id	sid	tid	token	lemma	pos
ap-4699	1	1	acta	acta	PROPN
ap-4699	1	2	polytechnica	polytechnica	PROPN
ap-4699	1	3	doi:10.14311	doi:10.14311	PROPN
ap-4699	1	4	/	/	SYM
ap-4699	1	5	ap.2018.58.0285	ap.2018.58.0285	PROPN
ap-4699	1	6	acta	acta	PROPN
ap-4699	1	7	polytechnica	polytechnica	PROPN
ap-4699	1	8	58(5):285–291	58(5):285–291	PROPN
ap-4699	1	9	,	,	PUNCT
ap-4699	1	10	2018	2018	NUM
ap-4699	1	11	©	©	PROPN
ap-4699	1	12	czech	czech	PROPN
ap-4699	1	13	technical	technical	PROPN
ap-4699	1	14	university	university	PROPN
ap-4699	1	15	in	in	ADP
ap-4699	1	16	prague	prague	PROPN
ap-4699	1	17	,	,	PUNCT
ap-4699	1	18	2018	2018	NUM
ap-4699	1	19	available	available	ADJ
ap-4699	1	20	online	online	ADV
ap-4699	1	21	at	at	ADP
ap-4699	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4699	1	23	minimal	minimal	ADJ
ap-4699	1	24	non	non	ADJ
ap-4699	1	25	-	-	ADJ
ap-4699	1	26	integer	integer	ADJ
ap-4699	1	27	alphabets	alphabet	NOUN
ap-4699	1	28	allowing	allow	VERB
ap-4699	1	29	parallel	parallel	ADJ
ap-4699	1	30	addition	addition	NOUN
ap-4699	1	31	jan	jan	PROPN
ap-4699	1	32	legerskýa	legerskýa	PROPN
ap-4699	1	33	,	,	PUNCT
ap-4699	1	34	b	b	PROPN
ap-4699	1	35	a	a	DET
ap-4699	1	36	research	research	NOUN
ap-4699	1	37	institute	institute	NOUN
ap-4699	1	38	for	for	ADP
ap-4699	1	39	symbolic	symbolic	ADJ
ap-4699	1	40	computation	computation	NOUN
ap-4699	1	41	,	,	PUNCT
ap-4699	1	42	johannes	johannes	PROPN
ap-4699	1	43	kepler	kepler	PROPN
ap-4699	1	44	university	university	PROPN
ap-4699	1	45	altenbergerstraße	altenbergerstraße	VERB
ap-4699	1	46	69	69	NUM
ap-4699	1	47	,	,	PUNCT
ap-4699	1	48	a-4040	a-4040	PROPN
ap-4699	1	49	linz	linz	PROPN
ap-4699	1	50	,	,	PUNCT
ap-4699	1	51	austria	austria	PROPN
ap-4699	1	52	b	b	PROPN
ap-4699	1	53	faculty	faculty	NOUN
ap-4699	1	54	of	of	ADP
ap-4699	1	55	nuclear	nuclear	ADJ
ap-4699	1	56	sciences	science	NOUN
ap-4699	1	57	and	and	CCONJ
ap-4699	1	58	physical	physical	ADJ
ap-4699	1	59	engineering	engineering	NOUN
ap-4699	1	60	,	,	PUNCT
ap-4699	1	61	czech	czech	PROPN
ap-4699	1	62	technical	technical	PROPN
ap-4699	1	63	university	university	PROPN
ap-4699	1	64	in	in	ADP
ap-4699	1	65	prague	prague	PROPN
ap-4699	1	66	trojanova	trojanova	X
ap-4699	1	67	13	13	NUM
ap-4699	1	68	,	,	PUNCT
ap-4699	1	69	120	120	NUM
ap-4699	1	70	00	00	NUM
ap-4699	1	71	praha	praha	PROPN
ap-4699	1	72	2	2	NUM
ap-4699	1	73	,	,	PUNCT
ap-4699	1	74	czech	czech	PROPN
ap-4699	1	75	republic	republic	NOUN
ap-4699	1	76	correspondence	correspondence	NOUN
ap-4699	1	77	:	:	PUNCT
ap-4699	1	78	jan.legersky@risc.jku.at	jan.legersky@risc.jku.at	VERB
ap-4699	1	79	abstract	abstract	NOUN
ap-4699	1	80	.	.	PUNCT
ap-4699	2	1	parallel	parallel	ADJ
ap-4699	2	2	addition	addition	NOUN
ap-4699	2	3	,	,	PUNCT
ap-4699	2	4	i.e.	i.e.	X
ap-4699	2	5	,	,	PUNCT
ap-4699	2	6	addition	addition	NOUN
ap-4699	2	7	with	with	ADP
ap-4699	2	8	limited	limited	ADJ
ap-4699	2	9	carry	carry	NOUN
ap-4699	2	10	propagation	propagation	NOUN
ap-4699	2	11	has	have	AUX
ap-4699	2	12	been	be	AUX
ap-4699	2	13	so	so	ADV
ap-4699	2	14	far	far	ADV
ap-4699	2	15	studied	study	VERB
ap-4699	2	16	for	for	ADP
ap-4699	2	17	complex	complex	ADJ
ap-4699	2	18	bases	basis	NOUN
ap-4699	2	19	and	and	CCONJ
ap-4699	2	20	integer	integer	NOUN
ap-4699	2	21	alphabets	alphabet	NOUN
ap-4699	2	22	.	.	PUNCT
ap-4699	3	1	we	we	PRON
ap-4699	3	2	focus	focus	VERB
ap-4699	3	3	on	on	ADP
ap-4699	3	4	alphabets	alphabet	NOUN
ap-4699	3	5	consisting	consist	VERB
ap-4699	3	6	of	of	ADP
ap-4699	3	7	integer	integer	NOUN
ap-4699	3	8	combinations	combination	NOUN
ap-4699	3	9	of	of	ADP
ap-4699	3	10	powers	power	NOUN
ap-4699	3	11	of	of	ADP
ap-4699	3	12	the	the	DET
ap-4699	3	13	base	base	NOUN
ap-4699	3	14	.	.	PUNCT
ap-4699	4	1	we	we	PRON
ap-4699	4	2	give	give	VERB
ap-4699	4	3	necessary	necessary	ADJ
ap-4699	4	4	conditions	condition	NOUN
ap-4699	4	5	on	on	ADP
ap-4699	4	6	the	the	DET
ap-4699	4	7	alphabet	alphabet	NOUN
ap-4699	4	8	allowing	allow	VERB
ap-4699	4	9	parallel	parallel	ADJ
ap-4699	4	10	addition	addition	NOUN
ap-4699	4	11	.	.	PUNCT
ap-4699	5	1	under	under	ADP
ap-4699	5	2	certain	certain	ADJ
ap-4699	5	3	assumptions	assumption	NOUN
ap-4699	5	4	,	,	PUNCT
ap-4699	5	5	we	we	PRON
ap-4699	5	6	prove	prove	VERB
ap-4699	5	7	the	the	DET
ap-4699	5	8	same	same	ADJ
ap-4699	5	9	lower	low	ADJ
ap-4699	5	10	bound	bind	VERB
ap-4699	5	11	on	on	ADP
ap-4699	5	12	the	the	DET
ap-4699	5	13	size	size	NOUN
ap-4699	5	14	of	of	ADP
ap-4699	5	15	the	the	DET
ap-4699	5	16	generalized	generalized	ADJ
ap-4699	5	17	alphabet	alphabet	NOUN
ap-4699	5	18	that	that	PRON
ap-4699	5	19	is	be	AUX
ap-4699	5	20	known	know	VERB
ap-4699	5	21	for	for	ADP
ap-4699	5	22	alphabets	alphabet	NOUN
ap-4699	5	23	consisting	consist	VERB
ap-4699	5	24	of	of	ADP
ap-4699	5	25	consecutive	consecutive	ADJ
ap-4699	5	26	integers	integer	NOUN
ap-4699	5	27	.	.	PUNCT
ap-4699	6	1	we	we	PRON
ap-4699	6	2	also	also	ADV
ap-4699	6	3	extend	extend	VERB
ap-4699	6	4	the	the	DET
ap-4699	6	5	characterization	characterization	NOUN
ap-4699	6	6	of	of	ADP
ap-4699	6	7	bases	basis	NOUN
ap-4699	6	8	allowing	allow	VERB
ap-4699	6	9	parallel	parallel	ADJ
ap-4699	6	10	addition	addition	NOUN
ap-4699	6	11	to	to	ADP
ap-4699	6	12	numeration	numeration	NOUN
ap-4699	6	13	systems	system	NOUN
ap-4699	6	14	with	with	ADP
ap-4699	6	15	non	non	ADJ
ap-4699	6	16	-	-	ADJ
ap-4699	6	17	integer	integer	ADJ
ap-4699	6	18	alphabets	alphabet	NOUN
ap-4699	6	19	.	.	PUNCT
ap-4699	7	1	keywords	keyword	NOUN
ap-4699	7	2	:	:	PUNCT
ap-4699	7	3	numeration	numeration	NOUN
ap-4699	7	4	system	system	NOUN
ap-4699	7	5	,	,	PUNCT
ap-4699	7	6	parallel	parallel	ADJ
ap-4699	7	7	addition	addition	NOUN
ap-4699	7	8	,	,	PUNCT
ap-4699	7	9	minimal	minimal	ADJ
ap-4699	7	10	alphabet	alphabet	NOUN
ap-4699	7	11	.	.	PUNCT
ap-4699	8	1	1	1	X
ap-4699	8	2	.	.	X
ap-4699	8	3	introduction	introduction	NOUN
ap-4699	8	4	the	the	DET
ap-4699	8	5	concept	concept	NOUN
ap-4699	8	6	of	of	ADP
ap-4699	8	7	parallel	parallel	ADJ
ap-4699	8	8	addition	addition	NOUN
ap-4699	8	9	in	in	ADP
ap-4699	8	10	a	a	DET
ap-4699	8	11	numeration	numeration	NOUN
ap-4699	8	12	system	system	NOUN
ap-4699	8	13	with	with	ADP
ap-4699	8	14	a	a	DET
ap-4699	8	15	base	base	NOUN
ap-4699	8	16	β	β	X
ap-4699	8	17	and	and	CCONJ
ap-4699	8	18	alphabet	alphabet	PROPN
ap-4699	8	19	a	a	PRON
ap-4699	8	20	was	be	AUX
ap-4699	8	21	introduced	introduce	VERB
ap-4699	8	22	by	by	ADP
ap-4699	8	23	a.	a.	NOUN
ap-4699	8	24	avizienis	avizienis	NOUN
ap-4699	9	1	[	[	X
ap-4699	9	2	1	1	NUM
ap-4699	9	3	]	]	PUNCT
ap-4699	9	4	.	.	PUNCT
ap-4699	10	1	the	the	DET
ap-4699	10	2	crucial	crucial	ADJ
ap-4699	10	3	difference	difference	NOUN
ap-4699	10	4	from	from	ADP
ap-4699	10	5	standard	standard	ADJ
ap-4699	10	6	addition	addition	NOUN
ap-4699	10	7	is	be	AUX
ap-4699	10	8	that	that	PRON
ap-4699	10	9	carry	carry	VERB
ap-4699	10	10	propagation	propagation	NOUN
ap-4699	10	11	is	be	AUX
ap-4699	10	12	limited	limit	VERB
ap-4699	10	13	and	and	CCONJ
ap-4699	10	14	hence	hence	ADV
ap-4699	10	15	an	an	DET
ap-4699	10	16	output	output	NOUN
ap-4699	10	17	digit	digit	NOUN
ap-4699	10	18	depends	depend	VERB
ap-4699	10	19	only	only	ADV
ap-4699	10	20	on	on	ADP
ap-4699	10	21	bounded	bounded	ADJ
ap-4699	10	22	number	number	NOUN
ap-4699	10	23	of	of	ADP
ap-4699	10	24	input	input	NOUN
ap-4699	10	25	digits	digit	NOUN
ap-4699	10	26	.	.	PUNCT
ap-4699	11	1	therefore	therefore	ADV
ap-4699	11	2	,	,	PUNCT
ap-4699	11	3	the	the	DET
ap-4699	11	4	whole	whole	ADJ
ap-4699	11	5	operation	operation	NOUN
ap-4699	11	6	can	can	AUX
ap-4699	11	7	run	run	VERB
ap-4699	11	8	in	in	ADP
ap-4699	11	9	constant	constant	ADJ
ap-4699	11	10	time	time	NOUN
ap-4699	11	11	in	in	ADP
ap-4699	11	12	parallel	parallel	NOUN
ap-4699	11	13	.	.	PUNCT
ap-4699	12	1	it	it	PRON
ap-4699	12	2	is	be	AUX
ap-4699	12	3	known	know	VERB
ap-4699	12	4	that	that	SCONJ
ap-4699	12	5	the	the	DET
ap-4699	12	6	alphabet	alphabet	NOUN
ap-4699	12	7	a	a	PRON
ap-4699	12	8	must	must	AUX
ap-4699	12	9	be	be	AUX
ap-4699	12	10	redundant	redundant	ADJ
ap-4699	13	1	[	[	X
ap-4699	13	2	2	2	NUM
ap-4699	13	3	]	]	PUNCT
ap-4699	13	4	,	,	PUNCT
ap-4699	13	5	otherwise	otherwise	ADV
ap-4699	13	6	parallel	parallel	ADJ
ap-4699	13	7	addition	addition	NOUN
ap-4699	13	8	is	be	AUX
ap-4699	13	9	not	not	PART
ap-4699	13	10	possible	possible	ADJ
ap-4699	13	11	.	.	PUNCT
ap-4699	14	1	necessary	necessary	ADJ
ap-4699	14	2	conditions	condition	NOUN
ap-4699	14	3	on	on	ADP
ap-4699	14	4	the	the	DET
ap-4699	14	5	base	base	NOUN
ap-4699	14	6	and	and	CCONJ
ap-4699	14	7	alphabet	alphabet	NOUN
ap-4699	14	8	were	be	AUX
ap-4699	14	9	further	far	ADV
ap-4699	14	10	studied	study	VERB
ap-4699	14	11	by	by	ADP
ap-4699	14	12	c.	c.	PROPN
ap-4699	14	13	frougny	frougny	PROPN
ap-4699	14	14	,	,	PUNCT
ap-4699	14	15	p.	p.	NOUN
ap-4699	14	16	heller	heller	PROPN
ap-4699	14	17	,	,	PUNCT
ap-4699	14	18	e.	e.	PROPN
ap-4699	14	19	pelantová	pelantová	PROPN
ap-4699	14	20	,	,	PUNCT
ap-4699	14	21	and	and	CCONJ
ap-4699	14	22	m.	m.	NOUN
ap-4699	14	23	svobodová	svobodová	PROPN
ap-4699	15	1	[	[	X
ap-4699	15	2	3–5	3–5	X
ap-4699	15	3	]	]	PUNCT
ap-4699	15	4	under	under	ADP
ap-4699	15	5	assumption	assumption	NOUN
ap-4699	15	6	that	that	SCONJ
ap-4699	15	7	the	the	DET
ap-4699	15	8	alphabet	alphabet	NOUN
ap-4699	15	9	a	a	DET
ap-4699	15	10	consists	consist	NOUN
ap-4699	15	11	of	of	ADP
ap-4699	15	12	consecutive	consecutive	ADJ
ap-4699	15	13	integers	integer	NOUN
ap-4699	15	14	containing	contain	VERB
ap-4699	15	15	0	0	NUM
ap-4699	15	16	.	.	PUNCT
ap-4699	16	1	it	it	PRON
ap-4699	16	2	was	be	AUX
ap-4699	16	3	shown	show	VERB
ap-4699	16	4	that	that	SCONJ
ap-4699	16	5	there	there	PRON
ap-4699	16	6	exists	exist	VERB
ap-4699	16	7	an	an	DET
ap-4699	16	8	integer	integer	NOUN
ap-4699	16	9	alphabet	alphabet	NOUN
ap-4699	16	10	allowing	allow	VERB
ap-4699	16	11	parallel	parallel	ADJ
ap-4699	16	12	addition	addition	NOUN
ap-4699	16	13	if	if	SCONJ
ap-4699	16	14	and	and	CCONJ
ap-4699	16	15	only	only	ADV
ap-4699	16	16	if	if	SCONJ
ap-4699	16	17	the	the	DET
ap-4699	16	18	base	base	NOUN
ap-4699	16	19	is	be	AUX
ap-4699	16	20	an	an	DET
ap-4699	16	21	algebraic	algebraic	ADJ
ap-4699	16	22	number	number	NOUN
ap-4699	16	23	with	with	ADP
ap-4699	16	24	no	no	DET
ap-4699	16	25	conjugates	conjugate	NOUN
ap-4699	16	26	of	of	ADP
ap-4699	16	27	modulus	modulus	NOUN
ap-4699	16	28	1	1	NUM
ap-4699	16	29	.	.	PUNCT
ap-4699	17	1	lower	low	ADJ
ap-4699	17	2	bounds	bound	NOUN
ap-4699	17	3	on	on	ADP
ap-4699	17	4	the	the	DET
ap-4699	17	5	size	size	NOUN
ap-4699	17	6	of	of	ADP
ap-4699	17	7	the	the	DET
ap-4699	17	8	alphabet	alphabet	NOUN
ap-4699	17	9	were	be	AUX
ap-4699	17	10	given	give	VERB
ap-4699	17	11	.	.	PUNCT
ap-4699	18	1	the	the	DET
ap-4699	18	2	main	main	ADJ
ap-4699	18	3	result	result	NOUN
ap-4699	18	4	of	of	ADP
ap-4699	18	5	this	this	DET
ap-4699	18	6	paper	paper	NOUN
ap-4699	18	7	is	be	AUX
ap-4699	18	8	generalization	generalization	NOUN
ap-4699	18	9	of	of	ADP
ap-4699	18	10	these	these	DET
ap-4699	18	11	results	result	NOUN
ap-4699	18	12	to	to	ADP
ap-4699	18	13	non	non	ADJ
ap-4699	18	14	-	-	ADJ
ap-4699	18	15	integer	integer	ADJ
ap-4699	18	16	alphabets	alphabet	NOUN
ap-4699	18	17	,	,	PUNCT
ap-4699	18	18	namely	namely	ADV
ap-4699	18	19	a	a	DET
ap-4699	18	20	⊂	⊂	PROPN
ap-4699	18	21	z[β	z[β	NOUN
ap-4699	18	22	]	]	PUNCT
ap-4699	18	23	.	.	PUNCT
ap-4699	19	1	such	such	ADJ
ap-4699	19	2	alphabets	alphabet	NOUN
ap-4699	19	3	might	might	AUX
ap-4699	19	4	have	have	VERB
ap-4699	19	5	elements	element	NOUN
ap-4699	19	6	smaller	small	ADJ
ap-4699	19	7	in	in	ADP
ap-4699	19	8	modulus	modulus	NOUN
ap-4699	19	9	comparing	compare	VERB
ap-4699	19	10	to	to	ADP
ap-4699	19	11	integer	integer	PROPN
ap-4699	19	12	ones	one	NOUN
ap-4699	19	13	.	.	PUNCT
ap-4699	20	1	this	this	PRON
ap-4699	20	2	is	be	AUX
ap-4699	20	3	useful	useful	ADJ
ap-4699	20	4	for	for	ADP
ap-4699	20	5	instance	instance	NOUN
ap-4699	20	6	in	in	ADP
ap-4699	20	7	online	online	ADJ
ap-4699	20	8	multiplication	multiplication	NOUN
ap-4699	20	9	and	and	CCONJ
ap-4699	20	10	division	division	NOUN
ap-4699	20	11	[	[	X
ap-4699	20	12	6	6	NUM
ap-4699	20	13	]	]	PUNCT
ap-4699	20	14	.	.	PUNCT
ap-4699	21	1	parallel	parallel	ADJ
ap-4699	21	2	addition	addition	NOUN
ap-4699	21	3	algorithms	algorithm	NOUN
ap-4699	21	4	that	that	PRON
ap-4699	21	5	use	use	VERB
ap-4699	21	6	non	non	ADJ
ap-4699	21	7	-	-	ADJ
ap-4699	21	8	integer	integer	ADJ
ap-4699	21	9	alphabets	alphabet	NOUN
ap-4699	21	10	are	be	AUX
ap-4699	21	11	discussed	discuss	VERB
ap-4699	21	12	in	in	ADP
ap-4699	21	13	[	[	X
ap-4699	21	14	7	7	NUM
ap-4699	21	15	]	]	PUNCT
ap-4699	21	16	.	.	PUNCT
ap-4699	22	1	the	the	DET
ap-4699	22	2	paper	paper	NOUN
ap-4699	22	3	[	[	X
ap-4699	22	4	8	8	X
ap-4699	22	5	]	]	PUNCT
ap-4699	22	6	discusses	discuss	VERB
ap-4699	22	7	consequences	consequence	NOUN
ap-4699	22	8	of	of	ADP
ap-4699	22	9	parallel	parallel	ADJ
ap-4699	22	10	addition	addition	NOUN
ap-4699	22	11	for	for	ADP
ap-4699	22	12	eventually	eventually	ADV
ap-4699	22	13	periodic	periodic	ADJ
ap-4699	22	14	representations	representation	NOUN
ap-4699	22	15	in	in	ADP
ap-4699	22	16	q(β	q(β	NOUN
ap-4699	22	17	)	)	PUNCT
ap-4699	22	18	.	.	PUNCT
ap-4699	23	1	this	this	DET
ap-4699	23	2	paper	paper	NOUN
ap-4699	23	3	is	be	AUX
ap-4699	23	4	organized	organize	VERB
ap-4699	23	5	as	as	SCONJ
ap-4699	23	6	follows	follow	VERB
ap-4699	23	7	:	:	PUNCT
ap-4699	23	8	in	in	ADP
ap-4699	23	9	section	section	NOUN
ap-4699	23	10	2	2	NUM
ap-4699	23	11	,	,	PUNCT
ap-4699	23	12	we	we	PRON
ap-4699	23	13	recall	recall	VERB
ap-4699	23	14	the	the	DET
ap-4699	23	15	necessary	necessary	ADJ
ap-4699	23	16	definitions	definition	NOUN
ap-4699	23	17	and	and	CCONJ
ap-4699	23	18	show	show	VERB
ap-4699	23	19	that	that	SCONJ
ap-4699	23	20	for	for	ADP
ap-4699	23	21	parallel	parallel	ADJ
ap-4699	23	22	addition	addition	NOUN
ap-4699	23	23	we	we	PRON
ap-4699	23	24	can	can	AUX
ap-4699	23	25	consider	consider	VERB
ap-4699	23	26	only	only	ADJ
ap-4699	23	27	bases	basis	NOUN
ap-4699	23	28	being	be	AUX
ap-4699	23	29	algebraic	algebraic	ADJ
ap-4699	23	30	numbers	number	NOUN
ap-4699	23	31	.	.	PUNCT
ap-4699	24	1	in	in	ADP
ap-4699	24	2	section	section	NOUN
ap-4699	24	3	3	3	NUM
ap-4699	24	4	,	,	PUNCT
ap-4699	24	5	we	we	PRON
ap-4699	24	6	prove	prove	VERB
ap-4699	24	7	that	that	SCONJ
ap-4699	24	8	if	if	SCONJ
ap-4699	24	9	(	(	PUNCT
ap-4699	24	10	β	β	X
ap-4699	24	11	,	,	PUNCT
ap-4699	24	12	a	a	PRON
ap-4699	24	13	)	)	PUNCT
ap-4699	24	14	allows	allow	VERB
ap-4699	24	15	parallel	parallel	ADJ
ap-4699	24	16	addition	addition	NOUN
ap-4699	24	17	and	and	CCONJ
ap-4699	24	18	β′	β′	NUM
ap-4699	24	19	is	be	AUX
ap-4699	24	20	a	a	DET
ap-4699	24	21	conjugate	conjugate	NOUN
ap-4699	24	22	of	of	ADP
ap-4699	24	23	β	β	NOUN
ap-4699	24	24	,	,	PUNCT
ap-4699	24	25	then	then	ADV
ap-4699	24	26	there	there	PRON
ap-4699	24	27	is	be	VERB
ap-4699	24	28	an	an	DET
ap-4699	24	29	alphabet	alphabet	NOUN
ap-4699	24	30	a′	a′	NOUN
ap-4699	24	31	such	such	ADJ
ap-4699	24	32	that	that	SCONJ
ap-4699	24	33	(	(	PUNCT
ap-4699	24	34	β′,a′	β′,a′	NOUN
ap-4699	24	35	)	)	PUNCT
ap-4699	24	36	allows	allow	VERB
ap-4699	24	37	parallel	parallel	ADJ
ap-4699	24	38	addition	addition	NOUN
ap-4699	24	39	.	.	PUNCT
ap-4699	25	1	if	if	SCONJ
ap-4699	25	2	a[β	a[β	PROPN
ap-4699	25	3	]	]	X
ap-4699	25	4	=	=	SYM
ap-4699	25	5	z[β	z[β	NOUN
ap-4699	25	6	]	]	PUNCT
ap-4699	25	7	,	,	PUNCT
ap-4699	25	8	we	we	PRON
ap-4699	25	9	show	show	VERB
ap-4699	25	10	that	that	SCONJ
ap-4699	25	11	a	a	PRON
ap-4699	25	12	must	must	AUX
ap-4699	25	13	contain	contain	VERB
ap-4699	25	14	all	all	DET
ap-4699	25	15	representatives	representative	NOUN
ap-4699	25	16	modulo	modulo	VERB
ap-4699	25	17	β	β	PROPN
ap-4699	25	18	and	and	CCONJ
ap-4699	25	19	β	β	X
ap-4699	25	20	−	−	NOUN
ap-4699	26	1	1	1	X
ap-4699	26	2	.	.	PUNCT
ap-4699	27	1	if	if	SCONJ
ap-4699	27	2	β	β	X
ap-4699	27	3	is	be	AUX
ap-4699	27	4	an	an	DET
ap-4699	27	5	algebraic	algebraic	ADJ
ap-4699	27	6	integer	integer	NOUN
ap-4699	27	7	,	,	PUNCT
ap-4699	27	8	a	a	DET
ap-4699	27	9	consequence	consequence	NOUN
ap-4699	27	10	is	be	AUX
ap-4699	27	11	the	the	DET
ap-4699	27	12	same	same	ADJ
ap-4699	27	13	lower	low	ADJ
ap-4699	27	14	bound	bind	VERB
ap-4699	27	15	on	on	ADP
ap-4699	27	16	the	the	DET
ap-4699	27	17	size	size	NOUN
ap-4699	27	18	of	of	ADP
ap-4699	27	19	a	a	DET
ap-4699	27	20	⊂	⊂	PROPN
ap-4699	27	21	z[β	z[β	NOUN
ap-4699	27	22	]	]	PUNCT
ap-4699	27	23	as	as	ADP
ap-4699	27	24	for	for	ADP
ap-4699	27	25	integer	integer	NOUN
ap-4699	27	26	alphabets	alphabet	NOUN
ap-4699	27	27	.	.	PUNCT
ap-4699	28	1	the	the	DET
ap-4699	28	2	assumption	assumption	NOUN
ap-4699	28	3	a[β	a[β	PROPN
ap-4699	28	4	]	]	X
ap-4699	28	5	=	=	SYM
ap-4699	28	6	z[β	z[β	X
ap-4699	28	7	]	]	PUNCT
ap-4699	28	8	or	or	CCONJ
ap-4699	28	9	existence	existence	NOUN
ap-4699	28	10	of	of	ADP
ap-4699	28	11	parallel	parallel	ADJ
ap-4699	28	12	addition	addition	NOUN
ap-4699	28	13	without	without	ADP
ap-4699	28	14	anticipation	anticipation	NOUN
ap-4699	28	15	implies	imply	VERB
ap-4699	28	16	that	that	SCONJ
ap-4699	28	17	β	β	PROPN
ap-4699	28	18	is	be	AUX
ap-4699	28	19	expanding	expand	VERB
ap-4699	28	20	,	,	PUNCT
ap-4699	28	21	i.e.	i.e.	X
ap-4699	28	22	,	,	PUNCT
ap-4699	28	23	all	all	DET
ap-4699	28	24	its	its	PRON
ap-4699	28	25	conjugates	conjugate	NOUN
ap-4699	28	26	are	be	AUX
ap-4699	28	27	greater	great	ADJ
ap-4699	28	28	than	than	ADP
ap-4699	28	29	one	one	NUM
ap-4699	28	30	in	in	ADP
ap-4699	28	31	modulus	modulus	NOUN
ap-4699	28	32	.	.	PUNCT
ap-4699	29	1	the	the	DET
ap-4699	29	2	key	key	ADJ
ap-4699	29	3	result	result	NOUN
ap-4699	29	4	from	from	ADP
ap-4699	29	5	[	[	X
ap-4699	29	6	3	3	NUM
ap-4699	29	7	]	]	PUNCT
ap-4699	29	8	is	be	AUX
ap-4699	29	9	generalized	generalize	VERB
ap-4699	29	10	to	to	ADP
ap-4699	29	11	a	a	DET
ap-4699	29	12	⊂	⊂	PROPN
ap-4699	29	13	z[β	z[β	X
ap-4699	29	14	]	]	PUNCT
ap-4699	29	15	in	in	ADP
ap-4699	29	16	section	section	NOUN
ap-4699	29	17	4	4	NUM
ap-4699	29	18	.	.	PUNCT
ap-4699	30	1	namely	namely	ADV
ap-4699	30	2	,	,	PUNCT
ap-4699	30	3	there	there	PRON
ap-4699	30	4	is	be	VERB
ap-4699	30	5	an	an	DET
ap-4699	30	6	alphabet	alphabet	NOUN
ap-4699	30	7	in	in	ADP
ap-4699	30	8	z[β	z[β	NOUN
ap-4699	30	9	]	]	PUNCT
ap-4699	30	10	allowing	allow	VERB
ap-4699	30	11	so	so	ADV
ap-4699	30	12	-	-	PUNCT
ap-4699	30	13	called	call	VERB
ap-4699	30	14	k	k	ADJ
ap-4699	30	15	-	-	PUNCT
ap-4699	30	16	block	block	ADJ
ap-4699	30	17	parallel	parallel	ADJ
ap-4699	30	18	addition	addition	NOUN
ap-4699	30	19	if	if	SCONJ
ap-4699	30	20	and	and	CCONJ
ap-4699	30	21	only	only	ADV
ap-4699	30	22	if	if	SCONJ
ap-4699	30	23	β	β	NOUN
ap-4699	30	24	is	be	AUX
ap-4699	30	25	an	an	DET
ap-4699	30	26	algebraic	algebraic	ADJ
ap-4699	30	27	number	number	NOUN
ap-4699	30	28	with	with	ADP
ap-4699	30	29	no	no	DET
ap-4699	30	30	conjugates	conjugate	NOUN
ap-4699	30	31	of	of	ADP
ap-4699	30	32	modulus	modulus	NOUN
ap-4699	30	33	one	one	NUM
ap-4699	30	34	.	.	PUNCT
ap-4699	31	1	2	2	X
ap-4699	31	2	.	.	X
ap-4699	31	3	preliminaries	preliminary	NOUN
ap-4699	31	4	the	the	DET
ap-4699	31	5	concept	concept	NOUN
ap-4699	31	6	of	of	ADP
ap-4699	31	7	positional	positional	ADJ
ap-4699	31	8	numeration	numeration	NOUN
ap-4699	31	9	systems	system	NOUN
ap-4699	31	10	with	with	ADP
ap-4699	31	11	integer	integer	NOUN
ap-4699	31	12	bases	basis	NOUN
ap-4699	31	13	and	and	CCONJ
ap-4699	31	14	digits	digit	NOUN
ap-4699	31	15	is	be	AUX
ap-4699	31	16	very	very	ADV
ap-4699	31	17	old	old	ADJ
ap-4699	31	18	and	and	CCONJ
ap-4699	31	19	can	can	AUX
ap-4699	31	20	be	be	AUX
ap-4699	31	21	easily	easily	ADV
ap-4699	31	22	generalized	generalize	VERB
ap-4699	31	23	:	:	PUNCT
ap-4699	31	24	definition	definition	NOUN
ap-4699	31	25	2.1	2.1	NUM
ap-4699	31	26	.	.	PUNCT
ap-4699	32	1	if	if	SCONJ
ap-4699	32	2	β	β	X
ap-4699	32	3	∈	∈	PROPN
ap-4699	32	4	c	c	NOUN
ap-4699	32	5	is	be	AUX
ap-4699	32	6	such	such	ADJ
ap-4699	32	7	that	that	SCONJ
ap-4699	32	8	|β|	|β|	PRON
ap-4699	32	9	>	>	X
ap-4699	32	10	1	1	NUM
ap-4699	32	11	and	and	CCONJ
ap-4699	32	12	a	a	DET
ap-4699	32	13	⊂	⊂	PROPN
ap-4699	32	14	c	c	PROPN
ap-4699	32	15	is	be	AUX
ap-4699	32	16	a	a	DET
ap-4699	32	17	finite	finite	ADJ
ap-4699	32	18	set	set	NOUN
ap-4699	32	19	containing	contain	VERB
ap-4699	32	20	0	0	NUM
ap-4699	32	21	,	,	PUNCT
ap-4699	32	22	then	then	ADV
ap-4699	32	23	the	the	DET
ap-4699	32	24	pair	pair	NOUN
ap-4699	32	25	(	(	PUNCT
ap-4699	32	26	β	β	X
ap-4699	32	27	,	,	PUNCT
ap-4699	32	28	a	a	PRON
ap-4699	32	29	)	)	PUNCT
ap-4699	32	30	is	be	AUX
ap-4699	32	31	called	call	VERB
ap-4699	32	32	a	a	DET
ap-4699	32	33	numeration	numeration	NOUN
ap-4699	32	34	system	system	NOUN
ap-4699	32	35	with	with	ADP
ap-4699	32	36	a	a	DET
ap-4699	32	37	base	base	NOUN
ap-4699	32	38	β	β	X
ap-4699	32	39	and	and	CCONJ
ap-4699	32	40	digit	digit	NOUN
ap-4699	32	41	set	set	VERB
ap-4699	32	42	a	a	PRON
ap-4699	32	43	,	,	PUNCT
ap-4699	32	44	usually	usually	ADV
ap-4699	32	45	called	call	VERB
ap-4699	32	46	an	an	DET
ap-4699	32	47	alphabet	alphabet	NOUN
ap-4699	32	48	.	.	PUNCT
ap-4699	33	1	numbers	number	NOUN
ap-4699	33	2	in	in	ADP
ap-4699	33	3	a	a	DET
ap-4699	33	4	numeration	numeration	NOUN
ap-4699	33	5	system	system	NOUN
ap-4699	33	6	(	(	PUNCT
ap-4699	33	7	β	β	X
ap-4699	33	8	,	,	PUNCT
ap-4699	33	9	a	a	PRON
ap-4699	33	10	)	)	PUNCT
ap-4699	33	11	are	be	AUX
ap-4699	33	12	represented	represent	VERB
ap-4699	33	13	in	in	ADP
ap-4699	33	14	the	the	DET
ap-4699	33	15	following	following	ADJ
ap-4699	33	16	way	way	NOUN
ap-4699	33	17	:	:	PUNCT
ap-4699	33	18	let	let	VERB
ap-4699	33	19	x	x	PRON
ap-4699	33	20	be	be	AUX
ap-4699	33	21	a	a	DET
ap-4699	33	22	complex	complex	ADJ
ap-4699	33	23	number	number	NOUN
ap-4699	33	24	and	and	CCONJ
ap-4699	33	25	xn	xn	NUM
ap-4699	33	26	,	,	PUNCT
ap-4699	33	27	xn−1	xn−1	PROPN
ap-4699	33	28	,	,	PUNCT
ap-4699	33	29	xn−2	xn−2	PROPN
ap-4699	33	30	,	,	PUNCT
ap-4699	33	31	.	.	PUNCT
ap-4699	33	32	.	.	PUNCT
ap-4699	33	33	.	.	PUNCT
ap-4699	34	1	∈	∈	PROPN
ap-4699	34	2	a	a	PRON
ap-4699	34	3	,	,	PUNCT
ap-4699	34	4	n	n	X
ap-4699	34	5	≥	≥	NOUN
ap-4699	34	6	0	0	NUM
ap-4699	34	7	.	.	PUNCT
ap-4699	35	1	we	we	PRON
ap-4699	35	2	say	say	VERB
ap-4699	35	3	that	that	SCONJ
ap-4699	35	4	ω0xnxn−1	ω0xnxn−1	PROPN
ap-4699	35	5	·	·	PUNCT
ap-4699	35	6	·	·	PUNCT
ap-4699	35	7	·	·	PUNCT
ap-4699	35	8	x1x0	x1x0	PROPN
ap-4699	35	9	•x−1x−2	•x−1x−2	NOUN
ap-4699	35	10	·	·	PUNCT
ap-4699	35	11	·	·	PUNCT
ap-4699	35	12	·	·	PUNCT
ap-4699	35	13	is	be	AUX
ap-4699	35	14	a	a	DET
ap-4699	35	15	(	(	PUNCT
ap-4699	35	16	β	β	X
ap-4699	35	17	,	,	PUNCT
ap-4699	35	18	a)representation	a)representation	NOUN
ap-4699	35	19	of	of	ADP
ap-4699	35	20	x	x	PRON
ap-4699	35	21	if	if	SCONJ
ap-4699	35	22	x	x	PRON
ap-4699	35	23	=	=	SYM
ap-4699	35	24	∑n	∑n	PROPN
ap-4699	35	25	j=−∞	j=−∞	PROPN
ap-4699	35	26	xjβ	xjβ	PROPN
ap-4699	35	27	j	j	PROPN
ap-4699	35	28	,	,	PUNCT
ap-4699	35	29	where	where	SCONJ
ap-4699	35	30	ω0	ω0	PROPN
ap-4699	35	31	denotes	denote	VERB
ap-4699	35	32	the	the	DET
ap-4699	35	33	left	leave	VERB
ap-4699	35	34	-	-	PUNCT
ap-4699	35	35	infinite	infinite	NOUN
ap-4699	35	36	sequence	sequence	NOUN
ap-4699	35	37	of	of	ADP
ap-4699	35	38	zeros	zero	NOUN
ap-4699	35	39	.	.	PUNCT
ap-4699	36	1	the	the	DET
ap-4699	36	2	set	set	NOUN
ap-4699	36	3	of	of	ADP
ap-4699	36	4	all	all	DET
ap-4699	36	5	numbers	number	NOUN
ap-4699	36	6	which	which	PRON
ap-4699	36	7	have	have	VERB
ap-4699	36	8	a	a	DET
ap-4699	36	9	(	(	PUNCT
ap-4699	36	10	β	β	X
ap-4699	36	11	,	,	PUNCT
ap-4699	36	12	a)representation	a)representation	NOUN
ap-4699	36	13	with	with	ADP
ap-4699	36	14	only	only	ADV
ap-4699	36	15	finitely	finitely	ADV
ap-4699	36	16	many	many	ADJ
ap-4699	36	17	non	non	ADJ
ap-4699	36	18	-	-	ADJ
ap-4699	36	19	zero	zero	ADJ
ap-4699	36	20	digits	digit	NOUN
ap-4699	36	21	is	be	AUX
ap-4699	36	22	denoted	denote	VERB
ap-4699	36	23	by	by	ADP
ap-4699	36	24	fina(β	fina(β	PROPN
ap-4699	36	25	)	)	PUNCT
ap-4699	36	26	:	:	PUNCT
ap-4699	36	27	=	=	PRON
ap-4699	36	28	{	{	PUNCT
ap-4699	36	29	n∑	n∑	NOUN
ap-4699	36	30	j=−m	j=−m	PROPN
ap-4699	36	31	xjβ	xjβ	PROPN
ap-4699	36	32	j	j	PROPN
ap-4699	36	33	:	:	PUNCT
ap-4699	36	34	n	n	CCONJ
ap-4699	36	35	,	,	PUNCT
ap-4699	36	36	m	m	VERB
ap-4699	36	37	∈	∈	PROPN
ap-4699	36	38	n	n	CCONJ
ap-4699	36	39	,	,	PUNCT
ap-4699	36	40	xj	xj	PROPN
ap-4699	36	41	∈	∈	PROPN
ap-4699	36	42	a	a	PRON
ap-4699	36	43	}	}	PUNCT
ap-4699	36	44	.	.	PUNCT
ap-4699	37	1	the	the	DET
ap-4699	37	2	set	set	NOUN
ap-4699	37	3	of	of	ADP
ap-4699	37	4	all	all	DET
ap-4699	37	5	numbers	number	NOUN
ap-4699	37	6	with	with	ADP
ap-4699	37	7	a	a	DET
ap-4699	37	8	finite	finite	NOUN
ap-4699	37	9	(	(	PUNCT
ap-4699	37	10	β	β	NOUN
ap-4699	37	11	,	,	PUNCT
ap-4699	37	12	a)-representation	a)-representation	NOUN
ap-4699	37	13	with	with	SCONJ
ap-4699	37	14	only	only	ADJ
ap-4699	37	15	non	non	ADJ
ap-4699	37	16	-	-	ADJ
ap-4699	37	17	negative	negative	ADJ
ap-4699	37	18	powers	power	NOUN
ap-4699	37	19	of	of	ADP
ap-4699	37	20	β	β	PROPN
ap-4699	37	21	is	be	AUX
ap-4699	37	22	denoted	denote	VERB
ap-4699	37	23	by	by	ADP
ap-4699	37	24	a[β	a[β	PROPN
ap-4699	37	25	]	]	PUNCT
ap-4699	37	26	:	:	PUNCT
ap-4699	37	27	=	=	SYM
ap-4699	37	28	{	{	PUNCT
ap-4699	37	29	n∑	n∑	PROPN
ap-4699	37	30	j=0	j=0	PROPN
ap-4699	37	31	xjβ	xjβ	PROPN
ap-4699	37	32	j	j	PROPN
ap-4699	37	33	:	:	PUNCT
ap-4699	37	34	n	n	CCONJ
ap-4699	37	35	∈	∈	PROPN
ap-4699	37	36	n	n	CCONJ
ap-4699	37	37	,	,	PUNCT
ap-4699	37	38	xj	xj	PROPN
ap-4699	37	39	∈	∈	PROPN
ap-4699	37	40	a	a	PRON
ap-4699	37	41	}	}	PUNCT
ap-4699	37	42	.	.	PUNCT
ap-4699	38	1	we	we	PRON
ap-4699	38	2	remark	remark	VERB
ap-4699	38	3	that	that	SCONJ
ap-4699	38	4	the	the	DET
ap-4699	38	5	definition	definition	NOUN
ap-4699	38	6	of	of	ADP
ap-4699	38	7	a[β	a[β	PROPN
ap-4699	38	8	]	]	PUNCT
ap-4699	38	9	is	be	AUX
ap-4699	38	10	analogous	analogous	ADJ
ap-4699	38	11	to	to	ADP
ap-4699	38	12	the	the	DET
ap-4699	38	13	one	one	NUM
ap-4699	38	14	of	of	ADP
ap-4699	38	15	z[β	z[β	NOUN
ap-4699	38	16	]	]	PUNCT
ap-4699	38	17	,	,	PUNCT
ap-4699	38	18	i.e.	i.e.	X
ap-4699	38	19	the	the	DET
ap-4699	38	20	smallest	small	ADJ
ap-4699	38	21	ring	ring	NOUN
ap-4699	38	22	containing	contain	VERB
ap-4699	38	23	z	z	PROPN
ap-4699	38	24	285	285	NUM
ap-4699	38	25	http://dx.doi.org/10.14311/ap.2018.58.0285	http://dx.doi.org/10.14311/ap.2018.58.0285	X
ap-4699	38	26	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4699	38	27	jan	jan	PROPN
ap-4699	38	28	legerský	legerský	PROPN
ap-4699	38	29	acta	acta	PROPN
ap-4699	38	30	polytechnica	polytechnica	PROPN
ap-4699	38	31	and	and	CCONJ
ap-4699	38	32	β	β	X
ap-4699	38	33	,	,	PUNCT
ap-4699	38	34	which	which	PRON
ap-4699	38	35	is	be	AUX
ap-4699	38	36	equivalent	equivalent	ADJ
ap-4699	38	37	to	to	ADP
ap-4699	38	38	the	the	DET
ap-4699	38	39	set	set	NOUN
ap-4699	38	40	of	of	ADP
ap-4699	38	41	all	all	DET
ap-4699	38	42	sums	sum	NOUN
ap-4699	38	43	of	of	ADP
ap-4699	38	44	powers	power	NOUN
ap-4699	38	45	of	of	ADP
ap-4699	38	46	β	β	PROPN
ap-4699	38	47	with	with	ADP
ap-4699	38	48	integer	integer	NOUN
ap-4699	38	49	coefficients	coefficient	NOUN
ap-4699	38	50	.	.	PUNCT
ap-4699	39	1	now	now	ADV
ap-4699	39	2	we	we	PRON
ap-4699	39	3	show	show	VERB
ap-4699	39	4	that	that	SCONJ
ap-4699	39	5	whenever	whenever	SCONJ
ap-4699	39	6	we	we	PRON
ap-4699	39	7	require	require	VERB
ap-4699	39	8	the	the	DET
ap-4699	39	9	alphabet	alphabet	NOUN
ap-4699	39	10	to	to	PART
ap-4699	39	11	be	be	AUX
ap-4699	39	12	finite	finite	ADJ
ap-4699	39	13	and	and	CCONJ
ap-4699	39	14	the	the	DET
ap-4699	39	15	sum	sum	NOUN
ap-4699	39	16	of	of	ADP
ap-4699	39	17	two	two	NUM
ap-4699	39	18	numbers	number	NOUN
ap-4699	39	19	with	with	ADP
ap-4699	39	20	finite	finite	PROPN
ap-4699	39	21	(	(	PUNCT
ap-4699	39	22	β	β	X
ap-4699	39	23	,	,	PUNCT
ap-4699	39	24	a)-representations	a)-representation	NOUN
ap-4699	39	25	to	to	PART
ap-4699	39	26	have	have	VERB
ap-4699	39	27	again	again	ADV
ap-4699	39	28	a	a	DET
ap-4699	39	29	finite	finite	NOUN
ap-4699	39	30	(	(	PUNCT
ap-4699	39	31	β	β	X
ap-4699	39	32	,	,	PUNCT
ap-4699	39	33	a)representation	a)representation	PROPN
ap-4699	39	34	(	(	PUNCT
ap-4699	39	35	which	which	PRON
ap-4699	39	36	is	be	AUX
ap-4699	39	37	the	the	DET
ap-4699	39	38	case	case	NOUN
ap-4699	39	39	of	of	ADP
ap-4699	39	40	parallel	parallel	ADJ
ap-4699	39	41	addition	addition	NOUN
ap-4699	39	42	)	)	PUNCT
ap-4699	39	43	,	,	PUNCT
ap-4699	39	44	then	then	ADV
ap-4699	39	45	we	we	PRON
ap-4699	39	46	can	can	AUX
ap-4699	39	47	consider	consider	VERB
ap-4699	39	48	only	only	ADJ
ap-4699	39	49	bases	basis	NOUN
ap-4699	39	50	which	which	PRON
ap-4699	39	51	are	be	AUX
ap-4699	39	52	algebraic	algebraic	ADJ
ap-4699	39	53	numbers	number	NOUN
ap-4699	39	54	.	.	PUNCT
ap-4699	40	1	lemma	lemma	PROPN
ap-4699	40	2	2.2	2.2	NUM
ap-4699	40	3	.	.	PUNCT
ap-4699	41	1	let	let	VERB
ap-4699	41	2	β	β	PRON
ap-4699	41	3	be	be	AUX
ap-4699	41	4	a	a	DET
ap-4699	41	5	complex	complex	ADJ
ap-4699	41	6	number	number	NOUN
ap-4699	41	7	such	such	ADJ
ap-4699	41	8	that	that	SCONJ
ap-4699	41	9	|β|	|β|	PRON
ap-4699	41	10	>	>	X
ap-4699	41	11	1	1	NUM
ap-4699	41	12	and	and	CCONJ
ap-4699	41	13	a	a	DET
ap-4699	41	14	⊂	⊂	PROPN
ap-4699	41	15	z[β	z[β	X
ap-4699	41	16	]	]	PUNCT
ap-4699	41	17	be	be	AUX
ap-4699	41	18	a	a	DET
ap-4699	41	19	finite	finite	ADJ
ap-4699	41	20	alphabet	alphabet	NOUN
ap-4699	41	21	with	with	ADP
ap-4699	41	22	0	0	NUM
ap-4699	41	23	∈	∈	PROPN
ap-4699	41	24	a	a	DET
ap-4699	41	25	and	and	CCONJ
ap-4699	41	26	1	1	NUM
ap-4699	41	27	∈	∈	PROPN
ap-4699	41	28	fina(β	fina(β	PROPN
ap-4699	41	29	)	)	PUNCT
ap-4699	41	30	.	.	PUNCT
ap-4699	42	1	if	if	SCONJ
ap-4699	42	2	n	n	NUM
ap-4699	42	3	⊂	⊂	PROPN
ap-4699	42	4	fina(β	fina(β	PROPN
ap-4699	42	5	)	)	PUNCT
ap-4699	42	6	,	,	PUNCT
ap-4699	42	7	then	then	ADV
ap-4699	42	8	β	β	X
ap-4699	42	9	is	be	AUX
ap-4699	42	10	an	an	DET
ap-4699	42	11	algebraic	algebraic	ADJ
ap-4699	42	12	number	number	NOUN
ap-4699	42	13	.	.	PUNCT
ap-4699	43	1	proof	proof	NOUN
ap-4699	43	2	.	.	PUNCT
ap-4699	44	1	since	since	SCONJ
ap-4699	44	2	a	a	DET
ap-4699	44	3	⊂	⊂	PROPN
ap-4699	44	4	z[β	z[β	NOUN
ap-4699	44	5	]	]	PUNCT
ap-4699	44	6	,	,	PUNCT
ap-4699	44	7	all	all	DET
ap-4699	44	8	digits	digit	NOUN
ap-4699	44	9	can	can	AUX
ap-4699	44	10	be	be	AUX
ap-4699	44	11	expressed	express	VERB
ap-4699	44	12	as	as	ADP
ap-4699	44	13	finite	finite	ADJ
ap-4699	44	14	integer	integer	NOUN
ap-4699	44	15	combinations	combination	NOUN
ap-4699	44	16	of	of	ADP
ap-4699	44	17	powers	power	NOUN
ap-4699	44	18	of	of	ADP
ap-4699	44	19	β	β	PROPN
ap-4699	44	20	.	.	PUNCT
ap-4699	45	1	let	let	VERB
ap-4699	45	2	d	d	PRON
ap-4699	45	3	be	be	AUX
ap-4699	45	4	the	the	DET
ap-4699	45	5	maximal	maximal	ADJ
ap-4699	45	6	exponent	exponent	NOUN
ap-4699	45	7	of	of	ADP
ap-4699	45	8	β	β	PROPN
ap-4699	45	9	occurring	occur	VERB
ap-4699	45	10	in	in	ADP
ap-4699	45	11	these	these	DET
ap-4699	45	12	expressions	expression	NOUN
ap-4699	45	13	and	and	CCONJ
ap-4699	45	14	c	c	NOUN
ap-4699	45	15	be	be	AUX
ap-4699	45	16	the	the	DET
ap-4699	45	17	maximal	maximal	ADJ
ap-4699	45	18	absolute	absolute	ADJ
ap-4699	45	19	value	value	NOUN
ap-4699	45	20	of	of	ADP
ap-4699	45	21	the	the	DET
ap-4699	45	22	integer	integer	NOUN
ap-4699	45	23	coefficients	coefficient	NOUN
ap-4699	45	24	of	of	ADP
ap-4699	45	25	all	all	DET
ap-4699	45	26	digits	digit	NOUN
ap-4699	45	27	in	in	ADP
ap-4699	45	28	a.	a.	NOUN
ap-4699	45	29	hence	hence	ADV
ap-4699	45	30	,	,	PUNCT
ap-4699	45	31	for	for	ADP
ap-4699	45	32	every	every	DET
ap-4699	45	33	n	n	PRON
ap-4699	45	34	∈	∈	PROPN
ap-4699	45	35	n	n	CCONJ
ap-4699	45	36	,	,	PUNCT
ap-4699	45	37	there	there	PRON
ap-4699	45	38	exist	exist	VERB
ap-4699	45	39	m	m	PRON
ap-4699	45	40	,	,	PUNCT
ap-4699	45	41	n	n	PROPN
ap-4699	45	42	∈	∈	PROPN
ap-4699	45	43	n	n	NOUN
ap-4699	45	44	and	and	CCONJ
ap-4699	45	45	a−m	a−m	NOUN
ap-4699	45	46	,	,	PUNCT
ap-4699	45	47	.	.	PUNCT
ap-4699	45	48	.	.	PUNCT
ap-4699	46	1	.	.	PUNCT
ap-4699	47	1	,	,	PUNCT
ap-4699	47	2	an	an	DET
ap-4699	47	3	∈	∈	PROPN
ap-4699	47	4	a	a	DET
ap-4699	47	5	,	,	PUNCT
ap-4699	47	6	where	where	SCONJ
ap-4699	47	7	ai	ai	VERB
ap-4699	47	8	=	=	PUNCT
ap-4699	47	9	∑d	∑d	PROPN
ap-4699	47	10	j=0	j=0	PROPN
ap-4699	47	11	αijβ	αijβ	PROPN
ap-4699	47	12	j	j	PROPN
ap-4699	47	13	with	with	ADP
ap-4699	47	14	αij	αij	PROPN
ap-4699	47	15	∈	∈	PROPN
ap-4699	47	16	z	z	PROPN
ap-4699	47	17	and	and	CCONJ
ap-4699	47	18	|αij	|αij	PROPN
ap-4699	47	19	|	|	ADV
ap-4699	47	20	≤	≤	NUM
ap-4699	47	21	c	c	X
ap-4699	47	22	,	,	PUNCT
ap-4699	47	23	such	such	ADJ
ap-4699	47	24	that	that	SCONJ
ap-4699	47	25	n	n	NOUN
ap-4699	47	26	=	=	SYM
ap-4699	47	27	n∑	n∑	PROPN
ap-4699	47	28	i=−m	i=−m	PROPN
ap-4699	47	29	aiβ	aiβ	VERB
ap-4699	47	30	i	i	PRON
ap-4699	47	31	=	=	SYM
ap-4699	47	32	n∑	n∑	PROPN
ap-4699	47	33	i=−m	i=−m	CCONJ
ap-4699	47	34	d∑	d∑	PROPN
ap-4699	47	35	j=0	j=0	PROPN
ap-4699	47	36	αijβ	αijβ	PROPN
ap-4699	47	37	i+j	i+j	NUM
ap-4699	47	38	.	.	PUNCT
ap-4699	48	1	suppose	suppose	VERB
ap-4699	48	2	for	for	ADP
ap-4699	48	3	contradiction	contradiction	NOUN
ap-4699	48	4	that	that	PRON
ap-4699	48	5	β	β	NOUN
ap-4699	48	6	is	be	AUX
ap-4699	48	7	transcendental	transcendental	ADJ
ap-4699	48	8	.	.	PUNCT
ap-4699	49	1	therefore	therefore	ADV
ap-4699	49	2	,	,	PUNCT
ap-4699	49	3	the	the	DET
ap-4699	49	4	corresponding	corresponding	ADJ
ap-4699	49	5	integer	integer	NOUN
ap-4699	49	6	coefficients	coefficient	NOUN
ap-4699	49	7	of	of	ADP
ap-4699	49	8	powers	power	NOUN
ap-4699	49	9	of	of	ADP
ap-4699	49	10	β	β	NOUN
ap-4699	49	11	on	on	ADP
ap-4699	49	12	the	the	DET
ap-4699	49	13	left	left	ADJ
ap-4699	49	14	hand	hand	NOUN
ap-4699	49	15	side	side	NOUN
ap-4699	49	16	and	and	CCONJ
ap-4699	49	17	on	on	ADP
ap-4699	49	18	the	the	DET
ap-4699	49	19	right	right	ADJ
ap-4699	49	20	hand	hand	NOUN
ap-4699	49	21	side	side	NOUN
ap-4699	49	22	must	must	AUX
ap-4699	49	23	be	be	AUX
ap-4699	49	24	equal	equal	ADJ
ap-4699	49	25	,	,	PUNCT
ap-4699	49	26	particularly∑	particularly∑	PROPN
ap-4699	49	27	i+j=0	i+j=0	VERB
ap-4699	49	28	0≤j≤d	0≤j≤d	PROPN
ap-4699	49	29	,	,	PUNCT
ap-4699	49	30	−m≤i≤n	−m≤i≤n	PROPN
ap-4699	49	31	αij	αij	NOUN
ap-4699	50	1	=	=	PUNCT
ap-4699	50	2	n.	n.	NOUN
ap-4699	50	3	this	this	PRON
ap-4699	50	4	is	be	AUX
ap-4699	50	5	a	a	DET
ap-4699	50	6	contradiction	contradiction	NOUN
ap-4699	50	7	,	,	PUNCT
ap-4699	50	8	since	since	SCONJ
ap-4699	50	9	the	the	DET
ap-4699	50	10	left	left	ADJ
ap-4699	50	11	hand	hand	NOUN
ap-4699	50	12	side	side	NOUN
ap-4699	50	13	is	be	AUX
ap-4699	50	14	bounded	bound	VERB
ap-4699	50	15	by	by	ADP
ap-4699	50	16	(	(	PUNCT
ap-4699	50	17	d+	d+	NOUN
ap-4699	50	18	1	1	NUM
ap-4699	50	19	)	)	PUNCT
ap-4699	50	20	·	·	PUNCT
ap-4699	51	1	c	c	X
ap-4699	51	2	,	,	PUNCT
ap-4699	51	3	whereas	whereas	SCONJ
ap-4699	51	4	n	n	PRON
ap-4699	51	5	can	can	AUX
ap-4699	51	6	be	be	AUX
ap-4699	51	7	arbitrarily	arbitrarily	ADV
ap-4699	51	8	large	large	ADJ
ap-4699	51	9	.	.	PUNCT
ap-4699	52	1	corollary	corollary	ADJ
ap-4699	52	2	2.3	2.3	NUM
ap-4699	52	3	.	.	PUNCT
ap-4699	53	1	let	let	VERB
ap-4699	53	2	β	β	PRON
ap-4699	53	3	be	be	AUX
ap-4699	53	4	a	a	DET
ap-4699	53	5	complex	complex	ADJ
ap-4699	53	6	number	number	NOUN
ap-4699	53	7	such	such	ADJ
ap-4699	53	8	that	that	SCONJ
ap-4699	53	9	|β|	|β|	PRON
ap-4699	53	10	>	>	X
ap-4699	53	11	1	1	NUM
ap-4699	53	12	and	and	CCONJ
ap-4699	53	13	let	let	VERB
ap-4699	53	14	a	a	DET
ap-4699	53	15	⊂	⊂	PROPN
ap-4699	53	16	z[β	z[β	X
ap-4699	53	17	]	]	PUNCT
ap-4699	53	18	be	be	AUX
ap-4699	53	19	a	a	DET
ap-4699	53	20	finite	finite	ADJ
ap-4699	53	21	alphabet	alphabet	NOUN
ap-4699	53	22	with	with	ADP
ap-4699	53	23	0	0	NUM
ap-4699	53	24	∈	∈	PROPN
ap-4699	53	25	a	a	PRON
ap-4699	53	26	and	and	CCONJ
ap-4699	53	27	1	1	NUM
ap-4699	53	28	∈	∈	PROPN
ap-4699	53	29	fina(β	fina(β	PROPN
ap-4699	53	30	)	)	PUNCT
ap-4699	53	31	,	,	PUNCT
ap-4699	53	32	resp	resp	NOUN
ap-4699	53	33	.	.	PUNCT
ap-4699	54	1	1	1	NUM
ap-4699	54	2	∈	∈	PROPN
ap-4699	54	3	a[β	a[β	PROPN
ap-4699	54	4	]	]	PUNCT
ap-4699	54	5	.	.	PUNCT
ap-4699	55	1	if	if	SCONJ
ap-4699	55	2	the	the	DET
ap-4699	55	3	set	set	NOUN
ap-4699	55	4	fina(β	fina(β	PROPN
ap-4699	55	5	)	)	PUNCT
ap-4699	55	6	,	,	PUNCT
ap-4699	55	7	resp	resp	NOUN
ap-4699	55	8	.	.	PUNCT
ap-4699	56	1	a[β	a[β	PROPN
ap-4699	56	2	]	]	PUNCT
ap-4699	56	3	,	,	PUNCT
ap-4699	56	4	is	be	AUX
ap-4699	56	5	closed	close	VERB
ap-4699	56	6	under	under	ADP
ap-4699	56	7	addition	addition	NOUN
ap-4699	56	8	,	,	PUNCT
ap-4699	56	9	then	then	ADV
ap-4699	56	10	β	β	X
ap-4699	56	11	is	be	AUX
ap-4699	56	12	an	an	DET
ap-4699	56	13	algebraic	algebraic	ADJ
ap-4699	56	14	number	number	NOUN
ap-4699	56	15	.	.	PUNCT
ap-4699	57	1	proof	proof	NOUN
ap-4699	57	2	.	.	PUNCT
ap-4699	58	1	the	the	DET
ap-4699	58	2	closedness	closedness	NOUN
ap-4699	58	3	of	of	ADP
ap-4699	58	4	fina(β	fina(β	PROPN
ap-4699	58	5	)	)	PUNCT
ap-4699	58	6	under	under	ADP
ap-4699	58	7	addition	addition	NOUN
ap-4699	58	8	and	and	CCONJ
ap-4699	58	9	1	1	NUM
ap-4699	58	10	∈	∈	PROPN
ap-4699	58	11	fina(β	fina(β	PROPN
ap-4699	58	12	)	)	PUNCT
ap-4699	58	13	implies	imply	VERB
ap-4699	58	14	n	n	PROPN
ap-4699	58	15	⊂	⊂	PROPN
ap-4699	58	16	fina(β	fina(β	PROPN
ap-4699	58	17	)	)	PUNCT
ap-4699	58	18	.	.	PUNCT
ap-4699	59	1	if	if	SCONJ
ap-4699	59	2	a[β	a[β	PROPN
ap-4699	59	3	]	]	PUNCT
ap-4699	59	4	is	be	AUX
ap-4699	59	5	closed	close	VERB
ap-4699	59	6	under	under	ADP
ap-4699	59	7	addition	addition	NOUN
ap-4699	59	8	and	and	CCONJ
ap-4699	59	9	1	1	NUM
ap-4699	59	10	∈	∈	PROPN
ap-4699	59	11	a[β	a[β	PROPN
ap-4699	59	12	]	]	X
ap-4699	59	13	⊂	⊂	PROPN
ap-4699	60	1	fina(β	fina(β	PROPN
ap-4699	60	2	)	)	PUNCT
ap-4699	60	3	,	,	PUNCT
ap-4699	60	4	then	then	ADV
ap-4699	60	5	n	n	PROPN
ap-4699	60	6	⊂	⊂	PROPN
ap-4699	60	7	a[β	a[β	PROPN
ap-4699	60	8	]	]	X
ap-4699	60	9	⊂	⊂	PROPN
ap-4699	61	1	fina(β	fina(β	PROPN
ap-4699	61	2	)	)	PUNCT
ap-4699	61	3	.	.	PUNCT
ap-4699	62	1	in	in	ADP
ap-4699	62	2	both	both	DET
ap-4699	62	3	cases	case	NOUN
ap-4699	62	4	,	,	PUNCT
ap-4699	62	5	lemma	lemma	PROPN
ap-4699	62	6	2.2	2.2	NUM
ap-4699	62	7	applies	applie	NOUN
ap-4699	62	8	.	.	PUNCT
ap-4699	63	1	the	the	DET
ap-4699	63	2	concept	concept	NOUN
ap-4699	63	3	of	of	ADP
ap-4699	63	4	parallelism	parallelism	NOUN
ap-4699	63	5	for	for	ADP
ap-4699	63	6	operations	operation	NOUN
ap-4699	63	7	on	on	ADP
ap-4699	63	8	representations	representation	NOUN
ap-4699	63	9	is	be	AUX
ap-4699	63	10	formalized	formalize	VERB
ap-4699	63	11	by	by	ADP
ap-4699	63	12	the	the	DET
ap-4699	63	13	following	follow	VERB
ap-4699	63	14	definition	definition	NOUN
ap-4699	63	15	.	.	PUNCT
ap-4699	64	1	definition	definition	NOUN
ap-4699	64	2	2.4	2.4	NUM
ap-4699	64	3	.	.	PUNCT
ap-4699	65	1	let	let	VERB
ap-4699	65	2	a	a	PRON
ap-4699	65	3	and	and	CCONJ
ap-4699	65	4	b	b	NOUN
ap-4699	65	5	be	be	AUX
ap-4699	65	6	alphabets	alphabet	NOUN
ap-4699	65	7	.	.	PUNCT
ap-4699	66	1	a	a	DET
ap-4699	66	2	function	function	NOUN
ap-4699	66	3	ϕ	ϕ	NOUN
ap-4699	66	4	:	:	PUNCT
ap-4699	66	5	bz	bz	PROPN
ap-4699	66	6	→	→	PUNCT
ap-4699	66	7	az	az	PROPN
ap-4699	66	8	is	be	AUX
ap-4699	66	9	said	say	VERB
ap-4699	66	10	to	to	PART
ap-4699	66	11	be	be	AUX
ap-4699	66	12	p	p	NOUN
ap-4699	66	13	-	-	ADJ
ap-4699	66	14	local	local	ADJ
ap-4699	66	15	if	if	SCONJ
ap-4699	66	16	there	there	PRON
ap-4699	66	17	exist	exist	VERB
ap-4699	66	18	r	r	NOUN
ap-4699	66	19	,	,	PUNCT
ap-4699	66	20	t	t	PROPN
ap-4699	66	21	∈	∈	PROPN
ap-4699	66	22	n	n	CCONJ
ap-4699	66	23	satisfying	satisfy	VERB
ap-4699	66	24	p	p	NOUN
ap-4699	66	25	=	=	PUNCT
ap-4699	66	26	r	r	NOUN
ap-4699	66	27	+	+	NOUN
ap-4699	66	28	t	t	NOUN
ap-4699	66	29	+	+	CCONJ
ap-4699	66	30	1	1	NUM
ap-4699	66	31	and	and	CCONJ
ap-4699	66	32	a	a	DET
ap-4699	66	33	function	function	NOUN
ap-4699	66	34	φ	φ	NOUN
ap-4699	66	35	:	:	PUNCT
ap-4699	66	36	bp	bp	PROPN
ap-4699	66	37	→	→	PUNCT
ap-4699	66	38	a	a	DET
ap-4699	66	39	such	such	ADJ
ap-4699	66	40	that	that	PRON
ap-4699	66	41	,	,	PUNCT
ap-4699	66	42	for	for	ADP
ap-4699	66	43	any	any	DET
ap-4699	66	44	w	w	NOUN
ap-4699	66	45	=	=	SYM
ap-4699	66	46	(	(	PUNCT
ap-4699	66	47	wj)j∈z	wj)j∈z	NUM
ap-4699	66	48	∈	∈	PROPN
ap-4699	66	49	bz	bz	PROPN
ap-4699	66	50	and	and	CCONJ
ap-4699	66	51	its	its	PRON
ap-4699	66	52	image	image	NOUN
ap-4699	66	53	z	z	NOUN
ap-4699	66	54	=	=	SYM
ap-4699	66	55	ϕ(w	ϕ(w	PROPN
ap-4699	66	56	)	)	PUNCT
ap-4699	66	57	=	=	SYM
ap-4699	66	58	(	(	PUNCT
ap-4699	66	59	zj)j∈z	zj)j∈z	NUM
ap-4699	66	60	∈	∈	PROPN
ap-4699	66	61	az	az	PROPN
ap-4699	66	62	,	,	PUNCT
ap-4699	66	63	we	we	PRON
ap-4699	66	64	have	have	VERB
ap-4699	66	65	zj	zj	NOUN
ap-4699	66	66	=	=	SYM
ap-4699	66	67	φ(wj+t	φ(wj+t	PROPN
ap-4699	66	68	,	,	PUNCT
ap-4699	66	69	·	·	PUNCT
ap-4699	66	70	·	·	PUNCT
ap-4699	66	71	·	·	PUNCT
ap-4699	66	72	,	,	PUNCT
ap-4699	66	73	wj−r	wj−r	PROPN
ap-4699	66	74	)	)	PUNCT
ap-4699	66	75	for	for	ADP
ap-4699	66	76	every	every	DET
ap-4699	66	77	j	j	PROPN
ap-4699	66	78	∈	∈	PROPN
ap-4699	66	79	z.	z.	PROPN
ap-4699	67	1	the	the	DET
ap-4699	67	2	parameter	parameter	PROPN
ap-4699	67	3	t	t	PROPN
ap-4699	67	4	,	,	PUNCT
ap-4699	67	5	resp	resp	NOUN
ap-4699	67	6	.	.	PUNCT
ap-4699	68	1	r	r	X
ap-4699	68	2	,	,	PUNCT
ap-4699	68	3	is	be	AUX
ap-4699	68	4	called	call	VERB
ap-4699	68	5	anticipation	anticipation	NOUN
ap-4699	68	6	,	,	PUNCT
ap-4699	68	7	resp	resp	NOUN
ap-4699	68	8	.	.	PUNCT
ap-4699	69	1	memory	memory	NOUN
ap-4699	69	2	.	.	PUNCT
ap-4699	70	1	in	in	ADP
ap-4699	70	2	other	other	ADJ
ap-4699	70	3	words	word	NOUN
ap-4699	70	4	,	,	PUNCT
ap-4699	70	5	every	every	DET
ap-4699	70	6	digit	digit	NOUN
ap-4699	70	7	can	can	AUX
ap-4699	70	8	by	by	SCONJ
ap-4699	70	9	determined	determine	VERB
ap-4699	70	10	from	from	ADP
ap-4699	70	11	only	only	ADV
ap-4699	70	12	limited	limited	ADJ
ap-4699	70	13	number	number	NOUN
ap-4699	70	14	of	of	ADP
ap-4699	70	15	neighboring	neighboring	NOUN
ap-4699	70	16	input	input	NOUN
ap-4699	70	17	digits	digit	NOUN
ap-4699	70	18	.	.	PUNCT
ap-4699	71	1	since	since	SCONJ
ap-4699	71	2	a	a	DET
ap-4699	71	3	(	(	PUNCT
ap-4699	71	4	β	β	X
ap-4699	71	5	,	,	PUNCT
ap-4699	71	6	a+a)-representation	a+a)-representation	NUM
ap-4699	71	7	of	of	ADP
ap-4699	71	8	sum	sum	NOUN
ap-4699	71	9	of	of	ADP
ap-4699	71	10	two	two	NUM
ap-4699	71	11	numbers	number	NOUN
ap-4699	71	12	can	can	AUX
ap-4699	71	13	be	be	AUX
ap-4699	71	14	easily	easily	ADV
ap-4699	71	15	obtained	obtain	VERB
ap-4699	71	16	by	by	ADP
ap-4699	71	17	digit	digit	NOUN
ap-4699	71	18	-	-	PUNCT
ap-4699	71	19	wise	wise	ADJ
ap-4699	71	20	addition	addition	NOUN
ap-4699	71	21	,	,	PUNCT
ap-4699	71	22	the	the	DET
ap-4699	71	23	crucial	crucial	ADJ
ap-4699	71	24	part	part	NOUN
ap-4699	71	25	of	of	ADP
ap-4699	71	26	parallel	parallel	ADJ
ap-4699	71	27	addition	addition	NOUN
ap-4699	71	28	is	be	AUX
ap-4699	71	29	conversion	conversion	NOUN
ap-4699	71	30	from	from	ADP
ap-4699	71	31	the	the	DET
ap-4699	71	32	alphabet	alphabet	NOUN
ap-4699	71	33	a+a	a+a	NUM
ap-4699	71	34	to	to	ADP
ap-4699	71	35	a.	a.	NOUN
ap-4699	71	36	definition	definition	NOUN
ap-4699	71	37	2.5	2.5	NUM
ap-4699	71	38	.	.	PUNCT
ap-4699	72	1	let	let	VERB
ap-4699	72	2	β	β	PRON
ap-4699	72	3	be	be	AUX
ap-4699	72	4	a	a	DET
ap-4699	72	5	base	base	NOUN
ap-4699	72	6	and	and	CCONJ
ap-4699	72	7	let	let	VERB
ap-4699	72	8	a	a	PRON
ap-4699	72	9	and	and	CCONJ
ap-4699	72	10	b	b	NOUN
ap-4699	72	11	be	be	VERB
ap-4699	72	12	alphabets	alphabet	NOUN
ap-4699	72	13	containing	contain	VERB
ap-4699	72	14	0	0	NUM
ap-4699	72	15	.	.	PUNCT
ap-4699	73	1	a	a	DET
ap-4699	73	2	function	function	NOUN
ap-4699	73	3	ϕ	ϕ	NOUN
ap-4699	73	4	:	:	PUNCT
ap-4699	73	5	bz	bz	PROPN
ap-4699	73	6	→	→	SYM
ap-4699	73	7	az	az	PROPN
ap-4699	73	8	such	such	ADJ
ap-4699	73	9	that	that	SCONJ
ap-4699	73	10	(	(	PUNCT
ap-4699	73	11	1	1	NUM
ap-4699	73	12	.	.	PUNCT
ap-4699	73	13	)	)	PUNCT
ap-4699	73	14	for	for	ADP
ap-4699	73	15	any	any	DET
ap-4699	73	16	w	w	NOUN
ap-4699	73	17	=	=	SYM
ap-4699	73	18	(	(	PUNCT
ap-4699	73	19	wj)j∈z	wj)j∈z	NUM
ap-4699	73	20	∈	∈	PROPN
ap-4699	73	21	bz	bz	X
ap-4699	73	22	with	with	ADP
ap-4699	73	23	finitely	finitely	ADV
ap-4699	73	24	many	many	ADJ
ap-4699	73	25	non	non	ADJ
ap-4699	73	26	-	-	ADJ
ap-4699	73	27	zero	zero	NUM
ap-4699	73	28	digits	digit	NOUN
ap-4699	73	29	,	,	PUNCT
ap-4699	73	30	z	z	NOUN
ap-4699	73	31	=	=	SYM
ap-4699	73	32	ϕ(w	ϕ(w	PROPN
ap-4699	73	33	)	)	PUNCT
ap-4699	73	34	=	=	SYM
ap-4699	73	35	(	(	PUNCT
ap-4699	73	36	zj)j∈z	zj)j∈z	NUM
ap-4699	73	37	∈	∈	NOUN
ap-4699	73	38	az	az	NOUN
ap-4699	73	39	has	have	VERB
ap-4699	73	40	only	only	ADV
ap-4699	73	41	finite	finite	VERB
ap-4699	73	42	number	number	NOUN
ap-4699	73	43	of	of	ADP
ap-4699	73	44	non	non	ADJ
ap-4699	73	45	-	-	ADJ
ap-4699	73	46	zero	zero	NUM
ap-4699	73	47	digits	digit	NOUN
ap-4699	73	48	,	,	PUNCT
ap-4699	73	49	and	and	CCONJ
ap-4699	73	50	(	(	PUNCT
ap-4699	73	51	2	2	NUM
ap-4699	73	52	.	.	PUNCT
ap-4699	73	53	)	)	PUNCT
ap-4699	73	54	∑	∑	PUNCT
ap-4699	74	1	j∈z	j∈z	PROPN
ap-4699	74	2	wjβ	wjβ	PROPN
ap-4699	74	3	j	j	PROPN
ap-4699	75	1	=	=	PUNCT
ap-4699	75	2	∑	∑	PUNCT
ap-4699	75	3	j∈z	j∈z	PROPN
ap-4699	75	4	zjβ	zjβ	PROPN
ap-4699	75	5	j	j	PROPN
ap-4699	75	6	,	,	PUNCT
ap-4699	75	7	is	be	AUX
ap-4699	75	8	called	call	VERB
ap-4699	75	9	a	a	DET
ap-4699	75	10	digit	digit	NOUN
ap-4699	75	11	set	set	NOUN
ap-4699	75	12	conversion	conversion	NOUN
ap-4699	75	13	in	in	ADP
ap-4699	75	14	the	the	DET
ap-4699	75	15	base	base	NOUN
ap-4699	75	16	β	β	NOUN
ap-4699	75	17	from	from	ADP
ap-4699	75	18	b	b	PROPN
ap-4699	75	19	to	to	PART
ap-4699	75	20	a.	a.	NOUN
ap-4699	75	21	such	such	DET
ap-4699	75	22	a	a	DET
ap-4699	75	23	conversion	conversion	NOUN
ap-4699	75	24	ϕ	ϕ	NOUN
ap-4699	75	25	is	be	AUX
ap-4699	75	26	said	say	VERB
ap-4699	75	27	to	to	PART
ap-4699	75	28	be	be	AUX
ap-4699	75	29	computable	computable	ADJ
ap-4699	75	30	in	in	ADP
ap-4699	75	31	parallel	parallel	NOUN
ap-4699	75	32	if	if	SCONJ
ap-4699	75	33	ϕ	ϕ	NOUN
ap-4699	75	34	is	be	AUX
ap-4699	75	35	a	a	DET
ap-4699	75	36	p	p	ADJ
ap-4699	75	37	-	-	PUNCT
ap-4699	75	38	local	local	ADJ
ap-4699	75	39	function	function	NOUN
ap-4699	75	40	for	for	ADP
ap-4699	75	41	some	some	DET
ap-4699	75	42	p	p	PROPN
ap-4699	75	43	∈	∈	PROPN
ap-4699	75	44	n.	n.	NOUN
ap-4699	75	45	parallel	parallel	NOUN
ap-4699	75	46	addition	addition	NOUN
ap-4699	75	47	in	in	ADP
ap-4699	75	48	a	a	DET
ap-4699	75	49	numeration	numeration	NOUN
ap-4699	75	50	system	system	NOUN
ap-4699	75	51	(	(	PUNCT
ap-4699	75	52	β	β	X
ap-4699	75	53	,	,	PUNCT
ap-4699	75	54	a	a	PRON
ap-4699	75	55	)	)	PUNCT
ap-4699	75	56	is	be	AUX
ap-4699	75	57	a	a	DET
ap-4699	75	58	digit	digit	NOUN
ap-4699	75	59	set	set	VERB
ap-4699	75	60	conversion	conversion	NOUN
ap-4699	75	61	in	in	ADP
ap-4699	75	62	the	the	DET
ap-4699	75	63	base	base	NOUN
ap-4699	75	64	β	β	NOUN
ap-4699	75	65	from	from	ADP
ap-4699	75	66	a	a	DET
ap-4699	75	67	+	+	NOUN
ap-4699	75	68	a	a	PRON
ap-4699	75	69	to	to	ADP
ap-4699	75	70	a	a	PRON
ap-4699	75	71	,	,	PUNCT
ap-4699	75	72	which	which	PRON
ap-4699	75	73	is	be	AUX
ap-4699	75	74	computable	computable	ADJ
ap-4699	75	75	in	in	ADP
ap-4699	75	76	parallel	parallel	NOUN
ap-4699	75	77	.	.	PUNCT
ap-4699	76	1	3	3	X
ap-4699	76	2	.	.	X
ap-4699	76	3	necessary	necessary	ADJ
ap-4699	76	4	conditions	condition	NOUN
ap-4699	76	5	on	on	ADP
ap-4699	76	6	alphabets	alphabet	NOUN
ap-4699	76	7	allowing	allow	VERB
ap-4699	76	8	parallel	parallel	ADJ
ap-4699	76	9	addition	addition	NOUN
ap-4699	76	10	in	in	ADP
ap-4699	76	11	a	a	DET
ap-4699	76	12	⊂	⊂	PROPN
ap-4699	76	13	z[β	z[β	NOUN
ap-4699	76	14	]	]	PUNCT
ap-4699	76	15	through	through	ADP
ap-4699	76	16	this	this	DET
ap-4699	76	17	section	section	NOUN
ap-4699	76	18	,	,	PUNCT
ap-4699	76	19	we	we	PRON
ap-4699	76	20	assume	assume	VERB
ap-4699	76	21	that	that	SCONJ
ap-4699	76	22	the	the	DET
ap-4699	76	23	base	base	NOUN
ap-4699	76	24	β	β	X
ap-4699	76	25	is	be	AUX
ap-4699	76	26	an	an	DET
ap-4699	76	27	algebraic	algebraic	ADJ
ap-4699	76	28	number	number	NOUN
ap-4699	76	29	and	and	CCONJ
ap-4699	76	30	the	the	DET
ap-4699	76	31	alphabet	alphabet	NOUN
ap-4699	76	32	a	a	PRON
ap-4699	76	33	is	be	AUX
ap-4699	76	34	a	a	DET
ap-4699	76	35	finite	finite	NOUN
ap-4699	76	36	subset	subset	NOUN
ap-4699	76	37	of	of	ADP
ap-4699	76	38	z[β	z[β	NOUN
ap-4699	76	39	]	]	PUNCT
ap-4699	76	40	such	such	ADJ
ap-4699	76	41	that	that	SCONJ
ap-4699	76	42	{	{	PUNCT
ap-4699	76	43	0	0	NUM
ap-4699	76	44	}	}	PUNCT
ap-4699	76	45	(	(	PUNCT
ap-4699	76	46	a.	a.	NOUN
ap-4699	76	47	the	the	DET
ap-4699	76	48	finiteness	finiteness	NOUN
ap-4699	76	49	of	of	ADP
ap-4699	76	50	the	the	DET
ap-4699	76	51	alphabet	alphabet	NOUN
ap-4699	76	52	is	be	AUX
ap-4699	76	53	a	a	DET
ap-4699	76	54	natural	natural	ADJ
ap-4699	76	55	assumption	assumption	NOUN
ap-4699	76	56	for	for	ADP
ap-4699	76	57	a	a	DET
ap-4699	76	58	practical	practical	ADJ
ap-4699	76	59	numeration	numeration	NOUN
ap-4699	76	60	system	system	NOUN
ap-4699	76	61	,	,	PUNCT
ap-4699	76	62	whereas	whereas	SCONJ
ap-4699	76	63	the	the	DET
ap-4699	76	64	requirement	requirement	NOUN
ap-4699	76	65	that	that	SCONJ
ap-4699	76	66	β	β	PROPN
ap-4699	76	67	is	be	AUX
ap-4699	76	68	an	an	DET
ap-4699	76	69	algebraic	algebraic	ADJ
ap-4699	76	70	number	number	NOUN
ap-4699	76	71	is	be	AUX
ap-4699	76	72	justified	justify	VERB
ap-4699	76	73	by	by	ADP
ap-4699	76	74	corollary	corollary	ADJ
ap-4699	76	75	2.3	2.3	NUM
ap-4699	76	76	.	.	PUNCT
ap-4699	77	1	we	we	PRON
ap-4699	77	2	recall	recall	VERB
ap-4699	77	3	that	that	PRON
ap-4699	77	4	for	for	ADP
ap-4699	77	5	an	an	DET
ap-4699	77	6	algebraic	algebraic	ADJ
ap-4699	77	7	number	number	NOUN
ap-4699	77	8	β	β	NOUN
ap-4699	77	9	,	,	PUNCT
ap-4699	77	10	if	if	SCONJ
ap-4699	77	11	α	α	X
ap-4699	77	12	,	,	PUNCT
ap-4699	77	13	γ	γ	PROPN
ap-4699	77	14	,	,	PUNCT
ap-4699	77	15	δ	δ	PROPN
ap-4699	77	16	are	be	AUX
ap-4699	77	17	elements	element	NOUN
ap-4699	77	18	of	of	ADP
ap-4699	77	19	z[β	z[β	NOUN
ap-4699	77	20	]	]	PUNCT
ap-4699	77	21	,	,	PUNCT
ap-4699	77	22	then	then	ADV
ap-4699	77	23	γ	γ	PROPN
ap-4699	77	24	is	be	AUX
ap-4699	77	25	congruent	congruent	ADJ
ap-4699	77	26	to	to	ADP
ap-4699	77	27	δ	δ	PROPN
ap-4699	77	28	modulo	modulo	VERB
ap-4699	77	29	α	α	NOUN
ap-4699	77	30	in	in	ADP
ap-4699	77	31	z[β	z[β	NOUN
ap-4699	77	32	]	]	PUNCT
ap-4699	77	33	,	,	PUNCT
ap-4699	77	34	denoted	denote	VERB
ap-4699	77	35	by	by	ADP
ap-4699	77	36	γ	γ	PROPN
ap-4699	77	37	≡α	≡α	PROPN
ap-4699	77	38	δ	δ	PROPN
ap-4699	77	39	,	,	PUNCT
ap-4699	77	40	if	if	SCONJ
ap-4699	77	41	there	there	PRON
ap-4699	77	42	exists	exist	VERB
ap-4699	77	43	ε	ε	PROPN
ap-4699	77	44	∈	∈	PROPN
ap-4699	77	45	z[β	z[β	PROPN
ap-4699	77	46	]	]	PUNCT
ap-4699	77	47	such	such	ADJ
ap-4699	77	48	that	that	SCONJ
ap-4699	77	49	γ	γ	PROPN
ap-4699	77	50	−	−	PROPN
ap-4699	77	51	δ	δ	PROPN
ap-4699	77	52	=	=	SYM
ap-4699	77	53	αε	αε	PROPN
ap-4699	77	54	.	.	NOUN
ap-4699	78	1	in	in	ADP
ap-4699	78	2	this	this	DET
ap-4699	78	3	section	section	NOUN
ap-4699	78	4	,	,	PUNCT
ap-4699	78	5	we	we	PRON
ap-4699	78	6	recall	recall	VERB
ap-4699	78	7	the	the	DET
ap-4699	78	8	known	know	VERB
ap-4699	78	9	results	result	NOUN
ap-4699	78	10	on	on	ADP
ap-4699	78	11	necessary	necessary	ADJ
ap-4699	78	12	properties	property	NOUN
ap-4699	78	13	of	of	ADP
ap-4699	78	14	integer	integer	NOUN
ap-4699	78	15	alphabets	alphabet	NOUN
ap-4699	78	16	allowing	allow	VERB
ap-4699	78	17	parallel	parallel	ADJ
ap-4699	78	18	addition	addition	NOUN
ap-4699	78	19	,	,	PUNCT
ap-4699	78	20	and	and	CCONJ
ap-4699	78	21	we	we	PRON
ap-4699	78	22	extend	extend	VERB
ap-4699	78	23	them	they	PRON
ap-4699	78	24	to	to	ADP
ap-4699	78	25	non	non	ADJ
ap-4699	78	26	-	-	ADJ
ap-4699	78	27	integer	integer	ADJ
ap-4699	78	28	alphabets	alphabet	NOUN
ap-4699	78	29	.	.	PUNCT
ap-4699	79	1	in	in	ADP
ap-4699	79	2	[	[	X
ap-4699	79	3	4	4	NUM
ap-4699	79	4	]	]	PUNCT
ap-4699	79	5	,	,	PUNCT
ap-4699	79	6	the	the	DET
ap-4699	79	7	following	follow	VERB
ap-4699	79	8	statement	statement	NOUN
ap-4699	79	9	is	be	AUX
ap-4699	79	10	proven	prove	VERB
ap-4699	79	11	:	:	PUNCT
ap-4699	79	12	theorem	theorem	ADJ
ap-4699	79	13	3.1	3.1	NUM
ap-4699	79	14	.	.	PUNCT
ap-4699	80	1	let	let	VERB
ap-4699	80	2	(	(	PUNCT
ap-4699	80	3	β	β	X
ap-4699	80	4	,	,	PUNCT
ap-4699	80	5	a	a	PRON
ap-4699	80	6	)	)	PUNCT
ap-4699	80	7	be	be	AUX
ap-4699	80	8	a	a	DET
ap-4699	80	9	numeration	numeration	NOUN
ap-4699	80	10	system	system	NOUN
ap-4699	80	11	such	such	ADJ
ap-4699	80	12	that	that	SCONJ
ap-4699	80	13	a	a	DET
ap-4699	80	14	⊂	⊂	PROPN
ap-4699	80	15	z[β	z[β	NOUN
ap-4699	80	16	]	]	PUNCT
ap-4699	80	17	.	.	PUNCT
ap-4699	81	1	if	if	SCONJ
ap-4699	81	2	there	there	PRON
ap-4699	81	3	exists	exist	VERB
ap-4699	81	4	a	a	DET
ap-4699	81	5	p	p	ADJ
ap-4699	81	6	-	-	PUNCT
ap-4699	81	7	local	local	ADJ
ap-4699	81	8	parallel	parallel	ADJ
ap-4699	81	9	addition	addition	NOUN
ap-4699	81	10	in	in	ADP
ap-4699	81	11	(	(	PUNCT
ap-4699	81	12	β	β	X
ap-4699	81	13	,	,	PUNCT
ap-4699	81	14	a	a	PRON
ap-4699	81	15	)	)	PUNCT
ap-4699	81	16	defined	define	VERB
ap-4699	81	17	by	by	ADP
ap-4699	81	18	a	a	DET
ap-4699	81	19	function	function	NOUN
ap-4699	81	20	φ	φ	NOUN
ap-4699	81	21	:	:	PUNCT
ap-4699	81	22	(	(	PUNCT
ap-4699	81	23	a+a)p	a+a)p	PROPN
ap-4699	81	24	→	→	SYM
ap-4699	81	25	a	a	PRON
ap-4699	81	26	,	,	PUNCT
ap-4699	81	27	then	then	ADV
ap-4699	81	28	φ(b	φ(b	ADP
ap-4699	81	29	,	,	PUNCT
ap-4699	81	30	.	.	PUNCT
ap-4699	81	31	.	.	PUNCT
ap-4699	82	1	.	.	PUNCT
ap-4699	83	1	,	,	PUNCT
ap-4699	83	2	b	b	X
ap-4699	83	3	)	)	PUNCT
ap-4699	83	4	≡β−1	≡β−1	NUM
ap-4699	83	5	b	b	NOUN
ap-4699	83	6	for	for	ADP
ap-4699	83	7	any	any	DET
ap-4699	83	8	b	b	PROPN
ap-4699	83	9	∈	∈	PROPN
ap-4699	83	10	a+a	a+a	NOUN
ap-4699	83	11	.	.	PUNCT
ap-4699	84	1	the	the	DET
ap-4699	84	2	same	same	ADJ
ap-4699	84	3	paper	paper	NOUN
ap-4699	84	4	explain	explain	VERB
ap-4699	84	5	that	that	SCONJ
ap-4699	84	6	,	,	PUNCT
ap-4699	84	7	when	when	SCONJ
ap-4699	84	8	considering	consider	VERB
ap-4699	84	9	only	only	ADV
ap-4699	84	10	integer	integer	NOUN
ap-4699	84	11	alphabets	alphabet	NOUN
ap-4699	84	12	a	a	DET
ap-4699	84	13	⊂	⊂	PROPN
ap-4699	84	14	z	z	PROPN
ap-4699	84	15	from	from	ADP
ap-4699	84	16	the	the	DET
ap-4699	84	17	perspective	perspective	NOUN
ap-4699	84	18	of	of	ADP
ap-4699	84	19	parallel	parallel	ADJ
ap-4699	84	20	addition	addition	NOUN
ap-4699	84	21	algorithms	algorithm	NOUN
ap-4699	84	22	,	,	PUNCT
ap-4699	84	23	all	all	DET
ap-4699	84	24	the	the	DET
ap-4699	84	25	numbers	number	NOUN
ap-4699	84	26	β	β	X
ap-4699	84	27	,	,	PUNCT
ap-4699	84	28	1	1	NUM
ap-4699	84	29	/	/	SYM
ap-4699	84	30	β	β	NOUN
ap-4699	84	31	,	,	PUNCT
ap-4699	84	32	and	and	CCONJ
ap-4699	84	33	their	their	PRON
ap-4699	84	34	algebraic	algebraic	ADJ
ap-4699	84	35	conjugates	conjugate	NOUN
ap-4699	84	36	behave	behave	VERB
ap-4699	84	37	analogously	analogously	ADV
ap-4699	84	38	:	:	PUNCT
ap-4699	84	39	parallel	parallel	ADJ
ap-4699	84	40	addition	addition	NOUN
ap-4699	84	41	algorithms	algorithm	NOUN
ap-4699	84	42	exist	exist	VERB
ap-4699	84	43	either	either	CCONJ
ap-4699	84	44	for	for	ADP
ap-4699	84	45	all	all	PRON
ap-4699	84	46	,	,	PUNCT
ap-4699	84	47	or	or	CCONJ
ap-4699	84	48	for	for	ADP
ap-4699	84	49	none	none	NOUN
ap-4699	84	50	of	of	ADP
ap-4699	84	51	them	they	PRON
ap-4699	84	52	.	.	PUNCT
ap-4699	85	1	this	this	DET
ap-4699	85	2	statement	statement	NOUN
ap-4699	85	3	can	can	AUX
ap-4699	85	4	be	be	AUX
ap-4699	85	5	extended	extend	VERB
ap-4699	85	6	to	to	ADP
ap-4699	85	7	non	non	ADJ
ap-4699	85	8	-	-	ADJ
ap-4699	85	9	integer	integer	ADJ
ap-4699	85	10	alphabets	alphabet	NOUN
ap-4699	85	11	as	as	ADV
ap-4699	85	12	well	well	ADV
ap-4699	85	13	–	–	PUNCT
ap-4699	85	14	the	the	DET
ap-4699	85	15	following	follow	VERB
ap-4699	85	16	lemma	lemma	PROPN
ap-4699	85	17	summarizes	summarize	NOUN
ap-4699	85	18	that	that	SCONJ
ap-4699	85	19	if	if	SCONJ
ap-4699	85	20	we	we	PRON
ap-4699	85	21	have	have	VERB
ap-4699	85	22	a	a	DET
ap-4699	85	23	parallel	parallel	ADJ
ap-4699	85	24	addition	addition	NOUN
ap-4699	85	25	algorithm	algorithm	NOUN
ap-4699	85	26	for	for	ADP
ap-4699	85	27	a	a	DET
ap-4699	85	28	base	base	NOUN
ap-4699	85	29	β	β	NOUN
ap-4699	85	30	,	,	PUNCT
ap-4699	85	31	then	then	ADV
ap-4699	85	32	we	we	PRON
ap-4699	85	33	easily	easily	ADV
ap-4699	85	34	obtain	obtain	VERB
ap-4699	85	35	such	such	DET
ap-4699	85	36	an	an	DET
ap-4699	85	37	algorithm	algorithm	NOUN
ap-4699	85	38	also	also	ADV
ap-4699	85	39	for	for	ADP
ap-4699	85	40	conjugates	conjugate	NOUN
ap-4699	85	41	of	of	ADP
ap-4699	85	42	β	β	NOUN
ap-4699	85	43	by	by	ADP
ap-4699	85	44	field	field	NOUN
ap-4699	85	45	isomorphism	isomorphism	NOUN
ap-4699	85	46	.	.	PUNCT
ap-4699	86	1	regarding	regard	VERB
ap-4699	86	2	the	the	DET
ap-4699	86	3	base	base	NOUN
ap-4699	86	4	1	1	NUM
ap-4699	86	5	/	/	SYM
ap-4699	86	6	β	β	NOUN
ap-4699	86	7	,	,	PUNCT
ap-4699	86	8	we	we	PRON
ap-4699	86	9	can	can	AUX
ap-4699	86	10	use	use	VERB
ap-4699	86	11	the	the	DET
ap-4699	86	12	equality	equality	NOUN
ap-4699	86	13	fina(β	fina(β	ADV
ap-4699	86	14	)	)	PUNCT
ap-4699	86	15	=	=	SYM
ap-4699	86	16	fina(1	fina(1	ADJ
ap-4699	86	17	/	/	SYM
ap-4699	86	18	β	β	NOUN
ap-4699	86	19	)	)	PUNCT
ap-4699	86	20	to	to	PART
ap-4699	86	21	transfer	transfer	VERB
ap-4699	86	22	the	the	DET
ap-4699	86	23	parallel	parallel	ADJ
ap-4699	86	24	addition	addition	NOUN
ap-4699	86	25	algorithm	algorithm	NOUN
ap-4699	86	26	,	,	PUNCT
ap-4699	86	27	and	and	CCONJ
ap-4699	86	28	thus	thus	ADV
ap-4699	86	29	in	in	ADP
ap-4699	86	30	fact	fact	NOUN
ap-4699	86	31	drop	drop	VERB
ap-4699	86	32	the	the	DET
ap-4699	86	33	requirement	requirement	NOUN
ap-4699	86	34	on	on	ADP
ap-4699	86	35	the	the	DET
ap-4699	86	36	base	base	NOUN
ap-4699	86	37	to	to	PART
ap-4699	86	38	be	be	AUX
ap-4699	86	39	greater	great	ADJ
ap-4699	86	40	than	than	ADP
ap-4699	86	41	1	1	NUM
ap-4699	86	42	in	in	ADP
ap-4699	86	43	modulus	modulus	NOUN
ap-4699	86	44	.	.	PUNCT
ap-4699	86	45	286	286	NUM
ap-4699	86	46	vol	vol	NOUN
ap-4699	86	47	.	.	PUNCT
ap-4699	87	1	58	58	NUM
ap-4699	87	2	no	no	INTJ
ap-4699	87	3	.	.	PUNCT
ap-4699	88	1	5/2018	5/2018	NUM
ap-4699	88	2	minimal	minimal	ADJ
ap-4699	88	3	non	non	ADJ
ap-4699	88	4	-	-	ADJ
ap-4699	88	5	integer	integer	ADJ
ap-4699	88	6	alphabets	alphabet	NOUN
ap-4699	88	7	allowing	allow	VERB
ap-4699	88	8	parallel	parallel	ADJ
ap-4699	88	9	addition	addition	NOUN
ap-4699	88	10	lemma	lemma	PROPN
ap-4699	88	11	3.2	3.2	NUM
ap-4699	88	12	.	.	PUNCT
ap-4699	89	1	let	let	VERB
ap-4699	89	2	(	(	PUNCT
ap-4699	89	3	β	β	X
ap-4699	89	4	,	,	PUNCT
ap-4699	89	5	a	a	PRON
ap-4699	89	6	)	)	PUNCT
ap-4699	89	7	be	be	AUX
ap-4699	89	8	a	a	DET
ap-4699	89	9	numeration	numeration	NOUN
ap-4699	89	10	system	system	NOUN
ap-4699	89	11	such	such	ADJ
ap-4699	89	12	that	that	SCONJ
ap-4699	89	13	a	a	DET
ap-4699	89	14	⊂	⊂	PROPN
ap-4699	89	15	z[β	z[β	X
ap-4699	89	16	]	]	PUNCT
ap-4699	89	17	and	and	CCONJ
ap-4699	89	18	β	β	X
ap-4699	89	19	is	be	AUX
ap-4699	89	20	an	an	DET
ap-4699	89	21	algebraic	algebraic	ADJ
ap-4699	89	22	number	number	NOUN
ap-4699	89	23	.	.	PUNCT
ap-4699	90	1	let	let	VERB
ap-4699	90	2	β′	β′	NOUN
ap-4699	90	3	be	be	AUX
ap-4699	90	4	a	a	DET
ap-4699	90	5	conjugate	conjugate	NOUN
ap-4699	90	6	of	of	ADP
ap-4699	90	7	β	β	PRON
ap-4699	90	8	such	such	ADJ
ap-4699	90	9	that	that	DET
ap-4699	90	10	|β′|	|β′|	NOUN
ap-4699	90	11	6=	6=	PRON
ap-4699	90	12	1	1	NUM
ap-4699	90	13	and	and	CCONJ
ap-4699	90	14	σ	σ	NOUN
ap-4699	90	15	:	:	PUNCT
ap-4699	90	16	q(β	q(β	PROPN
ap-4699	90	17	)	)	PUNCT
ap-4699	90	18	7→	7→	NUM
ap-4699	90	19	q(β′	q(β′	PROPN
ap-4699	90	20	)	)	PUNCT
ap-4699	90	21	be	be	AUX
ap-4699	90	22	the	the	DET
ap-4699	90	23	corresponding	corresponding	ADJ
ap-4699	90	24	field	field	NOUN
ap-4699	90	25	isomorphism	isomorphism	NOUN
ap-4699	90	26	.	.	PUNCT
ap-4699	91	1	if	if	SCONJ
ap-4699	91	2	there	there	PRON
ap-4699	91	3	is	be	VERB
ap-4699	91	4	a	a	DET
ap-4699	91	5	p	p	ADJ
ap-4699	91	6	-	-	PUNCT
ap-4699	91	7	local	local	ADJ
ap-4699	91	8	parallel	parallel	NOUN
ap-4699	91	9	addition	addition	NOUN
ap-4699	91	10	function	function	NOUN
ap-4699	91	11	ϕ	ϕ	NOUN
ap-4699	91	12	in	in	ADP
ap-4699	91	13	(	(	PUNCT
ap-4699	91	14	β	β	X
ap-4699	91	15	,	,	PUNCT
ap-4699	91	16	a	a	NOUN
ap-4699	91	17	)	)	PUNCT
ap-4699	91	18	,	,	PUNCT
ap-4699	91	19	then	then	ADV
ap-4699	91	20	there	there	PRON
ap-4699	91	21	exists	exist	VERB
ap-4699	91	22	a	a	DET
ap-4699	91	23	p	p	ADJ
ap-4699	91	24	-	-	PUNCT
ap-4699	91	25	local	local	ADJ
ap-4699	91	26	parallel	parallel	ADJ
ap-4699	91	27	addition	addition	NOUN
ap-4699	91	28	function	function	NOUN
ap-4699	91	29	ϕ′	ϕ′	PUNCT
ap-4699	91	30	in	in	ADP
ap-4699	91	31	(	(	PUNCT
ap-4699	91	32	β′,a′	β′,a′	NOUN
ap-4699	91	33	)	)	PUNCT
ap-4699	91	34	,	,	PUNCT
ap-4699	91	35	where	where	SCONJ
ap-4699	91	36	a′	a′	PROPN
ap-4699	91	37	=	=	SYM
ap-4699	91	38	{	{	PUNCT
ap-4699	91	39	σ(a	σ(a	PROPN
ap-4699	91	40	)	)	PUNCT
ap-4699	91	41	:	:	PUNCT
ap-4699	91	42	a	a	DET
ap-4699	91	43	∈	∈	PROPN
ap-4699	91	44	a	a	PRON
ap-4699	91	45	}	}	PUNCT
ap-4699	91	46	.	.	PUNCT
ap-4699	92	1	proof	proof	NOUN
ap-4699	92	2	.	.	PUNCT
ap-4699	93	1	let	let	VERB
ap-4699	93	2	φ	φ	PROPN
ap-4699	93	3	:	:	PUNCT
ap-4699	93	4	ap	ap	PROPN
ap-4699	93	5	→	→	PUNCT
ap-4699	93	6	a	a	DET
ap-4699	93	7	be	be	AUX
ap-4699	93	8	a	a	DET
ap-4699	93	9	mapping	mapping	NOUN
ap-4699	93	10	which	which	PRON
ap-4699	93	11	defines	define	VERB
ap-4699	93	12	ϕ	ϕ	NOUN
ap-4699	93	13	with	with	ADP
ap-4699	93	14	p	p	NOUN
ap-4699	93	15	=	=	SYM
ap-4699	93	16	r+t+1	r+t+1	PROPN
ap-4699	93	17	.	.	PUNCT
ap-4699	94	1	we	we	PRON
ap-4699	94	2	define	define	VERB
ap-4699	94	3	a	a	DET
ap-4699	94	4	mapping	mapping	NOUN
ap-4699	94	5	φ′	φ′	NUM
ap-4699	94	6	:	:	PUNCT
ap-4699	94	7	(	(	PUNCT
ap-4699	94	8	a′)p	a′)p	PROPN
ap-4699	94	9	→	→	SYM
ap-4699	94	10	a′	a′	NOUN
ap-4699	94	11	by	by	ADP
ap-4699	94	12	φ′(w′j+t	φ′(w′j+t	PROPN
ap-4699	94	13	,	,	PUNCT
ap-4699	94	14	.	.	PUNCT
ap-4699	94	15	.	.	PUNCT
ap-4699	95	1	.	.	PUNCT
ap-4699	96	1	,	,	PUNCT
ap-4699	96	2	w′j−r	w′j−r	NOUN
ap-4699	96	3	)	)	PUNCT
ap-4699	97	1	=	=	SYM
ap-4699	97	2	σ	σ	PROPN
ap-4699	97	3	(	(	PUNCT
ap-4699	97	4	φ(σ−1(w′j+t	φ(σ−1(w′j+t	PROPN
ap-4699	97	5	)	)	PUNCT
ap-4699	97	6	,	,	PUNCT
ap-4699	97	7	.	.	PUNCT
ap-4699	97	8	.	.	PUNCT
ap-4699	97	9	.	.	PUNCT
ap-4699	98	1	,	,	PUNCT
ap-4699	98	2	σ−1(w′j−r	σ−1(w′j−r	NOUN
ap-4699	98	3	)	)	PUNCT
ap-4699	98	4	)	)	PUNCT
ap-4699	98	5	)	)	PUNCT
ap-4699	98	6	.	.	PUNCT
ap-4699	99	1	next	next	ADV
ap-4699	99	2	,	,	PUNCT
ap-4699	99	3	we	we	PRON
ap-4699	99	4	define	define	VERB
ap-4699	99	5	a	a	DET
ap-4699	99	6	digit	digit	NOUN
ap-4699	99	7	set	set	VERB
ap-4699	99	8	conversion	conversion	NOUN
ap-4699	99	9	ϕ′	ϕ′	PROPN
ap-4699	99	10	:	:	PUNCT
ap-4699	99	11	(	(	PUNCT
ap-4699	99	12	a′	a′	PROPN
ap-4699	99	13	+	+	ADJ
ap-4699	99	14	a′)→	a′)→	NOUN
ap-4699	99	15	a′	a′	NOUN
ap-4699	99	16	by	by	ADP
ap-4699	99	17	ϕ′(w′	ϕ′(w′	PROPN
ap-4699	99	18	)	)	PUNCT
ap-4699	99	19	=	=	SYM
ap-4699	100	1	(	(	PUNCT
ap-4699	100	2	z′j)j∈z	z′j)j∈z	X
ap-4699	100	3	,	,	PUNCT
ap-4699	100	4	where	where	SCONJ
ap-4699	100	5	w′	w′	PROPN
ap-4699	100	6	=	=	SYM
ap-4699	100	7	(	(	PUNCT
ap-4699	100	8	w′j)j∈z	w′j)j∈z	X
ap-4699	100	9	and	and	CCONJ
ap-4699	100	10	z′j	z′j	NOUN
ap-4699	100	11	=	=	SYM
ap-4699	100	12	φ′(w′j+t	φ′(w′j+t	PROPN
ap-4699	100	13	,	,	PUNCT
ap-4699	100	14	.	.	PUNCT
ap-4699	100	15	.	.	PUNCT
ap-4699	101	1	.	.	PUNCT
ap-4699	102	1	,	,	PUNCT
ap-4699	102	2	w′j−r	w′j−r	NOUN
ap-4699	102	3	)	)	PUNCT
ap-4699	102	4	.	.	PUNCT
ap-4699	103	1	obviously	obviously	ADV
ap-4699	103	2	,	,	PUNCT
ap-4699	103	3	if	if	SCONJ
ap-4699	103	4	w′	w′	PROPN
ap-4699	103	5	has	have	VERB
ap-4699	103	6	only	only	ADV
ap-4699	103	7	finitely	finitely	ADV
ap-4699	103	8	many	many	ADJ
ap-4699	103	9	non	non	ADJ
ap-4699	103	10	-	-	ADJ
ap-4699	103	11	zero	zero	NUM
ap-4699	103	12	entries	entry	NOUN
ap-4699	103	13	,	,	PUNCT
ap-4699	103	14	then	then	ADV
ap-4699	103	15	there	there	PRON
ap-4699	103	16	is	be	VERB
ap-4699	103	17	only	only	ADV
ap-4699	103	18	finitely	finitely	ADV
ap-4699	103	19	many	many	ADJ
ap-4699	103	20	non	non	NOUN
ap-4699	103	21	-	-	NOUN
ap-4699	103	22	zeros	zero	NOUN
ap-4699	103	23	in	in	ADP
ap-4699	103	24	(	(	PUNCT
ap-4699	103	25	z′j)j∈z	z′j)j∈z	X
ap-4699	103	26	,	,	PUNCT
ap-4699	103	27	since	since	SCONJ
ap-4699	103	28	φ′(0	φ′(0	NOUN
ap-4699	103	29	,	,	PUNCT
ap-4699	103	30	.	.	PUNCT
ap-4699	103	31	.	.	PUNCT
ap-4699	104	1	.	.	PUNCT
ap-4699	105	1	,	,	PUNCT
ap-4699	105	2	0	0	X
ap-4699	105	3	)	)	PUNCT
ap-4699	106	1	=	=	SYM
ap-4699	106	2	σ	σ	PROPN
ap-4699	106	3	(	(	PUNCT
ap-4699	106	4	φ(σ−1(0	φ(σ−1(0	PROPN
ap-4699	106	5	)	)	PUNCT
ap-4699	106	6	,	,	PUNCT
ap-4699	106	7	.	.	PUNCT
ap-4699	106	8	.	.	PUNCT
ap-4699	106	9	.	.	PUNCT
ap-4699	106	10	,	,	PUNCT
ap-4699	106	11	σ−1(0	σ−1(0	PROPN
ap-4699	106	12	)	)	PUNCT
ap-4699	106	13	)	)	PUNCT
ap-4699	106	14	)	)	PUNCT
ap-4699	107	1	=	=	PUNCT
ap-4699	107	2	σ	σ	PROPN
ap-4699	107	3	(	(	PUNCT
ap-4699	107	4	φ(0	φ(0	ADJ
ap-4699	107	5	,	,	PUNCT
ap-4699	107	6	.	.	PUNCT
ap-4699	107	7	.	.	PUNCT
ap-4699	107	8	.	.	PUNCT
ap-4699	108	1	,	,	PUNCT
ap-4699	108	2	0	0	NUM
ap-4699	108	3	)	)	PUNCT
ap-4699	108	4	)	)	PUNCT
ap-4699	109	1	=	=	PUNCT
ap-4699	109	2	σ	σ	PROPN
ap-4699	109	3	(	(	PUNCT
ap-4699	109	4	0	0	NUM
ap-4699	109	5	)	)	PUNCT
ap-4699	109	6	=	=	SYM
ap-4699	110	1	0	0	X
ap-4699	110	2	.	.	PUNCT
ap-4699	111	1	the	the	DET
ap-4699	111	2	value	value	NOUN
ap-4699	111	3	of	of	ADP
ap-4699	111	4	the	the	DET
ap-4699	111	5	number	number	NOUN
ap-4699	111	6	represented	represent	VERB
ap-4699	111	7	by	by	ADP
ap-4699	111	8	w′	w′	PROPN
ap-4699	111	9	is	be	AUX
ap-4699	111	10	also	also	ADV
ap-4699	111	11	preserved	preserve	VERB
ap-4699	111	12	:	:	PUNCT
ap-4699	111	13	∑	∑	PUNCT
ap-4699	111	14	j∈z	j∈z	X
ap-4699	111	15	w′jβ	w′jβ	ADV
ap-4699	111	16	′j	′j	NOUN
ap-4699	111	17	=	=	PUNCT
ap-4699	111	18	∑	∑	PUNCT
ap-4699	111	19	j∈z	j∈z	PROPN
ap-4699	111	20	σ(wj)σ(β)j	σ(wj)σ(β)j	NUM
ap-4699	111	21	=	=	SYM
ap-4699	111	22	σ	σ	PROPN
ap-4699	111	23	(	(	PUNCT
ap-4699	111	24	∑	∑	INTJ
ap-4699	111	25	j∈z	j∈z	PROPN
ap-4699	111	26	wjβ	wjβ	PROPN
ap-4699	111	27	j	j	PROPN
ap-4699	111	28	)	)	PUNCT
ap-4699	112	1	=	=	SYM
ap-4699	112	2	σ	σ	PROPN
ap-4699	112	3	(	(	PUNCT
ap-4699	112	4	∑	∑	PROPN
ap-4699	112	5	j∈z	j∈z	PROPN
ap-4699	112	6	zjβ	zjβ	PROPN
ap-4699	112	7	j	j	PROPN
ap-4699	112	8	)	)	PUNCT
ap-4699	113	1	=	=	SYM
ap-4699	113	2	σ	σ	PROPN
ap-4699	113	3	(	(	PUNCT
ap-4699	113	4	∑	∑	PROPN
ap-4699	113	5	j∈z	j∈z	PROPN
ap-4699	113	6	φ	φ	PROPN
ap-4699	113	7	(	(	PUNCT
ap-4699	113	8	wj+t	wj+t	PROPN
ap-4699	113	9	,	,	PUNCT
ap-4699	113	10	.	.	PUNCT
ap-4699	113	11	.	.	PUNCT
ap-4699	113	12	.	.	PUNCT
ap-4699	114	1	,	,	PUNCT
ap-4699	114	2	wj−r	wj−r	NOUN
ap-4699	114	3	)	)	PUNCT
ap-4699	114	4	βj	βj	PROPN
ap-4699	114	5	)	)	PUNCT
ap-4699	115	1	=	=	PUNCT
ap-4699	115	2	∑	∑	PUNCT
ap-4699	115	3	j∈z	j∈z	PROPN
ap-4699	115	4	σ	σ	PROPN
ap-4699	115	5	(	(	PUNCT
ap-4699	115	6	φ	φ	PROPN
ap-4699	115	7	(	(	PUNCT
ap-4699	115	8	wj+t	wj+t	PROPN
ap-4699	115	9	,	,	PUNCT
ap-4699	115	10	.	.	PUNCT
ap-4699	115	11	.	.	PUNCT
ap-4699	116	1	.	.	PUNCT
ap-4699	117	1	,	,	PUNCT
ap-4699	117	2	wj−r	wj−r	PROPN
ap-4699	117	3	)	)	PUNCT
ap-4699	117	4	)	)	PUNCT
ap-4699	118	1	β′	β′	NUM
ap-4699	119	1	j	j	X
ap-4699	120	1	=	=	PUNCT
ap-4699	120	2	∑	∑	PUNCT
ap-4699	120	3	j∈z	j∈z	NOUN
ap-4699	120	4	z′jβ	z′jβ	PROPN
ap-4699	120	5	′j	′j	NOUN
ap-4699	120	6	,	,	PUNCT
ap-4699	120	7	where	where	SCONJ
ap-4699	120	8	wj	wj	PROPN
ap-4699	120	9	=	=	SYM
ap-4699	120	10	σ−1(w′j	σ−1(w′j	NUM
ap-4699	120	11	)	)	PUNCT
ap-4699	120	12	for	for	ADP
ap-4699	120	13	j	j	PROPN
ap-4699	120	14	∈	∈	PROPN
ap-4699	120	15	z	z	PROPN
ap-4699	120	16	and	and	CCONJ
ap-4699	120	17	ϕ((wj)j∈z	ϕ((wj)j∈z	NUM
ap-4699	120	18	)	)	PUNCT
ap-4699	120	19	=	=	SYM
ap-4699	120	20	(	(	PUNCT
ap-4699	120	21	zj)j∈z	zj)j∈z	NUM
ap-4699	120	22	.	.	PUNCT
ap-4699	121	1	next	next	ADV
ap-4699	121	2	,	,	PUNCT
ap-4699	121	3	it	it	PRON
ap-4699	121	4	is	be	AUX
ap-4699	121	5	shown	show	VERB
ap-4699	121	6	again	again	ADV
ap-4699	121	7	in	in	ADP
ap-4699	121	8	[	[	X
ap-4699	121	9	4	4	X
ap-4699	121	10	]	]	PUNCT
ap-4699	121	11	that	that	SCONJ
ap-4699	121	12	if	if	SCONJ
ap-4699	121	13	a	a	DET
ap-4699	121	14	base	base	NOUN
ap-4699	121	15	β	β	X
ap-4699	121	16	has	have	VERB
ap-4699	121	17	a	a	DET
ap-4699	121	18	real	real	ADJ
ap-4699	121	19	conjugate	conjugate	NOUN
ap-4699	121	20	greater	great	ADJ
ap-4699	121	21	than	than	ADP
ap-4699	121	22	one	one	NUM
ap-4699	121	23	,	,	PUNCT
ap-4699	121	24	then	then	ADV
ap-4699	121	25	there	there	PRON
ap-4699	121	26	are	be	VERB
ap-4699	121	27	some	some	DET
ap-4699	121	28	extra	extra	ADJ
ap-4699	121	29	requirements	requirement	NOUN
ap-4699	121	30	on	on	ADP
ap-4699	121	31	the	the	DET
ap-4699	121	32	alphabet	alphabet	NOUN
ap-4699	121	33	a	a	DET
ap-4699	121	34	⊂	⊂	PROPN
ap-4699	121	35	z[β	z[β	NOUN
ap-4699	121	36	]	]	PUNCT
ap-4699	121	37	.	.	PUNCT
ap-4699	122	1	the	the	DET
ap-4699	122	2	following	follow	VERB
ap-4699	122	3	lemma	lemma	PROPN
ap-4699	122	4	strengthens	strengthen	VERB
ap-4699	122	5	the	the	DET
ap-4699	122	6	results	result	NOUN
ap-4699	122	7	a	a	DET
ap-4699	122	8	bit	bit	NOUN
ap-4699	122	9	.	.	PUNCT
ap-4699	123	1	lemma	lemma	PROPN
ap-4699	123	2	3.3	3.3	NUM
ap-4699	123	3	.	.	PUNCT
ap-4699	124	1	let	let	VERB
ap-4699	124	2	(	(	PUNCT
ap-4699	124	3	β	β	X
ap-4699	124	4	,	,	PUNCT
ap-4699	124	5	a	a	PRON
ap-4699	124	6	)	)	PUNCT
ap-4699	124	7	be	be	AUX
ap-4699	124	8	a	a	DET
ap-4699	124	9	numeration	numeration	NOUN
ap-4699	124	10	system	system	NOUN
ap-4699	124	11	such	such	ADJ
ap-4699	124	12	that	that	SCONJ
ap-4699	124	13	a	a	DET
ap-4699	124	14	⊂	⊂	PROPN
ap-4699	124	15	z[β	z[β	X
ap-4699	124	16	]	]	PUNCT
ap-4699	124	17	and	and	CCONJ
ap-4699	124	18	1	1	NUM
ap-4699	124	19	<	<	X
ap-4699	124	20	β	β	X
ap-4699	124	21	∈	∈	PROPN
ap-4699	124	22	r.	r.	PROPN
ap-4699	124	23	let	let	VERB
ap-4699	124	24	λ	λ	X
ap-4699	124	25	=	=	SYM
ap-4699	124	26	mina	mina	PROPN
ap-4699	124	27	and	and	CCONJ
ap-4699	124	28	λ	λ	NOUN
ap-4699	124	29	=	=	NOUN
ap-4699	124	30	maxa	maxa	NOUN
ap-4699	124	31	.	.	PUNCT
ap-4699	125	1	if	if	SCONJ
ap-4699	125	2	there	there	PRON
ap-4699	125	3	exists	exist	VERB
ap-4699	125	4	a	a	DET
ap-4699	125	5	p	p	ADJ
ap-4699	125	6	-	-	PUNCT
ap-4699	125	7	local	local	ADJ
ap-4699	125	8	parallel	parallel	ADJ
ap-4699	125	9	addition	addition	NOUN
ap-4699	125	10	in	in	ADP
ap-4699	125	11	(	(	PUNCT
ap-4699	125	12	β	β	X
ap-4699	125	13	,	,	PUNCT
ap-4699	125	14	a	a	PRON
ap-4699	125	15	)	)	PUNCT
ap-4699	125	16	,	,	PUNCT
ap-4699	125	17	with	with	ADP
ap-4699	125	18	p	p	NOUN
ap-4699	125	19	=	=	PUNCT
ap-4699	125	20	r	r	NOUN
ap-4699	125	21	+	+	NOUN
ap-4699	125	22	t	t	NOUN
ap-4699	125	23	+	+	CCONJ
ap-4699	125	24	1	1	NUM
ap-4699	125	25	,	,	PUNCT
ap-4699	125	26	defined	define	VERB
ap-4699	125	27	by	by	ADP
ap-4699	125	28	a	a	DET
ap-4699	125	29	mapping	mapping	NOUN
ap-4699	125	30	φ	φ	NOUN
ap-4699	125	31	:	:	PUNCT
ap-4699	125	32	(	(	PUNCT
ap-4699	125	33	a+a)p	a+a)p	PROPN
ap-4699	125	34	→	→	SYM
ap-4699	125	35	a	a	X
ap-4699	125	36	,	,	PUNCT
ap-4699	125	37	then	then	ADV
ap-4699	125	38	:	:	PUNCT
ap-4699	125	39	(	(	PUNCT
ap-4699	125	40	1	1	X
ap-4699	125	41	.	.	PUNCT
ap-4699	125	42	)	)	PUNCT
ap-4699	126	1	φ(b	φ(b	ADV
ap-4699	126	2	,	,	PUNCT
ap-4699	126	3	.	.	PUNCT
ap-4699	126	4	.	.	PUNCT
ap-4699	126	5	.	.	PUNCT
ap-4699	127	1	,	,	PUNCT
ap-4699	127	2	b	b	X
ap-4699	127	3	)	)	PUNCT
ap-4699	127	4	6=	6=	ADP
ap-4699	127	5	λ	λ	NOUN
ap-4699	127	6	for	for	ADP
ap-4699	127	7	all	all	DET
ap-4699	127	8	b	b	PROPN
ap-4699	127	9	∈	∈	PROPN
ap-4699	128	1	a	a	PRON
ap-4699	128	2	+	+	NOUN
ap-4699	128	3	a	a	DET
ap-4699	128	4	such	such	ADJ
ap-4699	128	5	that	that	DET
ap-4699	128	6	b	b	X
ap-4699	128	7	>	>	X
ap-4699	128	8	λ	λ	X
ap-4699	128	9	∧	∧	PROPN
ap-4699	128	10	(	(	PUNCT
ap-4699	128	11	b	b	NOUN
ap-4699	128	12	≥	≥	NOUN
ap-4699	128	13	0	0	NUM
ap-4699	128	14	∨	∨	NUM
ap-4699	128	15	t	t	NOUN
ap-4699	128	16	=	=	SYM
ap-4699	128	17	0	0	NUM
ap-4699	128	18	)	)	PUNCT
ap-4699	128	19	,	,	PUNCT
ap-4699	128	20	(	(	PUNCT
ap-4699	128	21	2	2	NUM
ap-4699	128	22	.	.	PUNCT
ap-4699	128	23	)	)	PUNCT
ap-4699	129	1	φ(b	φ(b	ADV
ap-4699	129	2	,	,	PUNCT
ap-4699	129	3	.	.	PUNCT
ap-4699	129	4	.	.	PUNCT
ap-4699	129	5	.	.	PUNCT
ap-4699	130	1	,	,	PUNCT
ap-4699	130	2	b	b	X
ap-4699	130	3	)	)	PUNCT
ap-4699	130	4	6=	6=	ADP
ap-4699	130	5	λ	λ	NOUN
ap-4699	130	6	for	for	ADP
ap-4699	130	7	all	all	DET
ap-4699	130	8	b	b	PROPN
ap-4699	130	9	∈	∈	PROPN
ap-4699	131	1	a	a	PRON
ap-4699	131	2	+	+	NOUN
ap-4699	131	3	a	a	DET
ap-4699	131	4	such	such	ADJ
ap-4699	131	5	that	that	DET
ap-4699	131	6	b	b	NOUN
ap-4699	131	7	<	<	X
ap-4699	131	8	λ	λ	X
ap-4699	131	9	∧	∧	PROPN
ap-4699	131	10	(	(	PUNCT
ap-4699	131	11	b	b	NOUN
ap-4699	131	12	≤	≤	ADV
ap-4699	131	13	0	0	NUM
ap-4699	131	14	∨	∨	NUM
ap-4699	131	15	t	t	NOUN
ap-4699	131	16	=	=	SYM
ap-4699	131	17	0	0	NUM
ap-4699	131	18	)	)	PUNCT
ap-4699	131	19	,	,	PUNCT
ap-4699	131	20	(	(	PUNCT
ap-4699	131	21	3	3	X
ap-4699	131	22	.	.	PUNCT
ap-4699	131	23	)	)	PUNCT
ap-4699	132	1	if	if	SCONJ
ap-4699	132	2	λ	λ	PROPN
ap-4699	132	3	6=	6=	NUM
ap-4699	132	4	0	0	NUM
ap-4699	132	5	,	,	PUNCT
ap-4699	132	6	then	then	ADV
ap-4699	132	7	φ(λ	φ(λ	PROPN
ap-4699	132	8	,	,	PUNCT
ap-4699	132	9	.	.	PUNCT
ap-4699	132	10	.	.	PUNCT
ap-4699	133	1	.	.	PUNCT
ap-4699	134	1	,	,	PUNCT
ap-4699	134	2	λ	λ	X
ap-4699	134	3	)	)	PUNCT
ap-4699	134	4	6=	6=	PUNCT
ap-4699	135	1	λ	λ	PROPN
ap-4699	135	2	,	,	PUNCT
ap-4699	135	3	(	(	PUNCT
ap-4699	135	4	4	4	NUM
ap-4699	135	5	.	.	PUNCT
ap-4699	135	6	)	)	PUNCT
ap-4699	136	1	if	if	SCONJ
ap-4699	136	2	λ	λ	PROPN
ap-4699	136	3	6=	6=	NUM
ap-4699	136	4	0	0	NUM
ap-4699	136	5	,	,	PUNCT
ap-4699	136	6	then	then	ADV
ap-4699	136	7	φ(λ	φ(λ	PROPN
ap-4699	136	8	,	,	PUNCT
ap-4699	136	9	.	.	PUNCT
ap-4699	136	10	.	.	PUNCT
ap-4699	137	1	.	.	PUNCT
ap-4699	138	1	,	,	PUNCT
ap-4699	138	2	λ	λ	X
ap-4699	138	3	)	)	PUNCT
ap-4699	138	4	6=	6=	ADP
ap-4699	138	5	λ	λ	X
ap-4699	138	6	.	.	PUNCT
ap-4699	138	7	proof	proof	NOUN
ap-4699	138	8	.	.	PUNCT
ap-4699	139	1	(	(	PUNCT
ap-4699	139	2	1	1	X
ap-4699	139	3	.	.	PUNCT
ap-4699	139	4	):	):	PUNCT
ap-4699	139	5	let	let	VERB
ap-4699	139	6	b	b	X
ap-4699	139	7	∈	∈	PROPN
ap-4699	139	8	a+a	a+a	NUM
ap-4699	139	9	be	be	AUX
ap-4699	139	10	such	such	ADJ
ap-4699	139	11	that	that	DET
ap-4699	139	12	b	b	PROPN
ap-4699	139	13	>	>	X
ap-4699	139	14	λ	λ	PROPN
ap-4699	139	15	.	.	PROPN
ap-4699	140	1	assume	assume	PROPN
ap-4699	140	2	,	,	PUNCT
ap-4699	140	3	for	for	ADP
ap-4699	140	4	contradiction	contradiction	NOUN
ap-4699	140	5	,	,	PUNCT
ap-4699	140	6	that	that	SCONJ
ap-4699	140	7	φ(b	φ(b	ADV
ap-4699	140	8	,	,	PUNCT
ap-4699	140	9	.	.	PUNCT
ap-4699	140	10	.	.	PUNCT
ap-4699	141	1	.	.	PUNCT
ap-4699	142	1	,	,	PUNCT
ap-4699	142	2	b	b	X
ap-4699	142	3	)	)	PUNCT
ap-4699	142	4	=	=	SYM
ap-4699	142	5	λ	λ	X
ap-4699	142	6	.	.	PUNCT
ap-4699	143	1	we	we	PRON
ap-4699	143	2	follow	follow	VERB
ap-4699	143	3	the	the	DET
ap-4699	143	4	proof	proof	NOUN
ap-4699	143	5	of	of	ADP
ap-4699	143	6	claim	claim	NOUN
ap-4699	143	7	3.5	3.5	NUM
ap-4699	143	8	.	.	PUNCT
ap-4699	144	1	in	in	ADP
ap-4699	144	2	[	[	X
ap-4699	144	3	4	4	NUM
ap-4699	144	4	]	]	PUNCT
ap-4699	144	5	.	.	PUNCT
ap-4699	145	1	for	for	ADP
ap-4699	145	2	any	any	DET
ap-4699	145	3	n	n	PRON
ap-4699	145	4	∈	∈	PROPN
ap-4699	145	5	n	n	CCONJ
ap-4699	145	6	,	,	PUNCT
ap-4699	145	7	n	n	DET
ap-4699	145	8	≥	≥	NOUN
ap-4699	145	9	1	1	NUM
ap-4699	145	10	,	,	PUNCT
ap-4699	145	11	we	we	PRON
ap-4699	145	12	consider	consider	VERB
ap-4699	145	13	the	the	DET
ap-4699	145	14	number	number	NOUN
ap-4699	145	15	represented	represent	VERB
ap-4699	145	16	by	by	ADP
ap-4699	145	17	ω0	ω0	PROPN
ap-4699	145	18	b	b	PROPN
ap-4699	145	19	.	.	PUNCT
ap-4699	145	20	.	.	PUNCT
ap-4699	145	21	.	.	PUNCT
ap-4699	146	1	b︸	b︸	PROPN
ap-4699	146	2	︷︷	︷︷	PROPN
ap-4699	146	3	︸	︸	ADP
ap-4699	146	4	t	t	PROPN
ap-4699	146	5	b	b	PROPN
ap-4699	146	6	.	.	PUNCT
ap-4699	146	7	.	.	PUNCT
ap-4699	146	8	.	.	PUNCT
ap-4699	147	1	b︸	b︸	PROPN
ap-4699	147	2	︷︷	︷︷	PROPN
ap-4699	147	3	︸	︸	ADP
ap-4699	147	4	n	n	CCONJ
ap-4699	147	5	•	•	NUM
ap-4699	147	6	b	b	PROPN
ap-4699	147	7	.	.	PUNCT
ap-4699	147	8	.	.	PUNCT
ap-4699	147	9	.	.	PUNCT
ap-4699	148	1	b︸	b︸	PROPN
ap-4699	148	2	︷︷	︷︷	PROPN
ap-4699	148	3	︸	︸	ADP
ap-4699	148	4	r	r	NOUN
ap-4699	148	5	0ω	0ω	NOUN
ap-4699	148	6	.	.	PUNCT
ap-4699	149	1	its	its	PRON
ap-4699	149	2	representation	representation	NOUN
ap-4699	149	3	after	after	ADP
ap-4699	149	4	the	the	DET
ap-4699	149	5	digit	digit	NOUN
ap-4699	149	6	set	set	NOUN
ap-4699	149	7	conversion	conversion	NOUN
ap-4699	149	8	has	have	VERB
ap-4699	149	9	the	the	DET
ap-4699	149	10	form	form	NOUN
ap-4699	149	11	ω0wr+t	ω0wr+t	NUM
ap-4699	149	12	.	.	PUNCT
ap-4699	149	13	.	.	PUNCT
ap-4699	149	14	.	.	PUNCT
ap-4699	150	1	w1︸	w1︸	PROPN
ap-4699	150	2	︷︷	︷︷	PROPN
ap-4699	150	3	︸	︸	X
ap-4699	150	4	βnw	βnw	VERB
ap-4699	150	5	λ	λ	X
ap-4699	150	6	.	.	PUNCT
ap-4699	150	7	.	.	PUNCT
ap-4699	151	1	.	.	PUNCT
ap-4699	152	1	λ︸	λ︸	PROPN
ap-4699	152	2	︷︷	︷︷	PROPN
ap-4699	152	3	︸	︸	PUNCT
ap-4699	152	4	n	n	PROPN
ap-4699	152	5	•	•	NOUN
ap-4699	152	6	w̃1	w̃1	PROPN
ap-4699	152	7	.	.	PUNCT
ap-4699	152	8	.	.	PUNCT
ap-4699	152	9	.	.	PUNCT
ap-4699	153	1	w̃r+t0ω	w̃r+t0ω	PROPN
ap-4699	153	2	,	,	PUNCT
ap-4699	153	3	where	where	SCONJ
ap-4699	153	4	w	w	PROPN
ap-4699	153	5	=	=	SYM
ap-4699	153	6	∑r+t	∑r+t	PROPN
ap-4699	153	7	j=1	j=1	NOUN
ap-4699	153	8	wjβ	wjβ	ADP
ap-4699	153	9	j−1	j−1	PROPN
ap-4699	153	10	and	and	CCONJ
ap-4699	153	11	wj	wj	PROPN
ap-4699	153	12	,	,	PUNCT
ap-4699	153	13	w̃j	w̃j	VERB
ap-4699	153	14	∈	∈	PROPN
ap-4699	153	15	a.	a.	NOUN
ap-4699	153	16	since	since	SCONJ
ap-4699	153	17	both	both	DET
ap-4699	153	18	representations	representation	NOUN
ap-4699	153	19	have	have	VERB
ap-4699	153	20	the	the	DET
ap-4699	153	21	same	same	ADJ
ap-4699	153	22	value	value	NOUN
ap-4699	153	23	,	,	PUNCT
ap-4699	153	24	we	we	PRON
ap-4699	153	25	get	get	VERB
ap-4699	153	26	:	:	PUNCT
ap-4699	153	27	b	b	X
ap-4699	153	28	n+t−1∑	n+t−1∑	PROPN
ap-4699	153	29	j=−r	j=−r	PROPN
ap-4699	153	30	βj	βj	PUNCT
ap-4699	153	31	=	=	PUNCT
ap-4699	153	32	βnw	βnw	PROPN
ap-4699	154	1	+	+	CCONJ
ap-4699	154	2	λ	λ	PROPN
ap-4699	154	3	n−1∑	n−1∑	NUM
ap-4699	154	4	j=0	j=0	PROPN
ap-4699	154	5	βj	βj	PRON
ap-4699	154	6	+	+	PUNCT
ap-4699	154	7	r+t∑	r+t∑	NOUN
ap-4699	154	8	j=1	j=1	NOUN
ap-4699	154	9	w̃jβ	w̃jβ	VERB
ap-4699	154	10	−j	−j	NOUN
ap-4699	154	11	for	for	ADP
ap-4699	154	12	all	all	DET
ap-4699	154	13	n	n	PRON
ap-4699	154	14	≥	≥	NUM
ap-4699	154	15	1	1	NUM
ap-4699	154	16	.	.	PUNCT
ap-4699	154	17	corollary	corollary	ADJ
ap-4699	154	18	3.6	3.6	NUM
ap-4699	154	19	.	.	PUNCT
ap-4699	155	1	in	in	ADP
ap-4699	155	2	[	[	X
ap-4699	155	3	4	4	X
ap-4699	155	4	]	]	PUNCT
ap-4699	155	5	gives	gives	AUX
ap-4699	155	6	thatw	thatw	VERB
ap-4699	155	7	=	=	SYM
ap-4699	155	8	bβt−λ	bβt−λ	NOUN
ap-4699	155	9	β−1	β−1	PUNCT
ap-4699	155	10	.	.	PUNCT
ap-4699	156	1	thus	thus	ADV
ap-4699	156	2	,	,	PUNCT
ap-4699	156	3	b	b	X
ap-4699	156	4	−1∑	−1∑	PROPN
ap-4699	156	5	j=−r	j=−r	PROPN
ap-4699	156	6	βj	βj	NOUN
ap-4699	157	1	+	+	CCONJ
ap-4699	157	2	b	b	X
ap-4699	157	3	βn+t	βn+t	SYM
ap-4699	157	4	−	−	ADP
ap-4699	157	5	1	1	NUM
ap-4699	157	6	β	β	NOUN
ap-4699	157	7	−	−	NOUN
ap-4699	157	8	1	1	NUM
ap-4699	157	9	=	=	SYM
ap-4699	157	10	βn	βn	NOUN
ap-4699	157	11	bβt	bβt	NOUN
ap-4699	157	12	−	−	PROPN
ap-4699	157	13	λ	λ	INTJ
ap-4699	157	14	β	β	NOUN
ap-4699	157	15	−	−	NOUN
ap-4699	157	16	1	1	NUM
ap-4699	157	17	+	+	NUM
ap-4699	157	18	λ	λ	AUX
ap-4699	157	19	βn	βn	NOUN
ap-4699	157	20	−	−	PROPN
ap-4699	157	21	1	1	NUM
ap-4699	157	22	β	β	NOUN
ap-4699	157	23	−	−	NOUN
ap-4699	157	24	1	1	NUM
ap-4699	158	1	+	+	CCONJ
ap-4699	158	2	t+r∑	t+r∑	ADJ
ap-4699	158	3	j=1	j=1	NOUN
ap-4699	158	4	w̃jβ	w̃jβ	PROPN
ap-4699	158	5	−j	−j	NOUN
ap-4699	158	6	.	.	PUNCT
ap-4699	159	1	hence	hence	ADV
ap-4699	159	2	b	b	X
ap-4699	159	3	(	(	PUNCT
ap-4699	159	4	r∑	r∑	NOUN
ap-4699	159	5	j=1	j=1	NOUN
ap-4699	159	6	1	1	NUM
ap-4699	159	7	βj	βj	NOUN
ap-4699	159	8	+	+	NUM
ap-4699	159	9	−1	−1	NOUN
ap-4699	159	10	β	β	NOUN
ap-4699	159	11	−	−	NOUN
ap-4699	159	12	1	1	NUM
ap-4699	159	13	)	)	PUNCT
ap-4699	159	14	=	=	SYM
ap-4699	160	1	λ	λ	NOUN
ap-4699	160	2	−1	−1	NOUN
ap-4699	160	3	β	β	NOUN
ap-4699	160	4	−	−	NOUN
ap-4699	160	5	1	1	NUM
ap-4699	161	1	+	+	CCONJ
ap-4699	161	2	t+r∑	t+r∑	NOUN
ap-4699	161	3	j=1	j=1	NOUN
ap-4699	161	4	w̃j	w̃j	VERB
ap-4699	161	5	1	1	NUM
ap-4699	161	6	βj	βj	ADP
ap-4699	161	7	≥	≥	NOUN
ap-4699	161	8	λ	λ	X
ap-4699	161	9	(	(	PUNCT
ap-4699	161	10	−1	−1	NOUN
ap-4699	161	11	β	β	NOUN
ap-4699	161	12	−	−	NOUN
ap-4699	161	13	1	1	NUM
ap-4699	162	1	+	+	CCONJ
ap-4699	162	2	t+r∑	t+r∑	ADJ
ap-4699	162	3	j=1	j=1	NOUN
ap-4699	162	4	1	1	NUM
ap-4699	162	5	βj	βj	NOUN
ap-4699	162	6	)	)	PUNCT
ap-4699	162	7	.	.	PUNCT
ap-4699	163	1	using	use	VERB
ap-4699	163	2	1	1	NUM
ap-4699	163	3	β−1	β−1	PUNCT
ap-4699	163	4	=	=	SYM
ap-4699	163	5	∑∞	∑∞	NOUN
ap-4699	163	6	j=1	j=1	NOUN
ap-4699	163	7	1	1	NUM
ap-4699	163	8	βj	βj	INTJ
ap-4699	163	9	,	,	PUNCT
ap-4699	163	10	we	we	PRON
ap-4699	163	11	get	get	VERB
ap-4699	163	12	−	−	PROPN
ap-4699	163	13	b	b	NOUN
ap-4699	163	14	1	1	NUM
ap-4699	163	15	βr	βr	NUM
ap-4699	163	16	1	1	NUM
ap-4699	163	17	β	β	NOUN
ap-4699	163	18	−	−	NOUN
ap-4699	163	19	1	1	NUM
ap-4699	163	20	=	=	SYM
ap-4699	163	21	−b	−b	VERB
ap-4699	163	22	∞∑	∞∑	PROPN
ap-4699	163	23	j	j	PROPN
ap-4699	164	1	=	=	NOUN
ap-4699	164	2	r+1	r+1	PROPN
ap-4699	164	3	1	1	NUM
ap-4699	164	4	βj	βj	NOUN
ap-4699	164	5	≥	≥	NOUN
ap-4699	164	6	−λ	−λ	VERB
ap-4699	164	7	∞∑	∞∑	PROPN
ap-4699	164	8	j	j	PROPN
ap-4699	164	9	=	=	SYM
ap-4699	164	10	r+t+1	r+t+1	PROPN
ap-4699	164	11	1	1	NUM
ap-4699	164	12	βj	βj	NOUN
ap-4699	164	13	=	=	PUNCT
ap-4699	164	14	−λ	−λ	PROPN
ap-4699	164	15	1	1	NUM
ap-4699	164	16	βr+t	βr+t	PROPN
ap-4699	164	17	1	1	NUM
ap-4699	164	18	β	β	NOUN
ap-4699	164	19	−	−	PROPN
ap-4699	164	20	1	1	NUM
ap-4699	164	21	.	.	PUNCT
ap-4699	165	1	thus	thus	ADV
ap-4699	165	2	,	,	PUNCT
ap-4699	165	3	we	we	PRON
ap-4699	165	4	have	have	VERB
ap-4699	165	5	λ	λ	PROPN
ap-4699	165	6	≥	≥	NUM
ap-4699	165	7	bβt	bβt	NOUN
ap-4699	165	8	.	.	PUNCT
ap-4699	166	1	if	if	SCONJ
ap-4699	166	2	t	t	NOUN
ap-4699	166	3	=	=	SYM
ap-4699	166	4	0	0	NUM
ap-4699	166	5	,	,	PUNCT
ap-4699	166	6	then	then	ADV
ap-4699	166	7	it	it	PRON
ap-4699	166	8	contradicts	contradict	VERB
ap-4699	166	9	the	the	DET
ap-4699	166	10	assumption	assumption	NOUN
ap-4699	166	11	b	b	PROPN
ap-4699	166	12	>	>	X
ap-4699	166	13	λ	λ	PROPN
ap-4699	166	14	.	.	PUNCT
ap-4699	167	1	if	if	SCONJ
ap-4699	167	2	b	b	PROPN
ap-4699	167	3	≥	≥	X
ap-4699	167	4	0	0	NUM
ap-4699	167	5	,	,	PUNCT
ap-4699	167	6	then	then	ADV
ap-4699	167	7	λ	λ	X
ap-4699	167	8	≥	≥	PROPN
ap-4699	167	9	bβt	bβt	PROPN
ap-4699	167	10	≥	≥	PROPN
ap-4699	167	11	b	b	PROPN
ap-4699	167	12	since	since	SCONJ
ap-4699	167	13	β	β	PROPN
ap-4699	167	14	>	>	X
ap-4699	167	15	1	1	NUM
ap-4699	167	16	,	,	PUNCT
ap-4699	167	17	which	which	PRON
ap-4699	167	18	is	be	AUX
ap-4699	167	19	also	also	ADV
ap-4699	167	20	a	a	DET
ap-4699	167	21	contradiction	contradiction	NOUN
ap-4699	167	22	.	.	PUNCT
ap-4699	168	1	the	the	DET
ap-4699	168	2	proof	proof	NOUN
ap-4699	168	3	of	of	ADP
ap-4699	168	4	(	(	PUNCT
ap-4699	168	5	2	2	NUM
ap-4699	168	6	.	.	PUNCT
ap-4699	168	7	)	)	PUNCT
ap-4699	168	8	is	be	AUX
ap-4699	168	9	similar	similar	ADJ
ap-4699	168	10	.	.	PUNCT
ap-4699	169	1	for	for	ADP
ap-4699	169	2	(	(	PUNCT
ap-4699	169	3	3	3	NUM
ap-4699	169	4	.	.	PUNCT
ap-4699	169	5	)	)	PUNCT
ap-4699	170	1	and	and	CCONJ
ap-4699	170	2	(	(	PUNCT
ap-4699	170	3	4	4	NUM
ap-4699	170	4	.	.	NUM
ap-4699	170	5	)	)	PUNCT
ap-4699	170	6	,	,	PUNCT
ap-4699	170	7	see	see	VERB
ap-4699	170	8	[	[	X
ap-4699	170	9	4	4	NUM
ap-4699	170	10	]	]	PUNCT
ap-4699	170	11	.	.	PUNCT
ap-4699	171	1	now	now	ADV
ap-4699	171	2	we	we	PRON
ap-4699	171	3	can	can	AUX
ap-4699	171	4	conclude	conclude	VERB
ap-4699	171	5	with	with	ADP
ap-4699	171	6	the	the	DET
ap-4699	171	7	following	follow	VERB
ap-4699	171	8	statement	statement	NOUN
ap-4699	171	9	.	.	PUNCT
ap-4699	172	1	theorem	theorem	VERB
ap-4699	172	2	3.4	3.4	NUM
ap-4699	172	3	.	.	PUNCT
ap-4699	173	1	let	let	VERB
ap-4699	173	2	(	(	PUNCT
ap-4699	173	3	β	β	X
ap-4699	173	4	,	,	PUNCT
ap-4699	173	5	a	a	PRON
ap-4699	173	6	)	)	PUNCT
ap-4699	173	7	be	be	AUX
ap-4699	173	8	a	a	DET
ap-4699	173	9	numeration	numeration	NOUN
ap-4699	173	10	system	system	NOUN
ap-4699	173	11	such	such	ADJ
ap-4699	173	12	that	that	SCONJ
ap-4699	173	13	a	a	DET
ap-4699	173	14	⊂	⊂	PROPN
ap-4699	173	15	z[β	z[β	NOUN
ap-4699	173	16	]	]	PUNCT
ap-4699	173	17	,	,	PUNCT
ap-4699	173	18	β	β	X
ap-4699	173	19	is	be	AUX
ap-4699	173	20	an	an	DET
ap-4699	173	21	algebraic	algebraic	ADJ
ap-4699	173	22	number	number	NOUN
ap-4699	173	23	with	with	ADP
ap-4699	173	24	a	a	DET
ap-4699	173	25	positive	positive	ADJ
ap-4699	173	26	real	real	ADJ
ap-4699	173	27	conjugate	conjugate	NOUN
ap-4699	173	28	and	and	CCONJ
ap-4699	173	29	there	there	PRON
ap-4699	173	30	is	be	VERB
ap-4699	173	31	parallel	parallel	ADJ
ap-4699	173	32	addition	addition	NOUN
ap-4699	173	33	in	in	ADP
ap-4699	173	34	(	(	PUNCT
ap-4699	173	35	β	β	X
ap-4699	173	36	,	,	PUNCT
ap-4699	173	37	a	a	PRON
ap-4699	173	38	)	)	PUNCT
ap-4699	173	39	.	.	PUNCT
ap-4699	174	1	let	let	VERB
ap-4699	174	2	λ	λ	X
ap-4699	174	3	=	=	SYM
ap-4699	174	4	mina	mina	PROPN
ap-4699	174	5	and	and	CCONJ
ap-4699	174	6	λ	λ	NOUN
ap-4699	174	7	=	=	NOUN
ap-4699	174	8	maxa	maxa	NOUN
ap-4699	174	9	.	.	PUNCT
ap-4699	175	1	if	if	SCONJ
ap-4699	175	2	λ	λ	PROPN
ap-4699	175	3	≡β−1	≡β−1	NUM
ap-4699	175	4	λ	λ	PROPN
ap-4699	175	5	,	,	PUNCT
ap-4699	175	6	then	then	ADV
ap-4699	175	7	there	there	PRON
ap-4699	175	8	exists	exist	VERB
ap-4699	175	9	c	c	PROPN
ap-4699	175	10	∈	∈	PROPN
ap-4699	175	11	a	a	PRON
ap-4699	175	12	,	,	PUNCT
ap-4699	175	13	λ	λ	PROPN
ap-4699	175	14	6=	6=	PROPN
ap-4699	175	15	c	c	PROPN
ap-4699	175	16	6=	6=	PROPN
ap-4699	175	17	λ	λ	NOUN
ap-4699	175	18	such	such	ADJ
ap-4699	175	19	that	that	SCONJ
ap-4699	175	20	λ	λ	PROPN
ap-4699	175	21	≡β−1	≡β−1	PROPN
ap-4699	175	22	c	c	PROPN
ap-4699	175	23	≡β−1	≡β−1	NUM
ap-4699	175	24	λ	λ	PROPN
ap-4699	175	25	.	.	PUNCT
ap-4699	176	1	if	if	SCONJ
ap-4699	176	2	λ	λ	PROPN
ap-4699	176	3	6≡β−1	6≡β−1	NUM
ap-4699	176	4	λ	λ	NOUN
ap-4699	176	5	,	,	PUNCT
ap-4699	176	6	then	then	ADV
ap-4699	176	7	there	there	PRON
ap-4699	176	8	exist	exist	VERB
ap-4699	176	9	a	a	DET
ap-4699	176	10	,	,	PUNCT
ap-4699	176	11	b	b	PROPN
ap-4699	176	12	∈	∈	PROPN
ap-4699	176	13	a	a	PRON
ap-4699	176	14	,	,	PUNCT
ap-4699	176	15	a	a	PRON
ap-4699	176	16	6=	6=	NUM
ap-4699	176	17	λ	λ	PROPN
ap-4699	176	18	,	,	PUNCT
ap-4699	176	19	b	b	PROPN
ap-4699	176	20	6=	6=	ADP
ap-4699	176	21	λ	λ	NOUN
ap-4699	176	22	such	such	ADJ
ap-4699	176	23	that	that	SCONJ
ap-4699	176	24	a	a	DET
ap-4699	176	25	≡β−1	≡β−1	NUM
ap-4699	176	26	λ	λ	PROPN
ap-4699	176	27	and	and	CCONJ
ap-4699	176	28	b	b	PROPN
ap-4699	176	29	≡β−1	≡β−1	NUM
ap-4699	176	30	λ	λ	PROPN
ap-4699	176	31	.	.	PROPN
ap-4699	176	32	287	287	NUM
ap-4699	176	33	jan	jan	PROPN
ap-4699	176	34	legerský	legerský	PROPN
ap-4699	176	35	acta	acta	PROPN
ap-4699	176	36	polytechnica	polytechnica	PROPN
ap-4699	176	37	proof	proof	NOUN
ap-4699	176	38	.	.	PUNCT
ap-4699	177	1	by	by	ADP
ap-4699	177	2	lemma	lemma	PROPN
ap-4699	177	3	3.2	3.2	NUM
ap-4699	177	4	,	,	PUNCT
ap-4699	177	5	we	we	PRON
ap-4699	177	6	can	can	AUX
ap-4699	177	7	assume	assume	VERB
ap-4699	177	8	that	that	SCONJ
ap-4699	177	9	the	the	DET
ap-4699	177	10	base	base	NOUN
ap-4699	177	11	β	β	X
ap-4699	177	12	itself	itself	PRON
ap-4699	177	13	is	be	AUX
ap-4699	177	14	real	real	ADJ
ap-4699	177	15	and	and	CCONJ
ap-4699	177	16	greater	great	ADJ
ap-4699	177	17	than	than	ADP
ap-4699	177	18	one	one	NUM
ap-4699	177	19	.	.	PUNCT
ap-4699	178	1	let	let	VERB
ap-4699	178	2	φ	φ	PROPN
ap-4699	178	3	be	be	AUX
ap-4699	178	4	a	a	DET
ap-4699	178	5	mapping	mapping	NOUN
ap-4699	178	6	which	which	PRON
ap-4699	178	7	defines	define	VERB
ap-4699	178	8	the	the	DET
ap-4699	178	9	parallel	parallel	ADJ
ap-4699	178	10	addition	addition	NOUN
ap-4699	178	11	.	.	PUNCT
ap-4699	179	1	since	since	SCONJ
ap-4699	179	2	{	{	PUNCT
ap-4699	179	3	0	0	NUM
ap-4699	179	4	}	}	PUNCT
ap-4699	179	5	(	(	PUNCT
ap-4699	179	6	a	a	X
ap-4699	179	7	,	,	PUNCT
ap-4699	179	8	we	we	PRON
ap-4699	179	9	know	know	VERB
ap-4699	179	10	that	that	SCONJ
ap-4699	179	11	λ	λ	PROPN
ap-4699	179	12	>	>	X
ap-4699	179	13	0	0	PUNCT
ap-4699	179	14	or	or	CCONJ
ap-4699	179	15	λ	λ	X
ap-4699	179	16	<	<	X
ap-4699	179	17	0	0	X
ap-4699	179	18	.	.	PUNCT
ap-4699	179	19	assume	assume	VERB
ap-4699	179	20	that	that	SCONJ
ap-4699	179	21	λ	λ	PROPN
ap-4699	179	22	>	>	X
ap-4699	179	23	0	0	PROPN
ap-4699	179	24	,	,	PUNCT
ap-4699	179	25	the	the	DET
ap-4699	179	26	latter	latter	ADJ
ap-4699	179	27	one	one	NOUN
ap-4699	179	28	is	be	AUX
ap-4699	179	29	analogous	analogous	ADJ
ap-4699	179	30	.	.	PUNCT
ap-4699	180	1	by	by	ADP
ap-4699	180	2	theorem	theorem	ADJ
ap-4699	180	3	3.1	3.1	NUM
ap-4699	180	4	and	and	CCONJ
ap-4699	180	5	lemma	lemma	PROPN
ap-4699	180	6	3.3	3.3	NUM
ap-4699	180	7	,	,	PUNCT
ap-4699	180	8	λ	λ	PROPN
ap-4699	180	9	≡β−1	≡β−1	SYM
ap-4699	180	10	φ(λ	φ(λ	PROPN
ap-4699	180	11	,	,	PUNCT
ap-4699	180	12	.	.	PUNCT
ap-4699	180	13	.	.	PUNCT
ap-4699	180	14	.	.	PUNCT
ap-4699	181	1	,	,	PUNCT
ap-4699	181	2	λ	λ	X
ap-4699	181	3	)	)	PUNCT
ap-4699	181	4	∈	∈	PROPN
ap-4699	181	5	a	a	PRON
ap-4699	181	6	and	and	CCONJ
ap-4699	181	7	λ	λ	X
ap-4699	181	8	6=	6=	PROPN
ap-4699	181	9	φ(λ	φ(λ	PROPN
ap-4699	181	10	,	,	PUNCT
ap-4699	181	11	.	.	PUNCT
ap-4699	181	12	.	.	PUNCT
ap-4699	182	1	.	.	PUNCT
ap-4699	183	1	,	,	PUNCT
ap-4699	183	2	λ	λ	X
ap-4699	183	3	)	)	PUNCT
ap-4699	183	4	6=	6=	ADP
ap-4699	184	1	λ	λ	NOUN
ap-4699	184	2	.	.	PUNCT
ap-4699	184	3	hence	hence	ADV
ap-4699	184	4	,	,	PUNCT
ap-4699	184	5	φ(λ	φ(λ	PROPN
ap-4699	184	6	,	,	PUNCT
ap-4699	184	7	.	.	PUNCT
ap-4699	184	8	.	.	PUNCT
ap-4699	184	9	.	.	PUNCT
ap-4699	185	1	,	,	PUNCT
ap-4699	185	2	λ	λ	X
ap-4699	185	3	)	)	PUNCT
ap-4699	185	4	is	be	AUX
ap-4699	185	5	a	a	DET
ap-4699	185	6	digit	digit	NOUN
ap-4699	185	7	of	of	ADP
ap-4699	185	8	a	a	PRON
ap-4699	185	9	which	which	PRON
ap-4699	185	10	belongs	belong	VERB
ap-4699	185	11	to	to	ADP
ap-4699	185	12	the	the	DET
ap-4699	185	13	same	same	ADJ
ap-4699	185	14	congruence	congruence	NOUN
ap-4699	185	15	class	class	NOUN
ap-4699	185	16	as	as	ADP
ap-4699	185	17	λ	λ	PROPN
ap-4699	185	18	.	.	PUNCT
ap-4699	186	1	if	if	SCONJ
ap-4699	186	2	λ	λ	PROPN
ap-4699	186	3	≡β−1	≡β−1	NUM
ap-4699	186	4	λ	λ	PROPN
ap-4699	186	5	,	,	PUNCT
ap-4699	186	6	the	the	DET
ap-4699	186	7	claim	claim	NOUN
ap-4699	186	8	follows	follow	VERB
ap-4699	186	9	,	,	PUNCT
ap-4699	186	10	with	with	ADP
ap-4699	186	11	c	c	NOUN
ap-4699	186	12	=	=	SYM
ap-4699	186	13	φ(λ	φ(λ	PROPN
ap-4699	186	14	,	,	PUNCT
ap-4699	186	15	.	.	PUNCT
ap-4699	187	1	.	.	PUNCT
ap-4699	187	2	.	.	PUNCT
ap-4699	188	1	,	,	PUNCT
ap-4699	188	2	λ	λ	X
ap-4699	188	3	)	)	PUNCT
ap-4699	188	4	.	.	PUNCT
ap-4699	189	1	the	the	DET
ap-4699	189	2	case	case	NOUN
ap-4699	189	3	that	that	PRON
ap-4699	189	4	λ	λ	NOUN
ap-4699	189	5	6≡β−1	6≡β−1	NUM
ap-4699	189	6	λ	λ	NOUN
ap-4699	189	7	is	be	AUX
ap-4699	189	8	divided	divide	VERB
ap-4699	189	9	into	into	ADP
ap-4699	189	10	two	two	NUM
ap-4699	189	11	sub	sub	NOUN
ap-4699	189	12	-	-	NOUN
ap-4699	189	13	cases	case	NOUN
ap-4699	189	14	.	.	PUNCT
ap-4699	190	1	if	if	SCONJ
ap-4699	190	2	λ	λ	PROPN
ap-4699	190	3	6=	6=	NUM
ap-4699	190	4	0	0	NUM
ap-4699	190	5	,	,	PUNCT
ap-4699	190	6	then	then	ADV
ap-4699	190	7	we	we	PRON
ap-4699	190	8	have	have	VERB
ap-4699	190	9	λ	λ	PROPN
ap-4699	190	10	≡β−1	≡β−1	NUM
ap-4699	190	11	φ(λ	φ(λ	PROPN
ap-4699	190	12	,	,	PUNCT
ap-4699	190	13	.	.	PUNCT
ap-4699	190	14	.	.	PUNCT
ap-4699	191	1	.	.	PUNCT
ap-4699	192	1	,	,	PUNCT
ap-4699	192	2	λ	λ	X
ap-4699	192	3	)	)	PUNCT
ap-4699	192	4	∈	∈	PROPN
ap-4699	192	5	a	a	PRON
ap-4699	192	6	and	and	CCONJ
ap-4699	192	7	φ(λ	φ(λ	PROPN
ap-4699	192	8	,	,	PUNCT
ap-4699	192	9	.	.	PUNCT
ap-4699	192	10	.	.	PUNCT
ap-4699	193	1	.	.	PUNCT
ap-4699	194	1	,	,	PUNCT
ap-4699	194	2	λ	λ	X
ap-4699	194	3	)	)	PUNCT
ap-4699	194	4	6=	6=	PUNCT
ap-4699	194	5	λ	λ	NOUN
ap-4699	194	6	again	again	ADV
ap-4699	194	7	by	by	ADP
ap-4699	194	8	theorem	theorem	ADJ
ap-4699	194	9	3.1	3.1	NUM
ap-4699	194	10	and	and	CCONJ
ap-4699	194	11	lemma	lemma	PROPN
ap-4699	194	12	3.3	3.3	NUM
ap-4699	194	13	,	,	PUNCT
ap-4699	194	14	which	which	PRON
ap-4699	194	15	implies	imply	VERB
ap-4699	194	16	the	the	DET
ap-4699	194	17	statement	statement	NOUN
ap-4699	194	18	,	,	PUNCT
ap-4699	194	19	with	with	ADP
ap-4699	194	20	a	a	DET
ap-4699	194	21	=	=	SYM
ap-4699	194	22	φ(λ	φ(λ	PROPN
ap-4699	194	23	,	,	PUNCT
ap-4699	194	24	.	.	PUNCT
ap-4699	194	25	.	.	PUNCT
ap-4699	195	1	.	.	PUNCT
ap-4699	196	1	,	,	PUNCT
ap-4699	196	2	λ	λ	NOUN
ap-4699	196	3	)	)	PUNCT
ap-4699	196	4	and	and	CCONJ
ap-4699	196	5	b	b	X
ap-4699	196	6	=	=	SYM
ap-4699	196	7	φ(λ	φ(λ	PROPN
ap-4699	196	8	,	,	PUNCT
ap-4699	196	9	.	.	PUNCT
ap-4699	196	10	.	.	PUNCT
ap-4699	197	1	.	.	PUNCT
ap-4699	198	1	,	,	PUNCT
ap-4699	198	2	λ	λ	X
ap-4699	198	3	)	)	PUNCT
ap-4699	198	4	.	.	PUNCT
ap-4699	199	1	if	if	SCONJ
ap-4699	199	2	λ	λ	X
ap-4699	199	3	=	=	SYM
ap-4699	199	4	0	0	NUM
ap-4699	199	5	,	,	PUNCT
ap-4699	199	6	then	then	ADV
ap-4699	199	7	all	all	DET
ap-4699	199	8	elements	element	NOUN
ap-4699	199	9	of	of	ADP
ap-4699	199	10	a	a	DET
ap-4699	199	11	+	+	NOUN
ap-4699	199	12	λ	λ	NOUN
ap-4699	199	13	are	be	AUX
ap-4699	199	14	positive	positive	ADJ
ap-4699	199	15	.	.	PUNCT
ap-4699	200	1	suppose	suppose	VERB
ap-4699	200	2	,	,	PUNCT
ap-4699	200	3	for	for	ADP
ap-4699	200	4	contradiction	contradiction	NOUN
ap-4699	200	5	,	,	PUNCT
ap-4699	200	6	that	that	SCONJ
ap-4699	200	7	there	there	PRON
ap-4699	200	8	is	be	VERB
ap-4699	200	9	no	no	DET
ap-4699	200	10	nonzero	nonzero	ADJ
ap-4699	200	11	element	element	NOUN
ap-4699	200	12	of	of	ADP
ap-4699	200	13	the	the	DET
ap-4699	200	14	alphabet	alphabet	NOUN
ap-4699	200	15	a	a	DET
ap-4699	200	16	congruent	congruent	ADJ
ap-4699	200	17	to	to	ADP
ap-4699	200	18	0	0	NUM
ap-4699	200	19	modulo	modulo	NOUN
ap-4699	200	20	β−	β−	NOUN
ap-4699	201	1	1	1	X
ap-4699	201	2	.	.	PUNCT
ap-4699	201	3	let	let	VERB
ap-4699	201	4	k	k	PRON
ap-4699	201	5	be	be	AUX
ap-4699	201	6	the	the	DET
ap-4699	201	7	number	number	NOUN
ap-4699	201	8	of	of	ADP
ap-4699	201	9	congruence	congruence	PROPN
ap-4699	201	10	classes	class	NOUN
ap-4699	201	11	occurring	occur	VERB
ap-4699	201	12	in	in	ADP
ap-4699	201	13	a	a	PRON
ap-4699	201	14	and	and	CCONJ
ap-4699	201	15	let	let	VERB
ap-4699	201	16	r	r	PRON
ap-4699	201	17	be	be	AUX
ap-4699	201	18	a	a	DET
ap-4699	201	19	subset	subset	NOUN
ap-4699	201	20	of	of	ADP
ap-4699	201	21	a	a	DET
ap-4699	201	22	such	such	ADJ
ap-4699	201	23	that	that	SCONJ
ap-4699	201	24	there	there	PRON
ap-4699	201	25	is	be	VERB
ap-4699	201	26	exactly	exactly	ADV
ap-4699	201	27	one	one	NUM
ap-4699	201	28	representative	representative	NOUN
ap-4699	201	29	of	of	ADP
ap-4699	201	30	each	each	PRON
ap-4699	201	31	of	of	ADP
ap-4699	201	32	those	those	DET
ap-4699	201	33	k	k	PROPN
ap-4699	201	34	congruence	congruence	PROPN
ap-4699	201	35	classes	class	NOUN
ap-4699	201	36	.	.	PUNCT
ap-4699	202	1	for	for	ADP
ap-4699	202	2	d	d	PROPN
ap-4699	202	3	∈	∈	PROPN
ap-4699	202	4	λ	λ	X
ap-4699	202	5	+	+	NOUN
ap-4699	202	6	r	r	NOUN
ap-4699	202	7	,	,	PUNCT
ap-4699	202	8	the	the	DET
ap-4699	202	9	value	value	NOUN
ap-4699	202	10	φ(d	φ(d	NUM
ap-4699	202	11	,	,	PUNCT
ap-4699	202	12	.	.	PUNCT
ap-4699	202	13	.	.	PUNCT
ap-4699	202	14	.	.	PUNCT
ap-4699	203	1	,	,	PUNCT
ap-4699	203	2	d	d	X
ap-4699	203	3	)	)	PUNCT
ap-4699	203	4	∈	∈	PROPN
ap-4699	203	5	a	a	PRON
ap-4699	203	6	is	be	AUX
ap-4699	203	7	not	not	PART
ap-4699	203	8	congruent	congruent	ADJ
ap-4699	203	9	to	to	ADP
ap-4699	203	10	0	0	NUM
ap-4699	203	11	,	,	PUNCT
ap-4699	203	12	as	as	ADP
ap-4699	203	13	φ(d	φ(d	X
ap-4699	203	14	,	,	PUNCT
ap-4699	203	15	.	.	PUNCT
ap-4699	203	16	.	.	PUNCT
ap-4699	203	17	.	.	PUNCT
ap-4699	204	1	,	,	PUNCT
ap-4699	204	2	d	d	X
ap-4699	204	3	)	)	PUNCT
ap-4699	204	4	6=	6=	ADP
ap-4699	204	5	λ	λ	SYM
ap-4699	204	6	=	=	SYM
ap-4699	204	7	0	0	NUM
ap-4699	204	8	by	by	ADP
ap-4699	204	9	lemma	lemma	PROPN
ap-4699	204	10	3.3	3.3	NUM
ap-4699	204	11	and	and	CCONJ
ap-4699	204	12	the	the	DET
ap-4699	204	13	congruence	congruence	NOUN
ap-4699	204	14	class	class	NOUN
ap-4699	204	15	containing	contain	VERB
ap-4699	204	16	zero	zero	NUM
ap-4699	204	17	has	have	VERB
ap-4699	204	18	only	only	ADV
ap-4699	204	19	one	one	NUM
ap-4699	204	20	element	element	NOUN
ap-4699	204	21	in	in	ADP
ap-4699	204	22	a	a	PRON
ap-4699	204	23	,	,	PUNCT
ap-4699	204	24	by	by	ADP
ap-4699	204	25	the	the	DET
ap-4699	204	26	previous	previous	ADJ
ap-4699	204	27	assumption	assumption	NOUN
ap-4699	204	28	.	.	PUNCT
ap-4699	205	1	therefore	therefore	ADV
ap-4699	205	2	,	,	PUNCT
ap-4699	205	3	the	the	DET
ap-4699	205	4	values	value	NOUN
ap-4699	205	5	fj	fj	X
ap-4699	205	6	=	=	PUNCT
ap-4699	205	7	φ(dj	φ(dj	PROPN
ap-4699	205	8	,	,	PUNCT
ap-4699	205	9	.	.	PUNCT
ap-4699	205	10	.	.	PUNCT
ap-4699	205	11	.	.	PUNCT
ap-4699	206	1	,	,	PUNCT
ap-4699	206	2	dj	dj	X
ap-4699	206	3	)	)	PUNCT
ap-4699	206	4	∈	∈	PROPN
ap-4699	206	5	a	a	PRON
ap-4699	206	6	for	for	ADP
ap-4699	206	7	k	k	PROPN
ap-4699	206	8	distinct	distinct	ADJ
ap-4699	206	9	digits	digit	NOUN
ap-4699	206	10	dj	dj	VERB
ap-4699	207	1	=	=	SYM
ap-4699	208	1	λ	λ	NOUN
ap-4699	209	1	+	+	CCONJ
ap-4699	209	2	ej	ej	PROPN
ap-4699	209	3	∈	∈	PROPN
ap-4699	209	4	λ	λ	X
ap-4699	210	1	+	+	NOUN
ap-4699	210	2	r	r	NOUN
ap-4699	210	3	belong	belong	VERB
ap-4699	210	4	to	to	ADP
ap-4699	211	1	only	only	ADV
ap-4699	211	2	k	k	PROPN
ap-4699	211	3	−	−	PROPN
ap-4699	211	4	1	1	NUM
ap-4699	211	5	congruence	congruence	NOUN
ap-4699	211	6	classes	class	NOUN
ap-4699	211	7	modulo	modulo	VERB
ap-4699	211	8	β	β	X
ap-4699	211	9	−	−	NOUN
ap-4699	211	10	1	1	NUM
ap-4699	211	11	.	.	PUNCT
ap-4699	212	1	hence	hence	ADV
ap-4699	212	2	,	,	PUNCT
ap-4699	212	3	there	there	PRON
ap-4699	212	4	exist	exist	VERB
ap-4699	212	5	two	two	NUM
ap-4699	212	6	distinct	distinct	ADJ
ap-4699	212	7	elements	element	NOUN
ap-4699	212	8	d1	d1	PROPN
ap-4699	212	9	,	,	PUNCT
ap-4699	212	10	d2	d2	PROPN
ap-4699	212	11	∈	∈	PROPN
ap-4699	212	12	λ	λ	NOUN
ap-4699	213	1	+	+	NOUN
ap-4699	213	2	r	r	NOUN
ap-4699	213	3	such	such	ADJ
ap-4699	213	4	that	that	DET
ap-4699	213	5	f1	f1	PROPN
ap-4699	213	6	≡β−1	≡β−1	NUM
ap-4699	213	7	f2	f2	PROPN
ap-4699	213	8	.	.	PUNCT
ap-4699	214	1	due	due	ADP
ap-4699	214	2	to	to	ADP
ap-4699	214	3	fj	fj	PROPN
ap-4699	214	4	=	=	PUNCT
ap-4699	214	5	φ(dj	φ(dj	PROPN
ap-4699	214	6	,	,	PUNCT
ap-4699	214	7	.	.	PUNCT
ap-4699	214	8	.	.	PUNCT
ap-4699	214	9	.	.	PUNCT
ap-4699	215	1	,	,	PUNCT
ap-4699	215	2	dj	dj	X
ap-4699	215	3	)	)	PUNCT
ap-4699	215	4	≡β−1	≡β−1	X
ap-4699	215	5	dj	dj	NOUN
ap-4699	216	1	=	=	SYM
ap-4699	216	2	λ	λ	PROPN
ap-4699	216	3	+	+	CCONJ
ap-4699	216	4	ej	ej	PROPN
ap-4699	216	5	for	for	ADP
ap-4699	216	6	each	each	DET
ap-4699	216	7	j	j	NOUN
ap-4699	216	8	,	,	PUNCT
ap-4699	216	9	we	we	PRON
ap-4699	216	10	obtain	obtain	VERB
ap-4699	216	11	also	also	ADV
ap-4699	216	12	e1	e1	PROPN
ap-4699	216	13	≡β−1	≡β−1	NUM
ap-4699	216	14	e2	e2	NOUN
ap-4699	216	15	which	which	PRON
ap-4699	216	16	contradicts	contradict	VERB
ap-4699	216	17	the	the	DET
ap-4699	216	18	construction	construction	NOUN
ap-4699	216	19	of	of	ADP
ap-4699	216	20	the	the	DET
ap-4699	216	21	set	set	NOUN
ap-4699	216	22	r.	r.	PROPN
ap-4699	216	23	3.1	3.1	NUM
ap-4699	216	24	.	.	PUNCT
ap-4699	217	1	a[β	a[β	PROPN
ap-4699	217	2	]	]	PUNCT
ap-4699	217	3	closed	close	VERB
ap-4699	217	4	under	under	ADP
ap-4699	217	5	addition	addition	NOUN
ap-4699	217	6	in	in	ADP
ap-4699	217	7	order	order	NOUN
ap-4699	217	8	to	to	PART
ap-4699	217	9	express	express	VERB
ap-4699	217	10	the	the	DET
ap-4699	217	11	properties	property	NOUN
ap-4699	217	12	of	of	ADP
ap-4699	217	13	the	the	DET
ap-4699	217	14	alphabet	alphabet	NOUN
ap-4699	217	15	a	a	PRON
ap-4699	217	16	allowing	allow	VERB
ap-4699	217	17	parallel	parallel	ADJ
ap-4699	217	18	addition	addition	NOUN
ap-4699	217	19	in	in	ADP
ap-4699	217	20	terms	term	NOUN
ap-4699	217	21	of	of	ADP
ap-4699	217	22	representatives	representative	NOUN
ap-4699	217	23	modulo	modulo	VERB
ap-4699	217	24	β	β	PROPN
ap-4699	217	25	and	and	CCONJ
ap-4699	217	26	β	β	X
ap-4699	217	27	−	−	PROPN
ap-4699	217	28	1	1	NUM
ap-4699	217	29	,	,	PUNCT
ap-4699	217	30	we	we	PRON
ap-4699	217	31	restrict	restrict	VERB
ap-4699	217	32	ourselves	ourselves	PRON
ap-4699	217	33	to	to	ADP
ap-4699	217	34	alphabets	alphabet	NOUN
ap-4699	217	35	such	such	ADJ
ap-4699	217	36	that	that	SCONJ
ap-4699	217	37	a[β	a[β	PROPN
ap-4699	217	38	]	]	PUNCT
ap-4699	217	39	is	be	AUX
ap-4699	217	40	closed	close	VERB
ap-4699	217	41	under	under	ADP
ap-4699	217	42	addition	addition	NOUN
ap-4699	217	43	,	,	PUNCT
ap-4699	217	44	or	or	CCONJ
ap-4699	217	45	even	even	ADV
ap-4699	217	46	slightly	slightly	ADV
ap-4699	217	47	stronger	strong	ADJ
ap-4699	217	48	condition	condition	NOUN
ap-4699	217	49	that	that	SCONJ
ap-4699	217	50	a[β	a[β	PROPN
ap-4699	217	51	]	]	X
ap-4699	217	52	=	=	SYM
ap-4699	217	53	z[β	z[β	NOUN
ap-4699	217	54	]	]	PUNCT
ap-4699	217	55	.	.	PUNCT
ap-4699	218	1	the	the	DET
ap-4699	218	2	following	follow	VERB
ap-4699	218	3	theorem	theorem	ADJ
ap-4699	218	4	summarizes	summarize	NOUN
ap-4699	218	5	some	some	DET
ap-4699	218	6	consequences	consequence	NOUN
ap-4699	218	7	of	of	ADP
ap-4699	218	8	these	these	DET
ap-4699	218	9	assumptions	assumption	NOUN
ap-4699	218	10	.	.	PUNCT
ap-4699	219	1	theorem	theorem	VERB
ap-4699	219	2	3.5	3.5	NUM
ap-4699	219	3	.	.	PUNCT
ap-4699	220	1	let	let	VERB
ap-4699	220	2	(	(	PUNCT
ap-4699	220	3	β	β	X
ap-4699	220	4	,	,	PUNCT
ap-4699	220	5	a	a	PRON
ap-4699	220	6	)	)	PUNCT
ap-4699	220	7	be	be	AUX
ap-4699	220	8	a	a	DET
ap-4699	220	9	numeration	numeration	NOUN
ap-4699	220	10	system	system	NOUN
ap-4699	220	11	such	such	ADJ
ap-4699	220	12	that	that	SCONJ
ap-4699	220	13	a	a	DET
ap-4699	220	14	⊂	⊂	PROPN
ap-4699	220	15	z[β	z[β	X
ap-4699	220	16	]	]	PUNCT
ap-4699	220	17	and	and	CCONJ
ap-4699	220	18	1	1	NUM
ap-4699	220	19	∈	∈	PROPN
ap-4699	220	20	a[β	a[β	PROPN
ap-4699	220	21	]	]	PUNCT
ap-4699	220	22	.	.	PUNCT
ap-4699	221	1	the	the	DET
ap-4699	221	2	following	follow	VERB
ap-4699	221	3	statements	statement	NOUN
ap-4699	221	4	hold	hold	VERB
ap-4699	221	5	:	:	PUNCT
ap-4699	221	6	(	(	PUNCT
ap-4699	221	7	1	1	NUM
ap-4699	221	8	.	.	PUNCT
ap-4699	221	9	)	)	PUNCT
ap-4699	222	1	if	if	SCONJ
ap-4699	222	2	a[β	a[β	PROPN
ap-4699	222	3	]	]	PUNCT
ap-4699	222	4	is	be	AUX
ap-4699	222	5	closed	close	VERB
ap-4699	222	6	under	under	ADP
ap-4699	222	7	addition	addition	NOUN
ap-4699	222	8	,	,	PUNCT
ap-4699	222	9	then	then	ADV
ap-4699	222	10	n[β	n[β	VERB
ap-4699	222	11	]	]	PUNCT
ap-4699	222	12	⊂	⊂	PROPN
ap-4699	222	13	a[β	a[β	PROPN
ap-4699	222	14	]	]	PUNCT
ap-4699	222	15	.	.	PUNCT
ap-4699	223	1	(	(	PUNCT
ap-4699	223	2	2	2	NUM
ap-4699	223	3	.	.	PUNCT
ap-4699	223	4	)	)	PUNCT
ap-4699	224	1	a[β	a[β	PROPN
ap-4699	224	2	]	]	PUNCT
ap-4699	224	3	is	be	AUX
ap-4699	224	4	additive	additive	ADJ
ap-4699	224	5	abelian	abelian	ADJ
ap-4699	224	6	group	group	NOUN
ap-4699	224	7	if	if	SCONJ
ap-4699	224	8	and	and	CCONJ
ap-4699	224	9	only	only	ADV
ap-4699	224	10	if	if	SCONJ
ap-4699	224	11	a[β	a[β	PROPN
ap-4699	224	12	]	]	X
ap-4699	224	13	=	=	SYM
ap-4699	224	14	z[β	z[β	NOUN
ap-4699	224	15	]	]	PUNCT
ap-4699	224	16	.	.	PUNCT
ap-4699	225	1	(	(	PUNCT
ap-4699	225	2	3	3	NUM
ap-4699	225	3	.	.	PUNCT
ap-4699	225	4	)	)	PUNCT
ap-4699	226	1	if	if	SCONJ
ap-4699	226	2	n	n	X
ap-4699	226	3	⊂	⊂	PROPN
ap-4699	226	4	a[β	a[β	PROPN
ap-4699	226	5	]	]	PUNCT
ap-4699	226	6	,	,	PUNCT
ap-4699	226	7	then	then	ADV
ap-4699	226	8	β	β	PROPN
ap-4699	226	9	is	be	AUX
ap-4699	226	10	expanding	expand	VERB
ap-4699	226	11	,	,	PUNCT
ap-4699	226	12	i.e.	i.e.	X
ap-4699	226	13	,	,	PUNCT
ap-4699	226	14	β	β	X
ap-4699	226	15	is	be	AUX
ap-4699	226	16	an	an	DET
ap-4699	226	17	algebraic	algebraic	ADJ
ap-4699	226	18	number	number	NOUN
ap-4699	226	19	with	with	ADP
ap-4699	226	20	all	all	DET
ap-4699	226	21	conjugates	conjugate	NOUN
ap-4699	226	22	greater	great	ADJ
ap-4699	226	23	than	than	ADP
ap-4699	226	24	1	1	NUM
ap-4699	226	25	in	in	ADP
ap-4699	226	26	modulus	modulus	NOUN
ap-4699	226	27	.	.	PUNCT
ap-4699	227	1	proof	proof	NOUN
ap-4699	227	2	.	.	PUNCT
ap-4699	228	1	(	(	PUNCT
ap-4699	228	2	1	1	X
ap-4699	228	3	.	.	NUM
ap-4699	228	4	):	):	PUNCT
ap-4699	228	5	obviously	obviously	ADV
ap-4699	228	6	,	,	PUNCT
ap-4699	228	7	if	if	SCONJ
ap-4699	228	8	a[β	a[β	PROPN
ap-4699	228	9	]	]	PUNCT
ap-4699	228	10	is	be	AUX
ap-4699	228	11	closed	close	VERB
ap-4699	228	12	under	under	ADP
ap-4699	228	13	addition	addition	NOUN
ap-4699	228	14	,	,	PUNCT
ap-4699	228	15	then	then	ADV
ap-4699	228	16	n	n	PROPN
ap-4699	228	17	⊂	⊂	PROPN
ap-4699	228	18	a[β	a[β	PROPN
ap-4699	228	19	]	]	PUNCT
ap-4699	228	20	.	.	PUNCT
ap-4699	229	1	since	since	SCONJ
ap-4699	229	2	0	0	NUM
ap-4699	229	3	∈	∈	PROPN
ap-4699	229	4	a	a	PRON
ap-4699	229	5	by	by	ADP
ap-4699	229	6	our	our	PRON
ap-4699	229	7	general	general	ADJ
ap-4699	229	8	assumption	assumption	NOUN
ap-4699	229	9	,	,	PUNCT
ap-4699	229	10	also	also	ADV
ap-4699	229	11	β	β	X
ap-4699	229	12	·	·	PUNCT
ap-4699	229	13	a[β	a[β	PROPN
ap-4699	229	14	]	]	X
ap-4699	229	15	⊂	⊂	PROPN
ap-4699	229	16	a[β	a[β	PROPN
ap-4699	229	17	]	]	PUNCT
ap-4699	229	18	.	.	PUNCT
ap-4699	230	1	therefore	therefore	ADV
ap-4699	230	2	,	,	PUNCT
ap-4699	230	3	β	β	X
ap-4699	230	4	·	·	PUNCT
ap-4699	230	5	n	n	X
ap-4699	230	6	⊂	⊂	PROPN
ap-4699	230	7	a[β	a[β	PROPN
ap-4699	230	8	]	]	PUNCT
ap-4699	230	9	and	and	CCONJ
ap-4699	230	10	the	the	DET
ap-4699	230	11	claim	claim	NOUN
ap-4699	230	12	n[β	n[β	NOUN
ap-4699	230	13	]	]	PUNCT
ap-4699	230	14	⊂	⊂	PROPN
ap-4699	230	15	a[β	a[β	PROPN
ap-4699	230	16	]	]	PUNCT
ap-4699	230	17	follows	follow	VERB
ap-4699	230	18	by	by	ADP
ap-4699	230	19	induction	induction	NOUN
ap-4699	230	20	.	.	PUNCT
ap-4699	231	1	(	(	PUNCT
ap-4699	231	2	2	2	X
ap-4699	231	3	.	.	NUM
ap-4699	231	4	):	):	PUNCT
ap-4699	231	5	the	the	DET
ap-4699	231	6	assumption	assumption	NOUN
ap-4699	231	7	that	that	SCONJ
ap-4699	231	8	a[β	a[β	PROPN
ap-4699	231	9	]	]	PUNCT
ap-4699	231	10	is	be	AUX
ap-4699	231	11	closed	close	VERB
ap-4699	231	12	also	also	ADV
ap-4699	231	13	under	under	ADP
ap-4699	231	14	subtraction	subtraction	NOUN
ap-4699	231	15	and	and	CCONJ
ap-4699	231	16	(	(	PUNCT
ap-4699	231	17	1	1	NUM
ap-4699	231	18	.	.	PUNCT
ap-4699	231	19	)	)	PUNCT
ap-4699	231	20	imply	imply	VERB
ap-4699	231	21	that	that	SCONJ
ap-4699	231	22	z[β	z[β	NOUN
ap-4699	231	23	]	]	X
ap-4699	231	24	=	=	PUNCT
ap-4699	231	25	n[β]−	n[β]−	SYM
ap-4699	231	26	n[β	n[β	NOUN
ap-4699	231	27	]	]	PUNCT
ap-4699	231	28	⊂	⊂	PROPN
ap-4699	231	29	a[β	a[β	PROPN
ap-4699	231	30	]	]	PUNCT
ap-4699	231	31	,	,	PUNCT
ap-4699	231	32	and	and	CCONJ
ap-4699	231	33	obviously	obviously	ADV
ap-4699	231	34	a[β	a[β	PROPN
ap-4699	231	35	]	]	X
ap-4699	231	36	⊂	⊂	PROPN
ap-4699	231	37	z[β	z[β	PROPN
ap-4699	231	38	]	]	PUNCT
ap-4699	231	39	.	.	PUNCT
ap-4699	232	1	the	the	DET
ap-4699	232	2	opposite	opposite	ADJ
ap-4699	232	3	implication	implication	NOUN
ap-4699	232	4	is	be	AUX
ap-4699	232	5	trivial	trivial	ADJ
ap-4699	232	6	.	.	PUNCT
ap-4699	233	1	(	(	PUNCT
ap-4699	233	2	3	3	X
ap-4699	233	3	.	.	NUM
ap-4699	233	4	):	):	PUNCT
ap-4699	233	5	by	by	ADP
ap-4699	233	6	lemma	lemma	PROPN
ap-4699	233	7	2.2	2.2	NUM
ap-4699	233	8	,	,	PUNCT
ap-4699	233	9	β	β	X
ap-4699	233	10	is	be	AUX
ap-4699	233	11	an	an	DET
ap-4699	233	12	algebraic	algebraic	ADJ
ap-4699	233	13	number	number	NOUN
ap-4699	233	14	since	since	SCONJ
ap-4699	233	15	a[β	a[β	PROPN
ap-4699	233	16	]	]	X
ap-4699	233	17	⊂	⊂	PROPN
ap-4699	233	18	fina(β	fina(β	PROPN
ap-4699	233	19	)	)	PUNCT
ap-4699	233	20	.	.	PUNCT
ap-4699	234	1	the	the	DET
ap-4699	234	2	proof	proof	NOUN
ap-4699	234	3	that	that	SCONJ
ap-4699	234	4	β	β	NOUN
ap-4699	234	5	must	must	AUX
ap-4699	234	6	be	be	AUX
ap-4699	234	7	expanding	expand	VERB
ap-4699	234	8	is	be	AUX
ap-4699	234	9	based	base	VERB
ap-4699	234	10	on	on	ADP
ap-4699	234	11	the	the	DET
ap-4699	234	12	paper	paper	NOUN
ap-4699	234	13	of	of	ADP
ap-4699	234	14	s.	s.	PROPN
ap-4699	234	15	akiyama	akiyama	PROPN
ap-4699	234	16	and	and	CCONJ
ap-4699	234	17	t.	t.	PROPN
ap-4699	234	18	zäimi	zäimi	PROPN
ap-4699	235	1	[	[	X
ap-4699	235	2	9	9	NUM
ap-4699	235	3	]	]	PUNCT
ap-4699	235	4	.	.	PUNCT
ap-4699	236	1	let	let	VERB
ap-4699	236	2	β′	β′	PRON
ap-4699	236	3	be	be	AUX
ap-4699	236	4	an	an	DET
ap-4699	236	5	algebraic	algebraic	ADJ
ap-4699	236	6	conjugate	conjugate	NOUN
ap-4699	236	7	of	of	ADP
ap-4699	236	8	β	β	PROPN
ap-4699	236	9	and	and	CCONJ
ap-4699	236	10	σ	σ	NUM
ap-4699	236	11	:	:	PUNCT
ap-4699	236	12	q(β)→	q(β)→	SYM
ap-4699	236	13	q(β′	q(β′	NUM
ap-4699	236	14	)	)	PUNCT
ap-4699	236	15	be	be	VERB
ap-4699	236	16	the	the	DET
ap-4699	236	17	field	field	NOUN
ap-4699	236	18	isomorphism	isomorphism	NOUN
ap-4699	236	19	such	such	ADJ
ap-4699	236	20	that	that	SCONJ
ap-4699	236	21	σ(β	σ(β	PROPN
ap-4699	236	22	)	)	PUNCT
ap-4699	237	1	=	=	SYM
ap-4699	237	2	β′.	β′.	PROPN
ap-4699	237	3	since	since	SCONJ
ap-4699	237	4	n	n	PROPN
ap-4699	237	5	⊂	⊂	PROPN
ap-4699	237	6	a[β	a[β	PROPN
ap-4699	237	7	]	]	PUNCT
ap-4699	237	8	,	,	PUNCT
ap-4699	237	9	for	for	ADP
ap-4699	237	10	all	all	DET
ap-4699	237	11	n	n	PRON
ap-4699	237	12	∈	∈	PRON
ap-4699	237	13	n	n	AUX
ap-4699	237	14	there	there	PRON
ap-4699	237	15	exist	exist	VERB
ap-4699	237	16	a0	a0	NOUN
ap-4699	237	17	,	,	PUNCT
ap-4699	237	18	.	.	PUNCT
ap-4699	237	19	.	.	PUNCT
ap-4699	238	1	.	.	PUNCT
ap-4699	239	1	,	,	PUNCT
ap-4699	239	2	an	an	DET
ap-4699	239	3	∈	∈	PROPN
ap-4699	239	4	a	a	DET
ap-4699	239	5	such	such	ADJ
ap-4699	239	6	that	that	DET
ap-4699	239	7	n∑	n∑	PROPN
ap-4699	239	8	i=0	i=0	PROPN
ap-4699	240	1	aiβ	aiβ	INTJ
ap-4699	240	2	i	i	NOUN
ap-4699	240	3	=	=	SYM
ap-4699	240	4	n	n	PROPN
ap-4699	240	5	=	=	SYM
ap-4699	240	6	σ(n	σ(n	PROPN
ap-4699	240	7	)	)	PUNCT
ap-4699	241	1	=	=	SYM
ap-4699	242	1	n∑	n∑	NOUN
ap-4699	242	2	i=0	i=0	PROPN
ap-4699	242	3	σ(ai)(β′)i	σ(ai)(β′)i	INTJ
ap-4699	242	4	.	.	PUNCT
ap-4699	243	1	denoting	denote	VERB
ap-4699	243	2	m̃	m̃	PROPN
ap-4699	243	3	:	:	PUNCT
ap-4699	243	4	=	=	SYM
ap-4699	243	5	max{|σ(a)|	max{|σ(a)|	X
ap-4699	243	6	:	:	PUNCT
ap-4699	243	7	a	a	DET
ap-4699	243	8	∈	∈	PROPN
ap-4699	243	9	a	a	X
ap-4699	243	10	}	}	PUNCT
ap-4699	243	11	,	,	PUNCT
ap-4699	243	12	we	we	PRON
ap-4699	243	13	have	have	VERB
ap-4699	243	14	n	n	NOUN
ap-4699	243	15	=	=	SYM
ap-4699	243	16	|n|	|n|	NUM
ap-4699	243	17	≤	≤	NOUN
ap-4699	243	18	n∑	n∑	PUNCT
ap-4699	243	19	i=0	i=0	ADJ
ap-4699	243	20	|σ(ai)|	|σ(ai)|	X
ap-4699	243	21	·	·	PUNCT
ap-4699	243	22	|β′|i	|β′|i	PUNCT
ap-4699	243	23	≤	≤	NOUN
ap-4699	244	1	∞∑	∞∑	NUM
ap-4699	244	2	i=0	i=0	ADJ
ap-4699	244	3	|σ(ai)|	|σ(ai)|	X
ap-4699	244	4	·	·	PUNCT
ap-4699	244	5	|β′|i	|β′|i	PUNCT
ap-4699	244	6	≤	≤	PUNCT
ap-4699	244	7	m̃	m̃	PROPN
ap-4699	244	8	∞∑	∞∑	PROPN
ap-4699	244	9	i=0	i=0	PROPN
ap-4699	244	10	|β′|i	|β′|i	PUNCT
ap-4699	244	11	.	.	PUNCT
ap-4699	245	1	as	as	SCONJ
ap-4699	245	2	n	n	PRON
ap-4699	245	3	is	be	AUX
ap-4699	245	4	arbitrarily	arbitrarily	ADV
ap-4699	245	5	large	large	ADJ
ap-4699	245	6	,	,	PUNCT
ap-4699	245	7	the	the	DET
ap-4699	245	8	sum	sum	NOUN
ap-4699	245	9	on	on	ADP
ap-4699	245	10	the	the	DET
ap-4699	245	11	right	right	ADJ
ap-4699	245	12	side	side	NOUN
ap-4699	245	13	diverges	diverge	NOUN
ap-4699	245	14	,	,	PUNCT
ap-4699	245	15	which	which	PRON
ap-4699	245	16	implies	imply	VERB
ap-4699	245	17	that	that	SCONJ
ap-4699	245	18	|β′|	|β′|	NOUN
ap-4699	245	19	≥	≥	NOUN
ap-4699	245	20	1	1	NUM
ap-4699	245	21	.	.	PUNCT
ap-4699	246	1	thus	thus	ADV
ap-4699	246	2	,	,	PUNCT
ap-4699	246	3	all	all	DET
ap-4699	246	4	conjugates	conjugate	NOUN
ap-4699	246	5	of	of	ADP
ap-4699	246	6	β	β	NOUN
ap-4699	246	7	are	be	AUX
ap-4699	246	8	at	at	ADV
ap-4699	246	9	least	least	ADJ
ap-4699	246	10	one	one	NUM
ap-4699	246	11	in	in	ADP
ap-4699	246	12	modulus	modulus	NOUN
ap-4699	246	13	.	.	PUNCT
ap-4699	247	1	if	if	SCONJ
ap-4699	247	2	the	the	DET
ap-4699	247	3	degree	degree	NOUN
ap-4699	247	4	of	of	ADP
ap-4699	247	5	β	β	PROPN
ap-4699	247	6	is	be	AUX
ap-4699	247	7	one	one	NUM
ap-4699	247	8	,	,	PUNCT
ap-4699	247	9	the	the	DET
ap-4699	247	10	statement	statement	NOUN
ap-4699	247	11	is	be	AUX
ap-4699	247	12	obvious	obvious	ADJ
ap-4699	247	13	.	.	PUNCT
ap-4699	248	1	therefore	therefore	ADV
ap-4699	248	2	,	,	PUNCT
ap-4699	248	3	we	we	PRON
ap-4699	248	4	may	may	AUX
ap-4699	248	5	assume	assume	VERB
ap-4699	248	6	that	that	SCONJ
ap-4699	248	7	deg	deg	PROPN
ap-4699	248	8	β	β	X
ap-4699	248	9	≥	≥	NUM
ap-4699	248	10	2	2	X
ap-4699	248	11	.	.	PUNCT
ap-4699	248	12	suppose	suppose	VERB
ap-4699	248	13	for	for	ADP
ap-4699	248	14	contradiction	contradiction	NOUN
ap-4699	248	15	that	that	SCONJ
ap-4699	248	16	|β′|	|β′|	NOUN
ap-4699	248	17	=	=	SYM
ap-4699	248	18	1	1	NUM
ap-4699	248	19	for	for	ADP
ap-4699	248	20	an	an	DET
ap-4699	248	21	algebraic	algebraic	ADJ
ap-4699	248	22	conjugate	conjugate	NOUN
ap-4699	248	23	β′	β′	NUM
ap-4699	248	24	of	of	ADP
ap-4699	248	25	β	β	PROPN
ap-4699	248	26	.	.	PUNCT
ap-4699	249	1	the	the	DET
ap-4699	249	2	complex	complex	ADJ
ap-4699	249	3	conjugate	conjugate	NOUN
ap-4699	249	4	β′	β′	NUM
ap-4699	249	5	is	be	AUX
ap-4699	249	6	also	also	ADV
ap-4699	249	7	an	an	DET
ap-4699	249	8	algebraic	algebraic	ADJ
ap-4699	249	9	conjugate	conjugate	NOUN
ap-4699	249	10	of	of	ADP
ap-4699	249	11	β	β	X
ap-4699	249	12	.	.	PUNCT
ap-4699	250	1	take	take	VERB
ap-4699	250	2	any	any	DET
ap-4699	250	3	algebraic	algebraic	ADJ
ap-4699	250	4	conjugate	conjugate	ADJ
ap-4699	250	5	γ	γ	NOUN
ap-4699	250	6	of	of	ADP
ap-4699	250	7	β	β	PROPN
ap-4699	250	8	and	and	CCONJ
ap-4699	250	9	the	the	DET
ap-4699	250	10	isomorphism	isomorphism	NOUN
ap-4699	250	11	σ′	σ′	PROPN
ap-4699	250	12	:	:	PUNCT
ap-4699	250	13	q(β′)→	q(β′)→	PROPN
ap-4699	250	14	q(γ	q(γ	PROPN
ap-4699	250	15	)	)	PUNCT
ap-4699	250	16	given	give	VERB
ap-4699	250	17	by	by	ADP
ap-4699	250	18	σ′(β′	σ′(β′	NOUN
ap-4699	250	19	)	)	PUNCT
ap-4699	250	20	=	=	SYM
ap-4699	251	1	γ	γ	X
ap-4699	251	2	.	.	PROPN
ap-4699	252	1	now	now	ADV
ap-4699	252	2	1	1	NUM
ap-4699	252	3	γ	γ	X
ap-4699	252	4	=	=	SYM
ap-4699	252	5	1	1	NUM
ap-4699	252	6	σ′(β′	σ′(β′	NOUN
ap-4699	252	7	)	)	PUNCT
ap-4699	252	8	=	=	SYM
ap-4699	252	9	σ′	σ′	PROPN
ap-4699	252	10	(	(	PUNCT
ap-4699	252	11	1	1	NUM
ap-4699	252	12	β′	β′	NUM
ap-4699	252	13	)	)	PUNCT
ap-4699	253	1	=	=	SYM
ap-4699	253	2	σ′	σ′	PROPN
ap-4699	253	3	(	(	PUNCT
ap-4699	253	4	β′	β′	NUM
ap-4699	253	5	β′β′	β′β′	NOUN
ap-4699	253	6	)	)	PUNCT
ap-4699	253	7	=	=	SYM
ap-4699	253	8	σ′	σ′	PROPN
ap-4699	253	9	(	(	PUNCT
ap-4699	253	10	β′	β′	X
ap-4699	253	11	|β′|2	|β′|2	NOUN
ap-4699	253	12	)	)	PUNCT
ap-4699	254	1	=	=	SYM
ap-4699	254	2	σ′(β′	σ′(β′	NOUN
ap-4699	254	3	)	)	PUNCT
ap-4699	254	4	.	.	PUNCT
ap-4699	255	1	hence	hence	ADV
ap-4699	255	2	,	,	PUNCT
ap-4699	255	3	1	1	NUM
ap-4699	255	4	γ	γ	NOUN
ap-4699	255	5	is	be	AUX
ap-4699	255	6	also	also	ADV
ap-4699	255	7	an	an	DET
ap-4699	255	8	algebraic	algebraic	ADJ
ap-4699	255	9	conjugate	conjugate	NOUN
ap-4699	255	10	of	of	ADP
ap-4699	255	11	β	β	X
ap-4699	255	12	.	.	PUNCT
ap-4699	256	1	moreover,∣∣	moreover,∣∣	VERB
ap-4699	256	2	1	1	NUM
ap-4699	256	3	γ	γ	X
ap-4699	256	4	∣∣	∣∣	NUM
ap-4699	256	5	≥	≥	NOUN
ap-4699	256	6	1	1	NUM
ap-4699	256	7	and	and	CCONJ
ap-4699	256	8	|γ|	|γ|	PRON
ap-4699	256	9	≥	≥	NOUN
ap-4699	256	10	1	1	NUM
ap-4699	256	11	,	,	PUNCT
ap-4699	256	12	which	which	PRON
ap-4699	256	13	implies	imply	VERB
ap-4699	256	14	that	that	SCONJ
ap-4699	256	15	|γ|	|γ|	PROPN
ap-4699	256	16	=	=	SYM
ap-4699	256	17	1	1	X
ap-4699	256	18	.	.	PUNCT
ap-4699	257	1	we	we	PRON
ap-4699	257	2	may	may	AUX
ap-4699	257	3	choose	choose	VERB
ap-4699	257	4	γ	γ	X
ap-4699	257	5	=	=	SYM
ap-4699	257	6	β	β	PROPN
ap-4699	257	7	,	,	PUNCT
ap-4699	257	8	which	which	PRON
ap-4699	257	9	contradicts	contradict	VERB
ap-4699	257	10	|β|	|β|	PRON
ap-4699	257	11	>	>	X
ap-4699	257	12	1	1	NUM
ap-4699	257	13	.	.	PUNCT
ap-4699	258	1	thus	thus	ADV
ap-4699	258	2	all	all	DET
ap-4699	258	3	conjugates	conjugate	NOUN
ap-4699	258	4	of	of	ADP
ap-4699	258	5	β	β	NOUN
ap-4699	258	6	are	be	AUX
ap-4699	258	7	greater	great	ADJ
ap-4699	258	8	than	than	ADP
ap-4699	258	9	one	one	NUM
ap-4699	258	10	in	in	ADP
ap-4699	258	11	modulus	modulus	NOUN
ap-4699	258	12	,	,	PUNCT
ap-4699	258	13	i.e.	i.e.	X
ap-4699	258	14	,	,	PUNCT
ap-4699	258	15	β	β	X
ap-4699	258	16	is	be	AUX
ap-4699	258	17	an	an	DET
ap-4699	258	18	expanding	expand	VERB
ap-4699	258	19	algebraic	algebraic	ADJ
ap-4699	258	20	number	number	NOUN
ap-4699	258	21	.	.	PUNCT
ap-4699	259	1	let	let	VERB
ap-4699	259	2	us	we	PRON
ap-4699	259	3	remark	remark	VERB
ap-4699	259	4	that	that	SCONJ
ap-4699	259	5	the	the	DET
ap-4699	259	6	assumption	assumption	NOUN
ap-4699	259	7	on	on	ADP
ap-4699	259	8	a[β	a[β	PROPN
ap-4699	259	9	]	]	PUNCT
ap-4699	259	10	to	to	PART
ap-4699	259	11	be	be	AUX
ap-4699	259	12	closed	close	VERB
ap-4699	259	13	under	under	ADP
ap-4699	259	14	addition	addition	NOUN
ap-4699	259	15	is	be	AUX
ap-4699	259	16	satisfied	satisfied	ADJ
ap-4699	259	17	by	by	ADP
ap-4699	259	18	a	a	DET
ap-4699	259	19	wide	wide	ADJ
ap-4699	259	20	class	class	NOUN
ap-4699	259	21	of	of	ADP
ap-4699	259	22	numeration	numeration	NOUN
ap-4699	259	23	systems	system	NOUN
ap-4699	259	24	.	.	PUNCT
ap-4699	260	1	namely	namely	ADV
ap-4699	260	2	,	,	PUNCT
ap-4699	260	3	if	if	SCONJ
ap-4699	260	4	a	a	DET
ap-4699	260	5	numeration	numeration	NOUN
ap-4699	260	6	system	system	NOUN
ap-4699	260	7	(	(	PUNCT
ap-4699	260	8	β	β	X
ap-4699	260	9	,	,	PUNCT
ap-4699	260	10	a	a	PRON
ap-4699	260	11	)	)	PUNCT
ap-4699	260	12	allows	allow	VERB
ap-4699	260	13	p	p	ADJ
ap-4699	260	14	-	-	ADJ
ap-4699	260	15	local	local	ADJ
ap-4699	260	16	parallel	parallel	ADJ
ap-4699	260	17	addition	addition	NOUN
ap-4699	260	18	such	such	ADJ
ap-4699	260	19	that	that	SCONJ
ap-4699	260	20	p	p	PROPN
ap-4699	260	21	=	=	SYM
ap-4699	260	22	r+1	r+1	PROPN
ap-4699	260	23	,	,	PUNCT
ap-4699	260	24	i.e.	i.e.	X
ap-4699	260	25	,	,	PUNCT
ap-4699	260	26	there	there	PRON
ap-4699	260	27	is	be	VERB
ap-4699	260	28	no	no	DET
ap-4699	260	29	anticipation	anticipation	NOUN
ap-4699	260	30	,	,	PUNCT
ap-4699	260	31	then	then	ADV
ap-4699	260	32	a[β	a[β	PROPN
ap-4699	260	33	]	]	PUNCT
ap-4699	260	34	is	be	AUX
ap-4699	260	35	obviously	obviously	ADV
ap-4699	260	36	closed	close	VERB
ap-4699	260	37	under	under	ADP
ap-4699	260	38	addition	addition	NOUN
ap-4699	260	39	.	.	PUNCT
ap-4699	261	1	hence	hence	ADV
ap-4699	261	2	,	,	PUNCT
ap-4699	261	3	(	(	PUNCT
ap-4699	261	4	1	1	NUM
ap-4699	261	5	.	.	PUNCT
ap-4699	261	6	)	)	PUNCT
ap-4699	262	1	and	and	CCONJ
ap-4699	262	2	(	(	PUNCT
ap-4699	262	3	3	3	NUM
ap-4699	262	4	.	.	PUNCT
ap-4699	262	5	)	)	PUNCT
ap-4699	262	6	give	give	VERB
ap-4699	262	7	the	the	DET
ap-4699	262	8	following	follow	VERB
ap-4699	262	9	corollary	corollary	NOUN
ap-4699	262	10	.	.	PUNCT
ap-4699	263	1	corollary	corollary	ADJ
ap-4699	263	2	3.6	3.6	NUM
ap-4699	263	3	.	.	PUNCT
ap-4699	264	1	let	let	VERB
ap-4699	264	2	(	(	PUNCT
ap-4699	264	3	β	β	X
ap-4699	264	4	,	,	PUNCT
ap-4699	264	5	a	a	PRON
ap-4699	264	6	)	)	PUNCT
ap-4699	264	7	be	be	AUX
ap-4699	264	8	a	a	DET
ap-4699	264	9	numeration	numeration	NOUN
ap-4699	264	10	system	system	NOUN
ap-4699	264	11	such	such	ADJ
ap-4699	264	12	that	that	SCONJ
ap-4699	264	13	1	1	NUM
ap-4699	264	14	∈	∈	PROPN
ap-4699	264	15	a[β	a[β	PROPN
ap-4699	264	16	]	]	PUNCT
ap-4699	264	17	and	and	CCONJ
ap-4699	264	18	a	a	DET
ap-4699	264	19	⊂	⊂	PROPN
ap-4699	264	20	z[β	z[β	NOUN
ap-4699	264	21	]	]	PUNCT
ap-4699	264	22	.	.	PUNCT
ap-4699	265	1	if	if	SCONJ
ap-4699	265	2	(	(	PUNCT
ap-4699	265	3	β	β	X
ap-4699	265	4	,	,	PUNCT
ap-4699	265	5	a	a	PRON
ap-4699	265	6	)	)	PUNCT
ap-4699	265	7	allows	allow	VERB
ap-4699	265	8	parallel	parallel	ADJ
ap-4699	265	9	addition	addition	NOUN
ap-4699	265	10	without	without	ADP
ap-4699	265	11	anticipation	anticipation	NOUN
ap-4699	265	12	,	,	PUNCT
ap-4699	265	13	then	then	ADV
ap-4699	265	14	β	β	PROPN
ap-4699	265	15	is	be	AUX
ap-4699	265	16	expanding	expand	VERB
ap-4699	265	17	.	.	PUNCT
ap-4699	266	1	for	for	ADP
ap-4699	266	2	β	β	PRON
ap-4699	266	3	expanding	expand	VERB
ap-4699	266	4	,	,	PUNCT
ap-4699	266	5	lemma	lemma	PROPN
ap-4699	266	6	8	8	NUM
ap-4699	266	7	in	in	ADP
ap-4699	266	8	[	[	X
ap-4699	266	9	10	10	NUM
ap-4699	266	10	]	]	PUNCT
ap-4699	266	11	provides	provide	VERB
ap-4699	266	12	a	a	DET
ap-4699	266	13	so	so	ADV
ap-4699	266	14	called	call	VERB
ap-4699	266	15	weak	weak	ADJ
ap-4699	266	16	representation	representation	NOUN
ap-4699	266	17	of	of	ADP
ap-4699	266	18	zero	zero	NUM
ap-4699	266	19	property	property	NOUN
ap-4699	266	20	such	such	ADJ
ap-4699	266	21	that	that	SCONJ
ap-4699	266	22	the	the	DET
ap-4699	266	23	absolute	absolute	ADJ
ap-4699	266	24	term	term	NOUN
ap-4699	266	25	is	be	AUX
ap-4699	266	26	dominant	dominant	ADJ
ap-4699	266	27	,	,	PUNCT
ap-4699	266	28	and	and	CCONJ
ap-4699	266	29	hence	hence	ADV
ap-4699	266	30	parallel	parallel	ADJ
ap-4699	266	31	addition	addition	NOUN
ap-4699	266	32	in	in	ADP
ap-4699	266	33	the	the	DET
ap-4699	266	34	base	base	NOUN
ap-4699	266	35	β	β	NOUN
ap-4699	266	36	without	without	ADP
ap-4699	266	37	anticipation	anticipation	NOUN
ap-4699	266	38	is	be	AUX
ap-4699	266	39	obtained	obtain	VERB
ap-4699	266	40	for	for	ADP
ap-4699	266	41	some	some	DET
ap-4699	266	42	integer	integer	NOUN
ap-4699	266	43	alphabet	alphabet	NOUN
ap-4699	266	44	ai	be	AUX
ap-4699	266	45	nt	not	PART
ap-4699	266	46	according	accord	VERB
ap-4699	266	47	theorem	theorem	ADJ
ap-4699	266	48	4.3	4.3	NUM
ap-4699	266	49	.	.	PUNCT
ap-4699	267	1	in	in	ADP
ap-4699	267	2	[	[	X
ap-4699	267	3	5	5	NUM
ap-4699	267	4	]	]	PUNCT
ap-4699	267	5	.	.	PUNCT
ap-4699	268	1	288	288	NUM
ap-4699	268	2	vol	vol	NOUN
ap-4699	268	3	.	.	PUNCT
ap-4699	268	4	58	58	NUM
ap-4699	268	5	no	no	INTJ
ap-4699	268	6	.	.	PUNCT
ap-4699	269	1	5/2018	5/2018	NUM
ap-4699	269	2	minimal	minimal	ADJ
ap-4699	269	3	non	non	ADJ
ap-4699	269	4	-	-	ADJ
ap-4699	269	5	integer	integer	ADJ
ap-4699	269	6	alphabets	alphabet	NOUN
ap-4699	269	7	allowing	allow	VERB
ap-4699	269	8	parallel	parallel	ADJ
ap-4699	269	9	addition	addition	NOUN
ap-4699	269	10	in	in	ADP
ap-4699	269	11	what	what	PRON
ap-4699	269	12	follows	follow	VERB
ap-4699	269	13	,	,	PUNCT
ap-4699	269	14	we	we	PRON
ap-4699	269	15	assume	assume	VERB
ap-4699	269	16	a[β	a[β	PROPN
ap-4699	269	17	]	]	X
ap-4699	269	18	=	=	SYM
ap-4699	269	19	z[β	z[β	NOUN
ap-4699	269	20	]	]	PUNCT
ap-4699	269	21	,	,	PUNCT
ap-4699	269	22	although	although	SCONJ
ap-4699	269	23	the	the	DET
ap-4699	269	24	weaker	weak	ADJ
ap-4699	269	25	assumption	assumption	NOUN
ap-4699	269	26	,	,	PUNCT
ap-4699	269	27	a[β	a[β	PROPN
ap-4699	269	28	]	]	PUNCT
ap-4699	269	29	being	be	AUX
ap-4699	269	30	closed	close	VERB
ap-4699	269	31	under	under	ADP
ap-4699	269	32	addition	addition	NOUN
ap-4699	269	33	,	,	PUNCT
ap-4699	269	34	would	would	AUX
ap-4699	269	35	be	be	AUX
ap-4699	269	36	sufficient	sufficient	ADJ
ap-4699	269	37	.	.	PUNCT
ap-4699	270	1	the	the	DET
ap-4699	270	2	reason	reason	NOUN
ap-4699	270	3	is	be	AUX
ap-4699	270	4	that	that	PRON
ap-4699	270	5	subtraction	subtraction	NOUN
ap-4699	270	6	is	be	AUX
ap-4699	270	7	also	also	ADV
ap-4699	270	8	required	require	VERB
ap-4699	270	9	in	in	ADP
ap-4699	270	10	applications	application	NOUN
ap-4699	270	11	using	use	VERB
ap-4699	270	12	parallel	parallel	ADJ
ap-4699	270	13	addition	addition	NOUN
ap-4699	270	14	,	,	PUNCT
ap-4699	270	15	such	such	ADJ
ap-4699	270	16	as	as	ADP
ap-4699	270	17	on	on	ADP
ap-4699	270	18	-	-	PUNCT
ap-4699	270	19	line	line	NOUN
ap-4699	270	20	multiplication	multiplication	NOUN
ap-4699	270	21	and	and	CCONJ
ap-4699	270	22	division	division	NOUN
ap-4699	270	23	.	.	PUNCT
ap-4699	271	1	hence	hence	ADV
ap-4699	271	2	the	the	DET
ap-4699	271	3	assumption	assumption	NOUN
ap-4699	271	4	is	be	AUX
ap-4699	271	5	justified	justify	VERB
ap-4699	271	6	by	by	ADP
ap-4699	271	7	(	(	PUNCT
ap-4699	271	8	2	2	NUM
ap-4699	271	9	.	.	NUM
ap-4699	271	10	)	)	PUNCT
ap-4699	271	11	of	of	ADP
ap-4699	271	12	theorem	theorem	NOUN
ap-4699	271	13	3.5	3.5	NUM
ap-4699	271	14	.	.	PUNCT
ap-4699	272	1	let	let	VERB
ap-4699	272	2	us	we	PRON
ap-4699	272	3	mention	mention	VERB
ap-4699	272	4	that	that	PRON
ap-4699	272	5	,	,	PUNCT
ap-4699	272	6	for	for	ADP
ap-4699	272	7	instance	instance	NOUN
ap-4699	272	8	,	,	PUNCT
ap-4699	272	9	the	the	DET
ap-4699	272	10	numeration	numeration	NOUN
ap-4699	272	11	system	system	NOUN
ap-4699	272	12	(	(	PUNCT
ap-4699	272	13	2	2	NUM
ap-4699	272	14	,	,	PUNCT
ap-4699	272	15	{	{	PUNCT
ap-4699	272	16	0	0	NUM
ap-4699	272	17	,	,	PUNCT
ap-4699	272	18	1	1	NUM
ap-4699	272	19	,	,	PUNCT
ap-4699	272	20	2	2	NUM
ap-4699	272	21	}	}	PUNCT
ap-4699	272	22	)	)	PUNCT
ap-4699	272	23	allows	allow	VERB
ap-4699	272	24	parallel	parallel	ADJ
ap-4699	272	25	addition	addition	NOUN
ap-4699	272	26	,	,	PUNCT
ap-4699	272	27	but	but	CCONJ
ap-4699	272	28	(	(	PUNCT
ap-4699	272	29	2	2	NUM
ap-4699	272	30	,	,	PUNCT
ap-4699	272	31	{	{	PUNCT
ap-4699	272	32	0	0	NUM
ap-4699	272	33	,	,	PUNCT
ap-4699	272	34	1	1	NUM
ap-4699	272	35	,	,	PUNCT
ap-4699	272	36	2	2	NUM
ap-4699	272	37	}	}	PUNCT
ap-4699	272	38	)	)	PUNCT
ap-4699	272	39	is	be	AUX
ap-4699	272	40	obviously	obviously	ADV
ap-4699	272	41	not	not	PART
ap-4699	272	42	closed	close	VERB
ap-4699	272	43	under	under	ADP
ap-4699	272	44	subtraction	subtraction	NOUN
ap-4699	272	45	.	.	PUNCT
ap-4699	273	1	theorem	theorem	VERB
ap-4699	273	2	3.7	3.7	NUM
ap-4699	273	3	.	.	PUNCT
ap-4699	274	1	if	if	SCONJ
ap-4699	274	2	a	a	DET
ap-4699	274	3	numeration	numeration	NOUN
ap-4699	274	4	system	system	NOUN
ap-4699	274	5	(	(	PUNCT
ap-4699	274	6	β	β	X
ap-4699	274	7	,	,	PUNCT
ap-4699	274	8	a	a	PRON
ap-4699	274	9	)	)	PUNCT
ap-4699	274	10	with	with	ADP
ap-4699	274	11	a[β	a[β	PROPN
ap-4699	274	12	]	]	X
ap-4699	274	13	=	=	SYM
ap-4699	274	14	z[β	z[β	X
ap-4699	274	15	]	]	PUNCT
ap-4699	274	16	allows	allow	VERB
ap-4699	274	17	parallel	parallel	ADJ
ap-4699	274	18	addition	addition	NOUN
ap-4699	274	19	,	,	PUNCT
ap-4699	274	20	then	then	ADV
ap-4699	274	21	the	the	DET
ap-4699	274	22	alphabet	alphabet	NOUN
ap-4699	274	23	a	a	PRON
ap-4699	274	24	contains	contain	VERB
ap-4699	274	25	at	at	ADV
ap-4699	274	26	least	least	ADV
ap-4699	274	27	one	one	NUM
ap-4699	274	28	representative	representative	NOUN
ap-4699	274	29	of	of	ADP
ap-4699	274	30	each	each	DET
ap-4699	274	31	congruence	congruence	NOUN
ap-4699	274	32	class	class	NOUN
ap-4699	274	33	modulo	modulo	PROPN
ap-4699	274	34	β	β	X
ap-4699	274	35	and	and	CCONJ
ap-4699	274	36	modulo	modulo	PROPN
ap-4699	274	37	β	β	X
ap-4699	274	38	−	−	PROPN
ap-4699	274	39	1	1	NUM
ap-4699	274	40	in	in	ADP
ap-4699	274	41	z[β	z[β	NOUN
ap-4699	274	42	]	]	PUNCT
ap-4699	274	43	.	.	PUNCT
ap-4699	275	1	proof	proof	NOUN
ap-4699	275	2	.	.	PUNCT
ap-4699	276	1	let	let	VERB
ap-4699	276	2	x	x	NOUN
ap-4699	276	3	=	=	SYM
ap-4699	276	4	∑n	∑n	PROPN
ap-4699	276	5	i=0	i=0	PROPN
ap-4699	276	6	xiβ	xiβ	NOUN
ap-4699	276	7	i	i	PRON
ap-4699	276	8	be	be	VERB
ap-4699	276	9	an	an	DET
ap-4699	276	10	element	element	NOUN
ap-4699	276	11	of	of	ADP
ap-4699	276	12	z[β	z[β	NOUN
ap-4699	276	13	]	]	PUNCT
ap-4699	276	14	.	.	PUNCT
ap-4699	277	1	since	since	SCONJ
ap-4699	277	2	x0	x0	PROPN
ap-4699	277	3	∈	∈	PROPN
ap-4699	277	4	z	z	PROPN
ap-4699	277	5	⊂	⊂	PROPN
ap-4699	277	6	a[β	a[β	PROPN
ap-4699	277	7	]	]	PUNCT
ap-4699	277	8	,	,	PUNCT
ap-4699	277	9	we	we	PRON
ap-4699	277	10	have	have	VERB
ap-4699	277	11	x	x	PART
ap-4699	277	12	≡β	≡β	VERB
ap-4699	277	13	x0	x0	PROPN
ap-4699	277	14	=	=	SYM
ap-4699	277	15	n∑	n∑	PROPN
ap-4699	277	16	i=0	i=0	PROPN
ap-4699	278	1	aiβ	aiβ	INTJ
ap-4699	278	2	i	i	PRON
ap-4699	278	3	≡β	≡β	VERB
ap-4699	278	4	a0	a0	NOUN
ap-4699	278	5	,	,	PUNCT
ap-4699	278	6	where	where	SCONJ
ap-4699	278	7	ai	ai	VERB
ap-4699	278	8	∈	∈	PROPN
ap-4699	278	9	a.	a.	NOUN
ap-4699	278	10	hence	hence	ADV
ap-4699	278	11	,	,	PUNCT
ap-4699	278	12	for	for	ADP
ap-4699	278	13	any	any	DET
ap-4699	278	14	element	element	NOUN
ap-4699	278	15	x	x	SYM
ap-4699	278	16	∈	∈	PROPN
ap-4699	278	17	z[β	z[β	PROPN
ap-4699	278	18	]	]	PUNCT
ap-4699	278	19	,	,	PUNCT
ap-4699	278	20	there	there	PRON
ap-4699	278	21	is	be	VERB
ap-4699	278	22	a	a	DET
ap-4699	278	23	digit	digit	NOUN
ap-4699	278	24	a0	a0	PROPN
ap-4699	278	25	∈	∈	PROPN
ap-4699	278	26	a	a	DET
ap-4699	278	27	such	such	ADJ
ap-4699	278	28	that	that	SCONJ
ap-4699	278	29	x	x	SYM
ap-4699	278	30	≡β	≡β	ADJ
ap-4699	278	31	a0	a0	NOUN
ap-4699	278	32	.	.	PUNCT
ap-4699	279	1	in	in	ADP
ap-4699	279	2	order	order	NOUN
ap-4699	279	3	to	to	PART
ap-4699	279	4	prove	prove	VERB
ap-4699	279	5	that	that	SCONJ
ap-4699	279	6	there	there	PRON
ap-4699	279	7	is	be	VERB
ap-4699	279	8	an	an	DET
ap-4699	279	9	element	element	NOUN
ap-4699	279	10	of	of	ADP
ap-4699	279	11	a	a	DET
ap-4699	279	12	congruent	congruent	NOUN
ap-4699	279	13	to	to	ADP
ap-4699	279	14	x	x	PART
ap-4699	279	15	modulo	modulo	VERB
ap-4699	279	16	β	β	X
ap-4699	279	17	−	−	NOUN
ap-4699	279	18	1	1	NUM
ap-4699	279	19	,	,	PUNCT
ap-4699	279	20	we	we	PRON
ap-4699	279	21	use	use	VERB
ap-4699	279	22	the	the	DET
ap-4699	279	23	binomial	binomial	ADJ
ap-4699	279	24	theorem	theorem	NOUN
ap-4699	279	25	:	:	PUNCT
ap-4699	279	26	x	x	SYM
ap-4699	279	27	=	=	SYM
ap-4699	279	28	n∑	n∑	PROPN
ap-4699	280	1	i=0	i=0	PROPN
ap-4699	280	2	xiβ	xiβ	PROPN
ap-4699	280	3	i	i	PROPN
ap-4699	280	4	=	=	PROPN
ap-4699	280	5	n∑	n∑	PROPN
ap-4699	280	6	i=0	i=0	PROPN
ap-4699	280	7	xi(β	xi(β	PUNCT
ap-4699	280	8	−	−	PROPN
ap-4699	280	9	1	1	NUM
ap-4699	280	10	+	+	NUM
ap-4699	280	11	1)i	1)i	NUM
ap-4699	280	12	=	=	SYM
ap-4699	280	13	n∑	n∑	PROPN
ap-4699	280	14	i=0	i=0	PROPN
ap-4699	280	15	x′j(β	x′j(β	PROPN
ap-4699	280	16	−	−	PROPN
ap-4699	280	17	1)j	1)j	PROPN
ap-4699	280	18	,	,	PUNCT
ap-4699	280	19	for	for	ADP
ap-4699	280	20	some	some	DET
ap-4699	280	21	x′j	x′j	PROPN
ap-4699	280	22	∈	∈	PROPN
ap-4699	280	23	z.	z.	PROPN
ap-4699	281	1	hence	hence	ADV
ap-4699	281	2	x	x	PROPN
ap-4699	281	3	≡β−1	≡β−1	SYM
ap-4699	281	4	x	x	SYM
ap-4699	281	5	′	′	NOUN
ap-4699	281	6	0	0	NUM
ap-4699	282	1	=	=	SYM
ap-4699	282	2	n∑	n∑	PROPN
ap-4699	282	3	i=0	i=0	PROPN
ap-4699	282	4	aiβ	aiβ	VERB
ap-4699	282	5	i	i	PRON
ap-4699	282	6	,	,	PUNCT
ap-4699	282	7	for	for	ADP
ap-4699	282	8	some	some	DET
ap-4699	282	9	ai	ai	PROPN
ap-4699	282	10	∈	∈	NOUN
ap-4699	282	11	a.	a.	NOUN
ap-4699	282	12	we	we	PRON
ap-4699	282	13	prove	prove	VERB
ap-4699	282	14	by	by	ADP
ap-4699	282	15	induction	induction	NOUN
ap-4699	282	16	with	with	ADP
ap-4699	282	17	respect	respect	NOUN
ap-4699	282	18	to	to	ADP
ap-4699	282	19	n	n	PROPN
ap-4699	282	20	that	that	PRON
ap-4699	282	21	x′0	x′0	PROPN
ap-4699	282	22	≡β−1	≡β−1	VERB
ap-4699	282	23	a	a	PRON
ap-4699	282	24	for	for	ADP
ap-4699	282	25	some	some	PRON
ap-4699	282	26	a	a	DET
ap-4699	282	27	∈	∈	NOUN
ap-4699	282	28	a.	a.	NOUN
ap-4699	282	29	if	if	SCONJ
ap-4699	282	30	n	n	PROPN
ap-4699	282	31	=	=	SYM
ap-4699	282	32	0	0	NUM
ap-4699	282	33	,	,	PUNCT
ap-4699	282	34	then	then	ADV
ap-4699	282	35	x′0	x′0	ADP
ap-4699	283	1	=	=	SYM
ap-4699	283	2	a0	a0	PROPN
ap-4699	283	3	∈	∈	PROPN
ap-4699	283	4	a.	a.	NOUN
ap-4699	283	5	for	for	ADP
ap-4699	283	6	n+	n+	X
ap-4699	283	7	1	1	NUM
ap-4699	283	8	,	,	PUNCT
ap-4699	283	9	we	we	PRON
ap-4699	283	10	have	have	VERB
ap-4699	283	11	x′0	x′0	NOUN
ap-4699	284	1	=	=	SYM
ap-4699	284	2	n+1∑	n+1∑	PROPN
ap-4699	284	3	i=0	i=0	PROPN
ap-4699	284	4	aiβ	aiβ	NOUN
ap-4699	284	5	i	i	NOUN
ap-4699	284	6	=	=	PROPN
ap-4699	284	7	a0	a0	PROPN
ap-4699	284	8	+	+	CCONJ
ap-4699	284	9	(	(	PUNCT
ap-4699	284	10	β	β	NOUN
ap-4699	284	11	−	−	NOUN
ap-4699	284	12	1	1	NUM
ap-4699	284	13	)	)	PUNCT
ap-4699	284	14	n∑	n∑	NOUN
ap-4699	285	1	i=0	i=0	PROPN
ap-4699	285	2	ai+1β	ai+1β	ADJ
ap-4699	285	3	i	i	PRON
ap-4699	286	1	+	+	CCONJ
ap-4699	286	2	n∑	n∑	PROPN
ap-4699	286	3	i=0	i=0	PROPN
ap-4699	286	4	ai+1β	ai+1β	PROPN
ap-4699	287	1	i	i	PROPN
ap-4699	287	2	≡β−1	≡β−1	NUM
ap-4699	287	3	a0	a0	NOUN
ap-4699	287	4	+	+	CCONJ
ap-4699	287	5	a′	a′	PROPN
ap-4699	287	6	≡β−1	≡β−1	X
ap-4699	287	7	a	a	DET
ap-4699	287	8	∈	∈	PROPN
ap-4699	288	1	a	a	PRON
ap-4699	288	2	,	,	PUNCT
ap-4699	288	3	where	where	SCONJ
ap-4699	288	4	we	we	PRON
ap-4699	288	5	use	use	VERB
ap-4699	288	6	the	the	DET
ap-4699	288	7	induction	induction	NOUN
ap-4699	288	8	assumption∑n	assumption∑n	VERB
ap-4699	288	9	i=0	i=0	PROPN
ap-4699	288	10	ai+1β	ai+1β	ADJ
ap-4699	288	11	i	i	PROPN
ap-4699	288	12	≡β−1	≡β−1	VERB
ap-4699	288	13	a′	a′	NOUN
ap-4699	288	14	∈	∈	PROPN
ap-4699	288	15	a	a	PRON
ap-4699	288	16	and	and	CCONJ
ap-4699	288	17	the	the	DET
ap-4699	288	18	statement	statement	NOUN
ap-4699	288	19	of	of	ADP
ap-4699	288	20	theorem	theorem	ADJ
ap-4699	288	21	3.1	3.1	NUM
ap-4699	288	22	,	,	PUNCT
ap-4699	288	23	i.e	i.e	X
ap-4699	288	24	,	,	PUNCT
ap-4699	288	25	for	for	ADP
ap-4699	288	26	each	each	DET
ap-4699	288	27	digit	digit	NOUN
ap-4699	288	28	b	b	PROPN
ap-4699	288	29	∈	∈	PROPN
ap-4699	288	30	a	a	DET
ap-4699	288	31	+	+	NOUN
ap-4699	288	32	a	a	PRON
ap-4699	288	33	there	there	PRON
ap-4699	288	34	is	be	VERB
ap-4699	288	35	a	a	DET
ap-4699	288	36	digit	digit	NOUN
ap-4699	288	37	a	a	DET
ap-4699	288	38	∈	∈	NOUN
ap-4699	288	39	a	a	DET
ap-4699	288	40	such	such	ADJ
ap-4699	288	41	that	that	DET
ap-4699	288	42	b	b	PROPN
ap-4699	288	43	≡β−1	≡β−1	NUM
ap-4699	288	44	a.	a.	NOUN
ap-4699	288	45	3.2	3.2	NUM
ap-4699	288	46	.	.	PUNCT
ap-4699	289	1	lower	low	ADJ
ap-4699	289	2	bound	bind	VERB
ap-4699	289	3	on	on	ADP
ap-4699	289	4	#	#	SYM
ap-4699	289	5	a	a	PRON
ap-4699	289	6	when	when	SCONJ
ap-4699	289	7	deriving	derive	VERB
ap-4699	289	8	the	the	DET
ap-4699	289	9	minimal	minimal	ADJ
ap-4699	289	10	size	size	NOUN
ap-4699	289	11	of	of	ADP
ap-4699	289	12	alphabets	alphabet	NOUN
ap-4699	289	13	for	for	ADP
ap-4699	289	14	parallel	parallel	ADJ
ap-4699	289	15	addition	addition	NOUN
ap-4699	289	16	,	,	PUNCT
ap-4699	289	17	we	we	PRON
ap-4699	289	18	assume	assume	VERB
ap-4699	289	19	that	that	SCONJ
ap-4699	289	20	the	the	DET
ap-4699	289	21	base	base	NOUN
ap-4699	289	22	β	β	X
ap-4699	289	23	is	be	AUX
ap-4699	289	24	an	an	DET
ap-4699	289	25	algebraic	algebraic	ADJ
ap-4699	289	26	integer	integer	NOUN
ap-4699	289	27	(	(	PUNCT
ap-4699	289	28	in	in	ADP
ap-4699	289	29	this	this	DET
ap-4699	289	30	whole	whole	ADJ
ap-4699	289	31	subsection	subsection	NOUN
ap-4699	289	32	)	)	PUNCT
ap-4699	289	33	,	,	PUNCT
ap-4699	289	34	since	since	SCONJ
ap-4699	289	35	it	it	PRON
ap-4699	289	36	enables	enable	VERB
ap-4699	289	37	us	we	PRON
ap-4699	289	38	to	to	PART
ap-4699	289	39	count	count	VERB
ap-4699	289	40	the	the	DET
ap-4699	289	41	number	number	NOUN
ap-4699	289	42	of	of	ADP
ap-4699	289	43	congruence	congruence	NOUN
ap-4699	289	44	classes	class	NOUN
ap-4699	289	45	,	,	PUNCT
ap-4699	289	46	and	and	CCONJ
ap-4699	289	47	hence	hence	ADV
ap-4699	289	48	to	to	PART
ap-4699	289	49	provide	provide	VERB
ap-4699	289	50	an	an	DET
ap-4699	289	51	explicit	explicit	ADJ
ap-4699	289	52	lower	low	ADJ
ap-4699	289	53	bound	bind	VERB
ap-4699	289	54	on	on	ADP
ap-4699	289	55	the	the	DET
ap-4699	289	56	size	size	NOUN
ap-4699	289	57	of	of	ADP
ap-4699	289	58	alphabet	alphabet	NOUN
ap-4699	289	59	allowing	allow	VERB
ap-4699	289	60	parallel	parallel	ADJ
ap-4699	289	61	addition	addition	NOUN
ap-4699	289	62	.	.	PUNCT
ap-4699	290	1	in	in	ADP
ap-4699	290	2	what	what	PRON
ap-4699	290	3	follows	follow	VERB
ap-4699	290	4	,	,	PUNCT
ap-4699	290	5	the	the	DET
ap-4699	290	6	monic	monic	ADJ
ap-4699	290	7	minimal	minimal	ADJ
ap-4699	290	8	polynomial	polynomial	NOUN
ap-4699	290	9	of	of	ADP
ap-4699	290	10	an	an	DET
ap-4699	290	11	algebraic	algebraic	ADJ
ap-4699	290	12	integer	integer	NOUN
ap-4699	290	13	α	α	PROPN
ap-4699	290	14	is	be	AUX
ap-4699	290	15	denoted	denote	VERB
ap-4699	290	16	by	by	ADP
ap-4699	290	17	mα	mα	PROPN
ap-4699	290	18	.	.	PUNCT
ap-4699	291	1	let	let	VERB
ap-4699	291	2	d	d	PRON
ap-4699	291	3	be	be	AUX
ap-4699	291	4	the	the	DET
ap-4699	291	5	degree	degree	NOUN
ap-4699	291	6	of	of	ADP
ap-4699	291	7	β	β	X
ap-4699	291	8	.	.	PUNCT
ap-4699	292	1	it	it	PRON
ap-4699	292	2	is	be	AUX
ap-4699	292	3	well	well	ADV
ap-4699	292	4	known	know	VERB
ap-4699	292	5	that	that	SCONJ
ap-4699	292	6	z[β	z[β	NOUN
ap-4699	292	7	]	]	X
ap-4699	292	8	=	=	SYM
ap-4699	292	9	{	{	PUNCT
ap-4699	292	10	∑d−1	∑d−1	X
ap-4699	292	11	i=0	i=0	PROPN
ap-4699	292	12	xiβ	xiβ	PROPN
ap-4699	292	13	i	i	PRON
ap-4699	292	14	:	:	PUNCT
ap-4699	292	15	xi	xi	PROPN
ap-4699	292	16	∈	∈	PROPN
ap-4699	293	1	z	z	NOUN
ap-4699	293	2	}	}	PUNCT
ap-4699	293	3	if	if	SCONJ
ap-4699	293	4	and	and	CCONJ
ap-4699	293	5	only	only	ADV
ap-4699	293	6	if	if	SCONJ
ap-4699	293	7	β	β	NOUN
ap-4699	293	8	is	be	AUX
ap-4699	293	9	an	an	DET
ap-4699	293	10	algebraic	algebraic	ADJ
ap-4699	293	11	integer	integer	NOUN
ap-4699	293	12	.	.	PUNCT
ap-4699	294	1	hence	hence	ADV
ap-4699	294	2	,	,	PUNCT
ap-4699	294	3	there	there	PRON
ap-4699	294	4	is	be	VERB
ap-4699	294	5	an	an	DET
ap-4699	294	6	obvious	obvious	ADJ
ap-4699	294	7	bijection	bijection	NOUN
ap-4699	294	8	π	π	NOUN
ap-4699	294	9	:	:	PUNCT
ap-4699	294	10	z[β]→	z[β]→	PROPN
ap-4699	294	11	zd	zd	AUX
ap-4699	294	12	given	give	VERB
ap-4699	294	13	by	by	ADP
ap-4699	294	14	π(u	π(u	PROPN
ap-4699	294	15	)	)	PUNCT
ap-4699	295	1	=	=	PUNCT
ap-4699	295	2	(	(	PUNCT
ap-4699	295	3	u0	u0	ADJ
ap-4699	295	4	,	,	PUNCT
ap-4699	295	5	u1	u1	NOUN
ap-4699	295	6	,	,	PUNCT
ap-4699	295	7	·	·	PUNCT
ap-4699	295	8	·	·	PUNCT
ap-4699	295	9	·	·	PUNCT
ap-4699	295	10	,	,	PUNCT
ap-4699	295	11	ud−1)t	ud−1)t	PROPN
ap-4699	295	12	for	for	ADP
ap-4699	295	13	every	every	DET
ap-4699	295	14	u	u	NOUN
ap-4699	295	15	=	=	PROPN
ap-4699	295	16	∑d−1	∑d−1	X
ap-4699	295	17	i=0	i=0	PROPN
ap-4699	295	18	uiβ	uiβ	X
ap-4699	295	19	i	i	PROPN
ap-4699	295	20	∈	∈	PROPN
ap-4699	295	21	z[β	z[β	PROPN
ap-4699	295	22	]	]	PUNCT
ap-4699	295	23	.	.	PUNCT
ap-4699	296	1	moreover	moreover	ADV
ap-4699	296	2	,	,	PUNCT
ap-4699	296	3	the	the	DET
ap-4699	296	4	additive	additive	ADJ
ap-4699	296	5	group	group	NOUN
ap-4699	296	6	zd	zd	PROPN
ap-4699	296	7	can	can	AUX
ap-4699	296	8	be	be	AUX
ap-4699	296	9	equipped	equip	VERB
ap-4699	296	10	with	with	ADP
ap-4699	296	11	a	a	DET
ap-4699	296	12	multiplication	multiplication	NOUN
ap-4699	296	13	such	such	ADJ
ap-4699	296	14	that	that	SCONJ
ap-4699	296	15	π	π	PROPN
ap-4699	296	16	is	be	AUX
ap-4699	296	17	a	a	DET
ap-4699	296	18	ring	ring	NOUN
ap-4699	296	19	isomorphism	isomorphism	NOUN
ap-4699	296	20	.	.	PUNCT
ap-4699	297	1	in	in	ADP
ap-4699	297	2	order	order	NOUN
ap-4699	297	3	to	to	PART
ap-4699	297	4	do	do	AUX
ap-4699	297	5	that	that	PRON
ap-4699	297	6	,	,	PUNCT
ap-4699	297	7	we	we	PRON
ap-4699	297	8	recall	recall	VERB
ap-4699	297	9	the	the	DET
ap-4699	297	10	concept	concept	NOUN
ap-4699	297	11	of	of	ADP
ap-4699	297	12	companion	companion	NOUN
ap-4699	297	13	matrix	matrix	NOUN
ap-4699	297	14	.	.	PUNCT
ap-4699	298	1	definition	definition	NOUN
ap-4699	298	2	3.8	3.8	NUM
ap-4699	298	3	.	.	PUNCT
ap-4699	299	1	let	let	VERB
ap-4699	299	2	p(x	p(x	VERB
ap-4699	299	3	)	)	PUNCT
ap-4699	299	4	=	=	PUNCT
ap-4699	300	1	xd	xd	INTJ
ap-4699	301	1	+	+	CCONJ
ap-4699	301	2	pd−1x	pd−1x	NUM
ap-4699	301	3	d−1	d−1	PROPN
ap-4699	301	4	+	+	PROPN
ap-4699	301	5	·	·	PUNCT
ap-4699	301	6	·	·	PUNCT
ap-4699	301	7	·	·	PUNCT
ap-4699	302	1	+	+	NUM
ap-4699	302	2	p1x	p1x	NOUN
ap-4699	302	3	+	+	CCONJ
ap-4699	302	4	p0	p0	NOUN
ap-4699	302	5	∈	∈	PROPN
ap-4699	302	6	z[x	z[x	NOUN
ap-4699	302	7	]	]	PUNCT
ap-4699	302	8	be	be	VERB
ap-4699	302	9	a	a	DET
ap-4699	302	10	monic	monic	ADJ
ap-4699	302	11	polynomial	polynomial	NOUN
ap-4699	302	12	with	with	ADP
ap-4699	302	13	integer	integer	NOUN
ap-4699	302	14	coefficients	coefficient	NOUN
ap-4699	302	15	,	,	PUNCT
ap-4699	302	16	d	d	X
ap-4699	302	17	≥	≥	NUM
ap-4699	302	18	1	1	NUM
ap-4699	302	19	.	.	PUNCT
ap-4699	303	1	the	the	DET
ap-4699	303	2	matrix	matrix	NOUN
ap-4699	303	3	s	s	VERB
ap-4699	303	4	:	:	PUNCT
ap-4699	303	5	=	=	SYM
ap-4699	303	6			ADJ
ap-4699	303	7	0	0	NUM
ap-4699	303	8	0	0	NUM
ap-4699	303	9	·	·	PUNCT
ap-4699	303	10	·	·	PUNCT
ap-4699	303	11	·	·	PUNCT
ap-4699	303	12	0	0	NUM
ap-4699	304	1	−p0	−p0	PROPN
ap-4699	304	2	1	1	NUM
ap-4699	304	3	0	0	NUM
ap-4699	304	4	·	·	PUNCT
ap-4699	304	5	·	·	PUNCT
ap-4699	304	6	·	·	PUNCT
ap-4699	304	7	0	0	PUNCT
ap-4699	305	1	−p1	−p1	X
ap-4699	305	2	0	0	NUM
ap-4699	305	3	1	1	NUM
ap-4699	305	4	·	·	PUNCT
ap-4699	305	5	·	·	PUNCT
ap-4699	305	6	·	·	PUNCT
ap-4699	305	7	0	0	NUM
ap-4699	305	8	−p2	−p2	PROPN
ap-4699	305	9	...	...	PUNCT
ap-4699	305	10	.	.	PUNCT
ap-4699	305	11	.	.	PUNCT
ap-4699	305	12	.	.	PUNCT
ap-4699	306	1	...	...	PUNCT
ap-4699	307	1	0	0	NUM
ap-4699	307	2	0	0	NUM
ap-4699	307	3	·	·	PUNCT
ap-4699	307	4	·	·	PUNCT
ap-4699	307	5	·	·	PUNCT
ap-4699	307	6	1	1	NUM
ap-4699	307	7	−pd−1	−pd−1	NUM
ap-4699	307	8			PROPN
ap-4699	307	9	∈	∈	PROPN
ap-4699	307	10	zd×d	zd×d	PROPN
ap-4699	307	11	is	be	AUX
ap-4699	307	12	the	the	DET
ap-4699	307	13	companion	companion	NOUN
ap-4699	307	14	matrix	matrix	NOUN
ap-4699	307	15	of	of	ADP
ap-4699	307	16	the	the	DET
ap-4699	307	17	polynomial	polynomial	ADJ
ap-4699	307	18	p.	p.	NOUN
ap-4699	307	19	it	it	PRON
ap-4699	307	20	is	be	AUX
ap-4699	307	21	well	well	ADV
ap-4699	307	22	known	know	VERB
ap-4699	307	23	(	(	PUNCT
ap-4699	307	24	see	see	VERB
ap-4699	307	25	for	for	ADP
ap-4699	307	26	instance	instance	NOUN
ap-4699	307	27	[	[	X
ap-4699	307	28	11	11	NUM
ap-4699	307	29	]	]	PUNCT
ap-4699	307	30	)	)	PUNCT
ap-4699	307	31	that	that	SCONJ
ap-4699	307	32	the	the	DET
ap-4699	307	33	characteristic	characteristic	ADJ
ap-4699	307	34	polynomial	polynomial	NOUN
ap-4699	307	35	of	of	ADP
ap-4699	307	36	the	the	DET
ap-4699	307	37	companion	companion	NOUN
ap-4699	307	38	matrix	matrix	NOUN
ap-4699	307	39	s	s	VERB
ap-4699	307	40	is	be	AUX
ap-4699	307	41	p.	p.	NOUN
ap-4699	307	42	the	the	DET
ap-4699	307	43	matrix	matrix	NOUN
ap-4699	307	44	s	s	PART
ap-4699	307	45	is	be	AUX
ap-4699	307	46	also	also	ADV
ap-4699	307	47	root	root	NOUN
ap-4699	307	48	of	of	ADP
ap-4699	307	49	the	the	DET
ap-4699	307	50	polynomial	polynomial	ADJ
ap-4699	307	51	p.	p.	NOUN
ap-4699	307	52	the	the	DET
ap-4699	307	53	claim	claim	NOUN
ap-4699	307	54	of	of	ADP
ap-4699	307	55	the	the	DET
ap-4699	307	56	following	following	NOUN
ap-4699	307	57	theorem	theorem	NOUN
ap-4699	307	58	,	,	PUNCT
ap-4699	307	59	which	which	PRON
ap-4699	307	60	provides	provide	VERB
ap-4699	307	61	the	the	DET
ap-4699	307	62	required	require	VERB
ap-4699	307	63	multiplication	multiplication	NOUN
ap-4699	307	64	in	in	ADP
ap-4699	307	65	zd	zd	PROPN
ap-4699	307	66	,	,	PUNCT
ap-4699	307	67	is	be	AUX
ap-4699	307	68	discussed	discuss	VERB
ap-4699	307	69	in	in	ADP
ap-4699	307	70	[	[	X
ap-4699	307	71	12	12	NUM
ap-4699	307	72	]	]	PUNCT
ap-4699	307	73	.	.	PUNCT
ap-4699	308	1	these	these	DET
ap-4699	308	2	topics	topic	NOUN
ap-4699	308	3	are	be	AUX
ap-4699	308	4	also	also	ADV
ap-4699	308	5	more	more	ADV
ap-4699	308	6	elaborated	elaborate	VERB
ap-4699	308	7	in	in	ADP
ap-4699	308	8	[	[	X
ap-4699	308	9	13	13	NUM
ap-4699	308	10	,	,	PUNCT
ap-4699	308	11	14	14	NUM
ap-4699	308	12	]	]	PUNCT
ap-4699	308	13	.	.	PUNCT
ap-4699	309	1	theorem	theorem	ADJ
ap-4699	309	2	3.9	3.9	NUM
ap-4699	309	3	.	.	PUNCT
ap-4699	310	1	let	let	VERB
ap-4699	310	2	β	β	PRON
ap-4699	310	3	be	be	AUX
ap-4699	310	4	an	an	DET
ap-4699	310	5	algebraic	algebraic	ADJ
ap-4699	310	6	integer	integer	NOUN
ap-4699	310	7	of	of	ADP
ap-4699	310	8	degree	degree	NOUN
ap-4699	310	9	d	d	X
ap-4699	310	10	≥	≥	NOUN
ap-4699	310	11	1	1	NUM
ap-4699	310	12	and	and	CCONJ
ap-4699	310	13	let	let	VERB
ap-4699	310	14	s	s	PRON
ap-4699	310	15	be	be	AUX
ap-4699	310	16	the	the	DET
ap-4699	310	17	companion	companion	NOUN
ap-4699	310	18	matrix	matrix	NOUN
ap-4699	310	19	of	of	ADP
ap-4699	310	20	mβ	mβ	PROPN
ap-4699	310	21	.	.	PUNCT
ap-4699	311	1	if	if	SCONJ
ap-4699	311	2	the	the	DET
ap-4699	311	3	multiplication	multiplication	NOUN
ap-4699	311	4	�	�	X
ap-4699	311	5	β	β	PROPN
ap-4699	311	6	:	:	PUNCT
ap-4699	311	7	zd	zd	PROPN
ap-4699	311	8	×	×	PROPN
ap-4699	311	9	zd	zd	PROPN
ap-4699	311	10	→	→	SYM
ap-4699	311	11	zd	zd	PROPN
ap-4699	311	12	is	be	AUX
ap-4699	311	13	defined	define	VERB
ap-4699	311	14	by	by	ADP
ap-4699	311	15	u	u	NOUN
ap-4699	311	16	�	�	PROPN
ap-4699	311	17	β	β	NOUN
ap-4699	311	18	v	v	NOUN
ap-4699	311	19	:	:	PUNCT
ap-4699	311	20	=	=	SYM
ap-4699	311	21	(	(	PUNCT
ap-4699	311	22	d−1∑	d−1∑	PROPN
ap-4699	311	23	i=0	i=0	PROPN
ap-4699	311	24	uis	uis	PROPN
ap-4699	311	25	i	i	PROPN
ap-4699	311	26	)	)	PUNCT
ap-4699	311	27	·	·	PUNCT
ap-4699	312	1	v	v	X
ap-4699	312	2	for	for	ADP
ap-4699	312	3	all	all	DET
ap-4699	312	4	u	u	NOUN
ap-4699	312	5	=	=	PUNCT
ap-4699	312	6	(	(	PUNCT
ap-4699	312	7	u0	u0	ADJ
ap-4699	312	8	,	,	PUNCT
ap-4699	312	9	u1	u1	NOUN
ap-4699	312	10	,	,	PUNCT
ap-4699	312	11	·	·	PUNCT
ap-4699	312	12	·	·	PUNCT
ap-4699	312	13	·	·	PUNCT
ap-4699	312	14	,	,	PUNCT
ap-4699	312	15	ud−1)t	ud−1)t	PROPN
ap-4699	312	16	,	,	PUNCT
ap-4699	312	17	v	v	PROPN
ap-4699	312	18	∈	∈	PROPN
ap-4699	312	19	zd	zd	PROPN
ap-4699	312	20	,	,	PUNCT
ap-4699	312	21	then	then	ADV
ap-4699	312	22	(	(	PUNCT
ap-4699	312	23	zd,+,	zd,+,	NOUN
ap-4699	312	24	�	�	NOUN
ap-4699	312	25	β	β	X
ap-4699	312	26	)	)	PUNCT
ap-4699	312	27	is	be	AUX
ap-4699	312	28	a	a	DET
ap-4699	312	29	commutative	commutative	ADJ
ap-4699	312	30	ring	ring	NOUN
ap-4699	312	31	which	which	PRON
ap-4699	312	32	is	be	AUX
ap-4699	312	33	isomorphic	isomorphic	ADJ
ap-4699	312	34	to	to	ADP
ap-4699	312	35	z[β	z[β	NOUN
ap-4699	312	36	]	]	PUNCT
ap-4699	312	37	by	by	ADP
ap-4699	312	38	the	the	DET
ap-4699	312	39	mapping	mapping	NOUN
ap-4699	312	40	π	π	PROPN
ap-4699	312	41	.	.	PUNCT
ap-4699	313	1	one	one	NUM
ap-4699	313	2	of	of	ADP
ap-4699	313	3	the	the	DET
ap-4699	313	4	consequences	consequence	NOUN
ap-4699	313	5	is	be	AUX
ap-4699	313	6	the	the	DET
ap-4699	313	7	following	follow	VERB
ap-4699	313	8	lemma	lemma	PROPN
ap-4699	313	9	.	.	PUNCT
ap-4699	314	1	although	although	SCONJ
ap-4699	314	2	it	it	PRON
ap-4699	314	3	is	be	AUX
ap-4699	314	4	a	a	DET
ap-4699	314	5	known	know	VERB
ap-4699	314	6	result	result	NOUN
ap-4699	314	7	,	,	PUNCT
ap-4699	314	8	we	we	PRON
ap-4699	314	9	include	include	VERB
ap-4699	314	10	its	its	PRON
ap-4699	314	11	proof	proof	NOUN
ap-4699	314	12	here	here	ADV
ap-4699	314	13	,	,	PUNCT
ap-4699	314	14	to	to	PART
ap-4699	314	15	be	be	AUX
ap-4699	314	16	more	more	ADV
ap-4699	314	17	self	self	NOUN
ap-4699	314	18	-	-	PUNCT
ap-4699	314	19	contained	contain	VERB
ap-4699	314	20	.	.	PUNCT
ap-4699	315	1	let	let	VERB
ap-4699	315	2	us	we	PRON
ap-4699	315	3	recall	recall	VERB
ap-4699	315	4	that	that	PRON
ap-4699	315	5	for	for	ADP
ap-4699	315	6	a	a	DET
ap-4699	315	7	non	non	ADJ
ap-4699	315	8	-	-	ADJ
ap-4699	315	9	singular	singular	ADJ
ap-4699	315	10	integer	integer	NOUN
ap-4699	315	11	matrix	matrix	NOUN
ap-4699	315	12	m	m	NOUN
ap-4699	315	13	∈	∈	PROPN
ap-4699	315	14	zd×d	zd×d	PROPN
ap-4699	315	15	,	,	PUNCT
ap-4699	315	16	two	two	NUM
ap-4699	315	17	vectors	vector	NOUN
ap-4699	315	18	x	x	PRON
ap-4699	315	19	,	,	PUNCT
ap-4699	315	20	y	y	PROPN
ap-4699	315	21	∈	∈	PROPN
ap-4699	315	22	zd	zd	PROPN
ap-4699	315	23	are	be	AUX
ap-4699	315	24	congruent	congruent	ADJ
ap-4699	315	25	modulo	modulo	NOUN
ap-4699	315	26	m	m	VERB
ap-4699	315	27	in	in	ADP
ap-4699	315	28	zd	zd	PROPN
ap-4699	315	29	,	,	PUNCT
ap-4699	315	30	denoted	denote	VERB
ap-4699	315	31	by	by	ADP
ap-4699	315	32	x	x	PROPN
ap-4699	315	33	≡m	≡m	PROPN
ap-4699	315	34	y	y	PROPN
ap-4699	315	35	,	,	PUNCT
ap-4699	315	36	if	if	SCONJ
ap-4699	315	37	x−	x−	PROPN
ap-4699	315	38	y	y	PROPN
ap-4699	315	39	∈mzd	∈mzd	PROPN
ap-4699	315	40	.	.	PUNCT
ap-4699	316	1	lemma	lemma	PROPN
ap-4699	316	2	3.10	3.10	NUM
ap-4699	316	3	.	.	PUNCT
ap-4699	317	1	let	let	VERB
ap-4699	317	2	β	β	PRON
ap-4699	317	3	be	be	AUX
ap-4699	317	4	an	an	DET
ap-4699	317	5	algebraic	algebraic	ADJ
ap-4699	317	6	integer	integer	NOUN
ap-4699	317	7	of	of	ADP
ap-4699	317	8	degree	degree	NOUN
ap-4699	317	9	d	d	NOUN
ap-4699	317	10	and	and	CCONJ
ap-4699	317	11	α	α	PRON
ap-4699	317	12	∈	∈	PROPN
ap-4699	317	13	z[β	z[β	PROPN
ap-4699	317	14	]	]	PUNCT
ap-4699	317	15	be	be	VERB
ap-4699	317	16	such	such	ADJ
ap-4699	317	17	that	that	SCONJ
ap-4699	317	18	degα	degα	NOUN
ap-4699	317	19	=	=	PUNCT
ap-4699	317	20	deg	deg	NOUN
ap-4699	317	21	β	β	X
ap-4699	317	22	.	.	PUNCT
ap-4699	318	1	the	the	DET
ap-4699	318	2	number	number	NOUN
ap-4699	318	3	of	of	ADP
ap-4699	318	4	congruence	congruence	PROPN
ap-4699	318	5	classes	class	NOUN
ap-4699	318	6	modulo	modulo	VERB
ap-4699	318	7	α	α	NOUN
ap-4699	318	8	in	in	ADP
ap-4699	318	9	z[β	z[β	NOUN
ap-4699	318	10	]	]	PUNCT
ap-4699	318	11	is	be	AUX
ap-4699	318	12	|mα(0)|	|mα(0)|	NOUN
ap-4699	318	13	.	.	PUNCT
ap-4699	319	1	proof	proof	NOUN
ap-4699	319	2	.	.	PUNCT
ap-4699	320	1	the	the	DET
ap-4699	320	2	number	number	NOUN
ap-4699	320	3	α	α	PROPN
ap-4699	320	4	is	be	AUX
ap-4699	320	5	an	an	DET
ap-4699	320	6	algebraic	algebraic	ADJ
ap-4699	320	7	integer	integer	NOUN
ap-4699	320	8	,	,	PUNCT
ap-4699	320	9	since	since	SCONJ
ap-4699	320	10	it	it	PRON
ap-4699	320	11	is	be	AUX
ap-4699	320	12	well	well	ADV
ap-4699	320	13	known	know	VERB
ap-4699	320	14	that	that	DET
ap-4699	320	15	sum	sum	NOUN
ap-4699	320	16	and	and	CCONJ
ap-4699	320	17	product	product	NOUN
ap-4699	320	18	of	of	ADP
ap-4699	320	19	algebraic	algebraic	ADJ
ap-4699	320	20	integers	integer	NOUN
ap-4699	320	21	is	be	AUX
ap-4699	320	22	an	an	DET
ap-4699	320	23	algebraic	algebraic	ADJ
ap-4699	320	24	integer	integer	NOUN
ap-4699	320	25	.	.	PUNCT
ap-4699	321	1	let	let	VERB
ap-4699	321	2	γ	γ	NOUN
ap-4699	321	3	,	,	PUNCT
ap-4699	321	4	δ	δ	PROPN
ap-4699	321	5	∈	∈	PROPN
ap-4699	321	6	z[β	z[β	PROPN
ap-4699	321	7	]	]	PUNCT
ap-4699	321	8	and	and	CCONJ
ap-4699	321	9	let	let	VERB
ap-4699	321	10	s	s	PRON
ap-4699	321	11	be	be	AUX
ap-4699	321	12	the	the	DET
ap-4699	321	13	companion	companion	NOUN
ap-4699	321	14	matrix	matrix	NOUN
ap-4699	321	15	of	of	ADP
ap-4699	321	16	the	the	DET
ap-4699	321	17	minimal	minimal	ADJ
ap-4699	321	18	polynomial	polynomial	ADJ
ap-4699	321	19	mβ	mβ	NOUN
ap-4699	321	20	of	of	ADP
ap-4699	321	21	the	the	DET
ap-4699	321	22	algebraic	algebraic	ADJ
ap-4699	321	23	integer	integer	NOUN
ap-4699	321	24	β	β	X
ap-4699	321	25	.	.	PUNCT
ap-4699	322	1	let	let	VERB
ap-4699	322	2	π(α	π(α	NOUN
ap-4699	322	3	)	)	PUNCT
ap-4699	323	1	=	=	SYM
ap-4699	323	2	(	(	PUNCT
ap-4699	323	3	a0	a0	PROPN
ap-4699	323	4	,	,	PUNCT
ap-4699	323	5	a1	a1	PROPN
ap-4699	323	6	,	,	PUNCT
ap-4699	323	7	·	·	PUNCT
ap-4699	323	8	·	·	PUNCT
ap-4699	323	9	·	·	PUNCT
ap-4699	323	10	,	,	PUNCT
ap-4699	323	11	ad−1)t	ad−1)t	PROPN
ap-4699	323	12	,	,	PUNCT
ap-4699	323	13	with	with	ADP
ap-4699	323	14	α	α	NOUN
ap-4699	323	15	=	=	PUNCT
ap-4699	323	16	∑d−1	∑d−1	X
ap-4699	323	17	i=0	i=0	PROPN
ap-4699	323	18	aiβ	aiβ	PROPN
ap-4699	323	19	i.	i.	NOUN
ap-4699	323	20	if	if	SCONJ
ap-4699	323	21	we	we	PRON
ap-4699	323	22	set	set	VERB
ap-4699	323	23	sα	sα	ADV
ap-4699	323	24	:	:	PUNCT
ap-4699	323	25	=	=	NUM
ap-4699	323	26	∑d−1	∑d−1	X
ap-4699	323	27	i=0	i=0	PROPN
ap-4699	323	28	ais	ais	PROPN
ap-4699	323	29	i	i	PROPN
ap-4699	323	30	,	,	PUNCT
ap-4699	323	31	then	then	ADV
ap-4699	323	32	the	the	DET
ap-4699	323	33	congruences	congruence	NOUN
ap-4699	323	34	≡α	≡α	NOUN
ap-4699	323	35	in	in	ADP
ap-4699	323	36	z[β	z[β	NOUN
ap-4699	323	37	]	]	PUNCT
ap-4699	323	38	and	and	CCONJ
ap-4699	323	39	≡sα	≡sα	PROPN
ap-4699	323	40	in	in	ADP
ap-4699	323	41	zd	zd	PROPN
ap-4699	323	42	fulfill	fulfill	NOUN
ap-4699	323	43	:	:	PUNCT
ap-4699	323	44	γ	γ	PROPN
ap-4699	323	45	≡α	≡α	PROPN
ap-4699	323	46	δ	δ	PROPN
ap-4699	323	47	⇐	⇐	ADJ
ap-4699	323	48	⇒	⇒	PROPN
ap-4699	323	49	∃ε	∃ε	PROPN
ap-4699	323	50	∈	∈	PROPN
ap-4699	323	51	z[β	z[β	PROPN
ap-4699	323	52	]	]	X
ap-4699	323	53	:	:	PUNCT
ap-4699	323	54	γ	γ	PROPN
ap-4699	323	55	−	−	PROPN
ap-4699	323	56	δ	δ	PROPN
ap-4699	323	57	=	=	PUNCT
ap-4699	323	58	αε	αε	ADP
ap-4699	323	59	⇐	⇐	ADJ
ap-4699	323	60	⇒	⇒	PROPN
ap-4699	323	61	∃z	∃z	PROPN
ap-4699	323	62	=	=	PUNCT
ap-4699	323	63	π(ε	π(ε	PROPN
ap-4699	323	64	)	)	PUNCT
ap-4699	323	65	∈	∈	PROPN
ap-4699	323	66	zd	zd	PROPN
ap-4699	323	67	:	:	PUNCT
ap-4699	323	68	π(γ)−	π(γ)−	PROPN
ap-4699	323	69	π(δ	π(δ	PROPN
ap-4699	323	70	)	)	PUNCT
ap-4699	323	71	=	=	PUNCT
ap-4699	324	1	=	=	PUNCT
ap-4699	325	1	π(γ	π(γ	NUM
ap-4699	325	2	−	−	PROPN
ap-4699	325	3	δ	δ	PROPN
ap-4699	325	4	)	)	PUNCT
ap-4699	325	5	=	=	SYM
ap-4699	326	1	π(α)	π(α)	PROPN
ap-4699	326	2	�	�	PROPN
ap-4699	326	3	β	β	NOUN
ap-4699	326	4	z	z	NOUN
ap-4699	326	5	=	=	PUNCT
ap-4699	326	6	sα	sα	ADJ
ap-4699	326	7	·	·	PUNCT
ap-4699	326	8	z	z	X
ap-4699	326	9	⇐	⇐	PROPN
ap-4699	326	10	⇒	⇒	PROPN
ap-4699	326	11	π(γ	π(γ	PROPN
ap-4699	326	12	)	)	PUNCT
ap-4699	326	13	≡sα	≡sα	PROPN
ap-4699	326	14	π(δ	π(δ	PROPN
ap-4699	326	15	)	)	PUNCT
ap-4699	326	16	.	.	PUNCT
ap-4699	327	1	289	289	NUM
ap-4699	327	2	jan	jan	PROPN
ap-4699	327	3	legerský	legerský	PROPN
ap-4699	327	4	acta	acta	PROPN
ap-4699	327	5	polytechnica	polytechnica	PROPN
ap-4699	327	6	thus	thus	ADV
ap-4699	327	7	,	,	PUNCT
ap-4699	327	8	the	the	DET
ap-4699	327	9	number	number	NOUN
ap-4699	327	10	of	of	ADP
ap-4699	327	11	congruence	congruence	PROPN
ap-4699	327	12	classes	class	NOUN
ap-4699	327	13	modulo	modulo	VERB
ap-4699	327	14	α	α	NOUN
ap-4699	327	15	in	in	ADP
ap-4699	327	16	z[β	z[β	NOUN
ap-4699	327	17	]	]	PUNCT
ap-4699	327	18	equals	equal	VERB
ap-4699	327	19	the	the	DET
ap-4699	327	20	number	number	NOUN
ap-4699	327	21	of	of	ADP
ap-4699	327	22	congruence	congruence	PROPN
ap-4699	327	23	classes	class	NOUN
ap-4699	327	24	modulo	modulo	VERB
ap-4699	327	25	sα	sα	ADV
ap-4699	327	26	in	in	ADP
ap-4699	327	27	zd	zd	PROPN
ap-4699	327	28	,	,	PUNCT
ap-4699	327	29	which	which	PRON
ap-4699	327	30	is	be	AUX
ap-4699	327	31	known	know	VERB
ap-4699	327	32	to	to	PART
ap-4699	327	33	be	be	AUX
ap-4699	327	34	|detsα|	|detsα|	NOUN
ap-4699	327	35	.	.	PUNCT
ap-4699	327	36	to	to	PART
ap-4699	327	37	show	show	VERB
ap-4699	327	38	that	that	SCONJ
ap-4699	327	39	|detsα|	|detsα|	NOUN
ap-4699	327	40	=	=	SYM
ap-4699	327	41	|mα(0)|	|mα(0)|	NOUN
ap-4699	327	42	,	,	PUNCT
ap-4699	327	43	we	we	PRON
ap-4699	327	44	proceed	proceed	VERB
ap-4699	327	45	in	in	ADP
ap-4699	327	46	the	the	DET
ap-4699	327	47	following	following	ADJ
ap-4699	327	48	way	way	NOUN
ap-4699	327	49	:	:	PUNCT
ap-4699	327	50	the	the	DET
ap-4699	327	51	characteristic	characteristic	ADJ
ap-4699	327	52	polynomial	polynomial	NOUN
ap-4699	327	53	of	of	ADP
ap-4699	327	54	the	the	DET
ap-4699	327	55	companion	companion	NOUN
ap-4699	327	56	matrix	matrix	NOUN
ap-4699	327	57	s	s	VERB
ap-4699	327	58	is	be	AUX
ap-4699	327	59	the	the	DET
ap-4699	327	60	same	same	ADJ
ap-4699	327	61	as	as	ADP
ap-4699	327	62	the	the	DET
ap-4699	327	63	minimal	minimal	ADJ
ap-4699	327	64	polynomial	polynomial	NOUN
ap-4699	327	65	of	of	ADP
ap-4699	327	66	β	β	X
ap-4699	327	67	.	.	PUNCT
ap-4699	328	1	since	since	SCONJ
ap-4699	328	2	minimal	minimal	ADJ
ap-4699	328	3	polynomials	polynomial	NOUN
ap-4699	328	4	have	have	VERB
ap-4699	328	5	no	no	DET
ap-4699	328	6	multiple	multiple	ADJ
ap-4699	328	7	roots	root	NOUN
ap-4699	328	8	,	,	PUNCT
ap-4699	328	9	s	s	VERB
ap-4699	328	10	is	be	AUX
ap-4699	328	11	diagonalizable	diagonalizable	ADJ
ap-4699	328	12	over	over	ADP
ap-4699	328	13	c	c	NOUN
ap-4699	328	14	,	,	PUNCT
ap-4699	328	15	i.e.	i.e.	X
ap-4699	328	16	,	,	PUNCT
ap-4699	328	17	s	s	NOUN
ap-4699	328	18	=	=	NOUN
ap-4699	328	19	p−1dp	p−1dp	NOUN
ap-4699	328	20	where	where	SCONJ
ap-4699	328	21	d	d	NOUN
ap-4699	328	22	is	be	AUX
ap-4699	328	23	a	a	DET
ap-4699	328	24	diagonal	diagonal	ADJ
ap-4699	328	25	matrix	matrix	NOUN
ap-4699	328	26	with	with	ADP
ap-4699	328	27	the	the	DET
ap-4699	328	28	conjugates	conjugate	NOUN
ap-4699	328	29	of	of	ADP
ap-4699	328	30	β	β	NOUN
ap-4699	328	31	on	on	ADP
ap-4699	328	32	the	the	DET
ap-4699	328	33	diagonal	diagonal	ADJ
ap-4699	328	34	,	,	PUNCT
ap-4699	328	35	and	and	CCONJ
ap-4699	328	36	p	p	NOUN
ap-4699	328	37	is	be	AUX
ap-4699	328	38	a	a	DET
ap-4699	328	39	non	non	ADJ
ap-4699	328	40	-	-	ADJ
ap-4699	328	41	singular	singular	ADJ
ap-4699	328	42	complex	complex	ADJ
ap-4699	328	43	matrix	matrix	NOUN
ap-4699	328	44	.	.	PUNCT
ap-4699	329	1	the	the	DET
ap-4699	329	2	matrix	matrix	NOUN
ap-4699	329	3	sα	sα	VERB
ap-4699	329	4	is	be	AUX
ap-4699	329	5	also	also	ADV
ap-4699	329	6	diagonalized	diagonalize	VERB
ap-4699	329	7	by	by	ADP
ap-4699	329	8	p	p	X
ap-4699	329	9	:	:	PUNCT
ap-4699	329	10	sα	sα	ADJ
ap-4699	329	11	=	=	SYM
ap-4699	329	12	d−1∑	d−1∑	PROPN
ap-4699	329	13	i=0	i=0	PROPN
ap-4699	329	14	ais	ais	PROPN
ap-4699	330	1	i	i	NOUN
ap-4699	330	2	=	=	SYM
ap-4699	330	3	d−1∑	d−1∑	PROPN
ap-4699	330	4	i=0	i=0	X
ap-4699	330	5	ai(p−1dp	ai(p−1dp	X
ap-4699	330	6	)	)	PUNCT
ap-4699	330	7	i	i	PROPN
ap-4699	330	8	=	=	SYM
ap-4699	330	9	p−1	p−1	PROPN
ap-4699	330	10	(	(	PUNCT
ap-4699	330	11	d−1∑	d−1∑	PROPN
ap-4699	330	12	i=0	i=0	PROPN
ap-4699	330	13	aid	aid	NOUN
ap-4699	330	14	i	i	NOUN
ap-4699	330	15	)	)	PUNCT
ap-4699	330	16	︸	︸	X
ap-4699	330	17	︷︷	︷︷	NOUN
ap-4699	330	18	︸	︸	PRON
ap-4699	330	19	dα	dα	PROPN
ap-4699	330	20	p	p	NOUN
ap-4699	330	21	.	.	PUNCT
ap-4699	331	1	it	it	PRON
ap-4699	331	2	is	be	AUX
ap-4699	331	3	known	know	VERB
ap-4699	331	4	(	(	PUNCT
ap-4699	331	5	see	see	VERB
ap-4699	331	6	for	for	ADP
ap-4699	331	7	instance	instance	NOUN
ap-4699	331	8	[	[	X
ap-4699	331	9	15	15	NUM
ap-4699	331	10	]	]	PUNCT
ap-4699	331	11	)	)	PUNCT
ap-4699	331	12	that	that	SCONJ
ap-4699	331	13	if	if	SCONJ
ap-4699	331	14	σ	σ	NOUN
ap-4699	331	15	:	:	PUNCT
ap-4699	331	16	q(β)→	q(β)→	NUM
ap-4699	331	17	q(β′	q(β′	NUM
ap-4699	331	18	)	)	PUNCT
ap-4699	331	19	is	be	AUX
ap-4699	331	20	a	a	DET
ap-4699	331	21	field	field	NOUN
ap-4699	331	22	isomorphism	isomorphism	NOUN
ap-4699	331	23	and	and	CCONJ
ap-4699	331	24	α	α	PRON
ap-4699	331	25	∈	∈	PROPN
ap-4699	332	1	q(β	q(β	PROPN
ap-4699	332	2	)	)	PUNCT
ap-4699	332	3	,	,	PUNCT
ap-4699	332	4	then	then	ADV
ap-4699	332	5	σ(α	σ(α	PROPN
ap-4699	332	6	)	)	PUNCT
ap-4699	332	7	is	be	AUX
ap-4699	332	8	a	a	DET
ap-4699	332	9	conjugate	conjugate	NOUN
ap-4699	332	10	of	of	ADP
ap-4699	332	11	α	α	NOUN
ap-4699	332	12	,	,	PUNCT
ap-4699	332	13	and	and	CCONJ
ap-4699	332	14	we	we	PRON
ap-4699	332	15	obtain	obtain	VERB
ap-4699	332	16	all	all	DET
ap-4699	332	17	conjugates	conjugate	NOUN
ap-4699	332	18	of	of	ADP
ap-4699	332	19	α	α	NOUN
ap-4699	332	20	in	in	ADP
ap-4699	332	21	this	this	DET
ap-4699	332	22	way	way	NOUN
ap-4699	332	23	.	.	PUNCT
ap-4699	333	1	since	since	SCONJ
ap-4699	333	2	α	α	NOUN
ap-4699	333	3	=	=	PUNCT
ap-4699	333	4	∑d−1	∑d−1	X
ap-4699	333	5	i=0	i=0	PROPN
ap-4699	333	6	aiβ	aiβ	PROPN
ap-4699	333	7	i	i	PROPN
ap-4699	333	8	,	,	PUNCT
ap-4699	333	9	degα	degα	PROPN
ap-4699	333	10	=	=	PUNCT
ap-4699	333	11	deg	deg	PROPN
ap-4699	333	12	β	β	X
ap-4699	333	13	and	and	CCONJ
ap-4699	333	14	d	d	PROPN
ap-4699	333	15	has	have	VERB
ap-4699	333	16	conjugates	conjugate	NOUN
ap-4699	333	17	of	of	ADP
ap-4699	333	18	β	β	NOUN
ap-4699	333	19	on	on	ADP
ap-4699	333	20	the	the	DET
ap-4699	333	21	diagonal	diagonal	NOUN
ap-4699	333	22	,	,	PUNCT
ap-4699	333	23	the	the	DET
ap-4699	333	24	diagonal	diagonal	ADJ
ap-4699	333	25	elements	element	NOUN
ap-4699	333	26	of	of	ADP
ap-4699	333	27	the	the	DET
ap-4699	333	28	diagonal	diagonal	ADJ
ap-4699	333	29	matrix	matrix	NOUN
ap-4699	333	30	dα	dα	NOUN
ap-4699	333	31	are	be	AUX
ap-4699	333	32	precisely	precisely	ADV
ap-4699	333	33	all	all	DET
ap-4699	333	34	conjugates	conjugate	NOUN
ap-4699	333	35	of	of	ADP
ap-4699	333	36	α	α	NOUN
ap-4699	333	37	.	.	PUNCT
ap-4699	334	1	hence	hence	ADV
ap-4699	334	2	,	,	PUNCT
ap-4699	334	3	|detsα|	|detsα|	NOUN
ap-4699	334	4	equals	equal	VERB
ap-4699	334	5	absolute	absolute	ADJ
ap-4699	334	6	value	value	NOUN
ap-4699	334	7	of	of	ADP
ap-4699	334	8	the	the	DET
ap-4699	334	9	product	product	NOUN
ap-4699	334	10	of	of	ADP
ap-4699	334	11	all	all	DET
ap-4699	334	12	conjugates	conjugate	NOUN
ap-4699	334	13	of	of	ADP
ap-4699	334	14	α	α	NOUN
ap-4699	334	15	,	,	PUNCT
ap-4699	334	16	which	which	PRON
ap-4699	334	17	is	be	AUX
ap-4699	334	18	|mα(0)|	|mα(0)|	NOUN
ap-4699	334	19	.	.	PUNCT
ap-4699	335	1	finally	finally	ADV
ap-4699	335	2	,	,	PUNCT
ap-4699	335	3	we	we	PRON
ap-4699	335	4	put	put	VERB
ap-4699	335	5	together	together	ADV
ap-4699	335	6	the	the	DET
ap-4699	335	7	fact	fact	NOUN
ap-4699	335	8	that	that	SCONJ
ap-4699	335	9	the	the	DET
ap-4699	335	10	alphabet	alphabet	NOUN
ap-4699	335	11	a	a	PRON
ap-4699	335	12	for	for	ADP
ap-4699	335	13	parallel	parallel	ADJ
ap-4699	335	14	addition	addition	NOUN
ap-4699	335	15	in	in	ADP
ap-4699	335	16	base	base	NOUN
ap-4699	335	17	β	β	PROPN
ap-4699	335	18	contains	contain	VERB
ap-4699	335	19	all	all	DET
ap-4699	335	20	representatives	representative	NOUN
ap-4699	335	21	modulo	modulo	VERB
ap-4699	335	22	β	β	X
ap-4699	335	23	and	and	CCONJ
ap-4699	335	24	modulo	modulo	PROPN
ap-4699	335	25	β	β	X
ap-4699	335	26	−	−	PROPN
ap-4699	335	27	1	1	NUM
ap-4699	335	28	,	,	PUNCT
ap-4699	335	29	the	the	DET
ap-4699	335	30	derived	derive	VERB
ap-4699	335	31	formula	formula	NOUN
ap-4699	335	32	for	for	ADP
ap-4699	335	33	the	the	DET
ap-4699	335	34	number	number	NOUN
ap-4699	335	35	of	of	ADP
ap-4699	335	36	congruence	congruence	NOUN
ap-4699	335	37	classes	class	NOUN
ap-4699	335	38	,	,	PUNCT
ap-4699	335	39	and	and	CCONJ
ap-4699	335	40	also	also	ADV
ap-4699	335	41	specific	specific	ADJ
ap-4699	335	42	restrictions	restriction	NOUN
ap-4699	335	43	on	on	ADP
ap-4699	335	44	alphabets	alphabet	NOUN
ap-4699	335	45	for	for	ADP
ap-4699	335	46	parallel	parallel	ADJ
ap-4699	335	47	addition	addition	NOUN
ap-4699	335	48	in	in	ADP
ap-4699	335	49	a	a	DET
ap-4699	335	50	base	base	NOUN
ap-4699	335	51	with	with	ADP
ap-4699	335	52	some	some	DET
ap-4699	335	53	positive	positive	ADJ
ap-4699	335	54	real	real	ADJ
ap-4699	335	55	conjugate	conjugate	NOUN
ap-4699	335	56	.	.	PUNCT
ap-4699	336	1	theorem	theorem	NOUN
ap-4699	336	2	3.11	3.11	NUM
ap-4699	336	3	.	.	PUNCT
ap-4699	337	1	let	let	VERB
ap-4699	337	2	(	(	PUNCT
ap-4699	337	3	β	β	X
ap-4699	337	4	,	,	PUNCT
ap-4699	337	5	a	a	PRON
ap-4699	337	6	)	)	PUNCT
ap-4699	337	7	be	be	AUX
ap-4699	337	8	a	a	DET
ap-4699	337	9	numeration	numeration	NOUN
ap-4699	337	10	system	system	NOUN
ap-4699	337	11	such	such	ADJ
ap-4699	337	12	that	that	SCONJ
ap-4699	337	13	β	β	PROPN
ap-4699	337	14	is	be	AUX
ap-4699	337	15	an	an	DET
ap-4699	337	16	algebraic	algebraic	ADJ
ap-4699	337	17	integer	integer	NOUN
ap-4699	337	18	and	and	CCONJ
ap-4699	337	19	a[β	a[β	PROPN
ap-4699	337	20	]	]	X
ap-4699	337	21	=	=	SYM
ap-4699	337	22	z[β	z[β	NOUN
ap-4699	337	23	]	]	PUNCT
ap-4699	337	24	.	.	PUNCT
ap-4699	338	1	if	if	SCONJ
ap-4699	338	2	the	the	DET
ap-4699	338	3	numeration	numeration	NOUN
ap-4699	338	4	system	system	NOUN
ap-4699	338	5	(	(	PUNCT
ap-4699	338	6	β	β	X
ap-4699	338	7	,	,	PUNCT
ap-4699	338	8	a	a	PRON
ap-4699	338	9	)	)	PUNCT
ap-4699	338	10	allows	allow	VERB
ap-4699	338	11	parallel	parallel	ADJ
ap-4699	338	12	addition	addition	NOUN
ap-4699	338	13	,	,	PUNCT
ap-4699	338	14	then	then	ADV
ap-4699	338	15	#	#	SYM
ap-4699	338	16	a	a	DET
ap-4699	338	17	≥	≥	NUM
ap-4699	338	18	max	max	PROPN
ap-4699	338	19	{	{	PUNCT
ap-4699	338	20	|mβ(0)|	|mβ(0)|	PROPN
ap-4699	338	21	,	,	PUNCT
ap-4699	338	22	|mβ(1)|	|mβ(1)|	PROPN
ap-4699	338	23	}	}	PUNCT
ap-4699	338	24	.	.	PUNCT
ap-4699	339	1	moreover	moreover	ADV
ap-4699	339	2	,	,	PUNCT
ap-4699	339	3	if	if	SCONJ
ap-4699	339	4	β	β	PROPN
ap-4699	339	5	has	have	VERB
ap-4699	339	6	a	a	DET
ap-4699	339	7	positive	positive	ADJ
ap-4699	339	8	real	real	ADJ
ap-4699	339	9	conjugate	conjugate	NOUN
ap-4699	339	10	,	,	PUNCT
ap-4699	339	11	then	then	ADV
ap-4699	339	12	#	#	SYM
ap-4699	339	13	a	a	DET
ap-4699	339	14	≥	≥	NUM
ap-4699	339	15	max	max	PROPN
ap-4699	339	16	{	{	PUNCT
ap-4699	339	17	|mβ(0)|	|mβ(0)|	PROPN
ap-4699	339	18	,	,	PUNCT
ap-4699	339	19	|mβ(1)|+	|mβ(1)|+	PROPN
ap-4699	339	20	2	2	NUM
ap-4699	339	21	}	}	PUNCT
ap-4699	339	22	.	.	PUNCT
ap-4699	340	1	proof	proof	NOUN
ap-4699	340	2	.	.	PUNCT
ap-4699	341	1	by	by	ADP
ap-4699	341	2	theorem	theorem	NOUN
ap-4699	341	3	3.7	3.7	NUM
ap-4699	341	4	,	,	PUNCT
ap-4699	341	5	the	the	DET
ap-4699	341	6	alphabet	alphabet	NOUN
ap-4699	341	7	a	a	PRON
ap-4699	341	8	for	for	ADP
ap-4699	341	9	parallel	parallel	ADJ
ap-4699	341	10	addition	addition	NOUN
ap-4699	341	11	must	must	AUX
ap-4699	341	12	contain	contain	VERB
ap-4699	341	13	all	all	DET
ap-4699	341	14	representatives	representative	NOUN
ap-4699	341	15	modulo	modulo	VERB
ap-4699	341	16	β	β	X
ap-4699	341	17	and	and	CCONJ
ap-4699	341	18	modulo	modulo	PROPN
ap-4699	341	19	β	β	X
ap-4699	341	20	−	−	PROPN
ap-4699	341	21	1	1	NUM
ap-4699	341	22	in	in	ADP
ap-4699	341	23	z[β	z[β	NOUN
ap-4699	341	24	]	]	PUNCT
ap-4699	341	25	.	.	PUNCT
ap-4699	342	1	the	the	DET
ap-4699	342	2	numbers	number	NOUN
ap-4699	342	3	of	of	ADP
ap-4699	342	4	congruence	congruence	PROPN
ap-4699	342	5	classes	class	NOUN
ap-4699	342	6	are	be	AUX
ap-4699	342	7	|mβ(0)|	|mβ(0)|	NOUN
ap-4699	342	8	and	and	CCONJ
ap-4699	342	9	|mβ−1(0)|	|mβ−1(0)|	NOUN
ap-4699	342	10	,	,	PUNCT
ap-4699	342	11	respectively	respectively	ADV
ap-4699	342	12	,	,	PUNCT
ap-4699	342	13	by	by	ADP
ap-4699	342	14	lemma	lemma	PROPN
ap-4699	342	15	3.10	3.10	NUM
ap-4699	342	16	.	.	PUNCT
ap-4699	343	1	obviously	obviously	ADV
ap-4699	343	2	,	,	PUNCT
ap-4699	343	3	mβ−1(x	mβ−1(x	NOUN
ap-4699	343	4	)	)	PUNCT
ap-4699	344	1	=	=	SYM
ap-4699	344	2	mβ(x+	mβ(x+	NOUN
ap-4699	344	3	1	1	NUM
ap-4699	344	4	)	)	PUNCT
ap-4699	344	5	.	.	PUNCT
ap-4699	345	1	thus	thus	ADV
ap-4699	345	2	mβ−1(0	mβ−1(0	X
ap-4699	345	3	)	)	PUNCT
ap-4699	345	4	=	=	SYM
ap-4699	345	5	mβ(1	mβ(1	NOUN
ap-4699	345	6	)	)	PUNCT
ap-4699	345	7	.	.	PUNCT
ap-4699	346	1	theorem	theorem	VERB
ap-4699	346	2	3.4	3.4	NUM
ap-4699	346	3	ensures	ensure	VERB
ap-4699	346	4	that	that	SCONJ
ap-4699	346	5	if	if	SCONJ
ap-4699	346	6	the	the	DET
ap-4699	346	7	minimal	minimal	ADJ
ap-4699	346	8	and	and	CCONJ
ap-4699	346	9	maximal	maximal	ADJ
ap-4699	346	10	element	element	NOUN
ap-4699	346	11	of	of	ADP
ap-4699	346	12	a	a	PRON
ap-4699	346	13	are	be	AUX
ap-4699	346	14	congruent	congruent	ADJ
ap-4699	346	15	modulo	modulo	NOUN
ap-4699	346	16	β−1	β−1	ADP
ap-4699	346	17	,	,	PUNCT
ap-4699	346	18	then	then	ADV
ap-4699	346	19	there	there	PRON
ap-4699	346	20	are	be	VERB
ap-4699	346	21	at	at	ADV
ap-4699	346	22	least	least	ADJ
ap-4699	346	23	three	three	NUM
ap-4699	346	24	digits	digit	NOUN
ap-4699	346	25	of	of	ADP
ap-4699	346	26	a	a	PRON
ap-4699	346	27	in	in	ADP
ap-4699	346	28	this	this	DET
ap-4699	346	29	class	class	NOUN
ap-4699	346	30	.	.	PUNCT
ap-4699	347	1	otherwise	otherwise	ADV
ap-4699	347	2	,	,	PUNCT
ap-4699	347	3	the	the	DET
ap-4699	347	4	class	class	NOUN
ap-4699	347	5	of	of	ADP
ap-4699	347	6	the	the	DET
ap-4699	347	7	minimal	minimal	ADJ
ap-4699	347	8	and	and	CCONJ
ap-4699	347	9	also	also	ADV
ap-4699	347	10	the	the	DET
ap-4699	347	11	class	class	NOUN
ap-4699	347	12	of	of	ADP
ap-4699	347	13	the	the	DET
ap-4699	347	14	maximal	maximal	ADJ
ap-4699	347	15	element	element	NOUN
ap-4699	347	16	of	of	ADP
ap-4699	347	17	a	a	DET
ap-4699	347	18	have	have	NOUN
ap-4699	347	19	at	at	ADV
ap-4699	347	20	least	least	ADV
ap-4699	347	21	two	two	NUM
ap-4699	347	22	elements	element	NOUN
ap-4699	347	23	.	.	PUNCT
ap-4699	348	1	both	both	PRON
ap-4699	348	2	lead	lead	VERB
ap-4699	348	3	to	to	ADP
ap-4699	348	4	the	the	DET
ap-4699	348	5	conclusion	conclusion	NOUN
ap-4699	348	6	that	that	SCONJ
ap-4699	348	7	#	#	SYM
ap-4699	348	8	a	a	DET
ap-4699	348	9	≥	≥	NUM
ap-4699	348	10	|mβ(1)|+	|mβ(1)|+	NOUN
ap-4699	348	11	2	2	NUM
ap-4699	348	12	.	.	X
ap-4699	348	13	we	we	PRON
ap-4699	348	14	remark	remark	VERB
ap-4699	348	15	that	that	SCONJ
ap-4699	348	16	the	the	DET
ap-4699	348	17	obtained	obtain	VERB
ap-4699	348	18	bound	bind	VERB
ap-4699	348	19	is	be	AUX
ap-4699	348	20	basically	basically	ADV
ap-4699	348	21	the	the	DET
ap-4699	348	22	same	same	ADJ
ap-4699	348	23	as	as	ADP
ap-4699	348	24	the	the	DET
ap-4699	348	25	one	one	NOUN
ap-4699	348	26	for	for	ADP
ap-4699	348	27	integer	integer	NOUN
ap-4699	348	28	alphabets	alphabet	NOUN
ap-4699	348	29	in	in	ADP
ap-4699	348	30	[	[	X
ap-4699	348	31	4	4	NUM
ap-4699	348	32	]	]	PUNCT
ap-4699	348	33	.	.	PUNCT
ap-4699	349	1	4	4	X
ap-4699	349	2	.	.	NOUN
ap-4699	349	3	necessary	necessary	ADJ
ap-4699	349	4	and	and	CCONJ
ap-4699	349	5	sufficient	sufficient	ADJ
ap-4699	349	6	condition	condition	NOUN
ap-4699	349	7	on	on	ADP
ap-4699	349	8	bases	basis	NOUN
ap-4699	349	9	for	for	ADP
ap-4699	349	10	parallel	parallel	ADJ
ap-4699	349	11	addition	addition	NOUN
ap-4699	349	12	p.	p.	NOUN
ap-4699	349	13	kornerup	kornerup	NOUN
ap-4699	350	1	[	[	X
ap-4699	350	2	2	2	X
ap-4699	350	3	]	]	PUNCT
ap-4699	350	4	proposed	propose	VERB
ap-4699	350	5	a	a	DET
ap-4699	350	6	more	more	ADV
ap-4699	350	7	general	general	ADJ
ap-4699	350	8	concept	concept	NOUN
ap-4699	350	9	of	of	ADP
ap-4699	350	10	parallel	parallel	ADJ
ap-4699	350	11	addition	addition	NOUN
ap-4699	350	12	called	call	VERB
ap-4699	350	13	k	k	ADJ
ap-4699	350	14	-	-	PUNCT
ap-4699	350	15	block	block	ADJ
ap-4699	350	16	parallel	parallel	ADJ
ap-4699	350	17	addition	addition	NOUN
ap-4699	350	18	.	.	PUNCT
ap-4699	351	1	the	the	DET
ap-4699	351	2	idea	idea	NOUN
ap-4699	351	3	is	be	AUX
ap-4699	351	4	that	that	SCONJ
ap-4699	351	5	blocks	block	NOUN
ap-4699	351	6	of	of	ADP
ap-4699	351	7	k	k	NOUN
ap-4699	351	8	digits	digit	NOUN
ap-4699	351	9	are	be	AUX
ap-4699	351	10	considered	consider	VERB
ap-4699	351	11	as	as	ADP
ap-4699	351	12	one	one	NUM
ap-4699	351	13	digit	digit	NOUN
ap-4699	351	14	in	in	ADP
ap-4699	351	15	the	the	DET
ap-4699	351	16	new	new	ADJ
ap-4699	351	17	numeration	numeration	NOUN
ap-4699	351	18	system	system	NOUN
ap-4699	351	19	with	with	ADP
ap-4699	351	20	base	base	NOUN
ap-4699	351	21	being	be	AUX
ap-4699	351	22	the	the	DET
ap-4699	351	23	k	k	NOUN
ap-4699	351	24	-	-	PUNCT
ap-4699	351	25	th	th	VERB
ap-4699	351	26	power	power	NOUN
ap-4699	351	27	of	of	ADP
ap-4699	351	28	the	the	DET
ap-4699	351	29	original	original	ADJ
ap-4699	351	30	one	one	NUM
ap-4699	351	31	.	.	PUNCT
ap-4699	352	1	definition	definition	NOUN
ap-4699	352	2	4.1	4.1	NUM
ap-4699	352	3	.	.	PUNCT
ap-4699	353	1	for	for	ADP
ap-4699	353	2	a	a	DET
ap-4699	353	3	positive	positive	ADJ
ap-4699	353	4	integer	integer	NOUN
ap-4699	353	5	k	k	PROPN
ap-4699	353	6	,	,	PUNCT
ap-4699	353	7	the	the	DET
ap-4699	353	8	numeration	numeration	NOUN
ap-4699	353	9	system	system	NOUN
ap-4699	353	10	(	(	PUNCT
ap-4699	353	11	β	β	X
ap-4699	353	12	,	,	PUNCT
ap-4699	353	13	a	a	PRON
ap-4699	353	14	)	)	PUNCT
ap-4699	353	15	allows	allow	VERB
ap-4699	353	16	k	k	ADJ
ap-4699	353	17	-	-	ADJ
ap-4699	353	18	block	block	ADJ
ap-4699	353	19	parallel	parallel	ADJ
ap-4699	353	20	addition	addition	NOUN
ap-4699	353	21	if	if	SCONJ
ap-4699	353	22	there	there	PRON
ap-4699	353	23	exists	exist	VERB
ap-4699	353	24	parallel	parallel	ADJ
ap-4699	353	25	addition	addition	NOUN
ap-4699	353	26	in	in	ADP
ap-4699	353	27	(	(	PUNCT
ap-4699	353	28	βk	βk	NOUN
ap-4699	353	29	,	,	PUNCT
ap-4699	353	30	a(k	a(k	NUM
ap-4699	353	31	)	)	PUNCT
ap-4699	353	32	)	)	PUNCT
ap-4699	353	33	,	,	PUNCT
ap-4699	353	34	where	where	SCONJ
ap-4699	353	35	a(k	a(k	NUM
ap-4699	353	36	)	)	PUNCT
ap-4699	354	1	=	=	PRON
ap-4699	354	2	{	{	PUNCT
ap-4699	354	3	ak−1β	ak−1β	PROPN
ap-4699	354	4	k−1	k−1	PROPN
ap-4699	354	5	+	+	CCONJ
ap-4699	354	6	·	·	PUNCT
ap-4699	354	7	·	·	PUNCT
ap-4699	354	8	·	·	PUNCT
ap-4699	354	9	+	+	NUM
ap-4699	354	10	a1β	a1β	NOUN
ap-4699	354	11	+	+	CCONJ
ap-4699	354	12	a0	a0	PROPN
ap-4699	354	13	:	:	PUNCT
ap-4699	354	14	ai	ai	VERB
ap-4699	354	15	∈	∈	PROPN
ap-4699	354	16	a	a	PRON
ap-4699	354	17	}	}	PUNCT
ap-4699	354	18	.	.	PUNCT
ap-4699	355	1	we	we	PRON
ap-4699	355	2	remark	remark	VERB
ap-4699	355	3	that	that	SCONJ
ap-4699	355	4	1	1	NUM
ap-4699	355	5	-	-	PUNCT
ap-4699	355	6	block	block	NOUN
ap-4699	355	7	parallel	parallel	ADJ
ap-4699	355	8	addition	addition	NOUN
ap-4699	355	9	is	be	AUX
ap-4699	355	10	the	the	DET
ap-4699	355	11	same	same	ADJ
ap-4699	355	12	as	as	ADP
ap-4699	355	13	parallel	parallel	ADJ
ap-4699	355	14	addition	addition	NOUN
ap-4699	355	15	.	.	PUNCT
ap-4699	356	1	c.	c.	PROPN
ap-4699	356	2	frougny	frougny	PROPN
ap-4699	356	3	,	,	PUNCT
ap-4699	356	4	p.	p.	NOUN
ap-4699	356	5	heller	heller	PROPN
ap-4699	356	6	,	,	PUNCT
ap-4699	356	7	e.	e.	PROPN
ap-4699	356	8	pelantová	pelantová	PROPN
ap-4699	356	9	and	and	CCONJ
ap-4699	356	10	m.	m.	NOUN
ap-4699	356	11	svobodová	svobodová	PROPN
ap-4699	357	1	[	[	X
ap-4699	357	2	3	3	NUM
ap-4699	357	3	]	]	PUNCT
ap-4699	357	4	showed	show	VERB
ap-4699	357	5	that	that	SCONJ
ap-4699	357	6	for	for	ADP
ap-4699	357	7	a	a	DET
ap-4699	357	8	given	give	VERB
ap-4699	357	9	base	base	NOUN
ap-4699	357	10	β	β	NOUN
ap-4699	357	11	,	,	PUNCT
ap-4699	357	12	there	there	PRON
ap-4699	357	13	exists	exist	VERB
ap-4699	357	14	an	an	DET
ap-4699	357	15	integer	integer	NOUN
ap-4699	357	16	alphabet	alphabet	NOUN
ap-4699	357	17	a	a	DET
ap-4699	357	18	such	such	ADJ
ap-4699	357	19	that	that	SCONJ
ap-4699	357	20	(	(	PUNCT
ap-4699	357	21	β	β	X
ap-4699	357	22	,	,	PUNCT
ap-4699	357	23	a	a	PRON
ap-4699	357	24	)	)	PUNCT
ap-4699	357	25	allows	allow	VERB
ap-4699	357	26	parallel	parallel	ADJ
ap-4699	357	27	addition	addition	NOUN
ap-4699	357	28	if	if	SCONJ
ap-4699	357	29	and	and	CCONJ
ap-4699	357	30	only	only	ADV
ap-4699	357	31	if	if	SCONJ
ap-4699	357	32	β	β	NOUN
ap-4699	357	33	is	be	AUX
ap-4699	357	34	an	an	DET
ap-4699	357	35	algebraic	algebraic	ADJ
ap-4699	357	36	number	number	NOUN
ap-4699	357	37	with	with	ADP
ap-4699	357	38	no	no	DET
ap-4699	357	39	conjugates	conjugate	NOUN
ap-4699	357	40	of	of	ADP
ap-4699	357	41	modulus	modulus	NOUN
ap-4699	357	42	1	1	NUM
ap-4699	357	43	.	.	PUNCT
ap-4699	358	1	moreover	moreover	ADV
ap-4699	358	2	,	,	PUNCT
ap-4699	358	3	it	it	PRON
ap-4699	358	4	was	be	AUX
ap-4699	358	5	shown	show	VERB
ap-4699	358	6	that	that	SCONJ
ap-4699	358	7	the	the	DET
ap-4699	358	8	concept	concept	NOUN
ap-4699	358	9	of	of	ADP
ap-4699	358	10	k	k	ADJ
ap-4699	358	11	-	-	PUNCT
ap-4699	358	12	block	block	NOUN
ap-4699	358	13	parallel	parallel	ADJ
ap-4699	358	14	addition	addition	NOUN
ap-4699	358	15	does	do	AUX
ap-4699	358	16	not	not	PART
ap-4699	358	17	enlarge	enlarge	VERB
ap-4699	358	18	the	the	DET
ap-4699	358	19	class	class	NOUN
ap-4699	358	20	of	of	ADP
ap-4699	358	21	basis	basis	NOUN
ap-4699	358	22	allowing	allow	VERB
ap-4699	358	23	parallel	parallel	ADJ
ap-4699	358	24	addition	addition	NOUN
ap-4699	358	25	in	in	ADP
ap-4699	358	26	case	case	NOUN
ap-4699	358	27	of	of	ADP
ap-4699	358	28	integer	integer	NOUN
ap-4699	358	29	alphabets	alphabet	NOUN
ap-4699	358	30	.	.	PUNCT
ap-4699	359	1	we	we	PRON
ap-4699	359	2	prove	prove	VERB
ap-4699	359	3	an	an	DET
ap-4699	359	4	extension	extension	NOUN
ap-4699	359	5	of	of	ADP
ap-4699	359	6	these	these	DET
ap-4699	359	7	statements	statement	NOUN
ap-4699	359	8	also	also	ADV
ap-4699	359	9	to	to	ADP
ap-4699	359	10	alphabets	alphabet	NOUN
ap-4699	359	11	being	be	AUX
ap-4699	359	12	subsets	subset	NOUN
ap-4699	359	13	of	of	ADP
ap-4699	359	14	z[β	z[β	NOUN
ap-4699	359	15	]	]	PUNCT
ap-4699	359	16	in	in	ADP
ap-4699	359	17	theorem	theorem	NOUN
ap-4699	359	18	4.2	4.2	NUM
ap-4699	359	19	.	.	PUNCT
ap-4699	360	1	although	although	SCONJ
ap-4699	360	2	the	the	DET
ap-4699	360	3	k	k	ADJ
ap-4699	360	4	-	-	PUNCT
ap-4699	360	5	block	block	NOUN
ap-4699	360	6	concept	concept	NOUN
ap-4699	360	7	does	do	AUX
ap-4699	360	8	not	not	PART
ap-4699	360	9	enlarge	enlarge	VERB
ap-4699	360	10	the	the	DET
ap-4699	360	11	class	class	NOUN
ap-4699	360	12	of	of	ADP
ap-4699	360	13	bases	basis	NOUN
ap-4699	360	14	for	for	ADP
ap-4699	360	15	parallel	parallel	ADJ
ap-4699	360	16	addition	addition	NOUN
ap-4699	360	17	,	,	PUNCT
ap-4699	360	18	it	it	PRON
ap-4699	360	19	might	might	AUX
ap-4699	360	20	decrease	decrease	VERB
ap-4699	360	21	the	the	DET
ap-4699	360	22	minimal	minimal	ADJ
ap-4699	360	23	size	size	NOUN
ap-4699	360	24	of	of	ADP
ap-4699	360	25	the	the	DET
ap-4699	360	26	alphabet	alphabet	PROPN
ap-4699	360	27	.	.	PUNCT
ap-4699	361	1	theorem	theorem	VERB
ap-4699	361	2	4.2	4.2	NUM
ap-4699	361	3	.	.	PUNCT
ap-4699	362	1	let	let	VERB
ap-4699	362	2	β	β	PRON
ap-4699	362	3	be	be	AUX
ap-4699	362	4	a	a	DET
ap-4699	362	5	complex	complex	ADJ
ap-4699	362	6	number	number	NOUN
ap-4699	362	7	such	such	ADJ
ap-4699	362	8	that	that	SCONJ
ap-4699	362	9	|β|	|β|	PRON
ap-4699	362	10	>	>	X
ap-4699	362	11	1	1	X
ap-4699	362	12	.	.	PUNCT
ap-4699	363	1	there	there	PRON
ap-4699	363	2	exists	exist	VERB
ap-4699	363	3	an	an	DET
ap-4699	363	4	alphabet	alphabet	NOUN
ap-4699	363	5	a	a	DET
ap-4699	363	6	⊂	⊂	PROPN
ap-4699	363	7	z[β	z[β	X
ap-4699	363	8	]	]	PUNCT
ap-4699	363	9	with	with	ADP
ap-4699	363	10	0	0	NUM
ap-4699	363	11	∈	∈	PROPN
ap-4699	363	12	a	a	DET
ap-4699	363	13	and	and	CCONJ
ap-4699	363	14	1	1	NUM
ap-4699	363	15	∈	∈	PROPN
ap-4699	363	16	fina(β	fina(β	PROPN
ap-4699	363	17	)	)	PUNCT
ap-4699	363	18	which	which	PRON
ap-4699	363	19	allows	allow	VERB
ap-4699	363	20	k	k	ADJ
ap-4699	363	21	-	-	ADJ
ap-4699	363	22	block	block	ADJ
ap-4699	363	23	parallel	parallel	ADJ
ap-4699	363	24	addition	addition	NOUN
ap-4699	363	25	in	in	ADP
ap-4699	363	26	(	(	PUNCT
ap-4699	363	27	β	β	X
ap-4699	363	28	,	,	PUNCT
ap-4699	363	29	a	a	NOUN
ap-4699	363	30	)	)	PUNCT
ap-4699	363	31	for	for	ADP
ap-4699	363	32	some	some	DET
ap-4699	363	33	k	k	PROPN
ap-4699	363	34	∈	∈	PROPN
ap-4699	363	35	n	n	CCONJ
ap-4699	363	36	,	,	PUNCT
ap-4699	363	37	if	if	SCONJ
ap-4699	363	38	and	and	CCONJ
ap-4699	363	39	only	only	ADV
ap-4699	363	40	if	if	SCONJ
ap-4699	363	41	β	β	NOUN
ap-4699	363	42	is	be	AUX
ap-4699	363	43	an	an	DET
ap-4699	363	44	algebraic	algebraic	ADJ
ap-4699	363	45	number	number	NOUN
ap-4699	363	46	with	with	ADP
ap-4699	363	47	no	no	DET
ap-4699	363	48	conjugate	conjugate	NOUN
ap-4699	363	49	of	of	ADP
ap-4699	363	50	modulus	modulus	NOUN
ap-4699	363	51	1	1	NUM
ap-4699	363	52	.	.	PUNCT
ap-4699	364	1	if	if	SCONJ
ap-4699	364	2	this	this	PRON
ap-4699	364	3	is	be	AUX
ap-4699	364	4	the	the	DET
ap-4699	364	5	case	case	NOUN
ap-4699	364	6	,	,	PUNCT
ap-4699	364	7	then	then	ADV
ap-4699	364	8	there	there	PRON
ap-4699	364	9	also	also	ADV
ap-4699	364	10	exists	exist	VERB
ap-4699	364	11	an	an	DET
ap-4699	364	12	alphabet	alphabet	NOUN
ap-4699	364	13	in	in	ADP
ap-4699	364	14	z	z	NOUN
ap-4699	364	15	allowing	allow	VERB
ap-4699	364	16	1	1	NUM
ap-4699	364	17	-	-	PUNCT
ap-4699	364	18	block	block	NOUN
ap-4699	364	19	parallel	parallel	ADJ
ap-4699	364	20	addition	addition	NOUN
ap-4699	364	21	in	in	ADP
ap-4699	364	22	base	base	NOUN
ap-4699	364	23	β	β	NOUN
ap-4699	364	24	.	.	PUNCT
ap-4699	365	1	proof	proof	NOUN
ap-4699	365	2	.	.	PUNCT
ap-4699	366	1	if	if	SCONJ
ap-4699	366	2	the	the	DET
ap-4699	366	3	base	base	NOUN
ap-4699	366	4	β	β	X
ap-4699	366	5	is	be	AUX
ap-4699	366	6	an	an	DET
ap-4699	366	7	algebraic	algebraic	ADJ
ap-4699	366	8	number	number	NOUN
ap-4699	366	9	with	with	ADP
ap-4699	366	10	no	no	DET
ap-4699	366	11	conjugates	conjugate	NOUN
ap-4699	366	12	of	of	ADP
ap-4699	366	13	modulus	modulus	NOUN
ap-4699	366	14	1	1	NUM
ap-4699	366	15	,	,	PUNCT
ap-4699	366	16	then	then	ADV
ap-4699	366	17	[	[	X
ap-4699	366	18	5	5	NUM
ap-4699	366	19	]	]	PUNCT
ap-4699	366	20	provides	provide	VERB
ap-4699	366	21	a	a	DET
ap-4699	366	22	∈	∈	NOUN
ap-4699	366	23	n	n	NOUN
ap-4699	366	24	such	such	ADJ
ap-4699	366	25	that	that	SCONJ
ap-4699	366	26	the	the	DET
ap-4699	366	27	alphabet	alphabet	NOUN
ap-4699	366	28	a	a	X
ap-4699	366	29	=	=	X
ap-4699	366	30	{	{	PUNCT
ap-4699	366	31	−a,−a+	−a,−a+	X
ap-4699	366	32	1	1	NUM
ap-4699	366	33	,	,	PUNCT
ap-4699	366	34	.	.	PUNCT
ap-4699	366	35	.	.	PUNCT
ap-4699	367	1	.	.	PUNCT
ap-4699	368	1	,	,	PUNCT
ap-4699	368	2	0	0	NUM
ap-4699	368	3	,	,	PUNCT
ap-4699	368	4	.	.	PUNCT
ap-4699	368	5	.	.	PUNCT
ap-4699	369	1	.	.	PUNCT
ap-4699	370	1	,	,	PUNCT
ap-4699	370	2	a−1	a−1	PROPN
ap-4699	370	3	,	,	PUNCT
ap-4699	370	4	a	a	PRON
ap-4699	370	5	}	}	PUNCT
ap-4699	370	6	allows	allow	VERB
ap-4699	370	7	1	1	NUM
ap-4699	370	8	-	-	PUNCT
ap-4699	370	9	block	block	NOUN
ap-4699	370	10	parallel	parallel	ADJ
ap-4699	370	11	addition	addition	NOUN
ap-4699	370	12	.	.	PUNCT
ap-4699	371	1	obviously	obviously	ADV
ap-4699	371	2	,	,	PUNCT
ap-4699	371	3	0	0	NUM
ap-4699	371	4	∈	∈	PROPN
ap-4699	371	5	a	a	DET
ap-4699	371	6	⊂	⊂	PROPN
ap-4699	371	7	z	z	PROPN
ap-4699	371	8	and	and	CCONJ
ap-4699	371	9	1	1	NUM
ap-4699	371	10	∈	∈	PROPN
ap-4699	371	11	fina(β	fina(β	PROPN
ap-4699	371	12	)	)	PUNCT
ap-4699	371	13	.	.	PUNCT
ap-4699	372	1	for	for	ADP
ap-4699	372	2	the	the	DET
ap-4699	372	3	opposite	opposite	ADJ
ap-4699	372	4	implication	implication	NOUN
ap-4699	372	5	,	,	PUNCT
ap-4699	372	6	β	β	X
ap-4699	372	7	is	be	AUX
ap-4699	372	8	an	an	DET
ap-4699	372	9	algebraic	algebraic	ADJ
ap-4699	372	10	number	number	NOUN
ap-4699	372	11	by	by	ADP
ap-4699	372	12	corollary	corollary	ADJ
ap-4699	372	13	2.3	2.3	NUM
ap-4699	372	14	.	.	PUNCT
ap-4699	373	1	let	let	VERB
ap-4699	373	2	r	r	NOUN
ap-4699	373	3	,	,	PUNCT
ap-4699	373	4	s	s	NOUN
ap-4699	373	5	∈	∈	PROPN
ap-4699	373	6	n	n	NOUN
ap-4699	373	7	and	and	CCONJ
ap-4699	373	8	u−r	u−r	ADJ
ap-4699	373	9	,	,	PUNCT
ap-4699	373	10	.	.	PUNCT
ap-4699	373	11	.	.	PUNCT
ap-4699	374	1	.	.	PUNCT
ap-4699	375	1	,	,	PUNCT
ap-4699	375	2	us	us	PROPN
ap-4699	375	3	∈	∈	VERB
ap-4699	375	4	a	a	DET
ap-4699	375	5	be	be	AUX
ap-4699	375	6	such	such	ADJ
ap-4699	375	7	that	that	SCONJ
ap-4699	375	8	s∑	s∑	PROPN
ap-4699	375	9	j=−r	j=−r	PROPN
ap-4699	375	10	ujβ	ujβ	PROPN
ap-4699	375	11	j	j	PROPN
ap-4699	375	12	=	=	SYM
ap-4699	375	13	1	1	NUM
ap-4699	375	14	∈	∈	PROPN
ap-4699	375	15	fina(β	fina(β	PROPN
ap-4699	375	16	)	)	PUNCT
ap-4699	375	17	.	.	PUNCT
ap-4699	376	1	(	(	PUNCT
ap-4699	376	2	1	1	X
ap-4699	376	3	)	)	PUNCT
ap-4699	376	4	now	now	ADV
ap-4699	376	5	we	we	PRON
ap-4699	376	6	slightly	slightly	ADV
ap-4699	376	7	modify	modify	VERB
ap-4699	376	8	the	the	DET
ap-4699	376	9	proof	proof	NOUN
ap-4699	376	10	from	from	ADP
ap-4699	376	11	[	[	X
ap-4699	376	12	3	3	NUM
ap-4699	376	13	]	]	PUNCT
ap-4699	376	14	to	to	PART
ap-4699	376	15	show	show	VERB
ap-4699	376	16	that	that	SCONJ
ap-4699	376	17	if	if	SCONJ
ap-4699	376	18	β	β	NOUN
ap-4699	376	19	has	have	VERB
ap-4699	376	20	a	a	DET
ap-4699	376	21	conjugate	conjugate	NOUN
ap-4699	376	22	of	of	ADP
ap-4699	376	23	modulus	modulus	NOUN
ap-4699	376	24	1	1	NUM
ap-4699	376	25	,	,	PUNCT
ap-4699	376	26	then	then	ADV
ap-4699	376	27	there	there	PRON
ap-4699	376	28	is	be	VERB
ap-4699	376	29	no	no	DET
ap-4699	376	30	alphabet	alphabet	NOUN
ap-4699	376	31	in	in	ADP
ap-4699	376	32	z[β	z[β	NOUN
ap-4699	376	33	]	]	PUNCT
ap-4699	376	34	allowing	allow	VERB
ap-4699	376	35	block	block	NOUN
ap-4699	376	36	parallel	parallel	ADJ
ap-4699	376	37	addition	addition	NOUN
ap-4699	376	38	.	.	PUNCT
ap-4699	377	1	let	let	VERB
ap-4699	377	2	γ	γ	X
ap-4699	377	3	be	be	AUX
ap-4699	377	4	a	a	DET
ap-4699	377	5	conjugate	conjugate	NOUN
ap-4699	377	6	of	of	ADP
ap-4699	377	7	β	β	PRON
ap-4699	377	8	such	such	ADJ
ap-4699	377	9	that	that	SCONJ
ap-4699	377	10	|γ|	|γ|	PROPN
ap-4699	377	11	=	=	SYM
ap-4699	377	12	1	1	NUM
ap-4699	377	13	and	and	CCONJ
ap-4699	377	14	let	let	VERB
ap-4699	377	15	σ	σ	NOUN
ap-4699	377	16	:	:	PUNCT
ap-4699	377	17	q(β	q(β	PROPN
ap-4699	377	18	)	)	PUNCT
ap-4699	377	19	→	→	SYM
ap-4699	377	20	q(γ	q(γ	PROPN
ap-4699	377	21	)	)	PUNCT
ap-4699	377	22	be	be	VERB
ap-4699	377	23	the	the	DET
ap-4699	377	24	field	field	NOUN
ap-4699	377	25	isomorphism	isomorphism	NOUN
ap-4699	378	1	such	such	ADJ
ap-4699	378	2	that	that	SCONJ
ap-4699	378	3	σ(β	σ(β	PROPN
ap-4699	378	4	)	)	PUNCT
ap-4699	378	5	=	=	SYM
ap-4699	378	6	γ	γ	X
ap-4699	378	7	.	.	PUNCT
ap-4699	378	8	let	let	VERB
ap-4699	378	9	a′	a′	NOUN
ap-4699	378	10	:	:	PUNCT
ap-4699	378	11	=	=	SYM
ap-4699	378	12	{	{	PUNCT
ap-4699	378	13	σ(a	σ(a	PROPN
ap-4699	378	14	)	)	PUNCT
ap-4699	378	15	:	:	PUNCT
ap-4699	378	16	a	a	DET
ap-4699	378	17	∈	∈	PROPN
ap-4699	378	18	a	a	PRON
ap-4699	378	19	}	}	PUNCT
ap-4699	378	20	.	.	PUNCT
ap-4699	379	1	assume	assume	VERB
ap-4699	379	2	,	,	PUNCT
ap-4699	379	3	for	for	ADP
ap-4699	379	4	contradiction	contradiction	NOUN
ap-4699	379	5	,	,	PUNCT
ap-4699	379	6	that	that	SCONJ
ap-4699	379	7	there	there	PRON
ap-4699	379	8	are	be	VERB
ap-4699	379	9	k	k	NOUN
ap-4699	379	10	,	,	PUNCT
ap-4699	379	11	p	p	PROPN
ap-4699	379	12	∈	∈	PROPN
ap-4699	379	13	n	n	PRON
ap-4699	379	14	such	such	ADJ
ap-4699	379	15	that	that	SCONJ
ap-4699	379	16	there	there	PRON
ap-4699	379	17	exists	exist	VERB
ap-4699	379	18	p	p	ADJ
ap-4699	379	19	-	-	ADJ
ap-4699	379	20	local	local	ADJ
ap-4699	379	21	function	function	NOUN
ap-4699	379	22	performing	perform	VERB
ap-4699	379	23	k	k	ADJ
ap-4699	379	24	-	-	PUNCT
ap-4699	379	25	block	block	ADJ
ap-4699	379	26	parallel	parallel	ADJ
ap-4699	379	27	addition	addition	NOUN
ap-4699	379	28	on	on	ADP
ap-4699	379	29	(	(	PUNCT
ap-4699	379	30	β	β	X
ap-4699	379	31	,	,	PUNCT
ap-4699	379	32	a	a	PRON
ap-4699	379	33	)	)	PUNCT
ap-4699	379	34	.	.	PUNCT
ap-4699	380	1	we	we	PRON
ap-4699	380	2	denote	denote	VERB
ap-4699	380	3	s	s	VERB
ap-4699	380	4	:	:	PUNCT
ap-4699	380	5	=	=	SYM
ap-4699	380	6	max	max	X
ap-4699	380	7	{	{	PUNCT
ap-4699	380	8	∣∣∣∣pk−1∑	∣∣∣∣pk−1∑	X
ap-4699	380	9	j=0	j=0	PROPN
ap-4699	380	10	ajγ	ajγ	PROPN
ap-4699	380	11	j	j	PROPN
ap-4699	380	12	∣∣∣∣	∣∣∣∣	PROPN
ap-4699	380	13	:	:	PUNCT
ap-4699	380	14	aj	aj	PROPN
ap-4699	380	15	∈	∈	PROPN
ap-4699	380	16	a′	a′	PROPN
ap-4699	380	17	}	}	PUNCT
ap-4699	380	18	.	.	PUNCT
ap-4699	381	1	290	290	NUM
ap-4699	381	2	vol	vol	NOUN
ap-4699	381	3	.	.	PUNCT
ap-4699	382	1	58	58	NUM
ap-4699	382	2	no	no	INTJ
ap-4699	382	3	.	.	PUNCT
ap-4699	383	1	5/2018	5/2018	NUM
ap-4699	383	2	minimal	minimal	ADJ
ap-4699	383	3	non	non	ADJ
ap-4699	383	4	-	-	ADJ
ap-4699	383	5	integer	integer	ADJ
ap-4699	383	6	alphabets	alphabet	NOUN
ap-4699	383	7	allowing	allow	VERB
ap-4699	383	8	parallel	parallel	ADJ
ap-4699	383	9	addition	addition	NOUN
ap-4699	383	10	since	since	SCONJ
ap-4699	383	11	there	there	PRON
ap-4699	383	12	are	be	VERB
ap-4699	383	13	infinitely	infinitely	ADV
ap-4699	383	14	many	many	ADJ
ap-4699	383	15	j	j	NOUN
ap-4699	383	16	such	such	ADJ
ap-4699	383	17	that	that	PRON
ap-4699	383	18	re	re	VERB
ap-4699	383	19	γj	γj	NOUN
ap-4699	383	20	>	>	SYM
ap-4699	383	21	1	1	NUM
ap-4699	383	22	2	2	NUM
ap-4699	383	23	,	,	PUNCT
ap-4699	383	24	there	there	PRON
ap-4699	383	25	exists	exist	VERB
ap-4699	383	26	n	n	PRON
ap-4699	383	27	>	>	X
ap-4699	383	28	p	p	PROPN
ap-4699	383	29	and	and	CCONJ
ap-4699	383	30	indices	indice	VERB
ap-4699	383	31	0	0	NUM
ap-4699	383	32	≤	≤	NUM
ap-4699	383	33	j1	j1	X
ap-4699	383	34	<	<	X
ap-4699	383	35	·	·	PUNCT
ap-4699	383	36	·	·	PUNCT
ap-4699	383	37	·	·	PUNCT
ap-4699	383	38	<	<	X
ap-4699	383	39	jm	jm	PROPN
ap-4699	383	40	≤	≤	PROPN
ap-4699	383	41	kn	kn	PROPN
ap-4699	384	1	−	−	PROPN
ap-4699	384	2	1	1	NUM
ap-4699	384	3	satisfying	satisfy	VERB
ap-4699	384	4	ji+1	ji+1	NOUN
ap-4699	384	5	−	−	PROPN
ap-4699	385	1	ji	ji	INTJ
ap-4699	385	2	>	>	X
ap-4699	385	3	r	r	PROPN
ap-4699	385	4	+	+	SYM
ap-4699	385	5	s	s	NOUN
ap-4699	385	6	for	for	ADP
ap-4699	385	7	all	all	PRON
ap-4699	385	8	i	i	PRON
ap-4699	385	9	∈	∈	PROPN
ap-4699	385	10	{	{	PUNCT
ap-4699	385	11	1	1	NUM
ap-4699	385	12	,	,	PUNCT
ap-4699	385	13	.	.	PUNCT
ap-4699	385	14	.	.	PUNCT
ap-4699	385	15	.	.	PUNCT
ap-4699	386	1	,	,	PUNCT
ap-4699	386	2	m−	m−	PROPN
ap-4699	386	3	1	1	NUM
ap-4699	386	4	}	}	PUNCT
ap-4699	386	5	such	such	ADJ
ap-4699	386	6	that	that	SCONJ
ap-4699	386	7	2s	2s	PROPN
ap-4699	386	8	<	<	X
ap-4699	386	9	re	re	X
ap-4699	386	10	kn−1∑	kn−1∑	AUX
ap-4699	386	11	j=0	j=0	PROPN
ap-4699	386	12	εjγ	εjγ	VERB
ap-4699	386	13	j	j	PROPN
ap-4699	386	14	≤	≤	PROPN
ap-4699	386	15	∣∣∣∣kn−1∑	∣∣∣∣kn−1∑	NUM
ap-4699	386	16	j=0	j=0	PROPN
ap-4699	386	17	εjγ	εjγ	VERB
ap-4699	386	18	j	j	PROPN
ap-4699	386	19	∣∣∣∣	∣∣∣∣	PROPN
ap-4699	386	20	,	,	PUNCT
ap-4699	386	21	where	where	SCONJ
ap-4699	386	22	εj	εj	NOUN
ap-4699	386	23	=	=	NOUN
ap-4699	386	24	1	1	NUM
ap-4699	386	25	if	if	SCONJ
ap-4699	386	26	j	j	PROPN
ap-4699	386	27	=	=	SYM
ap-4699	386	28	ji	ji	PROPN
ap-4699	386	29	for	for	ADP
ap-4699	386	30	some	some	DET
ap-4699	386	31	i	i	PRON
ap-4699	386	32	∈	∈	PROPN
ap-4699	386	33	{	{	PUNCT
ap-4699	386	34	1	1	NUM
ap-4699	386	35	,	,	PUNCT
ap-4699	386	36	.	.	PUNCT
ap-4699	386	37	.	.	PUNCT
ap-4699	387	1	.	.	PUNCT
ap-4699	388	1	,	,	PUNCT
ap-4699	388	2	m	m	VERB
ap-4699	388	3	}	}	PUNCT
ap-4699	388	4	and	and	CCONJ
ap-4699	388	5	εj	εj	VERB
ap-4699	388	6	=	=	NOUN
ap-4699	388	7	0	0	PUNCT
ap-4699	389	1	otherwise	otherwise	ADV
ap-4699	389	2	.	.	PUNCT
ap-4699	390	1	by	by	ADP
ap-4699	390	2	using	use	VERB
ap-4699	390	3	the	the	DET
ap-4699	390	4	representation	representation	NOUN
ap-4699	390	5	(	(	PUNCT
ap-4699	390	6	1	1	NUM
ap-4699	390	7	)	)	PUNCT
ap-4699	390	8	of	of	ADP
ap-4699	390	9	1	1	NUM
ap-4699	390	10	and	and	CCONJ
ap-4699	390	11	the	the	DET
ap-4699	390	12	fact	fact	NOUN
ap-4699	390	13	that	that	SCONJ
ap-4699	390	14	ji+1	ji+1	PROPN
ap-4699	391	1	−	−	X
ap-4699	392	1	ji	ji	INTJ
ap-4699	392	2	>	>	X
ap-4699	392	3	r	r	PROPN
ap-4699	392	4	+	+	SYM
ap-4699	392	5	s	s	X
ap-4699	392	6	,	,	PUNCT
ap-4699	392	7	we	we	PRON
ap-4699	392	8	have∑kn−1	have∑kn−1	ADP
ap-4699	392	9	j=0	j=0	PROPN
ap-4699	392	10	εjγ	εjγ	VERB
ap-4699	392	11	j	j	PROPN
ap-4699	392	12	=	=	SYM
ap-4699	392	13	∑m	∑m	PROPN
ap-4699	392	14	i=1	i=1	PROPN
ap-4699	392	15	1	1	NUM
ap-4699	392	16	·	·	PUNCT
ap-4699	392	17	γji	γji	NOUN
ap-4699	392	18	=	=	NOUN
ap-4699	392	19	∑kn−1+s	∑kn−1+s	PRON
ap-4699	392	20	j=−r	j=−r	PROPN
ap-4699	392	21	v′jγ	v′jγ	ADP
ap-4699	392	22	j	j	PROPN
ap-4699	392	23	for	for	ADP
ap-4699	392	24	some	some	DET
ap-4699	392	25	v′j	v′j	NUM
ap-4699	392	26	∈	∈	PROPN
ap-4699	392	27	a′.	a′.	NOUN
ap-4699	392	28	hence	hence	ADV
ap-4699	392	29	2s	2s	X
ap-4699	392	30	<	<	X
ap-4699	392	31	t	t	X
ap-4699	392	32	:	:	PUNCT
ap-4699	392	33	=	=	SYM
ap-4699	392	34	max	max	PROPN
ap-4699	392	35	{	{	PUNCT
ap-4699	392	36	∣∣∣∣kn−1+s∑	∣∣∣∣kn−1+s∑	PROPN
ap-4699	392	37	j=−r	j=−r	PROPN
ap-4699	392	38	ajγ	ajγ	VERB
ap-4699	392	39	j	j	PROPN
ap-4699	392	40	∣∣∣∣	∣∣∣∣	PROPN
ap-4699	392	41	:	:	PUNCT
ap-4699	392	42	aj	aj	PROPN
ap-4699	392	43	∈	∈	PROPN
ap-4699	392	44	a′	a′	PROPN
ap-4699	392	45	}	}	PUNCT
ap-4699	392	46	.	.	PUNCT
ap-4699	393	1	let	let	VERB
ap-4699	393	2	x′	x′	PROPN
ap-4699	393	3	=	=	PUNCT
ap-4699	393	4	∑kn−1+s	∑kn−1+s	PRON
ap-4699	393	5	j=−r	j=−r	PROPN
ap-4699	393	6	σ(xj)γj	σ(xj)γj	NOUN
ap-4699	393	7	,	,	PUNCT
ap-4699	393	8	where	where	SCONJ
ap-4699	393	9	x−r	x−r	PROPN
ap-4699	393	10	,	,	PUNCT
ap-4699	393	11	.	.	PUNCT
ap-4699	393	12	.	.	PUNCT
ap-4699	394	1	.	.	PUNCT
ap-4699	395	1	,	,	PUNCT
ap-4699	395	2	xkn−1+s	xkn−1+s	PROPN
ap-4699	395	3	∈	∈	PROPN
ap-4699	395	4	a	a	PRON
ap-4699	395	5	,	,	PUNCT
ap-4699	395	6	be	be	AUX
ap-4699	395	7	such	such	ADJ
ap-4699	395	8	that	that	SCONJ
ap-4699	395	9	|x′|	|x′|	PROPN
ap-4699	395	10	=	=	PROPN
ap-4699	395	11	t	t	PROPN
ap-4699	395	12	.	.	PUNCT
ap-4699	396	1	let	let	VERB
ap-4699	396	2	x	x	PUNCT
ap-4699	397	1	=	=	NOUN
ap-4699	397	2	∑kn−1+s	∑kn−1+s	PRON
ap-4699	397	3	j=−r	j=−r	PROPN
ap-4699	397	4	xjβ	xjβ	PROPN
ap-4699	397	5	j	j	PROPN
ap-4699	397	6	,	,	PUNCT
ap-4699	397	7	i.e.	i.e.	X
ap-4699	397	8	,	,	PUNCT
ap-4699	397	9	x′	x′	PROPN
ap-4699	397	10	=	=	SYM
ap-4699	397	11	σ(x	σ(x	PROPN
ap-4699	397	12	)	)	PUNCT
ap-4699	397	13	.	.	PUNCT
ap-4699	398	1	since	since	SCONJ
ap-4699	398	2	there	there	PRON
ap-4699	398	3	is	be	VERB
ap-4699	398	4	a	a	DET
ap-4699	398	5	kblock	kblock	NOUN
ap-4699	398	6	parallel	parallel	ADJ
ap-4699	398	7	addition	addition	NOUN
ap-4699	398	8	in	in	ADP
ap-4699	398	9	(	(	PUNCT
ap-4699	398	10	β	β	X
ap-4699	398	11	,	,	PUNCT
ap-4699	398	12	a	a	NOUN
ap-4699	398	13	)	)	PUNCT
ap-4699	398	14	,	,	PUNCT
ap-4699	398	15	we	we	PRON
ap-4699	398	16	have	have	VERB
ap-4699	398	17	x+x	x+x	PROPN
ap-4699	398	18	=	=	SYM
ap-4699	398	19	k(n+p)−1+s∑	k(n+p)−1+s∑	PROPN
ap-4699	398	20	j	j	PROPN
ap-4699	398	21	=	=	PROPN
ap-4699	398	22	kn+s	kn+s	PROPN
ap-4699	398	23	zjβ	zjβ	NOUN
ap-4699	398	24	j+	j+	PUNCT
ap-4699	398	25	kn−1+s∑	kn−1+s∑	PRON
ap-4699	398	26	j=−r	j=−r	NOUN
ap-4699	398	27	zjβ	zjβ	NOUN
ap-4699	398	28	j+	j+	NUM
ap-4699	398	29	−r−1∑	−r−1∑	PROPN
ap-4699	398	30	j=−kp−r	j=−kp−r	NOUN
ap-4699	398	31	zjβ	zjβ	NOUN
ap-4699	398	32	j	j	PROPN
ap-4699	398	33	,	,	PUNCT
ap-4699	398	34	where	where	SCONJ
ap-4699	398	35	zj	zj	PROPN
ap-4699	398	36	∈	∈	PROPN
ap-4699	398	37	a.	a.	NOUN
ap-4699	398	38	we	we	PRON
ap-4699	398	39	denote	denote	VERB
ap-4699	398	40	z′j	z′j	NOUN
ap-4699	398	41	:	:	PUNCT
ap-4699	398	42	=	=	SYM
ap-4699	398	43	σ(zj	σ(zj	NOUN
ap-4699	398	44	)	)	PUNCT
ap-4699	398	45	.	.	PUNCT
ap-4699	399	1	hence	hence	ADV
ap-4699	399	2	,	,	PUNCT
ap-4699	399	3	z′j	z′j	PROPN
ap-4699	399	4	∈	∈	PROPN
ap-4699	399	5	a′	a′	PROPN
ap-4699	399	6	,	,	PUNCT
ap-4699	399	7	and	and	CCONJ
ap-4699	399	8	|x′|+	|x′|+	VERB
ap-4699	399	9	2s	2s	X
ap-4699	399	10	<	<	X
ap-4699	399	11	|x′|+	|x′|+	NOUN
ap-4699	399	12	|x′|	|x′|	NOUN
ap-4699	399	13	=	=	SYM
ap-4699	399	14	|x′	|x′	PROPN
ap-4699	399	15	+	+	CCONJ
ap-4699	399	16	x′|	x′|	SYM
ap-4699	399	17	≤	≤	NUM
ap-4699	399	18	∣∣∣∣k(n+p)−1+s∑	∣∣∣∣k(n+p)−1+s∑	PROPN
ap-4699	399	19	j	j	PROPN
ap-4699	399	20	=	=	PROPN
ap-4699	399	21	kn+s	kn+s	PROPN
ap-4699	399	22	z′jγ	z′jγ	NOUN
ap-4699	399	23	j	j	PROPN
ap-4699	399	24	∣∣∣∣+	∣∣∣∣+	PROPN
ap-4699	399	25	∣∣∣∣kn−1+s∑	∣∣∣∣kn−1+s∑	PROPN
ap-4699	399	26	j=−r	j=−r	PROPN
ap-4699	399	27	z′jγ	z′jγ	VERB
ap-4699	399	28	j	j	PROPN
ap-4699	399	29	∣∣∣∣+	∣∣∣∣+	PROPN
ap-4699	399	30	∣∣∣∣	∣∣∣∣	PROPN
ap-4699	399	31	−r−1∑	−r−1∑	PROPN
ap-4699	399	32	j=−kp−r	j=−kp−r	NOUN
ap-4699	399	33	z′jγ	z′jγ	NOUN
ap-4699	399	34	j	j	PROPN
ap-4699	399	35	∣∣∣∣	∣∣∣∣	PROPN
ap-4699	399	36	≤	≤	PUNCT
ap-4699	400	1	|γkn+s|s	|γkn+s|s	ADP
ap-4699	400	2	+	+	ADJ
ap-4699	400	3	|x′|+	|x′|+	NOUN
ap-4699	400	4	|γ−kp−r|s	|γ−kp−r|	NOUN
ap-4699	400	5	=	=	PUNCT
ap-4699	401	1	2s	2s	PROPN
ap-4699	401	2	+	+	CCONJ
ap-4699	401	3	|x′|	|x′|	PROPN
ap-4699	401	4	,	,	PUNCT
ap-4699	401	5	which	which	PRON
ap-4699	401	6	is	be	AUX
ap-4699	401	7	a	a	DET
ap-4699	401	8	contradiction	contradiction	NOUN
ap-4699	401	9	.	.	PUNCT
ap-4699	402	1	5	5	X
ap-4699	402	2	.	.	X
ap-4699	402	3	conclusion	conclusion	NOUN
ap-4699	402	4	we	we	PRON
ap-4699	402	5	have	have	AUX
ap-4699	402	6	shown	show	VERB
ap-4699	402	7	that	that	SCONJ
ap-4699	402	8	the	the	DET
ap-4699	402	9	necessary	necessary	ADJ
ap-4699	402	10	conditions	condition	NOUN
ap-4699	402	11	on	on	ADP
ap-4699	402	12	β	β	PRON
ap-4699	402	13	and	and	CCONJ
ap-4699	402	14	a	a	DET
ap-4699	402	15	allowing	allow	VERB
ap-4699	402	16	parallel	parallel	ADJ
ap-4699	402	17	addition	addition	NOUN
ap-4699	402	18	that	that	PRON
ap-4699	402	19	were	be	AUX
ap-4699	402	20	known	know	VERB
ap-4699	402	21	for	for	ADP
ap-4699	402	22	alphabets	alphabet	NOUN
ap-4699	402	23	consisting	consist	VERB
ap-4699	402	24	of	of	ADP
ap-4699	402	25	consecutive	consecutive	ADJ
ap-4699	402	26	integers	integer	NOUN
ap-4699	402	27	can	can	AUX
ap-4699	402	28	be	be	AUX
ap-4699	402	29	largely	largely	ADV
ap-4699	402	30	extended	extend	VERB
ap-4699	402	31	to	to	ADP
ap-4699	402	32	alphabets	alphabet	NOUN
ap-4699	402	33	a	a	DET
ap-4699	402	34	being	being	NOUN
ap-4699	402	35	subsets	subset	NOUN
ap-4699	402	36	of	of	ADP
ap-4699	402	37	z[β	z[β	NOUN
ap-4699	402	38	]	]	PUNCT
ap-4699	402	39	.	.	PUNCT
ap-4699	403	1	during	during	ADP
ap-4699	403	2	our	our	PRON
ap-4699	403	3	investigation	investigation	NOUN
ap-4699	403	4	,	,	PUNCT
ap-4699	403	5	we	we	PRON
ap-4699	403	6	also	also	ADV
ap-4699	403	7	considered	consider	VERB
ap-4699	403	8	even	even	ADV
ap-4699	403	9	more	more	ADV
ap-4699	403	10	general	general	ADJ
ap-4699	403	11	case	case	NOUN
ap-4699	403	12	:	:	PUNCT
ap-4699	403	13	β	β	X
ap-4699	403	14	∈	∈	PROPN
ap-4699	403	15	z[ω	z[ω	PROPN
ap-4699	403	16	]	]	PUNCT
ap-4699	403	17	and	and	CCONJ
ap-4699	403	18	a	a	DET
ap-4699	403	19	⊂	⊂	X
ap-4699	403	20	z[ω	z[ω	PROPN
ap-4699	403	21	]	]	PUNCT
ap-4699	403	22	,	,	PUNCT
ap-4699	403	23	where	where	SCONJ
ap-4699	403	24	ω	ω	PROPN
ap-4699	403	25	is	be	AUX
ap-4699	403	26	an	an	DET
ap-4699	403	27	algebraic	algebraic	ADJ
ap-4699	403	28	number	number	NOUN
ap-4699	403	29	.	.	PUNCT
ap-4699	404	1	clearly	clearly	ADV
ap-4699	404	2	,	,	PUNCT
ap-4699	404	3	z[β	z[β	X
ap-4699	404	4	]	]	X
ap-4699	404	5	⊂	⊂	PROPN
ap-4699	404	6	z[ω	z[ω	PROPN
ap-4699	404	7	]	]	PUNCT
ap-4699	404	8	,	,	PUNCT
ap-4699	404	9	but	but	CCONJ
ap-4699	404	10	if	if	SCONJ
ap-4699	404	11	z[β	z[β	NOUN
ap-4699	404	12	]	]	X
ap-4699	404	13	(	(	PUNCT
ap-4699	404	14	z[ω	z[ω	PROPN
ap-4699	404	15	]	]	PUNCT
ap-4699	404	16	,	,	PUNCT
ap-4699	404	17	then	then	ADV
ap-4699	404	18	congruences	congruence	VERB
ap-4699	404	19	modulo	modulo	PROPN
ap-4699	404	20	β	β	X
ap-4699	404	21	or	or	CCONJ
ap-4699	404	22	β	β	PRON
ap-4699	404	23	−	−	PROPN
ap-4699	404	24	1	1	NUM
ap-4699	404	25	behave	behave	VERB
ap-4699	404	26	differently	differently	ADV
ap-4699	404	27	in	in	ADP
ap-4699	404	28	z[ω	z[ω	NOUN
ap-4699	404	29	]	]	PUNCT
ap-4699	404	30	than	than	ADP
ap-4699	404	31	in	in	ADP
ap-4699	404	32	z[β	z[β	NOUN
ap-4699	404	33	]	]	PUNCT
ap-4699	404	34	.	.	PUNCT
ap-4699	405	1	due	due	ADP
ap-4699	405	2	to	to	ADP
ap-4699	405	3	this	this	DET
ap-4699	405	4	fact	fact	NOUN
ap-4699	405	5	,	,	PUNCT
ap-4699	405	6	theorem	theorem	VERB
ap-4699	405	7	3.7	3.7	NUM
ap-4699	405	8	does	do	AUX
ap-4699	405	9	not	not	PART
ap-4699	405	10	hold	hold	VERB
ap-4699	405	11	in	in	ADP
ap-4699	405	12	the	the	DET
ap-4699	405	13	z[ω	z[ω	NOUN
ap-4699	405	14	]	]	PUNCT
ap-4699	405	15	setting	setting	NOUN
ap-4699	405	16	,	,	PUNCT
ap-4699	405	17	see	see	VERB
ap-4699	405	18	a	a	DET
ap-4699	405	19	counterexample	counterexample	NOUN
ap-4699	405	20	:	:	PUNCT
ap-4699	405	21	let	let	VERB
ap-4699	405	22	ω	ω	NOUN
ap-4699	405	23	=	=	SYM
ap-4699	405	24	1	1	NUM
ap-4699	405	25	2	2	NUM
ap-4699	405	26	√	√	NUM
ap-4699	405	27	5	5	NUM
ap-4699	405	28	+	+	SYM
ap-4699	405	29	1	1	NUM
ap-4699	405	30	2	2	NUM
ap-4699	405	31	,	,	PUNCT
ap-4699	405	32	β	β	X
ap-4699	405	33	=	=	PUNCT
ap-4699	405	34	−2ω+1	−2ω+1	NOUN
ap-4699	405	35	=	=	PUNCT
ap-4699	406	1	−	−	PROPN
ap-4699	406	2	√	√	NUM
ap-4699	406	3	5	5	NUM
ap-4699	406	4	and	and	CCONJ
ap-4699	406	5	a	a	DET
ap-4699	406	6	=	=	X
ap-4699	406	7	{	{	PUNCT
ap-4699	406	8	−2,−1	−2,−1	PROPN
ap-4699	406	9	,	,	PUNCT
ap-4699	406	10	0	0	NUM
ap-4699	406	11	,	,	PUNCT
ap-4699	406	12	1	1	NUM
ap-4699	406	13	,	,	PUNCT
ap-4699	406	14	2	2	NUM
ap-4699	406	15	,	,	PUNCT
ap-4699	406	16	3	3	NUM
ap-4699	406	17	}	}	PUNCT
ap-4699	406	18	⊂	⊂	PROPN
ap-4699	406	19	z[ω	z[ω	PROPN
ap-4699	406	20	]	]	PUNCT
ap-4699	406	21	.	.	PUNCT
ap-4699	407	1	this	this	DET
ap-4699	407	2	numeration	numeration	NOUN
ap-4699	407	3	system	system	NOUN
ap-4699	407	4	allows	allow	VERB
ap-4699	407	5	parallel	parallel	ADJ
ap-4699	407	6	addition	addition	NOUN
ap-4699	407	7	,	,	PUNCT
ap-4699	407	8	but	but	CCONJ
ap-4699	407	9	−2	−2	PROPN
ap-4699	407	10	≡β−1	≡β−1	NUM
ap-4699	407	11	0	0	NUM
ap-4699	407	12	≡β−1	≡β−1	NUM
ap-4699	407	13	2	2	NUM
ap-4699	407	14	and	and	CCONJ
ap-4699	407	15	−1	−1	NOUN
ap-4699	407	16	≡β−1	≡β−1	NUM
ap-4699	407	17	1	1	NUM
ap-4699	407	18	≡β−1	≡β−1	NUM
ap-4699	407	19	3	3	NUM
ap-4699	407	20	,	,	PUNCT
ap-4699	407	21	with	with	ADP
ap-4699	407	22	≡β−1	≡β−1	NUM
ap-4699	407	23	in	in	ADP
ap-4699	407	24	z[ω	z[ω	NOUN
ap-4699	407	25	]	]	PUNCT
ap-4699	407	26	.	.	PUNCT
ap-4699	408	1	the	the	DET
ap-4699	408	2	minimal	minimal	ADJ
ap-4699	408	3	polynomial	polynomial	NOUN
ap-4699	408	4	of	of	ADP
ap-4699	408	5	β	β	PROPN
ap-4699	408	6	is	be	AUX
ap-4699	408	7	x2	x2	PROPN
ap-4699	408	8	−	−	PROPN
ap-4699	408	9	5	5	NUM
ap-4699	408	10	,	,	PUNCT
ap-4699	408	11	thus	thus	ADV
ap-4699	408	12	there	there	PRON
ap-4699	408	13	are	be	VERB
ap-4699	408	14	six	six	NUM
ap-4699	408	15	congruence	congruence	NOUN
ap-4699	408	16	classes	class	NOUN
ap-4699	408	17	modulo	modulo	VERB
ap-4699	408	18	β	β	X
ap-4699	408	19	−	−	PROPN
ap-4699	408	20	1	1	NUM
ap-4699	408	21	in	in	ADP
ap-4699	408	22	z[ω	z[ω	NOUN
ap-4699	408	23	]	]	PUNCT
ap-4699	408	24	,	,	PUNCT
ap-4699	408	25	i.e.	i.e.	X
ap-4699	408	26	,	,	PUNCT
ap-4699	408	27	a	a	PRON
ap-4699	408	28	does	do	AUX
ap-4699	408	29	not	not	PART
ap-4699	408	30	contain	contain	VERB
ap-4699	408	31	representatives	representative	NOUN
ap-4699	408	32	of	of	ADP
ap-4699	408	33	all	all	DET
ap-4699	408	34	congruence	congruence	PROPN
ap-4699	408	35	classes	class	NOUN
ap-4699	408	36	.	.	PUNCT
ap-4699	409	1	see	see	VERB
ap-4699	409	2	[	[	X
ap-4699	409	3	14	14	NUM
ap-4699	409	4	]	]	PUNCT
ap-4699	409	5	for	for	ADP
ap-4699	409	6	further	further	ADJ
ap-4699	409	7	elaboration	elaboration	NOUN
ap-4699	409	8	.	.	PUNCT
ap-4699	410	1	we	we	PRON
ap-4699	410	2	conjecture	conjecture	VERB
ap-4699	410	3	the	the	DET
ap-4699	410	4	converse	converse	NOUN
ap-4699	410	5	of	of	ADP
ap-4699	410	6	corollary	corollary	ADJ
ap-4699	410	7	3.6	3.6	NUM
ap-4699	410	8	,	,	PUNCT
ap-4699	410	9	that	that	PRON
ap-4699	410	10	is	be	AUX
ap-4699	410	11	:	:	PUNCT
ap-4699	410	12	let	let	VERB
ap-4699	410	13	(	(	PUNCT
ap-4699	410	14	β	β	X
ap-4699	410	15	,	,	PUNCT
ap-4699	410	16	a	a	PRON
ap-4699	410	17	)	)	PUNCT
ap-4699	410	18	be	be	AUX
ap-4699	410	19	a	a	DET
ap-4699	410	20	numeration	numeration	NOUN
ap-4699	410	21	system	system	NOUN
ap-4699	410	22	such	such	ADJ
ap-4699	410	23	that	that	SCONJ
ap-4699	410	24	1	1	NUM
ap-4699	410	25	∈	∈	PROPN
ap-4699	410	26	a[β	a[β	PROPN
ap-4699	410	27	]	]	PUNCT
ap-4699	410	28	and	and	CCONJ
ap-4699	410	29	a	a	DET
ap-4699	410	30	⊂	⊂	PROPN
ap-4699	410	31	z[β	z[β	NOUN
ap-4699	410	32	]	]	PUNCT
ap-4699	410	33	.	.	PUNCT
ap-4699	411	1	let	let	AUX
ap-4699	411	2	(	(	PUNCT
ap-4699	411	3	β	β	X
ap-4699	411	4	,	,	PUNCT
ap-4699	411	5	a	a	PRON
ap-4699	411	6	)	)	PUNCT
ap-4699	411	7	allow	allow	VERB
ap-4699	411	8	parallel	parallel	ADJ
ap-4699	411	9	addition	addition	NOUN
ap-4699	411	10	by	by	ADP
ap-4699	411	11	p	p	ADJ
ap-4699	411	12	-	-	ADJ
ap-4699	411	13	local	local	ADJ
ap-4699	411	14	function	function	NOUN
ap-4699	411	15	with	with	ADP
ap-4699	411	16	p	p	NOUN
ap-4699	411	17	=	=	PUNCT
ap-4699	411	18	r	r	NOUN
ap-4699	411	19	+	+	CCONJ
ap-4699	411	20	t+	t+	NOUN
ap-4699	411	21	1	1	NUM
ap-4699	411	22	.	.	PUNCT
ap-4699	412	1	if	if	SCONJ
ap-4699	412	2	β	β	PROPN
ap-4699	412	3	is	be	AUX
ap-4699	412	4	expanding	expand	VERB
ap-4699	412	5	,	,	PUNCT
ap-4699	412	6	then	then	ADV
ap-4699	412	7	the	the	DET
ap-4699	412	8	parallel	parallel	ADJ
ap-4699	412	9	addition	addition	NOUN
ap-4699	412	10	is	be	AUX
ap-4699	412	11	without	without	ADP
ap-4699	412	12	anticipation	anticipation	NOUN
ap-4699	412	13	,	,	PUNCT
ap-4699	412	14	i.e.	i.e.	X
ap-4699	412	15	,	,	PUNCT
ap-4699	412	16	t	t	NOUN
ap-4699	412	17	=	=	SYM
ap-4699	412	18	0	0	NUM
ap-4699	412	19	.	.	NOUN
ap-4699	413	1	6	6	NUM
ap-4699	413	2	.	.	PUNCT
ap-4699	413	3	acknowledgments	acknowledgment	NOUN
ap-4699	413	4	this	this	DET
ap-4699	413	5	work	work	NOUN
ap-4699	413	6	was	be	AUX
ap-4699	413	7	supported	support	VERB
ap-4699	413	8	by	by	ADP
ap-4699	413	9	gačr	gačr	NOUN
ap-4699	413	10	13	13	NUM
ap-4699	413	11	-	-	PUNCT
ap-4699	413	12	03538s	03538	NOUN
ap-4699	413	13	and	and	CCONJ
ap-4699	413	14	sgs	sgs	PROPN
ap-4699	413	15	17/193	17/193	NUM
ap-4699	413	16	/	/	SYM
ap-4699	413	17	ohk4/3t/14	ohk4/3t/14	NOUN
ap-4699	413	18	.	.	PUNCT
ap-4699	414	1	the	the	DET
ap-4699	414	2	author	author	NOUN
ap-4699	414	3	thanks	thank	NOUN
ap-4699	414	4	to	to	ADP
ap-4699	414	5	milena	milena	PROPN
ap-4699	414	6	svobodová	svobodová	PROPN
ap-4699	414	7	and	and	CCONJ
ap-4699	414	8	edita	edita	NOUN
ap-4699	414	9	pelantová	pelantová	NOUN
ap-4699	414	10	for	for	ADP
ap-4699	414	11	fruitful	fruitful	ADJ
ap-4699	414	12	discussions	discussion	NOUN
ap-4699	414	13	.	.	PUNCT
ap-4699	415	1	references	reference	NOUN
ap-4699	415	2	[	[	X
ap-4699	415	3	1	1	NUM
ap-4699	415	4	]	]	PUNCT
ap-4699	415	5	a.	a.	NOUN
ap-4699	415	6	avizienis	avizienis	PROPN
ap-4699	415	7	.	.	PUNCT
ap-4699	416	1	signed	sign	VERB
ap-4699	416	2	-	-	PUNCT
ap-4699	416	3	digit	digit	NOUN
ap-4699	416	4	number	number	NOUN
ap-4699	416	5	representations	representation	NOUN
ap-4699	416	6	for	for	ADP
ap-4699	416	7	fast	fast	ADJ
ap-4699	416	8	parallel	parallel	ADJ
ap-4699	416	9	arithmetic	arithmetic	NOUN
ap-4699	416	10	.	.	PUNCT
ap-4699	417	1	ieee	ieee	PROPN
ap-4699	417	2	trans	trans	PROPN
ap-4699	417	3	comput	comput	PROPN
ap-4699	417	4	10:389–400	10:389–400	PROPN
ap-4699	417	5	,	,	PUNCT
ap-4699	417	6	1961	1961	NUM
ap-4699	417	7	.	.	PUNCT
ap-4699	418	1	[	[	X
ap-4699	418	2	2	2	NUM
ap-4699	418	3	]	]	PUNCT
ap-4699	418	4	p.	p.	NOUN
ap-4699	418	5	kornerup	kornerup	NOUN
ap-4699	418	6	.	.	PUNCT
ap-4699	419	1	necessary	necessary	ADJ
ap-4699	419	2	and	and	CCONJ
ap-4699	419	3	sufficient	sufficient	ADJ
ap-4699	419	4	conditions	condition	NOUN
ap-4699	419	5	for	for	ADP
ap-4699	419	6	parallel	parallel	ADJ
ap-4699	419	7	,	,	PUNCT
ap-4699	419	8	constant	constant	ADJ
ap-4699	419	9	time	time	NOUN
ap-4699	419	10	conversion	conversion	NOUN
ap-4699	419	11	and	and	CCONJ
ap-4699	419	12	addition	addition	NOUN
ap-4699	419	13	.	.	PUNCT
ap-4699	420	1	proc	proc	NOUN
ap-4699	420	2	14th	14th	PROPN
ap-4699	420	3	ieee	ieee	NOUN
ap-4699	420	4	symp	symp	NOUN
ap-4699	420	5	on	on	ADP
ap-4699	420	6	comp	comp	NOUN
ap-4699	420	7	arith	arith	NOUN
ap-4699	420	8	pp	pp	ADV
ap-4699	420	9	.	.	PUNCT
ap-4699	421	1	152–155	152–155	NUM
ap-4699	421	2	,	,	PUNCT
ap-4699	421	3	1999	1999	NUM
ap-4699	421	4	.	.	PUNCT
ap-4699	422	1	[	[	X
ap-4699	422	2	3	3	X
ap-4699	422	3	]	]	X
ap-4699	422	4	c.	c.	PROPN
ap-4699	422	5	frougny	frougny	PROPN
ap-4699	422	6	,	,	PUNCT
ap-4699	422	7	p.	p.	NOUN
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ap-4699	423	8	-	-	PUNCT
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ap-4699	423	14	-	-	ADJ
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ap-4699	423	17	systems	system	NOUN
ap-4699	423	18	.	.	PUNCT
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ap-4699	424	5	,	,	PUNCT
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ap-4699	424	7	.	.	PUNCT
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ap-4699	425	2	4	4	NUM
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ap-4699	425	6	,	,	PUNCT
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ap-4699	426	5	,	,	PUNCT
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ap-4699	426	7	.	.	PUNCT
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ap-4699	427	2	5	5	NUM
ap-4699	427	3	]	]	PUNCT
ap-4699	427	4	c.	c.	PROPN
ap-4699	427	5	frougny	frougny	PROPN
ap-4699	427	6	,	,	PUNCT
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ap-4699	427	9	,	,	PUNCT
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ap-4699	427	11	svobodová	svobodová	PROPN
ap-4699	427	12	.	.	PUNCT
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ap-4699	428	3	in	in	ADP
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ap-4699	428	5	-	-	ADJ
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ap-4699	428	9	.	.	PUNCT
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ap-4699	429	5	,	,	PUNCT
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ap-4699	429	7	.	.	PUNCT
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ap-4699	430	2	6	6	NUM
ap-4699	430	3	]	]	PUNCT
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ap-4699	430	6	,	,	PUNCT
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ap-4699	430	8	frougny	frougny	PROPN
ap-4699	430	9	,	,	PUNCT
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ap-4699	430	15	.	.	PUNCT
ap-4699	431	1	on	on	ADP
ap-4699	431	2	-	-	PUNCT
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ap-4699	432	6	.	.	PUNCT
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ap-4699	433	2	.	.	PUNCT
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ap-4699	434	2	.	.	PUNCT
ap-4699	435	1	(	(	PUNCT
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ap-4699	435	6	.	.	PUNCT
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ap-4699	436	2	.	.	X
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ap-4699	436	4	,	,	PUNCT
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ap-4699	436	6	.	.	PUNCT
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ap-4699	438	9	.	.	PUNCT
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ap-4699	439	7	,	,	PUNCT
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ap-4699	439	9	.	.	PUNCT
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ap-4699	441	2	8	8	NUM
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ap-4699	442	9	.	.	PUNCT
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ap-4699	443	5	,	,	PUNCT
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ap-4699	443	7	.	.	PUNCT
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ap-4699	444	6	-	-	PUNCT
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ap-4699	444	8	-	-	SYM
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ap-4699	447	8	.	.	PUNCT
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ap-4699	448	12	.	.	PUNCT
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ap-4699	463	16	.	.	PUNCT
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ap-4699	468	12	acta	acta	PROPN
ap-4699	468	13	polytechnica	polytechnica	PROPN
ap-4699	468	14	58(5):285–291	58(5):285–291	PROPN
ap-4699	468	15	,	,	PUNCT
ap-4699	468	16	2018	2018	NUM
ap-4699	468	17	1	1	NUM
ap-4699	468	18	introduction	introduction	NOUN
ap-4699	468	19	2	2	NUM
ap-4699	468	20	preliminaries	preliminary	NOUN
ap-4699	468	21	3	3	NUM
ap-4699	468	22	necessary	necessary	ADJ
ap-4699	468	23	conditions	condition	NOUN
ap-4699	468	24	on	on	ADP
ap-4699	468	25	alphabets	alphabet	NOUN
ap-4699	468	26	allowing	allow	VERB
ap-4699	468	27	parallel	parallel	ADJ
ap-4699	468	28	addition	addition	NOUN
ap-4699	468	29	in	in	ADP
ap-4699	468	30	a	a	DET
ap-4699	468	31	subset	subset	NOUN
ap-4699	468	32	z[beta	z[beta	NOUN
ap-4699	468	33	]	]	X
ap-4699	468	34	3.1	3.1	NUM
ap-4699	468	35	a[beta	a[beta	NOUN
ap-4699	468	36	]	]	PUNCT
ap-4699	468	37	closed	close	VERB
ap-4699	468	38	under	under	ADP
ap-4699	468	39	addition	addition	NOUN
ap-4699	468	40	3.2	3.2	NUM
ap-4699	468	41	lower	lower	ADV
ap-4699	468	42	bound	bind	VERB
ap-4699	468	43	on	on	ADP
ap-4699	468	44	a	a	DET
ap-4699	468	45	4	4	NUM
ap-4699	468	46	necessary	necessary	ADJ
ap-4699	468	47	and	and	CCONJ
ap-4699	468	48	sufficient	sufficient	ADJ
ap-4699	468	49	condition	condition	NOUN
ap-4699	468	50	on	on	ADP
ap-4699	468	51	bases	basis	NOUN
ap-4699	468	52	for	for	ADP
ap-4699	468	53	parallel	parallel	ADJ
ap-4699	468	54	addition	addition	NOUN
ap-4699	468	55	5	5	NUM
ap-4699	468	56	conclusion	conclusion	NOUN
ap-4699	468	57	6	6	NUM
ap-4699	468	58	acknowledgments	acknowledgment	NOUN
ap-4699	468	59	references	reference	NOUN