	15	because	because	SCONJ
ejpam-4962	573	16	nj	nj	PROPN
ejpam-4962	573	17	≥	≥	PROPN
ejpam-4962	573	18	n(g	n(g	NUM
ejpam-4962	573	19	)	)	PUNCT
ejpam-4962	573	20	=	=	SYM
ejpam-4962	574	1	5	5	NUM
ejpam-4962	574	2	)	)	PUNCT
ejpam-4962	574	3	.	.	PUNCT
ejpam-4962	575	1	then	then	ADV
ejpam-4962	575	2	z	z	PROPN
ejpam-4962	575	3	/∈	/∈	PUNCT
ejpam-4962	575	4	ig({p	ig({p	PROPN
ejpam-4962	575	5	,	,	PUNCT
ejpam-4962	575	6	q	q	X
ejpam-4962	575	7	,	,	PUNCT
ejpam-4962	575	8	s	s	NOUN
ejpam-4962	575	9	}	}	PUNCT
ejpam-4962	575	10	)	)	PUNCT
ejpam-4962	575	11	,	,	PUNCT
ejpam-4962	575	12	a	a	DET
ejpam-4962	575	13	contradiction	contradiction	NOUN
ejpam-4962	575	14	.	.	PUNCT
ejpam-4962	576	1	suppose	suppose	VERB
ejpam-4962	576	2	|v	|v	PROPN
ejpam-4962	576	3	′′	′′	PROPN
ejpam-4962	576	4	1	1	NUM
ejpam-4962	576	5	|	|	ADV
ejpam-4962	576	6	=	=	NOUN
ejpam-4962	576	7	0	0	X
ejpam-4962	576	8	.	.	PUNCT
ejpam-4962	577	1	then	then	ADV
ejpam-4962	577	2	v	v	ADP
ejpam-4962	577	3	′′	′′	PROPN
ejpam-4962	577	4	2	2	NUM
ejpam-4962	577	5	⊆	⊆	NUM
ejpam-4962	577	6	snj	snj	NOUN
ejpam-4962	577	7	for	for	ADP
ejpam-4962	577	8	some	some	DET
ejpam-4962	577	9	j	j	PROPN
ejpam-4962	577	10	∈	∈	PROPN
ejpam-4962	577	11	{	{	PUNCT
ejpam-4962	577	12	1	1	NUM
ejpam-4962	577	13	,	,	PUNCT
ejpam-4962	577	14	2	2	NUM
ejpam-4962	577	15	,	,	PUNCT
ejpam-4962	577	16	.	.	PUNCT
ejpam-4962	577	17	.	.	PUNCT
ejpam-4962	577	18	.	.	PUNCT
ejpam-4962	578	1	,	,	PUNCT
ejpam-4962	578	2	k	k	X
ejpam-4962	578	3	}	}	PUNCT
ejpam-4962	578	4	.	.	PUNCT
ejpam-4962	579	1	let	let	VERB
ejpam-4962	579	2	w	w	NOUN
ejpam-4962	579	3	∈	∈	PROPN
ejpam-4962	579	4	snj	snj	ADJ
ejpam-4962	579	5	\	\	PROPN
ejpam-4962	579	6	v	v	ADP
ejpam-4962	579	7	′′	′′	PROPN
ejpam-4962	579	8	2	2	NUM
ejpam-4962	579	9	.	.	PUNCT
ejpam-4962	580	1	then	then	ADV
ejpam-4962	580	2	w	w	PROPN
ejpam-4962	580	3	∈	∈	PROPN
ejpam-4962	580	4	v	v	ADP
ejpam-4962	580	5	′′	′′	PROPN
ejpam-4962	580	6	0	0	NUM
ejpam-4962	580	7	\	\	NOUN
ejpam-4962	580	8	ng(v	ng(v	PUNCT
ejpam-4962	581	1	′′	′′	PROPN
ejpam-4962	581	2	2	2	NUM
ejpam-4962	581	3	)	)	PUNCT
ejpam-4962	581	4	,	,	PUNCT
ejpam-4962	581	5	a	a	DET
ejpam-4962	581	6	contradiction	contradiction	NOUN
ejpam-4962	581	7	.	.	PUNCT
ejpam-4962	582	1	hence	hence	ADV
ejpam-4962	582	2	,	,	PUNCT
ejpam-4962	582	3	γgr(g	γgr(g	PROPN
ejpam-4962	582	4	)	)	PUNCT
ejpam-4962	582	5	≥	≥	NOUN
ejpam-4962	582	6	n(g	n(g	NUM
ejpam-4962	582	7	)	)	PUNCT
ejpam-4962	583	1	+	+	CCONJ
ejpam-4962	583	2	1	1	X
ejpam-4962	583	3	.	.	PUNCT
ejpam-4962	583	4	the	the	DET
ejpam-4962	583	5	same	same	ADJ
ejpam-4962	583	6	argument	argument	NOUN
ejpam-4962	583	7	can	can	AUX
ejpam-4962	583	8	be	be	AUX
ejpam-4962	583	9	used	use	VERB
ejpam-4962	583	10	to	to	PART
ejpam-4962	583	11	show	show	VERB
ejpam-4962	583	12	that	that	SCONJ
ejpam-4962	583	13	γgr(g	γgr(g	PROPN
ejpam-4962	583	14	)	)	PUNCT
ejpam-4962	583	15	≥	≥	NOUN
ejpam-4962	583	16	6	6	NUM
ejpam-4962	583	17	if	if	SCONJ
ejpam-4962	583	18	6	6	NUM
ejpam-4962	583	19	≤	≤	NUM
ejpam-4962	583	20	n(g	n(g	NUM
ejpam-4962	583	21	)	)	PUNCT
ejpam-4962	584	1	+	+	CCONJ
ejpam-4962	584	2	1	1	X
ejpam-4962	584	3	.	.	PUNCT
ejpam-4962	584	4	accordingly	accordingly	ADV
ejpam-4962	584	5	,	,	PUNCT
ejpam-4962	584	6	γgr(g	γgr(g	PROPN
ejpam-4962	584	7	)	)	PUNCT
ejpam-4962	584	8	=	=	SYM
ejpam-4962	584	9	min{n(g	min{n(g	NOUN
ejpam-4962	584	10	)	)	PUNCT
ejpam-4962	585	1	+	+	CCONJ
ejpam-4962	585	2	1	1	NUM
ejpam-4962	585	3	,	,	PUNCT
ejpam-4962	585	4	6	6	NUM
ejpam-4962	585	5	}	}	PUNCT
ejpam-4962	585	6	.	.	PUNCT
ejpam-4962	586	1	example	example	NOUN
ejpam-4962	587	1	1	1	NUM
ejpam-4962	587	2	.	.	X
ejpam-4962	588	1	for	for	ADP
ejpam-4962	588	2	any	any	DET
ejpam-4962	588	3	two	two	NUM
ejpam-4962	588	4	integers	integer	NOUN
ejpam-4962	588	5	m	m	VERB
ejpam-4962	588	6	,	,	PUNCT
ejpam-4962	588	7	n	n	PRON
ejpam-4962	588	8	≥	≥	NOUN
ejpam-4962	588	9	2	2	NUM
ejpam-4962	588	10	,	,	PUNCT
ejpam-4962	588	11	γgr(km	γgr(km	NOUN
ejpam-4962	588	12	,	,	PUNCT
ejpam-4962	588	13	n	n	CCONJ
ejpam-4962	588	14	)	)	PUNCT
ejpam-4962	588	15	=	=	VERB
ejpam-4962	588	16	min{m+	min{m+	PROPN
ejpam-4962	588	17	1	1	NUM
ejpam-4962	588	18	,	,	PUNCT
ejpam-4962	588	19	n+	n+	NUM
ejpam-4962	588	20	1	1	NUM
ejpam-4962	588	21	,	,	PUNCT
ejpam-4962	588	22	6	6	NUM
ejpam-4962	588	23	}	}	PUNCT
ejpam-4962	588	24	.	.	PUNCT
ejpam-4962	589	1	the	the	DET
ejpam-4962	589	2	next	next	ADJ
ejpam-4962	589	3	result	result	NOUN
ejpam-4962	589	4	shows	show	VERB
ejpam-4962	589	5	that	that	SCONJ
ejpam-4962	589	6	every	every	DET
ejpam-4962	589	7	pair	pair	NOUN
ejpam-4962	589	8	of	of	ADP
ejpam-4962	589	9	positive	positive	ADJ
ejpam-4962	589	10	integers	integer	NOUN
ejpam-4962	589	11	(	(	PUNCT
ejpam-4962	589	12	both	both	CCONJ
ejpam-4962	589	13	at	at	ADV
ejpam-4962	589	14	least	least	ADJ
ejpam-4962	589	15	4	4	NUM
ejpam-4962	589	16	)	)	PUNCT
ejpam-4962	589	17	are	be	AUX
ejpam-4962	589	18	realizable	realizable	ADJ
ejpam-4962	589	19	as	as	ADP
ejpam-4962	589	20	the	the	DET
ejpam-4962	589	21	geodetic	geodetic	ADJ
ejpam-4962	589	22	domination	domination	NOUN
ejpam-4962	589	23	number	number	NOUN
ejpam-4962	589	24	and	and	CCONJ
ejpam-4962	589	25	geodetic	geodetic	ADJ
ejpam-4962	589	26	roman	roman	ADJ
ejpam-4962	589	27	domination	domination	NOUN
ejpam-4962	589	28	number	number	NOUN
ejpam-4962	589	29	of	of	ADP
ejpam-4962	589	30	a	a	DET
ejpam-4962	589	31	connected	connected	ADJ
ejpam-4962	589	32	graph	graph	NOUN
ejpam-4962	589	33	.	.	PUNCT
ejpam-4962	590	1	theorem	theorem	NOUN
ejpam-4962	590	2	6	6	NUM
ejpam-4962	590	3	.	.	PUNCT
ejpam-4962	591	1	let	let	VERB
ejpam-4962	591	2	a	a	PRON
ejpam-4962	591	3	and	and	CCONJ
ejpam-4962	591	4	b	b	NOUN
ejpam-4962	591	5	be	be	AUX
ejpam-4962	591	6	positive	positive	ADJ
ejpam-4962	591	7	integers	integer	NOUN
ejpam-4962	591	8	such	such	ADJ
ejpam-4962	591	9	that	that	SCONJ
ejpam-4962	591	10	4	4	NUM
ejpam-4962	591	11	≤	≤	NOUN
ejpam-4962	591	12	a	a	DET
ejpam-4962	591	13	≤	≤	NUM
ejpam-4962	591	14	b	b	NOUN
ejpam-4962	591	15	≤	≤	NUM
ejpam-4962	591	16	2a	2a	NUM
ejpam-4962	591	17	.	.	PUNCT
ejpam-4962	592	1	then	then	ADV
ejpam-4962	592	2	there	there	PRON
ejpam-4962	592	3	exists	exist	VERB
ejpam-4962	592	4	a	a	DET
ejpam-4962	592	5	connected	connected	ADJ
ejpam-4962	592	6	graph	graph	NOUN
ejpam-4962	592	7	g	g	ADP
ejpam-4962	592	8	such	such	ADJ
ejpam-4962	592	9	that	that	PRON
ejpam-4962	592	10	γg(g	γg(g	NOUN
ejpam-4962	592	11	)	)	PUNCT
ejpam-4962	592	12	=	=	SYM
ejpam-4962	592	13	a	a	PROPN
ejpam-4962	592	14	and	and	CCONJ
ejpam-4962	592	15	γgr(g	γgr(g	PROPN
ejpam-4962	592	16	)	)	PUNCT
ejpam-4962	592	17	=	=	SYM
ejpam-4962	592	18	b.	b.	PROPN
ejpam-4962	592	19	proof	proof	NOUN
ejpam-4962	592	20	.	.	PUNCT
ejpam-4962	593	1	consider	consider	VERB
ejpam-4962	593	2	the	the	DET
ejpam-4962	593	3	following	follow	VERB
ejpam-4962	593	4	cases	case	NOUN
ejpam-4962	593	5	:	:	PUNCT
ejpam-4962	593	6	case	case	NOUN
ejpam-4962	593	7	1	1	NUM
ejpam-4962	593	8	.	.	PUNCT
ejpam-4962	594	1	a	a	DET
ejpam-4962	594	2	=	=	X
ejpam-4962	594	3	b.	b.	PROPN
ejpam-4962	594	4	let	let	VERB
ejpam-4962	594	5	g	g	PROPN
ejpam-4962	594	6	=	=	SYM
ejpam-4962	594	7	ka	ka	PROPN
ejpam-4962	594	8	.	.	PUNCT
ejpam-4962	595	1	then	then	ADV
ejpam-4962	595	2	γg(g	γg(g	X
ejpam-4962	595	3	)	)	PUNCT
ejpam-4962	595	4	=	=	SYM
ejpam-4962	595	5	γgr(g	γgr(g	PROPN
ejpam-4962	595	6	)	)	PUNCT
ejpam-4962	595	7	=	=	SYM
ejpam-4962	595	8	a.	a.	PROPN
ejpam-4962	595	9	r.	r.	PROPN
ejpam-4962	595	10	fortosa	fortosa	PROPN
ejpam-4962	595	11	,	,	PUNCT
ejpam-4962	595	12	s.	s.	PROPN
ejpam-4962	595	13	canoy	canoy	PROPN
ejpam-4962	595	14	jr	jr	PROPN
ejpam-4962	595	15	.	.	PROPN
ejpam-4962	595	16	/	/	SYM
ejpam-4962	595	17	eur	eur	PROPN
ejpam-4962	595	18	.	.	PUNCT
ejpam-4962	596	1	j.	j.	PROPN
ejpam-4962	596	2	pure	pure	PROPN
ejpam-4962	596	3	appl	appl	PROPN
ejpam-4962	596	4	.	.	PROPN
ejpam-4962	596	5	math	math	PROPN
ejpam-4962	596	6	,	,	PUNCT
ejpam-4962	596	7	16	16	NUM
ejpam-4962	596	8	(	(	PUNCT
ejpam-4962	596	9	4	4	NUM
ejpam-4962	596	10	)	)	PUNCT
ejpam-4962	596	11	(	(	PUNCT
ejpam-4962	596	12	2023	2023	NUM
ejpam-4962	596	13	)	)	PUNCT
ejpam-4962	596	14	,	,	PUNCT
ejpam-4962	596	15	2368	2368	NUM
ejpam-4962	596	16	-	-	SYM
ejpam-4962	596	17	2383	2383	NUM
ejpam-4962	596	18	2378	2378	NUM
ejpam-4962	596	19	case	case	NOUN
ejpam-4962	596	20	2	2	NUM
ejpam-4962	596	21	.	.	PUNCT
ejpam-4962	597	1	a	a	DET
ejpam-4962	597	2	<	<	X
ejpam-4962	597	3	b.	b.	PROPN
ejpam-4962	597	4	subcase	subcase	PROPN
ejpam-4962	597	5	1	1	NUM
ejpam-4962	597	6	.	.	PUNCT
ejpam-4962	597	7	b	b	X
ejpam-4962	597	8	=	=	PRON
ejpam-4962	597	9	a+	a+	PUNCT
ejpam-4962	597	10	1	1	X
ejpam-4962	597	11	.	.	PUNCT
ejpam-4962	598	1	let	let	VERB
ejpam-4962	598	2	g	g	PROPN
ejpam-4962	598	3	=	=	SYM
ejpam-4962	598	4	k1,a	k1,a	PROPN
ejpam-4962	598	5	.	.	PUNCT
ejpam-4962	599	1	then	then	ADV
ejpam-4962	599	2	γg(g	γg(g	ADP
ejpam-4962	599	3	)	)	PUNCT
ejpam-4962	599	4	=	=	SYM
ejpam-4962	599	5	a	a	PROPN
ejpam-4962	599	6	and	and	CCONJ
ejpam-4962	599	7	γgr(g	γgr(g	PROPN
ejpam-4962	599	8	)	)	PUNCT
ejpam-4962	600	1	=	=	SYM
ejpam-4962	600	2	a+	a+	PUNCT
ejpam-4962	600	3	1	1	NUM
ejpam-4962	600	4	=	=	SYM
ejpam-4962	600	5	b.	b.	PROPN
ejpam-4962	600	6	subcase	subcase	NOUN
ejpam-4962	600	7	2	2	NUM
ejpam-4962	600	8	.	.	PUNCT
ejpam-4962	600	9	b	b	X
ejpam-4962	601	1	=	=	SYM
ejpam-4962	601	2	2a−	2a−	NUM
ejpam-4962	601	3	1	1	NUM
ejpam-4962	601	4	.	.	PUNCT
ejpam-4962	602	1	let	let	VERB
ejpam-4962	602	2	m	m	VERB
ejpam-4962	602	3	=	=	VERB
ejpam-4962	602	4	b−	b−	PROPN
ejpam-4962	602	5	a	a	DET
ejpam-4962	602	6	=	=	SYM
ejpam-4962	602	7	a−	a−	PROPN
ejpam-4962	602	8	1	1	NUM
ejpam-4962	602	9	and	and	CCONJ
ejpam-4962	602	10	let	let	VERB
ejpam-4962	602	11	g	g	PROPN
ejpam-4962	602	12	=	=	SYM
ejpam-4962	602	13	p3	p3	PROPN
ejpam-4962	602	14	m.	m.	NOUN
ejpam-4962	602	15	then	then	ADV
ejpam-4962	602	16	γg(p3m+1	γg(p3m+1	PROPN
ejpam-4962	602	17	)	)	PUNCT
ejpam-4962	603	1	=	=	PUNCT
ejpam-4962	604	1	m+	m+	NUM
ejpam-4962	605	1	1	1	NUM
ejpam-4962	605	2	=	=	SYM
ejpam-4962	605	3	a	a	NOUN
ejpam-4962	605	4	by	by	ADP
ejpam-4962	605	5	remark	remark	NOUN
ejpam-4962	605	6	2(ii	2(ii	NUM
ejpam-4962	605	7	)	)	PUNCT
ejpam-4962	605	8	,	,	PUNCT
ejpam-4962	605	9	and	and	CCONJ
ejpam-4962	605	10	by	by	ADP
ejpam-4962	605	11	proposition	proposition	NOUN
ejpam-4962	605	12	3(ii	3(ii	NUM
ejpam-4962	605	13	)	)	PUNCT
ejpam-4962	605	14	,	,	PUNCT
ejpam-4962	605	15	γgr(p3	γgr(p3	NOUN
ejpam-4962	605	16	m	m	VERB
ejpam-4962	605	17	)	)	PUNCT
ejpam-4962	605	18	=	=	SYM
ejpam-4962	605	19	2m+	2m+	NUM
ejpam-4962	605	20	1	1	NUM
ejpam-4962	605	21	=	=	SYM
ejpam-4962	605	22	2a−	2a−	NUM
ejpam-4962	605	23	1	1	NUM
ejpam-4962	605	24	=	=	SYM
ejpam-4962	605	25	b.	b.	PROPN
ejpam-4962	605	26	subcase	subcase	NOUN
ejpam-4962	605	27	3	3	NUM
ejpam-4962	605	28	.	.	X
ejpam-4962	606	1	b	b	X
ejpam-4962	606	2	=	=	SYM
ejpam-4962	606	3	2a	2a	NUM
ejpam-4962	606	4	.	.	PUNCT
ejpam-4962	607	1	let	let	VERB
ejpam-4962	607	2	g	g	PROPN
ejpam-4962	607	3	=	=	SYM
ejpam-4962	607	4	c3a	c3a	X
ejpam-4962	607	5	.	.	PROPN
ejpam-4962	607	6	then	then	ADV
ejpam-4962	607	7	γg(g	γg(g	NUM
ejpam-4962	607	8	)	)	PUNCT
ejpam-4962	607	9	=	=	SYM
ejpam-4962	607	10	γg(c3a	γg(c3a	X
ejpam-4962	607	11	)	)	PUNCT
ejpam-4962	607	12	=	=	PUNCT
ejpam-4962	607	13	⌈3a3	⌈3a3	PROPN
ejpam-4962	607	14	⌉	⌉	NOUN
ejpam-4962	607	15	=	=	PUNCT
ejpam-4962	607	16	a	a	PRON
ejpam-4962	607	17	by	by	ADP
ejpam-4962	607	18	remark	remark	NOUN
ejpam-4962	607	19	2(i	2(i	NUM
ejpam-4962	607	20	)	)	PUNCT
ejpam-4962	607	21	and	and	CCONJ
ejpam-4962	607	22	by	by	ADP
ejpam-4962	607	23	proposition	proposition	NOUN
ejpam-4962	607	24	3(i	3(i	NUM
ejpam-4962	607	25	)	)	PUNCT
ejpam-4962	607	26	,	,	PUNCT
ejpam-4962	607	27	γgr(g	γgr(g	PROPN
ejpam-4962	607	28	)	)	PUNCT
ejpam-4962	607	29	=	=	SYM
ejpam-4962	607	30	2a	2a	NUM
ejpam-4962	607	31	.	.	PUNCT
ejpam-4962	608	1	subcase	subcase	PROPN
ejpam-4962	608	2	4	4	NUM
ejpam-4962	608	3	.	.	PUNCT
ejpam-4962	608	4	a+	a+	PUNCT
ejpam-4962	609	1	2	2	NUM
ejpam-4962	609	2	≤	≤	NUM
ejpam-4962	609	3	b	b	NOUN
ejpam-4962	609	4	<	<	X
ejpam-4962	609	5	2a−	2a−	NUM
ejpam-4962	609	6	1	1	NUM
ejpam-4962	609	7	then	then	ADV
ejpam-4962	609	8	2a	2a	NUM
ejpam-4962	609	9	−	−	PROPN
ejpam-4962	609	10	b	b	NOUN
ejpam-4962	609	11	−	−	PROPN
ejpam-4962	609	12	1	1	NUM
ejpam-4962	609	13	≥	≥	NOUN
ejpam-4962	609	14	1	1	NUM
ejpam-4962	609	15	,	,	PUNCT
ejpam-4962	609	16	i.e.	i.e.	X
ejpam-4962	609	17	,	,	PUNCT
ejpam-4962	609	18	2a	2a	NUM
ejpam-4962	609	19	−	−	PROPN
ejpam-4962	609	20	b	b	SYM
ejpam-4962	609	21	≥	≥	NUM
ejpam-4962	609	22	2	2	NUM
ejpam-4962	609	23	.	.	PUNCT
ejpam-4962	610	1	let	let	VERB
ejpam-4962	610	2	m	m	VERB
ejpam-4962	610	3	=	=	SYM
ejpam-4962	610	4	b	b	X
ejpam-4962	610	5	−	−	NOUN
ejpam-4962	610	6	a.	a.	NOUN
ejpam-4962	610	7	consider	consider	VERB
ejpam-4962	610	8	the	the	DET
ejpam-4962	610	9	graph	graph	NOUN
ejpam-4962	610	10	g	g	NOUN
ejpam-4962	610	11	in	in	ADP
ejpam-4962	610	12	figure	figure	NOUN
ejpam-4962	610	13	1	1	NUM
ejpam-4962	610	14	obtained	obtain	VERB
ejpam-4962	610	15	from	from	ADP
ejpam-4962	610	16	p3m−2	p3m−2	PROPN
ejpam-4962	610	17	=	=	PUNCT
ejpam-4962	611	1	[	[	X
ejpam-4962	611	2	v1	v1	NOUN
ejpam-4962	611	3	,	,	PUNCT
ejpam-4962	611	4	v2	v2	PROPN
ejpam-4962	611	5	,	,	PUNCT
ejpam-4962	611	6	v3	v3	PROPN
ejpam-4962	611	7	,	,	PUNCT
ejpam-4962	611	8	v4	v4	PROPN
ejpam-4962	611	9	,	,	PUNCT
ejpam-4962	611	10	.	.	PUNCT
ejpam-4962	611	11	.	.	PUNCT
ejpam-4962	612	1	.	.	PUNCT
ejpam-4962	613	1	,	,	PUNCT
ejpam-4962	613	2	v3(m+1)−2	v3(m+1)−2	PROPN
ejpam-4962	613	3	]	]	PUNCT
ejpam-4962	613	4	by	by	ADP
ejpam-4962	613	5	adding	add	VERB
ejpam-4962	613	6	the	the	DET
ejpam-4962	613	7	edges	edge	NOUN
ejpam-4962	613	8	v3(m+1)−2wj	v3(m+1)−2wj	VERB
ejpam-4962	613	9	for	for	ADP
ejpam-4962	613	10	each	each	DET
ejpam-4962	613	11	j	j	PROPN
ejpam-4962	613	12	∈	∈	PROPN
ejpam-4962	613	13	{	{	PUNCT
ejpam-4962	613	14	1	1	NUM
ejpam-4962	613	15	,	,	PUNCT
ejpam-4962	613	16	2	2	NUM
ejpam-4962	613	17	,	,	PUNCT
ejpam-4962	613	18	.	.	PUNCT
ejpam-4962	613	19	.	.	PUNCT
ejpam-4962	613	20	.	.	PUNCT
ejpam-4962	614	1	,	,	PUNCT
ejpam-4962	614	2	2a	2a	NUM
ejpam-4962	614	3	−	−	NOUN
ejpam-4962	614	4	b	b	NOUN
ejpam-4962	614	5	−	−	NOUN
ejpam-4962	614	6	1	1	NUM
ejpam-4962	614	7	}	}	PUNCT
ejpam-4962	614	8	.	.	PUNCT
ejpam-4962	615	1	let	let	VERB
ejpam-4962	615	2	s1	s1	PROPN
ejpam-4962	615	3	=	=	SYM
ejpam-4962	615	4	{	{	PUNCT
ejpam-4962	615	5	v1	v1	PROPN
ejpam-4962	615	6	,	,	PUNCT
ejpam-4962	615	7	v4	v4	NOUN
ejpam-4962	615	8	,	,	PUNCT
ejpam-4962	615	9	.	.	PUNCT
ejpam-4962	615	10	.	.	PUNCT
ejpam-4962	616	1	.	.	PUNCT
ejpam-4962	617	1	,	,	PUNCT
ejpam-4962	617	2	v3(m+1)−2	v3(m+1)−2	PROPN
ejpam-4962	617	3	}	}	PUNCT
ejpam-4962	617	4	.	.	PUNCT
ejpam-4962	618	1	then	then	ADV
ejpam-4962	618	2	s1	s1	PROPN
ejpam-4962	618	3	is	be	AUX
ejpam-4962	618	4	a	a	DET
ejpam-4962	618	5	γg	γg	ADV
ejpam-4962	618	6	-	-	PUNCT
ejpam-4962	618	7	set	set	NOUN
ejpam-4962	618	8	in	in	ADP
ejpam-4962	618	9	p3(m+1)−2	p3(m+1)−2	PROPN
ejpam-4962	618	10	.	.	PUNCT
ejpam-4962	619	1	hence	hence	ADV
ejpam-4962	619	2	,	,	PUNCT
ejpam-4962	619	3	s	s	PART
ejpam-4962	619	4	=	=	NOUN
ejpam-4962	619	5	{	{	PUNCT
ejpam-4962	619	6	v1	v1	PROPN
ejpam-4962	619	7	,	,	PUNCT
ejpam-4962	619	8	v4	v4	NOUN
ejpam-4962	619	9	,	,	PUNCT
ejpam-4962	619	10	.	.	PUNCT
ejpam-4962	619	11	.	.	PUNCT
ejpam-4962	619	12	.	.	PUNCT
ejpam-4962	620	1	,	,	PUNCT
ejpam-4962	620	2	v3(m+1)−2	v3(m+1)−2	PROPN
ejpam-4962	620	3	,	,	PUNCT
ejpam-4962	620	4	w1	w1	NOUN
ejpam-4962	620	5	,	,	PUNCT
ejpam-4962	620	6	w2	w2	NOUN
ejpam-4962	620	7	,	,	PUNCT
ejpam-4962	620	8	.	.	PUNCT
ejpam-4962	620	9	.	.	PUNCT
ejpam-4962	620	10	.	.	PUNCT
ejpam-4962	621	1	,	,	PUNCT
ejpam-4962	621	2	w2a−b−1	w2a−b−1	NOUN
ejpam-4962	621	3	}	}	PUNCT
ejpam-4962	621	4	is	be	AUX
ejpam-4962	621	5	a	a	DET
ejpam-4962	621	6	γg	γg	ADV
ejpam-4962	621	7	-	-	PUNCT
ejpam-4962	621	8	set	set	NOUN
ejpam-4962	621	9	in	in	ADP
ejpam-4962	621	10	g	g	NOUN
ejpam-4962	621	11	and	and	CCONJ
ejpam-4962	621	12	γg(g	γg(g	NOUN
ejpam-4962	621	13	)	)	PUNCT
ejpam-4962	621	14	=	=	SYM
ejpam-4962	621	15	|s|	|s|	NOUN
ejpam-4962	621	16	=	=	SYM
ejpam-4962	621	17	(	(	PUNCT
ejpam-4962	621	18	m+1)+2a−b−1	m+1)+2a−b−1	NOUN
ejpam-4962	621	19	=	=	PUNCT
ejpam-4962	621	20	(	(	PUNCT
ejpam-4962	621	21	b−a+1)+2a−b−1	b−a+1)+2a−b−1	VERB
ejpam-4962	621	22	=	=	NOUN
ejpam-4962	621	23	a.	a.	NOUN
ejpam-4962	621	24	suppose	suppose	VERB
ejpam-4962	621	25	2a−b−1	2a−b−1	NUM
ejpam-4962	621	26	=	=	SYM
ejpam-4962	621	27	1	1	NUM
ejpam-4962	621	28	.	.	PUNCT
ejpam-4962	621	29	then	then	ADV
ejpam-4962	621	30	b	b	X
ejpam-4962	621	31	=	=	SYM
ejpam-4962	621	32	2a−2	2a−2	NUM
ejpam-4962	621	33	,	,	PUNCT
ejpam-4962	621	34	m	m	NOUN
ejpam-4962	621	35	=	=	SYM
ejpam-4962	622	1	a−2	a−2	PROPN
ejpam-4962	622	2	.	.	PUNCT
ejpam-4962	623	1	then	then	ADV
ejpam-4962	623	2	g	g	PROPN
ejpam-4962	623	3	=	=	SYM
ejpam-4962	623	4	p3(m+1)−1	p3(m+1)−1	PROPN
ejpam-4962	623	5	.	.	PUNCT
ejpam-4962	624	1	by	by	ADP
ejpam-4962	624	2	remark	remark	NOUN
ejpam-4962	624	3	2(ii	2(ii	NUM
ejpam-4962	624	4	)	)	PUNCT
ejpam-4962	624	5	,	,	PUNCT
ejpam-4962	624	6	γg(p3(m+1)−1	γg(p3(m+1)−1	VERB
ejpam-4962	624	7	)	)	PUNCT
ejpam-4962	624	8	=	=	PUNCT
ejpam-4962	624	9	m+2	m+2	NOUN
ejpam-4962	624	10	=	=	PUNCT
ejpam-4962	624	11	a	a	PRON
ejpam-4962	624	12	and	and	CCONJ
ejpam-4962	624	13	by	by	ADP
ejpam-4962	624	14	proposition	proposition	NOUN
ejpam-4962	624	15	3(ii	3(ii	NUM
ejpam-4962	624	16	)	)	PUNCT
ejpam-4962	624	17	,	,	PUNCT
ejpam-4962	624	18	γgr(g	γgr(g	PROPN
ejpam-4962	624	19	)	)	PUNCT
ejpam-4962	624	20	=	=	PUNCT
ejpam-4962	625	1	γgr(p3(m+1)−1	γgr(p3(m+1)−1	ADJ
ejpam-4962	625	2	)	)	PUNCT
ejpam-4962	625	3	=	=	SYM
ejpam-4962	625	4	2(m	2(m	NOUN
ejpam-4962	626	1	+	+	CCONJ
ejpam-4962	626	2	1	1	X
ejpam-4962	626	3	)	)	PUNCT
ejpam-4962	626	4	=	=	SYM
ejpam-4962	626	5	b.	b.	PROPN
ejpam-4962	626	6	next	next	ADV
ejpam-4962	626	7	,	,	PUNCT
ejpam-4962	626	8	suppose	suppose	VERB
ejpam-4962	626	9	that	that	SCONJ
ejpam-4962	626	10	2a−	2a−	PROPN
ejpam-4962	626	11	b−	b−	NOUN
ejpam-4962	626	12	1	1	NUM
ejpam-4962	626	13	≥	≥	NOUN
ejpam-4962	626	14	2	2	NUM
ejpam-4962	626	15	.	.	PUNCT
ejpam-4962	627	1	if	if	SCONJ
ejpam-4962	627	2	s1	s1	PROPN
ejpam-4962	627	3	∩	∩	ADJ
ejpam-4962	627	4	v2	v2	NOUN
ejpam-4962	627	5	=	=	NOUN
ejpam-4962	627	6	∅	∅	NOUN
ejpam-4962	627	7	,	,	PUNCT
ejpam-4962	627	8	then	then	ADV
ejpam-4962	627	9	γgr(g	γgr(g	PROPN
ejpam-4962	627	10	)	)	PUNCT
ejpam-4962	627	11	=	=	SYM
ejpam-4962	627	12	γgr(p3m+1	γgr(p3m+1	PROPN
ejpam-4962	627	13	)	)	PUNCT
ejpam-4962	627	14	+	+	CCONJ
ejpam-4962	628	1	2a−	2a−	NUM
ejpam-4962	628	2	b−	b−	NOUN
ejpam-4962	628	3	1	1	NUM
ejpam-4962	628	4	=	=	SYM
ejpam-4962	628	5	2(m+	2(m+	NUM
ejpam-4962	628	6	1	1	NUM
ejpam-4962	628	7	)	)	PUNCT
ejpam-4962	628	8	+	+	CCONJ
ejpam-4962	628	9	a−	a−	PROPN
ejpam-4962	628	10	(	(	PUNCT
ejpam-4962	628	11	m+	m+	NOUN
ejpam-4962	628	12	1	1	NUM
ejpam-4962	628	13	)	)	PUNCT
ejpam-4962	628	14	=	=	VERB
ejpam-4962	629	1	m+	m+	NUM
ejpam-4962	629	2	a	a	PRON
ejpam-4962	629	3	=	=	X
ejpam-4962	629	4	b.	b.	PROPN
ejpam-4962	629	5	suppose	suppose	VERB
ejpam-4962	629	6	s1	s1	NOUN
ejpam-4962	629	7	∩	∩	NOUN
ejpam-4962	629	8	v2	v2	PROPN
ejpam-4962	629	9	̸=	̸=	PROPN
ejpam-4962	629	10	∅.	∅.	NOUN
ejpam-4962	629	11	since	since	SCONJ
ejpam-4962	629	12	f	f	PROPN
ejpam-4962	629	13	is	be	AUX
ejpam-4962	629	14	a	a	DET
ejpam-4962	629	15	γgr	γgr	NOUN
ejpam-4962	629	16	-	-	PUNCT
ejpam-4962	629	17	function	function	NOUN
ejpam-4962	629	18	on	on	ADP
ejpam-4962	629	19	g	g	NOUN
ejpam-4962	629	20	,	,	PUNCT
ejpam-4962	629	21	|s1	|s1	ADV
ejpam-4962	629	22	∩	∩	NOUN
ejpam-4962	629	23	v2|	v2|	X
ejpam-4962	629	24	=	=	SYM
ejpam-4962	629	25	1	1	NUM
ejpam-4962	629	26	and	and	CCONJ
ejpam-4962	629	27	v3m+1	v3m+1	PROPN
ejpam-4962	629	28	∈	∈	PROPN
ejpam-4962	629	29	v0	v0	NOUN
ejpam-4962	629	30	.	.	PUNCT
ejpam-4962	630	1	let	let	VERB
ejpam-4962	630	2	v3m+2	v3m+2	PRON
ejpam-4962	630	3	∈	∈	PROPN
ejpam-4962	630	4	s1∩v2	s1∩v2	PROPN
ejpam-4962	630	5	.	.	PUNCT
ejpam-4962	631	1	it	it	PRON
ejpam-4962	631	2	is	be	AUX
ejpam-4962	631	3	routine	routine	ADJ
ejpam-4962	631	4	to	to	PART
ejpam-4962	631	5	show	show	VERB
ejpam-4962	631	6	that	that	SCONJ
ejpam-4962	631	7	g	g	NOUN
ejpam-4962	631	8	=	=	PUNCT
ejpam-4962	631	9	(	(	PUNCT
ejpam-4962	631	10	v	v	NUM
ejpam-4962	631	11	′	′	NUM
ejpam-4962	631	12	0	0	NUM
ejpam-4962	631	13	,	,	PUNCT
ejpam-4962	631	14	v	v	NOUN
ejpam-4962	631	15	′	′	NUM
ejpam-4962	631	16	1	1	NUM
ejpam-4962	631	17	,	,	PUNCT
ejpam-4962	631	18	v	v	NOUN
ejpam-4962	631	19	′	′	NUM
ejpam-4962	631	20	2	2	NUM
ejpam-4962	631	21	)	)	PUNCT
ejpam-4962	631	22	where	where	SCONJ
ejpam-4962	631	23	v	v	X
ejpam-4962	631	24	′	′	NOUN
ejpam-4962	631	25	1	1	NUM
ejpam-4962	631	26	=	=	SYM
ejpam-4962	631	27	v1	v1	NOUN
ejpam-4962	631	28	\	\	NOUN
ejpam-4962	631	29	(	(	PUNCT
ejpam-4962	631	30	s1	s1	PROPN
ejpam-4962	631	31	\{v3m+2	\{v3m+2	PROPN
ejpam-4962	631	32	}	}	PUNCT
ejpam-4962	631	33	)	)	PUNCT
ejpam-4962	631	34	,	,	PUNCT
ejpam-4962	631	35	v	v	X
ejpam-4962	631	36	′	′	NUM
ejpam-4962	631	37	2	2	NUM
ejpam-4962	631	38	=	=	SYM
ejpam-4962	631	39	v2	v2	PROPN
ejpam-4962	631	40	and	and	CCONJ
ejpam-4962	631	41	v	v	NOUN
ejpam-4962	631	42	′	′	NUM
ejpam-4962	631	43	0	0	NUM
ejpam-4962	632	1	=	=	SYM
ejpam-4962	632	2	v0	v0	NOUN
ejpam-4962	632	3	is	be	AUX
ejpam-4962	632	4	a	a	DET
ejpam-4962	632	5	γgr	γgr	NOUN
ejpam-4962	632	6	-	-	PUNCT
ejpam-4962	632	7	function	function	NOUN
ejpam-4962	632	8	on	on	ADP
ejpam-4962	632	9	p3m+2	p3m+2	PROPN
ejpam-4962	632	10	.	.	PUNCT
ejpam-4962	633	1	then	then	ADV
ejpam-4962	633	2	γgr(g	γgr(g	PROPN
ejpam-4962	633	3	)	)	PUNCT
ejpam-4962	633	4	=	=	SYM
ejpam-4962	633	5	γgr(p3m+2	γgr(p3m+2	PROPN
ejpam-4962	633	6	)	)	PUNCT
ejpam-4962	634	1	+	+	CCONJ
ejpam-4962	635	1	2a−	2a−	NUM
ejpam-4962	635	2	b−	b−	NOUN
ejpam-4962	635	3	2	2	NUM
ejpam-4962	635	4	=	=	SYM
ejpam-4962	635	5	2(m+	2(m+	NUM
ejpam-4962	635	6	1	1	NUM
ejpam-4962	635	7	)	)	PUNCT
ejpam-4962	635	8	+	+	CCONJ
ejpam-4962	635	9	a−	a−	PROPN
ejpam-4962	635	10	(	(	PUNCT
ejpam-4962	635	11	m+	m+	NOUN
ejpam-4962	635	12	2	2	NUM
ejpam-4962	635	13	)	)	PUNCT
ejpam-4962	635	14	=	=	SYM
ejpam-4962	636	1	b.	b.	PROPN
ejpam-4962	636	2	g	g	NOUN
ejpam-4962	636	3	:	:	PUNCT
ejpam-4962	636	4	v1	v1	VERB
ejpam-4962	636	5	v2	v2	PROPN
ejpam-4962	636	6	v3	v3	PROPN
ejpam-4962	636	7	v4	v4	PROPN
ejpam-4962	636	8	·	·	PUNCT
ejpam-4962	636	9	·	·	PUNCT
ejpam-4962	636	10	·	·	PUNCT
ejpam-4962	637	1	v3(m+1)−2	v3(m+1)−2	ADJ
ejpam-4962	637	2	w2a−b−1	w2a−b−1	NOUN
ejpam-4962	637	3	...	...	PUNCT
ejpam-4962	637	4	w1	w1	PROPN
ejpam-4962	637	5	w2	w2	PROPN
ejpam-4962	637	6	w3	w3	PROPN
ejpam-4962	637	7	w4	w4	PROPN
ejpam-4962	637	8	w5	w5	PROPN
ejpam-4962	637	9	figure	figure	NOUN
ejpam-4962	637	10	1	1	NUM
ejpam-4962	637	11	:	:	PUNCT
ejpam-4962	637	12	a	a	DET
ejpam-4962	637	13	graph	graph	NOUN
ejpam-4962	637	14	g	g	NOUN
ejpam-4962	637	15	with	with	ADP
ejpam-4962	637	16	γg(g	γg(g	NOUN
ejpam-4962	637	17	)	)	PUNCT
ejpam-4962	637	18	=	=	SYM
ejpam-4962	637	19	a	a	PROPN
ejpam-4962	637	20	and	and	CCONJ
ejpam-4962	637	21	γgr(g	γgr(g	PROPN
ejpam-4962	637	22	)	)	PUNCT
ejpam-4962	637	23	=	=	SYM
ejpam-4962	638	1	b	b	PROPN
ejpam-4962	638	2	this	this	PRON
ejpam-4962	638	3	proves	prove	VERB
ejpam-4962	638	4	the	the	DET
ejpam-4962	638	5	assertion	assertion	NOUN
ejpam-4962	638	6	.	.	PUNCT
ejpam-4962	639	1	corollary	corollary	ADJ
ejpam-4962	639	2	3	3	X
ejpam-4962	639	3	.	.	PUNCT
ejpam-4962	640	1	let	let	VERB
ejpam-4962	640	2	n	n	PRON
ejpam-4962	640	3	be	be	AUX
ejpam-4962	640	4	a	a	DET
ejpam-4962	640	5	positive	positive	ADJ
ejpam-4962	640	6	integer	integer	NOUN
ejpam-4962	640	7	with	with	ADP
ejpam-4962	640	8	n	n	PRON
ejpam-4962	640	9	≥	≥	NUM
ejpam-4962	640	10	2	2	NUM
ejpam-4962	640	11	.	.	PUNCT
ejpam-4962	641	1	then	then	ADV
ejpam-4962	641	2	there	there	PRON
ejpam-4962	641	3	exists	exist	VERB
ejpam-4962	641	4	a	a	DET
ejpam-4962	641	5	connected	connected	ADJ
ejpam-4962	641	6	graph	graph	NOUN
ejpam-4962	641	7	g	g	ADP
ejpam-4962	641	8	such	such	ADJ
ejpam-4962	641	9	that	that	DET
ejpam-4962	641	10	γgr(g	γgr(g	PROPN
ejpam-4962	641	11	)	)	PUNCT
ejpam-4962	641	12	−	−	NOUN
ejpam-4962	641	13	γg(g	γg(g	NOUN
ejpam-4962	641	14	)	)	PUNCT
ejpam-4962	641	15	=	=	VERB
ejpam-4962	642	1	n.	n.	NOUN
ejpam-4962	642	2	in	in	ADP
ejpam-4962	642	3	other	other	ADJ
ejpam-4962	642	4	words	word	NOUN
ejpam-4962	642	5	,	,	PUNCT
ejpam-4962	642	6	the	the	DET
ejpam-4962	642	7	difference	difference	NOUN
ejpam-4962	642	8	γgr(g	γgr(g	PROPN
ejpam-4962	642	9	)	)	PUNCT
ejpam-4962	642	10	−	−	NOUN
ejpam-4962	642	11	γg(g	γg(g	CCONJ
ejpam-4962	642	12	)	)	PUNCT
ejpam-4962	642	13	can	can	AUX
ejpam-4962	642	14	be	be	AUX
ejpam-4962	642	15	made	make	VERB
ejpam-4962	642	16	arbitrarily	arbitrarily	ADV
ejpam-4962	642	17	large	large	ADJ
ejpam-4962	642	18	.	.	PUNCT
ejpam-4962	643	1	proposition	proposition	NOUN
ejpam-4962	643	2	7	7	NUM
ejpam-4962	643	3	.	.	PUNCT
ejpam-4962	644	1	let	let	VERB
ejpam-4962	644	2	g	g	NOUN
ejpam-4962	644	3	and	and	CCONJ
ejpam-4962	644	4	h	h	NOUN
ejpam-4962	644	5	be	be	AUX
ejpam-4962	644	6	non	non	ADJ
ejpam-4962	644	7	-	-	ADJ
ejpam-4962	644	8	complete	complete	ADJ
ejpam-4962	644	9	graphs	graph	NOUN
ejpam-4962	644	10	.	.	PUNCT
ejpam-4962	645	1	then	then	ADV
ejpam-4962	645	2	3	3	NUM
ejpam-4962	645	3	≤	≤	NUM
ejpam-4962	645	4	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	645	5	)	)	PUNCT
ejpam-4962	645	6	≤	≤	NOUN
ejpam-4962	645	7	6	6	NUM
ejpam-4962	645	8	.	.	PUNCT
ejpam-4962	645	9	r.	r.	PROPN
ejpam-4962	645	10	fortosa	fortosa	PROPN
ejpam-4962	645	11	,	,	PUNCT
ejpam-4962	645	12	s.	s.	PROPN
ejpam-4962	645	13	canoy	canoy	PROPN
ejpam-4962	645	14	jr	jr	PROPN
ejpam-4962	645	15	.	.	PROPN
ejpam-4962	645	16	/	/	SYM
ejpam-4962	645	17	eur	eur	PROPN
ejpam-4962	645	18	.	.	PUNCT
ejpam-4962	646	1	j.	j.	PROPN
ejpam-4962	646	2	pure	pure	PROPN
ejpam-4962	646	3	appl	appl	PROPN
ejpam-4962	646	4	.	.	PROPN
ejpam-4962	646	5	math	math	PROPN
ejpam-4962	646	6	,	,	PUNCT
ejpam-4962	646	7	16	16	NUM
ejpam-4962	646	8	(	(	PUNCT
ejpam-4962	646	9	4	4	NUM
ejpam-4962	646	10	)	)	PUNCT
ejpam-4962	646	11	(	(	PUNCT
ejpam-4962	646	12	2023	2023	NUM
ejpam-4962	646	13	)	)	PUNCT
ejpam-4962	646	14	,	,	PUNCT
ejpam-4962	646	15	2368	2368	NUM
ejpam-4962	646	16	-	-	SYM
ejpam-4962	646	17	2383	2383	NUM
ejpam-4962	646	18	2379	2379	NUM
ejpam-4962	646	19	proof	proof	NOUN
ejpam-4962	646	20	.	.	PUNCT
ejpam-4962	647	1	since	since	SCONJ
ejpam-4962	647	2	g+h	g+h	PROPN
ejpam-4962	647	3	/∈	/∈	PUNCT
ejpam-4962	647	4	{	{	PUNCT
ejpam-4962	647	5	k1,k2	k1,k2	NOUN
ejpam-4962	647	6	}	}	PUNCT
ejpam-4962	647	7	,	,	PUNCT
ejpam-4962	647	8	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	647	9	)	)	PUNCT
ejpam-4962	647	10	≥	≥	NOUN
ejpam-4962	647	11	3	3	NUM
ejpam-4962	647	12	,	,	PUNCT
ejpam-4962	647	13	by	by	ADP
ejpam-4962	647	14	(	(	PUNCT
ejpam-4962	647	15	i	i	NOUN
ejpam-4962	647	16	)	)	PUNCT
ejpam-4962	647	17	and	and	CCONJ
ejpam-4962	647	18	(	(	PUNCT
ejpam-4962	647	19	ii	ii	NOUN
ejpam-4962	647	20	)	)	PUNCT
ejpam-4962	647	21	of	of	ADP
ejpam-4962	647	22	theorem	theorem	NOUN
ejpam-4962	647	23	2	2	NUM
ejpam-4962	647	24	.	.	PUNCT
ejpam-4962	647	25	pick	pick	PROPN
ejpam-4962	647	26	u	u	NOUN
ejpam-4962	647	27	,	,	PUNCT
ejpam-4962	647	28	v	v	PROPN
ejpam-4962	647	29	∈	∈	PROPN
ejpam-4962	647	30	v	v	NOUN
ejpam-4962	647	31	(	(	PUNCT
ejpam-4962	647	32	g	g	NOUN
ejpam-4962	647	33	)	)	PUNCT
ejpam-4962	647	34	and	and	CCONJ
ejpam-4962	647	35	x	x	X
ejpam-4962	647	36	,	,	PUNCT
ejpam-4962	647	37	y	y	PROPN
ejpam-4962	647	38	∈	∈	PROPN
ejpam-4962	647	39	v	v	ADP
ejpam-4962	647	40	(	(	PUNCT
ejpam-4962	647	41	h	h	NOUN
ejpam-4962	647	42	)	)	PUNCT
ejpam-4962	647	43	such	such	ADJ
ejpam-4962	647	44	that	that	PRON
ejpam-4962	647	45	uv	uv	NOUN
ejpam-4962	647	46	/∈	/∈	PUNCT
ejpam-4962	647	47	e(g	e(g	PROPN
ejpam-4962	647	48	)	)	PUNCT
ejpam-4962	647	49	and	and	CCONJ
ejpam-4962	647	50	xy	xy	PROPN
ejpam-4962	647	51	/∈	/∈	PUNCT
ejpam-4962	647	52	e(h	e(h	PROPN
ejpam-4962	647	53	)	)	PUNCT
ejpam-4962	647	54	.	.	PUNCT
ejpam-4962	648	1	let	let	VERB
ejpam-4962	648	2	v1	v1	VERB
ejpam-4962	648	3	=	=	SYM
ejpam-4962	648	4	{	{	PUNCT
ejpam-4962	648	5	v	v	NOUN
ejpam-4962	648	6	,	,	PUNCT
ejpam-4962	648	7	y	y	NOUN
ejpam-4962	648	8	}	}	PUNCT
ejpam-4962	648	9	,	,	PUNCT
ejpam-4962	648	10	v2	v2	PROPN
ejpam-4962	648	11	=	=	SYM
ejpam-4962	648	12	{	{	PUNCT
ejpam-4962	648	13	u	u	NOUN
ejpam-4962	648	14	,	,	PUNCT
ejpam-4962	648	15	x	x	NOUN
ejpam-4962	648	16	}	}	PUNCT
ejpam-4962	648	17	and	and	CCONJ
ejpam-4962	648	18	v0	v0	PROPN
ejpam-4962	648	19	=	=	SYM
ejpam-4962	648	20	v	v	PROPN
ejpam-4962	648	21	(	(	PUNCT
ejpam-4962	648	22	g+h	g+h	NOUN
ejpam-4962	648	23	)	)	PUNCT
ejpam-4962	648	24	\	\	PUNCT
ejpam-4962	649	1	(	(	PUNCT
ejpam-4962	649	2	v1	v1	VERB
ejpam-4962	649	3	∪	∪	NOUN
ejpam-4962	649	4	v2	v2	NOUN
ejpam-4962	649	5	)	)	PUNCT
ejpam-4962	649	6	.	.	PUNCT
ejpam-4962	650	1	then	then	ADV
ejpam-4962	650	2	f	f	PROPN
ejpam-4962	650	3	=	=	SYM
ejpam-4962	650	4	(	(	PUNCT
ejpam-4962	650	5	v0	v0	PROPN
ejpam-4962	650	6	,	,	PUNCT
ejpam-4962	650	7	v1	v1	NOUN
ejpam-4962	650	8	,	,	PUNCT
ejpam-4962	650	9	v2	v2	PROPN
ejpam-4962	650	10	)	)	PUNCT
ejpam-4962	650	11	is	be	AUX
ejpam-4962	650	12	a	a	DET
ejpam-4962	650	13	grdf	grdf	NOUN
ejpam-4962	650	14	on	on	ADP
ejpam-4962	650	15	g+h	g+h	PROPN
ejpam-4962	650	16	.	.	PUNCT
ejpam-4962	651	1	hence	hence	ADV
ejpam-4962	651	2	,	,	PUNCT
ejpam-4962	651	3	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	651	4	)	)	PUNCT
ejpam-4962	651	5	≤	≤	NUM
ejpam-4962	651	6	ωgr	ωgr	SYM
ejpam-4962	651	7	g+h(f	g+h(f	NOUN
ejpam-4962	651	8	)	)	PUNCT
ejpam-4962	651	9	=	=	SYM
ejpam-4962	652	1	6	6	X
ejpam-4962	652	2	.	.	PUNCT
ejpam-4962	652	3	lemma	lemma	PROPN
ejpam-4962	652	4	2	2	X
ejpam-4962	652	5	.	.	PUNCT
ejpam-4962	653	1	let	let	VERB
ejpam-4962	653	2	g	g	NOUN
ejpam-4962	653	3	and	and	CCONJ
ejpam-4962	653	4	h	h	NOUN
ejpam-4962	653	5	be	be	AUX
ejpam-4962	653	6	non	non	ADJ
ejpam-4962	653	7	-	-	ADJ
ejpam-4962	653	8	complete	complete	ADJ
ejpam-4962	653	9	graphs	graph	NOUN
ejpam-4962	653	10	and	and	CCONJ
ejpam-4962	653	11	let	let	VERB
ejpam-4962	653	12	s	s	PRON
ejpam-4962	653	13	=	=	VERB
ejpam-4962	653	14	sg∪sh	sg∪sh	PROPN
ejpam-4962	653	15	,	,	PUNCT
ejpam-4962	653	16	where	where	SCONJ
ejpam-4962	653	17	sg	sg	ADP
ejpam-4962	653	18	⊆	⊆	NUM
ejpam-4962	653	19	v	v	NOUN
ejpam-4962	653	20	(	(	PUNCT
ejpam-4962	653	21	g	g	NOUN
ejpam-4962	653	22	)	)	PUNCT
ejpam-4962	653	23	and	and	CCONJ
ejpam-4962	653	24	sh	sh	PROPN
ejpam-4962	653	25	⊆	⊆	NUM
ejpam-4962	653	26	v	v	NOUN
ejpam-4962	653	27	(	(	PUNCT
ejpam-4962	653	28	h	h	NOUN
ejpam-4962	653	29	)	)	PUNCT
ejpam-4962	653	30	,	,	PUNCT
ejpam-4962	653	31	be	be	AUX
ejpam-4962	653	32	a	a	DET
ejpam-4962	653	33	geodetic	geodetic	ADJ
ejpam-4962	653	34	set	set	NOUN
ejpam-4962	653	35	in	in	ADP
ejpam-4962	653	36	g+h	g+h	PROPN
ejpam-4962	653	37	.	.	PUNCT
ejpam-4962	654	1	then	then	ADV
ejpam-4962	654	2	each	each	PRON
ejpam-4962	654	3	of	of	ADP
ejpam-4962	654	4	the	the	DET
ejpam-4962	654	5	following	following	ADJ
ejpam-4962	654	6	statements	statement	NOUN
ejpam-4962	654	7	holds	hold	VERB
ejpam-4962	654	8	.	.	PUNCT
ejpam-4962	655	1	(	(	PUNCT
ejpam-4962	655	2	i	i	NOUN
ejpam-4962	655	3	)	)	PUNCT
ejpam-4962	655	4	if	if	SCONJ
ejpam-4962	655	5	|sg|	|sg|	NOUN
ejpam-4962	655	6	≥	≥	NOUN
ejpam-4962	655	7	2	2	NUM
ejpam-4962	655	8	and	and	CCONJ
ejpam-4962	655	9	|sh	|sh	ADP
ejpam-4962	655	10	|	|	ADV
ejpam-4962	655	11	≤	≤	ADJ
ejpam-4962	655	12	1	1	NUM
ejpam-4962	655	13	,	,	PUNCT
ejpam-4962	655	14	then	then	ADV
ejpam-4962	655	15	sg	sg	PROPN
ejpam-4962	655	16	is	be	AUX
ejpam-4962	655	17	a	a	DET
ejpam-4962	655	18	2	2	NUM
ejpam-4962	655	19	-	-	PUNCT
ejpam-4962	655	20	path	path	NOUN
ejpam-4962	655	21	closure	closure	NOUN
ejpam-4962	655	22	absorbing	absorb	VERB
ejpam-4962	655	23	set	set	VERB
ejpam-4962	655	24	in	in	ADP
ejpam-4962	655	25	g.	g.	PROPN
ejpam-4962	655	26	(	(	PUNCT
ejpam-4962	655	27	ii	ii	PROPN
ejpam-4962	655	28	)	)	PUNCT
ejpam-4962	655	29	if	if	SCONJ
ejpam-4962	655	30	|sh	|sh	ADP
ejpam-4962	655	31	|	|	ADV
ejpam-4962	655	32	≥	≥	NOUN
ejpam-4962	655	33	2	2	NUM
ejpam-4962	655	34	and	and	CCONJ
ejpam-4962	655	35	|sg|	|sg|	PROPN
ejpam-4962	655	36	≤	≤	NUM
ejpam-4962	655	37	1	1	NUM
ejpam-4962	655	38	,	,	PUNCT
ejpam-4962	655	39	then	then	ADV
ejpam-4962	655	40	sh	sh	PROPN
ejpam-4962	655	41	is	be	AUX
ejpam-4962	655	42	a	a	DET
ejpam-4962	655	43	2	2	NUM
ejpam-4962	655	44	-	-	PUNCT
ejpam-4962	655	45	path	path	NOUN
ejpam-4962	655	46	closure	closure	NOUN
ejpam-4962	655	47	absorbing	absorb	VERB
ejpam-4962	655	48	set	set	NOUN
ejpam-4962	655	49	in	in	ADP
ejpam-4962	655	50	h.	h.	PROPN
ejpam-4962	655	51	proof	proof	NOUN
ejpam-4962	655	52	.	.	PUNCT
ejpam-4962	656	1	suppose	suppose	VERB
ejpam-4962	656	2	|sg|	|sg|	NOUN
ejpam-4962	656	3	≥	≥	NOUN
ejpam-4962	656	4	2	2	NUM
ejpam-4962	656	5	and	and	CCONJ
ejpam-4962	656	6	|sh	|sh	ADP
ejpam-4962	656	7	|	|	ADV
ejpam-4962	656	8	≤	≤	ADJ
ejpam-4962	656	9	1	1	NUM
ejpam-4962	656	10	.	.	PUNCT
ejpam-4962	657	1	if	if	SCONJ
ejpam-4962	657	2	sg	sg	PROPN
ejpam-4962	657	3	=	=	SYM
ejpam-4962	657	4	v	v	NOUN
ejpam-4962	657	5	(	(	PUNCT
ejpam-4962	657	6	g	g	NOUN
ejpam-4962	657	7	)	)	PUNCT
ejpam-4962	657	8	,	,	PUNCT
ejpam-4962	657	9	then	then	ADV
ejpam-4962	657	10	we	we	PRON
ejpam-4962	657	11	are	be	AUX
ejpam-4962	657	12	done	do	VERB
ejpam-4962	657	13	.	.	PUNCT
ejpam-4962	658	1	suppose	suppose	VERB
ejpam-4962	658	2	that	that	SCONJ
ejpam-4962	658	3	sg	sg	PROPN
ejpam-4962	658	4	̸=	̸=	PROPN
ejpam-4962	658	5	v	v	NOUN
ejpam-4962	658	6	(	(	PUNCT
ejpam-4962	658	7	g	g	NOUN
ejpam-4962	658	8	)	)	PUNCT
ejpam-4962	658	9	and	and	CCONJ
ejpam-4962	658	10	let	let	VERB
ejpam-4962	658	11	v	v	NUM
ejpam-4962	658	12	∈	∈	PROPN
ejpam-4962	658	13	v	v	NOUN
ejpam-4962	658	14	(	(	PUNCT
ejpam-4962	658	15	g	g	NOUN
ejpam-4962	658	16	)	)	PUNCT
ejpam-4962	658	17	\sg	\sg	PROPN
ejpam-4962	658	18	.	.	PUNCT
ejpam-4962	659	1	since	since	SCONJ
ejpam-4962	659	2	s	s	PROPN
ejpam-4962	659	3	is	be	AUX
ejpam-4962	659	4	a	a	DET
ejpam-4962	659	5	geodetic	geodetic	ADJ
ejpam-4962	659	6	set	set	NOUN
ejpam-4962	659	7	in	in	ADP
ejpam-4962	659	8	g+h	g+h	PROPN
ejpam-4962	659	9	and	and	CCONJ
ejpam-4962	659	10	|sh	|sh	ADP
ejpam-4962	659	11	|	|	ADV
ejpam-4962	659	12	≤	≤	ADJ
ejpam-4962	659	13	1	1	NUM
ejpam-4962	659	14	,	,	PUNCT
ejpam-4962	659	15	there	there	PRON
ejpam-4962	659	16	exist	exist	VERB
ejpam-4962	659	17	p	p	PRON
ejpam-4962	659	18	,	,	PUNCT
ejpam-4962	659	19	q	q	PROPN
ejpam-4962	659	20	∈	∈	NOUN
ejpam-4962	659	21	sg	sg	ADP
ejpam-4962	659	22	such	such	ADJ
ejpam-4962	660	1	that	that	PRON
ejpam-4962	660	2	v	v	ADP
ejpam-4962	660	3	∈	∈	PROPN
ejpam-4962	660	4	ig+h(p	ig+h(p	PROPN
ejpam-4962	660	5	,	,	PUNCT
ejpam-4962	660	6	q	q	NOUN
ejpam-4962	660	7	)	)	PUNCT
ejpam-4962	660	8	.	.	PUNCT
ejpam-4962	661	1	this	this	PRON
ejpam-4962	661	2	implies	imply	VERB
ejpam-4962	661	3	that	that	SCONJ
ejpam-4962	661	4	dg(p	dg(p	NOUN
ejpam-4962	661	5	,	,	PUNCT
ejpam-4962	661	6	q	q	X
ejpam-4962	661	7	)	)	PUNCT
ejpam-4962	661	8	=	=	SYM
ejpam-4962	661	9	2	2	NUM
ejpam-4962	661	10	and	and	CCONJ
ejpam-4962	661	11	v	v	ADP
ejpam-4962	661	12	∈	∈	NOUN
ejpam-4962	661	13	ig(p	ig(p	NOUN
ejpam-4962	661	14	,	,	PUNCT
ejpam-4962	661	15	q	q	NOUN
ejpam-4962	661	16	)	)	PUNCT
ejpam-4962	661	17	.	.	PUNCT
ejpam-4962	662	1	hence	hence	ADV
ejpam-4962	662	2	,	,	PUNCT
ejpam-4962	662	3	sg	sg	PROPN
ejpam-4962	662	4	is	be	AUX
ejpam-4962	662	5	a	a	DET
ejpam-4962	662	6	2	2	NUM
ejpam-4962	662	7	-	-	PUNCT
ejpam-4962	662	8	path	path	NOUN
ejpam-4962	662	9	closure	closure	NOUN
ejpam-4962	662	10	absorbing	absorb	VERB
ejpam-4962	662	11	set	set	NOUN
ejpam-4962	662	12	in	in	ADP
ejpam-4962	662	13	g	g	NOUN
ejpam-4962	662	14	,	,	PUNCT
ejpam-4962	662	15	showing	show	VERB
ejpam-4962	662	16	that	that	SCONJ
ejpam-4962	662	17	(	(	PUNCT
ejpam-4962	662	18	i	i	NOUN
ejpam-4962	662	19	)	)	PUNCT
ejpam-4962	662	20	holds	hold	VERB
ejpam-4962	662	21	.	.	PUNCT
ejpam-4962	663	1	similarly	similarly	ADV
ejpam-4962	663	2	,	,	PUNCT
ejpam-4962	663	3	(	(	PUNCT
ejpam-4962	663	4	ii	ii	NOUN
ejpam-4962	663	5	)	)	PUNCT
ejpam-4962	663	6	holds	hold	VERB
ejpam-4962	663	7	.	.	PUNCT
ejpam-4962	664	1	theorem	theorem	NOUN
ejpam-4962	664	2	7	7	NUM
ejpam-4962	664	3	.	.	PUNCT
ejpam-4962	665	1	let	let	VERB
ejpam-4962	665	2	g	g	NOUN
ejpam-4962	665	3	and	and	CCONJ
ejpam-4962	665	4	h	h	NOUN
ejpam-4962	665	5	be	be	AUX
ejpam-4962	665	6	non	non	ADJ
ejpam-4962	665	7	-	-	ADJ
ejpam-4962	665	8	complete	complete	ADJ
ejpam-4962	665	9	graphs	graph	NOUN
ejpam-4962	665	10	.	.	PUNCT
ejpam-4962	666	1	then	then	ADV
ejpam-4962	666	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	666	3	)	)	PUNCT
ejpam-4962	666	4	=	=	SYM
ejpam-4962	667	1	3	3	NUM
ejpam-4962	667	2	if	if	SCONJ
ejpam-4962	667	3	and	and	CCONJ
ejpam-4962	667	4	only	only	ADV
ejpam-4962	667	5	if	if	SCONJ
ejpam-4962	667	6	g	g	PROPN
ejpam-4962	667	7	∈	∈	PROPN
ejpam-4962	667	8	{	{	PUNCT
ejpam-4962	667	9	k2,k2	k2,k2	PROPN
ejpam-4962	667	10	+	+	NUM
ejpam-4962	667	11	g1	g1	PROPN
ejpam-4962	667	12	}	}	PUNCT
ejpam-4962	667	13	or	or	CCONJ
ejpam-4962	667	14	h	h	NOUN
ejpam-4962	667	15	∈	∈	PROPN
ejpam-4962	667	16	{	{	PUNCT
ejpam-4962	667	17	k2,k2	k2,k2	PROPN
ejpam-4962	667	18	+	+	PROPN
ejpam-4962	667	19	h1	h1	PROPN
ejpam-4962	667	20	}	}	PUNCT
ejpam-4962	667	21	for	for	ADP
ejpam-4962	667	22	some	some	DET
ejpam-4962	667	23	graphs	graph	NOUN
ejpam-4962	667	24	g1	g1	NOUN
ejpam-4962	667	25	and	and	CCONJ
ejpam-4962	667	26	h1	h1	NOUN
ejpam-4962	667	27	.	.	PUNCT
ejpam-4962	668	1	proof	proof	NOUN
ejpam-4962	668	2	.	.	PUNCT
ejpam-4962	669	1	suppose	suppose	VERB
ejpam-4962	669	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	669	3	)	)	PUNCT
ejpam-4962	669	4	=	=	SYM
ejpam-4962	670	1	3	3	X
ejpam-4962	670	2	.	.	PUNCT
ejpam-4962	670	3	since	since	SCONJ
ejpam-4962	670	4	g	g	PROPN
ejpam-4962	670	5	and	and	CCONJ
ejpam-4962	670	6	h	h	NOUN
ejpam-4962	670	7	are	be	AUX
ejpam-4962	670	8	non	non	ADJ
ejpam-4962	670	9	-	-	ADJ
ejpam-4962	670	10	complete	complete	ADJ
ejpam-4962	670	11	graphs	graph	NOUN
ejpam-4962	670	12	and	and	CCONJ
ejpam-4962	670	13	g+h	g+h	PROPN
ejpam-4962	670	14	is	be	AUX
ejpam-4962	670	15	a	a	DET
ejpam-4962	670	16	connected	connected	ADJ
ejpam-4962	670	17	graph	graph	NOUN
ejpam-4962	670	18	,	,	PUNCT
ejpam-4962	670	19	g+h	g+h	PROPN
ejpam-4962	671	1	=	=	SYM
ejpam-4962	671	2	k2+f	k2+f	PROPN
ejpam-4962	671	3	for	for	ADP
ejpam-4962	671	4	some	some	DET
ejpam-4962	671	5	non	non	ADJ
ejpam-4962	671	6	-	-	ADJ
ejpam-4962	671	7	complete	complete	ADJ
ejpam-4962	671	8	graph	graph	NOUN
ejpam-4962	671	9	f	f	X
ejpam-4962	671	10	by	by	ADP
ejpam-4962	671	11	theorem	theorem	ADJ
ejpam-4962	671	12	2(iii	2(iii	NUM
ejpam-4962	671	13	)	)	PUNCT
ejpam-4962	671	14	.	.	PUNCT
ejpam-4962	672	1	let	let	VERB
ejpam-4962	672	2	k2	k2	PROPN
ejpam-4962	672	3	=	=	PUNCT
ejpam-4962	672	4	{	{	PUNCT
ejpam-4962	672	5	a	a	PROPN
ejpam-4962	672	6	,	,	PUNCT
ejpam-4962	672	7	b	b	NOUN
ejpam-4962	672	8	}	}	PUNCT
ejpam-4962	672	9	.	.	PUNCT
ejpam-4962	673	1	then	then	ADV
ejpam-4962	673	2	a	a	DET
ejpam-4962	673	3	,	,	PUNCT
ejpam-4962	673	4	b	b	PROPN
ejpam-4962	673	5	∈	∈	PROPN
ejpam-4962	673	6	v	v	NOUN
ejpam-4962	673	7	(	(	PUNCT
ejpam-4962	673	8	g	g	NOUN
ejpam-4962	673	9	)	)	PUNCT
ejpam-4962	673	10	or	or	CCONJ
ejpam-4962	673	11	a	a	PRON
ejpam-4962	673	12	,	,	PUNCT
ejpam-4962	673	13	b	b	PROPN
ejpam-4962	673	14	∈	∈	PROPN
ejpam-4962	673	15	v	v	NOUN
ejpam-4962	673	16	(	(	PUNCT
ejpam-4962	673	17	h	h	NOUN
ejpam-4962	673	18	)	)	PUNCT
ejpam-4962	673	19	.	.	PUNCT
ejpam-4962	674	1	we	we	PRON
ejpam-4962	674	2	may	may	AUX
ejpam-4962	674	3	assume	assume	VERB
ejpam-4962	674	4	that	that	SCONJ
ejpam-4962	674	5	a	a	DET
ejpam-4962	674	6	,	,	PUNCT
ejpam-4962	674	7	b	b	PROPN
ejpam-4962	674	8	∈	∈	PROPN
ejpam-4962	674	9	v	v	NOUN
ejpam-4962	674	10	(	(	PUNCT
ejpam-4962	674	11	g	g	NOUN
ejpam-4962	674	12	)	)	PUNCT
ejpam-4962	674	13	.	.	PUNCT
ejpam-4962	675	1	then	then	ADV
ejpam-4962	675	2	g	g	PROPN
ejpam-4962	675	3	=	=	PROPN
ejpam-4962	675	4	k2	k2	PROPN
ejpam-4962	675	5	or	or	CCONJ
ejpam-4962	675	6	g	g	PROPN
ejpam-4962	675	7	=	=	PROPN
ejpam-4962	675	8	k2	k2	PROPN
ejpam-4962	675	9	+	+	PROPN
ejpam-4962	675	10	g1	g1	PROPN
ejpam-4962	675	11	where	where	SCONJ
ejpam-4962	675	12	g1	g1	PROPN
ejpam-4962	675	13	=	=	SYM
ejpam-4962	675	14	⟨v	⟨v	PROPN
ejpam-4962	675	15	(	(	PUNCT
ejpam-4962	675	16	g	g	NOUN
ejpam-4962	675	17	)	)	PUNCT
ejpam-4962	675	18	\	\	NOUN
ejpam-4962	676	1	{	{	PUNCT
ejpam-4962	676	2	a	a	PRON
ejpam-4962	676	3	,	,	PUNCT
ejpam-4962	676	4	b}⟩.	b}⟩.	VERB
ejpam-4962	676	5	conversely	conversely	ADV
ejpam-4962	676	6	,	,	PUNCT
ejpam-4962	676	7	if	if	SCONJ
ejpam-4962	676	8	g	g	PROPN
ejpam-4962	676	9	=	=	SYM
ejpam-4962	676	10	k2	k2	PROPN
ejpam-4962	676	11	,	,	PUNCT
ejpam-4962	676	12	then	then	ADV
ejpam-4962	676	13	γgr(g	γgr(g	PROPN
ejpam-4962	676	14	+	+	CCONJ
ejpam-4962	676	15	h	h	X
ejpam-4962	676	16	)	)	PUNCT
ejpam-4962	676	17	=	=	SYM
ejpam-4962	677	1	3	3	X
ejpam-4962	677	2	.	.	X
ejpam-4962	678	1	if	if	SCONJ
ejpam-4962	678	2	g	g	PROPN
ejpam-4962	678	3	=	=	SYM
ejpam-4962	678	4	k2	k2	PROPN
ejpam-4962	678	5	+	+	CCONJ
ejpam-4962	678	6	g1	g1	PROPN
ejpam-4962	678	7	for	for	ADP
ejpam-4962	678	8	some	some	DET
ejpam-4962	678	9	graph	graph	NOUN
ejpam-4962	678	10	g1	g1	NOUN
ejpam-4962	678	11	,	,	PUNCT
ejpam-4962	678	12	then	then	ADV
ejpam-4962	678	13	g+h	g+h	PROPN
ejpam-4962	678	14	=	=	SYM
ejpam-4962	678	15	k2+(g1+h	k2+(g1+h	PROPN
ejpam-4962	678	16	)	)	PUNCT
ejpam-4962	678	17	.	.	PUNCT
ejpam-4962	679	1	by	by	ADP
ejpam-4962	679	2	theorem	theorem	ADJ
ejpam-4962	679	3	2(iii	2(iii	NUM
ejpam-4962	679	4	)	)	PUNCT
ejpam-4962	679	5	,	,	PUNCT
ejpam-4962	679	6	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	679	7	)	)	PUNCT
ejpam-4962	679	8	=	=	SYM
ejpam-4962	680	1	3	3	X
ejpam-4962	680	2	.	.	PUNCT
ejpam-4962	680	3	the	the	DET
ejpam-4962	680	4	same	same	ADJ
ejpam-4962	680	5	conclusion	conclusion	NOUN
ejpam-4962	680	6	holds	hold	VERB
ejpam-4962	680	7	when	when	SCONJ
ejpam-4962	680	8	h	h	PROPN
ejpam-4962	680	9	∈	∈	PROPN
ejpam-4962	680	10	{	{	PUNCT
ejpam-4962	680	11	k2,k2	k2,k2	PROPN
ejpam-4962	680	12	+	+	PROPN
ejpam-4962	680	13	h1	h1	PROPN
ejpam-4962	680	14	}	}	PUNCT
ejpam-4962	680	15	for	for	ADP
ejpam-4962	680	16	some	some	DET
ejpam-4962	680	17	graph	graph	NOUN
ejpam-4962	680	18	h1	h1	PROPN
ejpam-4962	680	19	.	.	PUNCT
ejpam-4962	680	20	theorem	theorem	ADJ
ejpam-4962	680	21	8	8	NUM
ejpam-4962	680	22	.	.	PUNCT
ejpam-4962	681	1	let	let	VERB
ejpam-4962	681	2	g	g	NOUN
ejpam-4962	681	3	and	and	CCONJ
ejpam-4962	681	4	h	h	NOUN
ejpam-4962	681	5	be	be	AUX
ejpam-4962	681	6	non	non	ADJ
ejpam-4962	681	7	-	-	ADJ
ejpam-4962	681	8	complete	complete	ADJ
ejpam-4962	681	9	graphs	graph	NOUN
ejpam-4962	681	10	.	.	PUNCT
ejpam-4962	682	1	then	then	ADV
ejpam-4962	682	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	682	3	)	)	PUNCT
ejpam-4962	682	4	=	=	SYM
ejpam-4962	683	1	3	3	NUM
ejpam-4962	683	2	if	if	SCONJ
ejpam-4962	683	3	and	and	CCONJ
ejpam-4962	683	4	only	only	ADV
ejpam-4962	683	5	if	if	SCONJ
ejpam-4962	683	6	ρ2(g	ρ2(g	NUM
ejpam-4962	683	7	)	)	PUNCT
ejpam-4962	683	8	=	=	SYM
ejpam-4962	683	9	2	2	NUM
ejpam-4962	683	10	or	or	CCONJ
ejpam-4962	683	11	ρ2(h	ρ2(h	NUM
ejpam-4962	683	12	)	)	PUNCT
ejpam-4962	683	13	=	=	SYM
ejpam-4962	683	14	2	2	X
ejpam-4962	683	15	.	.	PUNCT
ejpam-4962	684	1	proof	proof	NOUN
ejpam-4962	684	2	.	.	PUNCT
ejpam-4962	685	1	suppose	suppose	VERB
ejpam-4962	685	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	685	3	)	)	PUNCT
ejpam-4962	685	4	=	=	SYM
ejpam-4962	686	1	3	3	X
ejpam-4962	686	2	.	.	PUNCT
ejpam-4962	686	3	by	by	ADP
ejpam-4962	686	4	theorem	theorem	NOUN
ejpam-4962	686	5	7	7	NUM
ejpam-4962	686	6	,	,	PUNCT
ejpam-4962	686	7	ρ2(g	ρ2(g	NUM
ejpam-4962	686	8	)	)	PUNCT
ejpam-4962	686	9	=	=	SYM
ejpam-4962	686	10	2	2	NUM
ejpam-4962	686	11	or	or	CCONJ
ejpam-4962	686	12	ρ2(g	ρ2(g	NUM
ejpam-4962	686	13	)	)	PUNCT
ejpam-4962	686	14	=	=	SYM
ejpam-4962	686	15	2	2	X
ejpam-4962	686	16	.	.	PUNCT
ejpam-4962	686	17	conversely	conversely	ADV
ejpam-4962	686	18	,	,	PUNCT
ejpam-4962	686	19	suppose	suppose	VERB
ejpam-4962	686	20	that	that	SCONJ
ejpam-4962	686	21	ρ2(g	ρ2(g	X
ejpam-4962	686	22	)	)	PUNCT
ejpam-4962	686	23	=	=	SYM
ejpam-4962	686	24	2	2	NUM
ejpam-4962	686	25	say	say	VERB
ejpam-4962	686	26	s	s	PART
ejpam-4962	686	27	=	=	PUNCT
ejpam-4962	686	28	{	{	PUNCT
ejpam-4962	686	29	x	x	PROPN
ejpam-4962	686	30	,	,	PUNCT
ejpam-4962	686	31	y	y	PRON
ejpam-4962	686	32	}	}	PUNCT
ejpam-4962	686	33	is	be	AUX
ejpam-4962	686	34	a	a	DET
ejpam-4962	686	35	2	2	NUM
ejpam-4962	686	36	-	-	PUNCT
ejpam-4962	686	37	path	path	NOUN
ejpam-4962	686	38	closure	closure	NOUN
ejpam-4962	686	39	absorbing	absorb	VERB
ejpam-4962	686	40	set	set	VERB
ejpam-4962	686	41	in	in	ADP
ejpam-4962	686	42	g.	g.	PROPN
ejpam-4962	686	43	if	if	SCONJ
ejpam-4962	686	44	|v	|v	PROPN
ejpam-4962	686	45	(	(	PUNCT
ejpam-4962	686	46	g)|	g)|	NOUN
ejpam-4962	686	47	=	=	SYM
ejpam-4962	686	48	2	2	NUM
ejpam-4962	686	49	,	,	PUNCT
ejpam-4962	686	50	then	then	ADV
ejpam-4962	686	51	g	g	PROPN
ejpam-4962	686	52	=	=	PROPN
ejpam-4962	686	53	k2	k2	PROPN
ejpam-4962	686	54	.	.	PUNCT
ejpam-4962	686	55	suppose	suppose	VERB
ejpam-4962	686	56	g	g	PROPN
ejpam-4962	686	57	̸=	̸=	PROPN
ejpam-4962	686	58	k2	k2	PROPN
ejpam-4962	686	59	.	.	PUNCT
ejpam-4962	687	1	then	then	ADV
ejpam-4962	687	2	for	for	ADP
ejpam-4962	687	3	all	all	PRON
ejpam-4962	687	4	u	u	NOUN
ejpam-4962	687	5	∈	∈	PROPN
ejpam-4962	687	6	v	v	NOUN
ejpam-4962	687	7	(	(	PUNCT
ejpam-4962	687	8	g	g	NOUN
ejpam-4962	687	9	)	)	PUNCT
ejpam-4962	687	10	\	\	NOUN
ejpam-4962	688	1	{	{	PUNCT
ejpam-4962	688	2	x	x	NOUN
ejpam-4962	688	3	,	,	PUNCT
ejpam-4962	688	4	y	y	NOUN
ejpam-4962	688	5	}	}	PUNCT
ejpam-4962	688	6	,	,	PUNCT
ejpam-4962	688	7	dg(x	dg(x	X
ejpam-4962	688	8	,	,	PUNCT
ejpam-4962	688	9	y	y	NOUN
ejpam-4962	688	10	)	)	PUNCT
ejpam-4962	688	11	=	=	SYM
ejpam-4962	688	12	2	2	NUM
ejpam-4962	688	13	and	and	CCONJ
ejpam-4962	688	14	u	u	NOUN
ejpam-4962	688	15	∈	∈	PROPN
ejpam-4962	688	16	ig(x	ig(x	X
ejpam-4962	688	17	,	,	PUNCT
ejpam-4962	688	18	y	y	PROPN
ejpam-4962	688	19	)	)	PUNCT
ejpam-4962	688	20	.	.	PUNCT
ejpam-4962	689	1	this	this	PRON
ejpam-4962	689	2	implies	imply	VERB
ejpam-4962	689	3	that	that	SCONJ
ejpam-4962	689	4	g	g	PROPN
ejpam-4962	689	5	=	=	PRON
ejpam-4962	689	6	{	{	PUNCT
ejpam-4962	689	7	x	x	NOUN
ejpam-4962	689	8	,	,	PUNCT
ejpam-4962	689	9	y}+g1	y}+g1	PROPN
ejpam-4962	689	10	for	for	ADP
ejpam-4962	689	11	some	some	DET
ejpam-4962	689	12	graph	graph	NOUN
ejpam-4962	689	13	g1	g1	NOUN
ejpam-4962	689	14	.	.	PUNCT
ejpam-4962	690	1	by	by	ADP
ejpam-4962	690	2	theorem	theorem	ADJ
ejpam-4962	690	3	2(iii	2(iii	NUM
ejpam-4962	690	4	)	)	PUNCT
ejpam-4962	690	5	,	,	PUNCT
ejpam-4962	690	6	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	690	7	)	)	PUNCT
ejpam-4962	690	8	=	=	SYM
ejpam-4962	691	1	3	3	X
ejpam-4962	691	2	.	.	X
ejpam-4962	691	3	similarly	similarly	ADV
ejpam-4962	691	4	,	,	PUNCT
ejpam-4962	691	5	if	if	SCONJ
ejpam-4962	691	6	ρ2(h	ρ2(h	NUM
ejpam-4962	691	7	)	)	PUNCT
ejpam-4962	691	8	=	=	SYM
ejpam-4962	691	9	2	2	NUM
ejpam-4962	691	10	,	,	PUNCT
ejpam-4962	691	11	then	then	ADV
ejpam-4962	691	12	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	691	13	)	)	PUNCT
ejpam-4962	691	14	=	=	SYM
ejpam-4962	692	1	3	3	X
ejpam-4962	692	2	.	.	X
ejpam-4962	692	3	theorem	theorem	NOUN
ejpam-4962	692	4	9	9	NUM
ejpam-4962	692	5	.	.	PUNCT
ejpam-4962	693	1	let	let	VERB
ejpam-4962	693	2	g	g	NOUN
ejpam-4962	693	3	and	and	CCONJ
ejpam-4962	693	4	h	h	NOUN
ejpam-4962	693	5	be	be	AUX
ejpam-4962	693	6	non	non	ADJ
ejpam-4962	693	7	-	-	ADJ
ejpam-4962	693	8	complete	complete	ADJ
ejpam-4962	693	9	graphs	graph	NOUN
ejpam-4962	693	10	.	.	PUNCT
ejpam-4962	694	1	then	then	ADV
ejpam-4962	694	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	694	3	)	)	PUNCT
ejpam-4962	694	4	=	=	SYM
ejpam-4962	695	1	4	4	NUM
ejpam-4962	695	2	if	if	SCONJ
ejpam-4962	695	3	and	and	CCONJ
ejpam-4962	695	4	only	only	ADV
ejpam-4962	695	5	if	if	SCONJ
ejpam-4962	695	6	one	one	NUM
ejpam-4962	695	7	of	of	ADP
ejpam-4962	695	8	the	the	DET
ejpam-4962	695	9	following	follow	VERB
ejpam-4962	695	10	conditions	condition	NOUN
ejpam-4962	695	11	holds	hold	VERB
ejpam-4962	695	12	:	:	PUNCT
ejpam-4962	695	13	(	(	PUNCT
ejpam-4962	695	14	i	i	NOUN
ejpam-4962	695	15	)	)	PUNCT
ejpam-4962	695	16	ρ2(h	ρ2(h	X
ejpam-4962	695	17	)	)	PUNCT
ejpam-4962	695	18	̸=	̸=	PROPN
ejpam-4962	695	19	2	2	NUM
ejpam-4962	695	20	and	and	CCONJ
ejpam-4962	695	21	there	there	PRON
ejpam-4962	695	22	exists	exist	VERB
ejpam-4962	695	23	a	a	DET
ejpam-4962	695	24	ρ2	ρ2	NOUN
ejpam-4962	695	25	-	-	PUNCT
ejpam-4962	695	26	set	set	VERB
ejpam-4962	695	27	{	{	PUNCT
ejpam-4962	695	28	x	x	NOUN
ejpam-4962	695	29	,	,	PUNCT
ejpam-4962	695	30	y	y	PROPN
ejpam-4962	695	31	,	,	PUNCT
ejpam-4962	695	32	z	z	NOUN
ejpam-4962	695	33	}	}	PUNCT
ejpam-4962	695	34	in	in	ADP
ejpam-4962	695	35	g	g	PROPN
ejpam-4962	695	36	such	such	DET
ejpam-4962	695	37	that	that	DET
ejpam-4962	695	38	v	v	NOUN
ejpam-4962	695	39	(	(	PUNCT
ejpam-4962	695	40	g)\{x	g)\{x	PROPN
ejpam-4962	695	41	,	,	PUNCT
ejpam-4962	695	42	y	y	PROPN
ejpam-4962	695	43	,	,	PUNCT
ejpam-4962	695	44	z	z	NOUN
ejpam-4962	695	45	}	}	PUNCT
ejpam-4962	695	46	⊆	⊆	NUM
ejpam-4962	695	47	ng(x	ng(x	NUM
ejpam-4962	695	48	)	)	PUNCT
ejpam-4962	695	49	.	.	PUNCT
ejpam-4962	696	1	(	(	PUNCT
ejpam-4962	696	2	ii	ii	X
ejpam-4962	696	3	)	)	PUNCT
ejpam-4962	696	4	ρ2(g	ρ2(g	X
ejpam-4962	696	5	)	)	PUNCT
ejpam-4962	696	6	̸=	̸=	PROPN
ejpam-4962	696	7	2	2	NUM
ejpam-4962	696	8	and	and	CCONJ
ejpam-4962	696	9	there	there	PRON
ejpam-4962	696	10	exists	exist	VERB
ejpam-4962	696	11	a	a	DET
ejpam-4962	696	12	ρ2	ρ2	NOUN
ejpam-4962	696	13	-	-	PUNCT
ejpam-4962	696	14	set	set	VERB
ejpam-4962	696	15	{	{	PUNCT
ejpam-4962	696	16	x	x	NOUN
ejpam-4962	696	17	,	,	PUNCT
ejpam-4962	696	18	y	y	PROPN
ejpam-4962	696	19	,	,	PUNCT
ejpam-4962	696	20	z	z	NOUN
ejpam-4962	696	21	}	}	PUNCT
ejpam-4962	696	22	in	in	ADP
ejpam-4962	696	23	h	h	PROPN
ejpam-4962	696	24	such	such	ADJ
ejpam-4962	696	25	that	that	DET
ejpam-4962	696	26	v	v	NOUN
ejpam-4962	696	27	(	(	PUNCT
ejpam-4962	696	28	h)\{x	h)\{x	PROPN
ejpam-4962	696	29	,	,	PUNCT
ejpam-4962	696	30	y	y	PROPN
ejpam-4962	696	31	,	,	PUNCT
ejpam-4962	696	32	z	z	NOUN
ejpam-4962	696	33	}	}	PUNCT
ejpam-4962	696	34	⊆	⊆	NUM
ejpam-4962	696	35	nh(x	nh(x	NUM
ejpam-4962	696	36	)	)	PUNCT
ejpam-4962	696	37	.	.	PUNCT
ejpam-4962	697	1	r.	r.	PROPN
ejpam-4962	697	2	fortosa	fortosa	PROPN
ejpam-4962	697	3	,	,	PUNCT
ejpam-4962	697	4	s.	s.	PROPN
ejpam-4962	697	5	canoy	canoy	PROPN
ejpam-4962	697	6	jr	jr	PROPN
ejpam-4962	697	7	.	.	PROPN
ejpam-4962	697	8	/	/	SYM
ejpam-4962	697	9	eur	eur	PROPN
ejpam-4962	697	10	.	.	PUNCT
ejpam-4962	698	1	j.	j.	PROPN
ejpam-4962	698	2	pure	pure	PROPN
ejpam-4962	698	3	appl	appl	PROPN
ejpam-4962	698	4	.	.	PROPN
ejpam-4962	698	5	math	math	PROPN
ejpam-4962	698	6	,	,	PUNCT
ejpam-4962	698	7	16	16	NUM
ejpam-4962	698	8	(	(	PUNCT
ejpam-4962	698	9	4	4	NUM
ejpam-4962	698	10	)	)	PUNCT
ejpam-4962	698	11	(	(	PUNCT
ejpam-4962	698	12	2023	2023	NUM
ejpam-4962	698	13	)	)	PUNCT
ejpam-4962	698	14	,	,	PUNCT
ejpam-4962	698	15	2368	2368	NUM
ejpam-4962	698	16	-	-	SYM
ejpam-4962	698	17	2383	2383	NUM
ejpam-4962	698	18	2380	2380	NUM
ejpam-4962	698	19	proof	proof	NOUN
ejpam-4962	698	20	.	.	PUNCT
ejpam-4962	698	21	suppose	suppose	VERB
ejpam-4962	698	22	γgr(g	γgr(g	PROPN
ejpam-4962	698	23	+	+	CCONJ
ejpam-4962	698	24	h	h	NOUN
ejpam-4962	698	25	)	)	PUNCT
ejpam-4962	698	26	=	=	SYM
ejpam-4962	698	27	4	4	X
ejpam-4962	698	28	.	.	X
ejpam-4962	698	29	let	let	VERB
ejpam-4962	698	30	f	f	PROPN
ejpam-4962	698	31	=	=	SYM
ejpam-4962	698	32	(	(	PUNCT
ejpam-4962	698	33	v0	v0	PROPN
ejpam-4962	698	34	,	,	PUNCT
ejpam-4962	698	35	v1	v1	NOUN
ejpam-4962	698	36	,	,	PUNCT
ejpam-4962	698	37	v2	v2	PROPN
ejpam-4962	698	38	)	)	PUNCT
ejpam-4962	698	39	be	be	AUX
ejpam-4962	698	40	a	a	DET
ejpam-4962	698	41	γgr	γgr	NOUN
ejpam-4962	698	42	-	-	PUNCT
ejpam-4962	698	43	function	function	NOUN
ejpam-4962	698	44	on	on	ADP
ejpam-4962	698	45	g	g	PROPN
ejpam-4962	698	46	+	+	CCONJ
ejpam-4962	698	47	h.	h.	PROPN
ejpam-4962	698	48	then	then	ADV
ejpam-4962	698	49	|v1|	|v1|	VERB
ejpam-4962	698	50	+	+	CCONJ
ejpam-4962	698	51	2|v2|	2|v2|	NUM
ejpam-4962	698	52	=	=	SYM
ejpam-4962	698	53	4	4	X
ejpam-4962	698	54	.	.	PUNCT
ejpam-4962	698	55	suppose	suppose	VERB
ejpam-4962	698	56	|v2|	|v2|	NOUN
ejpam-4962	698	57	=	=	SYM
ejpam-4962	699	1	0	0	X
ejpam-4962	699	2	.	.	PUNCT
ejpam-4962	700	1	then	then	ADV
ejpam-4962	700	2	|v1|	|v1|	NOUN
ejpam-4962	700	3	=	=	SYM
ejpam-4962	700	4	|v	|v	PROPN
ejpam-4962	700	5	(	(	PUNCT
ejpam-4962	700	6	g	g	PROPN
ejpam-4962	700	7	+	+	CCONJ
ejpam-4962	700	8	h)|	h)|	NOUN
ejpam-4962	700	9	=	=	NOUN
ejpam-4962	700	10	4	4	NUM
ejpam-4962	700	11	.	.	PUNCT
ejpam-4962	701	1	since	since	SCONJ
ejpam-4962	701	2	g	g	PROPN
ejpam-4962	701	3	and	and	CCONJ
ejpam-4962	701	4	h	h	NOUN
ejpam-4962	701	5	are	be	AUX
ejpam-4962	701	6	non	non	ADJ
ejpam-4962	701	7	-	-	ADJ
ejpam-4962	701	8	complete	complete	ADJ
ejpam-4962	701	9	graphs	graph	NOUN
ejpam-4962	701	10	and	and	CCONJ
ejpam-4962	701	11	γgr(g	γgr(g	PROPN
ejpam-4962	702	1	+	+	NOUN
ejpam-4962	702	2	h	h	NOUN
ejpam-4962	702	3	)	)	PUNCT
ejpam-4962	702	4	̸=	̸=	PROPN
ejpam-4962	702	5	3	3	NUM
ejpam-4962	702	6	,	,	PUNCT
ejpam-4962	702	7	this	this	PRON
ejpam-4962	702	8	is	be	AUX
ejpam-4962	702	9	not	not	PART
ejpam-4962	702	10	possible	possible	ADJ
ejpam-4962	702	11	.	.	PUNCT
ejpam-4962	703	1	suppose	suppose	VERB
ejpam-4962	703	2	|v2|	|v2|	NOUN
ejpam-4962	703	3	=	=	SYM
ejpam-4962	703	4	2	2	NUM
ejpam-4962	703	5	,	,	PUNCT
ejpam-4962	703	6	say	say	VERB
ejpam-4962	703	7	v2	v2	NOUN
ejpam-4962	703	8	=	=	SYM
ejpam-4962	703	9	{	{	PUNCT
ejpam-4962	703	10	v	v	NOUN
ejpam-4962	703	11	,	,	PUNCT
ejpam-4962	703	12	w	w	NOUN
ejpam-4962	703	13	}	}	PUNCT
ejpam-4962	703	14	.	.	PUNCT
ejpam-4962	704	1	then	then	ADV
ejpam-4962	704	2	v2	v2	VERB
ejpam-4962	704	3	⊆	⊆	NUM
ejpam-4962	704	4	v	v	NOUN
ejpam-4962	704	5	(	(	PUNCT
ejpam-4962	704	6	g	g	NOUN
ejpam-4962	704	7	)	)	PUNCT
ejpam-4962	704	8	or	or	CCONJ
ejpam-4962	704	9	v2	v2	VERB
ejpam-4962	704	10	⊆	⊆	NUM
ejpam-4962	704	11	v	v	NOUN
ejpam-4962	704	12	(	(	PUNCT
ejpam-4962	704	13	h	h	NOUN
ejpam-4962	704	14	)	)	PUNCT
ejpam-4962	704	15	,	,	PUNCT
ejpam-4962	704	16	since	since	SCONJ
ejpam-4962	704	17	v2	v2	PROPN
ejpam-4962	704	18	is	be	AUX
ejpam-4962	704	19	a	a	DET
ejpam-4962	704	20	geodetic	geodetic	ADJ
ejpam-4962	704	21	set	set	NOUN
ejpam-4962	704	22	of	of	ADP
ejpam-4962	704	23	g	g	PROPN
ejpam-4962	704	24	+	+	CCONJ
ejpam-4962	704	25	h.	h.	PROPN
ejpam-4962	704	26	suppose	suppose	VERB
ejpam-4962	704	27	that	that	SCONJ
ejpam-4962	704	28	v2	v2	VERB
ejpam-4962	704	29	⊆	⊆	NUM
ejpam-4962	704	30	v	v	NOUN
ejpam-4962	704	31	(	(	PUNCT
ejpam-4962	704	32	g	g	NOUN
ejpam-4962	704	33	)	)	PUNCT
ejpam-4962	704	34	.	.	PUNCT
ejpam-4962	705	1	by	by	ADP
ejpam-4962	705	2	lemma	lemma	PROPN
ejpam-4962	705	3	2	2	NUM
ejpam-4962	705	4	,	,	PUNCT
ejpam-4962	705	5	v2	v2	PROPN
ejpam-4962	705	6	is	be	AUX
ejpam-4962	705	7	a	a	DET
ejpam-4962	705	8	2	2	NUM
ejpam-4962	705	9	-	-	PUNCT
ejpam-4962	705	10	path	path	NOUN
ejpam-4962	705	11	closure	closure	NOUN
ejpam-4962	705	12	absorbing	absorb	VERB
ejpam-4962	705	13	set	set	NOUN
ejpam-4962	705	14	.	.	PUNCT
ejpam-4962	706	1	hence	hence	ADV
ejpam-4962	706	2	,	,	PUNCT
ejpam-4962	706	3	ρ2(g	ρ2(g	X
ejpam-4962	706	4	)	)	PUNCT
ejpam-4962	706	5	=	=	SYM
ejpam-4962	706	6	2	2	X
ejpam-4962	706	7	.	.	PUNCT
ejpam-4962	706	8	by	by	ADP
ejpam-4962	706	9	theorem	theorem	NOUN
ejpam-4962	706	10	8	8	NUM
ejpam-4962	706	11	,	,	PUNCT
ejpam-4962	706	12	this	this	PRON
ejpam-4962	706	13	implies	imply	VERB
ejpam-4962	706	14	that	that	SCONJ
ejpam-4962	706	15	γgr(g	γgr(g	PROPN
ejpam-4962	706	16	+	+	CCONJ
ejpam-4962	706	17	h	h	NOUN
ejpam-4962	706	18	)	)	PUNCT
ejpam-4962	706	19	=	=	SYM
ejpam-4962	706	20	3	3	NUM
ejpam-4962	706	21	,	,	PUNCT
ejpam-4962	706	22	a	a	DET
ejpam-4962	706	23	contradiction	contradiction	NOUN
ejpam-4962	706	24	.	.	PUNCT
ejpam-4962	707	1	hence	hence	ADV
ejpam-4962	707	2	,	,	PUNCT
ejpam-4962	707	3	|v2|	|v2|	NOUN
ejpam-4962	707	4	=	=	SYM
ejpam-4962	707	5	1	1	NUM
ejpam-4962	707	6	and	and	CCONJ
ejpam-4962	707	7	|v1|	|v1|	NOUN
ejpam-4962	707	8	=	=	SYM
ejpam-4962	707	9	2	2	X
ejpam-4962	707	10	.	.	X
ejpam-4962	707	11	assume	assume	VERB
ejpam-4962	707	12	first	first	ADV
ejpam-4962	707	13	that	that	SCONJ
ejpam-4962	707	14	v2	v2	NOUN
ejpam-4962	707	15	=	=	SYM
ejpam-4962	707	16	{	{	PUNCT
ejpam-4962	707	17	x	x	NOUN
ejpam-4962	707	18	}	}	PUNCT
ejpam-4962	707	19	⊆	⊆	NUM
ejpam-4962	707	20	v	v	NOUN
ejpam-4962	707	21	(	(	PUNCT
ejpam-4962	707	22	g	g	NOUN
ejpam-4962	707	23	)	)	PUNCT
ejpam-4962	707	24	.	.	PUNCT
ejpam-4962	708	1	let	let	VERB
ejpam-4962	708	2	v1	v1	VERB
ejpam-4962	708	3	=	=	SYM
ejpam-4962	708	4	{	{	PUNCT
ejpam-4962	708	5	y	y	PROPN
ejpam-4962	708	6	,	,	PUNCT
ejpam-4962	708	7	z	z	NOUN
ejpam-4962	708	8	}	}	PUNCT
ejpam-4962	708	9	.	.	PUNCT
ejpam-4962	709	1	since	since	SCONJ
ejpam-4962	709	2	γgr(g	γgr(g	PROPN
ejpam-4962	709	3	+	+	PROPN
ejpam-4962	709	4	h	h	NOUN
ejpam-4962	709	5	)	)	PUNCT
ejpam-4962	709	6	̸=	̸=	PROPN
ejpam-4962	709	7	3	3	NUM
ejpam-4962	709	8	,	,	PUNCT
ejpam-4962	709	9	ρ2(g	ρ2(g	X
ejpam-4962	709	10	)	)	PUNCT
ejpam-4962	709	11	̸=	̸=	PROPN
ejpam-4962	709	12	2	2	NUM
ejpam-4962	709	13	and	and	CCONJ
ejpam-4962	709	14	ρ2(h	ρ2(h	NUM
ejpam-4962	709	15	)	)	PUNCT
ejpam-4962	709	16	̸=	̸=	PROPN
ejpam-4962	709	17	2	2	NUM
ejpam-4962	709	18	.	.	PUNCT
ejpam-4962	709	19	hence	hence	ADV
ejpam-4962	709	20	,	,	PUNCT
ejpam-4962	709	21	v1	v1	VERB
ejpam-4962	709	22	⊆	⊆	NUM
ejpam-4962	709	23	v	v	NOUN
ejpam-4962	709	24	(	(	PUNCT
ejpam-4962	709	25	g	g	NOUN
ejpam-4962	709	26	)	)	PUNCT
ejpam-4962	709	27	.	.	PUNCT
ejpam-4962	710	1	since	since	SCONJ
ejpam-4962	710	2	f	f	PROPN
ejpam-4962	710	3	is	be	AUX
ejpam-4962	710	4	a	a	DET
ejpam-4962	710	5	grdf	grdf	NOUN
ejpam-4962	710	6	on	on	ADP
ejpam-4962	710	7	g+h	g+h	PROPN
ejpam-4962	710	8	,	,	PUNCT
ejpam-4962	710	9	v	v	X
ejpam-4962	710	10	(	(	PUNCT
ejpam-4962	710	11	g	g	NOUN
ejpam-4962	710	12	)	)	PUNCT
ejpam-4962	710	13	\	\	NOUN
ejpam-4962	711	1	{	{	PUNCT
ejpam-4962	711	2	x	x	NOUN
ejpam-4962	711	3	,	,	PUNCT
ejpam-4962	711	4	y	y	PROPN
ejpam-4962	711	5	,	,	PUNCT
ejpam-4962	711	6	z	z	NOUN
ejpam-4962	711	7	}	}	PUNCT
ejpam-4962	711	8	⊆	⊆	NUM
ejpam-4962	711	9	ng(x	ng(x	NUM
ejpam-4962	711	10	)	)	PUNCT
ejpam-4962	711	11	and	and	CCONJ
ejpam-4962	711	12	{	{	PUNCT
ejpam-4962	711	13	x	x	NOUN
ejpam-4962	711	14	,	,	PUNCT
ejpam-4962	711	15	y	y	PROPN
ejpam-4962	711	16	,	,	PUNCT
ejpam-4962	711	17	z	z	NOUN
ejpam-4962	711	18	}	}	PUNCT
ejpam-4962	711	19	is	be	AUX
ejpam-4962	711	20	a	a	DET
ejpam-4962	711	21	ρ2	ρ2	NOUN
ejpam-4962	711	22	-	-	PUNCT
ejpam-4962	711	23	set	set	NOUN
ejpam-4962	711	24	in	in	ADP
ejpam-4962	711	25	g.	g.	PROPN
ejpam-4962	711	26	this	this	PRON
ejpam-4962	711	27	shows	show	VERB
ejpam-4962	711	28	that	that	SCONJ
ejpam-4962	711	29	(	(	PUNCT
ejpam-4962	711	30	i	i	NOUN
ejpam-4962	711	31	)	)	PUNCT
ejpam-4962	711	32	holds	hold	VERB
ejpam-4962	711	33	.	.	PUNCT
ejpam-4962	712	1	similarly	similarly	ADV
ejpam-4962	712	2	,	,	PUNCT
ejpam-4962	712	3	(	(	PUNCT
ejpam-4962	712	4	ii	ii	NOUN
ejpam-4962	712	5	)	)	PUNCT
ejpam-4962	712	6	holds	hold	VERB
ejpam-4962	712	7	if	if	SCONJ
ejpam-4962	712	8	v2	v2	PROPN
ejpam-4962	712	9	=	=	SYM
ejpam-4962	712	10	{	{	PUNCT
ejpam-4962	712	11	x	x	NOUN
ejpam-4962	712	12	}	}	PUNCT
ejpam-4962	712	13	⊆	⊆	NUM
ejpam-4962	712	14	v	v	NOUN
ejpam-4962	712	15	(	(	PUNCT
ejpam-4962	712	16	h	h	NOUN
ejpam-4962	712	17	)	)	PUNCT
ejpam-4962	712	18	.	.	PUNCT
ejpam-4962	713	1	conversely	conversely	ADV
ejpam-4962	713	2	,	,	PUNCT
ejpam-4962	713	3	suppose	suppose	VERB
ejpam-4962	713	4	(	(	PUNCT
ejpam-4962	713	5	i	i	NOUN
ejpam-4962	713	6	)	)	PUNCT
ejpam-4962	713	7	holds	hold	VERB
ejpam-4962	713	8	.	.	PUNCT
ejpam-4962	714	1	by	by	ADP
ejpam-4962	714	2	the	the	DET
ejpam-4962	714	3	preceding	precede	VERB
ejpam-4962	714	4	result	result	NOUN
ejpam-4962	714	5	,	,	PUNCT
ejpam-4962	714	6	γgr(g	γgr(g	PROPN
ejpam-4962	714	7	+	+	CCONJ
ejpam-4962	714	8	h	h	X
ejpam-4962	714	9	)	)	PUNCT
ejpam-4962	714	10	̸=	̸=	PROPN
ejpam-4962	714	11	3	3	NUM
ejpam-4962	714	12	.	.	PUNCT
ejpam-4962	715	1	thus	thus	ADV
ejpam-4962	715	2	,	,	PUNCT
ejpam-4962	715	3	γgr(g	γgr(g	PROPN
ejpam-4962	715	4	+	+	CCONJ
ejpam-4962	715	5	h	h	X
ejpam-4962	715	6	)	)	PUNCT
ejpam-4962	715	7	≥	≥	NOUN
ejpam-4962	715	8	4	4	NUM
ejpam-4962	715	9	.	.	PUNCT
ejpam-4962	715	10	let	let	VERB
ejpam-4962	715	11	v2	v2	VERB
ejpam-4962	715	12	=	=	SYM
ejpam-4962	715	13	{	{	PUNCT
ejpam-4962	715	14	x	x	NOUN
ejpam-4962	715	15	}	}	PUNCT
ejpam-4962	715	16	,	,	PUNCT
ejpam-4962	715	17	v1	v1	NOUN
ejpam-4962	715	18	=	=	SYM
ejpam-4962	715	19	{	{	PUNCT
ejpam-4962	715	20	y	y	PROPN
ejpam-4962	715	21	,	,	PUNCT
ejpam-4962	715	22	z	z	NOUN
ejpam-4962	715	23	}	}	PUNCT
ejpam-4962	715	24	and	and	CCONJ
ejpam-4962	715	25	v0	v0	PROPN
ejpam-4962	715	26	=	=	SYM
ejpam-4962	715	27	v	v	PROPN
ejpam-4962	715	28	(	(	PUNCT
ejpam-4962	715	29	g	g	PROPN
ejpam-4962	715	30	+	+	NOUN
ejpam-4962	715	31	h	h	NOUN
ejpam-4962	715	32	)	)	PUNCT
ejpam-4962	715	33	\	\	PUNCT
ejpam-4962	716	1	(	(	PUNCT
ejpam-4962	716	2	v1	v1	VERB
ejpam-4962	716	3	∪	∪	NOUN
ejpam-4962	716	4	v2	v2	NOUN
ejpam-4962	716	5	)	)	PUNCT
ejpam-4962	716	6	.	.	PUNCT
ejpam-4962	717	1	then	then	ADV
ejpam-4962	717	2	f	f	PROPN
ejpam-4962	717	3	=	=	SYM
ejpam-4962	717	4	(	(	PUNCT
ejpam-4962	717	5	v0	v0	PROPN
ejpam-4962	717	6	,	,	PUNCT
ejpam-4962	717	7	v1	v1	NOUN
ejpam-4962	717	8	,	,	PUNCT
ejpam-4962	717	9	v2	v2	PROPN
ejpam-4962	717	10	)	)	PUNCT
ejpam-4962	717	11	is	be	AUX
ejpam-4962	717	12	a	a	DET
ejpam-4962	717	13	grdf	grdf	NOUN
ejpam-4962	717	14	on	on	ADP
ejpam-4962	717	15	g	g	PROPN
ejpam-4962	717	16	+	+	PROPN
ejpam-4962	717	17	h.	h.	PROPN
ejpam-4962	717	18	hence	hence	ADV
ejpam-4962	717	19	,	,	PUNCT
ejpam-4962	717	20	γgr(g	γgr(g	PROPN
ejpam-4962	717	21	+	+	CCONJ
ejpam-4962	717	22	h	h	NOUN
ejpam-4962	717	23	)	)	PUNCT
ejpam-4962	717	24	≤	≤	NUM
ejpam-4962	717	25	ωgr	ωgr	SYM
ejpam-4962	717	26	g+h(f	g+h(f	NOUN
ejpam-4962	717	27	)	)	PUNCT
ejpam-4962	717	28	=	=	SYM
ejpam-4962	718	1	4	4	X
ejpam-4962	718	2	.	.	PUNCT
ejpam-4962	718	3	therefore	therefore	ADV
ejpam-4962	718	4	,	,	PUNCT
ejpam-4962	718	5	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	718	6	)	)	PUNCT
ejpam-4962	718	7	=	=	SYM
ejpam-4962	719	1	4	4	X
ejpam-4962	719	2	.	.	PUNCT
ejpam-4962	719	3	the	the	DET
ejpam-4962	719	4	same	same	ADJ
ejpam-4962	719	5	conclusion	conclusion	NOUN
ejpam-4962	719	6	holds	hold	VERB
ejpam-4962	719	7	if	if	SCONJ
ejpam-4962	719	8	(	(	PUNCT
ejpam-4962	719	9	ii	ii	NOUN
ejpam-4962	719	10	)	)	PUNCT
ejpam-4962	719	11	holds	hold	VERB
ejpam-4962	719	12	.	.	PUNCT
ejpam-4962	720	1	theorem	theorem	NOUN
ejpam-4962	720	2	10	10	NUM
ejpam-4962	720	3	.	.	PUNCT
ejpam-4962	721	1	let	let	VERB
ejpam-4962	721	2	g	g	NOUN
ejpam-4962	721	3	and	and	CCONJ
ejpam-4962	721	4	h	h	NOUN
ejpam-4962	721	5	be	be	AUX
ejpam-4962	721	6	non	non	ADJ
ejpam-4962	721	7	-	-	ADJ
ejpam-4962	721	8	complete	complete	ADJ
ejpam-4962	721	9	graphs	graph	NOUN
ejpam-4962	721	10	such	such	ADJ
ejpam-4962	721	11	that	that	DET
ejpam-4962	721	12	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	721	13	)	)	PUNCT
ejpam-4962	721	14	/∈	/∈	PUNCT
ejpam-4962	722	1	{	{	PUNCT
ejpam-4962	722	2	3	3	NUM
ejpam-4962	722	3	,	,	PUNCT
ejpam-4962	722	4	4	4	NUM
ejpam-4962	722	5	}	}	PUNCT
ejpam-4962	722	6	.	.	PUNCT
ejpam-4962	723	1	then	then	ADV
ejpam-4962	723	2	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	723	3	)	)	PUNCT
ejpam-4962	723	4	=	=	SYM
ejpam-4962	724	1	5	5	NUM
ejpam-4962	725	1	if	if	SCONJ
ejpam-4962	725	2	and	and	CCONJ
ejpam-4962	725	3	only	only	ADV
ejpam-4962	725	4	if	if	SCONJ
ejpam-4962	725	5	one	one	NUM
ejpam-4962	725	6	of	of	ADP
ejpam-4962	725	7	the	the	DET
ejpam-4962	725	8	following	follow	VERB
ejpam-4962	725	9	holds	hold	VERB
ejpam-4962	725	10	:	:	PUNCT
ejpam-4962	725	11	(	(	PUNCT
ejpam-4962	725	12	i	i	NOUN
ejpam-4962	725	13	)	)	PUNCT
ejpam-4962	725	14	γ(g	γ(g	PROPN
ejpam-4962	725	15	)	)	PUNCT
ejpam-4962	725	16	=	=	SYM
ejpam-4962	725	17	1	1	NUM
ejpam-4962	725	18	and	and	CCONJ
ejpam-4962	725	19	ρ2(h	ρ2(h	NUM
ejpam-4962	725	20	)	)	PUNCT
ejpam-4962	725	21	=	=	SYM
ejpam-4962	725	22	3	3	NUM
ejpam-4962	725	23	(	(	PUNCT
ejpam-4962	725	24	ii	ii	NOUN
ejpam-4962	725	25	)	)	PUNCT
ejpam-4962	725	26	γ(h	γ(h	NOUN
ejpam-4962	725	27	)	)	PUNCT
ejpam-4962	725	28	=	=	SYM
ejpam-4962	725	29	1	1	NUM
ejpam-4962	725	30	and	and	CCONJ
ejpam-4962	725	31	ρ2(g	ρ2(g	NUM
ejpam-4962	725	32	)	)	PUNCT
ejpam-4962	725	33	=	=	SYM
ejpam-4962	725	34	3	3	NUM
ejpam-4962	725	35	(	(	PUNCT
ejpam-4962	725	36	iii	iii	NOUN
ejpam-4962	725	37	)	)	PUNCT
ejpam-4962	725	38	there	there	PRON
ejpam-4962	725	39	exists	exist	VERB
ejpam-4962	725	40	nonadjacent	nonadjacent	ADJ
ejpam-4962	725	41	vertices	vertex	NOUN
ejpam-4962	725	42	v	v	ADP
ejpam-4962	725	43	,	,	PUNCT
ejpam-4962	725	44	w	w	PROPN
ejpam-4962	725	45	∈	∈	PROPN
ejpam-4962	725	46	v	v	ADP
ejpam-4962	725	47	(	(	PUNCT
ejpam-4962	725	48	g	g	NOUN
ejpam-4962	725	49	)	)	PUNCT
ejpam-4962	725	50	and	and	CCONJ
ejpam-4962	725	51	x	x	X
ejpam-4962	725	52	,	,	PUNCT
ejpam-4962	725	53	y	y	PROPN
ejpam-4962	725	54	∈	∈	PROPN
ejpam-4962	725	55	v	v	ADP
ejpam-4962	725	56	(	(	PUNCT
ejpam-4962	725	57	h	h	NOUN
ejpam-4962	725	58	)	)	PUNCT
ejpam-4962	725	59	such	such	ADJ
ejpam-4962	725	60	that	that	PRON
ejpam-4962	725	61	v	v	NOUN
ejpam-4962	725	62	(	(	PUNCT
ejpam-4962	725	63	g	g	NOUN
ejpam-4962	725	64	)	)	PUNCT
ejpam-4962	725	65	\	\	NOUN
ejpam-4962	725	66	{	{	PUNCT
ejpam-4962	725	67	v	v	NOUN
ejpam-4962	725	68	,	,	PUNCT
ejpam-4962	725	69	w	w	NOUN
ejpam-4962	725	70	}	}	PUNCT
ejpam-4962	725	71	⊆	⊆	NUM
ejpam-4962	725	72	ng(v	ng(v	NUM
ejpam-4962	725	73	)	)	PUNCT
ejpam-4962	725	74	.	.	PUNCT
ejpam-4962	726	1	(	(	PUNCT
ejpam-4962	726	2	iv	iv	X
ejpam-4962	726	3	)	)	PUNCT
ejpam-4962	726	4	there	there	PRON
ejpam-4962	726	5	exists	exist	VERB
ejpam-4962	726	6	nonadjacent	nonadjacent	ADJ
ejpam-4962	726	7	vertices	vertex	NOUN
ejpam-4962	726	8	v	v	ADP
ejpam-4962	726	9	,	,	PUNCT
ejpam-4962	726	10	w	w	PROPN
ejpam-4962	726	11	∈	∈	PROPN
ejpam-4962	726	12	v	v	ADP
ejpam-4962	726	13	(	(	PUNCT
ejpam-4962	726	14	g	g	NOUN
ejpam-4962	726	15	)	)	PUNCT
ejpam-4962	726	16	and	and	CCONJ
ejpam-4962	726	17	x	x	X
ejpam-4962	726	18	,	,	PUNCT
ejpam-4962	726	19	y	y	PROPN
ejpam-4962	726	20	∈	∈	PROPN
ejpam-4962	726	21	v	v	ADP
ejpam-4962	726	22	(	(	PUNCT
ejpam-4962	726	23	h	h	NOUN
ejpam-4962	726	24	)	)	PUNCT
ejpam-4962	726	25	such	such	ADJ
ejpam-4962	726	26	that	that	PRON
ejpam-4962	726	27	v	v	NOUN
ejpam-4962	726	28	(	(	PUNCT
ejpam-4962	726	29	h	h	NOUN
ejpam-4962	726	30	)	)	PUNCT
ejpam-4962	726	31	\	\	NOUN
ejpam-4962	726	32	{	{	PUNCT
ejpam-4962	726	33	x	x	NOUN
ejpam-4962	726	34	,	,	PUNCT
ejpam-4962	726	35	y	y	PROPN
ejpam-4962	726	36	}	}	PUNCT
ejpam-4962	726	37	⊆	⊆	NUM
ejpam-4962	726	38	nh(x	nh(x	NUM
ejpam-4962	726	39	)	)	PUNCT
ejpam-4962	726	40	.	.	PUNCT
ejpam-4962	727	1	(	(	PUNCT
ejpam-4962	727	2	v	v	X
ejpam-4962	727	3	)	)	PUNCT
ejpam-4962	727	4	there	there	PRON
ejpam-4962	727	5	exist	exist	VERB
ejpam-4962	727	6	v	v	ADP
ejpam-4962	727	7	,	,	PUNCT
ejpam-4962	727	8	w	w	PROPN
ejpam-4962	727	9	,	,	PUNCT
ejpam-4962	727	10	x	x	PRON
ejpam-4962	727	11	,	,	PUNCT
ejpam-4962	727	12	y	y	PROPN
ejpam-4962	727	13	∈	∈	PROPN
ejpam-4962	727	14	v	v	ADP
ejpam-4962	727	15	(	(	PUNCT
ejpam-4962	727	16	g	g	NOUN
ejpam-4962	727	17	)	)	PUNCT
ejpam-4962	728	1	such	such	ADJ
ejpam-4962	728	2	that	that	PRON
ejpam-4962	728	3	v	v	NOUN
ejpam-4962	728	4	(	(	PUNCT
ejpam-4962	728	5	g	g	NOUN
ejpam-4962	728	6	)	)	PUNCT
ejpam-4962	728	7	\	\	NOUN
ejpam-4962	728	8	{	{	PUNCT
ejpam-4962	728	9	v	v	NOUN
ejpam-4962	728	10	,	,	PUNCT
ejpam-4962	728	11	w	w	PROPN
ejpam-4962	728	12	,	,	PUNCT
ejpam-4962	728	13	x	x	NOUN
ejpam-4962	728	14	,	,	PUNCT
ejpam-4962	728	15	y	y	PROPN
ejpam-4962	728	16	}	}	PUNCT
ejpam-4962	728	17	⊆	⊆	NUM
ejpam-4962	728	18	ng(v	ng(v	NUM
ejpam-4962	728	19	)	)	PUNCT
ejpam-4962	728	20	.	.	PUNCT
ejpam-4962	729	1	(	(	PUNCT
ejpam-4962	729	2	vi	vi	X
ejpam-4962	729	3	)	)	PUNCT
ejpam-4962	729	4	there	there	PRON
ejpam-4962	729	5	exist	exist	VERB
ejpam-4962	729	6	v	v	ADP
ejpam-4962	729	7	,	,	PUNCT
ejpam-4962	729	8	w	w	PROPN
ejpam-4962	729	9	,	,	PUNCT
ejpam-4962	729	10	x	x	PRON
ejpam-4962	729	11	,	,	PUNCT
ejpam-4962	729	12	y	y	PROPN
ejpam-4962	729	13	∈	∈	PROPN
ejpam-4962	729	14	v	v	ADP
ejpam-4962	729	15	(	(	PUNCT
ejpam-4962	729	16	h	h	NOUN
ejpam-4962	729	17	)	)	PUNCT
ejpam-4962	729	18	such	such	ADJ
ejpam-4962	729	19	that	that	DET
ejpam-4962	729	20	v	v	NOUN
ejpam-4962	729	21	(	(	PUNCT
ejpam-4962	729	22	h	h	NOUN
ejpam-4962	729	23	)	)	PUNCT
ejpam-4962	729	24	\	\	NOUN
ejpam-4962	729	25	{	{	PUNCT
ejpam-4962	729	26	v	v	NOUN
ejpam-4962	729	27	,	,	PUNCT
ejpam-4962	729	28	w	w	PROPN
ejpam-4962	729	29	,	,	PUNCT
ejpam-4962	729	30	x	x	NOUN
ejpam-4962	729	31	,	,	PUNCT
ejpam-4962	729	32	y	y	PROPN
ejpam-4962	729	33	}	}	PUNCT
ejpam-4962	729	34	⊆	⊆	NUM
ejpam-4962	729	35	nh(v	nh(v	NOUN
ejpam-4962	729	36	)	)	PUNCT
ejpam-4962	729	37	.	.	PUNCT
ejpam-4962	730	1	(	(	PUNCT
ejpam-4962	730	2	vii	vii	PROPN
ejpam-4962	730	3	)	)	PUNCT
ejpam-4962	730	4	there	there	PRON
ejpam-4962	730	5	exist	exist	VERB
ejpam-4962	730	6	v	v	ADP
ejpam-4962	730	7	,	,	PUNCT
ejpam-4962	730	8	w	w	PROPN
ejpam-4962	730	9	,	,	PUNCT
ejpam-4962	730	10	x	x	SYM
ejpam-4962	730	11	∈	∈	NOUN
ejpam-4962	730	12	v	v	ADP
ejpam-4962	730	13	(	(	PUNCT
ejpam-4962	730	14	g	g	NOUN
ejpam-4962	730	15	)	)	PUNCT
ejpam-4962	731	1	such	such	ADJ
ejpam-4962	731	2	that	that	PRON
ejpam-4962	731	3	v	v	NOUN
ejpam-4962	731	4	(	(	PUNCT
ejpam-4962	731	5	g	g	NOUN
ejpam-4962	731	6	)	)	PUNCT
ejpam-4962	731	7	\	\	NOUN
ejpam-4962	731	8	{	{	PUNCT
ejpam-4962	731	9	v	v	NOUN
ejpam-4962	731	10	,	,	PUNCT
ejpam-4962	731	11	w	w	PROPN
ejpam-4962	731	12	,	,	PUNCT
ejpam-4962	731	13	x	x	NOUN
ejpam-4962	731	14	}	}	PUNCT
ejpam-4962	731	15	⊆	⊆	NUM
ejpam-4962	731	16	ng({v	ng({v	NUM
ejpam-4962	731	17	,	,	PUNCT
ejpam-4962	731	18	w	w	NOUN
ejpam-4962	731	19	}	}	PUNCT
ejpam-4962	731	20	)	)	PUNCT
ejpam-4962	731	21	.	.	PUNCT
ejpam-4962	732	1	(	(	PUNCT
ejpam-4962	732	2	viii	viii	NOUN
ejpam-4962	732	3	)	)	PUNCT
ejpam-4962	732	4	there	there	PRON
ejpam-4962	732	5	exist	exist	VERB
ejpam-4962	732	6	v	v	ADP
ejpam-4962	732	7	,	,	PUNCT
ejpam-4962	732	8	w	w	PROPN
ejpam-4962	732	9	,	,	PUNCT
ejpam-4962	732	10	x	x	SYM
ejpam-4962	732	11	∈	∈	NOUN
ejpam-4962	732	12	v	v	ADP
ejpam-4962	732	13	(	(	PUNCT
ejpam-4962	732	14	h	h	NOUN
ejpam-4962	732	15	)	)	PUNCT
ejpam-4962	732	16	such	such	ADJ
ejpam-4962	732	17	that	that	DET
ejpam-4962	732	18	v	v	NOUN
ejpam-4962	732	19	(	(	PUNCT
ejpam-4962	732	20	h	h	NOUN
ejpam-4962	732	21	)	)	PUNCT
ejpam-4962	732	22	\	\	NOUN
ejpam-4962	732	23	{	{	PUNCT
ejpam-4962	732	24	v	v	NOUN
ejpam-4962	732	25	,	,	PUNCT
ejpam-4962	732	26	w	w	PROPN
ejpam-4962	732	27	,	,	PUNCT
ejpam-4962	732	28	x	x	NOUN
ejpam-4962	732	29	}	}	PUNCT
ejpam-4962	732	30	⊆	⊆	NUM
ejpam-4962	732	31	nh({v	nh({v	NOUN
ejpam-4962	732	32	,	,	PUNCT
ejpam-4962	732	33	w	w	NOUN
ejpam-4962	732	34	}	}	PUNCT
ejpam-4962	732	35	)	)	PUNCT
ejpam-4962	732	36	.	.	PUNCT
ejpam-4962	733	1	proof	proof	NOUN
ejpam-4962	733	2	.	.	PUNCT
ejpam-4962	734	1	let	let	VERB
ejpam-4962	734	2	g	g	NOUN
ejpam-4962	734	3	and	and	CCONJ
ejpam-4962	734	4	h	h	NOUN
ejpam-4962	734	5	be	be	AUX
ejpam-4962	734	6	non	non	ADJ
ejpam-4962	734	7	-	-	ADJ
ejpam-4962	734	8	complete	complete	ADJ
ejpam-4962	734	9	graphs	graph	NOUN
ejpam-4962	734	10	such	such	ADJ
ejpam-4962	734	11	that	that	DET
ejpam-4962	734	12	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	734	13	)	)	PUNCT
ejpam-4962	734	14	̸=	̸=	PROPN
ejpam-4962	734	15	{	{	PUNCT
ejpam-4962	734	16	3	3	NUM
ejpam-4962	734	17	,	,	PUNCT
ejpam-4962	734	18	4	4	NUM
ejpam-4962	734	19	}	}	PUNCT
ejpam-4962	734	20	.	.	PUNCT
ejpam-4962	735	1	suppose	suppose	VERB
ejpam-4962	735	2	that	that	SCONJ
ejpam-4962	735	3	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	735	4	)	)	PUNCT
ejpam-4962	735	5	=	=	SYM
ejpam-4962	736	1	5	5	X
ejpam-4962	736	2	.	.	PUNCT
ejpam-4962	736	3	let	let	VERB
ejpam-4962	736	4	f	f	PROPN
ejpam-4962	736	5	=	=	SYM
ejpam-4962	736	6	(	(	PUNCT
ejpam-4962	736	7	v0	v0	PROPN
ejpam-4962	736	8	,	,	PUNCT
ejpam-4962	736	9	v1	v1	NOUN
ejpam-4962	736	10	,	,	PUNCT
ejpam-4962	736	11	v2	v2	PROPN
ejpam-4962	736	12	)	)	PUNCT
ejpam-4962	736	13	be	be	AUX
ejpam-4962	736	14	a	a	DET
ejpam-4962	736	15	γgr	γgr	NOUN
ejpam-4962	736	16	-	-	PUNCT
ejpam-4962	736	17	function	function	NOUN
ejpam-4962	736	18	ong+h	ong+h	PROPN
ejpam-4962	736	19	.	.	PUNCT
ejpam-4962	737	1	then	then	ADV
ejpam-4962	737	2	|v1|+2|v2|	|v1|+2|v2|	NUM
ejpam-4962	737	3	=	=	SYM
ejpam-4962	737	4	5	5	X
ejpam-4962	737	5	.	.	PUNCT
ejpam-4962	737	6	suppose	suppose	VERB
ejpam-4962	737	7	that	that	SCONJ
ejpam-4962	737	8	|v2|	|v2|	NOUN
ejpam-4962	737	9	=	=	SYM
ejpam-4962	737	10	0	0	X
ejpam-4962	737	11	.	.	PUNCT
ejpam-4962	738	1	then	then	ADV
ejpam-4962	738	2	|v1|	|v1|	NOUN
ejpam-4962	738	3	=	=	SYM
ejpam-4962	738	4	|v	|v	PROPN
ejpam-4962	738	5	(	(	PUNCT
ejpam-4962	738	6	g	g	PROPN
ejpam-4962	738	7	+	+	CCONJ
ejpam-4962	738	8	h)|	h)|	NOUN
ejpam-4962	738	9	=	=	ADJ
ejpam-4962	738	10	5	5	NUM
ejpam-4962	738	11	.	.	PUNCT
ejpam-4962	739	1	since	since	SCONJ
ejpam-4962	739	2	g	g	PROPN
ejpam-4962	739	3	and	and	CCONJ
ejpam-4962	739	4	h	h	NOUN
ejpam-4962	739	5	are	be	AUX
ejpam-4962	739	6	non	non	ADJ
ejpam-4962	739	7	-	-	ADJ
ejpam-4962	739	8	complete	complete	ADJ
ejpam-4962	739	9	graphs	graph	NOUN
ejpam-4962	739	10	and	and	CCONJ
ejpam-4962	739	11	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	739	12	)	)	PUNCT
ejpam-4962	740	1	̸=	̸=	PROPN
ejpam-4962	740	2	{	{	PUNCT
ejpam-4962	740	3	3	3	NUM
ejpam-4962	740	4	,	,	PUNCT
ejpam-4962	740	5	4	4	NUM
ejpam-4962	740	6	}	}	PUNCT
ejpam-4962	740	7	,	,	PUNCT
ejpam-4962	740	8	this	this	PRON
ejpam-4962	740	9	is	be	AUX
ejpam-4962	740	10	not	not	PART
ejpam-4962	740	11	possible	possible	ADJ
ejpam-4962	740	12	.	.	PUNCT
ejpam-4962	741	1	suppose	suppose	VERB
ejpam-4962	741	2	|v2|	|v2|	NOUN
ejpam-4962	741	3	=	=	SYM
ejpam-4962	741	4	1	1	X
ejpam-4962	741	5	,	,	PUNCT
ejpam-4962	741	6	say	say	VERB
ejpam-4962	741	7	v2	v2	NOUN
ejpam-4962	741	8	=	=	SYM
ejpam-4962	741	9	{	{	PUNCT
ejpam-4962	741	10	v	v	NOUN
ejpam-4962	741	11	}	}	PUNCT
ejpam-4962	741	12	.	.	PUNCT
ejpam-4962	742	1	then	then	ADV
ejpam-4962	742	2	|v1|	|v1|	NOUN
ejpam-4962	742	3	=	=	SYM
ejpam-4962	742	4	3	3	X
ejpam-4962	742	5	.	.	X
ejpam-4962	742	6	assume	assume	VERB
ejpam-4962	742	7	that	that	SCONJ
ejpam-4962	742	8	v2	v2	PROPN
ejpam-4962	742	9	=	=	SYM
ejpam-4962	742	10	{	{	PUNCT
ejpam-4962	742	11	v	v	NOUN
ejpam-4962	742	12	}	}	PUNCT
ejpam-4962	742	13	⊆	⊆	NUM
ejpam-4962	742	14	v	v	NOUN
ejpam-4962	742	15	(	(	PUNCT
ejpam-4962	742	16	g	g	NOUN
ejpam-4962	742	17	)	)	PUNCT
ejpam-4962	742	18	.	.	PUNCT
ejpam-4962	743	1	if	if	SCONJ
ejpam-4962	743	2	v1	v1	VERB
ejpam-4962	743	3	⊆	⊆	NUM
ejpam-4962	743	4	v	v	NOUN
ejpam-4962	743	5	(	(	PUNCT
ejpam-4962	743	6	h	h	NOUN
ejpam-4962	743	7	)	)	PUNCT
ejpam-4962	743	8	.	.	PUNCT
ejpam-4962	744	1	then	then	ADV
ejpam-4962	744	2	v	v	X
ejpam-4962	744	3	(	(	PUNCT
ejpam-4962	744	4	g	g	NOUN
ejpam-4962	744	5	)	)	PUNCT
ejpam-4962	744	6	\	\	NOUN
ejpam-4962	745	1	{	{	PUNCT
ejpam-4962	745	2	v	v	NOUN
ejpam-4962	745	3	}	}	PUNCT
ejpam-4962	745	4	⊆	⊆	NUM
ejpam-4962	745	5	ng(v	ng(v	NOUN
ejpam-4962	745	6	)	)	PUNCT
ejpam-4962	745	7	.	.	PUNCT
ejpam-4962	746	1	this	this	PRON
ejpam-4962	746	2	implies	imply	VERB
ejpam-4962	746	3	that	that	SCONJ
ejpam-4962	746	4	γ(g	γ(g	PROPN
ejpam-4962	746	5	)	)	PUNCT
ejpam-4962	746	6	=	=	PUNCT
ejpam-4962	747	1	1	1	X
ejpam-4962	747	2	.	.	PUNCT
ejpam-4962	747	3	since	since	SCONJ
ejpam-4962	747	4	v1	v1	NOUN
ejpam-4962	747	5	is	be	AUX
ejpam-4962	747	6	a	a	DET
ejpam-4962	747	7	2	2	NUM
ejpam-4962	747	8	-	-	PUNCT
ejpam-4962	747	9	path	path	NOUN
ejpam-4962	747	10	closure	closure	NOUN
ejpam-4962	747	11	absoring	absore	VERB
ejpam-4962	747	12	set	set	VERB
ejpam-4962	747	13	in	in	ADP
ejpam-4962	747	14	h	h	NOUN
ejpam-4962	747	15	and	and	CCONJ
ejpam-4962	747	16	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	747	17	)	)	PUNCT
ejpam-4962	747	18	̸=	̸=	PROPN
ejpam-4962	747	19	3	3	NUM
ejpam-4962	747	20	,	,	PUNCT
ejpam-4962	747	21	v1	v1	NOUN
ejpam-4962	747	22	is	be	AUX
ejpam-4962	747	23	a	a	DET
ejpam-4962	747	24	ρ2	ρ2	NOUN
ejpam-4962	747	25	-	-	PUNCT
ejpam-4962	747	26	set	set	NOUN
ejpam-4962	747	27	in	in	ADP
ejpam-4962	747	28	h.	h.	PROPN
ejpam-4962	747	29	hence	hence	ADV
ejpam-4962	747	30	,	,	PUNCT
ejpam-4962	747	31	ρ2(h	ρ2(h	X
ejpam-4962	747	32	)	)	PUNCT
ejpam-4962	747	33	=	=	SYM
ejpam-4962	747	34	3	3	NUM
ejpam-4962	747	35	and	and	CCONJ
ejpam-4962	747	36	(	(	PUNCT
ejpam-4962	747	37	i	i	NOUN
ejpam-4962	747	38	)	)	PUNCT
ejpam-4962	747	39	holds	hold	VERB
ejpam-4962	747	40	.	.	PUNCT
ejpam-4962	748	1	suppose	suppose	VERB
ejpam-4962	748	2	|v1	|v1	PROPN
ejpam-4962	748	3	∩	∩	PROPN
ejpam-4962	748	4	v	v	X
ejpam-4962	748	5	(	(	PUNCT
ejpam-4962	748	6	g)|	g)|	NOUN
ejpam-4962	748	7	=	=	SYM
ejpam-4962	748	8	1	1	NUM
ejpam-4962	748	9	,	,	PUNCT
ejpam-4962	748	10	say	say	VERB
ejpam-4962	748	11	w	w	PROPN
ejpam-4962	748	12	∈	∈	PROPN
ejpam-4962	748	13	v	v	ADP
ejpam-4962	748	14	∩	∩	ADJ
ejpam-4962	748	15	v	v	NOUN
ejpam-4962	748	16	(	(	PUNCT
ejpam-4962	748	17	g	g	NOUN
ejpam-4962	748	18	)	)	PUNCT
ejpam-4962	748	19	.	.	PUNCT
ejpam-4962	749	1	then	then	ADV
ejpam-4962	749	2	v	v	X
ejpam-4962	749	3	(	(	PUNCT
ejpam-4962	749	4	g	g	NOUN
ejpam-4962	749	5	)	)	PUNCT
ejpam-4962	749	6	\	\	NOUN
ejpam-4962	750	1	{	{	PUNCT
ejpam-4962	750	2	v	v	NOUN
ejpam-4962	750	3	,	,	PUNCT
ejpam-4962	750	4	w	w	NOUN
ejpam-4962	750	5	}	}	PUNCT
ejpam-4962	750	6	⊆	⊆	NUM
ejpam-4962	750	7	ng(v	ng(v	NUM
ejpam-4962	750	8	)	)	PUNCT
ejpam-4962	750	9	.	.	PUNCT
ejpam-4962	751	1	since	since	SCONJ
ejpam-4962	751	2	h	h	PROPN
ejpam-4962	751	3	is	be	AUX
ejpam-4962	751	4	non	non	ADJ
ejpam-4962	751	5	-	-	ADJ
ejpam-4962	751	6	complete	complete	ADJ
ejpam-4962	751	7	and	and	CCONJ
ejpam-4962	751	8	ρ2(h	ρ2(h	NUM
ejpam-4962	751	9	)	)	PUNCT
ejpam-4962	751	10	̸=	̸=	PROPN
ejpam-4962	751	11	2	2	NUM
ejpam-4962	751	12	,	,	PUNCT
ejpam-4962	751	13	vw	vw	PROPN
ejpam-4962	751	14	/∈	/∈	PUNCT
ejpam-4962	751	15	e(g	e(g	PROPN
ejpam-4962	751	16	)	)	PUNCT
ejpam-4962	751	17	.	.	PUNCT
ejpam-4962	752	1	since	since	SCONJ
ejpam-4962	752	2	ρ2(g	ρ2(g	NUM
ejpam-4962	752	3	)	)	PUNCT
ejpam-4962	752	4	̸=	̸=	PROPN
ejpam-4962	752	5	2	2	NUM
ejpam-4962	752	6	,	,	PUNCT
ejpam-4962	752	7	xy	xy	PROPN
ejpam-4962	752	8	/∈	/∈	PUNCT
ejpam-4962	752	9	e(h	e(h	PROPN
ejpam-4962	752	10	)	)	PUNCT
ejpam-4962	752	11	.	.	PUNCT
ejpam-4962	753	1	this	this	PRON
ejpam-4962	753	2	shows	show	VERB
ejpam-4962	753	3	that	that	SCONJ
ejpam-4962	753	4	(	(	PUNCT
ejpam-4962	753	5	iii	iii	NOUN
ejpam-4962	753	6	)	)	PUNCT
ejpam-4962	753	7	holds	hold	VERB
ejpam-4962	753	8	.	.	PUNCT
ejpam-4962	754	1	next	next	ADV
ejpam-4962	754	2	,	,	PUNCT
ejpam-4962	754	3	suppose	suppose	VERB
ejpam-4962	754	4	that	that	SCONJ
ejpam-4962	754	5	references	reference	NOUN
ejpam-4962	754	6	2381	2381	NUM
ejpam-4962	754	7	|v1	|v1	NUM
ejpam-4962	754	8	∩	∩	PROPN
ejpam-4962	754	9	v	v	X
ejpam-4962	754	10	(	(	PUNCT
ejpam-4962	754	11	g)|	g)|	VERB
ejpam-4962	754	12	≥	≥	NOUN
ejpam-4962	754	13	2	2	NUM
ejpam-4962	754	14	.	.	PUNCT
ejpam-4962	754	15	since	since	SCONJ
ejpam-4962	754	16	γgr(g+h	γgr(g+h	NOUN
ejpam-4962	754	17	)	)	PUNCT
ejpam-4962	754	18	̸=	̸=	PROPN
ejpam-4962	754	19	4	4	NUM
ejpam-4962	754	20	,	,	PUNCT
ejpam-4962	754	21	it	it	PRON
ejpam-4962	754	22	follows	follow	VERB
ejpam-4962	754	23	that	that	SCONJ
ejpam-4962	754	24	v1	v1	NOUN
ejpam-4962	754	25	⊆	⊆	NUM
ejpam-4962	754	26	v	v	NOUN
ejpam-4962	754	27	(	(	PUNCT
ejpam-4962	754	28	g	g	NOUN
ejpam-4962	754	29	)	)	PUNCT
ejpam-4962	754	30	,	,	PUNCT
ejpam-4962	754	31	i.e.	i.e.	X
ejpam-4962	754	32	|v1	|v1	PRON
ejpam-4962	754	33	∩	∩	ADJ
ejpam-4962	754	34	v	v	NOUN
ejpam-4962	754	35	(	(	PUNCT
ejpam-4962	754	36	g)|	g)|	NOUN
ejpam-4962	754	37	=	=	SYM
ejpam-4962	754	38	3	3	X
ejpam-4962	754	39	.	.	PUNCT
ejpam-4962	754	40	clearly	clearly	ADV
ejpam-4962	754	41	,	,	PUNCT
ejpam-4962	754	42	v	v	X
ejpam-4962	754	43	(	(	PUNCT
ejpam-4962	754	44	g	g	NOUN
ejpam-4962	754	45	)	)	PUNCT
ejpam-4962	754	46	\	\	NOUN
ejpam-4962	754	47	{	{	PUNCT
ejpam-4962	754	48	v	v	NOUN
ejpam-4962	754	49	,	,	PUNCT
ejpam-4962	754	50	w	w	PROPN
ejpam-4962	754	51	,	,	PUNCT
ejpam-4962	754	52	x	x	NOUN
ejpam-4962	754	53	,	,	PUNCT
ejpam-4962	754	54	y	y	PROPN
ejpam-4962	754	55	}	}	PUNCT
ejpam-4962	754	56	⊆	⊆	NUM
ejpam-4962	754	57	ng(v	ng(v	NOUN
ejpam-4962	754	58	)	)	PUNCT
ejpam-4962	754	59	,	,	PUNCT
ejpam-4962	754	60	showing	show	VERB
ejpam-4962	754	61	that	that	SCONJ
ejpam-4962	754	62	(	(	PUNCT
ejpam-4962	754	63	v	v	NOUN
ejpam-4962	754	64	)	)	PUNCT
ejpam-4962	754	65	holds	hold	NOUN
ejpam-4962	754	66	.	.	PUNCT
ejpam-4962	755	1	suppose	suppose	VERB
ejpam-4962	755	2	now	now	ADV
ejpam-4962	755	3	that	that	DET
ejpam-4962	755	4	|v2|	|v2|	NOUN
ejpam-4962	755	5	=	=	SYM
ejpam-4962	755	6	2	2	NUM
ejpam-4962	755	7	,	,	PUNCT
ejpam-4962	755	8	say	say	VERB
ejpam-4962	755	9	v2	v2	NOUN
ejpam-4962	755	10	=	=	SYM
ejpam-4962	755	11	{	{	PUNCT
ejpam-4962	755	12	v	v	NOUN
ejpam-4962	755	13	,	,	PUNCT
ejpam-4962	755	14	w	w	NOUN
ejpam-4962	755	15	}	}	PUNCT
ejpam-4962	755	16	.	.	PUNCT
ejpam-4962	756	1	then	then	ADV
ejpam-4962	756	2	|v1|	|v1|	NOUN
ejpam-4962	756	3	=	=	SYM
ejpam-4962	756	4	1	1	X
ejpam-4962	756	5	.	.	PUNCT
ejpam-4962	756	6	let	let	VERB
ejpam-4962	756	7	v1	v1	VERB
ejpam-4962	756	8	=	=	SYM
ejpam-4962	756	9	{	{	PUNCT
ejpam-4962	756	10	x	x	NOUN
ejpam-4962	756	11	}	}	PUNCT
ejpam-4962	756	12	.	.	PUNCT
ejpam-4962	757	1	assume	assume	VERB
ejpam-4962	757	2	that	that	SCONJ
ejpam-4962	757	3	v2∩v	v2∩v	PROPN
ejpam-4962	757	4	(	(	PUNCT
ejpam-4962	757	5	g	g	NOUN
ejpam-4962	757	6	)	)	PUNCT
ejpam-4962	757	7	̸=	̸=	NOUN
ejpam-4962	757	8	∅	∅	NOUN
ejpam-4962	757	9	,	,	PUNCT
ejpam-4962	757	10	say	say	VERB
ejpam-4962	757	11	v	v	NUM
ejpam-4962	757	12	∈	∈	PROPN
ejpam-4962	757	13	v	v	NOUN
ejpam-4962	757	14	(	(	PUNCT
ejpam-4962	757	15	g	g	NOUN
ejpam-4962	757	16	)	)	PUNCT
ejpam-4962	757	17	.	.	PUNCT
ejpam-4962	758	1	since	since	SCONJ
ejpam-4962	758	2	ρ2(g	ρ2(g	NUM
ejpam-4962	758	3	)	)	PUNCT
ejpam-4962	758	4	̸=	̸=	PROPN
ejpam-4962	758	5	2	2	NUM
ejpam-4962	758	6	and	and	CCONJ
ejpam-4962	758	7	ρ2(h	ρ2(h	NUM
ejpam-4962	758	8	)	)	PUNCT
ejpam-4962	758	9	̸=	̸=	PROPN
ejpam-4962	758	10	2	2	NUM
ejpam-4962	758	11	,	,	PUNCT
ejpam-4962	758	12	{	{	PUNCT
ejpam-4962	758	13	v	v	NOUN
ejpam-4962	758	14	,	,	PUNCT
ejpam-4962	758	15	w	w	PROPN
ejpam-4962	758	16	,	,	PUNCT
ejpam-4962	758	17	x	x	NOUN
ejpam-4962	758	18	}	}	PUNCT
ejpam-4962	758	19	⊆	⊆	NUM
ejpam-4962	758	20	v	v	NOUN
ejpam-4962	758	21	(	(	PUNCT
ejpam-4962	758	22	g	g	NOUN
ejpam-4962	758	23	)	)	PUNCT
ejpam-4962	758	24	.	.	PUNCT
ejpam-4962	759	1	hence	hence	ADV
ejpam-4962	759	2	{	{	PUNCT
ejpam-4962	759	3	v	v	NOUN
ejpam-4962	759	4	,	,	PUNCT
ejpam-4962	759	5	w	w	PROPN
ejpam-4962	759	6	,	,	PUNCT
ejpam-4962	759	7	x	x	PRON
ejpam-4962	759	8	}	}	PUNCT
ejpam-4962	759	9	is	be	AUX
ejpam-4962	759	10	a	a	DET
ejpam-4962	759	11	ρ2	ρ2	NOUN
ejpam-4962	759	12	-	-	PUNCT
ejpam-4962	759	13	set	set	NOUN
ejpam-4962	759	14	in	in	ADP
ejpam-4962	759	15	g	g	PROPN
ejpam-4962	759	16	and	and	CCONJ
ejpam-4962	759	17	v	v	NOUN
ejpam-4962	759	18	(	(	PUNCT
ejpam-4962	759	19	g	g	NOUN
ejpam-4962	759	20	)	)	PUNCT
ejpam-4962	759	21	\	\	NOUN
ejpam-4962	759	22	{	{	PUNCT
ejpam-4962	759	23	v	v	NOUN
ejpam-4962	759	24	,	,	PUNCT
ejpam-4962	759	25	w	w	PROPN
ejpam-4962	759	26	,	,	PUNCT
ejpam-4962	759	27	x	x	NOUN
ejpam-4962	759	28	}	}	PUNCT
ejpam-4962	759	29	⊆	⊆	NUM
ejpam-4962	759	30	ng({v	ng({v	NUM
ejpam-4962	759	31	,	,	PUNCT
ejpam-4962	759	32	w	w	NOUN
ejpam-4962	759	33	}	}	PUNCT
ejpam-4962	759	34	)	)	PUNCT
ejpam-4962	759	35	.	.	PUNCT
ejpam-4962	760	1	this	this	PRON
ejpam-4962	760	2	shows	show	VERB
ejpam-4962	760	3	that	that	SCONJ
ejpam-4962	760	4	(	(	PUNCT
ejpam-4962	760	5	vii	vii	PROPN
ejpam-4962	760	6	)	)	PUNCT
ejpam-4962	760	7	holds	hold	VERB
ejpam-4962	760	8	.	.	PUNCT
ejpam-4962	761	1	similarly	similarly	ADV
ejpam-4962	761	2	,	,	PUNCT
ejpam-4962	761	3	(	(	PUNCT
ejpam-4962	761	4	ii	ii	NOUN
ejpam-4962	761	5	)	)	PUNCT
ejpam-4962	761	6	or	or	CCONJ
ejpam-4962	761	7	(	(	PUNCT
ejpam-4962	761	8	iv	iv	X
ejpam-4962	761	9	)	)	PUNCT
ejpam-4962	761	10	or	or	CCONJ
ejpam-4962	761	11	(	(	PUNCT
ejpam-4962	761	12	vi	vi	NOUN
ejpam-4962	761	13	)	)	PUNCT
ejpam-4962	761	14	or	or	CCONJ
ejpam-4962	761	15	(	(	PUNCT
ejpam-4962	761	16	viii	viii	NOUN
ejpam-4962	761	17	)	)	PUNCT
ejpam-4962	761	18	holds	hold	NOUN
ejpam-4962	761	19	.	.	PUNCT
ejpam-4962	762	1	the	the	DET
ejpam-4962	762	2	converse	converse	NOUN
ejpam-4962	762	3	is	be	AUX
ejpam-4962	762	4	clear	clear	ADJ
ejpam-4962	762	5	.	.	PUNCT
ejpam-4962	763	1	conclusion	conclusion	NOUN
ejpam-4962	763	2	this	this	DET
ejpam-4962	763	3	study	study	NOUN
ejpam-4962	763	4	introduced	introduce	VERB
ejpam-4962	763	5	the	the	DET
ejpam-4962	763	6	notion	notion	NOUN
ejpam-4962	763	7	of	of	ADP
ejpam-4962	763	8	geodetic	geodetic	ADJ
ejpam-4962	763	9	roman	roman	ADJ
ejpam-4962	763	10	domination	domination	NOUN
ejpam-4962	763	11	.	.	PUNCT
ejpam-4962	764	1	some	some	DET
ejpam-4962	764	2	properties	property	NOUN
ejpam-4962	764	3	of	of	ADP
ejpam-4962	764	4	geodetic	geodetic	ADJ
ejpam-4962	764	5	roman	roman	ADJ
ejpam-4962	764	6	dominating	dominating	NOUN
ejpam-4962	764	7	functions	function	NOUN
ejpam-4962	764	8	were	be	AUX
ejpam-4962	764	9	explored	explore	VERB
ejpam-4962	764	10	and	and	CCONJ
ejpam-4962	764	11	the	the	DET
ejpam-4962	764	12	geodetic	geodetic	ADJ
ejpam-4962	764	13	roman	roman	ADJ
ejpam-4962	764	14	domination	domination	NOUN
ejpam-4962	764	15	numbers	number	NOUN
ejpam-4962	764	16	of	of	ADP
ejpam-4962	764	17	certain	certain	ADJ
ejpam-4962	764	18	graphs	graph	NOUN
ejpam-4962	764	19	were	be	AUX
ejpam-4962	764	20	determined	determine	VERB
ejpam-4962	764	21	.	.	PUNCT
ejpam-4962	765	1	it	it	PRON
ejpam-4962	765	2	was	be	AUX
ejpam-4962	765	3	also	also	ADV
ejpam-4962	765	4	shown	show	VERB
ejpam-4962	765	5	that	that	SCONJ
ejpam-4962	765	6	any	any	DET
ejpam-4962	765	7	pair	pair	NOUN
ejpam-4962	765	8	of	of	ADP
ejpam-4962	765	9	positive	positive	ADJ
ejpam-4962	765	10	integers	integer	NOUN
ejpam-4962	765	11	(	(	PUNCT
ejpam-4962	765	12	subject	subject	ADJ
ejpam-4962	765	13	to	to	ADP
ejpam-4962	765	14	a	a	DET
ejpam-4962	765	15	constraint	constraint	NOUN
ejpam-4962	765	16	)	)	PUNCT
ejpam-4962	765	17	are	be	AUX
ejpam-4962	765	18	realizable	realizable	ADJ
ejpam-4962	765	19	as	as	ADP
ejpam-4962	765	20	the	the	DET
ejpam-4962	765	21	geodetic	geodetic	ADJ
ejpam-4962	765	22	domination	domination	NOUN
ejpam-4962	765	23	number	number	NOUN
ejpam-4962	765	24	and	and	CCONJ
ejpam-4962	765	25	geodetic	geodetic	ADJ
ejpam-4962	765	26	roman	roman	ADJ
ejpam-4962	765	27	domination	domination	NOUN
ejpam-4962	765	28	number	number	NOUN
ejpam-4962	765	29	of	of	ADP
ejpam-4962	765	30	a	a	DET
ejpam-4962	765	31	connected	connected	ADJ
ejpam-4962	765	32	graph	graph	NOUN
ejpam-4962	765	33	.	.	PUNCT
ejpam-4962	766	1	this	this	DET
ejpam-4962	766	2	newly	newly	ADV
ejpam-4962	766	3	defined	define	VERB
ejpam-4962	766	4	variant	variant	NOUN
ejpam-4962	766	5	of	of	ADP
ejpam-4962	766	6	roman	roman	ADJ
ejpam-4962	766	7	domination	domination	NOUN
ejpam-4962	766	8	may	may	AUX
ejpam-4962	766	9	be	be	AUX
ejpam-4962	766	10	investigated	investigate	VERB
ejpam-4962	766	11	further	far	ADV
ejpam-4962	766	12	for	for	ADP
ejpam-4962	766	13	other	other	ADJ
ejpam-4962	766	14	graphs	graph	NOUN
ejpam-4962	766	15	including	include	VERB
ejpam-4962	766	16	those	those	DET
ejpam-4962	766	17	ones	one	NOUN
ejpam-4962	766	18	resulting	result	VERB
ejpam-4962	766	19	from	from	ADP
ejpam-4962	766	20	some	some	DET
ejpam-4962	766	21	binary	binary	ADJ
ejpam-4962	766	22	operations	operation	NOUN
ejpam-4962	766	23	of	of	ADP
ejpam-4962	766	24	graphs	graph	NOUN
ejpam-4962	766	25	.	.	PUNCT
ejpam-4962	767	1	one	one	PRON
ejpam-4962	767	2	may	may	AUX
ejpam-4962	767	3	also	also	ADV
ejpam-4962	767	4	try	try	VERB
ejpam-4962	767	5	exploring	explore	VERB
ejpam-4962	767	6	the	the	DET
ejpam-4962	767	7	relationship	relationship	NOUN
ejpam-4962	767	8	between	between	ADP
ejpam-4962	767	9	this	this	DET
ejpam-4962	767	10	variant	variant	NOUN
ejpam-4962	767	11	and	and	CCONJ
ejpam-4962	767	12	the	the	DET
ejpam-4962	767	13	other	other	ADJ
ejpam-4962	767	14	variations	variation	NOUN
ejpam-4962	767	15	of	of	ADP
ejpam-4962	767	16	roman	roman	ADJ
ejpam-4962	767	17	domination	domination	NOUN
ejpam-4962	767	18	.	.	PUNCT
ejpam-4962	768	1	acknowledgements	acknowledgement	NOUN
ejpam-4962	768	2	the	the	DET
ejpam-4962	768	3	authors	author	NOUN
ejpam-4962	768	4	would	would	AUX
ejpam-4962	768	5	like	like	VERB
ejpam-4962	768	6	to	to	PART
ejpam-4962	768	7	thank	thank	VERB
ejpam-4962	768	8	the	the	DET
ejpam-4962	768	9	referees	referee	NOUN
ejpam-4962	768	10	for	for	ADP
ejpam-4962	768	11	the	the	DET
ejpam-4962	768	12	invaluable	invaluable	ADJ
ejpam-4962	768	13	assistance	assistance	NOUN
ejpam-4962	768	14	they	they	PRON
ejpam-4962	768	15	gave	give	VERB
ejpam-4962	768	16	us	we	PRON
ejpam-4962	768	17	through	through	ADP
ejpam-4962	768	18	their	their	PRON
ejpam-4962	768	19	comments	comment	NOUN
ejpam-4962	768	20	and	and	CCONJ
ejpam-4962	768	21	suggestions	suggestion	NOUN
ejpam-4962	768	22	which	which	PRON
ejpam-4962	768	23	led	lead	VERB
ejpam-4962	768	24	to	to	ADP
ejpam-4962	768	25	the	the	DET
ejpam-4962	768	26	improvement	improvement	NOUN
ejpam-4962	768	27	of	of	ADP
ejpam-4962	768	28	the	the	DET
ejpam-4962	768	29	paper	paper	NOUN
ejpam-4962	768	30	.	.	PUNCT
ejpam-4962	769	1	the	the	DET
ejpam-4962	769	2	authors	author	NOUN
ejpam-4962	769	3	are	be	AUX
ejpam-4962	769	4	also	also	ADV
ejpam-4962	769	5	grateful	grateful	ADJ
ejpam-4962	769	6	to	to	ADP
ejpam-4962	769	7	the	the	DET
ejpam-4962	769	8	department	department	NOUN
ejpam-4962	769	9	of	of	ADP
ejpam-4962	769	10	science	science	NOUN
ejpam-4962	769	11	and	and	CCONJ
ejpam-4962	769	12	technology	technology	NOUN
ejpam-4962	769	13	accelerated	accelerate	VERB
ejpam-4962	769	14	science	science	NOUN
ejpam-4962	769	15	and	and	CCONJ
ejpam-4962	769	16	technology	technology	NOUN
ejpam-4962	769	17	human	human	ADJ
ejpam-4962	769	18	resource	resource	NOUN
ejpam-4962	769	19	development	development	NOUN
ejpam-4962	769	20	program	program	NOUN
ejpam-4962	769	21	(	(	PUNCT
ejpam-4962	769	22	dost	dost	NOUN
ejpam-4962	769	23	-	-	PUNCT
ejpam-4962	769	24	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-4962	769	25	and	and	CCONJ
ejpam-4962	769	26	msu	msu	PROPN
ejpam-4962	769	27	-	-	PUNCT
ejpam-4962	769	28	iligan	iligan	PROPN
ejpam-4962	769	29	institute	institute	PROPN
ejpam-4962	769	30	of	of	ADP
ejpam-4962	769	31	technology	technology	PROPN
ejpam-4962	769	32	,	,	PUNCT
ejpam-4962	769	33	philippines	philippine	NOUN
ejpam-4962	769	34	for	for	ADP
ejpam-4962	769	35	funding	fund	VERB
ejpam-4962	769	36	this	this	DET
ejpam-4962	769	37	research	research	NOUN
ejpam-4962	769	38	.	.	PUNCT
ejpam-4962	770	1	references	reference	NOUN
ejpam-4962	770	2	[	[	X
ejpam-4962	770	3	1	1	NUM
ejpam-4962	770	4	]	]	PUNCT
ejpam-4962	770	5	m.	m.	NOUN
ejpam-4962	770	6	adabi	adabi	PROPN
ejpam-4962	770	7	,	,	PUNCT
ejpam-4962	770	8	e.	e.	PROPN
ejpam-4962	770	9	ebrahimi	ebrahimi	PROPN
ejpam-4962	770	10	targhi	targhi	PROPN
ejpam-4962	770	11	,	,	PUNCT
ejpam-4962	770	12	n.	n.	PROPN
ejpam-4962	770	13	jafari	jafari	PROPN
ejpam-4962	770	14	rad	rad	PROPN
ejpam-4962	770	15	,	,	PUNCT
ejpam-4962	770	16	and	and	CCONJ
ejpam-4962	770	17	m.	m.	NOUN
ejpam-4962	770	18	saied	saie	VERB
ejpam-4962	770	19	moradi	moradi	NOUN
ejpam-4962	770	20	.	.	PUNCT
ejpam-4962	771	1	properties	property	NOUN
ejpam-4962	771	2	of	of	ADP
ejpam-4962	771	3	independent	independent	ADJ
ejpam-4962	771	4	roman	roman	ADJ
ejpam-4962	771	5	domination	domination	NOUN
ejpam-4962	771	6	in	in	ADP
ejpam-4962	771	7	graphs	graph	NOUN
ejpam-4962	771	8	.	.	PUNCT
ejpam-4962	772	1	australasian	australasian	ADJ
ejpam-4962	772	2	journal	journal	NOUN
ejpam-4962	772	3	of	of	ADP
ejpam-4962	772	4	combinatorics	combinatoric	NOUN
ejpam-4962	772	5	,	,	PUNCT
ejpam-4962	772	6	52:11–18	52:11–18	NUM
ejpam-4962	772	7	,	,	PUNCT
ejpam-4962	772	8	2012	2012	NUM
ejpam-4962	772	9	.	.	PUNCT
ejpam-4962	773	1	[	[	X
ejpam-4962	773	2	2	2	NUM
ejpam-4962	773	3	]	]	X
ejpam-4962	773	4	h.a	h.a	PROPN
ejpam-4962	773	5	.	.	PROPN
ejpam-4962	773	6	ahangar	ahangar	PROPN
ejpam-4962	773	7	,	,	PUNCT
ejpam-4962	773	8	m.a	m.a	PROPN
ejpam-4962	773	9	.	.	PROPN
ejpam-4962	773	10	henning	henning	PROPN
ejpam-4962	773	11	,	,	PUNCT
ejpam-4962	773	12	v.	v.	ADP
ejpam-4962	773	13	samodivkin	samodivkin	NOUN
ejpam-4962	773	14	,	,	PUNCT
ejpam-4962	773	15	and	and	CCONJ
ejpam-4962	773	16	i.g	i.g	PROPN
ejpam-4962	773	17	.	.	PROPN
ejpam-4962	773	18	yero	yero	PROPN
ejpam-4962	773	19	.	.	PUNCT
ejpam-4962	773	20	total	total	ADJ
ejpam-4962	773	21	roman	roman	ADJ
ejpam-4962	773	22	domination	domination	NOUN
ejpam-4962	773	23	in	in	ADP
ejpam-4962	773	24	graphs	graph	NOUN
ejpam-4962	773	25	.	.	PUNCT
ejpam-4962	774	1	applicable	applicable	ADJ
ejpam-4962	774	2	analysis	analysis	NOUN
ejpam-4962	774	3	and	and	CCONJ
ejpam-4962	774	4	discrete	discrete	ADJ
ejpam-4962	774	5	mathematics	mathematic	NOUN
ejpam-4962	774	6	,	,	PUNCT
ejpam-4962	774	7	10(2):501–517	10(2):501–517	PROPN
ejpam-4962	774	8	,	,	PUNCT
ejpam-4962	774	9	2016	2016	NUM
ejpam-4962	774	10	.	.	PUNCT
ejpam-4962	775	1	[	[	X
ejpam-4962	775	2	3	3	X
ejpam-4962	775	3	]	]	X
ejpam-4962	775	4	m.p	m.p	PROPN
ejpam-4962	775	5	.	.	PROPN
ejpam-4962	775	6	alvarez	alvarez	PROPN
ejpam-4962	775	7	-	-	PUNCT
ejpam-4962	775	8	ruiz	ruiz	PROPN
ejpam-4962	775	9	,	,	PUNCT
ejpam-4962	775	10	t.	t.	PROPN
ejpam-4962	775	11	mediavilla	mediavilla	PROPN
ejpam-4962	775	12	-	-	PUNCT
ejpam-4962	775	13	gradolph	gradolph	NOUN
ejpam-4962	775	14	,	,	PUNCT
ejpam-4962	775	15	s.m	s.m	PROPN
ejpam-4962	775	16	.	.	PROPN
ejpam-4962	775	17	sheikholeslami	sheikholeslami	PROPN
ejpam-4962	775	18	,	,	PUNCT
ejpam-4962	775	19	j.c	j.c	PROPN
ejpam-4962	775	20	.	.	PROPN
ejpam-4962	775	21	valenzuelatripodoro	valenzuelatripodoro	PROPN
ejpam-4962	775	22	,	,	PUNCT
ejpam-4962	775	23	and	and	CCONJ
ejpam-4962	775	24	i.g	i.g	PROPN
ejpam-4962	775	25	.	.	PROPN
ejpam-4962	775	26	yero	yero	PROPN
ejpam-4962	775	27	.	.	PUNCT
ejpam-4962	776	1	on	on	ADP
ejpam-4962	776	2	the	the	DET
ejpam-4962	776	3	strong	strong	ADJ
ejpam-4962	776	4	roman	roman	ADJ
ejpam-4962	776	5	domination	domination	NOUN
ejpam-4962	776	6	number	number	NOUN
ejpam-4962	776	7	of	of	ADP
ejpam-4962	776	8	graphs	graph	NOUN
ejpam-4962	776	9	.	.	PUNCT
ejpam-4962	777	1	discrete	discrete	ADJ
ejpam-4962	777	2	applied	apply	VERB
ejpam-4962	777	3	mathematics	mathematic	NOUN
ejpam-4962	777	4	,	,	PUNCT
ejpam-4962	777	5	231:44–59	231:44–59	NUM
ejpam-4962	777	6	,	,	PUNCT
ejpam-4962	777	7	2017	2017	NUM
ejpam-4962	777	8	.	.	PUNCT
ejpam-4962	778	1	[	[	X
ejpam-4962	778	2	4	4	X
ejpam-4962	778	3	]	]	PUNCT
ejpam-4962	778	4	s.	s.	PROPN
ejpam-4962	778	5	banerjee	banerjee	PROPN
ejpam-4962	778	6	,	,	PUNCT
ejpam-4962	778	7	j.m	j.m	PROPN
ejpam-4962	778	8	.	.	PROPN
ejpam-4962	778	9	keil	keil	PROPN
ejpam-4962	778	10	,	,	PUNCT
ejpam-4962	778	11	and	and	CCONJ
ejpam-4962	778	12	d.	d.	PROPN
ejpam-4962	778	13	pradhan	pradhan	PROPN
ejpam-4962	778	14	.	.	PUNCT
ejpam-4962	779	1	perfect	perfect	ADJ
ejpam-4962	779	2	roman	roman	ADJ
ejpam-4962	779	3	domination	domination	NOUN
ejpam-4962	779	4	in	in	ADP
ejpam-4962	779	5	graphs	graph	NOUN
ejpam-4962	779	6	.	.	PUNCT
ejpam-4962	780	1	theoretical	theoretical	ADJ
ejpam-4962	780	2	computer	computer	NOUN
ejpam-4962	780	3	science	science	NOUN
ejpam-4962	780	4	,	,	PUNCT
ejpam-4962	780	5	796:1–21	796:1–21	NUM
ejpam-4962	780	6	,	,	PUNCT
ejpam-4962	780	7	2019	2019	NUM
ejpam-4962	780	8	.	.	PUNCT
ejpam-4962	781	1	references	reference	NOUN
ejpam-4962	781	2	2382	2382	NUM
ejpam-4962	781	3	[	[	X
ejpam-4962	781	4	5	5	NUM
ejpam-4962	781	5	]	]	X
ejpam-4962	781	6	r.a	r.a	PROPN
ejpam-4962	781	7	.	.	PROPN
ejpam-4962	781	8	beeler	beeler	PROPN
ejpam-4962	781	9	,	,	PUNCT
ejpam-4962	781	10	t.w	t.w	PROPN
ejpam-4962	781	11	.	.	PROPN
ejpam-4962	781	12	haynes	haynes	PROPN
ejpam-4962	781	13	,	,	PUNCT
ejpam-4962	781	14	and	and	CCONJ
ejpam-4962	781	15	s.t	s.t	PROPN
ejpam-4962	781	16	.	.	PROPN
ejpam-4962	781	17	hedetnieme	hedetnieme	PROPN
ejpam-4962	781	18	.	.	PUNCT
ejpam-4962	782	1	double	double	ADJ
ejpam-4962	782	2	roman	roman	ADJ
ejpam-4962	782	3	domination	domination	NOUN
ejpam-4962	782	4	.	.	PUNCT
ejpam-4962	783	1	discrete	discrete	ADJ
ejpam-4962	783	2	applied	apply	VERB
ejpam-4962	783	3	mathematics	mathematic	NOUN
ejpam-4962	783	4	,	,	PUNCT
ejpam-4962	783	5	211:23–29	211:23–29	NUM
ejpam-4962	783	6	,	,	PUNCT
ejpam-4962	783	7	2016	2016	NUM
ejpam-4962	783	8	.	.	PUNCT
ejpam-4962	784	1	[	[	X
ejpam-4962	784	2	6	6	NUM
ejpam-4962	784	3	]	]	X
ejpam-4962	784	4	f.	f.	PROPN
ejpam-4962	784	5	buckley	buckley	PROPN
ejpam-4962	784	6	,	,	PUNCT
ejpam-4962	784	7	f.	f.	PROPN
ejpam-4962	784	8	harary	harary	PROPN
ejpam-4962	784	9	,	,	PUNCT
ejpam-4962	784	10	and	and	CCONJ
ejpam-4962	784	11	l.v	l.v	PROPN
ejpam-4962	784	12	.	.	PROPN
ejpam-4962	784	13	quintas	quinta	NOUN
ejpam-4962	784	14	.	.	PUNCT
ejpam-4962	785	1	extremal	extremal	ADJ
ejpam-4962	785	2	results	result	NOUN
ejpam-4962	785	3	on	on	ADP
ejpam-4962	785	4	the	the	DET
ejpam-4962	785	5	geodetic	geodetic	ADJ
ejpam-4962	785	6	number	number	NOUN
ejpam-4962	785	7	of	of	ADP
ejpam-4962	785	8	graphs	graph	NOUN
ejpam-4962	785	9	.	.	PUNCT
ejpam-4962	786	1	discrete	discrete	ADJ
ejpam-4962	786	2	applied	apply	VERB
ejpam-4962	786	3	mathematics	mathematic	NOUN
ejpam-4962	786	4	,	,	PUNCT
ejpam-4962	786	5	211:23–29	211:23–29	NUM
ejpam-4962	786	6	,	,	PUNCT
ejpam-4962	786	7	2016	2016	NUM
ejpam-4962	786	8	.	.	PUNCT
ejpam-4962	787	1	[	[	X
ejpam-4962	787	2	7	7	X
ejpam-4962	787	3	]	]	X
ejpam-4962	787	4	g.	g.	NOUN
ejpam-4962	787	5	cagaanan	cagaanan	PROPN
ejpam-4962	787	6	and	and	CCONJ
ejpam-4962	787	7	s.r	s.r	PROPN
ejpam-4962	787	8	.	.	PROPN
ejpam-4962	787	9	canoy	canoy	PROPN
ejpam-4962	787	10	jr	jr	PROPN
ejpam-4962	787	11	.	.	PROPN
ejpam-4962	787	12	on	on	ADP
ejpam-4962	787	13	the	the	DET
ejpam-4962	787	14	geodetic	geodetic	ADJ
ejpam-4962	787	15	covers	cover	NOUN
ejpam-4962	787	16	and	and	CCONJ
ejpam-4962	787	17	geodetic	geodetic	ADJ
ejpam-4962	787	18	bases	basis	NOUN
ejpam-4962	787	19	of	of	ADP
ejpam-4962	787	20	the	the	DET
ejpam-4962	787	21	composition	composition	NOUN
ejpam-4962	787	22	g[km	g[km	PROPN
ejpam-4962	787	23	]	]	PUNCT
ejpam-4962	787	24	.	.	PUNCT
ejpam-4962	788	1	ars	ars	PROPN
ejpam-4962	788	2	combinatoria	combinatoria	PROPN
ejpam-4962	788	3	,	,	PUNCT
ejpam-4962	788	4	79:33–45	79:33–45	NUM
ejpam-4962	788	5	,	,	PUNCT
ejpam-4962	788	6	2006	2006	NUM
ejpam-4962	788	7	.	.	PUNCT
ejpam-4962	789	1	[	[	X
ejpam-4962	789	2	8	8	NUM
ejpam-4962	789	3	]	]	PUNCT
ejpam-4962	789	4	g.	g.	NOUN
ejpam-4962	789	5	cagaanan	cagaanan	PROPN
ejpam-4962	789	6	and	and	CCONJ
ejpam-4962	789	7	s.r	s.r	PROPN
ejpam-4962	789	8	.	.	PROPN
ejpam-4962	789	9	canoy	canoy	PROPN
ejpam-4962	789	10	jr	jr	PROPN
ejpam-4962	789	11	.	.	PROPN
ejpam-4962	789	12	bounds	bound	VERB
ejpam-4962	789	13	for	for	ADP
ejpam-4962	789	14	the	the	DET
ejpam-4962	789	15	geodetic	geodetic	ADJ
ejpam-4962	789	16	number	number	NOUN
ejpam-4962	789	17	of	of	ADP
ejpam-4962	789	18	the	the	DET
ejpam-4962	789	19	cartesian	cartesian	ADJ
ejpam-4962	789	20	product	product	NOUN
ejpam-4962	789	21	of	of	ADP
ejpam-4962	789	22	graphs	graph	NOUN
ejpam-4962	789	23	.	.	PUNCT
ejpam-4962	790	1	utilitas	utilitas	PROPN
ejpam-4962	790	2	matematica	matematica	PROPN
ejpam-4962	790	3	,	,	PUNCT
ejpam-4962	790	4	79:91–98	79:91–98	NUM
ejpam-4962	790	5	,	,	PUNCT
ejpam-4962	790	6	2009	2009	NUM
ejpam-4962	790	7	.	.	PUNCT
ejpam-4962	791	1	[	[	X
ejpam-4962	791	2	9	9	NUM
ejpam-4962	791	3	]	]	X
ejpam-4962	791	4	g.	g.	PROPN
ejpam-4962	791	5	chartrand	chartrand	PROPN
ejpam-4962	791	6	,	,	PUNCT
ejpam-4962	791	7	f.	f.	PROPN
ejpam-4962	791	8	harary	harary	PROPN
ejpam-4962	791	9	,	,	PUNCT
ejpam-4962	791	10	and	and	CCONJ
ejpam-4962	791	11	p.	p.	PROPN
ejpam-4962	791	12	zhang	zhang	PROPN
ejpam-4962	791	13	.	.	PUNCT
ejpam-4962	792	1	on	on	ADP
ejpam-4962	792	2	the	the	DET
ejpam-4962	792	3	geodetic	geodetic	ADJ
ejpam-4962	792	4	number	number	NOUN
ejpam-4962	792	5	of	of	ADP
ejpam-4962	792	6	a	a	DET
ejpam-4962	792	7	graph	graph	NOUN
ejpam-4962	792	8	.	.	PUNCT
ejpam-4962	793	1	networks	network	NOUN
ejpam-4962	793	2	:	:	PUNCT
ejpam-4962	793	3	an	an	DET
ejpam-4962	793	4	international	international	ADJ
ejpam-4962	793	5	journal	journal	NOUN
ejpam-4962	793	6	,	,	PUNCT
ejpam-4962	793	7	39(1):1–6	39(1):1–6	NUM
ejpam-4962	793	8	,	,	PUNCT
ejpam-4962	793	9	2002	2002	NUM
ejpam-4962	793	10	.	.	PUNCT
ejpam-4962	794	1	[	[	X
ejpam-4962	794	2	10	10	NUM
ejpam-4962	794	3	]	]	X
ejpam-4962	794	4	m.	m.	NOUN
ejpam-4962	794	5	chellali	chellali	PROPN
ejpam-4962	794	6	,	,	PUNCT
ejpam-4962	794	7	t.w	t.w	PROPN
ejpam-4962	794	8	.	.	PROPN
ejpam-4962	794	9	haynes	haynes	PROPN
ejpam-4962	794	10	,	,	PUNCT
ejpam-4962	794	11	and	and	CCONJ
ejpam-4962	794	12	s.t	s.t	PROPN
ejpam-4962	794	13	.	.	PROPN
ejpam-4962	794	14	hedetnieme	hedetnieme	PROPN
ejpam-4962	794	15	.	.	PUNCT
ejpam-4962	795	1	roman	roman	NOUN
ejpam-4962	795	2	{	{	PUNCT
ejpam-4962	795	3	2}-domination	2}-domination	NUM
ejpam-4962	795	4	.	.	PUNCT
ejpam-4962	796	1	discrete	discrete	ADJ
ejpam-4962	796	2	applied	apply	VERB
ejpam-4962	796	3	math	math	NOUN
ejpam-4962	796	4	,	,	PUNCT
ejpam-4962	796	5	204:22–28	204:22–28	NUM
ejpam-4962	796	6	,	,	PUNCT
ejpam-4962	796	7	2016	2016	NUM
ejpam-4962	796	8	.	.	PUNCT
ejpam-4962	797	1	[	[	X
ejpam-4962	797	2	11	11	NUM
ejpam-4962	797	3	]	]	X
ejpam-4962	797	4	s.r	s.r	PROPN
ejpam-4962	797	5	.	.	PROPN
ejpam-4962	797	6	chellathurai	chellathurai	PROPN
ejpam-4962	797	7	and	and	CCONJ
ejpam-4962	797	8	s.p	s.p	PROPN
ejpam-4962	797	9	.	.	PROPN
ejpam-4962	797	10	vijaya	vijaya	PROPN
ejpam-4962	797	11	.	.	PUNCT
ejpam-4962	798	1	the	the	DET
ejpam-4962	798	2	geodetic	geodetic	ADJ
ejpam-4962	798	3	domination	domination	NOUN
ejpam-4962	798	4	number	number	NOUN
ejpam-4962	798	5	for	for	ADP
ejpam-4962	798	6	the	the	DET
ejpam-4962	798	7	product	product	NOUN
ejpam-4962	798	8	of	of	ADP
ejpam-4962	798	9	graphs	graph	NOUN
ejpam-4962	798	10	.	.	PUNCT
ejpam-4962	799	1	3(4):19–30	3(4):19–30	NUM
ejpam-4962	799	2	,	,	PUNCT
ejpam-4962	799	3	2014	2014	NUM
ejpam-4962	799	4	.	.	PUNCT
ejpam-4962	800	1	[	[	X
ejpam-4962	800	2	12	12	NUM
ejpam-4962	800	3	]	]	X
ejpam-4962	800	4	e.j	e.j	PROPN
ejpam-4962	800	5	.	.	PROPN
ejpam-4962	800	6	cockayne	cockayne	PROPN
ejpam-4962	800	7	,	,	PUNCT
ejpam-4962	800	8	p.a	p.a	PROPN
ejpam-4962	800	9	.	.	PROPN
ejpam-4962	800	10	deryer	deryer	PROPN
ejpam-4962	800	11	,	,	PUNCT
ejpam-4962	800	12	s.m	s.m	PROPN
ejpam-4962	800	13	.	.	PROPN
ejpam-4962	800	14	hedetnieme	hedetnieme	PROPN
ejpam-4962	800	15	,	,	PUNCT
ejpam-4962	800	16	and	and	CCONJ
ejpam-4962	800	17	s.t	s.t	PROPN
ejpam-4962	800	18	.	.	PROPN
ejpam-4962	800	19	hedetnieme	hedetnieme	PROPN
ejpam-4962	800	20	.	.	PUNCT
ejpam-4962	801	1	roman	roman	ADJ
ejpam-4962	801	2	domination	domination	NOUN
ejpam-4962	801	3	in	in	ADP
ejpam-4962	801	4	graphs	graph	NOUN
ejpam-4962	801	5	.	.	PUNCT
ejpam-4962	802	1	discrete	discrete	ADJ
ejpam-4962	802	2	mathematics	mathematic	NOUN
ejpam-4962	802	3	,	,	PUNCT
ejpam-4962	802	4	278(13):11–22	278(13):11–22	NUM
ejpam-4962	802	5	,	,	PUNCT
ejpam-4962	802	6	2004	2004	NUM
ejpam-4962	802	7	.	.	PUNCT
ejpam-4962	803	1	[	[	X
ejpam-4962	803	2	13	13	NUM
ejpam-4962	803	3	]	]	X
ejpam-4962	803	4	h.	h.	NOUN
ejpam-4962	803	5	escuardo	escuardo	PROPN
ejpam-4962	803	6	,	,	PUNCT
ejpam-4962	803	7	r.	r.	PROPN
ejpam-4962	803	8	hansberg	hansberg	PROPN
ejpam-4962	803	9	,	,	PUNCT
ejpam-4962	803	10	a.	a.	PROPN
ejpam-4962	803	11	jafari	jafari	PROPN
ejpam-4962	803	12	rad	rad	PROPN
ejpam-4962	803	13	,	,	PUNCT
ejpam-4962	803	14	and	and	CCONJ
ejpam-4962	803	15	l.	l.	PROPN
ejpam-4962	803	16	volkman	volkman	PROPN
ejpam-4962	803	17	.	.	PUNCT
ejpam-4962	804	1	geodetic	geodetic	ADJ
ejpam-4962	804	2	domination	domination	NOUN
ejpam-4962	804	3	in	in	ADP
ejpam-4962	804	4	graphs	graph	NOUN
ejpam-4962	804	5	.	.	PUNCT
ejpam-4962	805	1	jcmcc	jcmcc	ADJ
ejpam-4962	805	2	-	-	PUNCT
ejpam-4962	805	3	journal	journal	NOUN
ejpam-4962	805	4	of	of	ADP
ejpam-4962	805	5	combinatorial	combinatorial	ADJ
ejpam-4962	805	6	mathematics	mathematic	NOUN
ejpam-4962	805	7	and	and	CCONJ
ejpam-4962	805	8	combinatorial	combinatorial	ADJ
ejpam-4962	805	9	computing	computing	NOUN
ejpam-4962	805	10	,	,	PUNCT
ejpam-4962	805	11	pages	page	NOUN
ejpam-4962	805	12	77–89	77–89	NUM
ejpam-4962	805	13	,	,	PUNCT
ejpam-4962	805	14	2011	2011	NUM
ejpam-4962	805	15	.	.	PUNCT
ejpam-4962	806	1	[	[	X
ejpam-4962	806	2	14	14	NUM
ejpam-4962	806	3	]	]	PUNCT
ejpam-4962	806	4	a.	a.	NOUN
ejpam-4962	806	5	hansberg	hansberg	PROPN
ejpam-4962	806	6	and	and	CCONJ
ejpam-4962	806	7	l.	l.	PROPN
ejpam-4962	806	8	volkman	volkman	PROPN
ejpam-4962	806	9	.	.	PUNCT
ejpam-4962	807	1	on	on	ADP
ejpam-4962	807	2	the	the	DET
ejpam-4962	807	3	geodetic	geodetic	ADJ
ejpam-4962	807	4	and	and	CCONJ
ejpam-4962	807	5	geodetic	geodetic	ADJ
ejpam-4962	807	6	domination	domination	NOUN
ejpam-4962	807	7	numbers	number	NOUN
ejpam-4962	807	8	of	of	ADP
ejpam-4962	807	9	a	a	DET
ejpam-4962	807	10	graph	graph	NOUN
ejpam-4962	807	11	.	.	PUNCT
ejpam-4962	807	12	discrete	discrete	ADJ
ejpam-4962	807	13	mathematics	mathematic	NOUN
ejpam-4962	807	14	,	,	PUNCT
ejpam-4962	807	15	310(15	310(15	PROPN
ejpam-4962	807	16	-	-	SYM
ejpam-4962	807	17	16):2140–2146	16):2140–2146	NUM
ejpam-4962	807	18	,	,	PUNCT
ejpam-4962	807	19	2010	2010	NUM
ejpam-4962	807	20	.	.	PUNCT
ejpam-4962	808	1	[	[	X
ejpam-4962	808	2	15	15	NUM
ejpam-4962	808	3	]	]	X
ejpam-4962	808	4	m.a	m.a	PROPN
ejpam-4962	808	5	.	.	PROPN
ejpam-4962	808	6	henning	henning	PROPN
ejpam-4962	808	7	,	,	PUNCT
ejpam-4962	808	8	w.f	w.f	PROPN
ejpam-4962	808	9	.	.	PROPN
ejpam-4962	808	10	klostermeyer	klostermeyer	PROPN
ejpam-4962	808	11	,	,	PUNCT
ejpam-4962	808	12	and	and	CCONJ
ejpam-4962	808	13	g.	g.	PROPN
ejpam-4962	808	14	macgillivray	macgillivray	PROPN
ejpam-4962	808	15	.	.	PUNCT
ejpam-4962	809	1	perfect	perfect	ADJ
ejpam-4962	809	2	roman	roman	ADJ
ejpam-4962	809	3	domination	domination	NOUN
ejpam-4962	809	4	in	in	ADP
ejpam-4962	809	5	trees	tree	NOUN
ejpam-4962	809	6	.	.	PUNCT
ejpam-4962	810	1	discrete	discrete	ADJ
ejpam-4962	810	2	applied	apply	VERB
ejpam-4962	810	3	mathematics	mathematic	NOUN
ejpam-4962	810	4	,	,	PUNCT
ejpam-4962	810	5	236:235–245	236:235–245	NUM
ejpam-4962	810	6	,	,	PUNCT
ejpam-4962	810	7	2018	2018	NUM
ejpam-4962	810	8	.	.	PUNCT
ejpam-4962	811	1	[	[	X
ejpam-4962	811	2	16	16	NUM
ejpam-4962	811	3	]	]	PUNCT
ejpam-4962	811	4	k.	k.	PROPN
ejpam-4962	811	5	kammerling	kammerling	PROPN
ejpam-4962	811	6	and	and	CCONJ
ejpam-4962	811	7	l.	l.	PROPN
ejpam-4962	811	8	volkman	volkman	PROPN
ejpam-4962	811	9	.	.	PUNCT
ejpam-4962	812	1	roman	roman	ADJ
ejpam-4962	812	2	k	k	NOUN
ejpam-4962	812	3	-	-	PUNCT
ejpam-4962	812	4	domination	domination	NOUN
ejpam-4962	812	5	in	in	ADP
ejpam-4962	812	6	graphs	graph	NOUN
ejpam-4962	812	7	.	.	PUNCT
ejpam-4962	813	1	journal	journal	NOUN
ejpam-4962	813	2	of	of	ADP
ejpam-4962	813	3	the	the	DET
ejpam-4962	813	4	korean	korean	PROPN
ejpam-4962	813	5	mathematical	mathematical	ADJ
ejpam-4962	813	6	society	society	NOUN
ejpam-4962	813	7	,	,	PUNCT
ejpam-4962	813	8	46(6):1309–1318	46(6):1309–1318	PROPN
ejpam-4962	813	9	,	,	PUNCT
ejpam-4962	813	10	2009	2009	NUM
ejpam-4962	813	11	.	.	PUNCT
ejpam-4962	814	1	[	[	X
ejpam-4962	814	2	17	17	NUM
ejpam-4962	814	3	]	]	X
ejpam-4962	814	4	m.h	m.h	PROPN
ejpam-4962	814	5	.	.	PROPN
ejpam-4962	814	6	muddebiha	muddebiha	PROPN
ejpam-4962	814	7	and	and	CCONJ
ejpam-4962	814	8	sumangaladevi	sumangaladevi	ADJ
ejpam-4962	814	9	.	.	PUNCT
ejpam-4962	815	1	connected	connect	VERB
ejpam-4962	815	2	roman	roman	ADJ
ejpam-4962	815	3	domination	domination	NOUN
ejpam-4962	815	4	in	in	ADP
ejpam-4962	815	5	graphs	graph	NOUN
ejpam-4962	815	6	.	.	PUNCT
ejpam-4962	816	1	international	international	ADJ
ejpam-4962	816	2	journal	journal	PROPN
ejpam-4962	816	3	of	of	ADP
ejpam-4962	816	4	research	research	NOUN
ejpam-4962	816	5	and	and	CCONJ
ejpam-4962	816	6	engineering	engineering	NOUN
ejpam-4962	816	7	technology	technology	NOUN
ejpam-4962	816	8	,	,	PUNCT
ejpam-4962	816	9	2(10):333–340	2(10):333–340	NUM
ejpam-4962	816	10	,	,	PUNCT
ejpam-4962	816	11	2013	2013	NUM
ejpam-4962	816	12	.	.	PUNCT
ejpam-4962	817	1	[	[	X
ejpam-4962	817	2	18	18	NUM
ejpam-4962	817	3	]	]	X
ejpam-4962	817	4	j.j	j.j	PROPN
ejpam-4962	817	5	.	.	PROPN
ejpam-4962	817	6	mulloor	mulloor	PROPN
ejpam-4962	817	7	and	and	CCONJ
ejpam-4962	817	8	v.	v.	ADP
ejpam-4962	817	9	sangeetha	sangeetha	PROPN
ejpam-4962	817	10	.	.	PUNCT
ejpam-4962	818	1	restrained	restrain	VERB
ejpam-4962	818	2	geodetic	geodetic	ADJ
ejpam-4962	818	3	domination	domination	NOUN
ejpam-4962	818	4	in	in	ADP
ejpam-4962	818	5	graphs	graph	NOUN
ejpam-4962	818	6	.	.	PUNCT
ejpam-4962	819	1	discrete	discrete	ADJ
ejpam-4962	819	2	mathematics	mathematic	NOUN
ejpam-4962	819	3	,	,	PUNCT
ejpam-4962	819	4	algorithms	algorithm	NOUN
ejpam-4962	819	5	and	and	CCONJ
ejpam-4962	819	6	applications	application	NOUN
ejpam-4962	819	7	,	,	PUNCT
ejpam-4962	819	8	12(6):2050084	12(6):2050084	NUM
ejpam-4962	819	9	,	,	PUNCT
ejpam-4962	819	10	2020	2020	NUM
ejpam-4962	819	11	.	.	PUNCT
ejpam-4962	820	1	[	[	X
ejpam-4962	820	2	19	19	NUM
ejpam-4962	820	3	]	]	PUNCT
ejpam-4962	820	4	p.r.l	p.r.l	NOUN
ejpam-4962	820	5	.	.	PUNCT
ejpam-4962	821	1	pushpam	pushpam	NOUN
ejpam-4962	821	2	and	and	CCONJ
ejpam-4962	821	3	t.n.m	t.n.m	NOUN
ejpam-4962	821	4	.	.	PUNCT
ejpam-4962	822	1	malini	malini	PROPN
ejpam-4962	822	2	mai	mai	PROPN
ejpam-4962	822	3	.	.	PROPN
ejpam-4962	822	4	edge	edge	PROPN
ejpam-4962	822	5	roman	roman	ADJ
ejpam-4962	822	6	domination	domination	NOUN
ejpam-4962	822	7	in	in	ADP
ejpam-4962	822	8	graphs	graph	NOUN
ejpam-4962	822	9	.	.	PUNCT
ejpam-4962	823	1	journal	journal	NOUN
ejpam-4962	823	2	of	of	ADP
ejpam-4962	823	3	combinatorial	combinatorial	ADJ
ejpam-4962	823	4	mathematics	mathematic	NOUN
ejpam-4962	823	5	and	and	CCONJ
ejpam-4962	823	6	combinatorial	combinatorial	ADJ
ejpam-4962	823	7	computing	computing	NOUN
ejpam-4962	823	8	,	,	PUNCT
ejpam-4962	823	9	69:175–182	69:175–182	NOUN
ejpam-4962	823	10	,	,	PUNCT
ejpam-4962	823	11	2009	2009	NUM
ejpam-4962	823	12	.	.	PUNCT
ejpam-4962	824	1	[	[	X
ejpam-4962	824	2	20	20	NUM
ejpam-4962	824	3	]	]	X
ejpam-4962	824	4	c.s	c.s	PROPN
ejpam-4962	824	5	.	.	PROPN
ejpam-4962	824	6	revelle	revelle	PROPN
ejpam-4962	824	7	and	and	CCONJ
ejpam-4962	824	8	k.e	k.e	PROPN
ejpam-4962	824	9	.	.	PUNCT
ejpam-4962	825	1	rosing	rosing	PROPN
ejpam-4962	825	2	.	.	PUNCT
ejpam-4962	826	1	defendens	defenden	VERB
ejpam-4962	826	2	imperium	imperium	NOUN
ejpam-4962	826	3	romanum	romanum	NOUN
ejpam-4962	826	4	:	:	PUNCT
ejpam-4962	826	5	a	a	DET
ejpam-4962	826	6	classical	classical	ADJ
ejpam-4962	826	7	problem	problem	NOUN
ejpam-4962	826	8	in	in	ADP
ejpam-4962	826	9	military	military	ADJ
ejpam-4962	826	10	strategy	strategy	NOUN
ejpam-4962	826	11	.	.	PUNCT
ejpam-4962	827	1	american	american	PROPN
ejpam-4962	827	2	mathematical	mathematical	PROPN
ejpam-4962	827	3	monthly	monthly	PROPN
ejpam-4962	827	4	,	,	PUNCT
ejpam-4962	827	5	107(7):585–594	107(7):585–594	PROPN
ejpam-4962	827	6	,	,	PUNCT
ejpam-4962	827	7	2000	2000	NUM
ejpam-4962	827	8	.	.	PUNCT
ejpam-4962	828	1	references	reference	NOUN
ejpam-4962	828	2	2383	2383	NUM
ejpam-4962	829	1	[	[	X
ejpam-4962	829	2	21	21	NUM
ejpam-4962	829	3	]	]	X
ejpam-4962	829	4	c.j	c.j	PROPN
ejpam-4962	829	5	.	.	PROPN
ejpam-4962	829	6	saromines	saromine	NOUN
ejpam-4962	829	7	and	and	CCONJ
ejpam-4962	829	8	s.r	s.r	PROPN
ejpam-4962	829	9	.	.	PROPN
ejpam-4962	829	10	canoy	canoy	PROPN
ejpam-4962	829	11	jr	jr	PROPN
ejpam-4962	829	12	.	.	PROPN
ejpam-4962	829	13	geodetic	geodetic	ADJ
ejpam-4962	829	14	hop	hop	NOUN
ejpam-4962	829	15	dominating	dominating	NOUN
ejpam-4962	829	16	sets	set	NOUN
ejpam-4962	829	17	in	in	ADP
ejpam-4962	829	18	a	a	DET
ejpam-4962	829	19	graph	graph	NOUN
ejpam-4962	829	20	.	.	PUNCT
ejpam-4962	830	1	european	european	ADJ
ejpam-4962	830	2	journal	journal	PROPN
ejpam-4962	830	3	of	of	ADP
ejpam-4962	830	4	pure	pure	ADJ
ejpam-4962	830	5	and	and	CCONJ
ejpam-4962	830	6	applied	applied	ADJ
ejpam-4962	830	7	mathematics	mathematic	NOUN
ejpam-4962	830	8	,	,	PUNCT
ejpam-4962	830	9	16(1):5–17	16(1):5–17	NUM
ejpam-4962	830	10	,	,	PUNCT
ejpam-4962	830	11	2023	2023	NUM
ejpam-4962	830	12	.	.	PUNCT
ejpam-4962	831	1	[	[	X
ejpam-4962	831	2	22	22	NUM
ejpam-4962	831	3	]	]	PUNCT
ejpam-4962	831	4	i.	i.	PROPN
ejpam-4962	831	5	stewart	stewart	PROPN
ejpam-4962	831	6	.	.	PUNCT
ejpam-4962	832	1	defend	defend	VERB
ejpam-4962	832	2	the	the	DET
ejpam-4962	832	3	roman	roman	ADJ
ejpam-4962	832	4	empire	empire	NOUN
ejpam-4962	832	5	!	!	PUNCT
ejpam-4962	833	1	scientific	scientific	ADJ
ejpam-4962	833	2	american	american	PROPN
ejpam-4962	833	3	,	,	PUNCT
ejpam-4962	833	4	281(6):136–138	281(6):136–138	PROPN
ejpam-4962	833	5	,	,	PUNCT
ejpam-4962	833	6	1999	1999	NUM
ejpam-4962	833	7	.	.	PUNCT
ejpam-4962	834	1	[	[	X
ejpam-4962	834	2	23	23	NUM
ejpam-4962	834	3	]	]	X
ejpam-4962	834	4	p.a.p	p.a.p	PROPN
ejpam-4962	834	5	.	.	PROPN
ejpam-4962	834	6	sudhahar	sudhahar	PROPN
ejpam-4962	834	7	,	,	PUNCT
ejpam-4962	834	8	a.	a.	NOUN
ejpam-4962	834	9	ajitha	ajitha	PROPN
ejpam-4962	834	10	,	,	PUNCT
ejpam-4962	834	11	and	and	CCONJ
ejpam-4962	834	12	a.	a.	PROPN
ejpam-4962	834	13	subramanian	subramanian	PROPN
ejpam-4962	834	14	.	.	PUNCT
ejpam-4962	835	1	edge	edge	PROPN
ejpam-4962	835	2	geodetic	geodetic	ADJ
ejpam-4962	835	3	domination	domination	NOUN
ejpam-4962	835	4	number	number	NOUN
ejpam-4962	835	5	of	of	ADP
ejpam-4962	835	6	a	a	DET
ejpam-4962	835	7	graph	graph	NOUN
ejpam-4962	835	8	.	.	PUNCT
ejpam-4962	836	1	international	international	ADJ
ejpam-4962	836	2	journal	journal	NOUN
ejpam-4962	836	3	of	of	ADP
ejpam-4962	836	4	mathematics	mathematic	NOUN
ejpam-4962	836	5	and	and	CCONJ
ejpam-4962	836	6	its	its	PRON
ejpam-4962	836	7	applications	application	NOUN
ejpam-4962	836	8	,	,	PUNCT
ejpam-4962	836	9	4(3	4(3	NUM
ejpam-4962	836	10	-	-	PUNCT
ejpam-4962	836	11	b):45–50	b):45–50	NOUN
ejpam-4962	836	12	,	,	PUNCT
ejpam-4962	836	13	2016	2016	NUM
ejpam-4962	836	14	.	.	PUNCT
ejpam-4962	837	1	[	[	X
ejpam-4962	837	2	24	24	NUM
ejpam-4962	837	3	]	]	X
ejpam-4962	837	4	t.l	t.l	PROPN
ejpam-4962	837	5	.	.	NOUN
ejpam-4962	837	6	tacbobo	tacbobo	NOUN
ejpam-4962	837	7	,	,	PUNCT
ejpam-4962	837	8	f.p	f.p	PROPN
ejpam-4962	837	9	.	.	PROPN
ejpam-4962	837	10	jamil	jamil	PROPN
ejpam-4962	837	11	,	,	PUNCT
ejpam-4962	837	12	and	and	CCONJ
ejpam-4962	837	13	s.r	s.r	PROPN
ejpam-4962	837	14	.	.	PROPN
ejpam-4962	837	15	canoy	canoy	PROPN
ejpam-4962	837	16	jr	jr	PROPN
ejpam-4962	837	17	.	.	PROPN
ejpam-4962	837	18	monophonic	monophonic	ADJ
ejpam-4962	837	19	and	and	CCONJ
ejpam-4962	837	20	geodetic	geodetic	ADJ
ejpam-4962	837	21	domination	domination	NOUN
ejpam-4962	837	22	in	in	ADP
ejpam-4962	837	23	the	the	DET
ejpam-4962	837	24	join	join	NOUN
ejpam-4962	837	25	,	,	PUNCT
ejpam-4962	837	26	corona	corona	NOUN
ejpam-4962	837	27	and	and	CCONJ
ejpam-4962	837	28	composition	composition	NOUN
ejpam-4962	837	29	of	of	ADP
ejpam-4962	837	30	graphs	graph	NOUN
ejpam-4962	837	31	.	.	PUNCT
ejpam-4962	838	1	ars	ar	NOUN
ejpam-4962	838	2	combinatoria	combinatoria	PROPN
ejpam-4962	838	3	,	,	PUNCT
ejpam-4962	838	4	(	(	PUNCT
ejpam-4962	838	5	112):13–31	112):13–31	NUM
ejpam-4962	838	6	,	,	PUNCT
ejpam-4962	838	7	2013	2013	NUM
ejpam-4962	838	8	.	.	PUNCT