id	sid	tid	token	lemma	pos
ejpam-6321	1	1	european	european	PROPN
ejpam-6321	1	2	journal	journal	PROPN
ejpam-6321	1	3	of	of	ADP
ejpam-6321	1	4	pure	pure	ADJ
ejpam-6321	1	5	and	and	CCONJ
ejpam-6321	1	6	applied	applied	ADJ
ejpam-6321	1	7	mathematics	mathematic	NOUN
ejpam-6321	1	8	2025	2025	NUM
ejpam-6321	1	9	,	,	PUNCT
ejpam-6321	1	10	vol	vol	NOUN
ejpam-6321	1	11	.	.	PROPN
ejpam-6321	1	12	18	18	NUM
ejpam-6321	1	13	,	,	PUNCT
ejpam-6321	1	14	issue	issue	NOUN
ejpam-6321	1	15	3	3	NUM
ejpam-6321	1	16	,	,	PUNCT
ejpam-6321	1	17	article	article	NOUN
ejpam-6321	1	18	number	number	NOUN
ejpam-6321	1	19	6321	6321	NUM
ejpam-6321	1	20	issn	issn	PROPN
ejpam-6321	1	21	1307	1307	NUM
ejpam-6321	1	22	-	-	SYM
ejpam-6321	1	23	5543	5543	NUM
ejpam-6321	1	24	–	–	PUNCT
ejpam-6321	1	25	ejpam.com	ejpam.com	X
ejpam-6321	1	26	published	publish	VERB
ejpam-6321	1	27	by	by	ADP
ejpam-6321	1	28	new	new	PROPN
ejpam-6321	1	29	york	york	PROPN
ejpam-6321	1	30	business	business	PROPN
ejpam-6321	1	31	global	global	ADJ
ejpam-6321	1	32	common	common	ADJ
ejpam-6321	1	33	fixed	fix	VERB
ejpam-6321	1	34	points	point	NOUN
ejpam-6321	1	35	of	of	ADP
ejpam-6321	1	36	triplet	triplet	NOUN
ejpam-6321	1	37	mappings	mapping	NOUN
ejpam-6321	1	38	in	in	ADP
ejpam-6321	1	39	gm	gm	PROPN
ejpam-6321	1	40	-	-	PUNCT
ejpam-6321	1	41	spaces	space	NOUN
ejpam-6321	1	42	with	with	ADP
ejpam-6321	1	43	applications	application	NOUN
ejpam-6321	1	44	to	to	PART
ejpam-6321	1	45	ai	ai	VERB
ejpam-6321	1	46	convergence	convergence	NOUN
ejpam-6321	1	47	and	and	CCONJ
ejpam-6321	1	48	cryptographic	cryptographic	ADJ
ejpam-6321	1	49	consensus	consensus	NOUN
ejpam-6321	1	50	maha	maha	NOUN
ejpam-6321	1	51	noorwali1	noorwali1	PROPN
ejpam-6321	1	52	,	,	PUNCT
ejpam-6321	1	53	raed	raed	PROPN
ejpam-6321	1	54	hatamleh2	hatamleh2	PROPN
ejpam-6321	1	55	,	,	PUNCT
ejpam-6321	1	56	arif	arif	PROPN
ejpam-6321	1	57	mehmood3	mehmood3	PROPN
ejpam-6321	1	58	,	,	PUNCT
ejpam-6321	1	59	abdallah	abdallah	PROPN
ejpam-6321	1	60	al	al	PROPN
ejpam-6321	1	61	-	-	PUNCT
ejpam-6321	1	62	husban4	husban4	PROPN
ejpam-6321	1	63	,	,	PUNCT
ejpam-6321	1	64	khaled	khaled	PROPN
ejpam-6321	1	65	a.	a.	PROPN
ejpam-6321	1	66	aldwoah5	aldwoah5	PROPN
ejpam-6321	1	67	,	,	PUNCT
ejpam-6321	1	68	cris	cris	PROPN
ejpam-6321	1	69	l.	l.	PROPN
ejpam-6321	1	70	armada6,7	armada6,7	PROPN
ejpam-6321	1	71	,	,	PUNCT
ejpam-6321	1	72	jamil	jamil	PROPN
ejpam-6321	1	73	j.	j.	PROPN
ejpam-6321	1	74	hamja8	hamja8	PROPN
ejpam-6321	1	75	,	,	PUNCT
ejpam-6321	2	1	alaa	alaa	PROPN
ejpam-6321	2	2	m.	m.	PROPN
ejpam-6321	2	3	abd	abd	PROPN
ejpam-6321	2	4	el	el	PROPN
ejpam-6321	2	5	-	-	PROPN
ejpam-6321	2	6	latif9,∗	latif9,∗	PROPN
ejpam-6321	2	7	1	1	NUM
ejpam-6321	2	8	department	department	NOUN
ejpam-6321	2	9	of	of	ADP
ejpam-6321	2	10	mathematics	mathematic	NOUN
ejpam-6321	2	11	,	,	PUNCT
ejpam-6321	2	12	faculty	faculty	NOUN
ejpam-6321	2	13	of	of	ADP
ejpam-6321	2	14	sciences	science	NOUN
ejpam-6321	2	15	,	,	PUNCT
ejpam-6321	2	16	king	king	NOUN
ejpam-6321	2	17	abdulaziz	abdulaziz	PROPN
ejpam-6321	2	18	university	university	PROPN
ejpam-6321	2	19	,	,	PUNCT
ejpam-6321	2	20	p.o	p.o	PROPN
ejpam-6321	2	21	.	.	PROPN
ejpam-6321	2	22	box	box	PROPN
ejpam-6321	2	23	80203	80203	NUM
ejpam-6321	2	24	,	,	PUNCT
ejpam-6321	2	25	jeddah	jeddah	PROPN
ejpam-6321	2	26	21589	21589	NUM
ejpam-6321	2	27	,	,	PUNCT
ejpam-6321	2	28	15	15	NUM
ejpam-6321	2	29	saudi	saudi	PROPN
ejpam-6321	2	30	arabia	arabia	PROPN
ejpam-6321	2	31	2	2	NUM
ejpam-6321	2	32	department	department	NOUN
ejpam-6321	2	33	of	of	ADP
ejpam-6321	2	34	mathematics	mathematic	NOUN
ejpam-6321	2	35	,	,	PUNCT
ejpam-6321	2	36	faculty	faculty	NOUN
ejpam-6321	2	37	of	of	ADP
ejpam-6321	2	38	science	science	NOUN
ejpam-6321	2	39	,	,	PUNCT
ejpam-6321	2	40	jadara	jadara	PROPN
ejpam-6321	2	41	university	university	PROPN
ejpam-6321	2	42	,	,	PUNCT
ejpam-6321	2	43	p.o	p.o	PROPN
ejpam-6321	2	44	.	.	PROPN
ejpam-6321	2	45	box	box	PROPN
ejpam-6321	2	46	733	733	NUM
ejpam-6321	2	47	,	,	PUNCT
ejpam-6321	2	48	irbid	irbid	ADJ
ejpam-6321	2	49	21110	21110	NUM
ejpam-6321	2	50	,	,	PUNCT
ejpam-6321	2	51	jordan	jordan	PROPN
ejpam-6321	2	52	3	3	NUM
ejpam-6321	2	53	department	department	PROPN
ejpam-6321	2	54	of	of	ADP
ejpam-6321	2	55	mathematics	mathematics	PROPN
ejpam-6321	2	56	,	,	PUNCT
ejpam-6321	2	57	institute	institute	PROPN
ejpam-6321	2	58	of	of	ADP
ejpam-6321	2	59	numerical	numerical	PROPN
ejpam-6321	2	60	sciences	sciences	PROPN
ejpam-6321	2	61	,	,	PUNCT
ejpam-6321	2	62	gomal	gomal	ADJ
ejpam-6321	2	63	university	university	NOUN
ejpam-6321	2	64	,	,	PUNCT
ejpam-6321	2	65	dera	dera	PROPN
ejpam-6321	2	66	ismail	ismail	PROPN
ejpam-6321	2	67	khan	khan	PROPN
ejpam-6321	2	68	29050	29050	NUM
ejpam-6321	2	69	,	,	PUNCT
ejpam-6321	2	70	kpk	kpk	PROPN
ejpam-6321	2	71	,	,	PUNCT
ejpam-6321	2	72	pakistan	pakistan	PROPN
ejpam-6321	2	73	4	4	NUM
ejpam-6321	2	74	department	department	NOUN
ejpam-6321	2	75	of	of	ADP
ejpam-6321	2	76	mathematics	mathematic	NOUN
ejpam-6321	2	77	,	,	PUNCT
ejpam-6321	2	78	faculty	faculty	NOUN
ejpam-6321	2	79	of	of	ADP
ejpam-6321	2	80	science	science	NOUN
ejpam-6321	2	81	and	and	CCONJ
ejpam-6321	2	82	technology	technology	NOUN
ejpam-6321	2	83	,	,	PUNCT
ejpam-6321	2	84	irbid	irbid	VERB
ejpam-6321	2	85	national	national	ADJ
ejpam-6321	2	86	university	university	PROPN
ejpam-6321	2	87	,	,	PUNCT
ejpam-6321	2	88	p.o	p.o	PROPN
ejpam-6321	2	89	.	.	PROPN
ejpam-6321	2	90	box	box	PROPN
ejpam-6321	2	91	:	:	PUNCT
ejpam-6321	2	92	2600	2600	NUM
ejpam-6321	2	93	irbid	irbid	ADJ
ejpam-6321	2	94	,	,	PUNCT
ejpam-6321	2	95	jordan	jordan	PROPN
ejpam-6321	2	96	5	5	NUM
ejpam-6321	2	97	department	department	NOUN
ejpam-6321	2	98	of	of	ADP
ejpam-6321	2	99	mathematics	mathematic	NOUN
ejpam-6321	2	100	,	,	PUNCT
ejpam-6321	2	101	faculty	faculty	NOUN
ejpam-6321	2	102	of	of	ADP
ejpam-6321	2	103	science	science	NOUN
ejpam-6321	2	104	,	,	PUNCT
ejpam-6321	2	105	islamic	islamic	PROPN
ejpam-6321	2	106	university	university	PROPN
ejpam-6321	2	107	of	of	ADP
ejpam-6321	2	108	madinah	madinah	PROPN
ejpam-6321	2	109	,	,	PUNCT
ejpam-6321	2	110	saudia	saudia	PROPN
ejpam-6321	2	111	arabia	arabia	PROPN
ejpam-6321	2	112	6	6	NUM
ejpam-6321	2	113	vietnam	vietnam	PROPN
ejpam-6321	2	114	national	national	PROPN
ejpam-6321	2	115	university	university	PROPN
ejpam-6321	2	116	ho	ho	PROPN
ejpam-6321	2	117	chi	chi	PROPN
ejpam-6321	2	118	minh	minh	PROPN
ejpam-6321	2	119	city	city	PROPN
ejpam-6321	2	120	,	,	PUNCT
ejpam-6321	2	121	linh	linh	NOUN
ejpam-6321	2	122	trung	trung	VERB
ejpam-6321	2	123	ward	ward	NOUN
ejpam-6321	2	124	,	,	PUNCT
ejpam-6321	2	125	thu	thu	PROPN
ejpam-6321	2	126	duc	duc	PROPN
ejpam-6321	2	127	city	city	PROPN
ejpam-6321	2	128	,	,	PUNCT
ejpam-6321	2	129	ho	ho	PROPN
ejpam-6321	2	130	chi	chi	PROPN
ejpam-6321	2	131	minh	minh	PROPN
ejpam-6321	2	132	city	city	PROPN
ejpam-6321	2	133	,	,	PUNCT
ejpam-6321	2	134	vietnam	vietnam	PROPN
ejpam-6321	2	135	7	7	NUM
ejpam-6321	2	136	department	department	NOUN
ejpam-6321	2	137	of	of	ADP
ejpam-6321	2	138	applied	apply	VERB
ejpam-6321	2	139	mathematics	mathematic	NOUN
ejpam-6321	2	140	,	,	PUNCT
ejpam-6321	2	141	faculty	faculty	NOUN
ejpam-6321	2	142	of	of	ADP
ejpam-6321	2	143	applied	apply	VERB
ejpam-6321	2	144	science	science	NOUN
ejpam-6321	2	145	,	,	PUNCT
ejpam-6321	2	146	ho	ho	PROPN
ejpam-6321	2	147	chi	chi	PROPN
ejpam-6321	2	148	minh	minh	PROPN
ejpam-6321	2	149	city	city	PROPN
ejpam-6321	2	150	university	university	PROPN
ejpam-6321	2	151	of	of	ADP
ejpam-6321	2	152	technology	technology	NOUN
ejpam-6321	2	153	(	(	PUNCT
ejpam-6321	2	154	hcmut	hcmut	ADJ
ejpam-6321	2	155	)	)	PUNCT
ejpam-6321	2	156	,	,	PUNCT
ejpam-6321	2	157	268	268	NUM
ejpam-6321	2	158	ly	ly	ADP
ejpam-6321	2	159	thuong	thuong	NOUN
ejpam-6321	2	160	kiet	kiet	PROPN
ejpam-6321	2	161	,	,	PUNCT
ejpam-6321	2	162	ward	ward	VERB
ejpam-6321	2	163	14	14	NUM
ejpam-6321	2	164	,	,	PUNCT
ejpam-6321	2	165	district	district	NOUN
ejpam-6321	2	166	10	10	NUM
ejpam-6321	2	167	,	,	PUNCT
ejpam-6321	2	168	ho	ho	PROPN
ejpam-6321	2	169	chi	chi	PROPN
ejpam-6321	2	170	minh	minh	PROPN
ejpam-6321	2	171	city	city	PROPN
ejpam-6321	2	172	,	,	PUNCT
ejpam-6321	2	173	vietnam	vietnam	PROPN
ejpam-6321	2	174	8	8	NUM
ejpam-6321	2	175	mathematics	mathematic	NOUN
ejpam-6321	2	176	and	and	CCONJ
ejpam-6321	2	177	sciences	sciences	PROPN
ejpam-6321	2	178	department	department	PROPN
ejpam-6321	2	179	,	,	PUNCT
ejpam-6321	2	180	college	college	NOUN
ejpam-6321	2	181	of	of	ADP
ejpam-6321	2	182	arts	art	NOUN
ejpam-6321	2	183	and	and	CCONJ
ejpam-6321	2	184	sciences	science	NOUN
ejpam-6321	2	185	,	,	PUNCT
ejpam-6321	2	186	mindanao	mindanao	PROPN
ejpam-6321	2	187	state	state	PROPN
ejpam-6321	2	188	universitytawi	universitytawi	PROPN
ejpam-6321	2	189	-	-	PUNCT
ejpam-6321	2	190	tawi	tawi	NOUN
ejpam-6321	2	191	college	college	PROPN
ejpam-6321	2	192	of	of	ADP
ejpam-6321	2	193	technology	technology	NOUN
ejpam-6321	2	194	and	and	CCONJ
ejpam-6321	2	195	oceanography	oceanography	NOUN
ejpam-6321	2	196	,	,	PUNCT
ejpam-6321	2	197	7500	7500	NUM
ejpam-6321	2	198	philippine	philippine	ADJ
ejpam-6321	2	199	9	9	NUM
ejpam-6321	2	200	department	department	NOUN
ejpam-6321	2	201	of	of	ADP
ejpam-6321	2	202	mathematics	mathematics	PROPN
ejpam-6321	2	203	,	,	PUNCT
ejpam-6321	2	204	college	college	NOUN
ejpam-6321	2	205	of	of	ADP
ejpam-6321	2	206	science	science	NOUN
ejpam-6321	2	207	,	,	PUNCT
ejpam-6321	2	208	northern	northern	ADJ
ejpam-6321	2	209	border	border	NOUN
ejpam-6321	2	210	university	university	NOUN
ejpam-6321	2	211	,	,	PUNCT
ejpam-6321	2	212	arar	arar	NOUN
ejpam-6321	2	213	91431	91431	NUM
ejpam-6321	2	214	,	,	PUNCT
ejpam-6321	2	215	saudi	saudi	PROPN
ejpam-6321	2	216	arabia	arabia	PROPN
ejpam-6321	2	217	abstract	abstract	NOUN
ejpam-6321	2	218	.	.	PUNCT
ejpam-6321	3	1	this	this	DET
ejpam-6321	3	2	paper	paper	NOUN
ejpam-6321	3	3	investigates	investigate	VERB
ejpam-6321	3	4	the	the	DET
ejpam-6321	3	5	existence	existence	NOUN
ejpam-6321	3	6	and	and	CCONJ
ejpam-6321	3	7	uniqueness	uniqueness	NOUN
ejpam-6321	3	8	of	of	ADP
ejpam-6321	3	9	common	common	ADJ
ejpam-6321	3	10	fixed	fix	VERB
ejpam-6321	3	11	points	point	NOUN
ejpam-6321	3	12	for	for	ADP
ejpam-6321	3	13	three	three	NUM
ejpam-6321	3	14	selfmappings	selfmapping	NOUN
ejpam-6321	3	15	in	in	ADP
ejpam-6321	3	16	generalized	generalized	ADJ
ejpam-6321	3	17	metric	metric	ADJ
ejpam-6321	3	18	spaces	space	NOUN
ejpam-6321	3	19	(	(	PUNCT
ejpam-6321	3	20	gm	gm	NOUN
ejpam-6321	3	21	-	-	PUNCT
ejpam-6321	3	22	spaces	space	NOUN
ejpam-6321	3	23	)	)	PUNCT
ejpam-6321	3	24	,	,	PUNCT
ejpam-6321	3	25	establishing	establish	VERB
ejpam-6321	3	26	new	new	ADJ
ejpam-6321	3	27	results	result	NOUN
ejpam-6321	3	28	under	under	ADP
ejpam-6321	3	29	generalized	generalized	ADJ
ejpam-6321	3	30	contractive	contractive	ADJ
ejpam-6321	3	31	conditions	condition	NOUN
ejpam-6321	3	32	.	.	PUNCT
ejpam-6321	4	1	we	we	PRON
ejpam-6321	4	2	develop	develop	VERB
ejpam-6321	4	3	a	a	DET
ejpam-6321	4	4	comprehensive	comprehensive	ADJ
ejpam-6321	4	5	theoretical	theoretical	ADJ
ejpam-6321	4	6	framework	framework	NOUN
ejpam-6321	4	7	where	where	SCONJ
ejpam-6321	4	8	tripartite	tripartite	ADJ
ejpam-6321	4	9	mappings	mapping	NOUN
ejpam-6321	4	10	satisfy	satisfy	NOUN
ejpam-6321	4	11	inequalities	inequality	NOUN
ejpam-6321	4	12	involving	involve	VERB
ejpam-6321	4	13	combinations	combination	NOUN
ejpam-6321	4	14	of	of	ADP
ejpam-6321	4	15	distance	distance	NOUN
ejpam-6321	4	16	-	-	PUNCT
ejpam-6321	4	17	like	like	ADJ
ejpam-6321	4	18	terms	term	NOUN
ejpam-6321	4	19	formulated	formulate	VERB
ejpam-6321	4	20	through	through	ADP
ejpam-6321	4	21	minimum	minimum	ADJ
ejpam-6321	4	22	and	and	CCONJ
ejpam-6321	4	23	maximum	maximum	ADJ
ejpam-6321	4	24	comparisons	comparison	NOUN
ejpam-6321	4	25	.	.	PUNCT
ejpam-6321	5	1	the	the	DET
ejpam-6321	5	2	contractive	contractive	ADJ
ejpam-6321	5	3	conditions	condition	NOUN
ejpam-6321	5	4	are	be	AUX
ejpam-6321	5	5	governed	govern	VERB
ejpam-6321	5	6	by	by	ADP
ejpam-6321	5	7	carefully	carefully	ADV
ejpam-6321	5	8	chosen	choose	VERB
ejpam-6321	5	9	parameters	parameter	NOUN
ejpam-6321	5	10	that	that	PRON
ejpam-6321	5	11	determine	determine	VERB
ejpam-6321	5	12	whether	whether	SCONJ
ejpam-6321	5	13	the	the	DET
ejpam-6321	5	14	mappings	mapping	NOUN
ejpam-6321	5	15	admit	admit	VERB
ejpam-6321	5	16	a	a	DET
ejpam-6321	5	17	common	common	ADJ
ejpam-6321	5	18	fixed	fix	VERB
ejpam-6321	5	19	point	point	NOUN
ejpam-6321	5	20	or	or	CCONJ
ejpam-6321	5	21	a	a	DET
ejpam-6321	5	22	unique	unique	ADJ
ejpam-6321	5	23	common	common	ADJ
ejpam-6321	5	24	fixed	fix	VERB
ejpam-6321	5	25	point	point	NOUN
ejpam-6321	5	26	.	.	PUNCT
ejpam-6321	6	1	to	to	PART
ejpam-6321	6	2	demonstrate	demonstrate	VERB
ejpam-6321	6	3	the	the	DET
ejpam-6321	6	4	practical	practical	ADJ
ejpam-6321	6	5	applicability	applicability	NOUN
ejpam-6321	6	6	of	of	ADP
ejpam-6321	6	7	our	our	PRON
ejpam-6321	6	8	theory	theory	NOUN
ejpam-6321	6	9	,	,	PUNCT
ejpam-6321	6	10	we	we	PRON
ejpam-6321	6	11	present	present	VERB
ejpam-6321	6	12	a	a	DET
ejpam-6321	6	13	concrete	concrete	ADJ
ejpam-6321	6	14	example	example	NOUN
ejpam-6321	6	15	using	use	VERB
ejpam-6321	6	16	the	the	DET
ejpam-6321	6	17	interval	interval	NOUN
ejpam-6321	6	18	[	[	X
ejpam-6321	6	19	0	0	NUM
ejpam-6321	6	20	,	,	PUNCT
ejpam-6321	6	21	1	1	NUM
ejpam-6321	6	22	]	]	PUNCT
ejpam-6321	6	23	equipped	equip	VERB
ejpam-6321	6	24	with	with	ADP
ejpam-6321	6	25	the	the	DET
ejpam-6321	6	26	g	g	NOUN
ejpam-6321	6	27	-	-	PUNCT
ejpam-6321	6	28	metric	metric	ADJ
ejpam-6321	6	29	g(x	g(x	NOUN
ejpam-6321	6	30	,	,	PUNCT
ejpam-6321	6	31	y	y	PROPN
ejpam-6321	6	32	,	,	PUNCT
ejpam-6321	6	33	z	z	NOUN
ejpam-6321	6	34	)	)	PUNCT
ejpam-6321	6	35	=	=	SYM
ejpam-6321	6	36	max{|x	max{|x	PROPN
ejpam-6321	6	37	−	−	PROPN
ejpam-6321	6	38	y|	y|	NOUN
ejpam-6321	6	39	,	,	PUNCT
ejpam-6321	6	40	|y	|y	VERB
ejpam-6321	6	41	−	−	PROPN
ejpam-6321	6	42	z|	z|	PROPN
ejpam-6321	6	43	,	,	PUNCT
ejpam-6321	6	44	|z	|z	PROPN
ejpam-6321	6	45	−	−	PROPN
ejpam-6321	6	46	x|	x|	PROPN
ejpam-6321	6	47	}	}	PUNCT
ejpam-6321	6	48	,	,	PUNCT
ejpam-6321	6	49	where	where	SCONJ
ejpam-6321	6	50	piecewise	piecewise	NOUN
ejpam-6321	6	51	-	-	PUNCT
ejpam-6321	6	52	defined	define	VERB
ejpam-6321	6	53	self	self	NOUN
ejpam-6321	6	54	-	-	PUNCT
ejpam-6321	6	55	mappings	mapping	NOUN
ejpam-6321	6	56	are	be	AUX
ejpam-6321	6	57	shown	show	VERB
ejpam-6321	6	58	to	to	PART
ejpam-6321	6	59	satisfy	satisfy	VERB
ejpam-6321	6	60	all	all	DET
ejpam-6321	6	61	required	require	VERB
ejpam-6321	6	62	conditions	condition	NOUN
ejpam-6321	6	63	and	and	CCONJ
ejpam-6321	6	64	converge	converge	VERB
ejpam-6321	6	65	to	to	ADP
ejpam-6321	6	66	a	a	DET
ejpam-6321	6	67	unique	unique	ADJ
ejpam-6321	6	68	common	common	ADJ
ejpam-6321	6	69	fixed	fix	VERB
ejpam-6321	6	70	point	point	NOUN
ejpam-6321	6	71	.	.	PUNCT
ejpam-6321	7	1	beyond	beyond	ADP
ejpam-6321	7	2	theoretical	theoretical	ADJ
ejpam-6321	7	3	advancements	advancement	NOUN
ejpam-6321	7	4	,	,	PUNCT
ejpam-6321	7	5	our	our	PRON
ejpam-6321	7	6	results	result	NOUN
ejpam-6321	7	7	offer	offer	VERB
ejpam-6321	7	8	significant	significant	ADJ
ejpam-6321	7	9	applications	application	NOUN
ejpam-6321	7	10	in	in	ADP
ejpam-6321	7	11	artificial	artificial	ADJ
ejpam-6321	7	12	intelligence	intelligence	NOUN
ejpam-6321	7	13	,	,	PUNCT
ejpam-6321	7	14	particularly	particularly	ADV
ejpam-6321	7	15	in	in	ADP
ejpam-6321	7	16	analyzing	analyze	VERB
ejpam-6321	7	17	convergence	convergence	NOUN
ejpam-6321	7	18	of	of	ADP
ejpam-6321	7	19	multi	multi	ADJ
ejpam-6321	7	20	-	-	ADJ
ejpam-6321	7	21	layer	layer	ADJ
ejpam-6321	7	22	neural	neural	ADJ
ejpam-6321	7	23	architectures	architecture	NOUN
ejpam-6321	7	24	,	,	PUNCT
ejpam-6321	7	25	and	and	CCONJ
ejpam-6321	7	26	in	in	ADP
ejpam-6321	7	27	cryptography	cryptography	NOUN
ejpam-6321	7	28	for	for	ADP
ejpam-6321	7	29	designing	design	VERB
ejpam-6321	7	30	secure	secure	ADJ
ejpam-6321	7	31	iterative	iterative	NOUN
ejpam-6321	7	32	protocols	protocol	NOUN
ejpam-6321	7	33	.	.	PUNCT
ejpam-6321	8	1	the	the	DET
ejpam-6321	8	2	framework	framework	NOUN
ejpam-6321	8	3	presented	present	VERB
ejpam-6321	8	4	here	here	ADV
ejpam-6321	8	5	not	not	PART
ejpam-6321	8	6	only	only	ADV
ejpam-6321	8	7	generalizes	generalize	VERB
ejpam-6321	8	8	existing	exist	VERB
ejpam-6321	8	9	fixed	fix	VERB
ejpam-6321	8	10	-	-	PUNCT
ejpam-6321	8	11	point	point	NOUN
ejpam-6321	8	12	theorems	theorem	NOUN
ejpam-6321	8	13	but	but	CCONJ
ejpam-6321	8	14	also	also	ADV
ejpam-6321	8	15	provides	provide	VERB
ejpam-6321	8	16	verifiable	verifiable	ADJ
ejpam-6321	8	17	computational	computational	ADJ
ejpam-6321	8	18	methods	method	NOUN
ejpam-6321	8	19	for	for	ADP
ejpam-6321	8	20	stability	stability	NOUN
ejpam-6321	8	21	analysis	analysis	NOUN
ejpam-6321	8	22	in	in	ADP
ejpam-6321	8	23	both	both	CCONJ
ejpam-6321	8	24	mathematical	mathematical	ADJ
ejpam-6321	8	25	and	and	CCONJ
ejpam-6321	8	26	applied	applied	ADJ
ejpam-6321	8	27	contexts	context	NOUN
ejpam-6321	8	28	.	.	PUNCT
ejpam-6321	9	1	2020	2020	NUM
ejpam-6321	9	2	mathematics	mathematic	NOUN
ejpam-6321	9	3	subject	subject	NOUN
ejpam-6321	9	4	classifications	classification	NOUN
ejpam-6321	9	5	:	:	PUNCT
ejpam-6321	9	6	54a05	54a05	NUM
ejpam-6321	9	7	key	key	ADJ
ejpam-6321	9	8	words	word	NOUN
ejpam-6321	9	9	and	and	CCONJ
ejpam-6321	9	10	phrases	phrase	NOUN
ejpam-6321	9	11	:	:	PUNCT
ejpam-6321	9	12	contraction	contraction	NOUN
ejpam-6321	9	13	conditions	condition	NOUN
ejpam-6321	9	14	,	,	PUNCT
ejpam-6321	9	15	common	common	ADJ
ejpam-6321	9	16	fixed	fix	VERB
ejpam-6321	9	17	point	point	NOUN
ejpam-6321	9	18	,	,	PUNCT
ejpam-6321	9	19	gm	gm	PROPN
ejpam-6321	9	20	-	-	PUNCT
ejpam-6321	9	21	space	space	NOUN
ejpam-6321	9	22	,	,	PUNCT
ejpam-6321	9	23	artificial	artificial	ADJ
ejpam-6321	9	24	intelligence	intelligence	NOUN
ejpam-6321	9	25	,	,	PUNCT
ejpam-6321	9	26	cryptographic	cryptographic	ADJ
ejpam-6321	9	27	protocols	protocol	NOUN
ejpam-6321	9	28	,	,	PUNCT
ejpam-6321	9	29	g	g	NOUN
ejpam-6321	9	30	-	-	PUNCT
ejpam-6321	9	31	metric	metric	ADJ
ejpam-6321	9	32	convergence	convergence	NOUN
ejpam-6321	9	33	∗corresponding	∗corresponde	VERB
ejpam-6321	9	34	author	author	NOUN
ejpam-6321	9	35	.	.	PUNCT
ejpam-6321	10	1	doi	doi	NOUN
ejpam-6321	10	2	:	:	PUNCT
ejpam-6321	10	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6321	https://doi.org/10.29020/nybg.ejpam.v18i3.6321	ADJ
ejpam-6321	10	4	email	email	NOUN
ejpam-6321	10	5	addresses	address	VERB
ejpam-6321	10	6	:	:	PUNCT
ejpam-6321	10	7	(	(	PUNCT
ejpam-6321	10	8	mnorwali@kau.edu.sa	mnorwali@kau.edu.sa	NOUN
ejpam-6321	10	9	)	)	PUNCT
ejpam-6321	10	10	(	(	PUNCT
ejpam-6321	10	11	m.	m.	NOUN
ejpam-6321	10	12	noorwali	noorwali	PROPN
ejpam-6321	10	13	)	)	PUNCT
ejpam-6321	10	14	,	,	PUNCT
ejpam-6321	10	15	(	(	PUNCT
ejpam-6321	10	16	raed@jadara.edu.jo	raed@jadara.edu.jo	NOUN
ejpam-6321	10	17	)	)	PUNCT
ejpam-6321	10	18	(	(	PUNCT
ejpam-6321	10	19	r.	r.	PROPN
ejpam-6321	10	20	hatamleh	hatamleh	PROPN
ejpam-6321	10	21	)	)	PUNCT
ejpam-6321	10	22	,	,	PUNCT
ejpam-6321	10	23	(	(	PUNCT
ejpam-6321	10	24	mehdaniyal@gmail.com	mehdaniyal@gmail.com	X
ejpam-6321	10	25	)	)	PUNCT
ejpam-6321	10	26	(	(	PUNCT
ejpam-6321	10	27	a.	a.	PROPN
ejpam-6321	10	28	mehmood	mehmood	PROPN
ejpam-6321	10	29	)	)	PUNCT
ejpam-6321	10	30	,	,	PUNCT
ejpam-6321	10	31	(	(	PUNCT
ejpam-6321	10	32	dralhosban@inu.edu.jo	dralhosban@inu.edu.jo	NOUN
ejpam-6321	10	33	)	)	PUNCT
ejpam-6321	10	34	(	(	PUNCT
ejpam-6321	10	35	a.	a.	NOUN
ejpam-6321	10	36	alhusban	alhusban	PROPN
ejpam-6321	10	37	)	)	PUNCT
ejpam-6321	10	38	,	,	PUNCT
ejpam-6321	10	39	(	(	PUNCT
ejpam-6321	10	40	aldwoah@yahoo.com	aldwoah@yahoo.com	X
ejpam-6321	10	41	)	)	PUNCT
ejpam-6321	10	42	(	(	PUNCT
ejpam-6321	10	43	k.	k.	PROPN
ejpam-6321	10	44	a.	a.	PROPN
ejpam-6321	10	45	aldwoah	aldwoah	PROPN
ejpam-6321	10	46	)	)	PUNCT
ejpam-6321	10	47	,	,	PUNCT
ejpam-6321	10	48	(	(	PUNCT
ejpam-6321	10	49	cris.armada@hcmut.edu.vn	cris.armada@hcmut.edu.vn	NOUN
ejpam-6321	10	50	)	)	PUNCT
ejpam-6321	10	51	(	(	PUNCT
ejpam-6321	10	52	c.	c.	PROPN
ejpam-6321	10	53	l.	l.	PROPN
ejpam-6321	10	54	armada	armada	PROPN
ejpam-6321	10	55	)	)	PUNCT
ejpam-6321	10	56	,	,	PUNCT
ejpam-6321	10	57	(	(	PUNCT
ejpam-6321	10	58	jamilhamja@msutawi-tawi.edu.ph	jamilhamja@msutawi-tawi.edu.ph	PROPN
ejpam-6321	10	59	)	)	PUNCT
ejpam-6321	10	60	(	(	PUNCT
ejpam-6321	10	61	j.	j.	PROPN
ejpam-6321	10	62	j.	j.	PROPN
ejpam-6321	10	63	hamja	hamja	PROPN
ejpam-6321	10	64	)	)	PUNCT
ejpam-6321	10	65	,	,	PUNCT
ejpam-6321	10	66	(	(	PUNCT
ejpam-6321	10	67	alaa.ali@nbu.edu.sa	alaa.ali@nbu.edu.sa	PROPN
ejpam-6321	10	68	)	)	PUNCT
ejpam-6321	10	69	(	(	PUNCT
ejpam-6321	10	70	a.	a.	NOUN
ejpam-6321	10	71	m.	m.	PROPN
ejpam-6321	10	72	abd	abd	PROPN
ejpam-6321	10	73	el	el	PROPN
ejpam-6321	10	74	-	-	PROPN
ejpam-6321	10	75	latif	latif	PROPN
ejpam-6321	10	76	)	)	PUNCT
ejpam-6321	10	77	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6321	11	1	1	1	NUM
ejpam-6321	11	2	copyright	copyright	NOUN
ejpam-6321	11	3	:	:	PUNCT
ejpam-6321	11	4	©	©	PROPN
ejpam-6321	11	5	2025	2025	NUM
ejpam-6321	11	6	the	the	DET
ejpam-6321	11	7	author(s	author(s	NOUN
ejpam-6321	11	8	)	)	PUNCT
ejpam-6321	11	9	.	.	PUNCT
ejpam-6321	12	1	(	(	PUNCT
ejpam-6321	12	2	cc	cc	NOUN
ejpam-6321	12	3	by	by	ADP
ejpam-6321	12	4	-	-	PUNCT
ejpam-6321	12	5	nc	nc	PROPN
ejpam-6321	12	6	4.0	4.0	NUM
ejpam-6321	12	7	)	)	PUNCT
ejpam-6321	12	8	m.	m.	NOUN
ejpam-6321	12	9	noorwali	noorwali	PROPN
ejpam-6321	12	10	et	et	PROPN
ejpam-6321	12	11	al	al	PROPN
ejpam-6321	12	12	.	.	PUNCT
ejpam-6321	12	13	/	/	SYM
ejpam-6321	12	14	eur	eur	PROPN
ejpam-6321	12	15	.	.	PUNCT
ejpam-6321	13	1	j.	j.	PROPN
ejpam-6321	13	2	pure	pure	PROPN
ejpam-6321	13	3	appl	appl	PROPN
ejpam-6321	13	4	.	.	PROPN
ejpam-6321	13	5	math	math	PROPN
ejpam-6321	13	6	,	,	PUNCT
ejpam-6321	13	7	18	18	NUM
ejpam-6321	13	8	(	(	PUNCT
ejpam-6321	13	9	3	3	NUM
ejpam-6321	13	10	)	)	PUNCT
ejpam-6321	13	11	(	(	PUNCT
ejpam-6321	13	12	2025	2025	NUM
ejpam-6321	13	13	)	)	PUNCT
ejpam-6321	13	14	,	,	PUNCT
ejpam-6321	13	15	6321	6321	NUM
ejpam-6321	13	16	2	2	NUM
ejpam-6321	13	17	of	of	ADP
ejpam-6321	13	18	27	27	NUM
ejpam-6321	13	19	1	1	NUM
ejpam-6321	13	20	.	.	PUNCT
ejpam-6321	14	1	introduction	introduction	NOUN
ejpam-6321	14	2	fixed	fix	VERB
ejpam-6321	14	3	point	point	NOUN
ejpam-6321	14	4	theory	theory	NOUN
ejpam-6321	14	5	(	(	PUNCT
ejpam-6321	14	6	fpt	fpt	PROPN
ejpam-6321	14	7	)	)	PUNCT
ejpam-6321	14	8	is	be	AUX
ejpam-6321	14	9	a	a	DET
ejpam-6321	14	10	branch	branch	NOUN
ejpam-6321	14	11	of	of	ADP
ejpam-6321	14	12	mathematics	mathematic	NOUN
ejpam-6321	14	13	that	that	PRON
ejpam-6321	14	14	deals	deal	VERB
ejpam-6321	14	15	with	with	ADP
ejpam-6321	14	16	the	the	DET
ejpam-6321	14	17	existence	existence	NOUN
ejpam-6321	14	18	and	and	CCONJ
ejpam-6321	14	19	uniqueness	uniqueness	NOUN
ejpam-6321	14	20	of	of	ADP
ejpam-6321	14	21	fixed	fix	VERB
ejpam-6321	14	22	points	point	NOUN
ejpam-6321	14	23	for	for	ADP
ejpam-6321	14	24	various	various	ADJ
ejpam-6321	14	25	types	type	NOUN
ejpam-6321	14	26	of	of	ADP
ejpam-6321	14	27	mappings	mapping	NOUN
ejpam-6321	14	28	or	or	CCONJ
ejpam-6321	14	29	functions	function	NOUN
ejpam-6321	14	30	.	.	PUNCT
ejpam-6321	15	1	fpt	fpt	PROPN
ejpam-6321	15	2	plays	play	VERB
ejpam-6321	15	3	a	a	DET
ejpam-6321	15	4	crucial	crucial	ADJ
ejpam-6321	15	5	role	role	NOUN
ejpam-6321	15	6	in	in	ADP
ejpam-6321	15	7	various	various	ADJ
ejpam-6321	15	8	branches	branch	NOUN
ejpam-6321	15	9	of	of	ADP
ejpam-6321	15	10	mathematics	mathematic	NOUN
ejpam-6321	15	11	,	,	PUNCT
ejpam-6321	15	12	including	include	VERB
ejpam-6321	15	13	analysis	analysis	NOUN
ejpam-6321	15	14	,	,	PUNCT
ejpam-6321	15	15	topology	topology	NOUN
ejpam-6321	15	16	,	,	PUNCT
ejpam-6321	15	17	and	and	CCONJ
ejpam-6321	15	18	functional	functional	ADJ
ejpam-6321	15	19	analysis	analysis	NOUN
ejpam-6321	15	20	.	.	PUNCT
ejpam-6321	16	1	it	it	PRON
ejpam-6321	16	2	has	have	VERB
ejpam-6321	16	3	numerous	numerous	ADJ
ejpam-6321	16	4	applications	application	NOUN
ejpam-6321	16	5	in	in	ADP
ejpam-6321	16	6	diverse	diverse	ADJ
ejpam-6321	16	7	fields	field	NOUN
ejpam-6321	16	8	such	such	ADJ
ejpam-6321	16	9	as	as	ADP
ejpam-6321	16	10	economics	economic	NOUN
ejpam-6321	16	11	,	,	PUNCT
ejpam-6321	16	12	physics	physics	NOUN
ejpam-6321	16	13	,	,	PUNCT
ejpam-6321	16	14	engineering	engineering	NOUN
ejpam-6321	16	15	,	,	PUNCT
ejpam-6321	16	16	computer	computer	NOUN
ejpam-6321	16	17	science	science	NOUN
ejpam-6321	16	18	,	,	PUNCT
ejpam-6321	16	19	and	and	CCONJ
ejpam-6321	16	20	more	more	ADV
ejpam-6321	16	21	recently	recently	ADV
ejpam-6321	16	22	,	,	PUNCT
ejpam-6321	16	23	artificial	artificial	ADJ
ejpam-6321	16	24	intelligence	intelligence	NOUN
ejpam-6321	16	25	(	(	PUNCT
ejpam-6321	16	26	ai	ai	NOUN
ejpam-6321	16	27	)	)	PUNCT
ejpam-6321	16	28	and	and	CCONJ
ejpam-6321	16	29	cryptography	cryptography	NOUN
ejpam-6321	16	30	.	.	PUNCT
ejpam-6321	17	1	in	in	ADP
ejpam-6321	17	2	ai	ai	PROPN
ejpam-6321	17	3	,	,	PUNCT
ejpam-6321	17	4	fpt	fpt	PROPN
ejpam-6321	17	5	supports	support	VERB
ejpam-6321	17	6	the	the	DET
ejpam-6321	17	7	theoretical	theoretical	ADJ
ejpam-6321	17	8	convergence	convergence	NOUN
ejpam-6321	17	9	of	of	ADP
ejpam-6321	17	10	learning	learn	VERB
ejpam-6321	17	11	dynamics	dynamic	NOUN
ejpam-6321	17	12	in	in	ADP
ejpam-6321	17	13	deep	deep	ADJ
ejpam-6321	17	14	neural	neural	ADJ
ejpam-6321	17	15	networks	network	NOUN
ejpam-6321	17	16	,	,	PUNCT
ejpam-6321	17	17	especially	especially	ADV
ejpam-6321	17	18	within	within	ADP
ejpam-6321	17	19	recurrent	recurrent	ADJ
ejpam-6321	17	20	and	and	CCONJ
ejpam-6321	17	21	iterative	iterative	NOUN
ejpam-6321	17	22	architectures	architecture	NOUN
ejpam-6321	17	23	.	.	PUNCT
ejpam-6321	18	1	in	in	ADP
ejpam-6321	18	2	cryptographic	cryptographic	ADJ
ejpam-6321	18	3	systems	system	NOUN
ejpam-6321	18	4	,	,	PUNCT
ejpam-6321	18	5	fixed	fix	VERB
ejpam-6321	18	6	point	point	NOUN
ejpam-6321	18	7	theorems	theorem	NOUN
ejpam-6321	18	8	ensure	ensure	VERB
ejpam-6321	18	9	consistency	consistency	NOUN
ejpam-6321	18	10	and	and	CCONJ
ejpam-6321	18	11	convergence	convergence	NOUN
ejpam-6321	18	12	in	in	ADP
ejpam-6321	18	13	multi	multi	ADJ
ejpam-6321	18	14	-	-	ADJ
ejpam-6321	18	15	party	party	ADJ
ejpam-6321	18	16	secure	secure	ADJ
ejpam-6321	18	17	computations	computation	NOUN
ejpam-6321	18	18	and	and	CCONJ
ejpam-6321	18	19	iterative	iterative	NOUN
ejpam-6321	18	20	encryption	encryption	NOUN
ejpam-6321	18	21	/	/	SYM
ejpam-6321	18	22	decryption	decryption	NOUN
ejpam-6321	18	23	protocols	protocol	NOUN
ejpam-6321	18	24	.	.	PUNCT
ejpam-6321	19	1	within	within	ADP
ejpam-6321	19	2	the	the	DET
ejpam-6321	19	3	context	context	NOUN
ejpam-6321	19	4	of	of	ADP
ejpam-6321	19	5	fpt	fpt	PROPN
ejpam-6321	19	6	,	,	PUNCT
ejpam-6321	19	7	stefan	stefan	PROPN
ejpam-6321	19	8	banach	banach	ADV
ejpam-6321	19	9	introduced	introduce	VERB
ejpam-6321	19	10	the	the	DET
ejpam-6321	19	11	“	"	PUNCT
ejpam-6321	19	12	banach	banach	NOUN
ejpam-6321	19	13	contraction	contraction	NOUN
ejpam-6321	19	14	principle	principle	NOUN
ejpam-6321	19	15	(	(	PUNCT
ejpam-6321	19	16	bcp	bcp	PROPN
ejpam-6321	19	17	)	)	PUNCT
ejpam-6321	19	18	”	"	PUNCT
ejpam-6321	19	19	,	,	PUNCT
ejpam-6321	19	20	which	which	PRON
ejpam-6321	19	21	states	state	VERB
ejpam-6321	19	22	that	that	SCONJ
ejpam-6321	19	23	a	a	DET
ejpam-6321	19	24	single	single	ADV
ejpam-6321	19	25	-	-	PUNCT
ejpam-6321	19	26	valued	value	VERB
ejpam-6321	19	27	contractive	contractive	ADJ
ejpam-6321	19	28	-	-	PUNCT
ejpam-6321	19	29	type	type	NOUN
ejpam-6321	19	30	mapping	mapping	NOUN
ejpam-6321	19	31	in	in	ADP
ejpam-6321	19	32	a	a	DET
ejpam-6321	19	33	complete	complete	ADJ
ejpam-6321	19	34	metric	metric	ADJ
ejpam-6321	19	35	space	space	NOUN
ejpam-6321	19	36	has	have	VERB
ejpam-6321	19	37	a	a	DET
ejpam-6321	19	38	unique	unique	ADJ
ejpam-6321	19	39	fixed	fix	VERB
ejpam-6321	19	40	point	point	NOUN
ejpam-6321	19	41	.	.	PUNCT
ejpam-6321	20	1	our	our	PRON
ejpam-6321	20	2	objective	objective	NOUN
ejpam-6321	20	3	is	be	AUX
ejpam-6321	20	4	to	to	PART
ejpam-6321	20	5	extend	extend	VERB
ejpam-6321	20	6	this	this	DET
ejpam-6321	20	7	principle	principle	NOUN
ejpam-6321	20	8	to	to	ADP
ejpam-6321	20	9	gm	gm	PROPN
ejpam-6321	20	10	-	-	PUNCT
ejpam-6321	20	11	spaces	space	NOUN
ejpam-6321	20	12	,	,	PUNCT
ejpam-6321	20	13	which	which	PRON
ejpam-6321	20	14	offer	offer	VERB
ejpam-6321	20	15	a	a	DET
ejpam-6321	20	16	broader	broad	ADJ
ejpam-6321	20	17	framework	framework	NOUN
ejpam-6321	20	18	.	.	PUNCT
ejpam-6321	21	1	our	our	PRON
ejpam-6321	21	2	approach	approach	NOUN
ejpam-6321	21	3	involves	involve	VERB
ejpam-6321	21	4	examining	examine	VERB
ejpam-6321	21	5	existing	exist	VERB
ejpam-6321	21	6	research	research	NOUN
ejpam-6321	21	7	articles	article	NOUN
ejpam-6321	21	8	in	in	ADP
ejpam-6321	21	9	the	the	DET
ejpam-6321	21	10	gmspace	gmspace	NOUN
ejpam-6321	21	11	domain	domain	NOUN
ejpam-6321	21	12	to	to	PART
ejpam-6321	21	13	identify	identify	VERB
ejpam-6321	21	14	any	any	DET
ejpam-6321	21	15	shortcomings	shortcoming	NOUN
ejpam-6321	21	16	in	in	ADP
ejpam-6321	21	17	current	current	ADJ
ejpam-6321	21	18	results	result	NOUN
ejpam-6321	21	19	.	.	PUNCT
ejpam-6321	22	1	by	by	ADP
ejpam-6321	22	2	accurately	accurately	ADV
ejpam-6321	22	3	pinpointing	pinpoint	VERB
ejpam-6321	22	4	these	these	DET
ejpam-6321	22	5	issues	issue	NOUN
ejpam-6321	22	6	,	,	PUNCT
ejpam-6321	22	7	we	we	PRON
ejpam-6321	22	8	aim	aim	VERB
ejpam-6321	22	9	to	to	PART
ejpam-6321	22	10	establish	establish	VERB
ejpam-6321	22	11	a	a	DET
ejpam-6321	22	12	more	more	ADV
ejpam-6321	22	13	comprehensive	comprehensive	ADJ
ejpam-6321	22	14	solution	solution	NOUN
ejpam-6321	22	15	.	.	PUNCT
ejpam-6321	23	1	ultimately	ultimately	ADV
ejpam-6321	23	2	,	,	PUNCT
ejpam-6321	23	3	our	our	PRON
ejpam-6321	23	4	goal	goal	NOUN
ejpam-6321	23	5	is	be	AUX
ejpam-6321	23	6	to	to	PART
ejpam-6321	23	7	refine	refine	VERB
ejpam-6321	23	8	and	and	CCONJ
ejpam-6321	23	9	strengthen	strengthen	VERB
ejpam-6321	23	10	the	the	DET
ejpam-6321	23	11	theoretical	theoretical	ADJ
ejpam-6321	23	12	foundation	foundation	NOUN
ejpam-6321	23	13	of	of	ADP
ejpam-6321	23	14	gm	gm	PROPN
ejpam-6321	23	15	-	-	PUNCT
ejpam-6321	23	16	spaces	space	NOUN
ejpam-6321	23	17	,	,	PUNCT
ejpam-6321	23	18	utilizing	utilize	VERB
ejpam-6321	23	19	the	the	DET
ejpam-6321	23	20	bcp	bcp	NOUN
ejpam-6321	23	21	as	as	ADP
ejpam-6321	23	22	a	a	DET
ejpam-6321	23	23	guiding	guide	VERB
ejpam-6321	23	24	principle	principle	NOUN
ejpam-6321	23	25	while	while	SCONJ
ejpam-6321	23	26	addressing	address	VERB
ejpam-6321	23	27	limitations	limitation	NOUN
ejpam-6321	23	28	to	to	PART
ejpam-6321	23	29	achieve	achieve	VERB
ejpam-6321	23	30	a	a	DET
ejpam-6321	23	31	more	more	ADV
ejpam-6321	23	32	universally	universally	ADV
ejpam-6321	23	33	applicable	applicable	ADJ
ejpam-6321	23	34	outcome	outcome	NOUN
ejpam-6321	23	35	,	,	PUNCT
ejpam-6321	23	36	particularly	particularly	ADV
ejpam-6321	23	37	in	in	ADP
ejpam-6321	23	38	ai	ai	ADP
ejpam-6321	23	39	and	and	CCONJ
ejpam-6321	23	40	cryptographic	cryptographic	ADJ
ejpam-6321	23	41	applications	application	NOUN
ejpam-6321	23	42	.	.	PUNCT
ejpam-6321	24	1	in	in	ADP
ejpam-6321	24	2	1941	1941	NUM
ejpam-6321	24	3	,	,	PUNCT
ejpam-6321	24	4	kakutani	kakutani	X
ejpam-6321	24	5	[	[	X
ejpam-6321	24	6	1	1	X
ejpam-6321	24	7	]	]	PUNCT
ejpam-6321	24	8	modified	modify	VERB
ejpam-6321	24	9	a	a	DET
ejpam-6321	24	10	generalization	generalization	NOUN
ejpam-6321	24	11	of	of	ADP
ejpam-6321	24	12	brouwers	brouwer	NOUN
ejpam-6321	24	13	fp	fp	PROPN
ejpam-6321	24	14	-	-	PUNCT
ejpam-6321	24	15	theorem	theorem	ADJ
ejpam-6321	24	16	.	.	PUNCT
ejpam-6321	25	1	the	the	DET
ejpam-6321	25	2	theory	theory	NOUN
ejpam-6321	25	3	originated	originate	VERB
ejpam-6321	25	4	from	from	ADP
ejpam-6321	25	5	the	the	DET
ejpam-6321	25	6	brouwer	brouwer	PROPN
ejpam-6321	25	7	fp	fp	PROPN
ejpam-6321	25	8	-	-	PUNCT
ejpam-6321	25	9	theorem	theorem	NOUN
ejpam-6321	25	10	proposed	propose	VERB
ejpam-6321	25	11	by	by	ADP
ejpam-6321	25	12	mathematician	mathematician	ADJ
ejpam-6321	25	13	l.e.j	l.e.j	PROPN
ejpam-6321	25	14	.	.	PUNCT
ejpam-6321	25	15	brouwer	brouwer	PROPN
ejpam-6321	25	16	in	in	ADP
ejpam-6321	25	17	1910	1910	NUM
ejpam-6321	25	18	.	.	PUNCT
ejpam-6321	26	1	this	this	PRON
ejpam-6321	26	2	theorem	theorem	VERB
ejpam-6321	26	3	states	state	NOUN
ejpam-6321	26	4	that	that	SCONJ
ejpam-6321	26	5	any	any	DET
ejpam-6321	26	6	continuous	continuous	ADJ
ejpam-6321	26	7	function	function	NOUN
ejpam-6321	26	8	from	from	ADP
ejpam-6321	26	9	a	a	DET
ejpam-6321	26	10	closed	closed	ADJ
ejpam-6321	26	11	ball	ball	NOUN
ejpam-6321	26	12	to	to	ADP
ejpam-6321	26	13	itself	itself	PRON
ejpam-6321	26	14	must	must	AUX
ejpam-6321	26	15	have	have	VERB
ejpam-6321	26	16	at	at	ADV
ejpam-6321	26	17	least	least	ADV
ejpam-6321	26	18	one	one	NUM
ejpam-6321	26	19	fp	fp	NOUN
ejpam-6321	26	20	.	.	PROPN
ejpam-6321	26	21	jungck	jungck	PROPN
ejpam-6321	26	22	and	and	CCONJ
ejpam-6321	26	23	rhoades	rhoade	NOUN
ejpam-6321	26	24	[	[	X
ejpam-6321	26	25	2	2	NUM
ejpam-6321	26	26	]	]	PUNCT
ejpam-6321	26	27	proved	prove	VERB
ejpam-6321	26	28	some	some	DET
ejpam-6321	26	29	fp	fp	NOUN
ejpam-6321	26	30	-	-	PUNCT
ejpam-6321	26	31	theorems	theorem	NOUN
ejpam-6321	26	32	for	for	ADP
ejpam-6321	26	33	compatible	compatible	ADJ
ejpam-6321	26	34	maps	map	NOUN
ejpam-6321	26	35	.	.	PUNCT
ejpam-6321	27	1	minak	minak	NOUN
ejpam-6321	27	2	et	et	PROPN
ejpam-6321	27	3	al	al	PROPN
ejpam-6321	27	4	.	.	PUNCT
ejpam-6321	28	1	[	[	X
ejpam-6321	28	2	3	3	X
ejpam-6321	28	3	]	]	PUNCT
ejpam-6321	28	4	described	describe	VERB
ejpam-6321	28	5	a	a	DET
ejpam-6321	28	6	new	new	ADJ
ejpam-6321	28	7	approach	approach	NOUN
ejpam-6321	28	8	to	to	ADP
ejpam-6321	28	9	fp	fp	NOUN
ejpam-6321	28	10	-	-	PUNCT
ejpam-6321	28	11	theorems	theorem	NOUN
ejpam-6321	28	12	for	for	ADP
ejpam-6321	28	13	multivalued	multivalued	ADJ
ejpam-6321	28	14	contractive	contractive	ADJ
ejpam-6321	28	15	mappings	mapping	NOUN
ejpam-6321	28	16	.	.	PUNCT
ejpam-6321	29	1	metric	metric	ADJ
ejpam-6321	29	2	spaces	space	NOUN
ejpam-6321	29	3	(	(	PUNCT
ejpam-6321	29	4	m	m	NOUN
ejpam-6321	29	5	-	-	NOUN
ejpam-6321	29	6	spaces	space	NOUN
ejpam-6321	29	7	)	)	PUNCT
ejpam-6321	29	8	provide	provide	VERB
ejpam-6321	29	9	a	a	DET
ejpam-6321	29	10	fundamental	fundamental	ADJ
ejpam-6321	29	11	framework	framework	NOUN
ejpam-6321	29	12	in	in	ADP
ejpam-6321	29	13	mathematics	mathematic	NOUN
ejpam-6321	29	14	for	for	ADP
ejpam-6321	29	15	studying	study	VERB
ejpam-6321	29	16	distance	distance	NOUN
ejpam-6321	29	17	and	and	CCONJ
ejpam-6321	29	18	convergence	convergence	NOUN
ejpam-6321	29	19	.	.	PUNCT
ejpam-6321	30	1	sullivan	sullivan	PROPN
ejpam-6321	31	1	[	[	X
ejpam-6321	31	2	4	4	X
ejpam-6321	31	3	]	]	PUNCT
ejpam-6321	31	4	showed	show	VERB
ejpam-6321	31	5	a	a	DET
ejpam-6321	31	6	characterization	characterization	NOUN
ejpam-6321	31	7	of	of	ADP
ejpam-6321	31	8	complete	complete	ADJ
ejpam-6321	31	9	m	m	NOUN
ejpam-6321	31	10	-	-	NOUN
ejpam-6321	31	11	spaces	space	NOUN
ejpam-6321	31	12	.	.	PUNCT
ejpam-6321	32	1	george	george	PROPN
ejpam-6321	32	2	and	and	CCONJ
ejpam-6321	32	3	veeramani	veeramani	NOUN
ejpam-6321	33	1	[	[	X
ejpam-6321	33	2	5	5	NUM
ejpam-6321	33	3	]	]	PUNCT
ejpam-6321	33	4	worked	work	VERB
ejpam-6321	33	5	on	on	ADP
ejpam-6321	33	6	some	some	DET
ejpam-6321	33	7	results	result	NOUN
ejpam-6321	33	8	in	in	ADP
ejpam-6321	33	9	fuzzy	fuzzy	ADJ
ejpam-6321	33	10	m	m	NOUN
ejpam-6321	33	11	-	-	NOUN
ejpam-6321	33	12	spaces	space	NOUN
ejpam-6321	33	13	.	.	PUNCT
ejpam-6321	34	1	bhaskar	bhaskar	NOUN
ejpam-6321	34	2	and	and	CCONJ
ejpam-6321	34	3	lakshmikantham	lakshmikantham	VERB
ejpam-6321	34	4	[	[	X
ejpam-6321	34	5	6	6	NUM
ejpam-6321	34	6	]	]	PUNCT
ejpam-6321	34	7	defined	define	VERB
ejpam-6321	34	8	fp	fp	NOUN
ejpam-6321	34	9	-	-	PUNCT
ejpam-6321	34	10	theorems	theorem	NOUN
ejpam-6321	34	11	in	in	ADP
ejpam-6321	34	12	partially	partially	ADV
ejpam-6321	34	13	ordered	order	VERB
ejpam-6321	34	14	m	m	NOUN
ejpam-6321	34	15	-	-	NOUN
ejpam-6321	34	16	spaces	space	NOUN
ejpam-6321	34	17	and	and	CCONJ
ejpam-6321	34	18	applications	application	NOUN
ejpam-6321	34	19	.	.	PUNCT
ejpam-6321	35	1	kutbi	kutbi	PROPN
ejpam-6321	35	2	et	et	PROPN
ejpam-6321	35	3	al	al	PROPN
ejpam-6321	35	4	.	.	PUNCT
ejpam-6321	36	1	[	[	X
ejpam-6321	36	2	7	7	X
ejpam-6321	36	3	]	]	PUNCT
ejpam-6321	36	4	proved	prove	VERB
ejpam-6321	36	5	cfp	cfp	PROPN
ejpam-6321	36	6	results	result	NOUN
ejpam-6321	36	7	for	for	ADP
ejpam-6321	36	8	mappings	mapping	NOUN
ejpam-6321	36	9	with	with	ADP
ejpam-6321	36	10	rational	rational	ADJ
ejpam-6321	36	11	expressions	expression	NOUN
ejpam-6321	36	12	.	.	PUNCT
ejpam-6321	37	1	in	in	ADP
ejpam-6321	37	2	2022	2022	NUM
ejpam-6321	37	3	,	,	PUNCT
ejpam-6321	37	4	yaseen	yaseen	PROPN
ejpam-6321	37	5	et	et	PROPN
ejpam-6321	37	6	al	al	PROPN
ejpam-6321	37	7	.	.	PUNCT
ejpam-6321	38	1	[	[	X
ejpam-6321	38	2	8	8	NUM
ejpam-6321	38	3	]	]	PUNCT
ejpam-6321	38	4	addressed	address	VERB
ejpam-6321	38	5	new	new	ADJ
ejpam-6321	38	6	results	result	NOUN
ejpam-6321	38	7	of	of	ADP
ejpam-6321	38	8	fp	fp	NOUN
ejpam-6321	38	9	-	-	PUNCT
ejpam-6321	38	10	theorems	theorem	NOUN
ejpam-6321	38	11	in	in	ADP
ejpam-6321	38	12	complete	complete	ADJ
ejpam-6321	38	13	m	m	NOUN
ejpam-6321	38	14	-	-	NOUN
ejpam-6321	38	15	spaces	space	NOUN
ejpam-6321	38	16	.	.	PUNCT
ejpam-6321	39	1	next	next	ADV
ejpam-6321	39	2	,	,	PUNCT
ejpam-6321	39	3	having	having	AUX
ejpam-6321	39	4	explored	explore	VERB
ejpam-6321	39	5	the	the	DET
ejpam-6321	39	6	concepts	concept	NOUN
ejpam-6321	39	7	within	within	ADP
ejpam-6321	39	8	fpt	fpt	PROPN
ejpam-6321	39	9	and	and	CCONJ
ejpam-6321	39	10	m	m	NOUN
ejpam-6321	39	11	-	-	NOUN
ejpam-6321	39	12	spaces	space	NOUN
ejpam-6321	39	13	,	,	PUNCT
ejpam-6321	39	14	we	we	PRON
ejpam-6321	39	15	delve	delve	VERB
ejpam-6321	39	16	into	into	ADP
ejpam-6321	39	17	the	the	DET
ejpam-6321	39	18	concepts	concept	NOUN
ejpam-6321	39	19	specific	specific	ADJ
ejpam-6321	39	20	to	to	ADP
ejpam-6321	39	21	gm	gm	NOUN
ejpam-6321	39	22	-	-	PUNCT
ejpam-6321	39	23	spaces	space	NOUN
ejpam-6321	39	24	.	.	PUNCT
ejpam-6321	40	1	to	to	PART
ejpam-6321	40	2	achieve	achieve	VERB
ejpam-6321	40	3	this	this	PRON
ejpam-6321	40	4	,	,	PUNCT
ejpam-6321	40	5	we	we	PRON
ejpam-6321	40	6	examine	examine	VERB
ejpam-6321	40	7	existing	exist	VERB
ejpam-6321	40	8	research	research	NOUN
ejpam-6321	40	9	carried	carry	VERB
ejpam-6321	40	10	out	out	ADP
ejpam-6321	40	11	in	in	ADP
ejpam-6321	40	12	this	this	DET
ejpam-6321	40	13	specific	specific	ADJ
ejpam-6321	40	14	area	area	NOUN
ejpam-6321	40	15	,	,	PUNCT
ejpam-6321	40	16	which	which	PRON
ejpam-6321	40	17	provides	provide	VERB
ejpam-6321	40	18	the	the	DET
ejpam-6321	40	19	foundation	foundation	NOUN
ejpam-6321	40	20	to	to	PART
ejpam-6321	40	21	establish	establish	VERB
ejpam-6321	40	22	the	the	DET
ejpam-6321	40	23	most	most	ADV
ejpam-6321	40	24	comprehensive	comprehensive	ADJ
ejpam-6321	40	25	outcomes	outcome	NOUN
ejpam-6321	40	26	.	.	PUNCT
ejpam-6321	41	1	rhoades	rhoade	NOUN
ejpam-6321	42	1	[	[	X
ejpam-6321	42	2	9	9	NUM
ejpam-6321	42	3	]	]	PUNCT
ejpam-6321	42	4	formulated	formulate	VERB
ejpam-6321	42	5	a	a	DET
ejpam-6321	42	6	fp	fp	ADV
ejpam-6321	42	7	-	-	PUNCT
ejpam-6321	42	8	theorem	theorem	NOUN
ejpam-6321	42	9	for	for	ADP
ejpam-6321	42	10	generalized	generalized	ADJ
ejpam-6321	42	11	metric	metric	ADJ
ejpam-6321	42	12	spaces	space	NOUN
ejpam-6321	42	13	(	(	PUNCT
ejpam-6321	42	14	gm	gm	NOUN
ejpam-6321	42	15	-	-	PUNCT
ejpam-6321	42	16	spaces	space	NOUN
ejpam-6321	42	17	)	)	PUNCT
ejpam-6321	42	18	.	.	PUNCT
ejpam-6321	43	1	in	in	ADP
ejpam-6321	43	2	2006	2006	NUM
ejpam-6321	43	3	,	,	PUNCT
ejpam-6321	43	4	mustafa	mustafa	NOUN
ejpam-6321	43	5	and	and	CCONJ
ejpam-6321	43	6	sims	sim	NOUN
ejpam-6321	43	7	[	[	X
ejpam-6321	43	8	10	10	NUM
ejpam-6321	43	9	]	]	PUNCT
ejpam-6321	43	10	proposed	propose	VERB
ejpam-6321	43	11	the	the	DET
ejpam-6321	43	12	idea	idea	NOUN
ejpam-6321	43	13	of	of	ADP
ejpam-6321	43	14	generalized	generalized	ADJ
ejpam-6321	43	15	metric	metric	ADJ
ejpam-6321	43	16	spaces	space	NOUN
ejpam-6321	43	17	.	.	PUNCT
ejpam-6321	44	1	azam	azam	PROPN
ejpam-6321	44	2	and	and	CCONJ
ejpam-6321	44	3	arshad	arshad	VERB
ejpam-6321	45	1	[	[	X
ejpam-6321	45	2	11	11	NUM
ejpam-6321	45	3	]	]	PUNCT
ejpam-6321	45	4	looked	look	VERB
ejpam-6321	45	5	into	into	ADP
ejpam-6321	45	6	a	a	DET
ejpam-6321	45	7	kannan	kannan	PROPN
ejpam-6321	45	8	fp	fp	PROPN
ejpam-6321	45	9	-	-	PUNCT
ejpam-6321	45	10	theorem	theorem	NOUN
ejpam-6321	45	11	on	on	ADP
ejpam-6321	45	12	gm	gm	PROPN
ejpam-6321	45	13	-	-	PUNCT
ejpam-6321	45	14	spaces	space	NOUN
ejpam-6321	45	15	.	.	PUNCT
ejpam-6321	46	1	sarma	sarma	NOUN
ejpam-6321	46	2	et	et	NOUN
ejpam-6321	46	3	al	al	PROPN
ejpam-6321	46	4	.	.	PUNCT
ejpam-6321	47	1	[	[	X
ejpam-6321	47	2	12	12	NUM
ejpam-6321	47	3	]	]	PUNCT
ejpam-6321	47	4	flourished	flourish	VERB
ejpam-6321	47	5	a	a	DET
ejpam-6321	47	6	contraction	contraction	NOUN
ejpam-6321	47	7	over	over	ADP
ejpam-6321	47	8	gm	gm	PROPN
ejpam-6321	47	9	-	-	PUNCT
ejpam-6321	47	10	spaces	space	NOUN
ejpam-6321	47	11	.	.	PUNCT
ejpam-6321	48	1	mihet	mihet	PROPN
ejpam-6321	49	1	[	[	X
ejpam-6321	49	2	13	13	NUM
ejpam-6321	49	3	]	]	PUNCT
ejpam-6321	49	4	worked	work	VERB
ejpam-6321	49	5	on	on	ADP
ejpam-6321	49	6	kannan	kannan	PROPN
ejpam-6321	49	7	fp	fp	PROPN
ejpam-6321	49	8	-	-	PROPN
ejpam-6321	49	9	principle	principle	NOUN
ejpam-6321	49	10	in	in	ADP
ejpam-6321	49	11	gm	gm	PROPN
ejpam-6321	49	12	-	-	PUNCT
ejpam-6321	49	13	spaces	space	NOUN
ejpam-6321	49	14	.	.	PUNCT
ejpam-6321	50	1	das	das	PROPN
ejpam-6321	50	2	and	and	CCONJ
ejpam-6321	50	3	dey	dey	VERB
ejpam-6321	51	1	[	[	X
ejpam-6321	51	2	14	14	NUM
ejpam-6321	51	3	]	]	PUNCT
ejpam-6321	51	4	addressed	address	VERB
ejpam-6321	51	5	fp	fp	NOUN
ejpam-6321	51	6	of	of	ADP
ejpam-6321	51	7	contractive	contractive	ADJ
ejpam-6321	51	8	mappings	mapping	NOUN
ejpam-6321	51	9	in	in	ADP
ejpam-6321	51	10	gm	gm	PROPN
ejpam-6321	51	11	-	-	PUNCT
ejpam-6321	51	12	spaces	space	NOUN
ejpam-6321	51	13	.	.	PUNCT
ejpam-6321	52	1	shatanawi	shatanawi	ADJ
ejpam-6321	53	1	[	[	X
ejpam-6321	53	2	15	15	NUM
ejpam-6321	53	3	]	]	PUNCT
ejpam-6321	53	4	formulated	formulate	VERB
ejpam-6321	53	5	a	a	DET
ejpam-6321	53	6	coupled	couple	VERB
ejpam-6321	53	7	fp	fp	NOUN
ejpam-6321	53	8	-	-	PUNCT
ejpam-6321	53	9	theorems	theorem	NOUN
ejpam-6321	53	10	in	in	ADP
ejpam-6321	53	11	gm	gm	PROPN
ejpam-6321	53	12	-	-	PUNCT
ejpam-6321	53	13	spaces	space	NOUN
ejpam-6321	53	14	.	.	PUNCT
ejpam-6321	54	1	abbas	abbas	PROPN
ejpam-6321	54	2	et	et	PROPN
ejpam-6321	54	3	al	al	PROPN
ejpam-6321	54	4	.	.	PUNCT
ejpam-6321	55	1	[	[	X
ejpam-6321	55	2	?	?	X
ejpam-6321	55	3	]	]	PUNCT
ejpam-6321	55	4	defined	define	VERB
ejpam-6321	55	5	cfp	cfp	PROPN
ejpam-6321	55	6	results	result	NOUN
ejpam-6321	55	7	for	for	ADP
ejpam-6321	55	8	three	three	NUM
ejpam-6321	55	9	maps	map	NOUN
ejpam-6321	55	10	in	in	ADP
ejpam-6321	55	11	gm	gm	NOUN
ejpam-6321	55	12	-	-	PUNCT
ejpam-6321	55	13	space	space	NOUN
ejpam-6321	55	14	.	.	PUNCT
ejpam-6321	56	1	abbas	abbas	PROPN
ejpam-6321	56	2	et	et	PROPN
ejpam-6321	56	3	al	al	PROPN
ejpam-6321	56	4	.	.	PUNCT
ejpam-6321	57	1	[	[	X
ejpam-6321	57	2	?	?	X
ejpam-6321	57	3	]	]	PUNCT
ejpam-6321	57	4	examined	examine	VERB
ejpam-6321	57	5	coupled	couple	VERB
ejpam-6321	57	6	cfp	cfp	PROPN
ejpam-6321	57	7	results	result	NOUN
ejpam-6321	57	8	in	in	ADP
ejpam-6321	57	9	two	two	NUM
ejpam-6321	57	10	gm	gm	NOUN
ejpam-6321	57	11	-	-	PUNCT
ejpam-6321	57	12	spaces	space	NOUN
ejpam-6321	57	13	.	.	PUNCT
ejpam-6321	58	1	di	di	NOUN
ejpam-6321	58	2	bari	bari	NOUN
ejpam-6321	58	3	and	and	CCONJ
ejpam-6321	58	4	vetro	vetro	VERB
ejpam-6321	58	5	[	[	X
ejpam-6321	58	6	16	16	NUM
ejpam-6321	58	7	]	]	PUNCT
ejpam-6321	58	8	explored	explore	VERB
ejpam-6321	58	9	cf	cf	NOUN
ejpam-6321	58	10	-	-	PUNCT
ejpam-6321	58	11	points	point	NOUN
ejpam-6321	58	12	in	in	ADP
ejpam-6321	58	13	gm	gm	PROPN
ejpam-6321	58	14	-	-	PUNCT
ejpam-6321	58	15	spaces	space	NOUN
ejpam-6321	58	16	.	.	PUNCT
ejpam-6321	59	1	romaguera	romaguera	NOUN
ejpam-6321	60	1	[	[	X
ejpam-6321	60	2	17	17	NUM
ejpam-6321	60	3	]	]	PUNCT
ejpam-6321	60	4	established	establish	VERB
ejpam-6321	60	5	fp	fp	NOUN
ejpam-6321	60	6	-	-	PUNCT
ejpam-6321	60	7	theorems	theorem	NOUN
ejpam-6321	60	8	for	for	ADP
ejpam-6321	60	9	generalized	generalized	ADJ
ejpam-6321	60	10	contractions	contraction	NOUN
ejpam-6321	60	11	on	on	ADP
ejpam-6321	60	12	partial	partial	ADJ
ejpam-6321	60	13	m	m	NOUN
ejpam-6321	60	14	-	-	NOUN
ejpam-6321	60	15	spaces	space	NOUN
ejpam-6321	60	16	.	.	PUNCT
ejpam-6321	61	1	mohanta	mohanta	NOUN
ejpam-6321	61	2	and	and	CCONJ
ejpam-6321	61	3	mohanta	mohanta	ADJ
ejpam-6321	61	4	[	[	X
ejpam-6321	61	5	18	18	NUM
ejpam-6321	61	6	]	]	PUNCT
ejpam-6321	61	7	tried	try	VERB
ejpam-6321	61	8	to	to	PART
ejpam-6321	61	9	investigate	investigate	VERB
ejpam-6321	61	10	a	a	DET
ejpam-6321	61	11	cfp	cfp	NOUN
ejpam-6321	61	12	theorem	theorem	NOUN
ejpam-6321	61	13	in	in	ADP
ejpam-6321	61	14	gmspaces	gmspace	NOUN
ejpam-6321	61	15	.	.	PUNCT
ejpam-6321	62	1	gugnani	gugnani	PROPN
ejpam-6321	62	2	et	et	PROPN
ejpam-6321	62	3	al	al	PROPN
ejpam-6321	62	4	.	.	PUNCT
ejpam-6321	63	1	[	[	X
ejpam-6321	63	2	19	19	NUM
ejpam-6321	63	3	]	]	PUNCT
ejpam-6321	63	4	described	describe	VERB
ejpam-6321	63	5	cfp	cfp	PROPN
ejpam-6321	63	6	results	result	NOUN
ejpam-6321	63	7	in	in	ADP
ejpam-6321	63	8	gm	gm	PROPN
ejpam-6321	63	9	-	-	PUNCT
ejpam-6321	63	10	spaces	space	NOUN
ejpam-6321	63	11	and	and	CCONJ
ejpam-6321	63	12	its	its	PRON
ejpam-6321	63	13	applications	application	NOUN
ejpam-6321	63	14	.	.	PUNCT
ejpam-6321	64	1	aydi	aydi	VERB
ejpam-6321	64	2	[	[	X
ejpam-6321	64	3	20	20	NUM
ejpam-6321	64	4	]	]	PUNCT
ejpam-6321	64	5	discussed	discuss	VERB
ejpam-6321	64	6	a	a	DET
ejpam-6321	64	7	cfp	cfp	NOUN
ejpam-6321	64	8	of	of	ADP
ejpam-6321	64	9	integral	integral	ADJ
ejpam-6321	64	10	type	type	NOUN
ejpam-6321	64	11	contraction	contraction	NOUN
ejpam-6321	64	12	in	in	ADP
ejpam-6321	64	13	gm	gm	PROPN
ejpam-6321	64	14	-	-	PUNCT
ejpam-6321	64	15	spaces	space	NOUN
ejpam-6321	64	16	.	.	PUNCT
ejpam-6321	65	1	kadelburg	kadelburg	NOUN
ejpam-6321	65	2	and	and	CCONJ
ejpam-6321	65	3	radenovic	radenovic	ADJ
ejpam-6321	66	1	[	[	X
ejpam-6321	66	2	21	21	NUM
ejpam-6321	66	3	]	]	PUNCT
ejpam-6321	66	4	worked	work	VERB
ejpam-6321	66	5	on	on	ADP
ejpam-6321	66	6	gm	gm	PROPN
ejpam-6321	66	7	-	-	PUNCT
ejpam-6321	66	8	spaces	space	NOUN
ejpam-6321	66	9	.	.	PUNCT
ejpam-6321	67	1	later	later	ADV
ejpam-6321	67	2	on	on	ADV
ejpam-6321	67	3	,	,	PUNCT
ejpam-6321	67	4	in	in	ADP
ejpam-6321	67	5	2015	2015	NUM
ejpam-6321	67	6	,	,	PUNCT
ejpam-6321	67	7	jleli	jleli	ADJ
ejpam-6321	67	8	and	and	CCONJ
ejpam-6321	67	9	samet	samet	VERB
ejpam-6321	68	1	[	[	X
ejpam-6321	68	2	22	22	NUM
ejpam-6321	68	3	]	]	PUNCT
ejpam-6321	68	4	provided	provide	VERB
ejpam-6321	68	5	a	a	DET
ejpam-6321	68	6	gm	gm	NOUN
ejpam-6321	68	7	-	-	PUNCT
ejpam-6321	68	8	space	space	NOUN
ejpam-6321	68	9	and	and	CCONJ
ejpam-6321	68	10	proved	prove	VERB
ejpam-6321	68	11	its	its	PRON
ejpam-6321	68	12	related	relate	VERB
ejpam-6321	68	13	fptheorems	fptheorem	NOUN
ejpam-6321	68	14	.	.	PUNCT
ejpam-6321	69	1	abed	abe	VERB
ejpam-6321	69	2	and	and	CCONJ
ejpam-6321	69	3	luaibi	luaibi	NOUN
ejpam-6321	70	1	[	[	X
ejpam-6321	70	2	23	23	NUM
ejpam-6321	70	3	]	]	PUNCT
ejpam-6321	70	4	addressed	address	VERB
ejpam-6321	70	5	two	two	NUM
ejpam-6321	70	6	fp	fp	NOUN
ejpam-6321	70	7	-	-	PUNCT
ejpam-6321	70	8	theorems	theorem	NOUN
ejpam-6321	70	9	in	in	ADP
ejpam-6321	70	10	gm	gm	PROPN
ejpam-6321	70	11	-	-	PUNCT
ejpam-6321	70	12	spaces	space	NOUN
ejpam-6321	70	13	.	.	PUNCT
ejpam-6321	71	1	lin	lin	PROPN
ejpam-6321	71	2	and	and	CCONJ
ejpam-6321	71	3	yun	yun	PROPN
ejpam-6321	72	1	[	[	X
ejpam-6321	72	2	24	24	NUM
ejpam-6321	72	3	]	]	PUNCT
ejpam-6321	72	4	examined	examine	VERB
ejpam-6321	72	5	gm	gm	PROPN
ejpam-6321	72	6	-	-	PUNCT
ejpam-6321	72	7	spaces	space	NOUN
ejpam-6321	72	8	and	and	CCONJ
ejpam-6321	72	9	mappings	mapping	NOUN
ejpam-6321	72	10	.	.	PUNCT
ejpam-6321	73	1	peng	peng	PROPN
ejpam-6321	73	2	and	and	CCONJ
ejpam-6321	73	3	sun	sun	PROPN
ejpam-6321	74	1	[	[	X
ejpam-6321	74	2	25	25	NUM
ejpam-6321	74	3	]	]	PUNCT
ejpam-6321	74	4	explored	explore	VERB
ejpam-6321	74	5	a	a	DET
ejpam-6321	74	6	study	study	NOUN
ejpam-6321	74	7	on	on	ADP
ejpam-6321	74	8	symmetric	symmetric	ADJ
ejpam-6321	74	9	products	product	NOUN
ejpam-6321	74	10	of	of	ADP
ejpam-6321	74	11	gm	gm	PROPN
ejpam-6321	74	12	-	-	PUNCT
ejpam-6321	74	13	spaces	space	NOUN
ejpam-6321	74	14	.	.	PUNCT
ejpam-6321	75	1	our	our	PRON
ejpam-6321	75	2	approach	approach	NOUN
ejpam-6321	75	3	involves	involve	VERB
ejpam-6321	75	4	utilizing	utilize	VERB
ejpam-6321	75	5	generalized	generalized	ADJ
ejpam-6321	75	6	outcomes	outcome	NOUN
ejpam-6321	75	7	through	through	ADP
ejpam-6321	75	8	the	the	DET
ejpam-6321	75	9	application	application	NOUN
ejpam-6321	75	10	of	of	ADP
ejpam-6321	75	11	maximum	maximum	ADJ
ejpam-6321	75	12	and	and	CCONJ
ejpam-6321	75	13	minimum	minimum	ADJ
ejpam-6321	75	14	type	type	NOUN
ejpam-6321	75	15	contractions	contraction	NOUN
ejpam-6321	75	16	within	within	ADP
ejpam-6321	75	17	gm	gm	PROPN
ejpam-6321	75	18	-	-	PUNCT
ejpam-6321	75	19	spaces	space	NOUN
ejpam-6321	75	20	without	without	ADP
ejpam-6321	75	21	continuity	continuity	NOUN
ejpam-6321	75	22	.	.	PUNCT
ejpam-6321	76	1	this	this	DET
ejpam-6321	76	2	approach	approach	NOUN
ejpam-6321	76	3	aims	aim	VERB
ejpam-6321	76	4	to	to	PART
ejpam-6321	76	5	significantly	significantly	ADV
ejpam-6321	76	6	enhance	enhance	VERB
ejpam-6321	76	7	the	the	DET
ejpam-6321	76	8	effectiveness	effectiveness	NOUN
ejpam-6321	76	9	of	of	ADP
ejpam-6321	76	10	our	our	PRON
ejpam-6321	76	11	results	result	NOUN
ejpam-6321	76	12	.	.	PUNCT
ejpam-6321	77	1	furthermore	furthermore	ADV
ejpam-6321	77	2	,	,	PUNCT
ejpam-6321	77	3	our	our	PRON
ejpam-6321	77	4	work	work	NOUN
ejpam-6321	77	5	endeavors	endeavor	NOUN
ejpam-6321	77	6	m.	m.	PROPN
ejpam-6321	77	7	noorwali	noorwali	PROPN
ejpam-6321	77	8	et	et	PROPN
ejpam-6321	77	9	al	al	PROPN
ejpam-6321	77	10	.	.	PUNCT
ejpam-6321	77	11	/	/	SYM
ejpam-6321	77	12	eur	eur	PROPN
ejpam-6321	77	13	.	.	PUNCT
ejpam-6321	78	1	j.	j.	PROPN
ejpam-6321	78	2	pure	pure	PROPN
ejpam-6321	78	3	appl	appl	PROPN
ejpam-6321	78	4	.	.	PROPN
ejpam-6321	78	5	math	math	PROPN
ejpam-6321	78	6	,	,	PUNCT
ejpam-6321	78	7	18	18	NUM
ejpam-6321	78	8	(	(	PUNCT
ejpam-6321	78	9	3	3	NUM
ejpam-6321	78	10	)	)	PUNCT
ejpam-6321	78	11	(	(	PUNCT
ejpam-6321	78	12	2025	2025	NUM
ejpam-6321	78	13	)	)	PUNCT
ejpam-6321	78	14	,	,	PUNCT
ejpam-6321	78	15	6321	6321	NUM
ejpam-6321	78	16	3	3	NUM
ejpam-6321	78	17	of	of	ADP
ejpam-6321	78	18	27	27	NUM
ejpam-6321	78	19	to	to	PART
ejpam-6321	78	20	build	build	VERB
ejpam-6321	78	21	upon	upon	SCONJ
ejpam-6321	78	22	the	the	DET
ejpam-6321	78	23	foundation	foundation	NOUN
ejpam-6321	78	24	of	of	ADP
ejpam-6321	78	25	the	the	DET
ejpam-6321	78	26	latest	late	ADJ
ejpam-6321	78	27	published	publish	VERB
ejpam-6321	78	28	results	result	NOUN
ejpam-6321	78	29	,	,	PUNCT
ejpam-6321	78	30	thereby	thereby	ADV
ejpam-6321	78	31	producing	produce	VERB
ejpam-6321	78	32	an	an	DET
ejpam-6321	78	33	expanded	expand	VERB
ejpam-6321	78	34	and	and	CCONJ
ejpam-6321	78	35	enriched	enrich	VERB
ejpam-6321	78	36	version	version	NOUN
ejpam-6321	78	37	of	of	ADP
ejpam-6321	78	38	its	its	PRON
ejpam-6321	78	39	findings	finding	NOUN
ejpam-6321	78	40	.	.	PUNCT
ejpam-6321	79	1	the	the	DET
ejpam-6321	79	2	significance	significance	NOUN
ejpam-6321	79	3	of	of	ADP
ejpam-6321	79	4	our	our	PRON
ejpam-6321	79	5	study	study	NOUN
ejpam-6321	79	6	lies	lie	VERB
ejpam-6321	79	7	in	in	ADP
ejpam-6321	79	8	its	its	PRON
ejpam-6321	79	9	capacity	capacity	NOUN
ejpam-6321	79	10	to	to	PART
ejpam-6321	79	11	extend	extend	VERB
ejpam-6321	79	12	and	and	CCONJ
ejpam-6321	79	13	enrich	enrich	VERB
ejpam-6321	79	14	existing	exist	VERB
ejpam-6321	79	15	results	result	NOUN
ejpam-6321	79	16	from	from	ADP
ejpam-6321	79	17	the	the	DET
ejpam-6321	79	18	realm	realm	NOUN
ejpam-6321	79	19	of	of	ADP
ejpam-6321	79	20	fpt	fpt	PROPN
ejpam-6321	79	21	.	.	PUNCT
ejpam-6321	80	1	through	through	ADP
ejpam-6321	80	2	our	our	PRON
ejpam-6321	80	3	exploration	exploration	NOUN
ejpam-6321	80	4	,	,	PUNCT
ejpam-6321	80	5	we	we	PRON
ejpam-6321	80	6	endeavor	endeavor	VERB
ejpam-6321	80	7	to	to	PART
ejpam-6321	80	8	enhance	enhance	VERB
ejpam-6321	80	9	our	our	PRON
ejpam-6321	80	10	understanding	understanding	NOUN
ejpam-6321	80	11	of	of	ADP
ejpam-6321	80	12	fp	fp	ADJ
ejpam-6321	80	13	principles	principle	NOUN
ejpam-6321	80	14	by	by	ADP
ejpam-6321	80	15	introducing	introduce	VERB
ejpam-6321	80	16	new	new	ADJ
ejpam-6321	80	17	dimensions	dimension	NOUN
ejpam-6321	80	18	.	.	PUNCT
ejpam-6321	81	1	our	our	PRON
ejpam-6321	81	2	main	main	ADJ
ejpam-6321	81	3	objective	objective	NOUN
ejpam-6321	81	4	is	be	AUX
ejpam-6321	81	5	to	to	PART
ejpam-6321	81	6	establish	establish	VERB
ejpam-6321	81	7	specific	specific	ADJ
ejpam-6321	81	8	conditions	condition	NOUN
ejpam-6321	81	9	under	under	ADP
ejpam-6321	81	10	which	which	PRON
ejpam-6321	81	11	fixed	fix	VERB
ejpam-6321	81	12	points	point	NOUN
ejpam-6321	81	13	uniquely	uniquely	ADV
ejpam-6321	81	14	exist	exist	VERB
ejpam-6321	81	15	.	.	PUNCT
ejpam-6321	82	1	to	to	PART
ejpam-6321	82	2	achieve	achieve	VERB
ejpam-6321	82	3	this	this	PRON
ejpam-6321	82	4	,	,	PUNCT
ejpam-6321	82	5	we	we	PRON
ejpam-6321	82	6	exploit	exploit	VERB
ejpam-6321	82	7	the	the	DET
ejpam-6321	82	8	power	power	NOUN
ejpam-6321	82	9	of	of	ADP
ejpam-6321	82	10	three	three	NUM
ejpam-6321	82	11	self	self	NOUN
ejpam-6321	82	12	-	-	PUNCT
ejpam-6321	82	13	mappings	mapping	NOUN
ejpam-6321	82	14	that	that	PRON
ejpam-6321	82	15	fulfill	fulfill	VERB
ejpam-6321	82	16	the	the	DET
ejpam-6321	82	17	criteria	criterion	NOUN
ejpam-6321	82	18	of	of	ADP
ejpam-6321	82	19	generalized	generalized	ADJ
ejpam-6321	82	20	contractive	contractive	ADJ
ejpam-6321	82	21	conditions	condition	NOUN
ejpam-6321	82	22	within	within	ADP
ejpam-6321	82	23	a	a	DET
ejpam-6321	82	24	gm	gm	NOUN
ejpam-6321	82	25	-	-	PUNCT
ejpam-6321	82	26	space	space	NOUN
ejpam-6321	82	27	.	.	PUNCT
ejpam-6321	83	1	this	this	DET
ejpam-6321	83	2	investigation	investigation	NOUN
ejpam-6321	83	3	is	be	AUX
ejpam-6321	83	4	undertaken	undertake	VERB
ejpam-6321	83	5	with	with	ADP
ejpam-6321	83	6	the	the	DET
ejpam-6321	83	7	goal	goal	NOUN
ejpam-6321	83	8	of	of	ADP
ejpam-6321	83	9	contributing	contribute	VERB
ejpam-6321	83	10	valuable	valuable	ADJ
ejpam-6321	83	11	insights	insight	NOUN
ejpam-6321	83	12	that	that	PRON
ejpam-6321	83	13	further	far	ADV
ejpam-6321	83	14	illuminate	illuminate	VERB
ejpam-6321	83	15	the	the	DET
ejpam-6321	83	16	existence	existence	NOUN
ejpam-6321	83	17	and	and	CCONJ
ejpam-6321	83	18	uniqueness	uniqueness	NOUN
ejpam-6321	83	19	of	of	ADP
ejpam-6321	83	20	fixed	fix	VERB
ejpam-6321	83	21	points	point	NOUN
ejpam-6321	83	22	,	,	PUNCT
ejpam-6321	83	23	while	while	SCONJ
ejpam-6321	83	24	simultaneously	simultaneously	ADV
ejpam-6321	83	25	advancing	advance	VERB
ejpam-6321	83	26	the	the	DET
ejpam-6321	83	27	frontiers	frontier	NOUN
ejpam-6321	83	28	of	of	ADP
ejpam-6321	83	29	gm	gm	PROPN
ejpam-6321	83	30	-	-	PUNCT
ejpam-6321	83	31	space	space	NOUN
ejpam-6321	83	32	theory	theory	NOUN
ejpam-6321	83	33	.	.	PUNCT
ejpam-6321	84	1	moreover	moreover	ADV
ejpam-6321	84	2	,	,	PUNCT
ejpam-6321	84	3	these	these	DET
ejpam-6321	84	4	insights	insight	NOUN
ejpam-6321	84	5	provide	provide	VERB
ejpam-6321	84	6	solid	solid	ADJ
ejpam-6321	84	7	theoretical	theoretical	ADJ
ejpam-6321	84	8	underpinnings	underpinning	NOUN
ejpam-6321	84	9	for	for	ADP
ejpam-6321	84	10	reliable	reliable	ADJ
ejpam-6321	84	11	iterative	iterative	NOUN
ejpam-6321	84	12	convergence	convergence	NOUN
ejpam-6321	84	13	in	in	ADP
ejpam-6321	84	14	ai	ai	PRON
ejpam-6321	84	15	algorithms	algorithms	NOUN
ejpam-6321	84	16	and	and	CCONJ
ejpam-6321	84	17	consistency	consistency	NOUN
ejpam-6321	84	18	guarantees	guarantee	NOUN
ejpam-6321	84	19	in	in	ADP
ejpam-6321	84	20	cryptographic	cryptographic	ADJ
ejpam-6321	84	21	communication	communication	NOUN
ejpam-6321	84	22	protocols	protocol	NOUN
ejpam-6321	84	23	.	.	PUNCT
ejpam-6321	85	1	2	2	X
ejpam-6321	85	2	.	.	X
ejpam-6321	85	3	preliminaries	preliminary	NOUN
ejpam-6321	85	4	this	this	DET
ejpam-6321	85	5	section	section	NOUN
ejpam-6321	85	6	is	be	AUX
ejpam-6321	85	7	devoted	devote	VERB
ejpam-6321	85	8	to	to	ADP
ejpam-6321	85	9	some	some	DET
ejpam-6321	85	10	fundamental	fundamental	ADJ
ejpam-6321	85	11	definitions	definition	NOUN
ejpam-6321	85	12	,	,	PUNCT
ejpam-6321	85	13	which	which	PRON
ejpam-6321	85	14	are	be	AUX
ejpam-6321	85	15	necessary	necessary	ADJ
ejpam-6321	85	16	for	for	ADP
ejpam-6321	85	17	the	the	DET
ejpam-6321	85	18	upcoming	upcoming	ADJ
ejpam-6321	85	19	sections	section	NOUN
ejpam-6321	85	20	.	.	PUNCT
ejpam-6321	86	1	2.1	2.1	NUM
ejpam-6321	86	2	.	.	PUNCT
ejpam-6321	87	1	generalized	generalize	VERB
ejpam-6321	87	2	metric	metric	ADJ
ejpam-6321	87	3	space	space	NOUN
ejpam-6321	87	4	this	this	DET
ejpam-6321	87	5	section	section	NOUN
ejpam-6321	87	6	explores	explore	NOUN
ejpam-6321	87	7	fixed	fix	VERB
ejpam-6321	87	8	point	point	NOUN
ejpam-6321	87	9	theory	theory	NOUN
ejpam-6321	87	10	in	in	ADP
ejpam-6321	87	11	generalized	generalized	ADJ
ejpam-6321	87	12	metric	metric	ADJ
ejpam-6321	87	13	spaces	space	NOUN
ejpam-6321	87	14	(	(	PUNCT
ejpam-6321	87	15	gm	gm	NOUN
ejpam-6321	87	16	-	-	PUNCT
ejpam-6321	87	17	spaces	space	NOUN
ejpam-6321	87	18	)	)	PUNCT
ejpam-6321	87	19	,	,	PUNCT
ejpam-6321	87	20	extending	extend	VERB
ejpam-6321	87	21	concepts	concept	NOUN
ejpam-6321	87	22	like	like	ADP
ejpam-6321	87	23	the	the	DET
ejpam-6321	87	24	banach	banach	NOUN
ejpam-6321	87	25	contraction	contraction	NOUN
ejpam-6321	87	26	principle	principle	NOUN
ejpam-6321	87	27	(	(	PUNCT
ejpam-6321	87	28	bcp	bcp	PROPN
ejpam-6321	87	29	)	)	PUNCT
ejpam-6321	87	30	to	to	ADP
ejpam-6321	87	31	a	a	DET
ejpam-6321	87	32	broader	broad	ADJ
ejpam-6321	87	33	context	context	NOUN
ejpam-6321	87	34	.	.	PUNCT
ejpam-6321	88	1	gm	gm	PROPN
ejpam-6321	88	2	-	-	PUNCT
ejpam-6321	88	3	spaces	space	NOUN
ejpam-6321	88	4	is	be	AUX
ejpam-6321	88	5	defined	define	VERB
ejpam-6321	88	6	and	and	CCONJ
ejpam-6321	88	7	introduce	introduce	VERB
ejpam-6321	88	8	g	g	NOUN
ejpam-6321	88	9	-	-	PUNCT
ejpam-6321	88	10	cauchy	cauchy	ADJ
ejpam-6321	88	11	sequences	sequence	NOUN
ejpam-6321	88	12	,	,	PUNCT
ejpam-6321	88	13	convergence	convergence	NOUN
ejpam-6321	88	14	,	,	PUNCT
ejpam-6321	88	15	and	and	CCONJ
ejpam-6321	88	16	completeness	completeness	NOUN
ejpam-6321	88	17	in	in	ADP
ejpam-6321	88	18	these	these	DET
ejpam-6321	88	19	spaces	space	NOUN
ejpam-6321	88	20	.	.	PUNCT
ejpam-6321	89	1	result	result	VERB
ejpam-6321	89	2	on	on	ADP
ejpam-6321	89	3	gm	gm	PROPN
ejpam-6321	89	4	space	space	NOUN
ejpam-6321	89	5	is	be	AUX
ejpam-6321	89	6	also	also	ADV
ejpam-6321	89	7	given	give	VERB
ejpam-6321	89	8	.	.	PUNCT
ejpam-6321	90	1	definition	definition	NOUN
ejpam-6321	90	2	1	1	NUM
ejpam-6321	90	3	.	.	PUNCT
ejpam-6321	91	1	[	[	X
ejpam-6321	91	2	2	2	X
ejpam-6321	91	3	]	]	PUNCT
ejpam-6321	91	4	suppose	suppose	VERB
ejpam-6321	91	5	a	a	DET
ejpam-6321	91	6	non	non	ADJ
ejpam-6321	91	7	-	-	ADJ
ejpam-6321	91	8	empty	empty	ADJ
ejpam-6321	91	9	set	set	NOUN
ejpam-6321	91	10	x	x	PUNCT
ejpam-6321	91	11	and	and	CCONJ
ejpam-6321	91	12	let	let	VERB
ejpam-6321	91	13	g	g	PRON
ejpam-6321	91	14	be	be	AUX
ejpam-6321	91	15	a	a	DET
ejpam-6321	91	16	function	function	NOUN
ejpam-6321	91	17	that	that	PRON
ejpam-6321	91	18	operates	operate	VERB
ejpam-6321	91	19	on	on	ADP
ejpam-6321	91	20	triples	triple	NOUN
ejpam-6321	91	21	of	of	ADP
ejpam-6321	91	22	elements	element	NOUN
ejpam-6321	91	23	from	from	ADP
ejpam-6321	91	24	x	x	PUNCT
ejpam-6321	91	25	denoted	denote	VERB
ejpam-6321	91	26	as	as	ADP
ejpam-6321	91	27	g	g	PROPN
ejpam-6321	91	28	:	:	PUNCT
ejpam-6321	91	29	x	x	PROPN
ejpam-6321	91	30	×x	×x	X
ejpam-6321	91	31	×x	×x	X
ejpam-6321	91	32	→	→	PUNCT
ejpam-6321	91	33	[	[	X
ejpam-6321	91	34	0,∞	0,∞	NOUN
ejpam-6321	91	35	)	)	PUNCT
ejpam-6321	91	36	.	.	PUNCT
ejpam-6321	92	1	we	we	PRON
ejpam-6321	92	2	will	will	AUX
ejpam-6321	92	3	call	call	VERB
ejpam-6321	92	4	this	this	DET
ejpam-6321	92	5	structure	structure	NOUN
ejpam-6321	92	6	a	a	DET
ejpam-6321	92	7	generalized	generalized	ADJ
ejpam-6321	92	8	metric	metric	ADJ
ejpam-6321	92	9	space	space	NOUN
ejpam-6321	92	10	(	(	PUNCT
ejpam-6321	92	11	gm	gm	NOUN
ejpam-6321	92	12	-	-	PUNCT
ejpam-6321	92	13	space	space	NOUN
ejpam-6321	92	14	)	)	PUNCT
ejpam-6321	92	15	if	if	SCONJ
ejpam-6321	92	16	the	the	DET
ejpam-6321	92	17	following	follow	VERB
ejpam-6321	92	18	axioms	axiom	NOUN
ejpam-6321	92	19	hold	hold	VERB
ejpam-6321	92	20	true	true	ADJ
ejpam-6321	92	21	:	:	PUNCT
ejpam-6321	92	22	(	(	PUNCT
ejpam-6321	92	23	i	i	NOUN
ejpam-6321	92	24	)	)	PUNCT
ejpam-6321	92	25	g(x1	g(x1	NOUN
ejpam-6321	92	26	,	,	PUNCT
ejpam-6321	92	27	x2	x2	PROPN
ejpam-6321	92	28	,	,	PUNCT
ejpam-6321	92	29	x3	x3	ADJ
ejpam-6321	92	30	)	)	PUNCT
ejpam-6321	92	31	=	=	SYM
ejpam-6321	92	32	0	0	NUM
ejpam-6321	93	1	iff	iff	VERB
ejpam-6321	93	2	x1	x1	PROPN
ejpam-6321	93	3	=	=	PUNCT
ejpam-6321	93	4	x2	x2	PROPN
ejpam-6321	93	5	=	=	SYM
ejpam-6321	93	6	x3	x3	PROPN
ejpam-6321	93	7	,	,	PUNCT
ejpam-6321	93	8	(	(	PUNCT
ejpam-6321	93	9	ii	ii	NOUN
ejpam-6321	93	10	)	)	PUNCT
ejpam-6321	93	11	0	0	PUNCT
ejpam-6321	94	1	<	<	X
ejpam-6321	94	2	g(x1	g(x1	PROPN
ejpam-6321	94	3	,	,	PUNCT
ejpam-6321	94	4	x1	x1	PROPN
ejpam-6321	94	5	,	,	PUNCT
ejpam-6321	94	6	x2	x2	PROPN
ejpam-6321	94	7	)	)	PUNCT
ejpam-6321	94	8	∀	∀	PUNCT
ejpam-6321	95	1	x1	x1	ADJ
ejpam-6321	95	2	,	,	PUNCT
ejpam-6321	95	3	x2	x2	PROPN
ejpam-6321	95	4	∈	∈	PROPN
ejpam-6321	95	5	x	x	X
ejpam-6321	95	6	,	,	PUNCT
ejpam-6321	95	7	with	with	ADP
ejpam-6321	95	8	x1	x1	PROPN
ejpam-6321	95	9	̸=	̸=	PROPN
ejpam-6321	95	10	x2	x2	PROPN
ejpam-6321	95	11	,	,	PUNCT
ejpam-6321	95	12	(	(	PUNCT
ejpam-6321	95	13	iii	iii	NOUN
ejpam-6321	95	14	)	)	PUNCT
ejpam-6321	95	15	g(x1	g(x1	NOUN
ejpam-6321	95	16	,	,	PUNCT
ejpam-6321	95	17	x1	x1	PROPN
ejpam-6321	95	18	,	,	PUNCT
ejpam-6321	95	19	x2	x2	PROPN
ejpam-6321	95	20	)	)	PUNCT
ejpam-6321	95	21	≤	≤	NOUN
ejpam-6321	95	22	g(x1	g(x1	NOUN
ejpam-6321	95	23	,	,	PUNCT
ejpam-6321	95	24	x2	x2	PROPN
ejpam-6321	95	25	,	,	PUNCT
ejpam-6321	95	26	x3	x3	ADJ
ejpam-6321	95	27	)	)	PUNCT
ejpam-6321	95	28	∀	∀	PUNCT
ejpam-6321	96	1	x1	x1	ADJ
ejpam-6321	96	2	,	,	PUNCT
ejpam-6321	96	3	x2	x2	PROPN
ejpam-6321	96	4	,	,	PUNCT
ejpam-6321	96	5	x3	x3	PROPN
ejpam-6321	96	6	∈	∈	PROPN
ejpam-6321	97	1	x1	x1	PROPN
ejpam-6321	97	2	,	,	PUNCT
ejpam-6321	97	3	with	with	ADP
ejpam-6321	97	4	x1	x1	PROPN
ejpam-6321	97	5	̸=	̸=	PROPN
ejpam-6321	97	6	x2	x2	PROPN
ejpam-6321	97	7	,	,	PUNCT
ejpam-6321	97	8	(	(	PUNCT
ejpam-6321	97	9	iv	iv	X
ejpam-6321	97	10	)	)	PUNCT
ejpam-6321	97	11	g(x1	g(x1	NOUN
ejpam-6321	97	12	,	,	PUNCT
ejpam-6321	97	13	x2	x2	PROPN
ejpam-6321	97	14	,	,	PUNCT
ejpam-6321	97	15	x3	x3	ADJ
ejpam-6321	97	16	)	)	PUNCT
ejpam-6321	98	1	=	=	SYM
ejpam-6321	98	2	g{p(x1	g{p(x1	NOUN
ejpam-6321	98	3	,	,	PUNCT
ejpam-6321	98	4	x2	x2	PROPN
ejpam-6321	98	5	,	,	PUNCT
ejpam-6321	98	6	x3	x3	ADJ
ejpam-6321	98	7	)	)	PUNCT
ejpam-6321	98	8	}	}	PUNCT
ejpam-6321	98	9	where	where	SCONJ
ejpam-6321	98	10	,	,	PUNCT
ejpam-6321	98	11	p	p	PRON
ejpam-6321	98	12	is	be	AUX
ejpam-6321	98	13	a	a	DET
ejpam-6321	98	14	permutation	permutation	NOUN
ejpam-6321	98	15	of	of	ADP
ejpam-6321	98	16	x1	x1	PROPN
ejpam-6321	98	17	,	,	PUNCT
ejpam-6321	98	18	x2	x2	PROPN
ejpam-6321	98	19	,	,	PUNCT
ejpam-6321	98	20	x3	x3	PROPN
ejpam-6321	98	21	.	.	PUNCT
ejpam-6321	99	1	(	(	PUNCT
ejpam-6321	99	2	symmetry	symmetry	NOUN
ejpam-6321	99	3	)	)	PUNCT
ejpam-6321	99	4	.	.	PUNCT
ejpam-6321	100	1	(	(	PUNCT
ejpam-6321	100	2	v	v	NOUN
ejpam-6321	100	3	)	)	PUNCT
ejpam-6321	100	4	g(x1	g(x1	NOUN
ejpam-6321	100	5	,	,	PUNCT
ejpam-6321	100	6	x2	x2	PROPN
ejpam-6321	100	7	,	,	PUNCT
ejpam-6321	100	8	x3	x3	ADJ
ejpam-6321	100	9	)	)	PUNCT
ejpam-6321	100	10	≤	≤	NOUN
ejpam-6321	100	11	g(x1	g(x1	NOUN
ejpam-6321	100	12	,	,	PUNCT
ejpam-6321	100	13	x1	x1	PROPN
ejpam-6321	100	14	,	,	PUNCT
ejpam-6321	100	15	x1	x1	PROPN
ejpam-6321	100	16	)	)	PUNCT
ejpam-6321	101	1	+	+	NOUN
ejpam-6321	101	2	g(x1	g(x1	NOUN
ejpam-6321	101	3	,	,	PUNCT
ejpam-6321	101	4	x2	x2	PROPN
ejpam-6321	101	5	,	,	PUNCT
ejpam-6321	101	6	x3	x3	ADJ
ejpam-6321	101	7	)	)	PUNCT
ejpam-6321	101	8	∀	∀	PUNCT
ejpam-6321	102	1	x1	x1	ADJ
ejpam-6321	102	2	,	,	PUNCT
ejpam-6321	102	3	x2	x2	PROPN
ejpam-6321	102	4	,	,	PUNCT
ejpam-6321	102	5	x3	x3	PROPN
ejpam-6321	102	6	∈	∈	PROPN
ejpam-6321	102	7	x.	x.	NOUN
ejpam-6321	102	8	a	a	DET
ejpam-6321	102	9	gm	gm	PROPN
ejpam-6321	102	10	is	be	AUX
ejpam-6321	102	11	symmetric	symmetric	ADJ
ejpam-6321	102	12	if	if	SCONJ
ejpam-6321	102	13	g(x1	g(x1	NOUN
ejpam-6321	102	14	,	,	PUNCT
ejpam-6321	102	15	x2	x2	PROPN
ejpam-6321	102	16	,	,	PUNCT
ejpam-6321	102	17	x2	x2	PROPN
ejpam-6321	102	18	)	)	PUNCT
ejpam-6321	102	19	=	=	SYM
ejpam-6321	102	20	g(x2	g(x2	NOUN
ejpam-6321	102	21	,	,	PUNCT
ejpam-6321	102	22	x1	x1	PROPN
ejpam-6321	102	23	,	,	PUNCT
ejpam-6321	102	24	x1	x1	NUM
ejpam-6321	102	25	)	)	PUNCT
ejpam-6321	102	26	∀	∀	PUNCT
ejpam-6321	103	1	x1	x1	ADJ
ejpam-6321	103	2	,	,	PUNCT
ejpam-6321	103	3	x2	x2	PROPN
ejpam-6321	103	4	∈	∈	PROPN
ejpam-6321	104	1	x	x	X
ejpam-6321	104	2	,	,	PUNCT
ejpam-6321	104	3	then	then	ADV
ejpam-6321	104	4	(	(	PUNCT
ejpam-6321	104	5	x	x	X
ejpam-6321	104	6	,	,	PUNCT
ejpam-6321	104	7	g	g	NOUN
ejpam-6321	104	8	)	)	PUNCT
ejpam-6321	104	9	is	be	AUX
ejpam-6321	104	10	called	call	VERB
ejpam-6321	104	11	a	a	DET
ejpam-6321	104	12	gmspace	gmspace	NOUN
ejpam-6321	104	13	.	.	PUNCT
ejpam-6321	105	1	definition	definition	NOUN
ejpam-6321	105	2	2	2	NUM
ejpam-6321	105	3	.	.	PUNCT
ejpam-6321	106	1	[	[	X
ejpam-6321	106	2	2	2	NUM
ejpam-6321	106	3	]	]	X
ejpam-6321	106	4	let	let	VERB
ejpam-6321	106	5	(	(	PUNCT
ejpam-6321	106	6	x	x	NOUN
ejpam-6321	106	7	,	,	PUNCT
ejpam-6321	106	8	g	g	NOUN
ejpam-6321	106	9	)	)	PUNCT
ejpam-6321	106	10	be	be	AUX
ejpam-6321	106	11	a	a	DET
ejpam-6321	106	12	gm	gm	NOUN
ejpam-6321	106	13	-	-	PUNCT
ejpam-6321	106	14	space	space	NOUN
ejpam-6321	106	15	and	and	CCONJ
ejpam-6321	106	16	{	{	PUNCT
ejpam-6321	106	17	xȷ	xȷ	PROPN
ejpam-6321	106	18	}	}	PUNCT
ejpam-6321	106	19	be	be	AUX
ejpam-6321	106	20	a	a	DET
ejpam-6321	106	21	sequence	sequence	NOUN
ejpam-6321	106	22	in	in	ADP
ejpam-6321	106	23	x.	x.	NOUN
ejpam-6321	106	24	then	then	ADV
ejpam-6321	106	25	,	,	PUNCT
ejpam-6321	106	26	(	(	PUNCT
ejpam-6321	106	27	i	i	NOUN
ejpam-6321	106	28	)	)	PUNCT
ejpam-6321	106	29	a	a	DET
ejpam-6321	106	30	sequence	sequence	NOUN
ejpam-6321	106	31	{	{	PUNCT
ejpam-6321	106	32	xȷ	xȷ	PROPN
ejpam-6321	106	33	}	}	PUNCT
ejpam-6321	106	34	is	be	AUX
ejpam-6321	106	35	considered	consider	VERB
ejpam-6321	106	36	a	a	DET
ejpam-6321	106	37	g	g	NOUN
ejpam-6321	106	38	-	-	PUNCT
ejpam-6321	106	39	cauchy	cauchy	ADJ
ejpam-6321	106	40	sequence	sequence	NOUN
ejpam-6321	106	41	in	in	ADP
ejpam-6321	106	42	the	the	DET
ejpam-6321	106	43	gm	gm	NOUN
ejpam-6321	106	44	-	-	PUNCT
ejpam-6321	106	45	space	space	NOUN
ejpam-6321	106	46	x	x	NOUN
ejpam-6321	106	47	if	if	SCONJ
ejpam-6321	106	48	,	,	PUNCT
ejpam-6321	106	49	for	for	ADP
ejpam-6321	106	50	any	any	DET
ejpam-6321	106	51	ε	ε	PROPN
ejpam-6321	106	52	>	>	X
ejpam-6321	106	53	0	0	PROPN
ejpam-6321	106	54	,	,	PUNCT
ejpam-6321	106	55	∃n0	∃n0	NOUN
ejpam-6321	106	56	=	=	PUNCT
ejpam-6321	106	57	n	n	NOUN
ejpam-6321	106	58	so	so	ADV
ejpam-6321	106	59	that	that	SCONJ
ejpam-6321	106	60	g(xȷ	g(xȷ	NOUN
ejpam-6321	106	61	,	,	PUNCT
ejpam-6321	106	62	xm	xm	PROPN
ejpam-6321	106	63	,	,	PUNCT
ejpam-6321	106	64	xl	xl	PROPN
ejpam-6321	106	65	)	)	PUNCT
ejpam-6321	106	66	<	<	X
ejpam-6321	106	67	ε	ε	PROPN
ejpam-6321	106	68	∀ȷ	∀ȷ	PROPN
ejpam-6321	106	69	,	,	PUNCT
ejpam-6321	106	70	m	m	PROPN
ejpam-6321	106	71	,	,	PUNCT
ejpam-6321	106	72	l	l	PROPN
ejpam-6321	106	73	≥	≥	PROPN
ejpam-6321	106	74	n0	n0	NUM
ejpam-6321	106	75	.	.	PUNCT
ejpam-6321	107	1	(	(	PUNCT
ejpam-6321	107	2	ii	ii	NOUN
ejpam-6321	107	3	)	)	PUNCT
ejpam-6321	107	4	a	a	DET
ejpam-6321	107	5	sequence	sequence	NOUN
ejpam-6321	107	6	{	{	PUNCT
ejpam-6321	107	7	xȷ	xȷ	PROPN
ejpam-6321	107	8	}	}	PUNCT
ejpam-6321	107	9	in	in	ADP
ejpam-6321	107	10	the	the	DET
ejpam-6321	107	11	gm	gm	NOUN
ejpam-6321	107	12	-	-	PUNCT
ejpam-6321	107	13	space	space	NOUN
ejpam-6321	107	14	x	x	PUNCT
ejpam-6321	107	15	is	be	AUX
ejpam-6321	107	16	said	say	VERB
ejpam-6321	107	17	to	to	PART
ejpam-6321	107	18	converge	converge	VERB
ejpam-6321	107	19	to	to	ADP
ejpam-6321	107	20	an	an	DET
ejpam-6321	107	21	element	element	NOUN
ejpam-6321	107	22	xinx	xinx	NOUN
ejpam-6321	108	1	if	if	SCONJ
ejpam-6321	108	2	∀	∀	NOUN
ejpam-6321	108	3	any	any	PRON
ejpam-6321	108	4	given	give	VERB
ejpam-6321	108	5	ε	ε	PROPN
ejpam-6321	108	6	>	>	X
ejpam-6321	108	7	0	0	PUNCT
ejpam-6321	108	8	∈	∈	PROPN
ejpam-6321	108	9	r	r	NOUN
ejpam-6321	108	10	,	,	PUNCT
ejpam-6321	108	11	∃n0	∃n0	NOUN
ejpam-6321	108	12	=	=	PUNCT
ejpam-6321	108	13	n	n	NOUN
ejpam-6321	108	14	so	so	ADV
ejpam-6321	108	15	that	that	SCONJ
ejpam-6321	108	16	g(xȷ	g(xȷ	NOUN
ejpam-6321	108	17	,	,	PUNCT
ejpam-6321	108	18	xm	xm	PROPN
ejpam-6321	108	19	,	,	PUNCT
ejpam-6321	108	20	xl	xl	PROPN
ejpam-6321	108	21	)	)	PUNCT
ejpam-6321	108	22	<	<	X
ejpam-6321	108	23	ε	ε	PROPN
ejpam-6321	108	24	,	,	PUNCT
ejpam-6321	108	25	whenever	whenever	SCONJ
ejpam-6321	108	26	m	m	PROPN
ejpam-6321	108	27	≥	≥	PROPN
ejpam-6321	108	28	n0	n0	NUM
ejpam-6321	108	29	.	.	PUNCT
ejpam-6321	108	30	(	(	PUNCT
ejpam-6321	108	31	iii	iii	X
ejpam-6321	108	32	)	)	PUNCT
ejpam-6321	108	33	the	the	DET
ejpam-6321	108	34	gm	gm	NOUN
ejpam-6321	108	35	-	-	PUNCT
ejpam-6321	108	36	space	space	NOUN
ejpam-6321	108	37	(	(	PUNCT
ejpam-6321	108	38	x	x	NOUN
ejpam-6321	108	39	,	,	PUNCT
ejpam-6321	108	40	g	g	NOUN
ejpam-6321	108	41	)	)	PUNCT
ejpam-6321	108	42	is	be	AUX
ejpam-6321	108	43	classified	classify	VERB
ejpam-6321	108	44	as	as	ADV
ejpam-6321	108	45	complete	complete	ADJ
ejpam-6321	108	46	if	if	SCONJ
ejpam-6321	108	47	every	every	DET
ejpam-6321	108	48	g	g	NOUN
ejpam-6321	108	49	-	-	PUNCT
ejpam-6321	108	50	cauchy	cauchy	ADJ
ejpam-6321	108	51	sequence	sequence	NOUN
ejpam-6321	108	52	is	be	AUX
ejpam-6321	108	53	g	g	NOUN
ejpam-6321	108	54	-	-	PUNCT
ejpam-6321	108	55	convergent	convergent	NOUN
ejpam-6321	108	56	in	in	ADP
ejpam-6321	108	57	x.	x.	NOUN
ejpam-6321	108	58	proposition	proposition	NOUN
ejpam-6321	108	59	1	1	NUM
ejpam-6321	108	60	.	.	PUNCT
ejpam-6321	109	1	[	[	X
ejpam-6321	109	2	2	2	NUM
ejpam-6321	109	3	]	]	X
ejpam-6321	109	4	let	let	VERB
ejpam-6321	109	5	(	(	PUNCT
ejpam-6321	109	6	x	x	NOUN
ejpam-6321	109	7	,	,	PUNCT
ejpam-6321	109	8	g	g	NOUN
ejpam-6321	109	9	)	)	PUNCT
ejpam-6321	109	10	be	be	AUX
ejpam-6321	109	11	a	a	DET
ejpam-6321	109	12	gm	gm	NOUN
ejpam-6321	109	13	-	-	PUNCT
ejpam-6321	109	14	space	space	NOUN
ejpam-6321	109	15	,	,	PUNCT
ejpam-6321	109	16	then	then	ADV
ejpam-6321	109	17	for	for	ADP
ejpam-6321	109	18	any	any	DET
ejpam-6321	109	19	x1	x1	PROPN
ejpam-6321	109	20	,	,	PUNCT
ejpam-6321	109	21	x2	x2	PROPN
ejpam-6321	109	22	,	,	PUNCT
ejpam-6321	109	23	x3	x3	PROPN
ejpam-6321	109	24	∈	∈	PROPN
ejpam-6321	109	25	x	x	INTJ
ejpam-6321	109	26	it	it	PRON
ejpam-6321	109	27	followes	follow	VERB
ejpam-6321	109	28	that	that	SCONJ
ejpam-6321	109	29	:	:	PUNCT
ejpam-6321	109	30	(	(	PUNCT
ejpam-6321	109	31	i	i	NOUN
ejpam-6321	109	32	)	)	PUNCT
ejpam-6321	109	33	if	if	SCONJ
ejpam-6321	109	34	g(x1	g(x1	NOUN
ejpam-6321	109	35	,	,	PUNCT
ejpam-6321	109	36	x2	x2	PROPN
ejpam-6321	109	37	,	,	PUNCT
ejpam-6321	109	38	x3	x3	ADJ
ejpam-6321	109	39	)	)	PUNCT
ejpam-6321	110	1	=	=	SYM
ejpam-6321	110	2	0	0	NUM
ejpam-6321	110	3	,	,	PUNCT
ejpam-6321	110	4	then	then	ADV
ejpam-6321	110	5	x1	x1	PROPN
ejpam-6321	110	6	=	=	PUNCT
ejpam-6321	111	1	x2	x2	PROPN
ejpam-6321	111	2	=	=	SYM
ejpam-6321	111	3	x3	x3	PROPN
ejpam-6321	111	4	,	,	PUNCT
ejpam-6321	111	5	m.	m.	PROPN
ejpam-6321	111	6	noorwali	noorwali	PROPN
ejpam-6321	111	7	et	et	PROPN
ejpam-6321	111	8	al	al	PROPN
ejpam-6321	111	9	.	.	PUNCT
ejpam-6321	111	10	/	/	SYM
ejpam-6321	111	11	eur	eur	PROPN
ejpam-6321	111	12	.	.	PUNCT
ejpam-6321	112	1	j.	j.	PROPN
ejpam-6321	112	2	pure	pure	PROPN
ejpam-6321	112	3	appl	appl	PROPN
ejpam-6321	112	4	.	.	PROPN
ejpam-6321	112	5	math	math	PROPN
ejpam-6321	112	6	,	,	PUNCT
ejpam-6321	112	7	18	18	NUM
ejpam-6321	112	8	(	(	PUNCT
ejpam-6321	112	9	3	3	NUM
ejpam-6321	112	10	)	)	PUNCT
ejpam-6321	112	11	(	(	PUNCT
ejpam-6321	112	12	2025	2025	NUM
ejpam-6321	112	13	)	)	PUNCT
ejpam-6321	112	14	,	,	PUNCT
ejpam-6321	112	15	6321	6321	NUM
ejpam-6321	112	16	4	4	NUM
ejpam-6321	112	17	of	of	ADP
ejpam-6321	112	18	27	27	NUM
ejpam-6321	112	19	(	(	PUNCT
ejpam-6321	112	20	ii	ii	NOUN
ejpam-6321	112	21	)	)	PUNCT
ejpam-6321	112	22	g(x1	g(x1	NOUN
ejpam-6321	112	23	,	,	PUNCT
ejpam-6321	112	24	x2	x2	PROPN
ejpam-6321	112	25	,	,	PUNCT
ejpam-6321	112	26	x3	x3	ADJ
ejpam-6321	112	27	)	)	PUNCT
ejpam-6321	112	28	≤	≤	NOUN
ejpam-6321	112	29	g(x1	g(x1	NOUN
ejpam-6321	112	30	,	,	PUNCT
ejpam-6321	112	31	x1	x1	PROPN
ejpam-6321	112	32	,	,	PUNCT
ejpam-6321	112	33	x3	x3	ADJ
ejpam-6321	112	34	)	)	PUNCT
ejpam-6321	112	35	,	,	PUNCT
ejpam-6321	112	36	(	(	PUNCT
ejpam-6321	112	37	iii	iii	X
ejpam-6321	112	38	)	)	PUNCT
ejpam-6321	112	39	g(x1	g(x1	NOUN
ejpam-6321	112	40	,	,	PUNCT
ejpam-6321	112	41	x2	x2	PROPN
ejpam-6321	112	42	,	,	PUNCT
ejpam-6321	112	43	x2	x2	PROPN
ejpam-6321	112	44	)	)	PUNCT
ejpam-6321	112	45	≤	≤	NOUN
ejpam-6321	112	46	2g(x2	2g(x2	NUM
ejpam-6321	112	47	,	,	PUNCT
ejpam-6321	112	48	x1	x1	PROPN
ejpam-6321	112	49	,	,	PUNCT
ejpam-6321	112	50	x1	x1	PROPN
ejpam-6321	112	51	)	)	PUNCT
ejpam-6321	112	52	,	,	PUNCT
ejpam-6321	112	53	(	(	PUNCT
ejpam-6321	112	54	iv	iv	X
ejpam-6321	112	55	)	)	PUNCT
ejpam-6321	112	56	g(x1	g(x1	NOUN
ejpam-6321	112	57	,	,	PUNCT
ejpam-6321	112	58	x2	x2	PROPN
ejpam-6321	112	59	,	,	PUNCT
ejpam-6321	112	60	x3	x3	ADJ
ejpam-6321	112	61	)	)	PUNCT
ejpam-6321	112	62	≤	≤	NOUN
ejpam-6321	112	63	g(x1	g(x1	NOUN
ejpam-6321	112	64	,	,	PUNCT
ejpam-6321	112	65	x	x	NOUN
ejpam-6321	112	66	,	,	PUNCT
ejpam-6321	112	67	x3	x3	ADJ
ejpam-6321	112	68	)	)	PUNCT
ejpam-6321	113	1	+	+	ADP
ejpam-6321	113	2	g(x	g(x	NOUN
ejpam-6321	113	3	,	,	PUNCT
ejpam-6321	113	4	x2	x2	PROPN
ejpam-6321	113	5	,	,	PUNCT
ejpam-6321	113	6	x3	x3	ADJ
ejpam-6321	113	7	)	)	PUNCT
ejpam-6321	113	8	,	,	PUNCT
ejpam-6321	113	9	(	(	PUNCT
ejpam-6321	113	10	v	v	NOUN
ejpam-6321	113	11	)	)	PUNCT
ejpam-6321	113	12	g(x1	g(x1	NOUN
ejpam-6321	113	13	,	,	PUNCT
ejpam-6321	113	14	x2	x2	PROPN
ejpam-6321	113	15	,	,	PUNCT
ejpam-6321	113	16	x3	x3	ADJ
ejpam-6321	113	17	)	)	PUNCT
ejpam-6321	113	18	≤	≤	NUM
ejpam-6321	113	19	2	2	NUM
ejpam-6321	113	20	3(g(x1	3(g(x1	NUM
ejpam-6321	113	21	,	,	PUNCT
ejpam-6321	113	22	x2	x2	PROPN
ejpam-6321	113	23	,	,	PUNCT
ejpam-6321	113	24	x	x	X
ejpam-6321	113	25	)	)	PUNCT
ejpam-6321	113	26	+	+	NOUN
ejpam-6321	113	27	g(x1	g(x1	NOUN
ejpam-6321	113	28	,	,	PUNCT
ejpam-6321	113	29	x	x	NOUN
ejpam-6321	113	30	,	,	PUNCT
ejpam-6321	113	31	x3	x3	ADJ
ejpam-6321	113	32	)	)	PUNCT
ejpam-6321	114	1	+	+	ADP
ejpam-6321	114	2	g(x	g(x	NOUN
ejpam-6321	114	3	,	,	PUNCT
ejpam-6321	114	4	x2	x2	PROPN
ejpam-6321	114	5	,	,	PUNCT
ejpam-6321	114	6	x3	x3	ADJ
ejpam-6321	114	7	)	)	PUNCT
ejpam-6321	114	8	)	)	PUNCT
ejpam-6321	114	9	,	,	PUNCT
ejpam-6321	114	10	(	(	PUNCT
ejpam-6321	114	11	vi	vi	NOUN
ejpam-6321	114	12	)	)	PUNCT
ejpam-6321	114	13	g(x1	g(x1	NOUN
ejpam-6321	114	14	,	,	PUNCT
ejpam-6321	114	15	x2	x2	PROPN
ejpam-6321	114	16	,	,	PUNCT
ejpam-6321	114	17	x3	x3	ADJ
ejpam-6321	114	18	)	)	PUNCT
ejpam-6321	114	19	≤	≤	NOUN
ejpam-6321	114	20	(	(	PUNCT
ejpam-6321	114	21	g(x1	g(x1	NOUN
ejpam-6321	114	22	,	,	PUNCT
ejpam-6321	114	23	x	x	X
ejpam-6321	114	24	,	,	PUNCT
ejpam-6321	114	25	x	x	X
ejpam-6321	114	26	)	)	PUNCT
ejpam-6321	114	27	+	+	NOUN
ejpam-6321	114	28	g(x2	g(x2	NOUN
ejpam-6321	114	29	,	,	PUNCT
ejpam-6321	114	30	x	x	NOUN
ejpam-6321	114	31	,	,	PUNCT
ejpam-6321	114	32	x	x	X
ejpam-6321	114	33	)	)	PUNCT
ejpam-6321	115	1	+	+	ADJ
ejpam-6321	115	2	g(x3	g(x3	ADJ
ejpam-6321	115	3	,	,	PUNCT
ejpam-6321	115	4	x	x	NOUN
ejpam-6321	115	5	,	,	PUNCT
ejpam-6321	115	6	x	x	NOUN
ejpam-6321	115	7	)	)	PUNCT
ejpam-6321	115	8	)	)	PUNCT
ejpam-6321	115	9	,	,	PUNCT
ejpam-6321	115	10	(	(	PUNCT
ejpam-6321	115	11	vii	vii	PROPN
ejpam-6321	115	12	)	)	PUNCT
ejpam-6321	115	13	|g(x1	|g(x1	PROPN
ejpam-6321	115	14	,	,	PUNCT
ejpam-6321	115	15	x2	x2	PROPN
ejpam-6321	115	16	,	,	PUNCT
ejpam-6321	115	17	x3)−g(x1	x3)−g(x1	PROPN
ejpam-6321	115	18	,	,	PUNCT
ejpam-6321	115	19	x2	x2	PROPN
ejpam-6321	115	20	,	,	PUNCT
ejpam-6321	115	21	x)|	x)|	PROPN
ejpam-6321	115	22	≤	≤	PROPN
ejpam-6321	115	23	max{g(x	max{g(x	PROPN
ejpam-6321	115	24	,	,	PUNCT
ejpam-6321	115	25	x3	x3	ADJ
ejpam-6321	115	26	,	,	PUNCT
ejpam-6321	115	27	x3	x3	ADJ
ejpam-6321	115	28	)	)	PUNCT
ejpam-6321	115	29	,	,	PUNCT
ejpam-6321	115	30	g(x3	g(x3	NOUN
ejpam-6321	115	31	,	,	PUNCT
ejpam-6321	115	32	x	x	NOUN
ejpam-6321	115	33	,	,	PUNCT
ejpam-6321	115	34	x	x	NOUN
ejpam-6321	115	35	)	)	PUNCT
ejpam-6321	115	36	}	}	PUNCT
ejpam-6321	115	37	,	,	PUNCT
ejpam-6321	115	38	(	(	PUNCT
ejpam-6321	115	39	viii	viii	NOUN
ejpam-6321	115	40	)	)	PUNCT
ejpam-6321	115	41	|g(x1	|g(x1	PROPN
ejpam-6321	115	42	,	,	PUNCT
ejpam-6321	115	43	x2	x2	PROPN
ejpam-6321	115	44	,	,	PUNCT
ejpam-6321	115	45	x3)−g(x1	x3)−g(x1	PROPN
ejpam-6321	115	46	,	,	PUNCT
ejpam-6321	115	47	x2	x2	PROPN
ejpam-6321	115	48	,	,	PUNCT
ejpam-6321	115	49	x)|	x)|	PROPN
ejpam-6321	115	50	≤	≤	PUNCT
ejpam-6321	115	51	g(x1	g(x1	NOUN
ejpam-6321	115	52	,	,	PUNCT
ejpam-6321	115	53	x	x	NOUN
ejpam-6321	115	54	,	,	PUNCT
ejpam-6321	115	55	x3	x3	ADJ
ejpam-6321	115	56	)	)	PUNCT
ejpam-6321	115	57	,	,	PUNCT
ejpam-6321	115	58	(	(	PUNCT
ejpam-6321	115	59	ix	ix	ADJ
ejpam-6321	115	60	)	)	PUNCT
ejpam-6321	115	61	|g(x1	|g(x1	ADJ
ejpam-6321	115	62	,	,	PUNCT
ejpam-6321	115	63	x2	x2	PROPN
ejpam-6321	115	64	,	,	PUNCT
ejpam-6321	115	65	x3)−g(q	x3)−g(q	PROPN
ejpam-6321	115	66	,	,	PUNCT
ejpam-6321	115	67	x3	x3	ADJ
ejpam-6321	115	68	,	,	PUNCT
ejpam-6321	115	69	x3)|	x3)|	PROPN
ejpam-6321	115	70	≤	≤	NUM
ejpam-6321	115	71	max{g(x1	max{g(x1	NOUN
ejpam-6321	115	72	,	,	PUNCT
ejpam-6321	115	73	x3	x3	ADJ
ejpam-6321	115	74	,	,	PUNCT
ejpam-6321	115	75	x3	x3	ADJ
ejpam-6321	115	76	)	)	PUNCT
ejpam-6321	115	77	,	,	PUNCT
ejpam-6321	115	78	g(x3	g(x3	NUM
ejpam-6321	115	79	,	,	PUNCT
ejpam-6321	115	80	x1	x1	PROPN
ejpam-6321	115	81	,	,	PUNCT
ejpam-6321	115	82	x1	x1	PROPN
ejpam-6321	115	83	)	)	PUNCT
ejpam-6321	115	84	}	}	PUNCT
ejpam-6321	115	85	,	,	PUNCT
ejpam-6321	115	86	(	(	PUNCT
ejpam-6321	115	87	x	x	X
ejpam-6321	115	88	)	)	PUNCT
ejpam-6321	115	89	|g(x1	|g(x1	ADJ
ejpam-6321	115	90	,	,	PUNCT
ejpam-6321	115	91	x2	x2	PROPN
ejpam-6321	115	92	,	,	PUNCT
ejpam-6321	115	93	x2)−g(x2	x2)−g(x2	PROPN
ejpam-6321	115	94	,	,	PUNCT
ejpam-6321	115	95	x1	x1	PROPN
ejpam-6321	115	96	,	,	PUNCT
ejpam-6321	115	97	x1)|	x1)|	PROPN
ejpam-6321	115	98	≤	≤	NUM
ejpam-6321	115	99	max{g(x2	max{g(x2	NOUN
ejpam-6321	115	100	,	,	PUNCT
ejpam-6321	115	101	x1	x1	PROPN
ejpam-6321	115	102	,	,	PUNCT
ejpam-6321	115	103	x1	x1	PROPN
ejpam-6321	115	104	)	)	PUNCT
ejpam-6321	115	105	,	,	PUNCT
ejpam-6321	115	106	g(x1	g(x1	NOUN
ejpam-6321	115	107	,	,	PUNCT
ejpam-6321	115	108	x2	x2	PROPN
ejpam-6321	115	109	,	,	PUNCT
ejpam-6321	115	110	x2	x2	PROPN
ejpam-6321	115	111	)	)	PUNCT
ejpam-6321	115	112	}	}	PUNCT
ejpam-6321	115	113	.	.	PUNCT
ejpam-6321	116	1	3	3	X
ejpam-6321	116	2	.	.	X
ejpam-6321	116	3	main	main	ADJ
ejpam-6321	116	4	results	result	NOUN
ejpam-6321	116	5	this	this	DET
ejpam-6321	116	6	section	section	NOUN
ejpam-6321	116	7	explores	explore	VERB
ejpam-6321	116	8	the	the	DET
ejpam-6321	116	9	existence	existence	NOUN
ejpam-6321	116	10	of	of	ADP
ejpam-6321	116	11	common	common	ADJ
ejpam-6321	116	12	fixed	fix	VERB
ejpam-6321	116	13	points	point	NOUN
ejpam-6321	116	14	(	(	PUNCT
ejpam-6321	116	15	cfp	cfp	NOUN
ejpam-6321	116	16	)	)	PUNCT
ejpam-6321	116	17	and	and	CCONJ
ejpam-6321	116	18	unique	unique	ADJ
ejpam-6321	116	19	common	common	ADJ
ejpam-6321	116	20	fixed	fix	VERB
ejpam-6321	116	21	points	point	NOUN
ejpam-6321	116	22	(	(	PUNCT
ejpam-6321	116	23	ucfp	ucfp	PROPN
ejpam-6321	116	24	)	)	PUNCT
ejpam-6321	116	25	for	for	ADP
ejpam-6321	116	26	three	three	NUM
ejpam-6321	116	27	self	self	NOUN
ejpam-6321	116	28	-	-	PUNCT
ejpam-6321	116	29	mappings	mapping	NOUN
ejpam-6321	116	30	in	in	ADP
ejpam-6321	116	31	gm	gm	PROPN
ejpam-6321	116	32	-	-	PUNCT
ejpam-6321	116	33	spaces	space	NOUN
ejpam-6321	116	34	.	.	PUNCT
ejpam-6321	117	1	theorems	theorem	NOUN
ejpam-6321	117	2	1	1	NUM
ejpam-6321	117	3	and	and	CCONJ
ejpam-6321	117	4	2	2	NUM
ejpam-6321	117	5	provide	provide	VERB
ejpam-6321	117	6	conditions	condition	NOUN
ejpam-6321	117	7	under	under	ADP
ejpam-6321	117	8	which	which	PRON
ejpam-6321	117	9	these	these	DET
ejpam-6321	117	10	mappings	mapping	NOUN
ejpam-6321	117	11	possess	possess	VERB
ejpam-6321	117	12	a	a	DET
ejpam-6321	117	13	cfp	cfp	NOUN
ejpam-6321	117	14	and	and	CCONJ
ejpam-6321	117	15	ucfp	ucfp	NOUN
ejpam-6321	117	16	,	,	PUNCT
ejpam-6321	117	17	involving	involve	VERB
ejpam-6321	117	18	inequalities	inequality	NOUN
ejpam-6321	117	19	with	with	ADP
ejpam-6321	117	20	constants	constant	NOUN
ejpam-6321	117	21	an	an	DET
ejpam-6321	117	22	example	example	NOUN
ejpam-6321	117	23	using	use	VERB
ejpam-6321	117	24	the	the	DET
ejpam-6321	117	25	gm	gm	NOUN
ejpam-6321	117	26	-	-	PUNCT
ejpam-6321	117	27	metric	metric	ADJ
ejpam-6321	117	28	on	on	ADP
ejpam-6321	117	29	the	the	DET
ejpam-6321	117	30	interval	interval	NOUN
ejpam-6321	117	31	[	[	X
ejpam-6321	117	32	0,1	0,1	NUM
ejpam-6321	117	33	]	]	PUNCT
ejpam-6321	117	34	demonstrates	demonstrate	VERB
ejpam-6321	117	35	the	the	DET
ejpam-6321	117	36	conditions	condition	NOUN
ejpam-6321	117	37	required	require	VERB
ejpam-6321	117	38	for	for	ADP
ejpam-6321	117	39	the	the	DET
ejpam-6321	117	40	existence	existence	NOUN
ejpam-6321	117	41	of	of	ADP
ejpam-6321	117	42	a	a	DET
ejpam-6321	117	43	ucfp	ucfp	NOUN
ejpam-6321	117	44	,	,	PUNCT
ejpam-6321	117	45	contributing	contribute	VERB
ejpam-6321	117	46	to	to	ADP
ejpam-6321	117	47	the	the	DET
ejpam-6321	117	48	fixed	fix	VERB
ejpam-6321	117	49	point	point	NOUN
ejpam-6321	117	50	theory	theory	NOUN
ejpam-6321	117	51	in	in	ADP
ejpam-6321	117	52	gm	gm	PROPN
ejpam-6321	117	53	-	-	PUNCT
ejpam-6321	117	54	spaces	space	NOUN
ejpam-6321	117	55	.	.	PUNCT
ejpam-6321	118	1	theorem	theorem	NOUN
ejpam-6321	118	2	1	1	NUM
ejpam-6321	118	3	.	.	PUNCT
ejpam-6321	119	1	let	let	AUX
ejpam-6321	119	2	(	(	PUNCT
ejpam-6321	119	3	x	x	NOUN
ejpam-6321	119	4	,	,	PUNCT
ejpam-6321	119	5	g	g	NOUN
ejpam-6321	119	6	)	)	PUNCT
ejpam-6321	119	7	be	be	AUX
ejpam-6321	119	8	a	a	DET
ejpam-6321	119	9	gm	gm	NOUN
ejpam-6321	119	10	-	-	PUNCT
ejpam-6321	119	11	space	space	NOUN
ejpam-6321	119	12	and	and	CCONJ
ejpam-6321	119	13	f	f	NOUN
ejpam-6321	119	14	:	:	PUNCT
ejpam-6321	120	1	x	x	X
ejpam-6321	120	2	×x	×x	X
ejpam-6321	120	3	×x	×x	X
ejpam-6321	120	4	→	→	SYM
ejpam-6321	120	5	x	x	PART
ejpam-6321	120	6	be	be	AUX
ejpam-6321	120	7	a	a	DET
ejpam-6321	120	8	mapping	mapping	NOUN
ejpam-6321	120	9	satisfying	satisfying	ADJ
ejpam-6321	120	10	:	:	PUNCT
ejpam-6321	120	11	g(f1x1	g(f1x1	NOUN
ejpam-6321	120	12	,	,	PUNCT
ejpam-6321	120	13	f2x2	f2x2	NOUN
ejpam-6321	120	14	,	,	PUNCT
ejpam-6321	120	15	f3x3	f3x3	X
ejpam-6321	120	16	)	)	PUNCT
ejpam-6321	120	17	≤	≤	NOUN
ejpam-6321	121	1	α1g(x1	α1g(x1	PROPN
ejpam-6321	121	2	,	,	PUNCT
ejpam-6321	121	3	x2	x2	PROPN
ejpam-6321	121	4	,	,	PUNCT
ejpam-6321	121	5	x3	x3	ADJ
ejpam-6321	121	6	)	)	PUNCT
ejpam-6321	122	1	+	+	CCONJ
ejpam-6321	122	2	α2g(x1	α2g(x1	PROPN
ejpam-6321	122	3	,	,	PUNCT
ejpam-6321	122	4	x2	x2	PROPN
ejpam-6321	122	5	,	,	PUNCT
ejpam-6321	122	6	f2x2	f2x2	X
ejpam-6321	122	7	)	)	PUNCT
ejpam-6321	122	8	+	+	CCONJ
ejpam-6321	122	9	α3g(f1x1	α3g(f1x1	PROPN
ejpam-6321	122	10	,	,	PUNCT
ejpam-6321	122	11	x2	x2	PROPN
ejpam-6321	122	12	,	,	PUNCT
ejpam-6321	122	13	x2	x2	PROPN
ejpam-6321	122	14	)	)	PUNCT
ejpam-6321	122	15	+	+	X
ejpam-6321	123	1	α4[g(f1x1	α4[g(f1x1	PROPN
ejpam-6321	123	2	,	,	PUNCT
ejpam-6321	123	3	x1	x1	PROPN
ejpam-6321	123	4	,	,	PUNCT
ejpam-6321	123	5	x1	x1	PROPN
ejpam-6321	123	6	)	)	PUNCT
ejpam-6321	124	1	+	+	NUM
ejpam-6321	124	2	g(x2	g(x2	NOUN
ejpam-6321	124	3	,	,	PUNCT
ejpam-6321	124	4	f2x2	f2x2	NOUN
ejpam-6321	124	5	,	,	PUNCT
ejpam-6321	124	6	x2	x2	PROPN
ejpam-6321	124	7	)	)	PUNCT
ejpam-6321	125	1	+	+	ADJ
ejpam-6321	125	2	g(x3	g(x3	ADJ
ejpam-6321	125	3	,	,	PUNCT
ejpam-6321	125	4	x3	x3	ADJ
ejpam-6321	125	5	,	,	PUNCT
ejpam-6321	125	6	f3x3	f3x3	X
ejpam-6321	125	7	)	)	PUNCT
ejpam-6321	125	8	]	]	PUNCT
ejpam-6321	126	1	+	+	CCONJ
ejpam-6321	126	2	α5min	α5min	NUM
ejpam-6321	126	3	{	{	PUNCT
ejpam-6321	126	4	g(x1	g(x1	NOUN
ejpam-6321	126	5	,	,	PUNCT
ejpam-6321	126	6	x2	x2	PROPN
ejpam-6321	126	7	,	,	PUNCT
ejpam-6321	126	8	f2x2	f2x2	NOUN
ejpam-6321	126	9	)	)	PUNCT
ejpam-6321	126	10	,	,	PUNCT
ejpam-6321	126	11	g(f1x1	g(f1x1	PROPN
ejpam-6321	126	12	,	,	PUNCT
ejpam-6321	126	13	f1x1	f1x1	NOUN
ejpam-6321	126	14	,	,	PUNCT
ejpam-6321	126	15	x2	x2	PROPN
ejpam-6321	126	16	)	)	PUNCT
ejpam-6321	126	17	,	,	PUNCT
ejpam-6321	126	18	g(f1x1	g(f1x1	PROPN
ejpam-6321	126	19	,	,	PUNCT
ejpam-6321	126	20	x2	x2	PROPN
ejpam-6321	126	21	,	,	PUNCT
ejpam-6321	126	22	x2	x2	PROPN
ejpam-6321	126	23	)	)	PUNCT
ejpam-6321	126	24	,	,	PUNCT
ejpam-6321	126	25	g(f2x2	g(f2x2	NOUN
ejpam-6321	126	26	,	,	PUNCT
ejpam-6321	126	27	f2x2	f2x2	NOUN
ejpam-6321	126	28	,	,	PUNCT
ejpam-6321	126	29	x3	x3	ADJ
ejpam-6321	126	30	)	)	PUNCT
ejpam-6321	126	31	}	}	PUNCT
ejpam-6321	126	32	(	(	PUNCT
ejpam-6321	126	33	1	1	X
ejpam-6321	126	34	)	)	PUNCT
ejpam-6321	126	35	for	for	ADP
ejpam-6321	126	36	all	all	DET
ejpam-6321	126	37	x1	x1	PROPN
ejpam-6321	126	38	,	,	PUNCT
ejpam-6321	126	39	x2	x2	PROPN
ejpam-6321	126	40	,	,	PUNCT
ejpam-6321	126	41	x3	x3	PROPN
ejpam-6321	126	42	∈	∈	PROPN
ejpam-6321	126	43	x	x	X
ejpam-6321	126	44	and	and	CCONJ
ejpam-6321	126	45	α1	α1	PROPN
ejpam-6321	126	46	,	,	PUNCT
ejpam-6321	126	47	α2	α2	ADJ
ejpam-6321	126	48	,	,	PUNCT
ejpam-6321	126	49	α3	α3	PROPN
ejpam-6321	126	50	,	,	PUNCT
ejpam-6321	126	51	α4	α4	NOUN
ejpam-6321	126	52	≥	≥	NOUN
ejpam-6321	126	53	0	0	NUM
ejpam-6321	126	54	with	with	ADP
ejpam-6321	126	55	α1	α1	PROPN
ejpam-6321	126	56	,	,	PUNCT
ejpam-6321	126	57	α2	α2	ADJ
ejpam-6321	126	58	,	,	PUNCT
ejpam-6321	126	59	α3	α3	NOUN
ejpam-6321	126	60	,	,	PUNCT
ejpam-6321	126	61	α4	α4	VERB
ejpam-6321	126	62	<	<	X
ejpam-6321	126	63	1	1	NUM
ejpam-6321	126	64	and	and	CCONJ
ejpam-6321	126	65	(	(	PUNCT
ejpam-6321	126	66	α1	α1	PROPN
ejpam-6321	126	67	+	+	CCONJ
ejpam-6321	126	68	α2	α2	ADJ
ejpam-6321	126	69	+	+	CCONJ
ejpam-6321	126	70	α3	α3	ADJ
ejpam-6321	126	71	+	+	CCONJ
ejpam-6321	126	72	α4	α4	NOUN
ejpam-6321	126	73	)	)	PUNCT
ejpam-6321	126	74	<	<	X
ejpam-6321	127	1	1	1	X
ejpam-6321	127	2	.	.	PUNCT
ejpam-6321	127	3	then	then	ADV
ejpam-6321	127	4	,	,	PUNCT
ejpam-6321	127	5	the	the	DET
ejpam-6321	127	6	three	three	NUM
ejpam-6321	127	7	individual	individual	ADJ
ejpam-6321	127	8	self	self	NOUN
ejpam-6321	127	9	-	-	PUNCT
ejpam-6321	127	10	mappings	mapping	NOUN
ejpam-6321	127	11	referred	refer	VERB
ejpam-6321	127	12	as	as	ADP
ejpam-6321	127	13	f1	f1	NOUN
ejpam-6321	127	14	,	,	PUNCT
ejpam-6321	127	15	f2	f2	PROPN
ejpam-6321	127	16	and	and	CCONJ
ejpam-6321	127	17	f3	f3	PROPN
ejpam-6321	127	18	possess	possess	VERB
ejpam-6321	127	19	a	a	DET
ejpam-6321	127	20	cfp	cfp	NOUN
ejpam-6321	127	21	in	in	ADP
ejpam-6321	127	22	x.	x.	PROPN
ejpam-6321	127	23	moreover	moreover	ADV
ejpam-6321	127	24	,	,	PUNCT
ejpam-6321	127	25	if	if	SCONJ
ejpam-6321	127	26	(	(	PUNCT
ejpam-6321	127	27	α1	α1	PROPN
ejpam-6321	127	28	+	+	CCONJ
ejpam-6321	127	29	α2	α2	ADJ
ejpam-6321	127	30	+	+	CCONJ
ejpam-6321	127	31	α3	α3	ADJ
ejpam-6321	127	32	)	)	PUNCT
ejpam-6321	127	33	<	<	X
ejpam-6321	127	34	1	1	NUM
ejpam-6321	127	35	,	,	PUNCT
ejpam-6321	127	36	the	the	DET
ejpam-6321	127	37	three	three	NUM
ejpam-6321	127	38	individual	individual	ADJ
ejpam-6321	127	39	self	self	NOUN
ejpam-6321	127	40	-	-	PUNCT
ejpam-6321	127	41	mappings	mapping	NOUN
ejpam-6321	127	42	referred	refer	VERB
ejpam-6321	127	43	as	as	ADP
ejpam-6321	127	44	f1	f1	NOUN
ejpam-6321	127	45	,	,	PUNCT
ejpam-6321	127	46	f2	f2	PROPN
ejpam-6321	127	47	and	and	CCONJ
ejpam-6321	127	48	f3	f3	PROPN
ejpam-6321	127	49	possess	possess	VERB
ejpam-6321	127	50	a	a	DET
ejpam-6321	127	51	ucfp	ucfp	NOUN
ejpam-6321	127	52	in	in	ADP
ejpam-6321	127	53	x.	x.	NOUN
ejpam-6321	127	54	proof	proof	NOUN
ejpam-6321	127	55	.	.	PUNCT
ejpam-6321	128	1	fix	fix	VERB
ejpam-6321	128	2	x0	x0	PROPN
ejpam-6321	128	3	∈	∈	PROPN
ejpam-6321	128	4	x	x	X
ejpam-6321	128	5	,	,	PUNCT
ejpam-6321	128	6	and	and	CCONJ
ejpam-6321	128	7	{	{	PUNCT
ejpam-6321	128	8	xk	xk	NOUN
ejpam-6321	128	9	}	}	PUNCT
ejpam-6321	128	10	be	be	AUX
ejpam-6321	128	11	a	a	DET
ejpam-6321	128	12	sequence	sequence	NOUN
ejpam-6321	128	13	in	in	ADP
ejpam-6321	128	14	x.	x.	NOUN
ejpam-6321	129	1	now	now	ADV
ejpam-6321	129	2	we	we	PRON
ejpam-6321	129	3	define	define	VERB
ejpam-6321	129	4	some	some	DET
ejpam-6321	129	5	iterative	iterative	NOUN
ejpam-6321	129	6	sequences	sequence	NOUN
ejpam-6321	129	7	in	in	ADP
ejpam-6321	129	8	x	x	X
ejpam-6321	129	9	such	such	ADJ
ejpam-6321	129	10	that	that	PRON
ejpam-6321	129	11	x(3k+1	x(3k+1	CCONJ
ejpam-6321	129	12	)	)	PUNCT
ejpam-6321	129	13	=	=	SYM
ejpam-6321	130	1	f1x3k	f1x3k	NUM
ejpam-6321	130	2	,	,	PUNCT
ejpam-6321	130	3	x(3k+2	x(3k+2	NOUN
ejpam-6321	130	4	)	)	PUNCT
ejpam-6321	130	5	=	=	PUNCT
ejpam-6321	131	1	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	131	2	)	)	PUNCT
ejpam-6321	131	3	and	and	CCONJ
ejpam-6321	131	4	x(3k+3	x(3k+3	NUM
ejpam-6321	131	5	)	)	PUNCT
ejpam-6321	132	1	=	=	SYM
ejpam-6321	132	2	f3x(3k+2	f3x(3k+2	NUM
ejpam-6321	132	3	)	)	PUNCT
ejpam-6321	132	4	∀	∀	PUNCT
ejpam-6321	133	1	k	k	X
ejpam-6321	133	2	≥	≥	NOUN
ejpam-6321	133	3	0	0	NUM
ejpam-6321	133	4	.	.	PUNCT
ejpam-6321	134	1	now	now	ADV
ejpam-6321	134	2	,	,	PUNCT
ejpam-6321	134	3	by	by	ADP
ejpam-6321	134	4	using	use	VERB
ejpam-6321	134	5	(	(	PUNCT
ejpam-6321	134	6	1	1	X
ejpam-6321	134	7	)	)	PUNCT
ejpam-6321	134	8	we	we	PRON
ejpam-6321	134	9	have	have	VERB
ejpam-6321	134	10	,	,	PUNCT
ejpam-6321	134	11	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	134	12	)	)	PUNCT
ejpam-6321	134	13	,	,	PUNCT
ejpam-6321	134	14	x(3k+2	x(3k+2	NOUN
ejpam-6321	134	15	)	)	PUNCT
ejpam-6321	134	16	,	,	PUNCT
ejpam-6321	134	17	x(3k+3	x(3k+3	NUM
ejpam-6321	134	18	)	)	PUNCT
ejpam-6321	134	19	)	)	PUNCT
ejpam-6321	135	1	=	=	PUNCT
ejpam-6321	135	2	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	135	3	,	,	PUNCT
ejpam-6321	135	4	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	135	5	)	)	PUNCT
ejpam-6321	135	6	,	,	PUNCT
ejpam-6321	135	7	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	135	8	)	)	PUNCT
ejpam-6321	135	9	)	)	PUNCT
ejpam-6321	135	10	≤	≤	NUM
ejpam-6321	135	11	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	135	12	,	,	PUNCT
ejpam-6321	135	13	x(3k+1	x(3k+1	PRON
ejpam-6321	135	14	)	)	PUNCT
ejpam-6321	135	15	,	,	PUNCT
ejpam-6321	135	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	135	17	)	)	PUNCT
ejpam-6321	135	18	)	)	PUNCT
ejpam-6321	136	1	+	+	CCONJ
ejpam-6321	136	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	136	3	,	,	PUNCT
ejpam-6321	136	4	x(3k+1	x(3k+1	PRON
ejpam-6321	136	5	)	)	PUNCT
ejpam-6321	136	6	,	,	PUNCT
ejpam-6321	136	7	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	136	8	)	)	PUNCT
ejpam-6321	136	9	)	)	PUNCT
ejpam-6321	137	1	+	+	CCONJ
ejpam-6321	137	2	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	137	3	,	,	PUNCT
ejpam-6321	137	4	x(3k+1	x(3k+1	ADV
ejpam-6321	137	5	)	)	PUNCT
ejpam-6321	137	6	,	,	PUNCT
ejpam-6321	137	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	137	8	)	)	PUNCT
ejpam-6321	137	9	)	)	PUNCT
ejpam-6321	138	1	+	+	CCONJ
ejpam-6321	138	2	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	138	3	,	,	PUNCT
ejpam-6321	138	4	x3k	x3k	NOUN
ejpam-6321	138	5	,	,	PUNCT
ejpam-6321	138	6	x3k	x3k	PUNCT
ejpam-6321	138	7	)	)	PUNCT
ejpam-6321	138	8	+	+	SYM
ejpam-6321	138	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	138	10	)	)	PUNCT
ejpam-6321	138	11	,	,	PUNCT
ejpam-6321	138	12	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	138	13	)	)	PUNCT
ejpam-6321	138	14	,	,	PUNCT
ejpam-6321	138	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	138	16	)	)	PUNCT
ejpam-6321	138	17	)	)	PUNCT
ejpam-6321	139	1	+	+	PUNCT
ejpam-6321	139	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	139	3	)	)	PUNCT
ejpam-6321	139	4	,	,	PUNCT
ejpam-6321	139	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	139	6	)	)	PUNCT
ejpam-6321	139	7	,	,	PUNCT
ejpam-6321	139	8	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	139	9	)	)	PUNCT
ejpam-6321	139	10	)	)	PUNCT
ejpam-6321	139	11	]	]	PUNCT
ejpam-6321	140	1	+	+	CCONJ
ejpam-6321	140	2	α5min	α5min	NUM
ejpam-6321	140	3	{	{	PUNCT
ejpam-6321	140	4	g(x3k	g(x3k	NOUN
ejpam-6321	140	5	,	,	PUNCT
ejpam-6321	140	6	x(3k+1	x(3k+1	PRON
ejpam-6321	140	7	)	)	PUNCT
ejpam-6321	140	8	,	,	PUNCT
ejpam-6321	140	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	140	10	)	)	PUNCT
ejpam-6321	140	11	)	)	PUNCT
ejpam-6321	140	12	,	,	PUNCT
ejpam-6321	140	13	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	140	14	,	,	PUNCT
ejpam-6321	140	15	f1x3k	f1x3k	NUM
ejpam-6321	140	16	,	,	PUNCT
ejpam-6321	140	17	x(3k+1	x(3k+1	NUM
ejpam-6321	140	18	)	)	PUNCT
ejpam-6321	140	19	)	)	PUNCT
ejpam-6321	140	20	,	,	PUNCT
ejpam-6321	140	21	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	140	22	,	,	PUNCT
ejpam-6321	140	23	x(3k+1	x(3k+1	ADV
ejpam-6321	140	24	)	)	PUNCT
ejpam-6321	140	25	,	,	PUNCT
ejpam-6321	140	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	140	27	)	)	PUNCT
ejpam-6321	140	28	)	)	PUNCT
ejpam-6321	140	29	,	,	PUNCT
ejpam-6321	140	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	140	31	)	)	PUNCT
ejpam-6321	140	32	,	,	PUNCT
ejpam-6321	140	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	140	34	)	)	PUNCT
ejpam-6321	140	35	,	,	PUNCT
ejpam-6321	140	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	140	37	)	)	PUNCT
ejpam-6321	140	38	)	)	PUNCT
ejpam-6321	140	39	}	}	PUNCT
ejpam-6321	141	1	=	=	SYM
ejpam-6321	141	2	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	141	3	,	,	PUNCT
ejpam-6321	141	4	x(3k+1	x(3k+1	PRON
ejpam-6321	141	5	)	)	PUNCT
ejpam-6321	141	6	,	,	PUNCT
ejpam-6321	141	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	141	8	)	)	PUNCT
ejpam-6321	141	9	)	)	PUNCT
ejpam-6321	142	1	+	+	CCONJ
ejpam-6321	142	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	142	3	,	,	PUNCT
ejpam-6321	142	4	x(3k+1	x(3k+1	CCONJ
ejpam-6321	142	5	)	)	PUNCT
ejpam-6321	142	6	,	,	PUNCT
ejpam-6321	142	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	142	8	)	)	PUNCT
ejpam-6321	142	9	)	)	PUNCT
ejpam-6321	143	1	+	+	CCONJ
ejpam-6321	143	2	α3g(x3k	α3g(x3k	NOUN
ejpam-6321	143	3	,	,	PUNCT
ejpam-6321	143	4	x(3k+1	x(3k+1	ADV
ejpam-6321	143	5	)	)	PUNCT
ejpam-6321	143	6	,	,	PUNCT
ejpam-6321	143	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	143	8	)	)	PUNCT
ejpam-6321	143	9	)	)	PUNCT
ejpam-6321	144	1	+	+	CCONJ
ejpam-6321	144	2	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	144	3	,	,	PUNCT
ejpam-6321	144	4	x3k	x3k	NOUN
ejpam-6321	144	5	,	,	PUNCT
ejpam-6321	144	6	x3k	x3k	PUNCT
ejpam-6321	144	7	)	)	PUNCT
ejpam-6321	144	8	+	+	SYM
ejpam-6321	144	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	144	10	)	)	PUNCT
ejpam-6321	144	11	,	,	PUNCT
ejpam-6321	144	12	x(3k+1	x(3k+1	CCONJ
ejpam-6321	144	13	)	)	PUNCT
ejpam-6321	144	14	,	,	PUNCT
ejpam-6321	144	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	144	16	)	)	PUNCT
ejpam-6321	144	17	)	)	PUNCT
ejpam-6321	145	1	+	+	PUNCT
ejpam-6321	145	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	145	3	)	)	PUNCT
ejpam-6321	145	4	,	,	PUNCT
ejpam-6321	145	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	145	6	)	)	PUNCT
ejpam-6321	145	7	,	,	PUNCT
ejpam-6321	145	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	145	9	)	)	PUNCT
ejpam-6321	145	10	)	)	PUNCT
ejpam-6321	145	11	]	]	PUNCT
ejpam-6321	146	1	+	+	CCONJ
ejpam-6321	146	2	α5min	α5min	NUM
ejpam-6321	146	3	{	{	PUNCT
ejpam-6321	146	4	g(x3k	g(x3k	NOUN
ejpam-6321	146	5	,	,	PUNCT
ejpam-6321	146	6	x(3k+1	x(3k+1	PRON
ejpam-6321	146	7	)	)	PUNCT
ejpam-6321	146	8	,	,	PUNCT
ejpam-6321	146	9	x(3k+1	x(3k+1	CCONJ
ejpam-6321	146	10	)	)	PUNCT
ejpam-6321	146	11	)	)	PUNCT
ejpam-6321	146	12	,	,	PUNCT
ejpam-6321	146	13	g(x3k	g(x3k	NOUN
ejpam-6321	146	14	,	,	PUNCT
ejpam-6321	146	15	x3k	x3k	NOUN
ejpam-6321	146	16	,	,	PUNCT
ejpam-6321	146	17	x(3k+1	x(3k+1	ADV
ejpam-6321	146	18	)	)	PUNCT
ejpam-6321	146	19	)	)	PUNCT
ejpam-6321	146	20	,	,	PUNCT
ejpam-6321	146	21	g(x3k	g(x3k	NOUN
ejpam-6321	146	22	,	,	PUNCT
ejpam-6321	146	23	x(3k+1	x(3k+1	PRON
ejpam-6321	146	24	)	)	PUNCT
ejpam-6321	146	25	,	,	PUNCT
ejpam-6321	146	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	146	27	)	)	PUNCT
ejpam-6321	146	28	)	)	PUNCT
ejpam-6321	146	29	,	,	PUNCT
ejpam-6321	146	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	146	31	)	)	PUNCT
ejpam-6321	146	32	,	,	PUNCT
ejpam-6321	146	33	x(3k+1	x(3k+1	CCONJ
ejpam-6321	146	34	)	)	PUNCT
ejpam-6321	146	35	,	,	PUNCT
ejpam-6321	146	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	146	37	)	)	PUNCT
ejpam-6321	146	38	)	)	PUNCT
ejpam-6321	146	39	}	}	PUNCT
ejpam-6321	146	40	m.	m.	PROPN
ejpam-6321	146	41	noorwali	noorwali	PROPN
ejpam-6321	146	42	et	et	PROPN
ejpam-6321	146	43	al	al	PROPN
ejpam-6321	146	44	.	.	PUNCT
ejpam-6321	146	45	/	/	SYM
ejpam-6321	146	46	eur	eur	PROPN
ejpam-6321	146	47	.	.	PUNCT
ejpam-6321	147	1	j.	j.	PROPN
ejpam-6321	147	2	pure	pure	PROPN
ejpam-6321	147	3	appl	appl	PROPN
ejpam-6321	147	4	.	.	PROPN
ejpam-6321	147	5	math	math	PROPN
ejpam-6321	147	6	,	,	PUNCT
ejpam-6321	147	7	18	18	NUM
ejpam-6321	147	8	(	(	PUNCT
ejpam-6321	147	9	3	3	NUM
ejpam-6321	147	10	)	)	PUNCT
ejpam-6321	147	11	(	(	PUNCT
ejpam-6321	147	12	2025	2025	NUM
ejpam-6321	147	13	)	)	PUNCT
ejpam-6321	147	14	,	,	PUNCT
ejpam-6321	147	15	6321	6321	NUM
ejpam-6321	147	16	5	5	NUM
ejpam-6321	147	17	of	of	ADP
ejpam-6321	147	18	27	27	NUM
ejpam-6321	147	19	after	after	ADP
ejpam-6321	147	20	applying	apply	VERB
ejpam-6321	147	21	the	the	DET
ejpam-6321	147	22	simplification	simplification	NOUN
ejpam-6321	147	23	process	process	NOUN
ejpam-6321	147	24	utilizing	utilize	VERB
ejpam-6321	147	25	the	the	DET
ejpam-6321	147	26	definition	definition	NOUN
ejpam-6321	147	27	of	of	ADP
ejpam-6321	147	28	gm	gm	PROPN
ejpam-6321	147	29	-	-	PUNCT
ejpam-6321	147	30	space	space	NOUN
ejpam-6321	147	31	we	we	PRON
ejpam-6321	147	32	have	have	AUX
ejpam-6321	147	33	achieved	achieve	VERB
ejpam-6321	147	34	the	the	DET
ejpam-6321	147	35	following	following	ADJ
ejpam-6321	147	36	result	result	NOUN
ejpam-6321	147	37	,	,	PUNCT
ejpam-6321	147	38	≤	≤	NUM
ejpam-6321	147	39	(	(	PUNCT
ejpam-6321	147	40	α1	α1	PROPN
ejpam-6321	147	41	+	+	CCONJ
ejpam-6321	147	42	α2)g(x3k	α2)g(x3k	ADJ
ejpam-6321	147	43	,	,	PUNCT
ejpam-6321	147	44	x(3k+1	x(3k+1	ADV
ejpam-6321	147	45	)	)	PUNCT
ejpam-6321	147	46	,	,	PUNCT
ejpam-6321	147	47	x(3k+2	x(3k+2	NOUN
ejpam-6321	147	48	)	)	PUNCT
ejpam-6321	147	49	)	)	PUNCT
ejpam-6321	148	1	+	+	CCONJ
ejpam-6321	149	1	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	149	2	,	,	PUNCT
ejpam-6321	149	3	x(3k+1	x(3k+1	PRON
ejpam-6321	149	4	)	)	PUNCT
ejpam-6321	149	5	,	,	PUNCT
ejpam-6321	149	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	149	7	)	)	PUNCT
ejpam-6321	149	8	)	)	PUNCT
ejpam-6321	150	1	+	+	ADP
ejpam-6321	150	2	g(x3k	g(x3k	NOUN
ejpam-6321	150	3	,	,	PUNCT
ejpam-6321	150	4	x(3k+1	x(3k+1	PRON
ejpam-6321	150	5	)	)	PUNCT
ejpam-6321	150	6	,	,	PUNCT
ejpam-6321	150	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	150	8	)	)	PUNCT
ejpam-6321	150	9	)	)	PUNCT
ejpam-6321	151	1	+	+	PUNCT
ejpam-6321	151	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	151	3	)	)	PUNCT
ejpam-6321	151	4	,	,	PUNCT
ejpam-6321	151	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	151	6	)	)	PUNCT
ejpam-6321	151	7	,	,	PUNCT
ejpam-6321	151	8	x(3k+3	x(3k+3	NUM
ejpam-6321	151	9	)	)	PUNCT
ejpam-6321	151	10	)	)	PUNCT
ejpam-6321	151	11	]	]	PUNCT
ejpam-6321	152	1	(	(	PUNCT
ejpam-6321	152	2	1−	1−	NUM
ejpam-6321	152	3	α4)g(x(3k+1	α4)g(x(3k+1	VERB
ejpam-6321	152	4	)	)	PUNCT
ejpam-6321	152	5	,	,	PUNCT
ejpam-6321	152	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	152	7	)	)	PUNCT
ejpam-6321	152	8	,	,	PUNCT
ejpam-6321	152	9	x(3k+3	x(3k+3	NUM
ejpam-6321	152	10	)	)	PUNCT
ejpam-6321	152	11	)	)	PUNCT
ejpam-6321	153	1	≤	≤	NOUN
ejpam-6321	153	2	(	(	PUNCT
ejpam-6321	153	3	α1	α1	PROPN
ejpam-6321	153	4	+	+	CCONJ
ejpam-6321	153	5	α2	α2	ADJ
ejpam-6321	153	6	+	+	CCONJ
ejpam-6321	153	7	2α4)g(x3k	2α4)g(x3k	NUM
ejpam-6321	153	8	,	,	PUNCT
ejpam-6321	153	9	x(3k+1	x(3k+1	ADV
ejpam-6321	153	10	)	)	PUNCT
ejpam-6321	153	11	,	,	PUNCT
ejpam-6321	153	12	x(3k+2	x(3k+2	NUM
ejpam-6321	153	13	)	)	PUNCT
ejpam-6321	153	14	)	)	PUNCT
ejpam-6321	153	15	≤	≤	NOUN
ejpam-6321	153	16	(	(	PUNCT
ejpam-6321	153	17	α1	α1	PROPN
ejpam-6321	153	18	+	+	CCONJ
ejpam-6321	153	19	α2	α2	ADJ
ejpam-6321	153	20	+	+	CCONJ
ejpam-6321	153	21	2α4	2α4	NUM
ejpam-6321	153	22	)	)	PUNCT
ejpam-6321	153	23	(	(	PUNCT
ejpam-6321	153	24	1−	1−	NUM
ejpam-6321	153	25	α4	α4	NOUN
ejpam-6321	153	26	)	)	PUNCT
ejpam-6321	153	27	g(x3k	g(x3k	NOUN
ejpam-6321	153	28	,	,	PUNCT
ejpam-6321	153	29	x(3k+1	x(3k+1	PRON
ejpam-6321	153	30	)	)	PUNCT
ejpam-6321	153	31	,	,	PUNCT
ejpam-6321	153	32	x(3k+2	x(3k+2	NOUN
ejpam-6321	153	33	)	)	PUNCT
ejpam-6321	153	34	)	)	PUNCT
ejpam-6321	153	35	,	,	PUNCT
ejpam-6321	153	36	taking	take	VERB
ejpam-6321	153	37	ρ1	ρ1	NOUN
ejpam-6321	153	38	=	=	SYM
ejpam-6321	153	39	(	(	PUNCT
ejpam-6321	153	40	α1	α1	PROPN
ejpam-6321	153	41	+	+	CCONJ
ejpam-6321	153	42	α2	α2	ADJ
ejpam-6321	154	1	+	+	CCONJ
ejpam-6321	155	1	2α4	2α4	NUM
ejpam-6321	155	2	)	)	PUNCT
ejpam-6321	155	3	(	(	PUNCT
ejpam-6321	155	4	1−	1−	NUM
ejpam-6321	155	5	α4	α4	NOUN
ejpam-6321	155	6	)	)	PUNCT
ejpam-6321	155	7	<	<	X
ejpam-6321	155	8	1	1	NUM
ejpam-6321	155	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	155	10	)	)	PUNCT
ejpam-6321	155	11	,	,	PUNCT
ejpam-6321	155	12	x(3k+2	x(3k+2	NOUN
ejpam-6321	155	13	)	)	PUNCT
ejpam-6321	155	14	,	,	PUNCT
ejpam-6321	155	15	x(3k+3	x(3k+3	NUM
ejpam-6321	155	16	)	)	PUNCT
ejpam-6321	155	17	)	)	PUNCT
ejpam-6321	156	1	≤	≤	NUM
ejpam-6321	156	2	ρ1g(x3k	ρ1g(x3k	NOUN
ejpam-6321	156	3	,	,	PUNCT
ejpam-6321	156	4	x(3k+1	x(3k+1	ADV
ejpam-6321	156	5	)	)	PUNCT
ejpam-6321	156	6	,	,	PUNCT
ejpam-6321	156	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	156	8	)	)	PUNCT
ejpam-6321	156	9	)	)	PUNCT
ejpam-6321	156	10	(	(	PUNCT
ejpam-6321	156	11	2	2	X
ejpam-6321	156	12	)	)	PUNCT
ejpam-6321	156	13	now	now	ADV
ejpam-6321	156	14	,	,	PUNCT
ejpam-6321	156	15	we	we	PRON
ejpam-6321	156	16	find	find	VERB
ejpam-6321	156	17	iteration	iteration	NOUN
ejpam-6321	156	18	of	of	ADP
ejpam-6321	156	19	a	a	DET
ejpam-6321	156	20	contraction	contraction	NOUN
ejpam-6321	156	21	.	.	PUNCT
ejpam-6321	157	1	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	157	2	)	)	PUNCT
ejpam-6321	157	3	,	,	PUNCT
ejpam-6321	157	4	x(3k+3	x(3k+3	NUM
ejpam-6321	157	5	)	)	PUNCT
ejpam-6321	157	6	,	,	PUNCT
ejpam-6321	157	7	x(3k+4	x(3k+4	NUM
ejpam-6321	157	8	)	)	PUNCT
ejpam-6321	157	9	)	)	PUNCT
ejpam-6321	158	1	=	=	SYM
ejpam-6321	158	2	g(f1x(3k+1	g(f1x(3k+1	X
ejpam-6321	158	3	)	)	PUNCT
ejpam-6321	158	4	,	,	PUNCT
ejpam-6321	158	5	f2x(3k+2	f2x(3k+2	NUM
ejpam-6321	158	6	)	)	PUNCT
ejpam-6321	158	7	,	,	PUNCT
ejpam-6321	158	8	f3x(3k+3	f3x(3k+3	NUM
ejpam-6321	158	9	)	)	PUNCT
ejpam-6321	158	10	)	)	PUNCT
ejpam-6321	158	11	≤	≤	NUM
ejpam-6321	158	12	α1g(x(3k+1	α1g(x(3k+1	PROPN
ejpam-6321	158	13	)	)	PUNCT
ejpam-6321	158	14	,	,	PUNCT
ejpam-6321	158	15	x(3k+2	x(3k+2	NOUN
ejpam-6321	158	16	)	)	PUNCT
ejpam-6321	158	17	,	,	PUNCT
ejpam-6321	158	18	x(3k+3	x(3k+3	NUM
ejpam-6321	158	19	)	)	PUNCT
ejpam-6321	158	20	)	)	PUNCT
ejpam-6321	159	1	+	+	CCONJ
ejpam-6321	159	2	α2g(x(3k+2	α2g(x(3k+2	PROPN
ejpam-6321	159	3	)	)	PUNCT
ejpam-6321	159	4	,	,	PUNCT
ejpam-6321	159	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	159	6	)	)	PUNCT
ejpam-6321	159	7	,	,	PUNCT
ejpam-6321	159	8	f2x(3k+2	f2x(3k+2	NOUN
ejpam-6321	159	9	)	)	PUNCT
ejpam-6321	159	10	)	)	PUNCT
ejpam-6321	160	1	+	+	CCONJ
ejpam-6321	160	2	α3g(f1x(3k+1	α3g(f1x(3k+1	NUM
ejpam-6321	160	3	)	)	PUNCT
ejpam-6321	160	4	,	,	PUNCT
ejpam-6321	160	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	160	6	)	)	PUNCT
ejpam-6321	160	7	,	,	PUNCT
ejpam-6321	160	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	160	9	)	)	PUNCT
ejpam-6321	160	10	)	)	PUNCT
ejpam-6321	161	1	+	+	CCONJ
ejpam-6321	161	2	α4[g(f1x(3k+1	α4[g(f1x(3k+1	X
ejpam-6321	161	3	)	)	PUNCT
ejpam-6321	161	4	,	,	PUNCT
ejpam-6321	161	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	161	6	)	)	PUNCT
ejpam-6321	161	7	,	,	PUNCT
ejpam-6321	161	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	161	9	)	)	PUNCT
ejpam-6321	161	10	)	)	PUNCT
ejpam-6321	162	1	+	+	PUNCT
ejpam-6321	162	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	162	3	)	)	PUNCT
ejpam-6321	162	4	,	,	PUNCT
ejpam-6321	162	5	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	162	6	)	)	PUNCT
ejpam-6321	162	7	,	,	PUNCT
ejpam-6321	162	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	162	9	)	)	PUNCT
ejpam-6321	162	10	)	)	PUNCT
ejpam-6321	163	1	+	+	PUNCT
ejpam-6321	163	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	163	3	)	)	PUNCT
ejpam-6321	163	4	,	,	PUNCT
ejpam-6321	163	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	163	6	)	)	PUNCT
ejpam-6321	163	7	,	,	PUNCT
ejpam-6321	163	8	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	163	9	)	)	PUNCT
ejpam-6321	163	10	)	)	PUNCT
ejpam-6321	163	11	]	]	PUNCT
ejpam-6321	164	1	+	+	CCONJ
ejpam-6321	164	2	α5min	α5min	NUM
ejpam-6321	164	3	{	{	PUNCT
ejpam-6321	164	4	g(x(3k+1	g(x(3k+1	PROPN
ejpam-6321	164	5	)	)	PUNCT
ejpam-6321	164	6	,	,	PUNCT
ejpam-6321	164	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	164	8	)	)	PUNCT
ejpam-6321	164	9	,	,	PUNCT
ejpam-6321	164	10	f2x(3k+2	f2x(3k+2	NOUN
ejpam-6321	164	11	)	)	PUNCT
ejpam-6321	164	12	)	)	PUNCT
ejpam-6321	164	13	,	,	PUNCT
ejpam-6321	164	14	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	164	15	)	)	PUNCT
ejpam-6321	164	16	,	,	PUNCT
ejpam-6321	164	17	f1x(3k+1	f1x(3k+1	NOUN
ejpam-6321	164	18	)	)	PUNCT
ejpam-6321	164	19	,	,	PUNCT
ejpam-6321	164	20	x(3k+2	x(3k+2	NOUN
ejpam-6321	164	21	)	)	PUNCT
ejpam-6321	164	22	)	)	PUNCT
ejpam-6321	164	23	,	,	PUNCT
ejpam-6321	164	24	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	164	25	)	)	PUNCT
ejpam-6321	164	26	,	,	PUNCT
ejpam-6321	164	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	164	28	)	)	PUNCT
ejpam-6321	164	29	,	,	PUNCT
ejpam-6321	164	30	x(3k+2	x(3k+2	NOUN
ejpam-6321	164	31	)	)	PUNCT
ejpam-6321	164	32	)	)	PUNCT
ejpam-6321	164	33	,	,	PUNCT
ejpam-6321	164	34	g(f2x(3k+2	g(f2x(3k+2	NOUN
ejpam-6321	164	35	)	)	PUNCT
ejpam-6321	164	36	,	,	PUNCT
ejpam-6321	164	37	f2x(3k+2	f2x(3k+2	NUM
ejpam-6321	164	38	)	)	PUNCT
ejpam-6321	164	39	,	,	PUNCT
ejpam-6321	164	40	x(3k+3	x(3k+3	NUM
ejpam-6321	164	41	)	)	PUNCT
ejpam-6321	164	42	)	)	PUNCT
ejpam-6321	164	43	}	}	PUNCT
ejpam-6321	164	44	=	=	SYM
ejpam-6321	164	45	α1g(x(3k+1	α1g(x(3k+1	PROPN
ejpam-6321	164	46	)	)	PUNCT
ejpam-6321	164	47	,	,	PUNCT
ejpam-6321	164	48	x(3k+2	x(3k+2	NOUN
ejpam-6321	164	49	)	)	PUNCT
ejpam-6321	164	50	,	,	PUNCT
ejpam-6321	164	51	x(3k+3	x(3k+3	NUM
ejpam-6321	164	52	)	)	PUNCT
ejpam-6321	164	53	)	)	PUNCT
ejpam-6321	165	1	+	+	CCONJ
ejpam-6321	165	2	α2g(x(3k+2	α2g(x(3k+2	PROPN
ejpam-6321	165	3	)	)	PUNCT
ejpam-6321	165	4	,	,	PUNCT
ejpam-6321	165	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	165	6	)	)	PUNCT
ejpam-6321	165	7	,	,	PUNCT
ejpam-6321	165	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	165	9	)	)	PUNCT
ejpam-6321	165	10	)	)	PUNCT
ejpam-6321	166	1	+	+	CCONJ
ejpam-6321	166	2	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	166	3	)	)	PUNCT
ejpam-6321	166	4	,	,	PUNCT
ejpam-6321	166	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	166	6	)	)	PUNCT
ejpam-6321	166	7	,	,	PUNCT
ejpam-6321	166	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	166	9	)	)	PUNCT
ejpam-6321	166	10	)	)	PUNCT
ejpam-6321	167	1	+	+	CCONJ
ejpam-6321	168	1	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	168	2	)	)	PUNCT
ejpam-6321	168	3	,	,	PUNCT
ejpam-6321	168	4	x(3k+1	x(3k+1	CCONJ
ejpam-6321	168	5	)	)	PUNCT
ejpam-6321	168	6	,	,	PUNCT
ejpam-6321	168	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	168	8	)	)	PUNCT
ejpam-6321	168	9	)	)	PUNCT
ejpam-6321	169	1	+	+	PUNCT
ejpam-6321	169	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	169	3	)	)	PUNCT
ejpam-6321	169	4	,	,	PUNCT
ejpam-6321	169	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	169	6	)	)	PUNCT
ejpam-6321	169	7	,	,	PUNCT
ejpam-6321	169	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	169	9	)	)	PUNCT
ejpam-6321	169	10	)	)	PUNCT
ejpam-6321	170	1	+	+	PUNCT
ejpam-6321	170	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	170	3	)	)	PUNCT
ejpam-6321	170	4	,	,	PUNCT
ejpam-6321	170	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	170	6	)	)	PUNCT
ejpam-6321	170	7	,	,	PUNCT
ejpam-6321	170	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	170	9	)	)	PUNCT
ejpam-6321	170	10	)	)	PUNCT
ejpam-6321	170	11	]	]	PUNCT
ejpam-6321	171	1	+	+	CCONJ
ejpam-6321	171	2	α5min	α5min	NUM
ejpam-6321	171	3	{	{	PUNCT
ejpam-6321	171	4	g(x(3k+1	g(x(3k+1	PROPN
ejpam-6321	171	5	)	)	PUNCT
ejpam-6321	171	6	,	,	PUNCT
ejpam-6321	171	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	8	)	)	PUNCT
ejpam-6321	171	9	,	,	PUNCT
ejpam-6321	171	10	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	11	)	)	PUNCT
ejpam-6321	171	12	)	)	PUNCT
ejpam-6321	171	13	,	,	PUNCT
ejpam-6321	171	14	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	171	15	)	)	PUNCT
ejpam-6321	171	16	,	,	PUNCT
ejpam-6321	171	17	x(3k+1	x(3k+1	CCONJ
ejpam-6321	171	18	)	)	PUNCT
ejpam-6321	171	19	,	,	PUNCT
ejpam-6321	171	20	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	21	)	)	PUNCT
ejpam-6321	171	22	)	)	PUNCT
ejpam-6321	171	23	,	,	PUNCT
ejpam-6321	171	24	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	171	25	)	)	PUNCT
ejpam-6321	171	26	,	,	PUNCT
ejpam-6321	171	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	28	)	)	PUNCT
ejpam-6321	171	29	,	,	PUNCT
ejpam-6321	171	30	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	31	)	)	PUNCT
ejpam-6321	171	32	)	)	PUNCT
ejpam-6321	171	33	,	,	PUNCT
ejpam-6321	171	34	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	171	35	)	)	PUNCT
ejpam-6321	171	36	,	,	PUNCT
ejpam-6321	171	37	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	38	)	)	PUNCT
ejpam-6321	171	39	,	,	PUNCT
ejpam-6321	171	40	x(3k+3	x(3k+3	NUM
ejpam-6321	171	41	)	)	PUNCT
ejpam-6321	171	42	)	)	PUNCT
ejpam-6321	171	43	}	}	PUNCT
ejpam-6321	171	44	after	after	ADP
ejpam-6321	171	45	applying	apply	VERB
ejpam-6321	171	46	the	the	DET
ejpam-6321	171	47	simplification	simplification	NOUN
ejpam-6321	171	48	process	process	NOUN
ejpam-6321	171	49	utilizing	utilize	VERB
ejpam-6321	171	50	the	the	DET
ejpam-6321	171	51	definition	definition	NOUN
ejpam-6321	171	52	of	of	ADP
ejpam-6321	171	53	gm	gm	PROPN
ejpam-6321	171	54	-	-	PUNCT
ejpam-6321	171	55	space	space	NOUN
ejpam-6321	171	56	we	we	PRON
ejpam-6321	171	57	have	have	AUX
ejpam-6321	171	58	achieved	achieve	VERB
ejpam-6321	171	59	the	the	DET
ejpam-6321	171	60	following	following	ADJ
ejpam-6321	171	61	result	result	NOUN
ejpam-6321	171	62	,	,	PUNCT
ejpam-6321	171	63	≤	≤	NUM
ejpam-6321	171	64	(	(	PUNCT
ejpam-6321	171	65	α1	α1	PROPN
ejpam-6321	171	66	+	+	X
ejpam-6321	171	67	α2)g(x(3k+1	α2)g(x(3k+1	NOUN
ejpam-6321	171	68	)	)	PUNCT
ejpam-6321	171	69	,	,	PUNCT
ejpam-6321	171	70	x(3k+2	x(3k+2	NOUN
ejpam-6321	171	71	)	)	PUNCT
ejpam-6321	171	72	,	,	PUNCT
ejpam-6321	171	73	x(3k+3	x(3k+3	NUM
ejpam-6321	171	74	)	)	PUNCT
ejpam-6321	171	75	)	)	PUNCT
ejpam-6321	172	1	+	+	CCONJ
ejpam-6321	173	1	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	173	2	)	)	PUNCT
ejpam-6321	173	3	,	,	PUNCT
ejpam-6321	173	4	x(3k+2	x(3k+2	NOUN
ejpam-6321	173	5	)	)	PUNCT
ejpam-6321	173	6	,	,	PUNCT
ejpam-6321	173	7	x(3k+3	x(3k+3	NUM
ejpam-6321	173	8	)	)	PUNCT
ejpam-6321	173	9	)	)	PUNCT
ejpam-6321	174	1	+	+	PUNCT
ejpam-6321	174	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	174	3	)	)	PUNCT
ejpam-6321	174	4	,	,	PUNCT
ejpam-6321	174	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	174	6	)	)	PUNCT
ejpam-6321	174	7	,	,	PUNCT
ejpam-6321	174	8	x(3k+3	x(3k+3	NUM
ejpam-6321	174	9	)	)	PUNCT
ejpam-6321	174	10	)	)	PUNCT
ejpam-6321	175	1	+	+	PUNCT
ejpam-6321	175	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	175	3	)	)	PUNCT
ejpam-6321	175	4	,	,	PUNCT
ejpam-6321	175	5	x(3k+3	x(3k+3	NUM
ejpam-6321	175	6	)	)	PUNCT
ejpam-6321	175	7	,	,	PUNCT
ejpam-6321	175	8	x(3k+4	x(3k+4	NUM
ejpam-6321	175	9	)	)	PUNCT
ejpam-6321	175	10	)	)	PUNCT
ejpam-6321	175	11	]	]	PUNCT
ejpam-6321	176	1	(	(	PUNCT
ejpam-6321	176	2	1−	1−	NUM
ejpam-6321	176	3	α4)g(x(3k+2	α4)g(x(3k+2	NUM
ejpam-6321	176	4	)	)	PUNCT
ejpam-6321	176	5	,	,	PUNCT
ejpam-6321	176	6	x(3k+3	x(3k+3	NUM
ejpam-6321	176	7	)	)	PUNCT
ejpam-6321	176	8	,	,	PUNCT
ejpam-6321	176	9	x(3k+4	x(3k+4	NUM
ejpam-6321	176	10	)	)	PUNCT
ejpam-6321	176	11	)	)	PUNCT
ejpam-6321	177	1	≤	≤	NUM
ejpam-6321	177	2	(	(	PUNCT
ejpam-6321	177	3	α1	α1	PROPN
ejpam-6321	177	4	+	+	CCONJ
ejpam-6321	177	5	α2	α2	ADJ
ejpam-6321	177	6	+	+	CCONJ
ejpam-6321	177	7	2α4)g(x(3k+1	2α4)g(x(3k+1	NUM
ejpam-6321	177	8	)	)	PUNCT
ejpam-6321	177	9	,	,	PUNCT
ejpam-6321	177	10	x(3k+2	x(3k+2	NOUN
ejpam-6321	177	11	)	)	PUNCT
ejpam-6321	177	12	,	,	PUNCT
ejpam-6321	177	13	x(3k+3	x(3k+3	NUM
ejpam-6321	177	14	)	)	PUNCT
ejpam-6321	177	15	)	)	PUNCT
ejpam-6321	177	16	≤	≤	NOUN
ejpam-6321	177	17	(	(	PUNCT
ejpam-6321	177	18	α1	α1	PROPN
ejpam-6321	177	19	+	+	CCONJ
ejpam-6321	177	20	α2	α2	ADJ
ejpam-6321	177	21	+	+	CCONJ
ejpam-6321	177	22	2α4	2α4	NUM
ejpam-6321	177	23	)	)	PUNCT
ejpam-6321	177	24	(	(	PUNCT
ejpam-6321	177	25	1−	1−	NUM
ejpam-6321	177	26	α4	α4	NOUN
ejpam-6321	177	27	)	)	PUNCT
ejpam-6321	177	28	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	177	29	)	)	PUNCT
ejpam-6321	177	30	,	,	PUNCT
ejpam-6321	177	31	x(3k+2	x(3k+2	NOUN
ejpam-6321	177	32	)	)	PUNCT
ejpam-6321	177	33	,	,	PUNCT
ejpam-6321	177	34	x(3k+3	x(3k+3	NUM
ejpam-6321	177	35	)	)	PUNCT
ejpam-6321	177	36	)	)	PUNCT
ejpam-6321	177	37	,	,	PUNCT
ejpam-6321	177	38	taking	take	VERB
ejpam-6321	177	39	ρ2	ρ2	NOUN
ejpam-6321	177	40	=	=	SYM
ejpam-6321	177	41	(	(	PUNCT
ejpam-6321	177	42	α1	α1	PROPN
ejpam-6321	177	43	+	+	CCONJ
ejpam-6321	177	44	α2	α2	ADJ
ejpam-6321	177	45	+	+	CCONJ
ejpam-6321	177	46	2α4	2α4	NUM
ejpam-6321	177	47	)	)	PUNCT
ejpam-6321	177	48	(	(	PUNCT
ejpam-6321	177	49	1−	1−	NUM
ejpam-6321	177	50	α4	α4	NOUN
ejpam-6321	177	51	)	)	PUNCT
ejpam-6321	177	52	<	<	X
ejpam-6321	177	53	1	1	NUM
ejpam-6321	177	54	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	177	55	)	)	PUNCT
ejpam-6321	177	56	,	,	PUNCT
ejpam-6321	177	57	x(3k+3	x(3k+3	NUM
ejpam-6321	177	58	)	)	PUNCT
ejpam-6321	177	59	,	,	PUNCT
ejpam-6321	177	60	x(3k+4	x(3k+4	NUM
ejpam-6321	177	61	)	)	PUNCT
ejpam-6321	177	62	)	)	PUNCT
ejpam-6321	177	63	≤	≤	NOUN
ejpam-6321	177	64	ρ2g(x(3k+1	ρ2g(x(3k+1	NOUN
ejpam-6321	177	65	)	)	PUNCT
ejpam-6321	177	66	,	,	PUNCT
ejpam-6321	177	67	x(3k+2	x(3k+2	NOUN
ejpam-6321	177	68	)	)	PUNCT
ejpam-6321	177	69	,	,	PUNCT
ejpam-6321	177	70	x(3k+3	x(3k+3	NUM
ejpam-6321	177	71	)	)	PUNCT
ejpam-6321	177	72	)	)	PUNCT
ejpam-6321	178	1	(	(	PUNCT
ejpam-6321	178	2	3	3	X
ejpam-6321	178	3	)	)	PUNCT
ejpam-6321	178	4	from	from	ADP
ejpam-6321	178	5	both	both	DET
ejpam-6321	178	6	cases	case	NOUN
ejpam-6321	178	7	we	we	PRON
ejpam-6321	178	8	get	get	VERB
ejpam-6321	178	9	that	that	PRON
ejpam-6321	178	10	,	,	PUNCT
ejpam-6321	178	11	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	178	12	)	)	PUNCT
ejpam-6321	178	13	,	,	PUNCT
ejpam-6321	178	14	x(3k+2	x(3k+2	NOUN
ejpam-6321	178	15	)	)	PUNCT
ejpam-6321	178	16	,	,	PUNCT
ejpam-6321	178	17	x(3k+3	x(3k+3	NUM
ejpam-6321	178	18	)	)	PUNCT
ejpam-6321	178	19	)	)	PUNCT
ejpam-6321	179	1	≤	≤	NUM
ejpam-6321	179	2	ρg(x3k	ρg(x3k	NOUN
ejpam-6321	179	3	,	,	PUNCT
ejpam-6321	179	4	x(3k+1	x(3k+1	ADV
ejpam-6321	179	5	)	)	PUNCT
ejpam-6321	179	6	,	,	PUNCT
ejpam-6321	179	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	179	8	)	)	PUNCT
ejpam-6321	179	9	)	)	PUNCT
ejpam-6321	179	10	,	,	PUNCT
ejpam-6321	179	11	∴	∴	NOUN
ejpam-6321	179	12	ρ1	ρ1	NOUN
ejpam-6321	179	13	=	=	SYM
ejpam-6321	179	14	ρ2	ρ2	NOUN
ejpam-6321	179	15	=	=	SYM
ejpam-6321	179	16	ρ	ρ	PROPN
ejpam-6321	179	17	inductively	inductively	ADV
ejpam-6321	179	18	we	we	PRON
ejpam-6321	179	19	have	have	AUX
ejpam-6321	179	20	,	,	PUNCT
ejpam-6321	179	21	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	179	22	)	)	PUNCT
ejpam-6321	179	23	,	,	PUNCT
ejpam-6321	179	24	x(3k+2	x(3k+2	NOUN
ejpam-6321	179	25	)	)	PUNCT
ejpam-6321	179	26	,	,	PUNCT
ejpam-6321	179	27	x(3k+3	x(3k+3	NUM
ejpam-6321	179	28	)	)	PUNCT
ejpam-6321	179	29	)	)	PUNCT
ejpam-6321	180	1	≤	≤	NOUN
ejpam-6321	180	2	ρ3k+1g(x0	ρ3k+1g(x0	NUM
ejpam-6321	180	3	,	,	PUNCT
ejpam-6321	180	4	x1	x1	PROPN
ejpam-6321	180	5	,	,	PUNCT
ejpam-6321	180	6	x2	x2	PROPN
ejpam-6321	180	7	)	)	PUNCT
ejpam-6321	180	8	m.	m.	NOUN
ejpam-6321	180	9	noorwali	noorwali	PROPN
ejpam-6321	180	10	et	et	PROPN
ejpam-6321	180	11	al	al	PROPN
ejpam-6321	180	12	.	.	PUNCT
ejpam-6321	180	13	/	/	SYM
ejpam-6321	180	14	eur	eur	PROPN
ejpam-6321	180	15	.	.	PUNCT
ejpam-6321	181	1	j.	j.	PROPN
ejpam-6321	181	2	pure	pure	PROPN
ejpam-6321	181	3	appl	appl	PROPN
ejpam-6321	181	4	.	.	PROPN
ejpam-6321	181	5	math	math	PROPN
ejpam-6321	181	6	,	,	PUNCT
ejpam-6321	181	7	18	18	NUM
ejpam-6321	181	8	(	(	PUNCT
ejpam-6321	181	9	3	3	NUM
ejpam-6321	181	10	)	)	PUNCT
ejpam-6321	181	11	(	(	PUNCT
ejpam-6321	181	12	2025	2025	NUM
ejpam-6321	181	13	)	)	PUNCT
ejpam-6321	181	14	,	,	PUNCT
ejpam-6321	181	15	6321	6321	NUM
ejpam-6321	181	16	6	6	NUM
ejpam-6321	181	17	of	of	ADP
ejpam-6321	181	18	27	27	NUM
ejpam-6321	181	19	next	next	ADV
ejpam-6321	181	20	we	we	PRON
ejpam-6321	181	21	will	will	AUX
ejpam-6321	181	22	show	show	VERB
ejpam-6321	181	23	that	that	SCONJ
ejpam-6321	181	24	{	{	PUNCT
ejpam-6321	181	25	xk	xk	ADJ
ejpam-6321	181	26	}	}	PUNCT
ejpam-6321	181	27	is	be	AUX
ejpam-6321	181	28	a	a	DET
ejpam-6321	181	29	g	g	NOUN
ejpam-6321	181	30	-	-	PUNCT
ejpam-6321	181	31	cauchy	cauchy	ADJ
ejpam-6321	181	32	sequence	sequence	NOUN
ejpam-6321	181	33	in	in	ADP
ejpam-6321	181	34	x	x	NOUN
ejpam-6321	181	35	,	,	PUNCT
ejpam-6321	181	36	for	for	ADP
ejpam-6321	181	37	any	any	DET
ejpam-6321	181	38	j	j	PROPN
ejpam-6321	181	39	,	,	PUNCT
ejpam-6321	181	40	n	n	CCONJ
ejpam-6321	181	41	,	,	PUNCT
ejpam-6321	181	42	k	k	PROPN
ejpam-6321	181	43	with	with	ADP
ejpam-6321	181	44	j	j	PROPN
ejpam-6321	181	45	>	>	X
ejpam-6321	181	46	n	n	PROPN
ejpam-6321	181	47	>	>	X
ejpam-6321	181	48	k.	k.	PROPN
ejpam-6321	181	49	g(xk	g(xk	PROPN
ejpam-6321	181	50	,	,	PUNCT
ejpam-6321	181	51	xn	xn	PROPN
ejpam-6321	181	52	,	,	PUNCT
ejpam-6321	181	53	xj	xj	PROPN
ejpam-6321	181	54	)	)	PUNCT
ejpam-6321	181	55	≤	≤	NOUN
ejpam-6321	181	56	g(xk	g(xk	NOUN
ejpam-6321	181	57	,	,	PUNCT
ejpam-6321	181	58	x(k+1	x(k+1	NUM
ejpam-6321	181	59	)	)	PUNCT
ejpam-6321	181	60	,	,	PUNCT
ejpam-6321	181	61	x(k+1	x(k+1	NOUN
ejpam-6321	181	62	)	)	PUNCT
ejpam-6321	181	63	)	)	PUNCT
ejpam-6321	182	1	+	+	CCONJ
ejpam-6321	182	2	g(x(k+1	g(x(k+1	NOUN
ejpam-6321	182	3	)	)	PUNCT
ejpam-6321	182	4	,	,	PUNCT
ejpam-6321	182	5	x(k+2	x(k+2	NOUN
ejpam-6321	182	6	)	)	PUNCT
ejpam-6321	182	7	,	,	PUNCT
ejpam-6321	182	8	x(k+2	x(k+2	NOUN
ejpam-6321	182	9	)	)	PUNCT
ejpam-6321	182	10	)	)	PUNCT
ejpam-6321	183	1	+	+	CCONJ
ejpam-6321	183	2	......	......	PUNCT
ejpam-6321	183	3	+	+	ADJ
ejpam-6321	183	4	g(x(j−1	g(x(j−1	PROPN
ejpam-6321	183	5	)	)	PUNCT
ejpam-6321	183	6	,	,	PUNCT
ejpam-6321	183	7	x(j−1	x(j−1	PROPN
ejpam-6321	183	8	)	)	PUNCT
ejpam-6321	183	9	,	,	PUNCT
ejpam-6321	183	10	xj	xj	PROPN
ejpam-6321	183	11	)	)	PUNCT
ejpam-6321	183	12	≤	≤	NOUN
ejpam-6321	183	13	g(xk	g(xk	NOUN
ejpam-6321	183	14	,	,	PUNCT
ejpam-6321	183	15	x(k+1	x(k+1	NUM
ejpam-6321	183	16	)	)	PUNCT
ejpam-6321	183	17	,	,	PUNCT
ejpam-6321	183	18	x(k+2	x(k+2	NOUN
ejpam-6321	183	19	)	)	PUNCT
ejpam-6321	183	20	)	)	PUNCT
ejpam-6321	184	1	+	+	CCONJ
ejpam-6321	184	2	g(x(k+1	g(x(k+1	NOUN
ejpam-6321	184	3	)	)	PUNCT
ejpam-6321	184	4	,	,	PUNCT
ejpam-6321	184	5	x(k+2	x(k+2	NUM
ejpam-6321	184	6	)	)	PUNCT
ejpam-6321	184	7	,	,	PUNCT
ejpam-6321	184	8	x(k+3	x(k+3	NUM
ejpam-6321	184	9	)	)	PUNCT
ejpam-6321	184	10	)	)	PUNCT
ejpam-6321	185	1	+	+	CCONJ
ejpam-6321	185	2	.......	.......	PUNCT
ejpam-6321	186	1	+	+	PROPN
ejpam-6321	186	2	g(x(j−2	g(x(j−2	PROPN
ejpam-6321	186	3	)	)	PUNCT
ejpam-6321	186	4	,	,	PUNCT
ejpam-6321	186	5	x(j−1	x(j−1	PROPN
ejpam-6321	186	6	)	)	PUNCT
ejpam-6321	186	7	,	,	PUNCT
ejpam-6321	186	8	xj	xj	PROPN
ejpam-6321	186	9	)	)	PUNCT
ejpam-6321	186	10	≤	≤	PUNCT
ejpam-6321	187	1	[	[	X
ejpam-6321	187	2	yk	yk	X
ejpam-6321	187	3	+	+	X
ejpam-6321	187	4	y(k+1	y(k+1	NOUN
ejpam-6321	187	5	)	)	PUNCT
ejpam-6321	188	1	+	+	CCONJ
ejpam-6321	188	2	........	........	PUNCT
ejpam-6321	188	3	+	+	NUM
ejpam-6321	188	4	y(j−2)]g(x0	y(j−2)]g(x0	PROPN
ejpam-6321	188	5	,	,	PUNCT
ejpam-6321	188	6	x1	x1	PROPN
ejpam-6321	188	7	,	,	PUNCT
ejpam-6321	188	8	x2	x2	PROPN
ejpam-6321	188	9	)	)	PUNCT
ejpam-6321	188	10	this	this	PRON
ejpam-6321	188	11	implies	imply	VERB
ejpam-6321	188	12	that	that	SCONJ
ejpam-6321	188	13	,	,	PUNCT
ejpam-6321	188	14	g(xk	g(xk	PROPN
ejpam-6321	188	15	,	,	PUNCT
ejpam-6321	188	16	xn	xn	PROPN
ejpam-6321	188	17	,	,	PUNCT
ejpam-6321	188	18	xj	xj	PROPN
ejpam-6321	188	19	)	)	PUNCT
ejpam-6321	188	20	≤	≤	PROPN
ejpam-6321	188	21	yk	yk	PROPN
ejpam-6321	188	22	(	(	PUNCT
ejpam-6321	188	23	1−	1−	NUM
ejpam-6321	188	24	y	y	NOUN
ejpam-6321	188	25	)	)	PUNCT
ejpam-6321	188	26	g(x0	g(x0	NOUN
ejpam-6321	188	27	,	,	PUNCT
ejpam-6321	188	28	x1	x1	PROPN
ejpam-6321	188	29	,	,	PUNCT
ejpam-6321	188	30	x2	x2	PROPN
ejpam-6321	188	31	)	)	PUNCT
ejpam-6321	188	32	the	the	DET
ejpam-6321	188	33	same	same	ADJ
ejpam-6321	188	34	hold	hold	NOUN
ejpam-6321	188	35	if	if	SCONJ
ejpam-6321	188	36	j	j	PROPN
ejpam-6321	188	37	=	=	SYM
ejpam-6321	188	38	n	n	PROPN
ejpam-6321	188	39	>	>	X
ejpam-6321	188	40	k	k	NOUN
ejpam-6321	188	41	and	and	CCONJ
ejpam-6321	188	42	if	if	SCONJ
ejpam-6321	188	43	j	j	PROPN
ejpam-6321	188	44	>	>	X
ejpam-6321	188	45	n	n	PROPN
ejpam-6321	188	46	=	=	SYM
ejpam-6321	189	1	k	k	NOUN
ejpam-6321	189	2	we	we	PRON
ejpam-6321	189	3	have	have	VERB
ejpam-6321	189	4	,	,	PUNCT
ejpam-6321	189	5	g(xk	g(xk	PROPN
ejpam-6321	189	6	,	,	PUNCT
ejpam-6321	189	7	xn	xn	PROPN
ejpam-6321	189	8	,	,	PUNCT
ejpam-6321	189	9	xj	xj	PROPN
ejpam-6321	189	10	)	)	PUNCT
ejpam-6321	189	11	≤	≤	ADJ
ejpam-6321	189	12	yk−1	yk−1	PROPN
ejpam-6321	189	13	(	(	PUNCT
ejpam-6321	189	14	1−	1−	NUM
ejpam-6321	189	15	y	y	NOUN
ejpam-6321	189	16	)	)	PUNCT
ejpam-6321	189	17	g(x0	g(x0	NOUN
ejpam-6321	189	18	,	,	PUNCT
ejpam-6321	189	19	x1	x1	PROPN
ejpam-6321	189	20	,	,	PUNCT
ejpam-6321	189	21	x2	x2	PROPN
ejpam-6321	189	22	)	)	PUNCT
ejpam-6321	189	23	(	(	PUNCT
ejpam-6321	189	24	4	4	X
ejpam-6321	189	25	)	)	PUNCT
ejpam-6321	189	26	if	if	SCONJ
ejpam-6321	189	27	we	we	PRON
ejpam-6321	189	28	take	take	VERB
ejpam-6321	189	29	the	the	DET
ejpam-6321	189	30	limit	limit	NOUN
ejpam-6321	189	31	as	as	ADP
ejpam-6321	189	32	k	k	PROPN
ejpam-6321	189	33	,	,	PUNCT
ejpam-6321	189	34	n	n	CCONJ
ejpam-6321	189	35	,	,	PUNCT
ejpam-6321	189	36	j	j	PROPN
ejpam-6321	189	37	→∞	→∞	PROPN
ejpam-6321	189	38	we	we	PRON
ejpam-6321	189	39	getg(xk	getg(xk	VERB
ejpam-6321	189	40	,	,	PUNCT
ejpam-6321	189	41	xn	xn	PROPN
ejpam-6321	189	42	,	,	PUNCT
ejpam-6321	189	43	xj)→	xj)→	PROPN
ejpam-6321	189	44	0	0	NUM
ejpam-6321	189	45	.	.	PUNCT
ejpam-6321	190	1	hence	hence	ADV
ejpam-6321	190	2	{	{	PUNCT
ejpam-6321	190	3	xk	xk	ADJ
ejpam-6321	190	4	}	}	PUNCT
ejpam-6321	190	5	is	be	AUX
ejpam-6321	190	6	a	a	DET
ejpam-6321	190	7	g	g	NOUN
ejpam-6321	190	8	-	-	PUNCT
ejpam-6321	190	9	cauchy	cauchy	ADJ
ejpam-6321	190	10	sequence	sequence	NOUN
ejpam-6321	190	11	.	.	PUNCT
ejpam-6321	191	1	by	by	ADP
ejpam-6321	191	2	g	g	NOUN
ejpam-6321	191	3	-	-	PUNCT
ejpam-6321	191	4	completeness	completeness	NOUN
ejpam-6321	191	5	of	of	ADP
ejpam-6321	191	6	x	x	NOUN
ejpam-6321	191	7	,	,	PUNCT
ejpam-6321	191	8	there	there	PRON
ejpam-6321	191	9	exists	exist	VERB
ejpam-6321	191	10	℘	℘	PROPN
ejpam-6321	191	11	∈	∈	PROPN
ejpam-6321	191	12	x	x	PUNCT
ejpam-6321	191	13	such	such	ADJ
ejpam-6321	191	14	that	that	SCONJ
ejpam-6321	191	15	{	{	PUNCT
ejpam-6321	191	16	xk	xk	ADJ
ejpam-6321	191	17	}	}	PUNCT
ejpam-6321	191	18	converges	converge	VERB
ejpam-6321	191	19	to	to	ADP
ejpam-6321	191	20	℘	℘	PROPN
ejpam-6321	191	21	as	as	SCONJ
ejpam-6321	191	22	k	k	PROPN
ejpam-6321	191	23	→∞.	→∞.	X
ejpam-6321	191	24	we	we	PRON
ejpam-6321	191	25	have	have	VERB
ejpam-6321	191	26	to	to	PART
ejpam-6321	191	27	show	show	VERB
ejpam-6321	191	28	that	that	DET
ejpam-6321	191	29	f1℘	f1℘	NOUN
ejpam-6321	191	30	=	=	PUNCT
ejpam-6321	191	31	℘	℘	PROPN
ejpam-6321	191	32	by	by	ADP
ejpam-6321	191	33	contrary	contrary	ADJ
ejpam-6321	191	34	case	case	NOUN
ejpam-6321	191	35	let	let	VERB
ejpam-6321	191	36	f1℘	f1℘	PROPN
ejpam-6321	191	37	̸=	̸=	PROPN
ejpam-6321	191	38	℘.	℘.	PROPN
ejpam-6321	191	39	g(f1℘	g(f1℘	PROPN
ejpam-6321	191	40	,	,	PUNCT
ejpam-6321	191	41	x(3k+2	x(3k+2	NOUN
ejpam-6321	191	42	)	)	PUNCT
ejpam-6321	191	43	,	,	PUNCT
ejpam-6321	191	44	x(3k+3	x(3k+3	NUM
ejpam-6321	191	45	)	)	PUNCT
ejpam-6321	191	46	)	)	PUNCT
ejpam-6321	192	1	=	=	SYM
ejpam-6321	192	2	g(f1℘	g(f1℘	PROPN
ejpam-6321	192	3	,	,	PUNCT
ejpam-6321	192	4	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	192	5	)	)	PUNCT
ejpam-6321	192	6	,	,	PUNCT
ejpam-6321	192	7	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	192	8	)	)	PUNCT
ejpam-6321	192	9	)	)	PUNCT
ejpam-6321	192	10	≤	≤	NOUN
ejpam-6321	192	11	α1g(℘	α1g(℘	NOUN
ejpam-6321	192	12	,	,	PUNCT
ejpam-6321	192	13	x(3k+1	x(3k+1	PRON
ejpam-6321	192	14	)	)	PUNCT
ejpam-6321	192	15	,	,	PUNCT
ejpam-6321	192	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	192	17	)	)	PUNCT
ejpam-6321	192	18	)	)	PUNCT
ejpam-6321	193	1	+	+	CCONJ
ejpam-6321	193	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	193	3	,	,	PUNCT
ejpam-6321	193	4	x(3k+1	x(3k+1	PRON
ejpam-6321	193	5	)	)	PUNCT
ejpam-6321	193	6	,	,	PUNCT
ejpam-6321	193	7	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	193	8	)	)	PUNCT
ejpam-6321	193	9	)	)	PUNCT
ejpam-6321	194	1	+	+	CCONJ
ejpam-6321	195	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	195	2	,	,	PUNCT
ejpam-6321	195	3	x(3k+1	x(3k+1	ADV
ejpam-6321	195	4	)	)	PUNCT
ejpam-6321	195	5	,	,	PUNCT
ejpam-6321	195	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	195	7	)	)	PUNCT
ejpam-6321	195	8	)	)	PUNCT
ejpam-6321	196	1	+	+	CCONJ
ejpam-6321	196	2	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	196	3	,	,	PUNCT
ejpam-6321	196	4	x3k	x3k	NOUN
ejpam-6321	196	5	,	,	PUNCT
ejpam-6321	196	6	x3k	x3k	PUNCT
ejpam-6321	196	7	)	)	PUNCT
ejpam-6321	197	1	+	+	SYM
ejpam-6321	197	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	197	3	)	)	PUNCT
ejpam-6321	197	4	,	,	PUNCT
ejpam-6321	197	5	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	197	6	)	)	PUNCT
ejpam-6321	197	7	,	,	PUNCT
ejpam-6321	197	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	197	9	)	)	PUNCT
ejpam-6321	197	10	)	)	PUNCT
ejpam-6321	198	1	+	+	PUNCT
ejpam-6321	198	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	198	3	)	)	PUNCT
ejpam-6321	198	4	,	,	PUNCT
ejpam-6321	198	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	198	6	)	)	PUNCT
ejpam-6321	198	7	,	,	PUNCT
ejpam-6321	198	8	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	198	9	)	)	PUNCT
ejpam-6321	198	10	)	)	PUNCT
ejpam-6321	198	11	]	]	PUNCT
ejpam-6321	199	1	+	+	CCONJ
ejpam-6321	199	2	α5min	α5min	NUM
ejpam-6321	199	3	{	{	PUNCT
ejpam-6321	199	4	g(℘	g(℘	PROPN
ejpam-6321	199	5	,	,	PUNCT
ejpam-6321	199	6	x(3k+1	x(3k+1	PRON
ejpam-6321	199	7	)	)	PUNCT
ejpam-6321	199	8	,	,	PUNCT
ejpam-6321	199	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	199	10	)	)	PUNCT
ejpam-6321	199	11	)	)	PUNCT
ejpam-6321	199	12	,	,	PUNCT
ejpam-6321	199	13	g(f1℘	g(f1℘	PROPN
ejpam-6321	199	14	,	,	PUNCT
ejpam-6321	199	15	f1℘	f1℘	PROPN
ejpam-6321	199	16	,	,	PUNCT
ejpam-6321	199	17	x(3k+1	x(3k+1	ADV
ejpam-6321	199	18	)	)	PUNCT
ejpam-6321	199	19	)	)	PUNCT
ejpam-6321	199	20	,	,	PUNCT
ejpam-6321	199	21	g(f1℘	g(f1℘	PROPN
ejpam-6321	199	22	,	,	PUNCT
ejpam-6321	199	23	x(3k+1	x(3k+1	ADV
ejpam-6321	199	24	)	)	PUNCT
ejpam-6321	199	25	,	,	PUNCT
ejpam-6321	199	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	199	27	)	)	PUNCT
ejpam-6321	199	28	)	)	PUNCT
ejpam-6321	199	29	,	,	PUNCT
ejpam-6321	199	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	199	31	)	)	PUNCT
ejpam-6321	199	32	,	,	PUNCT
ejpam-6321	199	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	199	34	)	)	PUNCT
ejpam-6321	199	35	,	,	PUNCT
ejpam-6321	199	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	199	37	)	)	PUNCT
ejpam-6321	199	38	)	)	PUNCT
ejpam-6321	199	39	}	}	PUNCT
ejpam-6321	199	40	=	=	SYM
ejpam-6321	200	1	α1g(℘	α1g(℘	NOUN
ejpam-6321	200	2	,	,	PUNCT
ejpam-6321	200	3	x(3k+1	x(3k+1	PRON
ejpam-6321	200	4	)	)	PUNCT
ejpam-6321	200	5	,	,	PUNCT
ejpam-6321	200	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	200	7	)	)	PUNCT
ejpam-6321	200	8	)	)	PUNCT
ejpam-6321	201	1	+	+	CCONJ
ejpam-6321	201	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	201	3	,	,	PUNCT
ejpam-6321	201	4	x(3k+1	x(3k+1	PRON
ejpam-6321	201	5	)	)	PUNCT
ejpam-6321	201	6	,	,	PUNCT
ejpam-6321	201	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	201	8	)	)	PUNCT
ejpam-6321	201	9	)	)	PUNCT
ejpam-6321	202	1	+	+	CCONJ
ejpam-6321	203	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	203	2	,	,	PUNCT
ejpam-6321	203	3	x(3k+1	x(3k+1	ADV
ejpam-6321	203	4	)	)	PUNCT
ejpam-6321	203	5	,	,	PUNCT
ejpam-6321	203	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	203	7	)	)	PUNCT
ejpam-6321	203	8	)	)	PUNCT
ejpam-6321	204	1	+	+	CCONJ
ejpam-6321	204	2	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	204	3	,	,	PUNCT
ejpam-6321	204	4	x3k	x3k	NOUN
ejpam-6321	204	5	,	,	PUNCT
ejpam-6321	204	6	x3k	x3k	PUNCT
ejpam-6321	204	7	)	)	PUNCT
ejpam-6321	205	1	+	+	SYM
ejpam-6321	205	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	205	3	)	)	PUNCT
ejpam-6321	205	4	,	,	PUNCT
ejpam-6321	205	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	205	6	)	)	PUNCT
ejpam-6321	205	7	,	,	PUNCT
ejpam-6321	205	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	205	9	)	)	PUNCT
ejpam-6321	205	10	)	)	PUNCT
ejpam-6321	206	1	+	+	PUNCT
ejpam-6321	206	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	206	3	)	)	PUNCT
ejpam-6321	206	4	,	,	PUNCT
ejpam-6321	206	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	206	6	)	)	PUNCT
ejpam-6321	206	7	,	,	PUNCT
ejpam-6321	206	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	206	9	)	)	PUNCT
ejpam-6321	206	10	)	)	PUNCT
ejpam-6321	206	11	]	]	PUNCT
ejpam-6321	207	1	+	+	CCONJ
ejpam-6321	207	2	α5min	α5min	NUM
ejpam-6321	207	3	{	{	PUNCT
ejpam-6321	207	4	g(℘	g(℘	PROPN
ejpam-6321	207	5	,	,	PUNCT
ejpam-6321	207	6	x(3k+1	x(3k+1	PRON
ejpam-6321	207	7	)	)	PUNCT
ejpam-6321	207	8	,	,	PUNCT
ejpam-6321	207	9	x(3k+1	x(3k+1	CCONJ
ejpam-6321	207	10	)	)	PUNCT
ejpam-6321	207	11	)	)	PUNCT
ejpam-6321	207	12	,	,	PUNCT
ejpam-6321	207	13	g(f1℘	g(f1℘	PROPN
ejpam-6321	207	14	,	,	PUNCT
ejpam-6321	207	15	f1℘	f1℘	PROPN
ejpam-6321	207	16	,	,	PUNCT
ejpam-6321	207	17	x(3k+1	x(3k+1	ADV
ejpam-6321	207	18	)	)	PUNCT
ejpam-6321	207	19	)	)	PUNCT
ejpam-6321	207	20	,	,	PUNCT
ejpam-6321	207	21	g(℘	g(℘	PROPN
ejpam-6321	207	22	,	,	PUNCT
ejpam-6321	207	23	x(3k+1	x(3k+1	PRON
ejpam-6321	207	24	)	)	PUNCT
ejpam-6321	207	25	,	,	PUNCT
ejpam-6321	207	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	207	27	)	)	PUNCT
ejpam-6321	207	28	)	)	PUNCT
ejpam-6321	207	29	,	,	PUNCT
ejpam-6321	207	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	207	31	)	)	PUNCT
ejpam-6321	207	32	,	,	PUNCT
ejpam-6321	207	33	x(3k+1	x(3k+1	CCONJ
ejpam-6321	207	34	)	)	PUNCT
ejpam-6321	207	35	,	,	PUNCT
ejpam-6321	207	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	207	37	)	)	PUNCT
ejpam-6321	207	38	)	)	PUNCT
ejpam-6321	207	39	}	}	PUNCT
ejpam-6321	207	40	after	after	ADP
ejpam-6321	207	41	simplification	simplification	NOUN
ejpam-6321	207	42	process	process	NOUN
ejpam-6321	207	43	we	we	PRON
ejpam-6321	207	44	get	get	VERB
ejpam-6321	207	45	,	,	PUNCT
ejpam-6321	207	46	g(f1℘	g(f1℘	PROPN
ejpam-6321	207	47	,	,	PUNCT
ejpam-6321	207	48	x(3k+2	x(3k+2	NOUN
ejpam-6321	207	49	)	)	PUNCT
ejpam-6321	207	50	,	,	PUNCT
ejpam-6321	207	51	x(3k+3	x(3k+3	NUM
ejpam-6321	207	52	)	)	PUNCT
ejpam-6321	207	53	)	)	PUNCT
ejpam-6321	207	54	≤	≤	NOUN
ejpam-6321	207	55	α1g(℘	α1g(℘	NOUN
ejpam-6321	207	56	,	,	PUNCT
ejpam-6321	207	57	x(3k+1	x(3k+1	PRON
ejpam-6321	207	58	)	)	PUNCT
ejpam-6321	207	59	,	,	PUNCT
ejpam-6321	207	60	x(3k+2	x(3k+2	NOUN
ejpam-6321	207	61	)	)	PUNCT
ejpam-6321	207	62	)	)	PUNCT
ejpam-6321	208	1	+	+	CCONJ
ejpam-6321	208	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	208	3	,	,	PUNCT
ejpam-6321	208	4	x(3k+1	x(3k+1	PRON
ejpam-6321	208	5	)	)	PUNCT
ejpam-6321	208	6	,	,	PUNCT
ejpam-6321	208	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	208	8	)	)	PUNCT
ejpam-6321	208	9	)	)	PUNCT
ejpam-6321	209	1	+	+	CCONJ
ejpam-6321	210	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	210	2	,	,	PUNCT
ejpam-6321	210	3	x(3k+1	x(3k+1	ADV
ejpam-6321	210	4	)	)	PUNCT
ejpam-6321	210	5	,	,	PUNCT
ejpam-6321	210	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	210	7	)	)	PUNCT
ejpam-6321	210	8	)	)	PUNCT
ejpam-6321	211	1	+	+	CCONJ
ejpam-6321	211	2	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	211	3	,	,	PUNCT
ejpam-6321	211	4	x3k	x3k	NOUN
ejpam-6321	211	5	,	,	PUNCT
ejpam-6321	211	6	x3k	x3k	PUNCT
ejpam-6321	211	7	)	)	PUNCT
ejpam-6321	212	1	+	+	SYM
ejpam-6321	212	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	212	3	)	)	PUNCT
ejpam-6321	212	4	,	,	PUNCT
ejpam-6321	212	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	212	6	)	)	PUNCT
ejpam-6321	212	7	,	,	PUNCT
ejpam-6321	212	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	212	9	)	)	PUNCT
ejpam-6321	212	10	)	)	PUNCT
ejpam-6321	213	1	+	+	PUNCT
ejpam-6321	213	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	213	3	)	)	PUNCT
ejpam-6321	213	4	,	,	PUNCT
ejpam-6321	213	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	213	6	)	)	PUNCT
ejpam-6321	213	7	,	,	PUNCT
ejpam-6321	213	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	213	9	)	)	PUNCT
ejpam-6321	213	10	)	)	PUNCT
ejpam-6321	213	11	]	]	PUNCT
ejpam-6321	213	12	applying	apply	VERB
ejpam-6321	213	13	limn→∞	limn→∞	PROPN
ejpam-6321	213	14	on	on	ADP
ejpam-6321	213	15	both	both	DET
ejpam-6321	213	16	sides	side	NOUN
ejpam-6321	213	17	,	,	PUNCT
ejpam-6321	213	18	we	we	PRON
ejpam-6321	213	19	get	get	VERB
ejpam-6321	213	20	,	,	PUNCT
ejpam-6321	213	21	g(f1℘	g(f1℘	PROPN
ejpam-6321	213	22	,	,	PUNCT
ejpam-6321	213	23	℘	℘	PROPN
ejpam-6321	213	24	,	,	PUNCT
ejpam-6321	213	25	℘	℘	NOUN
ejpam-6321	213	26	)	)	PUNCT
ejpam-6321	213	27	≤	≤	NOUN
ejpam-6321	213	28	α1g(℘	α1g(℘	NOUN
ejpam-6321	213	29	,	,	PUNCT
ejpam-6321	213	30	℘	℘	PROPN
ejpam-6321	213	31	,	,	PUNCT
ejpam-6321	213	32	℘)+α2g(℘	℘)+α2g(℘	NOUN
ejpam-6321	213	33	,	,	PUNCT
ejpam-6321	213	34	℘	℘	PROPN
ejpam-6321	213	35	,	,	PUNCT
ejpam-6321	213	36	℘)+α3g(f1℘	℘)+α3g(f1℘	NOUN
ejpam-6321	213	37	,	,	PUNCT
ejpam-6321	213	38	℘	℘	PROPN
ejpam-6321	213	39	,	,	PUNCT
ejpam-6321	213	40	℘)+α4[g(f1℘	℘)+α4[g(f1℘	PROPN
ejpam-6321	213	41	,	,	PUNCT
ejpam-6321	213	42	℘	℘	PROPN
ejpam-6321	213	43	,	,	PUNCT
ejpam-6321	213	44	℘)+g(℘	℘)+g(℘	PROPN
ejpam-6321	213	45	,	,	PUNCT
ejpam-6321	213	46	℘	℘	PROPN
ejpam-6321	213	47	,	,	PUNCT
ejpam-6321	213	48	℘)+g(℘	℘)+g(℘	PROPN
ejpam-6321	213	49	,	,	PUNCT
ejpam-6321	213	50	℘	℘	NOUN
ejpam-6321	213	51	,	,	PUNCT
ejpam-6321	213	52	℘	℘	NUM
ejpam-6321	213	53	)	)	PUNCT
ejpam-6321	213	54	]	]	PUNCT
ejpam-6321	213	55	≤	≤	NUM
ejpam-6321	213	56	(	(	PUNCT
ejpam-6321	213	57	α3	α3	PROPN
ejpam-6321	213	58	+	+	CCONJ
ejpam-6321	213	59	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	213	60	,	,	PUNCT
ejpam-6321	213	61	℘	℘	PROPN
ejpam-6321	213	62	,	,	PUNCT
ejpam-6321	213	63	℘	℘	NOUN
ejpam-6321	213	64	)	)	PUNCT
ejpam-6321	213	65	g(f1℘	g(f1℘	PROPN
ejpam-6321	213	66	,	,	PUNCT
ejpam-6321	213	67	℘	℘	PROPN
ejpam-6321	213	68	,	,	PUNCT
ejpam-6321	213	69	℘)−	℘)−	PROPN
ejpam-6321	213	70	(	(	PUNCT
ejpam-6321	213	71	α3+α4)g(f1℘	α3+α4)g(f1℘	PROPN
ejpam-6321	213	72	,	,	PUNCT
ejpam-6321	213	73	℘	℘	PROPN
ejpam-6321	213	74	,	,	PUNCT
ejpam-6321	213	75	℘	℘	NOUN
ejpam-6321	213	76	)	)	PUNCT
ejpam-6321	213	77	≤	≤	NOUN
ejpam-6321	213	78	0	0	NUM
ejpam-6321	213	79	,	,	PUNCT
ejpam-6321	213	80	is	be	AUX
ejpam-6321	213	81	a	a	DET
ejpam-6321	213	82	contradiction	contradiction	NOUN
ejpam-6321	213	83	.	.	PUNCT
ejpam-6321	214	1	since	since	SCONJ
ejpam-6321	214	2	(	(	PUNCT
ejpam-6321	214	3	1−α3−α4	1−α3−α4	NUM
ejpam-6321	214	4	)	)	PUNCT
ejpam-6321	214	5	̸=	̸=	PROPN
ejpam-6321	214	6	0	0	NUM
ejpam-6321	214	7	,	,	PUNCT
ejpam-6321	214	8	therefore	therefore	ADV
ejpam-6321	214	9	,	,	PUNCT
ejpam-6321	214	10	g(f1℘	g(f1℘	PROPN
ejpam-6321	214	11	,	,	PUNCT
ejpam-6321	214	12	℘	℘	PROPN
ejpam-6321	214	13	,	,	PUNCT
ejpam-6321	214	14	℘	℘	NUM
ejpam-6321	214	15	)	)	PUNCT
ejpam-6321	214	16	=	=	SYM
ejpam-6321	214	17	0	0	X
ejpam-6321	214	18	.	.	PUNCT
ejpam-6321	214	19	thus	thus	ADV
ejpam-6321	214	20	,	,	PUNCT
ejpam-6321	214	21	f1℘	f1℘	PROPN
ejpam-6321	214	22	=	=	SYM
ejpam-6321	214	23	℘	℘	PROPN
ejpam-6321	214	24	(	(	PUNCT
ejpam-6321	214	25	5	5	NUM
ejpam-6321	214	26	)	)	PUNCT
ejpam-6321	214	27	next	next	ADV
ejpam-6321	214	28	,	,	PUNCT
ejpam-6321	214	29	we	we	PRON
ejpam-6321	214	30	have	have	VERB
ejpam-6321	214	31	to	to	PART
ejpam-6321	214	32	show	show	VERB
ejpam-6321	214	33	that	that	SCONJ
ejpam-6321	214	34	f2℘	f2℘	NOUN
ejpam-6321	214	35	=	=	PUNCT
ejpam-6321	214	36	℘	℘	PROPN
ejpam-6321	214	37	by	by	ADP
ejpam-6321	214	38	contrary	contrary	ADJ
ejpam-6321	214	39	case	case	NOUN
ejpam-6321	214	40	let	let	VERB
ejpam-6321	214	41	f2℘	f2℘	NOUN
ejpam-6321	214	42	̸=	̸=	PROPN
ejpam-6321	214	43	℘.	℘.	PROPN
ejpam-6321	214	44	then	then	ADV
ejpam-6321	214	45	from	from	ADP
ejpam-6321	214	46	(	(	PUNCT
ejpam-6321	214	47	1	1	X
ejpam-6321	214	48	)	)	PUNCT
ejpam-6321	214	49	we	we	PRON
ejpam-6321	214	50	have	have	VERB
ejpam-6321	214	51	that	that	PRON
ejpam-6321	214	52	,	,	PUNCT
ejpam-6321	214	53	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	214	54	)	)	PUNCT
ejpam-6321	214	55	,	,	PUNCT
ejpam-6321	214	56	f2℘	f2℘	NOUN
ejpam-6321	214	57	,	,	PUNCT
ejpam-6321	214	58	x(3k+3	x(3k+3	NUM
ejpam-6321	214	59	)	)	PUNCT
ejpam-6321	214	60	)	)	PUNCT
ejpam-6321	215	1	=	=	PUNCT
ejpam-6321	215	2	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	215	3	,	,	PUNCT
ejpam-6321	215	4	f2℘	f2℘	NOUN
ejpam-6321	215	5	,	,	PUNCT
ejpam-6321	215	6	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	215	7	)	)	PUNCT
ejpam-6321	215	8	)	)	PUNCT
ejpam-6321	216	1	≤	≤	NUM
ejpam-6321	216	2	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	216	3	,	,	PUNCT
ejpam-6321	216	4	℘	℘	PROPN
ejpam-6321	216	5	,	,	PUNCT
ejpam-6321	216	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	216	7	)	)	PUNCT
ejpam-6321	216	8	)	)	PUNCT
ejpam-6321	217	1	+	+	CCONJ
ejpam-6321	217	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	217	3	,	,	PUNCT
ejpam-6321	217	4	℘	℘	NOUN
ejpam-6321	217	5	,	,	PUNCT
ejpam-6321	217	6	f2℘	f2℘	NOUN
ejpam-6321	217	7	)	)	PUNCT
ejpam-6321	217	8	+	+	CCONJ
ejpam-6321	217	9	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	217	10	,	,	PUNCT
ejpam-6321	217	11	℘	℘	NOUN
ejpam-6321	217	12	,	,	PUNCT
ejpam-6321	217	13	℘	℘	NUM
ejpam-6321	217	14	)	)	PUNCT
ejpam-6321	217	15	+	+	SYM
ejpam-6321	217	16	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	217	17	,	,	PUNCT
ejpam-6321	217	18	x3k	x3k	NOUN
ejpam-6321	217	19	,	,	PUNCT
ejpam-6321	217	20	x3k	x3k	PUNCT
ejpam-6321	217	21	)	)	PUNCT
ejpam-6321	217	22	+	+	SYM
ejpam-6321	217	23	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	217	24	)	)	PUNCT
ejpam-6321	217	25	,	,	PUNCT
ejpam-6321	217	26	f2℘	f2℘	NOUN
ejpam-6321	217	27	,	,	PUNCT
ejpam-6321	217	28	℘	℘	PROPN
ejpam-6321	217	29	)	)	PUNCT
ejpam-6321	217	30	+	+	NOUN
ejpam-6321	217	31	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	217	32	)	)	PUNCT
ejpam-6321	217	33	,	,	PUNCT
ejpam-6321	217	34	x(3k+2	x(3k+2	NOUN
ejpam-6321	217	35	)	)	PUNCT
ejpam-6321	217	36	,	,	PUNCT
ejpam-6321	217	37	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	217	38	)	)	PUNCT
ejpam-6321	217	39	)	)	PUNCT
ejpam-6321	217	40	]	]	PUNCT
ejpam-6321	218	1	+	+	CCONJ
ejpam-6321	218	2	α5min	α5min	NUM
ejpam-6321	218	3	{	{	PUNCT
ejpam-6321	218	4	g(x3k	g(x3k	NOUN
ejpam-6321	218	5	,	,	PUNCT
ejpam-6321	218	6	℘	℘	NOUN
ejpam-6321	218	7	,	,	PUNCT
ejpam-6321	218	8	f2℘	f2℘	NOUN
ejpam-6321	218	9	)	)	PUNCT
ejpam-6321	218	10	,	,	PUNCT
ejpam-6321	218	11	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	218	12	,	,	PUNCT
ejpam-6321	218	13	f1x3k	f1x3k	NUM
ejpam-6321	218	14	,	,	PUNCT
ejpam-6321	218	15	℘	℘	PROPN
ejpam-6321	218	16	)	)	PUNCT
ejpam-6321	218	17	,	,	PUNCT
ejpam-6321	218	18	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	218	19	,	,	PUNCT
ejpam-6321	218	20	℘	℘	PROPN
ejpam-6321	218	21	,	,	PUNCT
ejpam-6321	218	22	℘	℘	PROPN
ejpam-6321	218	23	)	)	PUNCT
ejpam-6321	218	24	,	,	PUNCT
ejpam-6321	218	25	g(f2℘	g(f2℘	PROPN
ejpam-6321	218	26	,	,	PUNCT
ejpam-6321	218	27	f2℘	f2℘	NOUN
ejpam-6321	218	28	,	,	PUNCT
ejpam-6321	218	29	x(3k+2	x(3k+2	NOUN
ejpam-6321	218	30	)	)	PUNCT
ejpam-6321	218	31	)	)	PUNCT
ejpam-6321	218	32	}	}	PUNCT
ejpam-6321	218	33	m.	m.	PROPN
ejpam-6321	218	34	noorwali	noorwali	PROPN
ejpam-6321	218	35	et	et	PROPN
ejpam-6321	218	36	al	al	PROPN
ejpam-6321	218	37	.	.	PUNCT
ejpam-6321	218	38	/	/	SYM
ejpam-6321	218	39	eur	eur	PROPN
ejpam-6321	218	40	.	.	PUNCT
ejpam-6321	219	1	j.	j.	PROPN
ejpam-6321	219	2	pure	pure	PROPN
ejpam-6321	219	3	appl	appl	PROPN
ejpam-6321	219	4	.	.	PROPN
ejpam-6321	219	5	math	math	PROPN
ejpam-6321	219	6	,	,	PUNCT
ejpam-6321	219	7	18	18	NUM
ejpam-6321	219	8	(	(	PUNCT
ejpam-6321	219	9	3	3	NUM
ejpam-6321	219	10	)	)	PUNCT
ejpam-6321	219	11	(	(	PUNCT
ejpam-6321	219	12	2025	2025	NUM
ejpam-6321	219	13	)	)	PUNCT
ejpam-6321	219	14	,	,	PUNCT
ejpam-6321	219	15	6321	6321	NUM
ejpam-6321	219	16	7	7	NUM
ejpam-6321	219	17	of	of	ADP
ejpam-6321	219	18	27	27	NUM
ejpam-6321	219	19	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	219	20	)	)	PUNCT
ejpam-6321	219	21	,	,	PUNCT
ejpam-6321	219	22	f2℘	f2℘	NOUN
ejpam-6321	219	23	,	,	PUNCT
ejpam-6321	219	24	x(3k+3	x(3k+3	NUM
ejpam-6321	219	25	)	)	PUNCT
ejpam-6321	219	26	)	)	PUNCT
ejpam-6321	220	1	=	=	SYM
ejpam-6321	220	2	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	220	3	,	,	PUNCT
ejpam-6321	220	4	℘	℘	PROPN
ejpam-6321	220	5	,	,	PUNCT
ejpam-6321	220	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	220	7	)	)	PUNCT
ejpam-6321	220	8	)	)	PUNCT
ejpam-6321	221	1	+	+	CCONJ
ejpam-6321	221	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	221	3	,	,	PUNCT
ejpam-6321	221	4	℘	℘	NOUN
ejpam-6321	221	5	,	,	PUNCT
ejpam-6321	221	6	f2℘	f2℘	NOUN
ejpam-6321	221	7	)	)	PUNCT
ejpam-6321	221	8	+	+	SYM
ejpam-6321	221	9	α3g(x3k	α3g(x3k	NOUN
ejpam-6321	221	10	,	,	PUNCT
ejpam-6321	221	11	℘	℘	NOUN
ejpam-6321	221	12	,	,	PUNCT
ejpam-6321	221	13	℘	℘	NUM
ejpam-6321	221	14	)	)	PUNCT
ejpam-6321	221	15	+	+	NUM
ejpam-6321	221	16	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	221	17	,	,	PUNCT
ejpam-6321	221	18	x3k	x3k	NOUN
ejpam-6321	221	19	,	,	PUNCT
ejpam-6321	221	20	x3k	x3k	PUNCT
ejpam-6321	221	21	)	)	PUNCT
ejpam-6321	221	22	+	+	SYM
ejpam-6321	221	23	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	221	24	)	)	PUNCT
ejpam-6321	221	25	,	,	PUNCT
ejpam-6321	221	26	f2℘	f2℘	NOUN
ejpam-6321	221	27	,	,	PUNCT
ejpam-6321	221	28	℘	℘	PROPN
ejpam-6321	221	29	)	)	PUNCT
ejpam-6321	221	30	+	+	NOUN
ejpam-6321	221	31	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	221	32	)	)	PUNCT
ejpam-6321	221	33	,	,	PUNCT
ejpam-6321	221	34	x(3k+2	x(3k+2	NOUN
ejpam-6321	221	35	)	)	PUNCT
ejpam-6321	221	36	,	,	PUNCT
ejpam-6321	221	37	x(3k+2	x(3k+2	NOUN
ejpam-6321	221	38	)	)	PUNCT
ejpam-6321	221	39	)	)	PUNCT
ejpam-6321	221	40	]	]	PUNCT
ejpam-6321	221	41	+	+	CCONJ
ejpam-6321	221	42	α5min	α5min	NUM
ejpam-6321	221	43	{	{	PUNCT
ejpam-6321	221	44	g(x3k	g(x3k	NOUN
ejpam-6321	221	45	,	,	PUNCT
ejpam-6321	221	46	℘	℘	NOUN
ejpam-6321	221	47	,	,	PUNCT
ejpam-6321	221	48	f2℘	f2℘	NOUN
ejpam-6321	221	49	)	)	PUNCT
ejpam-6321	221	50	,	,	PUNCT
ejpam-6321	221	51	g(x3k	g(x3k	NOUN
ejpam-6321	221	52	,	,	PUNCT
ejpam-6321	221	53	x3k	x3k	NOUN
ejpam-6321	221	54	,	,	PUNCT
ejpam-6321	221	55	℘	℘	PROPN
ejpam-6321	221	56	)	)	PUNCT
ejpam-6321	221	57	,	,	PUNCT
ejpam-6321	221	58	g(x3k	g(x3k	NOUN
ejpam-6321	221	59	,	,	PUNCT
ejpam-6321	221	60	℘	℘	PROPN
ejpam-6321	221	61	,	,	PUNCT
ejpam-6321	221	62	℘	℘	NUM
ejpam-6321	221	63	)	)	PUNCT
ejpam-6321	221	64	,	,	PUNCT
ejpam-6321	221	65	g(f2℘	g(f2℘	PROPN
ejpam-6321	221	66	,	,	PUNCT
ejpam-6321	221	67	f2℘	f2℘	NOUN
ejpam-6321	221	68	,	,	PUNCT
ejpam-6321	221	69	x(3k+2	x(3k+2	NOUN
ejpam-6321	221	70	)	)	PUNCT
ejpam-6321	221	71	)	)	PUNCT
ejpam-6321	221	72	}	}	PUNCT
ejpam-6321	221	73	applying	apply	VERB
ejpam-6321	221	74	limn→∞	limn→∞	PROPN
ejpam-6321	221	75	on	on	ADP
ejpam-6321	221	76	both	both	DET
ejpam-6321	221	77	sides	side	NOUN
ejpam-6321	221	78	,	,	PUNCT
ejpam-6321	221	79	we	we	PRON
ejpam-6321	221	80	get	get	VERB
ejpam-6321	221	81	,	,	PUNCT
ejpam-6321	221	82	g(℘	g(℘	NOUN
ejpam-6321	221	83	,	,	PUNCT
ejpam-6321	221	84	f2℘	f2℘	NOUN
ejpam-6321	221	85	,	,	PUNCT
ejpam-6321	221	86	℘	℘	PROPN
ejpam-6321	221	87	)	)	PUNCT
ejpam-6321	221	88	≤	≤	NOUN
ejpam-6321	221	89	α1g(℘	α1g(℘	NOUN
ejpam-6321	221	90	,	,	PUNCT
ejpam-6321	221	91	℘	℘	NOUN
ejpam-6321	221	92	,	,	PUNCT
ejpam-6321	221	93	℘	℘	NUM
ejpam-6321	221	94	)	)	PUNCT
ejpam-6321	221	95	+	+	NUM
ejpam-6321	221	96	α2g(℘	α2g(℘	NOUN
ejpam-6321	221	97	,	,	PUNCT
ejpam-6321	221	98	℘	℘	NOUN
ejpam-6321	221	99	,	,	PUNCT
ejpam-6321	221	100	f2℘	f2℘	NOUN
ejpam-6321	221	101	)	)	PUNCT
ejpam-6321	222	1	+	+	CCONJ
ejpam-6321	222	2	α3g(℘	α3g(℘	NUM
ejpam-6321	222	3	,	,	PUNCT
ejpam-6321	222	4	℘	℘	NOUN
ejpam-6321	222	5	,	,	PUNCT
ejpam-6321	222	6	℘)+	℘)+	NOUN
ejpam-6321	222	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	222	8	,	,	PUNCT
ejpam-6321	222	9	℘	℘	NOUN
ejpam-6321	222	10	,	,	PUNCT
ejpam-6321	222	11	℘	℘	NUM
ejpam-6321	222	12	)	)	PUNCT
ejpam-6321	222	13	+	+	NOUN
ejpam-6321	222	14	g(℘	g(℘	NOUN
ejpam-6321	222	15	,	,	PUNCT
ejpam-6321	222	16	f2℘	f2℘	NOUN
ejpam-6321	222	17	,	,	PUNCT
ejpam-6321	222	18	℘	℘	PROPN
ejpam-6321	222	19	)	)	PUNCT
ejpam-6321	222	20	+	+	NOUN
ejpam-6321	222	21	g(℘	g(℘	NOUN
ejpam-6321	222	22	,	,	PUNCT
ejpam-6321	222	23	℘	℘	NOUN
ejpam-6321	222	24	,	,	PUNCT
ejpam-6321	222	25	℘	℘	NUM
ejpam-6321	222	26	)	)	PUNCT
ejpam-6321	222	27	]	]	PUNCT
ejpam-6321	223	1	+	+	CCONJ
ejpam-6321	223	2	α5min	α5min	NUM
ejpam-6321	223	3	{	{	PUNCT
ejpam-6321	223	4	g(℘	g(℘	PROPN
ejpam-6321	223	5	,	,	PUNCT
ejpam-6321	223	6	℘	℘	NOUN
ejpam-6321	223	7	,	,	PUNCT
ejpam-6321	223	8	f2℘	f2℘	NOUN
ejpam-6321	223	9	)	)	PUNCT
ejpam-6321	223	10	,	,	PUNCT
ejpam-6321	223	11	g(℘	g(℘	PROPN
ejpam-6321	223	12	,	,	PUNCT
ejpam-6321	223	13	℘	℘	NOUN
ejpam-6321	223	14	,	,	PUNCT
ejpam-6321	223	15	℘	℘	PROPN
ejpam-6321	223	16	)	)	PUNCT
ejpam-6321	223	17	,	,	PUNCT
ejpam-6321	223	18	g(℘	g(℘	PROPN
ejpam-6321	223	19	,	,	PUNCT
ejpam-6321	223	20	℘	℘	NOUN
ejpam-6321	223	21	,	,	PUNCT
ejpam-6321	223	22	℘	℘	PROPN
ejpam-6321	223	23	)	)	PUNCT
ejpam-6321	223	24	,	,	PUNCT
ejpam-6321	223	25	g(f2℘	g(f2℘	PROPN
ejpam-6321	223	26	,	,	PUNCT
ejpam-6321	223	27	f2℘	f2℘	NOUN
ejpam-6321	223	28	,	,	PUNCT
ejpam-6321	223	29	℘	℘	NUM
ejpam-6321	223	30	)	)	PUNCT
ejpam-6321	223	31	}	}	PUNCT
ejpam-6321	223	32	≤	≤	NOUN
ejpam-6321	223	33	(	(	PUNCT
ejpam-6321	223	34	α2	α2	ADJ
ejpam-6321	223	35	+	+	CCONJ
ejpam-6321	223	36	α4)g(f2℘	α4)g(f2℘	NOUN
ejpam-6321	223	37	,	,	PUNCT
ejpam-6321	223	38	℘	℘	PROPN
ejpam-6321	223	39	,	,	PUNCT
ejpam-6321	223	40	℘	℘	NUM
ejpam-6321	223	41	)	)	PUNCT
ejpam-6321	223	42	g(f2℘	g(f2℘	PROPN
ejpam-6321	223	43	,	,	PUNCT
ejpam-6321	223	44	℘	℘	PROPN
ejpam-6321	223	45	,	,	PUNCT
ejpam-6321	223	46	℘)−	℘)−	NOUN
ejpam-6321	223	47	(	(	PUNCT
ejpam-6321	223	48	α2+α4)g(f2℘	α2+α4)g(f2℘	PROPN
ejpam-6321	223	49	,	,	PUNCT
ejpam-6321	223	50	℘	℘	NOUN
ejpam-6321	223	51	,	,	PUNCT
ejpam-6321	223	52	℘	℘	NOUN
ejpam-6321	223	53	)	)	PUNCT
ejpam-6321	223	54	≤	≤	NOUN
ejpam-6321	223	55	0	0	NUM
ejpam-6321	223	56	,	,	PUNCT
ejpam-6321	223	57	is	be	AUX
ejpam-6321	223	58	a	a	DET
ejpam-6321	223	59	contradiction	contradiction	NOUN
ejpam-6321	223	60	.	.	PUNCT
ejpam-6321	224	1	since	since	SCONJ
ejpam-6321	224	2	(	(	PUNCT
ejpam-6321	224	3	1−α2−α4	1−α2−α4	NUM
ejpam-6321	224	4	)	)	PUNCT
ejpam-6321	224	5	̸=	̸=	PROPN
ejpam-6321	224	6	0	0	NUM
ejpam-6321	224	7	,	,	PUNCT
ejpam-6321	224	8	therefore	therefore	ADV
ejpam-6321	224	9	,	,	PUNCT
ejpam-6321	224	10	g(f2℘	g(f2℘	PROPN
ejpam-6321	224	11	,	,	PUNCT
ejpam-6321	224	12	℘	℘	PROPN
ejpam-6321	224	13	,	,	PUNCT
ejpam-6321	224	14	℘	℘	NUM
ejpam-6321	224	15	)	)	PUNCT
ejpam-6321	224	16	=	=	SYM
ejpam-6321	224	17	0	0	X
ejpam-6321	224	18	.	.	PUNCT
ejpam-6321	224	19	thus	thus	ADV
ejpam-6321	224	20	,	,	PUNCT
ejpam-6321	224	21	f2℘	f2℘	NOUN
ejpam-6321	224	22	=	=	PUNCT
ejpam-6321	224	23	℘	℘	PROPN
ejpam-6321	224	24	(	(	PUNCT
ejpam-6321	224	25	6	6	NUM
ejpam-6321	224	26	)	)	PUNCT
ejpam-6321	224	27	next	next	ADV
ejpam-6321	224	28	,	,	PUNCT
ejpam-6321	224	29	we	we	PRON
ejpam-6321	224	30	have	have	VERB
ejpam-6321	224	31	to	to	PART
ejpam-6321	224	32	show	show	VERB
ejpam-6321	224	33	that	that	SCONJ
ejpam-6321	224	34	f3℘	f3℘	NOUN
ejpam-6321	224	35	=	=	SYM
ejpam-6321	224	36	℘	℘	PROPN
ejpam-6321	224	37	by	by	ADP
ejpam-6321	224	38	contrary	contrary	ADJ
ejpam-6321	224	39	case	case	NOUN
ejpam-6321	224	40	let	let	VERB
ejpam-6321	224	41	f3℘	f3℘	PROPN
ejpam-6321	224	42	̸=	̸=	PROPN
ejpam-6321	224	43	℘.	℘.	PROPN
ejpam-6321	224	44	then	then	ADV
ejpam-6321	224	45	from	from	ADP
ejpam-6321	224	46	(	(	PUNCT
ejpam-6321	224	47	1	1	X
ejpam-6321	224	48	)	)	PUNCT
ejpam-6321	224	49	we	we	PRON
ejpam-6321	224	50	have	have	VERB
ejpam-6321	224	51	that	that	PRON
ejpam-6321	224	52	,	,	PUNCT
ejpam-6321	224	53	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	224	54	)	)	PUNCT
ejpam-6321	224	55	,	,	PUNCT
ejpam-6321	224	56	x(3k+2	x(3k+2	NOUN
ejpam-6321	224	57	)	)	PUNCT
ejpam-6321	224	58	,	,	PUNCT
ejpam-6321	224	59	f3℘	f3℘	X
ejpam-6321	224	60	)	)	PUNCT
ejpam-6321	224	61	=	=	SYM
ejpam-6321	224	62	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	224	63	,	,	PUNCT
ejpam-6321	224	64	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	224	65	)	)	PUNCT
ejpam-6321	224	66	,	,	PUNCT
ejpam-6321	224	67	f3℘	f3℘	NOUN
ejpam-6321	224	68	)	)	PUNCT
ejpam-6321	224	69	≤	≤	NUM
ejpam-6321	224	70	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	224	71	,	,	PUNCT
ejpam-6321	224	72	x(3k+1	x(3k+1	PRON
ejpam-6321	224	73	)	)	PUNCT
ejpam-6321	224	74	,	,	PUNCT
ejpam-6321	224	75	℘	℘	PROPN
ejpam-6321	224	76	)	)	PUNCT
ejpam-6321	224	77	+	+	NUM
ejpam-6321	224	78	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	224	79	,	,	PUNCT
ejpam-6321	224	80	x(3k+1	x(3k+1	PRON
ejpam-6321	224	81	)	)	PUNCT
ejpam-6321	224	82	,	,	PUNCT
ejpam-6321	224	83	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	224	84	)	)	PUNCT
ejpam-6321	224	85	)	)	PUNCT
ejpam-6321	225	1	+	+	CCONJ
ejpam-6321	225	2	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	225	3	,	,	PUNCT
ejpam-6321	225	4	x(3k+1	x(3k+1	ADV
ejpam-6321	225	5	)	)	PUNCT
ejpam-6321	225	6	,	,	PUNCT
ejpam-6321	225	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	225	8	)	)	PUNCT
ejpam-6321	225	9	)	)	PUNCT
ejpam-6321	226	1	+	+	CCONJ
ejpam-6321	226	2	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	226	3	,	,	PUNCT
ejpam-6321	226	4	x3k	x3k	NOUN
ejpam-6321	226	5	,	,	PUNCT
ejpam-6321	226	6	x3k	x3k	PUNCT
ejpam-6321	226	7	)	)	PUNCT
ejpam-6321	226	8	+	+	SYM
ejpam-6321	226	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	226	10	)	)	PUNCT
ejpam-6321	226	11	,	,	PUNCT
ejpam-6321	226	12	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	226	13	)	)	PUNCT
ejpam-6321	226	14	,	,	PUNCT
ejpam-6321	226	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	226	16	)	)	PUNCT
ejpam-6321	226	17	)	)	PUNCT
ejpam-6321	227	1	+	+	PUNCT
ejpam-6321	227	2	g(℘	g(℘	NOUN
ejpam-6321	227	3	,	,	PUNCT
ejpam-6321	227	4	℘	℘	NOUN
ejpam-6321	227	5	,	,	PUNCT
ejpam-6321	227	6	f3℘	f3℘	NOUN
ejpam-6321	227	7	)	)	PUNCT
ejpam-6321	227	8	]	]	PUNCT
ejpam-6321	228	1	+	+	CCONJ
ejpam-6321	228	2	α5min	α5min	NUM
ejpam-6321	228	3	{	{	PUNCT
ejpam-6321	228	4	g(x3k	g(x3k	NOUN
ejpam-6321	228	5	,	,	PUNCT
ejpam-6321	228	6	x(3k+1	x(3k+1	PRON
ejpam-6321	228	7	)	)	PUNCT
ejpam-6321	228	8	,	,	PUNCT
ejpam-6321	228	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	228	10	)	)	PUNCT
ejpam-6321	228	11	)	)	PUNCT
ejpam-6321	228	12	,	,	PUNCT
ejpam-6321	228	13	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	228	14	,	,	PUNCT
ejpam-6321	228	15	f1x3k	f1x3k	NUM
ejpam-6321	228	16	,	,	PUNCT
ejpam-6321	228	17	x(3k+1	x(3k+1	NUM
ejpam-6321	228	18	)	)	PUNCT
ejpam-6321	228	19	)	)	PUNCT
ejpam-6321	228	20	,	,	PUNCT
ejpam-6321	228	21	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	228	22	,	,	PUNCT
ejpam-6321	228	23	x(3k+1	x(3k+1	ADV
ejpam-6321	228	24	)	)	PUNCT
ejpam-6321	228	25	,	,	PUNCT
ejpam-6321	228	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	228	27	)	)	PUNCT
ejpam-6321	228	28	)	)	PUNCT
ejpam-6321	228	29	,	,	PUNCT
ejpam-6321	228	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	228	31	)	)	PUNCT
ejpam-6321	228	32	,	,	PUNCT
ejpam-6321	228	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	228	34	)	)	PUNCT
ejpam-6321	228	35	,	,	PUNCT
ejpam-6321	228	36	x(3k	x(3k	X
ejpam-6321	228	37	+	+	ADJ
ejpam-6321	228	38	2	2	NUM
ejpam-6321	228	39	)	)	PUNCT
ejpam-6321	228	40	)	)	PUNCT
ejpam-6321	228	41	}	}	PUNCT
ejpam-6321	228	42	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	228	43	)	)	PUNCT
ejpam-6321	228	44	,	,	PUNCT
ejpam-6321	228	45	x(3k+2	x(3k+2	NOUN
ejpam-6321	228	46	)	)	PUNCT
ejpam-6321	228	47	,	,	PUNCT
ejpam-6321	228	48	f3℘	f3℘	X
ejpam-6321	228	49	)	)	PUNCT
ejpam-6321	228	50	=	=	SYM
ejpam-6321	228	51	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	228	52	,	,	PUNCT
ejpam-6321	228	53	x(3k+1	x(3k+1	PRON
ejpam-6321	228	54	)	)	PUNCT
ejpam-6321	228	55	,	,	PUNCT
ejpam-6321	228	56	℘	℘	PROPN
ejpam-6321	228	57	)	)	PUNCT
ejpam-6321	228	58	+	+	NUM
ejpam-6321	228	59	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	228	60	,	,	PUNCT
ejpam-6321	228	61	x(3k+1	x(3k+1	CCONJ
ejpam-6321	228	62	)	)	PUNCT
ejpam-6321	228	63	,	,	PUNCT
ejpam-6321	228	64	f2x(3k+1))+	f2x(3k+1))+	NOUN
ejpam-6321	228	65	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	228	66	,	,	PUNCT
ejpam-6321	228	67	x(3k+1	x(3k+1	ADV
ejpam-6321	228	68	)	)	PUNCT
ejpam-6321	228	69	,	,	PUNCT
ejpam-6321	228	70	x(3k+1	x(3k+1	CCONJ
ejpam-6321	228	71	)	)	PUNCT
ejpam-6321	228	72	)	)	PUNCT
ejpam-6321	229	1	+	+	CCONJ
ejpam-6321	229	2	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	229	3	,	,	PUNCT
ejpam-6321	229	4	x3k	x3k	NOUN
ejpam-6321	229	5	,	,	PUNCT
ejpam-6321	229	6	x3k	x3k	PUNCT
ejpam-6321	229	7	)	)	PUNCT
ejpam-6321	229	8	+	+	SYM
ejpam-6321	229	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	229	10	)	)	PUNCT
ejpam-6321	229	11	,	,	PUNCT
ejpam-6321	229	12	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	229	13	)	)	PUNCT
ejpam-6321	229	14	,	,	PUNCT
ejpam-6321	229	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	229	16	)	)	PUNCT
ejpam-6321	229	17	)	)	PUNCT
ejpam-6321	230	1	+	+	PUNCT
ejpam-6321	230	2	g(℘	g(℘	NOUN
ejpam-6321	230	3	,	,	PUNCT
ejpam-6321	230	4	℘	℘	NOUN
ejpam-6321	230	5	,	,	PUNCT
ejpam-6321	230	6	f3℘	f3℘	NOUN
ejpam-6321	230	7	)	)	PUNCT
ejpam-6321	230	8	]	]	PUNCT
ejpam-6321	231	1	+	+	CCONJ
ejpam-6321	231	2	α5min	α5min	NUM
ejpam-6321	231	3	{	{	PUNCT
ejpam-6321	231	4	g(x3k	g(x3k	NOUN
ejpam-6321	231	5	,	,	PUNCT
ejpam-6321	231	6	x(3k+1	x(3k+1	PRON
ejpam-6321	231	7	)	)	PUNCT
ejpam-6321	231	8	,	,	PUNCT
ejpam-6321	231	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	231	10	)	)	PUNCT
ejpam-6321	231	11	)	)	PUNCT
ejpam-6321	231	12	,	,	PUNCT
ejpam-6321	231	13	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	231	14	,	,	PUNCT
ejpam-6321	231	15	f1x3k	f1x3k	NUM
ejpam-6321	231	16	,	,	PUNCT
ejpam-6321	231	17	x(3k+1	x(3k+1	NUM
ejpam-6321	231	18	)	)	PUNCT
ejpam-6321	231	19	)	)	PUNCT
ejpam-6321	231	20	,	,	PUNCT
ejpam-6321	231	21	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	231	22	,	,	PUNCT
ejpam-6321	231	23	x(3k+1	x(3k+1	ADV
ejpam-6321	231	24	)	)	PUNCT
ejpam-6321	231	25	,	,	PUNCT
ejpam-6321	231	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	231	27	)	)	PUNCT
ejpam-6321	231	28	)	)	PUNCT
ejpam-6321	231	29	,	,	PUNCT
ejpam-6321	231	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	231	31	)	)	PUNCT
ejpam-6321	231	32	,	,	PUNCT
ejpam-6321	231	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	231	34	)	)	PUNCT
ejpam-6321	231	35	,	,	PUNCT
ejpam-6321	231	36	x(3k	x(3k	X
ejpam-6321	231	37	+	+	ADJ
ejpam-6321	231	38	2	2	NUM
ejpam-6321	231	39	)	)	PUNCT
ejpam-6321	231	40	)	)	PUNCT
ejpam-6321	231	41	}	}	PUNCT
ejpam-6321	231	42	applying	apply	VERB
ejpam-6321	231	43	limn→∞	limn→∞	PROPN
ejpam-6321	231	44	on	on	ADP
ejpam-6321	231	45	both	both	DET
ejpam-6321	231	46	sides	side	NOUN
ejpam-6321	231	47	,	,	PUNCT
ejpam-6321	231	48	we	we	PRON
ejpam-6321	231	49	get	get	VERB
ejpam-6321	231	50	,	,	PUNCT
ejpam-6321	231	51	g(℘	g(℘	PROPN
ejpam-6321	231	52	,	,	PUNCT
ejpam-6321	231	53	℘	℘	NOUN
ejpam-6321	231	54	,	,	PUNCT
ejpam-6321	231	55	f3℘	f3℘	NOUN
ejpam-6321	231	56	)	)	PUNCT
ejpam-6321	231	57	≤	≤	NOUN
ejpam-6321	231	58	α1g(℘	α1g(℘	NOUN
ejpam-6321	231	59	,	,	PUNCT
ejpam-6321	231	60	℘	℘	NOUN
ejpam-6321	231	61	,	,	PUNCT
ejpam-6321	231	62	℘	℘	NUM
ejpam-6321	231	63	)	)	PUNCT
ejpam-6321	231	64	+	+	NUM
ejpam-6321	231	65	α2g(℘	α2g(℘	NOUN
ejpam-6321	231	66	,	,	PUNCT
ejpam-6321	231	67	℘	℘	NOUN
ejpam-6321	231	68	,	,	PUNCT
ejpam-6321	231	69	℘	℘	NUM
ejpam-6321	231	70	)	)	PUNCT
ejpam-6321	231	71	+	+	NUM
ejpam-6321	231	72	α3g(℘	α3g(℘	NUM
ejpam-6321	231	73	,	,	PUNCT
ejpam-6321	231	74	℘	℘	NOUN
ejpam-6321	231	75	,	,	PUNCT
ejpam-6321	231	76	℘)+	℘)+	NOUN
ejpam-6321	231	77	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	231	78	,	,	PUNCT
ejpam-6321	231	79	℘	℘	NOUN
ejpam-6321	231	80	,	,	PUNCT
ejpam-6321	231	81	℘	℘	NUM
ejpam-6321	231	82	)	)	PUNCT
ejpam-6321	231	83	+	+	NOUN
ejpam-6321	231	84	g(℘	g(℘	NOUN
ejpam-6321	231	85	,	,	PUNCT
ejpam-6321	231	86	℘	℘	NOUN
ejpam-6321	231	87	,	,	PUNCT
ejpam-6321	231	88	℘	℘	NUM
ejpam-6321	231	89	)	)	PUNCT
ejpam-6321	231	90	+	+	NOUN
ejpam-6321	231	91	g(℘	g(℘	NOUN
ejpam-6321	231	92	,	,	PUNCT
ejpam-6321	231	93	℘	℘	NOUN
ejpam-6321	231	94	,	,	PUNCT
ejpam-6321	231	95	f3℘	f3℘	NOUN
ejpam-6321	231	96	)	)	PUNCT
ejpam-6321	231	97	]	]	PUNCT
ejpam-6321	232	1	+	+	CCONJ
ejpam-6321	232	2	α5min	α5min	NUM
ejpam-6321	232	3	{	{	PUNCT
ejpam-6321	232	4	g(℘	g(℘	PROPN
ejpam-6321	232	5	,	,	PUNCT
ejpam-6321	232	6	℘	℘	NOUN
ejpam-6321	232	7	,	,	PUNCT
ejpam-6321	232	8	℘	℘	PROPN
ejpam-6321	232	9	)	)	PUNCT
ejpam-6321	232	10	,	,	PUNCT
ejpam-6321	232	11	g(℘	g(℘	PROPN
ejpam-6321	232	12	,	,	PUNCT
ejpam-6321	232	13	℘	℘	NOUN
ejpam-6321	232	14	,	,	PUNCT
ejpam-6321	232	15	℘	℘	PROPN
ejpam-6321	232	16	)	)	PUNCT
ejpam-6321	232	17	,	,	PUNCT
ejpam-6321	232	18	g(℘	g(℘	PROPN
ejpam-6321	232	19	,	,	PUNCT
ejpam-6321	232	20	℘	℘	NOUN
ejpam-6321	232	21	,	,	PUNCT
ejpam-6321	232	22	℘	℘	PROPN
ejpam-6321	232	23	)	)	PUNCT
ejpam-6321	232	24	,	,	PUNCT
ejpam-6321	232	25	g(℘	g(℘	PROPN
ejpam-6321	232	26	,	,	PUNCT
ejpam-6321	232	27	℘	℘	NOUN
ejpam-6321	232	28	,	,	PUNCT
ejpam-6321	232	29	℘	℘	NUM
ejpam-6321	232	30	)	)	PUNCT
ejpam-6321	232	31	}	}	PUNCT
ejpam-6321	232	32	≤	≤	NUM
ejpam-6321	232	33	α4g(℘	α4g(℘	NOUN
ejpam-6321	232	34	,	,	PUNCT
ejpam-6321	232	35	℘	℘	PROPN
ejpam-6321	232	36	,	,	PUNCT
ejpam-6321	232	37	f3℘	f3℘	NOUN
ejpam-6321	232	38	)	)	PUNCT
ejpam-6321	232	39	g(℘	g(℘	PROPN
ejpam-6321	232	40	,	,	PUNCT
ejpam-6321	232	41	℘	℘	NOUN
ejpam-6321	232	42	,	,	PUNCT
ejpam-6321	232	43	f3℘)−α4g(℘	f3℘)−α4g(℘	NOUN
ejpam-6321	232	44	,	,	PUNCT
ejpam-6321	232	45	℘	℘	PROPN
ejpam-6321	232	46	,	,	PUNCT
ejpam-6321	232	47	f3℘	f3℘	NOUN
ejpam-6321	232	48	)	)	PUNCT
ejpam-6321	232	49	≤	≤	NOUN
ejpam-6321	232	50	0	0	NUM
ejpam-6321	232	51	,	,	PUNCT
ejpam-6321	232	52	is	be	AUX
ejpam-6321	232	53	a	a	DET
ejpam-6321	232	54	contradiction	contradiction	NOUN
ejpam-6321	232	55	.	.	PUNCT
ejpam-6321	233	1	since	since	SCONJ
ejpam-6321	233	2	(	(	PUNCT
ejpam-6321	233	3	1−α4	1−α4	NUM
ejpam-6321	233	4	)	)	PUNCT
ejpam-6321	233	5	̸=	̸=	PROPN
ejpam-6321	233	6	0	0	NUM
ejpam-6321	233	7	,	,	PUNCT
ejpam-6321	233	8	therefore	therefore	ADV
ejpam-6321	233	9	,	,	PUNCT
ejpam-6321	233	10	g(℘	g(℘	PROPN
ejpam-6321	233	11	,	,	PUNCT
ejpam-6321	233	12	℘	℘	NOUN
ejpam-6321	233	13	,	,	PUNCT
ejpam-6321	233	14	f3℘	f3℘	NOUN
ejpam-6321	233	15	)	)	PUNCT
ejpam-6321	233	16	=	=	SYM
ejpam-6321	233	17	0	0	X
ejpam-6321	233	18	.	.	PUNCT
ejpam-6321	234	1	thus	thus	ADV
ejpam-6321	234	2	,	,	PUNCT
ejpam-6321	234	3	f3℘	f3℘	NOUN
ejpam-6321	234	4	=	=	SYM
ejpam-6321	234	5	℘	℘	PROPN
ejpam-6321	234	6	(	(	PUNCT
ejpam-6321	234	7	7	7	NUM
ejpam-6321	234	8	)	)	PUNCT
ejpam-6321	234	9	thus	thus	ADV
ejpam-6321	234	10	from	from	ADP
ejpam-6321	234	11	(	(	PUNCT
ejpam-6321	234	12	5	5	NUM
ejpam-6321	234	13	)	)	PUNCT
ejpam-6321	234	14	,	,	PUNCT
ejpam-6321	234	15	(	(	PUNCT
ejpam-6321	234	16	6	6	NUM
ejpam-6321	234	17	)	)	PUNCT
ejpam-6321	234	18	and	and	CCONJ
ejpam-6321	234	19	(	(	PUNCT
ejpam-6321	234	20	7	7	NUM
ejpam-6321	234	21	)	)	PUNCT
ejpam-6321	234	22	,	,	PUNCT
ejpam-6321	234	23	it	it	PRON
ejpam-6321	234	24	is	be	AUX
ejpam-6321	234	25	proved	prove	VERB
ejpam-6321	234	26	that	that	SCONJ
ejpam-6321	234	27	℘	℘	PROPN
ejpam-6321	234	28	is	be	AUX
ejpam-6321	234	29	a	a	DET
ejpam-6321	234	30	cfp	cfp	NOUN
ejpam-6321	234	31	of	of	ADP
ejpam-6321	234	32	f1	f1	NOUN
ejpam-6321	234	33	,	,	PUNCT
ejpam-6321	234	34	f2	f2	PROPN
ejpam-6321	234	35	and	and	CCONJ
ejpam-6321	234	36	f3	f3	ADJ
ejpam-6321	234	37	,	,	PUNCT
ejpam-6321	234	38	such	such	ADJ
ejpam-6321	234	39	that	that	DET
ejpam-6321	234	40	f1℘	f1℘	PROPN
ejpam-6321	234	41	=	=	X
ejpam-6321	234	42	f2℘	f2℘	NOUN
ejpam-6321	234	43	=	=	SYM
ejpam-6321	234	44	f3℘	f3℘	NOUN
ejpam-6321	234	45	=	=	SYM
ejpam-6321	234	46	℘	℘	PROPN
ejpam-6321	234	47	uniqueness	uniqueness	NOUN
ejpam-6321	234	48	:	:	PUNCT
ejpam-6321	234	49	suppose	suppose	VERB
ejpam-6321	234	50	that	that	SCONJ
ejpam-6321	234	51	℘⋄	℘⋄	ADJ
ejpam-6321	234	52	∈	∈	NOUN
ejpam-6321	234	53	x	x	PUNCT
ejpam-6321	234	54	be	be	AUX
ejpam-6321	234	55	the	the	DET
ejpam-6321	234	56	other	other	ADJ
ejpam-6321	234	57	cfp	cfp	PROPN
ejpam-6321	234	58	of	of	ADP
ejpam-6321	234	59	f1	f1	NOUN
ejpam-6321	234	60	,	,	PUNCT
ejpam-6321	234	61	f2	f2	PROPN
ejpam-6321	234	62	and	and	CCONJ
ejpam-6321	234	63	f3	f3	ADJ
ejpam-6321	234	64	,	,	PUNCT
ejpam-6321	234	65	so	so	SCONJ
ejpam-6321	234	66	that	that	SCONJ
ejpam-6321	234	67	f1℘	f1℘	PROPN
ejpam-6321	234	68	⋄	⋄	NOUN
ejpam-6321	234	69	=	=	PUNCT
ejpam-6321	234	70	f2℘	f2℘	PROPN
ejpam-6321	234	71	⋄	⋄	NOUN
ejpam-6321	234	72	=	=	PUNCT
ejpam-6321	234	73	f3℘	f3℘	X
ejpam-6321	234	74	⋄	⋄	NOUN
ejpam-6321	234	75	=	=	SYM
ejpam-6321	234	76	℘⋄	℘⋄	PROPN
ejpam-6321	234	77	m.	m.	NOUN
ejpam-6321	234	78	noorwali	noorwali	PROPN
ejpam-6321	234	79	et	et	PROPN
ejpam-6321	234	80	al	al	PROPN
ejpam-6321	234	81	.	.	PUNCT
ejpam-6321	234	82	/	/	SYM
ejpam-6321	234	83	eur	eur	PROPN
ejpam-6321	234	84	.	.	PUNCT
ejpam-6321	235	1	j.	j.	PROPN
ejpam-6321	235	2	pure	pure	PROPN
ejpam-6321	235	3	appl	appl	PROPN
ejpam-6321	235	4	.	.	PROPN
ejpam-6321	235	5	math	math	PROPN
ejpam-6321	235	6	,	,	PUNCT
ejpam-6321	235	7	18	18	NUM
ejpam-6321	235	8	(	(	PUNCT
ejpam-6321	235	9	3	3	NUM
ejpam-6321	235	10	)	)	PUNCT
ejpam-6321	235	11	(	(	PUNCT
ejpam-6321	235	12	2025	2025	NUM
ejpam-6321	235	13	)	)	PUNCT
ejpam-6321	235	14	,	,	PUNCT
ejpam-6321	235	15	6321	6321	NUM
ejpam-6321	235	16	8	8	NUM
ejpam-6321	235	17	of	of	ADP
ejpam-6321	235	18	27	27	NUM
ejpam-6321	235	19	then	then	ADV
ejpam-6321	235	20	from	from	ADP
ejpam-6321	235	21	(	(	PUNCT
ejpam-6321	235	22	1	1	X
ejpam-6321	235	23	)	)	PUNCT
ejpam-6321	235	24	we	we	PRON
ejpam-6321	235	25	have	have	VERB
ejpam-6321	235	26	that	that	PRON
ejpam-6321	235	27	,	,	PUNCT
ejpam-6321	235	28	g(℘	g(℘	PROPN
ejpam-6321	235	29	,	,	PUNCT
ejpam-6321	235	30	℘⋄	℘⋄	NUM
ejpam-6321	235	31	,	,	PUNCT
ejpam-6321	235	32	℘⋄	℘⋄	NUM
ejpam-6321	235	33	)	)	PUNCT
ejpam-6321	235	34	=	=	SYM
ejpam-6321	235	35	g(f1℘	g(f1℘	NOUN
ejpam-6321	235	36	,	,	PUNCT
ejpam-6321	235	37	f2℘	f2℘	PROPN
ejpam-6321	235	38	⋄	⋄	PROPN
ejpam-6321	235	39	,	,	PUNCT
ejpam-6321	235	40	f3℘	f3℘	PROPN
ejpam-6321	235	41	⋄	⋄	NOUN
ejpam-6321	235	42	)	)	PUNCT
ejpam-6321	236	1	≤	≤	NOUN
ejpam-6321	236	2	α1g(℘	α1g(℘	NOUN
ejpam-6321	236	3	,	,	PUNCT
ejpam-6321	236	4	℘⋄	℘⋄	X
ejpam-6321	236	5	,	,	PUNCT
ejpam-6321	236	6	℘⋄	℘⋄	NUM
ejpam-6321	236	7	)	)	PUNCT
ejpam-6321	236	8	+	+	NUM
ejpam-6321	236	9	α2g(℘	α2g(℘	NOUN
ejpam-6321	236	10	,	,	PUNCT
ejpam-6321	236	11	℘⋄	℘⋄	NUM
ejpam-6321	236	12	,	,	PUNCT
ejpam-6321	236	13	f2℘	f2℘	PROPN
ejpam-6321	236	14	⋄	⋄	PROPN
ejpam-6321	236	15	)	)	PUNCT
ejpam-6321	237	1	+	+	CCONJ
ejpam-6321	238	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	238	2	,	,	PUNCT
ejpam-6321	238	3	℘	℘	PROPN
ejpam-6321	238	4	⋄	⋄	PROPN
ejpam-6321	238	5	,	,	PUNCT
ejpam-6321	238	6	℘⋄	℘⋄	NUM
ejpam-6321	238	7	)	)	PUNCT
ejpam-6321	239	1	+	+	NUM
ejpam-6321	239	2	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	239	3	,	,	PUNCT
ejpam-6321	239	4	℘	℘	PROPN
ejpam-6321	239	5	,	,	PUNCT
ejpam-6321	239	6	℘	℘	NUM
ejpam-6321	239	7	)	)	PUNCT
ejpam-6321	239	8	+	+	NOUN
ejpam-6321	239	9	g(℘⋄	g(℘⋄	NUM
ejpam-6321	239	10	,	,	PUNCT
ejpam-6321	239	11	f2℘	f2℘	PROPN
ejpam-6321	239	12	⋄	⋄	PROPN
ejpam-6321	239	13	,	,	PUNCT
ejpam-6321	239	14	℘⋄	℘⋄	NUM
ejpam-6321	239	15	)	)	PUNCT
ejpam-6321	240	1	+	+	NOUN
ejpam-6321	240	2	g(℘⋄	g(℘⋄	NUM
ejpam-6321	240	3	,	,	PUNCT
ejpam-6321	240	4	℘⋄	℘⋄	NUM
ejpam-6321	240	5	,	,	PUNCT
ejpam-6321	240	6	f3℘	f3℘	PROPN
ejpam-6321	240	7	⋄	⋄	NOUN
ejpam-6321	240	8	)	)	PUNCT
ejpam-6321	240	9	]	]	PUNCT
ejpam-6321	241	1	+	+	CCONJ
ejpam-6321	241	2	α5min	α5min	NUM
ejpam-6321	241	3	{	{	PUNCT
ejpam-6321	241	4	g(℘	g(℘	PROPN
ejpam-6321	241	5	,	,	PUNCT
ejpam-6321	241	6	℘⋄	℘⋄	NUM
ejpam-6321	241	7	,	,	PUNCT
ejpam-6321	241	8	f2℘	f2℘	PROPN
ejpam-6321	241	9	⋄	⋄	PROPN
ejpam-6321	241	10	)	)	PUNCT
ejpam-6321	241	11	,	,	PUNCT
ejpam-6321	241	12	g(f1℘	g(f1℘	PROPN
ejpam-6321	241	13	,	,	PUNCT
ejpam-6321	241	14	f1℘	f1℘	PROPN
ejpam-6321	241	15	,	,	PUNCT
ejpam-6321	241	16	℘	℘	PROPN
ejpam-6321	241	17	⋄	⋄	PROPN
ejpam-6321	241	18	)	)	PUNCT
ejpam-6321	241	19	,	,	PUNCT
ejpam-6321	241	20	g(f1℘	g(f1℘	PROPN
ejpam-6321	241	21	,	,	PUNCT
ejpam-6321	241	22	℘	℘	PROPN
ejpam-6321	241	23	⋄	⋄	PROPN
ejpam-6321	241	24	,	,	PUNCT
ejpam-6321	241	25	℘⋄	℘⋄	NUM
ejpam-6321	241	26	)	)	PUNCT
ejpam-6321	241	27	,	,	PUNCT
ejpam-6321	241	28	g(f2℘	g(f2℘	PROPN
ejpam-6321	241	29	⋄	⋄	PROPN
ejpam-6321	241	30	,	,	PUNCT
ejpam-6321	241	31	f2℘	f2℘	PROPN
ejpam-6321	241	32	⋄	⋄	PROPN
ejpam-6321	241	33	,	,	PUNCT
ejpam-6321	241	34	℘⋄	℘⋄	NUM
ejpam-6321	241	35	)	)	PUNCT
ejpam-6321	241	36	}	}	PUNCT
ejpam-6321	241	37	≤	≤	NUM
ejpam-6321	241	38	α1g(℘	α1g(℘	NOUN
ejpam-6321	241	39	,	,	PUNCT
ejpam-6321	241	40	℘⋄	℘⋄	X
ejpam-6321	241	41	,	,	PUNCT
ejpam-6321	241	42	℘⋄	℘⋄	NUM
ejpam-6321	241	43	)	)	PUNCT
ejpam-6321	241	44	+	+	NUM
ejpam-6321	241	45	α2g(℘	α2g(℘	NOUN
ejpam-6321	241	46	,	,	PUNCT
ejpam-6321	241	47	℘⋄	℘⋄	X
ejpam-6321	241	48	,	,	PUNCT
ejpam-6321	241	49	℘⋄	℘⋄	NUM
ejpam-6321	241	50	)	)	PUNCT
ejpam-6321	241	51	+	+	CCONJ
ejpam-6321	241	52	α3g(℘	α3g(℘	NUM
ejpam-6321	241	53	,	,	PUNCT
ejpam-6321	241	54	℘⋄	℘⋄	X
ejpam-6321	241	55	,	,	PUNCT
ejpam-6321	241	56	℘⋄	℘⋄	NUM
ejpam-6321	241	57	)	)	PUNCT
ejpam-6321	241	58	+	+	CCONJ
ejpam-6321	241	59	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	241	60	,	,	PUNCT
ejpam-6321	241	61	℘	℘	NOUN
ejpam-6321	241	62	,	,	PUNCT
ejpam-6321	241	63	℘	℘	NUM
ejpam-6321	241	64	)	)	PUNCT
ejpam-6321	241	65	+	+	NOUN
ejpam-6321	241	66	g(℘⋄	g(℘⋄	NUM
ejpam-6321	241	67	,	,	PUNCT
ejpam-6321	241	68	℘⋄	℘⋄	NUM
ejpam-6321	241	69	,	,	PUNCT
ejpam-6321	241	70	℘⋄	℘⋄	NUM
ejpam-6321	241	71	)	)	PUNCT
ejpam-6321	242	1	+	+	NOUN
ejpam-6321	242	2	g(℘⋄	g(℘⋄	NUM
ejpam-6321	242	3	,	,	PUNCT
ejpam-6321	242	4	℘⋄	℘⋄	NUM
ejpam-6321	242	5	,	,	PUNCT
ejpam-6321	242	6	℘⋄	℘⋄	NUM
ejpam-6321	242	7	)	)	PUNCT
ejpam-6321	242	8	]	]	PUNCT
ejpam-6321	243	1	+	+	CCONJ
ejpam-6321	243	2	α5min	α5min	NUM
ejpam-6321	243	3	{	{	PUNCT
ejpam-6321	243	4	g(℘	g(℘	PROPN
ejpam-6321	243	5	,	,	PUNCT
ejpam-6321	243	6	℘⋄	℘⋄	NUM
ejpam-6321	243	7	,	,	PUNCT
ejpam-6321	243	8	℘⋄	℘⋄	NUM
ejpam-6321	243	9	)	)	PUNCT
ejpam-6321	243	10	,	,	PUNCT
ejpam-6321	243	11	g(℘	g(℘	PROPN
ejpam-6321	243	12	,	,	PUNCT
ejpam-6321	243	13	℘	℘	NOUN
ejpam-6321	243	14	,	,	PUNCT
ejpam-6321	243	15	℘⋄	℘⋄	NUM
ejpam-6321	243	16	)	)	PUNCT
ejpam-6321	243	17	,	,	PUNCT
ejpam-6321	243	18	g(℘	g(℘	PROPN
ejpam-6321	243	19	,	,	PUNCT
ejpam-6321	243	20	℘⋄	℘⋄	NUM
ejpam-6321	243	21	,	,	PUNCT
ejpam-6321	243	22	℘⋄	℘⋄	NUM
ejpam-6321	243	23	)	)	PUNCT
ejpam-6321	243	24	,	,	PUNCT
ejpam-6321	243	25	g(℘⋄	g(℘⋄	NUM
ejpam-6321	243	26	,	,	PUNCT
ejpam-6321	243	27	℘⋄	℘⋄	X
ejpam-6321	243	28	,	,	PUNCT
ejpam-6321	243	29	℘⋄	℘⋄	NUM
ejpam-6321	243	30	)	)	PUNCT
ejpam-6321	243	31	}	}	PUNCT
ejpam-6321	243	32	≤	≤	NUM
ejpam-6321	243	33	α1g(℘	α1g(℘	NOUN
ejpam-6321	243	34	,	,	PUNCT
ejpam-6321	243	35	℘⋄	℘⋄	X
ejpam-6321	243	36	,	,	PUNCT
ejpam-6321	243	37	℘⋄	℘⋄	NUM
ejpam-6321	243	38	)	)	PUNCT
ejpam-6321	243	39	+	+	NUM
ejpam-6321	243	40	α2g(℘	α2g(℘	NOUN
ejpam-6321	243	41	,	,	PUNCT
ejpam-6321	243	42	℘⋄	℘⋄	X
ejpam-6321	243	43	,	,	PUNCT
ejpam-6321	243	44	℘⋄	℘⋄	NUM
ejpam-6321	243	45	)	)	PUNCT
ejpam-6321	243	46	+	+	CCONJ
ejpam-6321	243	47	α3g(℘	α3g(℘	NUM
ejpam-6321	243	48	,	,	PUNCT
ejpam-6321	243	49	℘⋄	℘⋄	X
ejpam-6321	243	50	,	,	PUNCT
ejpam-6321	243	51	℘⋄	℘⋄	NUM
ejpam-6321	243	52	)	)	PUNCT
ejpam-6321	243	53	(	(	PUNCT
ejpam-6321	243	54	α1	α1	PROPN
ejpam-6321	243	55	+	+	CCONJ
ejpam-6321	243	56	α2	α2	ADJ
ejpam-6321	243	57	+	+	CCONJ
ejpam-6321	243	58	α3)g(℘	α3)g(℘	PROPN
ejpam-6321	243	59	,	,	PUNCT
ejpam-6321	243	60	℘⋄	℘⋄	NUM
ejpam-6321	243	61	,	,	PUNCT
ejpam-6321	243	62	℘⋄)(1−	℘⋄)(1−	PROPN
ejpam-6321	243	63	α1	α1	PROPN
ejpam-6321	243	64	−	−	PROPN
ejpam-6321	243	65	α2	α2	PROPN
ejpam-6321	243	66	−	−	PROPN
ejpam-6321	243	67	α3)g(℘	α3)g(℘	PROPN
ejpam-6321	243	68	,	,	PUNCT
ejpam-6321	243	69	℘⋄	℘⋄	X
ejpam-6321	243	70	,	,	PUNCT
ejpam-6321	243	71	℘⋄	℘⋄	NUM
ejpam-6321	243	72	)	)	PUNCT
ejpam-6321	243	73	≤	≤	NOUN
ejpam-6321	243	74	0	0	NUM
ejpam-6321	243	75	,	,	PUNCT
ejpam-6321	243	76	is	be	AUX
ejpam-6321	243	77	a	a	DET
ejpam-6321	243	78	contradiction	contradiction	NOUN
ejpam-6321	243	79	.	.	PUNCT
ejpam-6321	244	1	since	since	SCONJ
ejpam-6321	244	2	,	,	PUNCT
ejpam-6321	244	3	(	(	PUNCT
ejpam-6321	244	4	1−	1−	NUM
ejpam-6321	244	5	α1	α1	PROPN
ejpam-6321	244	6	−	−	PROPN
ejpam-6321	244	7	α2	α2	ADJ
ejpam-6321	244	8	−	−	PROPN
ejpam-6321	244	9	α3	α3	NOUN
ejpam-6321	244	10	)	)	PUNCT
ejpam-6321	244	11	̸=	̸=	PROPN
ejpam-6321	244	12	0	0	NUM
ejpam-6321	244	13	,	,	PUNCT
ejpam-6321	244	14	therefore	therefore	ADV
ejpam-6321	244	15	g(℘	g(℘	NOUN
ejpam-6321	244	16	,	,	PUNCT
ejpam-6321	244	17	℘⋄	℘⋄	NUM
ejpam-6321	244	18	,	,	PUNCT
ejpam-6321	244	19	℘⋄	℘⋄	NUM
ejpam-6321	244	20	)	)	PUNCT
ejpam-6321	244	21	=	=	SYM
ejpam-6321	244	22	0	0	X
ejpam-6321	244	23	.	.	PUNCT
ejpam-6321	244	24	thus	thus	ADV
ejpam-6321	244	25	,	,	PUNCT
ejpam-6321	244	26	℘	℘	PROPN
ejpam-6321	244	27	=	=	SYM
ejpam-6321	244	28	℘⋄	℘⋄	NOUN
ejpam-6321	244	29	it	it	PRON
ejpam-6321	244	30	is	be	AUX
ejpam-6321	244	31	proved	prove	VERB
ejpam-6321	244	32	that	that	SCONJ
ejpam-6321	244	33	the	the	DET
ejpam-6321	244	34	three	three	NUM
ejpam-6321	244	35	individual	individual	ADJ
ejpam-6321	244	36	self	self	NOUN
ejpam-6321	244	37	-	-	PUNCT
ejpam-6321	244	38	mappings	mapping	NOUN
ejpam-6321	244	39	referred	refer	VERB
ejpam-6321	244	40	as	as	ADP
ejpam-6321	244	41	f1	f1	NOUN
ejpam-6321	244	42	,	,	PUNCT
ejpam-6321	244	43	f2	f2	PROPN
ejpam-6321	244	44	and	and	CCONJ
ejpam-6321	244	45	f3	f3	PROPN
ejpam-6321	244	46	possess	possess	VERB
ejpam-6321	244	47	a	a	DET
ejpam-6321	244	48	ucfp	ucfp	NOUN
ejpam-6321	244	49	in	in	ADP
ejpam-6321	244	50	x.	x.	NOUN
ejpam-6321	244	51	the	the	DET
ejpam-6321	244	52	algorithm	algorithm	NOUN
ejpam-6321	244	53	3	3	NUM
ejpam-6321	244	54	implements	implement	VERB
ejpam-6321	244	55	the	the	DET
ejpam-6321	244	56	iterative	iterative	ADJ
ejpam-6321	244	57	fixed	fix	VERB
ejpam-6321	244	58	-	-	PUNCT
ejpam-6321	244	59	point	point	NOUN
ejpam-6321	244	60	computation	computation	NOUN
ejpam-6321	244	61	scheme	scheme	NOUN
ejpam-6321	244	62	described	describe	VERB
ejpam-6321	244	63	in	in	ADP
ejpam-6321	244	64	theorem	theorem	NOUN
ejpam-6321	244	65	1	1	NUM
ejpam-6321	244	66	by	by	ADP
ejpam-6321	244	67	sequentially	sequentially	ADV
ejpam-6321	244	68	applying	apply	VERB
ejpam-6321	244	69	the	the	DET
ejpam-6321	244	70	three	three	NUM
ejpam-6321	244	71	contractive	contractive	ADJ
ejpam-6321	244	72	mappings	mapping	NOUN
ejpam-6321	244	73	f1	f1	NOUN
ejpam-6321	244	74	,	,	PUNCT
ejpam-6321	244	75	f2	f2	PROPN
ejpam-6321	244	76	,	,	PUNCT
ejpam-6321	244	77	and	and	CCONJ
ejpam-6321	244	78	f3	f3	ADJ
ejpam-6321	244	79	while	while	SCONJ
ejpam-6321	244	80	monitoring	monitor	VERB
ejpam-6321	244	81	convergence	convergence	NOUN
ejpam-6321	244	82	through	through	ADP
ejpam-6321	244	83	the	the	DET
ejpam-6321	244	84	g	g	NOUN
ejpam-6321	244	85	-	-	PUNCT
ejpam-6321	244	86	metric	metric	ADJ
ejpam-6321	244	87	condition	condition	NOUN
ejpam-6321	244	88	g(x3k	g(x3k	NOUN
ejpam-6321	244	89	,	,	PUNCT
ejpam-6321	244	90	x3k+1	x3k+1	ADV
ejpam-6321	244	91	,	,	PUNCT
ejpam-6321	244	92	x3k+2	x3k+2	PUNCT
ejpam-6321	244	93	)	)	PUNCT
ejpam-6321	244	94	<	<	X
ejpam-6321	244	95	tolerance	tolerance	NOUN
ejpam-6321	244	96	.	.	PUNCT
ejpam-6321	245	1	algorithm	algorithm	NOUN
ejpam-6321	245	2	1	1	NUM
ejpam-6321	245	3	fixed	fix	VERB
ejpam-6321	245	4	-	-	PUNCT
ejpam-6321	245	5	point	point	NOUN
ejpam-6321	245	6	computation	computation	NOUN
ejpam-6321	245	7	for	for	ADP
ejpam-6321	245	8	tripartite	tripartite	ADJ
ejpam-6321	245	9	mappings	mapping	NOUN
ejpam-6321	245	10	require	require	VERB
ejpam-6321	245	11	:	:	PUNCT
ejpam-6321	245	12	x	x	X
ejpam-6321	245	13	:	:	PUNCT
ejpam-6321	245	14	g	g	ADJ
ejpam-6321	245	15	-	-	PUNCT
ejpam-6321	245	16	metric	metric	ADJ
ejpam-6321	245	17	space	space	NOUN
ejpam-6321	245	18	,	,	PUNCT
ejpam-6321	245	19	f1	f1	NOUN
ejpam-6321	245	20	,	,	PUNCT
ejpam-6321	245	21	f2	f2	PROPN
ejpam-6321	245	22	,	,	PUNCT
ejpam-6321	245	23	f3	f3	PROPN
ejpam-6321	245	24	:	:	PUNCT
ejpam-6321	245	25	contractive	contractive	ADJ
ejpam-6321	245	26	mappings	mapping	NOUN
ejpam-6321	245	27	require	require	VERB
ejpam-6321	245	28	:	:	PUNCT
ejpam-6321	245	29	α1	α1	PROPN
ejpam-6321	245	30	,	,	PUNCT
ejpam-6321	245	31	α2	α2	ADJ
ejpam-6321	245	32	,	,	PUNCT
ejpam-6321	245	33	α3	α3	NOUN
ejpam-6321	245	34	,	,	PUNCT
ejpam-6321	245	35	α4	α4	NOUN
ejpam-6321	245	36	,	,	PUNCT
ejpam-6321	245	37	α5	α5	NOUN
ejpam-6321	245	38	:	:	PUNCT
ejpam-6321	245	39	theorem	theorem	ADJ
ejpam-6321	245	40	parameters	parameter	NOUN
ejpam-6321	245	41	ensure	ensure	VERB
ejpam-6321	245	42	:	:	PUNCT
ejpam-6321	245	43	common	common	ADJ
ejpam-6321	245	44	fixed	fix	VERB
ejpam-6321	245	45	point	point	NOUN
ejpam-6321	245	46	℘	℘	PROPN
ejpam-6321	245	47	1	1	NUM
ejpam-6321	245	48	:	:	PUNCT
ejpam-6321	245	49	initialize	initialize	VERB
ejpam-6321	245	50	x0	x0	PROPN
ejpam-6321	245	51	∈	∈	PROPN
ejpam-6321	245	52	x	x	PUNCT
ejpam-6321	245	53	2	2	NUM
ejpam-6321	245	54	:	:	PUNCT
ejpam-6321	245	55	for	for	ADP
ejpam-6321	245	56	k	k	PROPN
ejpam-6321	245	57	=	=	SYM
ejpam-6321	245	58	0	0	PROPN
ejpam-6321	245	59	to	to	AUX
ejpam-6321	245	60	max	max	PROPN
ejpam-6321	245	61	iterations	iteration	NOUN
ejpam-6321	245	62	do	do	VERB
ejpam-6321	245	63	3	3	NUM
ejpam-6321	245	64	:	:	PUNCT
ejpam-6321	245	65	x3k+1	x3k+1	PROPN
ejpam-6321	245	66	←	←	PROPN
ejpam-6321	245	67	f1(x3k	f1(x3k	PROPN
ejpam-6321	245	68	)	)	PUNCT
ejpam-6321	245	69	4	4	NUM
ejpam-6321	245	70	:	:	PUNCT
ejpam-6321	245	71	x3k+2	x3k+2	PUNCT
ejpam-6321	246	1	←	←	PROPN
ejpam-6321	246	2	f2(x3k+1	f2(x3k+1	PROPN
ejpam-6321	246	3	)	)	PUNCT
ejpam-6321	246	4	5	5	NUM
ejpam-6321	246	5	:	:	PUNCT
ejpam-6321	246	6	x3k+3	x3k+3	PROPN
ejpam-6321	246	7	←	←	PROPN
ejpam-6321	246	8	f3(x3k+2	f3(x3k+2	PROPN
ejpam-6321	246	9	)	)	PUNCT
ejpam-6321	246	10	6	6	NUM
ejpam-6321	246	11	:	:	PUNCT
ejpam-6321	246	12	compute	compute	PROPN
ejpam-6321	246	13	gk	gk	PROPN
ejpam-6321	246	14	←	←	PROPN
ejpam-6321	246	15	g(x3k	g(x3k	PROPN
ejpam-6321	246	16	,	,	PUNCT
ejpam-6321	246	17	x3k+1	x3k+1	ADV
ejpam-6321	246	18	,	,	PUNCT
ejpam-6321	246	19	x3k+2	x3k+2	PUNCT
ejpam-6321	246	20	)	)	PUNCT
ejpam-6321	246	21	7	7	NUM
ejpam-6321	246	22	:	:	PUNCT
ejpam-6321	246	23	if	if	SCONJ
ejpam-6321	246	24	gk	gk	PROPN
ejpam-6321	246	25	<	<	X
ejpam-6321	246	26	tolerance	tolerance	NOUN
ejpam-6321	246	27	then	then	ADV
ejpam-6321	246	28	8	8	NUM
ejpam-6321	246	29	:	:	PUNCT
ejpam-6321	246	30	return	return	VERB
ejpam-6321	246	31	x3k+3	x3k+3	NOUN
ejpam-6321	247	1	▷	▷	ADP
ejpam-6321	247	2	common	common	ADJ
ejpam-6321	247	3	fixed	fix	VERB
ejpam-6321	247	4	point	point	NOUN
ejpam-6321	247	5	found	find	VERB
ejpam-6321	247	6	9	9	NUM
ejpam-6321	247	7	:	:	PUNCT
ejpam-6321	247	8	end	end	VERB
ejpam-6321	247	9	if	if	SCONJ
ejpam-6321	247	10	10	10	NUM
ejpam-6321	247	11	:	:	PUNCT
ejpam-6321	247	12	end	end	VERB
ejpam-6321	247	13	for	for	ADP
ejpam-6321	247	14	figure	figure	NOUN
ejpam-6321	247	15	1	1	NUM
ejpam-6321	247	16	illustrates	illustrate	VERB
ejpam-6321	247	17	the	the	DET
ejpam-6321	247	18	convergence	convergence	NOUN
ejpam-6321	247	19	of	of	ADP
ejpam-6321	247	20	the	the	DET
ejpam-6321	247	21	g	g	NOUN
ejpam-6321	247	22	-	-	PUNCT
ejpam-6321	247	23	metric	metric	ADJ
ejpam-6321	247	24	during	during	ADP
ejpam-6321	247	25	fixed	fix	VERB
ejpam-6321	247	26	-	-	PUNCT
ejpam-6321	247	27	point	point	NOUN
ejpam-6321	247	28	iterations	iteration	NOUN
ejpam-6321	247	29	,	,	PUNCT
ejpam-6321	247	30	validating	validate	VERB
ejpam-6321	247	31	theorem	theorem	NOUN
ejpam-6321	247	32	1	1	NUM
ejpam-6321	247	33	under	under	ADP
ejpam-6321	247	34	contractive	contractive	ADJ
ejpam-6321	247	35	conditions	condition	NOUN
ejpam-6321	247	36	.	.	PUNCT
ejpam-6321	248	1	m.	m.	NOUN
ejpam-6321	248	2	noorwali	noorwali	PROPN
ejpam-6321	248	3	et	et	PROPN
ejpam-6321	248	4	al	al	PROPN
ejpam-6321	248	5	.	.	PUNCT
ejpam-6321	248	6	/	/	SYM
ejpam-6321	248	7	eur	eur	PROPN
ejpam-6321	248	8	.	.	PUNCT
ejpam-6321	249	1	j.	j.	PROPN
ejpam-6321	249	2	pure	pure	PROPN
ejpam-6321	249	3	appl	appl	PROPN
ejpam-6321	249	4	.	.	PROPN
ejpam-6321	249	5	math	math	PROPN
ejpam-6321	249	6	,	,	PUNCT
ejpam-6321	249	7	18	18	NUM
ejpam-6321	249	8	(	(	PUNCT
ejpam-6321	249	9	3	3	NUM
ejpam-6321	249	10	)	)	PUNCT
ejpam-6321	249	11	(	(	PUNCT
ejpam-6321	249	12	2025	2025	NUM
ejpam-6321	249	13	)	)	PUNCT
ejpam-6321	249	14	,	,	PUNCT
ejpam-6321	249	15	6321	6321	NUM
ejpam-6321	249	16	9	9	NUM
ejpam-6321	249	17	of	of	ADP
ejpam-6321	249	18	27	27	NUM
ejpam-6321	249	19	figure	figure	NOUN
ejpam-6321	249	20	1	1	NUM
ejpam-6321	249	21	:	:	PUNCT
ejpam-6321	249	22	g	g	NOUN
ejpam-6321	249	23	-	-	PUNCT
ejpam-6321	249	24	metric	metric	ADJ
ejpam-6321	249	25	convergence	convergence	NOUN
ejpam-6321	249	26	curve	curve	NOUN
ejpam-6321	249	27	validating	validate	VERB
ejpam-6321	249	28	theorem	theorem	NOUN
ejpam-6321	249	29	1	1	NUM
ejpam-6321	249	30	for	for	ADP
ejpam-6321	249	31	tripartite	tripartite	ADJ
ejpam-6321	249	32	mappings	mapping	NOUN
ejpam-6321	249	33	the	the	DET
ejpam-6321	249	34	fixed	fix	VERB
ejpam-6321	249	35	-	-	PUNCT
ejpam-6321	249	36	point	point	NOUN
ejpam-6321	249	37	framework	framework	NOUN
ejpam-6321	249	38	established	establish	VERB
ejpam-6321	249	39	in	in	ADP
ejpam-6321	249	40	theorem	theorem	NOUN
ejpam-6321	249	41	1	1	NUM
ejpam-6321	249	42	,	,	PUNCT
ejpam-6321	249	43	and	and	CCONJ
ejpam-6321	249	44	implemented	implement	VERB
ejpam-6321	249	45	through	through	ADP
ejpam-6321	249	46	algorithm	algorithm	NOUN
ejpam-6321	249	47	3	3	NUM
ejpam-6321	249	48	,	,	PUNCT
ejpam-6321	249	49	finds	find	VERB
ejpam-6321	249	50	significant	significant	ADJ
ejpam-6321	249	51	applications	application	NOUN
ejpam-6321	249	52	in	in	ADP
ejpam-6321	249	53	both	both	DET
ejpam-6321	249	54	artificial	artificial	ADJ
ejpam-6321	249	55	intelligence	intelligence	NOUN
ejpam-6321	249	56	and	and	CCONJ
ejpam-6321	249	57	cryptographic	cryptographic	ADJ
ejpam-6321	249	58	protocols	protocol	NOUN
ejpam-6321	249	59	.	.	PUNCT
ejpam-6321	250	1	in	in	ADP
ejpam-6321	250	2	deep	deep	ADJ
ejpam-6321	250	3	learning	learning	NOUN
ejpam-6321	250	4	architectures	architecture	NOUN
ejpam-6321	250	5	,	,	PUNCT
ejpam-6321	250	6	the	the	DET
ejpam-6321	250	7	mappings	mapping	NOUN
ejpam-6321	250	8	f1	f1	NOUN
ejpam-6321	250	9	,	,	PUNCT
ejpam-6321	250	10	f2	f2	PROPN
ejpam-6321	250	11	,	,	PUNCT
ejpam-6321	250	12	and	and	CCONJ
ejpam-6321	250	13	f3	f3	PROPN
ejpam-6321	250	14	may	may	AUX
ejpam-6321	250	15	be	be	AUX
ejpam-6321	250	16	interpreted	interpret	VERB
ejpam-6321	250	17	as	as	ADP
ejpam-6321	250	18	a	a	DET
ejpam-6321	250	19	feature	feature	NOUN
ejpam-6321	250	20	extraction	extraction	NOUN
ejpam-6321	250	21	layer	layer	NOUN
ejpam-6321	250	22	,	,	PUNCT
ejpam-6321	250	23	a	a	DET
ejpam-6321	250	24	regularization	regularization	NOUN
ejpam-6321	250	25	layer	layer	NOUN
ejpam-6321	250	26	,	,	PUNCT
ejpam-6321	250	27	and	and	CCONJ
ejpam-6321	250	28	an	an	DET
ejpam-6321	250	29	output	output	NOUN
ejpam-6321	250	30	projection	projection	NOUN
ejpam-6321	250	31	layer	layer	NOUN
ejpam-6321	250	32	,	,	PUNCT
ejpam-6321	250	33	respectively	respectively	ADV
ejpam-6321	250	34	.	.	PUNCT
ejpam-6321	251	1	the	the	DET
ejpam-6321	251	2	contraction	contraction	NOUN
ejpam-6321	251	3	parameters	parameter	NOUN
ejpam-6321	251	4	αi	αi	PROPN
ejpam-6321	251	5	serve	serve	VERB
ejpam-6321	251	6	to	to	PART
ejpam-6321	251	7	regulate	regulate	VERB
ejpam-6321	251	8	the	the	DET
ejpam-6321	251	9	stability	stability	NOUN
ejpam-6321	251	10	of	of	ADP
ejpam-6321	251	11	these	these	DET
ejpam-6321	251	12	layers	layer	NOUN
ejpam-6321	251	13	,	,	PUNCT
ejpam-6321	251	14	ensuring	ensure	VERB
ejpam-6321	251	15	that	that	SCONJ
ejpam-6321	251	16	the	the	DET
ejpam-6321	251	17	iterative	iterative	NOUN
ejpam-6321	251	18	composition	composition	NOUN
ejpam-6321	251	19	converges	converge	VERB
ejpam-6321	251	20	to	to	ADP
ejpam-6321	251	21	a	a	DET
ejpam-6321	251	22	fixed	fix	VERB
ejpam-6321	251	23	activation	activation	NOUN
ejpam-6321	251	24	pattern	pattern	NOUN
ejpam-6321	251	25	as	as	SCONJ
ejpam-6321	251	26	guaranteed	guarantee	VERB
ejpam-6321	251	27	by	by	ADP
ejpam-6321	251	28	the	the	DET
ejpam-6321	251	29	theoretical	theoretical	ADJ
ejpam-6321	251	30	result	result	NOUN
ejpam-6321	251	31	.	.	PUNCT
ejpam-6321	252	1	in	in	ADP
ejpam-6321	252	2	cryptography	cryptography	NOUN
ejpam-6321	252	3	,	,	PUNCT
ejpam-6321	252	4	especially	especially	ADV
ejpam-6321	252	5	within	within	ADP
ejpam-6321	252	6	secure	secure	ADJ
ejpam-6321	252	7	multi	multi	ADJ
ejpam-6321	252	8	-	-	ADJ
ejpam-6321	252	9	party	party	ADJ
ejpam-6321	252	10	computation	computation	NOUN
ejpam-6321	252	11	,	,	PUNCT
ejpam-6321	252	12	the	the	DET
ejpam-6321	252	13	same	same	ADJ
ejpam-6321	252	14	mappings	mapping	NOUN
ejpam-6321	252	15	can	can	AUX
ejpam-6321	252	16	be	be	AUX
ejpam-6321	252	17	understood	understand	VERB
ejpam-6321	252	18	as	as	SCONJ
ejpam-6321	252	19	follows	follow	VERB
ejpam-6321	252	20	:	:	PUNCT
ejpam-6321	252	21	f1	f1	NOUN
ejpam-6321	252	22	represents	represent	VERB
ejpam-6321	252	23	the	the	DET
ejpam-6321	252	24	encryption	encryption	NOUN
ejpam-6321	252	25	transformation	transformation	NOUN
ejpam-6321	252	26	,	,	PUNCT
ejpam-6321	252	27	f2	f2	PROPN
ejpam-6321	252	28	corresponds	correspond	VERB
ejpam-6321	252	29	to	to	ADP
ejpam-6321	252	30	the	the	DET
ejpam-6321	252	31	decryption	decryption	NOUN
ejpam-6321	252	32	process	process	NOUN
ejpam-6321	252	33	,	,	PUNCT
ejpam-6321	252	34	and	and	CCONJ
ejpam-6321	252	35	f3	f3	PROPN
ejpam-6321	252	36	performs	perform	VERB
ejpam-6321	252	37	key	key	ADJ
ejpam-6321	252	38	mixing	mixing	NOUN
ejpam-6321	252	39	.	.	PUNCT
ejpam-6321	253	1	the	the	DET
ejpam-6321	253	2	g	g	NOUN
ejpam-6321	253	3	-	-	PUNCT
ejpam-6321	253	4	metric	metric	ADJ
ejpam-6321	253	5	quantifies	quantifie	NOUN
ejpam-6321	253	6	the	the	DET
ejpam-6321	253	7	maximum	maximum	ADJ
ejpam-6321	253	8	deviation	deviation	NOUN
ejpam-6321	253	9	between	between	ADP
ejpam-6321	253	10	transformed	transform	VERB
ejpam-6321	253	11	message	message	NOUN
ejpam-6321	253	12	states	state	NOUN
ejpam-6321	253	13	.	.	PUNCT
ejpam-6321	254	1	convergence	convergence	NOUN
ejpam-6321	254	2	to	to	ADP
ejpam-6321	254	3	a	a	DET
ejpam-6321	254	4	fixed	fix	VERB
ejpam-6321	254	5	point	point	NOUN
ejpam-6321	254	6	under	under	ADP
ejpam-6321	254	7	this	this	DET
ejpam-6321	254	8	metric	metric	NOUN
ejpam-6321	254	9	reflects	reflect	VERB
ejpam-6321	254	10	the	the	DET
ejpam-6321	254	11	achievement	achievement	NOUN
ejpam-6321	254	12	of	of	ADP
ejpam-6321	254	13	a	a	DET
ejpam-6321	254	14	secure	secure	ADJ
ejpam-6321	254	15	and	and	CCONJ
ejpam-6321	254	16	synchronized	synchronize	VERB
ejpam-6321	254	17	outcome	outcome	NOUN
ejpam-6321	254	18	across	across	ADP
ejpam-6321	254	19	all	all	DET
ejpam-6321	254	20	parties	party	NOUN
ejpam-6321	254	21	involved	involve	VERB
ejpam-6321	254	22	.	.	PUNCT
ejpam-6321	255	1	thus	thus	ADV
ejpam-6321	255	2	,	,	PUNCT
ejpam-6321	255	3	theorem	theorem	VERB
ejpam-6321	255	4	1	1	NUM
ejpam-6321	255	5	and	and	CCONJ
ejpam-6321	255	6	algorithm	algorithm	PROPN
ejpam-6321	255	7	3	3	NUM
ejpam-6321	255	8	jointly	jointly	ADV
ejpam-6321	255	9	ensure	ensure	VERB
ejpam-6321	255	10	the	the	DET
ejpam-6321	255	11	validity	validity	NOUN
ejpam-6321	255	12	and	and	CCONJ
ejpam-6321	255	13	practical	practical	ADJ
ejpam-6321	255	14	effectiveness	effectiveness	NOUN
ejpam-6321	255	15	of	of	ADP
ejpam-6321	255	16	this	this	DET
ejpam-6321	255	17	unified	unify	VERB
ejpam-6321	255	18	framework	framework	NOUN
ejpam-6321	255	19	in	in	ADP
ejpam-6321	255	20	both	both	PRON
ejpam-6321	255	21	ai	ai	ADJ
ejpam-6321	255	22	and	and	CCONJ
ejpam-6321	255	23	cryptographic	cryptographic	ADJ
ejpam-6321	255	24	systems	system	NOUN
ejpam-6321	255	25	.	.	PUNCT
ejpam-6321	255	26	example	example	NOUN
ejpam-6321	256	1	1	1	NUM
ejpam-6321	256	2	.	.	PUNCT
ejpam-6321	256	3	let	let	VERB
ejpam-6321	256	4	x	x	PUNCT
ejpam-6321	256	5	=	=	PUNCT
ejpam-6321	257	1	[	[	X
ejpam-6321	257	2	0	0	NUM
ejpam-6321	257	3	,	,	PUNCT
ejpam-6321	257	4	1	1	NUM
ejpam-6321	257	5	]	]	PUNCT
ejpam-6321	257	6	with	with	ADP
ejpam-6321	257	7	g	g	NOUN
ejpam-6321	257	8	-	-	PUNCT
ejpam-6321	257	9	metric	metric	NOUN
ejpam-6321	257	10	defined	define	VERB
ejpam-6321	257	11	as	as	ADP
ejpam-6321	257	12	g(x	g(x	PROPN
ejpam-6321	257	13	,	,	PUNCT
ejpam-6321	257	14	y	y	PROPN
ejpam-6321	257	15	,	,	PUNCT
ejpam-6321	257	16	z	z	NOUN
ejpam-6321	257	17	)	)	PUNCT
ejpam-6321	257	18	=	=	SYM
ejpam-6321	257	19	max{|x−	max{|x−	PROPN
ejpam-6321	257	20	y|	y|	NOUN
ejpam-6321	257	21	,	,	PUNCT
ejpam-6321	257	22	|y	|y	VERB
ejpam-6321	257	23	−	−	PROPN
ejpam-6321	257	24	z|	z|	PROPN
ejpam-6321	257	25	,	,	PUNCT
ejpam-6321	257	26	|z	|z	PROPN
ejpam-6321	257	27	−	−	PROPN
ejpam-6321	257	28	x|	x|	PROPN
ejpam-6321	257	29	}	}	PUNCT
ejpam-6321	257	30	.	.	PUNCT
ejpam-6321	258	1	consider	consider	VERB
ejpam-6321	258	2	the	the	DET
ejpam-6321	258	3	piecewise	piecewise	NOUN
ejpam-6321	258	4	mapping	mapping	NOUN
ejpam-6321	258	5	:	:	PUNCT
ejpam-6321	258	6	fi(x	fi(x	NUM
ejpam-6321	258	7	)	)	PUNCT
ejpam-6321	259	1	=	=	PRON
ejpam-6321	259	2	{	{	PUNCT
ejpam-6321	259	3	x	x	SYM
ejpam-6321	259	4	4	4	NUM
ejpam-6321	259	5	+	+	SYM
ejpam-6321	259	6	1	1	NUM
ejpam-6321	259	7	2	2	NUM
ejpam-6321	259	8	x	x	SYM
ejpam-6321	259	9	∈	∈	NOUN
ejpam-6321	259	10	[	[	X
ejpam-6321	259	11	0	0	NUM
ejpam-6321	259	12	,	,	PUNCT
ejpam-6321	259	13	1	1	NUM
ejpam-6321	259	14	]	]	SYM
ejpam-6321	259	15	3x+1	3x+1	PROPN
ejpam-6321	259	16	5	5	NUM
ejpam-6321	259	17	x	x	SYM
ejpam-6321	259	18	>	>	X
ejpam-6321	259	19	1	1	NUM
ejpam-6321	259	20	(	(	PUNCT
ejpam-6321	259	21	i	i	NOUN
ejpam-6321	259	22	=	=	NOUN
ejpam-6321	259	23	1	1	NUM
ejpam-6321	259	24	,	,	PUNCT
ejpam-6321	259	25	2	2	NUM
ejpam-6321	259	26	,	,	PUNCT
ejpam-6321	259	27	3	3	X
ejpam-6321	259	28	)	)	PUNCT
ejpam-6321	259	29	this	this	DET
ejpam-6321	259	30	mapping	mapping	NOUN
ejpam-6321	259	31	satisfies	satisfy	VERB
ejpam-6321	259	32	the	the	DET
ejpam-6321	259	33	contractive	contractive	ADJ
ejpam-6321	259	34	conditions	condition	NOUN
ejpam-6321	259	35	of	of	ADP
ejpam-6321	259	36	theorem	theorem	NOUN
ejpam-6321	259	37	1	1	NUM
ejpam-6321	259	38	,	,	PUNCT
ejpam-6321	259	39	with	with	ADP
ejpam-6321	259	40	parameters	parameter	NOUN
ejpam-6321	259	41	:	:	PUNCT
ejpam-6321	259	42	α1	α1	PROPN
ejpam-6321	259	43	=	=	SYM
ejpam-6321	259	44	0.2	0.2	NUM
ejpam-6321	259	45	,	,	PUNCT
ejpam-6321	259	46	α2	α2	NOUN
ejpam-6321	259	47	=	=	SYM
ejpam-6321	259	48	0.15	0.15	NUM
ejpam-6321	259	49	,	,	PUNCT
ejpam-6321	259	50	α3	α3	NOUN
ejpam-6321	259	51	=	=	SYM
ejpam-6321	259	52	0.1	0.1	NUM
ejpam-6321	259	53	,	,	PUNCT
ejpam-6321	259	54	α4	α4	NOUN
ejpam-6321	259	55	=	=	NOUN
ejpam-6321	259	56	0.05	0.05	NUM
ejpam-6321	259	57	,	,	PUNCT
ejpam-6321	259	58	α5	α5	NOUN
ejpam-6321	259	59	=	=	PUNCT
ejpam-6321	259	60	0.02	0.02	NUM
ejpam-6321	259	61	the	the	DET
ejpam-6321	259	62	fixed	fix	VERB
ejpam-6321	259	63	-	-	PUNCT
ejpam-6321	259	64	point	point	NOUN
ejpam-6321	259	65	iteration	iteration	NOUN
ejpam-6321	259	66	applied	apply	VERB
ejpam-6321	259	67	to	to	ADP
ejpam-6321	259	68	this	this	DET
ejpam-6321	259	69	setup	setup	NOUN
ejpam-6321	259	70	yields	yield	VERB
ejpam-6321	259	71	a	a	DET
ejpam-6321	259	72	unique	unique	ADJ
ejpam-6321	259	73	common	common	ADJ
ejpam-6321	259	74	fixed	fix	VERB
ejpam-6321	259	75	point	point	NOUN
ejpam-6321	259	76	at	at	ADP
ejpam-6321	259	77	℘	℘	PROPN
ejpam-6321	259	78	=	=	SYM
ejpam-6321	259	79	2	2	NUM
ejpam-6321	259	80	3	3	NUM
ejpam-6321	259	81	,	,	PUNCT
ejpam-6321	259	82	confirming	confirm	VERB
ejpam-6321	259	83	the	the	DET
ejpam-6321	259	84	theorems	theorem	NOUN
ejpam-6321	259	85	theoretical	theoretical	ADJ
ejpam-6321	259	86	predictions	prediction	NOUN
ejpam-6321	259	87	through	through	ADP
ejpam-6321	259	88	constructive	constructive	ADJ
ejpam-6321	259	89	computation	computation	NOUN
ejpam-6321	259	90	.	.	PUNCT
ejpam-6321	260	1	algorithm	algorithm	NOUN
ejpam-6321	260	2	2	2	NUM
ejpam-6321	260	3	example	example	NOUN
ejpam-6321	260	4	implementation	implementation	NOUN
ejpam-6321	260	5	for	for	ADP
ejpam-6321	260	6	fi(x	fi(x	NUM
ejpam-6321	260	7	)	)	PUNCT
ejpam-6321	260	8	in	in	ADP
ejpam-6321	260	9	theorem	theorem	ADJ
ejpam-6321	260	10	1	1	NUM
ejpam-6321	260	11	1	1	NUM
ejpam-6321	260	12	:	:	PUNCT
ejpam-6321	260	13	function	function	NOUN
ejpam-6321	260	14	computefixedpoint(x0	computefixedpoint(x0	NOUN
ejpam-6321	260	15	,	,	PUNCT
ejpam-6321	260	16	tol	tol	NOUN
ejpam-6321	260	17	)	)	PUNCT
ejpam-6321	260	18	2	2	NUM
ejpam-6321	260	19	:	:	PUNCT
ejpam-6321	260	20	k	k	PROPN
ejpam-6321	260	21	←	←	PROPN
ejpam-6321	260	22	0	0	PROPN
ejpam-6321	260	23	3	3	NUM
ejpam-6321	260	24	:	:	PUNCT
ejpam-6321	260	25	repeat	repeat	VERB
ejpam-6321	260	26	4	4	NUM
ejpam-6321	260	27	:	:	PUNCT
ejpam-6321	260	28	xk+1	xk+1	PROPN
ejpam-6321	260	29	←	←	PROPN
ejpam-6321	260	30	xk	xk	PROPN
ejpam-6321	260	31	4	4	NUM
ejpam-6321	260	32	+	+	CCONJ
ejpam-6321	260	33	1	1	NUM
ejpam-6321	260	34	2	2	NUM
ejpam-6321	260	35	5	5	NUM
ejpam-6321	260	36	:	:	PUNCT
ejpam-6321	260	37	error	error	NOUN
ejpam-6321	260	38	←	←	NOUN
ejpam-6321	260	39	|xk+1	|xk+1	VERB
ejpam-6321	260	40	−	−	PROPN
ejpam-6321	260	41	xk|	xk|	PROPN
ejpam-6321	260	42	6	6	NUM
ejpam-6321	260	43	:	:	PUNCT
ejpam-6321	261	1	k	k	PROPN
ejpam-6321	261	2	←	←	PROPN
ejpam-6321	261	3	k	k	PROPN
ejpam-6321	261	4	+	+	PROPN
ejpam-6321	261	5	1	1	NUM
ejpam-6321	261	6	7	7	NUM
ejpam-6321	261	7	:	:	PUNCT
ejpam-6321	261	8	until	until	ADP
ejpam-6321	261	9	error	error	NOUN
ejpam-6321	261	10	<	<	X
ejpam-6321	261	11	tol	tol	NOUN
ejpam-6321	261	12	8	8	NUM
ejpam-6321	261	13	:	:	PUNCT
ejpam-6321	261	14	return	return	VERB
ejpam-6321	261	15	xk	xk	PROPN
ejpam-6321	261	16	9	9	NUM
ejpam-6321	261	17	:	:	PUNCT
ejpam-6321	261	18	end	end	NOUN
ejpam-6321	261	19	function	function	PROPN
ejpam-6321	261	20	m.	m.	NOUN
ejpam-6321	261	21	noorwali	noorwali	PROPN
ejpam-6321	261	22	et	et	PROPN
ejpam-6321	261	23	al	al	PROPN
ejpam-6321	261	24	.	.	PUNCT
ejpam-6321	261	25	/	/	SYM
ejpam-6321	261	26	eur	eur	PROPN
ejpam-6321	261	27	.	.	PUNCT
ejpam-6321	262	1	j.	j.	PROPN
ejpam-6321	262	2	pure	pure	PROPN
ejpam-6321	262	3	appl	appl	PROPN
ejpam-6321	262	4	.	.	PROPN
ejpam-6321	262	5	math	math	PROPN
ejpam-6321	262	6	,	,	PUNCT
ejpam-6321	262	7	18	18	NUM
ejpam-6321	262	8	(	(	PUNCT
ejpam-6321	262	9	3	3	NUM
ejpam-6321	262	10	)	)	PUNCT
ejpam-6321	262	11	(	(	PUNCT
ejpam-6321	262	12	2025	2025	NUM
ejpam-6321	262	13	)	)	PUNCT
ejpam-6321	262	14	,	,	PUNCT
ejpam-6321	262	15	6321	6321	NUM
ejpam-6321	262	16	10	10	NUM
ejpam-6321	262	17	of	of	ADP
ejpam-6321	262	18	27	27	NUM
ejpam-6321	262	19	this	this	DET
ejpam-6321	262	20	work	work	NOUN
ejpam-6321	262	21	builds	build	VERB
ejpam-6321	262	22	on	on	ADP
ejpam-6321	262	23	the	the	DET
ejpam-6321	262	24	fixed	fix	VERB
ejpam-6321	262	25	-	-	PUNCT
ejpam-6321	262	26	point	point	NOUN
ejpam-6321	262	27	framework	framework	NOUN
ejpam-6321	262	28	established	establish	VERB
ejpam-6321	262	29	in	in	ADP
ejpam-6321	262	30	theorem	theorem	NOUN
ejpam-6321	262	31	1	1	NUM
ejpam-6321	262	32	,	,	PUNCT
ejpam-6321	262	33	with	with	ADP
ejpam-6321	262	34	the	the	DET
ejpam-6321	262	35	algorithmic	algorithmic	ADJ
ejpam-6321	262	36	implementation	implementation	NOUN
ejpam-6321	262	37	provided	provide	VERB
ejpam-6321	262	38	above	above	ADV
ejpam-6321	262	39	and	and	CCONJ
ejpam-6321	262	40	its	its	PRON
ejpam-6321	262	41	correctness	correctness	NOUN
ejpam-6321	262	42	illustrated	illustrate	VERB
ejpam-6321	262	43	by	by	ADP
ejpam-6321	262	44	example	example	NOUN
ejpam-6321	262	45	1	1	NUM
ejpam-6321	262	46	and	and	CCONJ
ejpam-6321	262	47	algorithm	algorithm	NOUN
ejpam-6321	262	48	3	3	NUM
ejpam-6321	262	49	.	.	PUNCT
ejpam-6321	263	1	the	the	DET
ejpam-6321	263	2	applications	application	NOUN
ejpam-6321	263	3	in	in	ADP
ejpam-6321	263	4	artificial	artificial	ADJ
ejpam-6321	263	5	intelligence	intelligence	NOUN
ejpam-6321	263	6	and	and	CCONJ
ejpam-6321	263	7	cryptographic	cryptographic	ADJ
ejpam-6321	263	8	protocols	protocol	NOUN
ejpam-6321	263	9	demonstrate	demonstrate	VERB
ejpam-6321	263	10	the	the	DET
ejpam-6321	263	11	broader	broad	ADJ
ejpam-6321	263	12	relevance	relevance	NOUN
ejpam-6321	263	13	and	and	CCONJ
ejpam-6321	263	14	utility	utility	NOUN
ejpam-6321	263	15	of	of	ADP
ejpam-6321	263	16	the	the	DET
ejpam-6321	263	17	theoretical	theoretical	ADJ
ejpam-6321	263	18	results	result	NOUN
ejpam-6321	263	19	.	.	PUNCT
ejpam-6321	264	1	figure	figure	VERB
ejpam-6321	264	2	2	2	NUM
ejpam-6321	264	3	:	:	PUNCT
ejpam-6321	264	4	visual	visual	ADJ
ejpam-6321	264	5	verification	verification	NOUN
ejpam-6321	264	6	of	of	ADP
ejpam-6321	264	7	fixed	fix	VERB
ejpam-6321	264	8	-	-	PUNCT
ejpam-6321	264	9	point	point	NOUN
ejpam-6321	264	10	convergence	convergence	NOUN
ejpam-6321	264	11	for	for	ADP
ejpam-6321	264	12	the	the	DET
ejpam-6321	264	13	mapping	mapping	NOUN
ejpam-6321	264	14	in	in	ADP
ejpam-6321	264	15	example	example	NOUN
ejpam-6321	264	16	1	1	NUM
ejpam-6321	264	17	figure	figure	NOUN
ejpam-6321	264	18	2	2	NUM
ejpam-6321	264	19	confirms	confirm	VERB
ejpam-6321	264	20	the	the	DET
ejpam-6321	264	21	convergence	convergence	NOUN
ejpam-6321	264	22	of	of	ADP
ejpam-6321	264	23	the	the	DET
ejpam-6321	264	24	fixed	fix	VERB
ejpam-6321	264	25	-	-	PUNCT
ejpam-6321	264	26	point	point	NOUN
ejpam-6321	264	27	iteration	iteration	NOUN
ejpam-6321	264	28	in	in	ADP
ejpam-6321	264	29	example	example	NOUN
ejpam-6321	264	30	1	1	NUM
ejpam-6321	264	31	,	,	PUNCT
ejpam-6321	264	32	aligning	align	VERB
ejpam-6321	264	33	with	with	ADP
ejpam-6321	264	34	the	the	DET
ejpam-6321	264	35	theoretical	theoretical	ADJ
ejpam-6321	264	36	result	result	NOUN
ejpam-6321	264	37	from	from	ADP
ejpam-6321	264	38	theorem	theorem	ADJ
ejpam-6321	264	39	1	1	NUM
ejpam-6321	264	40	.	.	PUNCT
ejpam-6321	264	41	theorem	theorem	NOUN
ejpam-6321	264	42	2	2	NUM
ejpam-6321	264	43	.	.	X
ejpam-6321	265	1	let	let	AUX
ejpam-6321	265	2	(	(	PUNCT
ejpam-6321	265	3	x	x	NOUN
ejpam-6321	265	4	,	,	PUNCT
ejpam-6321	265	5	g	g	NOUN
ejpam-6321	265	6	)	)	PUNCT
ejpam-6321	265	7	be	be	AUX
ejpam-6321	265	8	a	a	DET
ejpam-6321	265	9	gm	gm	NOUN
ejpam-6321	265	10	-	-	PUNCT
ejpam-6321	265	11	space	space	NOUN
ejpam-6321	265	12	and	and	CCONJ
ejpam-6321	265	13	f	f	NOUN
ejpam-6321	265	14	:	:	PUNCT
ejpam-6321	266	1	x	x	X
ejpam-6321	266	2	×x	×x	X
ejpam-6321	266	3	×x	×x	X
ejpam-6321	266	4	→	→	SYM
ejpam-6321	266	5	x	x	PART
ejpam-6321	266	6	be	be	AUX
ejpam-6321	266	7	a	a	DET
ejpam-6321	266	8	mapping	mapping	NOUN
ejpam-6321	266	9	satisfying	satisfying	ADJ
ejpam-6321	266	10	:	:	PUNCT
ejpam-6321	266	11	g(f1x1	g(f1x1	NOUN
ejpam-6321	266	12	,	,	PUNCT
ejpam-6321	266	13	f2x2	f2x2	NOUN
ejpam-6321	266	14	,	,	PUNCT
ejpam-6321	266	15	f3x3	f3x3	X
ejpam-6321	266	16	)	)	PUNCT
ejpam-6321	266	17	≤	≤	NOUN
ejpam-6321	267	1	α1g(x1	α1g(x1	PROPN
ejpam-6321	267	2	,	,	PUNCT
ejpam-6321	267	3	x2	x2	PROPN
ejpam-6321	267	4	,	,	PUNCT
ejpam-6321	267	5	x3	x3	ADJ
ejpam-6321	267	6	)	)	PUNCT
ejpam-6321	268	1	+	+	CCONJ
ejpam-6321	268	2	α2g(x1	α2g(x1	PROPN
ejpam-6321	268	3	,	,	PUNCT
ejpam-6321	268	4	x2	x2	PROPN
ejpam-6321	268	5	,	,	PUNCT
ejpam-6321	268	6	f2x2	f2x2	X
ejpam-6321	268	7	)	)	PUNCT
ejpam-6321	268	8	+	+	CCONJ
ejpam-6321	268	9	α3g(f1x1	α3g(f1x1	PROPN
ejpam-6321	268	10	,	,	PUNCT
ejpam-6321	268	11	x2	x2	PROPN
ejpam-6321	268	12	,	,	PUNCT
ejpam-6321	268	13	x2	x2	PROPN
ejpam-6321	268	14	)	)	PUNCT
ejpam-6321	268	15	+	+	X
ejpam-6321	269	1	α4[g(f1x1	α4[g(f1x1	PROPN
ejpam-6321	269	2	,	,	PUNCT
ejpam-6321	269	3	x1	x1	PROPN
ejpam-6321	269	4	,	,	PUNCT
ejpam-6321	269	5	x1	x1	PROPN
ejpam-6321	269	6	)	)	PUNCT
ejpam-6321	270	1	+	+	NUM
ejpam-6321	270	2	g(x2	g(x2	NOUN
ejpam-6321	270	3	,	,	PUNCT
ejpam-6321	270	4	f2x2	f2x2	NOUN
ejpam-6321	270	5	,	,	PUNCT
ejpam-6321	270	6	x2	x2	PROPN
ejpam-6321	270	7	)	)	PUNCT
ejpam-6321	271	1	+	+	ADJ
ejpam-6321	271	2	g(x3	g(x3	ADJ
ejpam-6321	271	3	,	,	PUNCT
ejpam-6321	271	4	x3	x3	ADJ
ejpam-6321	271	5	,	,	PUNCT
ejpam-6321	271	6	f3x3	f3x3	X
ejpam-6321	271	7	)	)	PUNCT
ejpam-6321	271	8	]	]	PUNCT
ejpam-6321	272	1	+	+	CCONJ
ejpam-6321	272	2	α5max	α5max	PROPN
ejpam-6321	272	3	{	{	PUNCT
ejpam-6321	272	4	g(x1	g(x1	NOUN
ejpam-6321	272	5	,	,	PUNCT
ejpam-6321	272	6	x2	x2	PROPN
ejpam-6321	272	7	,	,	PUNCT
ejpam-6321	272	8	f2x2	f2x2	NOUN
ejpam-6321	272	9	)	)	PUNCT
ejpam-6321	272	10	,	,	PUNCT
ejpam-6321	272	11	g(f1x1	g(f1x1	PROPN
ejpam-6321	272	12	,	,	PUNCT
ejpam-6321	272	13	f1x1	f1x1	NOUN
ejpam-6321	272	14	,	,	PUNCT
ejpam-6321	272	15	x2	x2	PROPN
ejpam-6321	272	16	)	)	PUNCT
ejpam-6321	272	17	,	,	PUNCT
ejpam-6321	272	18	g(f1x1	g(f1x1	PROPN
ejpam-6321	272	19	,	,	PUNCT
ejpam-6321	272	20	x2	x2	PROPN
ejpam-6321	272	21	,	,	PUNCT
ejpam-6321	272	22	x2	x2	PROPN
ejpam-6321	272	23	)	)	PUNCT
ejpam-6321	272	24	,	,	PUNCT
ejpam-6321	272	25	g(f2x2	g(f2x2	NOUN
ejpam-6321	272	26	,	,	PUNCT
ejpam-6321	272	27	f2x2	f2x2	CCONJ
ejpam-6321	272	28	,	,	PUNCT
ejpam-6321	272	29	x3	x3	ADJ
ejpam-6321	272	30	)	)	PUNCT
ejpam-6321	272	31	g(f1x1	g(f1x1	NOUN
ejpam-6321	272	32	,	,	PUNCT
ejpam-6321	272	33	x1	x1	PROPN
ejpam-6321	272	34	,	,	PUNCT
ejpam-6321	272	35	x1	x1	PROPN
ejpam-6321	272	36	)	)	PUNCT
ejpam-6321	272	37	,	,	PUNCT
ejpam-6321	272	38	g(x2	g(x2	NOUN
ejpam-6321	272	39	,	,	PUNCT
ejpam-6321	272	40	f2x2	f2x2	X
ejpam-6321	272	41	,	,	PUNCT
ejpam-6321	272	42	x2	x2	PROPN
ejpam-6321	272	43	)	)	PUNCT
ejpam-6321	272	44	,	,	PUNCT
ejpam-6321	272	45	g(x3	g(x3	NOUN
ejpam-6321	272	46	,	,	PUNCT
ejpam-6321	272	47	x3	x3	ADJ
ejpam-6321	272	48	,	,	PUNCT
ejpam-6321	272	49	f3x3	f3x3	X
ejpam-6321	272	50	)	)	PUNCT
ejpam-6321	272	51	}	}	PUNCT
ejpam-6321	272	52	(	(	PUNCT
ejpam-6321	272	53	8)	8)	NUM
ejpam-6321	272	54	for	for	ADP
ejpam-6321	272	55	all	all	DET
ejpam-6321	272	56	x1	x1	PROPN
ejpam-6321	272	57	,	,	PUNCT
ejpam-6321	272	58	x2	x2	PROPN
ejpam-6321	272	59	,	,	PUNCT
ejpam-6321	272	60	x3	x3	PROPN
ejpam-6321	272	61	∈	∈	PROPN
ejpam-6321	272	62	x	x	X
ejpam-6321	272	63	and	and	CCONJ
ejpam-6321	272	64	α1	α1	PROPN
ejpam-6321	272	65	,	,	PUNCT
ejpam-6321	272	66	α2	α2	ADJ
ejpam-6321	272	67	,	,	PUNCT
ejpam-6321	272	68	α3	α3	NOUN
ejpam-6321	272	69	,	,	PUNCT
ejpam-6321	272	70	α4	α4	NOUN
ejpam-6321	272	71	,	,	PUNCT
ejpam-6321	272	72	α5	α5	X
ejpam-6321	272	73	≥	≥	NOUN
ejpam-6321	272	74	0	0	NUM
ejpam-6321	272	75	with	with	ADP
ejpam-6321	272	76	α1	α1	PROPN
ejpam-6321	272	77	,	,	PUNCT
ejpam-6321	272	78	α2	α2	ADJ
ejpam-6321	272	79	,	,	PUNCT
ejpam-6321	272	80	α3	α3	NOUN
ejpam-6321	272	81	,	,	PUNCT
ejpam-6321	272	82	α4	α4	NOUN
ejpam-6321	272	83	,	,	PUNCT
ejpam-6321	272	84	α5	α5	X
ejpam-6321	272	85	<	<	X
ejpam-6321	272	86	1	1	NUM
ejpam-6321	272	87	and	and	CCONJ
ejpam-6321	272	88	(	(	PUNCT
ejpam-6321	272	89	α1	α1	PROPN
ejpam-6321	272	90	+	+	CCONJ
ejpam-6321	272	91	α2	α2	ADJ
ejpam-6321	272	92	+	+	CCONJ
ejpam-6321	272	93	α3	α3	ADJ
ejpam-6321	272	94	+	+	CCONJ
ejpam-6321	272	95	α4	α4	NOUN
ejpam-6321	272	96	+	+	CCONJ
ejpam-6321	272	97	α5	α5	NOUN
ejpam-6321	272	98	)	)	PUNCT
ejpam-6321	272	99	<	<	X
ejpam-6321	272	100	1	1	X
ejpam-6321	272	101	.	.	PUNCT
ejpam-6321	272	102	then	then	ADV
ejpam-6321	272	103	,	,	PUNCT
ejpam-6321	272	104	the	the	DET
ejpam-6321	272	105	three	three	NUM
ejpam-6321	272	106	individual	individual	ADJ
ejpam-6321	272	107	self	self	NOUN
ejpam-6321	272	108	-	-	PUNCT
ejpam-6321	272	109	mappings	mapping	NOUN
ejpam-6321	272	110	referred	refer	VERB
ejpam-6321	272	111	as	as	ADP
ejpam-6321	272	112	f1	f1	NOUN
ejpam-6321	272	113	,	,	PUNCT
ejpam-6321	272	114	f2	f2	PROPN
ejpam-6321	272	115	and	and	CCONJ
ejpam-6321	272	116	f3	f3	PROPN
ejpam-6321	272	117	possess	possess	VERB
ejpam-6321	272	118	a	a	DET
ejpam-6321	272	119	cfp	cfp	NOUN
ejpam-6321	272	120	in	in	ADP
ejpam-6321	272	121	x.	x.	PROPN
ejpam-6321	272	122	moreover	moreover	ADV
ejpam-6321	272	123	,	,	PUNCT
ejpam-6321	272	124	if	if	SCONJ
ejpam-6321	272	125	(	(	PUNCT
ejpam-6321	272	126	α1	α1	PROPN
ejpam-6321	272	127	+	+	ADV
ejpam-6321	272	128	α2	α2	ADJ
ejpam-6321	272	129	+	+	ADJ
ejpam-6321	272	130	α3	α3	ADJ
ejpam-6321	272	131	+	+	NOUN
ejpam-6321	272	132	α4	α4	NOUN
ejpam-6321	272	133	)	)	PUNCT
ejpam-6321	272	134	<	<	X
ejpam-6321	272	135	1	1	NUM
ejpam-6321	272	136	,	,	PUNCT
ejpam-6321	272	137	the	the	DET
ejpam-6321	272	138	three	three	NUM
ejpam-6321	272	139	individual	individual	ADJ
ejpam-6321	272	140	self	self	NOUN
ejpam-6321	272	141	-	-	PUNCT
ejpam-6321	272	142	mappings	mapping	NOUN
ejpam-6321	272	143	referred	refer	VERB
ejpam-6321	272	144	as	as	ADP
ejpam-6321	272	145	f1	f1	NOUN
ejpam-6321	272	146	,	,	PUNCT
ejpam-6321	272	147	f2	f2	PROPN
ejpam-6321	272	148	and	and	CCONJ
ejpam-6321	272	149	f3	f3	PROPN
ejpam-6321	272	150	possess	possess	VERB
ejpam-6321	272	151	a	a	DET
ejpam-6321	272	152	ucfp	ucfp	NOUN
ejpam-6321	272	153	in	in	ADP
ejpam-6321	272	154	x.	x.	NOUN
ejpam-6321	272	155	proof	proof	NOUN
ejpam-6321	272	156	.	.	PUNCT
ejpam-6321	273	1	fix	fix	VERB
ejpam-6321	273	2	x0	x0	PROPN
ejpam-6321	273	3	∈	∈	PROPN
ejpam-6321	273	4	x	x	X
ejpam-6321	273	5	,	,	PUNCT
ejpam-6321	273	6	and	and	CCONJ
ejpam-6321	273	7	{	{	PUNCT
ejpam-6321	273	8	xk	xk	NOUN
ejpam-6321	273	9	}	}	PUNCT
ejpam-6321	273	10	be	be	AUX
ejpam-6321	273	11	a	a	DET
ejpam-6321	273	12	sequence	sequence	NOUN
ejpam-6321	273	13	in	in	ADP
ejpam-6321	273	14	x.	x.	NOUN
ejpam-6321	274	1	now	now	ADV
ejpam-6321	274	2	we	we	PRON
ejpam-6321	274	3	define	define	VERB
ejpam-6321	274	4	some	some	DET
ejpam-6321	274	5	iterative	iterative	NOUN
ejpam-6321	274	6	sequences	sequence	NOUN
ejpam-6321	274	7	in	in	ADP
ejpam-6321	274	8	x	x	X
ejpam-6321	274	9	such	such	ADJ
ejpam-6321	274	10	that	that	PRON
ejpam-6321	274	11	x(3k+1	x(3k+1	CCONJ
ejpam-6321	274	12	)	)	PUNCT
ejpam-6321	274	13	=	=	SYM
ejpam-6321	275	1	f1x3k	f1x3k	NUM
ejpam-6321	275	2	,	,	PUNCT
ejpam-6321	275	3	x(3k+2	x(3k+2	NOUN
ejpam-6321	275	4	)	)	PUNCT
ejpam-6321	275	5	=	=	PUNCT
ejpam-6321	276	1	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	276	2	)	)	PUNCT
ejpam-6321	276	3	and	and	CCONJ
ejpam-6321	276	4	x(3k+3	x(3k+3	NUM
ejpam-6321	276	5	)	)	PUNCT
ejpam-6321	277	1	=	=	SYM
ejpam-6321	277	2	f3x(3k+2	f3x(3k+2	NUM
ejpam-6321	277	3	)	)	PUNCT
ejpam-6321	277	4	∀	∀	PUNCT
ejpam-6321	278	1	k	k	X
ejpam-6321	278	2	≥	≥	NOUN
ejpam-6321	278	3	0	0	NUM
ejpam-6321	278	4	.	.	PUNCT
ejpam-6321	279	1	now	now	ADV
ejpam-6321	279	2	,	,	PUNCT
ejpam-6321	279	3	by	by	ADP
ejpam-6321	279	4	using	use	VERB
ejpam-6321	279	5	(	(	PUNCT
ejpam-6321	279	6	8)	8)	NUM
ejpam-6321	279	7	we	we	PRON
ejpam-6321	279	8	have	have	VERB
ejpam-6321	279	9	,	,	PUNCT
ejpam-6321	279	10	m.	m.	PROPN
ejpam-6321	279	11	noorwali	noorwali	PROPN
ejpam-6321	279	12	et	et	PROPN
ejpam-6321	279	13	al	al	PROPN
ejpam-6321	279	14	.	.	PUNCT
ejpam-6321	279	15	/	/	SYM
ejpam-6321	279	16	eur	eur	PROPN
ejpam-6321	279	17	.	.	PUNCT
ejpam-6321	280	1	j.	j.	PROPN
ejpam-6321	280	2	pure	pure	PROPN
ejpam-6321	280	3	appl	appl	PROPN
ejpam-6321	280	4	.	.	PROPN
ejpam-6321	280	5	math	math	PROPN
ejpam-6321	280	6	,	,	PUNCT
ejpam-6321	280	7	18	18	NUM
ejpam-6321	280	8	(	(	PUNCT
ejpam-6321	280	9	3	3	NUM
ejpam-6321	280	10	)	)	PUNCT
ejpam-6321	280	11	(	(	PUNCT
ejpam-6321	280	12	2025	2025	NUM
ejpam-6321	280	13	)	)	PUNCT
ejpam-6321	280	14	,	,	PUNCT
ejpam-6321	280	15	6321	6321	NUM
ejpam-6321	280	16	11	11	NUM
ejpam-6321	280	17	of	of	ADP
ejpam-6321	280	18	27	27	NUM
ejpam-6321	280	19	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	280	20	)	)	PUNCT
ejpam-6321	280	21	,	,	PUNCT
ejpam-6321	280	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	280	23	)	)	PUNCT
ejpam-6321	280	24	,	,	PUNCT
ejpam-6321	280	25	x(3k+3	x(3k+3	NUM
ejpam-6321	280	26	)	)	PUNCT
ejpam-6321	280	27	)	)	PUNCT
ejpam-6321	281	1	=	=	PUNCT
ejpam-6321	281	2	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	281	3	,	,	PUNCT
ejpam-6321	281	4	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	281	5	)	)	PUNCT
ejpam-6321	281	6	,	,	PUNCT
ejpam-6321	281	7	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	281	8	)	)	PUNCT
ejpam-6321	281	9	)	)	PUNCT
ejpam-6321	281	10	≤	≤	NUM
ejpam-6321	281	11	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	281	12	,	,	PUNCT
ejpam-6321	281	13	x(3k+1	x(3k+1	PRON
ejpam-6321	281	14	)	)	PUNCT
ejpam-6321	281	15	,	,	PUNCT
ejpam-6321	281	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	281	17	)	)	PUNCT
ejpam-6321	281	18	)	)	PUNCT
ejpam-6321	282	1	+	+	CCONJ
ejpam-6321	282	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	282	3	,	,	PUNCT
ejpam-6321	282	4	x(3k+1	x(3k+1	PRON
ejpam-6321	282	5	)	)	PUNCT
ejpam-6321	282	6	,	,	PUNCT
ejpam-6321	282	7	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	282	8	)	)	PUNCT
ejpam-6321	282	9	)	)	PUNCT
ejpam-6321	283	1	+	+	CCONJ
ejpam-6321	283	2	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	283	3	,	,	PUNCT
ejpam-6321	283	4	x(3k+1	x(3k+1	ADV
ejpam-6321	283	5	)	)	PUNCT
ejpam-6321	283	6	,	,	PUNCT
ejpam-6321	283	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	283	8	)	)	PUNCT
ejpam-6321	283	9	)	)	PUNCT
ejpam-6321	284	1	+	+	CCONJ
ejpam-6321	284	2	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	284	3	,	,	PUNCT
ejpam-6321	284	4	x3k	x3k	NOUN
ejpam-6321	284	5	,	,	PUNCT
ejpam-6321	284	6	x3k	x3k	PUNCT
ejpam-6321	284	7	)	)	PUNCT
ejpam-6321	284	8	+	+	SYM
ejpam-6321	284	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	284	10	)	)	PUNCT
ejpam-6321	284	11	,	,	PUNCT
ejpam-6321	284	12	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	284	13	)	)	PUNCT
ejpam-6321	284	14	,	,	PUNCT
ejpam-6321	284	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	284	16	)	)	PUNCT
ejpam-6321	284	17	)	)	PUNCT
ejpam-6321	285	1	+	+	PUNCT
ejpam-6321	285	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	285	3	)	)	PUNCT
ejpam-6321	285	4	,	,	PUNCT
ejpam-6321	285	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	285	6	)	)	PUNCT
ejpam-6321	285	7	,	,	PUNCT
ejpam-6321	285	8	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	285	9	)	)	PUNCT
ejpam-6321	285	10	)	)	PUNCT
ejpam-6321	285	11	]	]	PUNCT
ejpam-6321	286	1	+	+	CCONJ
ejpam-6321	286	2	α5max	α5max	PROPN
ejpam-6321	286	3			PART
ejpam-6321	286	4	g(x3k	g(x3k	NOUN
ejpam-6321	286	5	,	,	PUNCT
ejpam-6321	286	6	x(3k+1	x(3k+1	PRON
ejpam-6321	286	7	)	)	PUNCT
ejpam-6321	286	8	,	,	PUNCT
ejpam-6321	286	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	286	10	)	)	PUNCT
ejpam-6321	286	11	)	)	PUNCT
ejpam-6321	286	12	,	,	PUNCT
ejpam-6321	286	13	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	286	14	,	,	PUNCT
ejpam-6321	286	15	f1x3k	f1x3k	NUM
ejpam-6321	286	16	,	,	PUNCT
ejpam-6321	286	17	x(3k+1	x(3k+1	NUM
ejpam-6321	286	18	)	)	PUNCT
ejpam-6321	286	19	)	)	PUNCT
ejpam-6321	286	20	,	,	PUNCT
ejpam-6321	286	21	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	286	22	,	,	PUNCT
ejpam-6321	286	23	x(3k+1	x(3k+1	ADV
ejpam-6321	286	24	)	)	PUNCT
ejpam-6321	286	25	,	,	PUNCT
ejpam-6321	286	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	286	27	)	)	PUNCT
ejpam-6321	286	28	)	)	PUNCT
ejpam-6321	286	29	,	,	PUNCT
ejpam-6321	286	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	286	31	)	)	PUNCT
ejpam-6321	286	32	,	,	PUNCT
ejpam-6321	286	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	286	34	)	)	PUNCT
ejpam-6321	286	35	,	,	PUNCT
ejpam-6321	286	36	x(3k+2	x(3k+2	NUM
ejpam-6321	286	37	)	)	PUNCT
ejpam-6321	286	38	)	)	PUNCT
ejpam-6321	286	39	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	286	40	,	,	PUNCT
ejpam-6321	286	41	x3k	x3k	NOUN
ejpam-6321	286	42	,	,	PUNCT
ejpam-6321	286	43	x3k	x3k	NOUN
ejpam-6321	286	44	)	)	PUNCT
ejpam-6321	286	45	,	,	PUNCT
ejpam-6321	286	46	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	286	47	)	)	PUNCT
ejpam-6321	286	48	,	,	PUNCT
ejpam-6321	286	49	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	286	50	)	)	PUNCT
ejpam-6321	286	51	,	,	PUNCT
ejpam-6321	286	52	x(3k+1	x(3k+1	CCONJ
ejpam-6321	286	53	)	)	PUNCT
ejpam-6321	286	54	)	)	PUNCT
ejpam-6321	286	55	,	,	PUNCT
ejpam-6321	286	56	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	286	57	)	)	PUNCT
ejpam-6321	286	58	,	,	PUNCT
ejpam-6321	286	59	x(3k+2	x(3k+2	NOUN
ejpam-6321	286	60	)	)	PUNCT
ejpam-6321	286	61	,	,	PUNCT
ejpam-6321	286	62	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	286	63	)	)	PUNCT
ejpam-6321	286	64	)	)	PUNCT
ejpam-6321	287	1			ADJ
ejpam-6321	287	2	=	=	SYM
ejpam-6321	287	3	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	287	4	,	,	PUNCT
ejpam-6321	287	5	x(3k+1	x(3k+1	PRON
ejpam-6321	287	6	)	)	PUNCT
ejpam-6321	287	7	,	,	PUNCT
ejpam-6321	287	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	287	9	)	)	PUNCT
ejpam-6321	287	10	)	)	PUNCT
ejpam-6321	288	1	+	+	CCONJ
ejpam-6321	288	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	288	3	,	,	PUNCT
ejpam-6321	288	4	x(3k+1	x(3k+1	CCONJ
ejpam-6321	288	5	)	)	PUNCT
ejpam-6321	288	6	,	,	PUNCT
ejpam-6321	288	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	288	8	)	)	PUNCT
ejpam-6321	288	9	)	)	PUNCT
ejpam-6321	289	1	+	+	CCONJ
ejpam-6321	289	2	α3g(x3k	α3g(x3k	NOUN
ejpam-6321	289	3	,	,	PUNCT
ejpam-6321	289	4	x(3k+1	x(3k+1	ADV
ejpam-6321	289	5	)	)	PUNCT
ejpam-6321	289	6	,	,	PUNCT
ejpam-6321	289	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	289	8	)	)	PUNCT
ejpam-6321	289	9	)	)	PUNCT
ejpam-6321	290	1	+	+	CCONJ
ejpam-6321	290	2	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	290	3	,	,	PUNCT
ejpam-6321	290	4	x3k	x3k	NOUN
ejpam-6321	290	5	,	,	PUNCT
ejpam-6321	290	6	x3k	x3k	PUNCT
ejpam-6321	290	7	)	)	PUNCT
ejpam-6321	290	8	+	+	SYM
ejpam-6321	290	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	290	10	)	)	PUNCT
ejpam-6321	290	11	,	,	PUNCT
ejpam-6321	290	12	x(3k+1	x(3k+1	CCONJ
ejpam-6321	290	13	)	)	PUNCT
ejpam-6321	290	14	,	,	PUNCT
ejpam-6321	290	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	290	16	)	)	PUNCT
ejpam-6321	290	17	)	)	PUNCT
ejpam-6321	291	1	+	+	PUNCT
ejpam-6321	291	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	291	3	)	)	PUNCT
ejpam-6321	291	4	,	,	PUNCT
ejpam-6321	291	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	291	6	)	)	PUNCT
ejpam-6321	291	7	,	,	PUNCT
ejpam-6321	291	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	291	9	)	)	PUNCT
ejpam-6321	291	10	)	)	PUNCT
ejpam-6321	291	11	]	]	PUNCT
ejpam-6321	292	1	+	+	CCONJ
ejpam-6321	292	2	α5max	α5max	PROPN
ejpam-6321	292	3			PART
ejpam-6321	292	4	g(x3k	g(x3k	NOUN
ejpam-6321	292	5	,	,	PUNCT
ejpam-6321	292	6	x(3k+1	x(3k+1	PRON
ejpam-6321	292	7	)	)	PUNCT
ejpam-6321	292	8	,	,	PUNCT
ejpam-6321	292	9	x(3k+1	x(3k+1	CCONJ
ejpam-6321	292	10	)	)	PUNCT
ejpam-6321	292	11	)	)	PUNCT
ejpam-6321	292	12	,	,	PUNCT
ejpam-6321	292	13	g(x3k	g(x3k	NOUN
ejpam-6321	292	14	,	,	PUNCT
ejpam-6321	292	15	x3k	x3k	NOUN
ejpam-6321	292	16	,	,	PUNCT
ejpam-6321	292	17	x(3k+1	x(3k+1	ADV
ejpam-6321	292	18	)	)	PUNCT
ejpam-6321	292	19	)	)	PUNCT
ejpam-6321	292	20	,	,	PUNCT
ejpam-6321	292	21	g(x3k	g(x3k	NOUN
ejpam-6321	292	22	,	,	PUNCT
ejpam-6321	292	23	x(3k+1	x(3k+1	PRON
ejpam-6321	292	24	)	)	PUNCT
ejpam-6321	292	25	,	,	PUNCT
ejpam-6321	292	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	292	27	)	)	PUNCT
ejpam-6321	292	28	)	)	PUNCT
ejpam-6321	292	29	,	,	PUNCT
ejpam-6321	292	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	292	31	)	)	PUNCT
ejpam-6321	292	32	,	,	PUNCT
ejpam-6321	292	33	x(3k+1	x(3k+1	CCONJ
ejpam-6321	292	34	)	)	PUNCT
ejpam-6321	292	35	,	,	PUNCT
ejpam-6321	292	36	x(3k+2	x(3k+2	NUM
ejpam-6321	292	37	)	)	PUNCT
ejpam-6321	292	38	)	)	PUNCT
ejpam-6321	292	39	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	292	40	,	,	PUNCT
ejpam-6321	292	41	x3k	x3k	NOUN
ejpam-6321	292	42	,	,	PUNCT
ejpam-6321	292	43	x3k	x3k	NOUN
ejpam-6321	292	44	)	)	PUNCT
ejpam-6321	292	45	,	,	PUNCT
ejpam-6321	292	46	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	292	47	)	)	PUNCT
ejpam-6321	292	48	,	,	PUNCT
ejpam-6321	292	49	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	292	50	)	)	PUNCT
ejpam-6321	292	51	,	,	PUNCT
ejpam-6321	292	52	x(3k+1	x(3k+1	CCONJ
ejpam-6321	292	53	)	)	PUNCT
ejpam-6321	292	54	)	)	PUNCT
ejpam-6321	292	55	,	,	PUNCT
ejpam-6321	292	56	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	292	57	)	)	PUNCT
ejpam-6321	292	58	,	,	PUNCT
ejpam-6321	292	59	x(3k+2	x(3k+2	NOUN
ejpam-6321	292	60	)	)	PUNCT
ejpam-6321	292	61	,	,	PUNCT
ejpam-6321	292	62	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	292	63	)	)	PUNCT
ejpam-6321	292	64	)	)	PUNCT
ejpam-6321	292	65			ADP
ejpam-6321	292	66	after	after	ADP
ejpam-6321	292	67	applying	apply	VERB
ejpam-6321	292	68	the	the	DET
ejpam-6321	292	69	simplification	simplification	NOUN
ejpam-6321	292	70	process	process	NOUN
ejpam-6321	292	71	utilizing	utilize	VERB
ejpam-6321	292	72	the	the	DET
ejpam-6321	292	73	definition	definition	NOUN
ejpam-6321	292	74	of	of	ADP
ejpam-6321	292	75	gm	gm	PROPN
ejpam-6321	292	76	-	-	PUNCT
ejpam-6321	292	77	space	space	NOUN
ejpam-6321	292	78	we	we	PRON
ejpam-6321	292	79	have	have	AUX
ejpam-6321	292	80	achieved	achieve	VERB
ejpam-6321	292	81	the	the	DET
ejpam-6321	292	82	following	following	ADJ
ejpam-6321	292	83	result	result	NOUN
ejpam-6321	292	84	,	,	PUNCT
ejpam-6321	292	85	≤	≤	NUM
ejpam-6321	292	86	(	(	PUNCT
ejpam-6321	292	87	α1	α1	PROPN
ejpam-6321	292	88	+	+	CCONJ
ejpam-6321	292	89	α2)g(x3k	α2)g(x3k	ADJ
ejpam-6321	292	90	,	,	PUNCT
ejpam-6321	292	91	x(3k+1	x(3k+1	ADV
ejpam-6321	292	92	)	)	PUNCT
ejpam-6321	292	93	,	,	PUNCT
ejpam-6321	292	94	x(3k+2	x(3k+2	NOUN
ejpam-6321	292	95	)	)	PUNCT
ejpam-6321	292	96	)	)	PUNCT
ejpam-6321	293	1	+	+	CCONJ
ejpam-6321	294	1	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	294	2	,	,	PUNCT
ejpam-6321	294	3	x(3k+1	x(3k+1	PRON
ejpam-6321	294	4	)	)	PUNCT
ejpam-6321	294	5	,	,	PUNCT
ejpam-6321	294	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	294	7	)	)	PUNCT
ejpam-6321	294	8	)	)	PUNCT
ejpam-6321	295	1	+	+	ADP
ejpam-6321	295	2	g(x3k	g(x3k	NOUN
ejpam-6321	295	3	,	,	PUNCT
ejpam-6321	295	4	x(3k+1	x(3k+1	PRON
ejpam-6321	295	5	)	)	PUNCT
ejpam-6321	295	6	,	,	PUNCT
ejpam-6321	295	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	295	8	)	)	PUNCT
ejpam-6321	295	9	)	)	PUNCT
ejpam-6321	296	1	+	+	PUNCT
ejpam-6321	296	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	296	3	)	)	PUNCT
ejpam-6321	296	4	,	,	PUNCT
ejpam-6321	296	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	6	)	)	PUNCT
ejpam-6321	296	7	,	,	PUNCT
ejpam-6321	296	8	x(3k+3))]+	x(3k+3))]+	PROPN
ejpam-6321	296	9	α5max	α5max	PROPN
ejpam-6321	296	10	{	{	PUNCT
ejpam-6321	296	11	g(x3k	g(x3k	NOUN
ejpam-6321	296	12	,	,	PUNCT
ejpam-6321	296	13	x(3k+1	x(3k+1	PRON
ejpam-6321	296	14	)	)	PUNCT
ejpam-6321	296	15	,	,	PUNCT
ejpam-6321	296	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	17	)	)	PUNCT
ejpam-6321	296	18	)	)	PUNCT
ejpam-6321	296	19	,	,	PUNCT
ejpam-6321	296	20	g(x3k	g(x3k	NOUN
ejpam-6321	296	21	,	,	PUNCT
ejpam-6321	296	22	x(3k+1	x(3k+1	ADV
ejpam-6321	296	23	,	,	PUNCT
ejpam-6321	296	24	x(3k+2	x(3k+2	NUM
ejpam-6321	296	25	)	)	PUNCT
ejpam-6321	296	26	)	)	PUNCT
ejpam-6321	296	27	,	,	PUNCT
ejpam-6321	296	28	g(x3k	g(x3k	NOUN
ejpam-6321	296	29	,	,	PUNCT
ejpam-6321	296	30	x(3k+1	x(3k+1	PRON
ejpam-6321	296	31	)	)	PUNCT
ejpam-6321	296	32	,	,	PUNCT
ejpam-6321	296	33	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	34	)	)	PUNCT
ejpam-6321	296	35	)	)	PUNCT
ejpam-6321	296	36	,	,	PUNCT
ejpam-6321	296	37	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	296	38	)	)	PUNCT
ejpam-6321	296	39	,	,	PUNCT
ejpam-6321	296	40	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	41	)	)	PUNCT
ejpam-6321	296	42	,	,	PUNCT
ejpam-6321	296	43	x(3k+3	x(3k+3	NUM
ejpam-6321	296	44	)	)	PUNCT
ejpam-6321	296	45	)	)	PUNCT
ejpam-6321	296	46	}	}	PUNCT
ejpam-6321	296	47	(	(	PUNCT
ejpam-6321	296	48	1−	1−	NUM
ejpam-6321	296	49	α4)g(x(3k+1	α4)g(x(3k+1	VERB
ejpam-6321	296	50	)	)	PUNCT
ejpam-6321	296	51	,	,	PUNCT
ejpam-6321	296	52	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	53	)	)	PUNCT
ejpam-6321	296	54	,	,	PUNCT
ejpam-6321	296	55	x(3k+3	x(3k+3	NUM
ejpam-6321	296	56	)	)	PUNCT
ejpam-6321	296	57	)	)	PUNCT
ejpam-6321	296	58	≤	≤	NOUN
ejpam-6321	296	59	(	(	PUNCT
ejpam-6321	296	60	α1	α1	PROPN
ejpam-6321	296	61	+	+	CCONJ
ejpam-6321	296	62	α2	α2	ADJ
ejpam-6321	296	63	+	+	CCONJ
ejpam-6321	296	64	2α4)g(x3k	2α4)g(x3k	NUM
ejpam-6321	296	65	,	,	PUNCT
ejpam-6321	296	66	x(3k+1	x(3k+1	ADV
ejpam-6321	296	67	)	)	PUNCT
ejpam-6321	296	68	,	,	PUNCT
ejpam-6321	296	69	x(3k+2	x(3k+2	NOUN
ejpam-6321	296	70	)	)	PUNCT
ejpam-6321	296	71	)	)	PUNCT
ejpam-6321	297	1	+	+	CCONJ
ejpam-6321	297	2	α5max	α5max	PROPN
ejpam-6321	297	3	{	{	PUNCT
ejpam-6321	297	4	g(x3k	g(x3k	NOUN
ejpam-6321	297	5	,	,	PUNCT
ejpam-6321	297	6	x(3k+1	x(3k+1	PRON
ejpam-6321	297	7	)	)	PUNCT
ejpam-6321	297	8	,	,	PUNCT
ejpam-6321	297	9	x(3k+2	x(3k+2	NOUN
ejpam-6321	297	10	)	)	PUNCT
ejpam-6321	297	11	)	)	PUNCT
ejpam-6321	297	12	,	,	PUNCT
ejpam-6321	297	13	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	297	14	)	)	PUNCT
ejpam-6321	297	15	,	,	PUNCT
ejpam-6321	297	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	297	17	)	)	PUNCT
ejpam-6321	297	18	,	,	PUNCT
ejpam-6321	297	19	x(3k+3	x(3k+3	NUM
ejpam-6321	297	20	)	)	PUNCT
ejpam-6321	297	21	)	)	PUNCT
ejpam-6321	297	22	}	}	PUNCT
ejpam-6321	297	23	(	(	PUNCT
ejpam-6321	297	24	9	9	X
ejpam-6321	297	25	)	)	PUNCT
ejpam-6321	297	26	now	now	ADV
ejpam-6321	297	27	,	,	PUNCT
ejpam-6321	297	28	there	there	PRON
ejpam-6321	297	29	are	be	VERB
ejpam-6321	297	30	two	two	NUM
ejpam-6321	297	31	cases	case	NOUN
ejpam-6321	297	32	:	:	PUNCT
ejpam-6321	297	33	(	(	PUNCT
ejpam-6321	297	34	i	i	NOUN
ejpam-6321	297	35	)	)	PUNCT
ejpam-6321	297	36	.	.	PUNCT
ejpam-6321	298	1	if	if	SCONJ
ejpam-6321	298	2	g(x3k	g(x3k	NOUN
ejpam-6321	298	3	,	,	PUNCT
ejpam-6321	298	4	x(3k+1	x(3k+1	PRON
ejpam-6321	298	5	)	)	PUNCT
ejpam-6321	298	6	,	,	PUNCT
ejpam-6321	298	7	x(3k+2	x(3k+2	NUM
ejpam-6321	298	8	)	)	PUNCT
ejpam-6321	298	9	)	)	PUNCT
ejpam-6321	298	10	is	be	AUX
ejpam-6321	298	11	a	a	DET
ejpam-6321	298	12	maximum	maximum	ADJ
ejpam-6321	298	13	term	term	NOUN
ejpam-6321	298	14	in	in	ADP
ejpam-6321	298	15	(	(	PUNCT
ejpam-6321	298	16	9	9	NUM
ejpam-6321	298	17	)	)	PUNCT
ejpam-6321	298	18	then	then	ADV
ejpam-6321	298	19	we	we	PRON
ejpam-6321	298	20	can	can	AUX
ejpam-6321	298	21	write	write	VERB
ejpam-6321	298	22	,	,	PUNCT
ejpam-6321	298	23	(	(	PUNCT
ejpam-6321	298	24	1−α4)g(x(3k+1	1−α4)g(x(3k+1	NUM
ejpam-6321	298	25	)	)	PUNCT
ejpam-6321	298	26	,	,	PUNCT
ejpam-6321	298	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	298	28	)	)	PUNCT
ejpam-6321	298	29	,	,	PUNCT
ejpam-6321	298	30	x(3k+3	x(3k+3	NUM
ejpam-6321	298	31	)	)	PUNCT
ejpam-6321	298	32	)	)	PUNCT
ejpam-6321	299	1	≤	≤	NOUN
ejpam-6321	299	2	(	(	PUNCT
ejpam-6321	299	3	α1+α2	α1+α2	NOUN
ejpam-6321	299	4	+	+	NOUN
ejpam-6321	299	5	2α4)g(x3k	2α4)g(x3k	NUM
ejpam-6321	299	6	,	,	PUNCT
ejpam-6321	299	7	x(3k+1	x(3k+1	ADV
ejpam-6321	299	8	)	)	PUNCT
ejpam-6321	299	9	,	,	PUNCT
ejpam-6321	299	10	x(3k+2))+α5g(x3k	x(3k+2))+α5g(x3k	NUM
ejpam-6321	299	11	,	,	PUNCT
ejpam-6321	299	12	x(3k+1	x(3k+1	ADV
ejpam-6321	299	13	)	)	PUNCT
ejpam-6321	299	14	,	,	PUNCT
ejpam-6321	299	15	x(3k+2	x(3k+2	NOUN
ejpam-6321	299	16	)	)	PUNCT
ejpam-6321	299	17	)	)	PUNCT
ejpam-6321	300	1	(	(	PUNCT
ejpam-6321	300	2	α1	α1	PROPN
ejpam-6321	300	3	+	+	CCONJ
ejpam-6321	300	4	α2	α2	ADJ
ejpam-6321	300	5	+	+	CCONJ
ejpam-6321	300	6	2α4	2α4	NUM
ejpam-6321	300	7	+	+	CCONJ
ejpam-6321	300	8	α5)g(x3k	α5)g(x3k	X
ejpam-6321	300	9	,	,	PUNCT
ejpam-6321	300	10	x(3k+1	x(3k+1	PRON
ejpam-6321	300	11	)	)	PUNCT
ejpam-6321	300	12	,	,	PUNCT
ejpam-6321	300	13	x(3k+2	x(3k+2	NUM
ejpam-6321	300	14	)	)	PUNCT
ejpam-6321	300	15	)	)	PUNCT
ejpam-6321	300	16	≤	≤	NOUN
ejpam-6321	300	17	(	(	PUNCT
ejpam-6321	300	18	α1	α1	PROPN
ejpam-6321	300	19	+	+	CCONJ
ejpam-6321	300	20	α2	α2	ADJ
ejpam-6321	301	1	+	+	CCONJ
ejpam-6321	301	2	2α4	2α4	NUM
ejpam-6321	302	1	+	+	SYM
ejpam-6321	302	2	α5	α5	NOUN
ejpam-6321	302	3	)	)	PUNCT
ejpam-6321	302	4	(	(	PUNCT
ejpam-6321	302	5	1−	1−	NUM
ejpam-6321	302	6	α4	α4	NOUN
ejpam-6321	302	7	)	)	PUNCT
ejpam-6321	302	8	g(x3k	g(x3k	NOUN
ejpam-6321	302	9	,	,	PUNCT
ejpam-6321	302	10	x(3k+1	x(3k+1	PRON
ejpam-6321	302	11	)	)	PUNCT
ejpam-6321	302	12	,	,	PUNCT
ejpam-6321	302	13	x(3k+2	x(3k+2	NOUN
ejpam-6321	302	14	)	)	PUNCT
ejpam-6321	302	15	)	)	PUNCT
ejpam-6321	302	16	,	,	PUNCT
ejpam-6321	302	17	where	where	SCONJ
ejpam-6321	302	18	γ1	γ1	PROPN
ejpam-6321	302	19	=	=	SYM
ejpam-6321	302	20	(	(	PUNCT
ejpam-6321	302	21	α1	α1	PROPN
ejpam-6321	302	22	+	+	CCONJ
ejpam-6321	302	23	α2	α2	ADJ
ejpam-6321	302	24	+	+	CCONJ
ejpam-6321	302	25	2α4	2α4	NUM
ejpam-6321	302	26	+	+	SYM
ejpam-6321	302	27	α5	α5	NOUN
ejpam-6321	302	28	)	)	PUNCT
ejpam-6321	302	29	(	(	PUNCT
ejpam-6321	302	30	1−	1−	NUM
ejpam-6321	302	31	α4	α4	NOUN
ejpam-6321	302	32	)	)	PUNCT
ejpam-6321	302	33	<	<	X
ejpam-6321	302	34	1	1	NUM
ejpam-6321	302	35	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	302	36	)	)	PUNCT
ejpam-6321	302	37	,	,	PUNCT
ejpam-6321	302	38	x(3k+2	x(3k+2	NOUN
ejpam-6321	302	39	)	)	PUNCT
ejpam-6321	302	40	,	,	PUNCT
ejpam-6321	302	41	x(3k+3	x(3k+3	NUM
ejpam-6321	302	42	)	)	PUNCT
ejpam-6321	302	43	)	)	PUNCT
ejpam-6321	302	44	≤	≤	NUM
ejpam-6321	302	45	γ1g(x3k	γ1g(x3k	NOUN
ejpam-6321	302	46	,	,	PUNCT
ejpam-6321	302	47	x(3k+1	x(3k+1	ADV
ejpam-6321	302	48	)	)	PUNCT
ejpam-6321	302	49	,	,	PUNCT
ejpam-6321	302	50	x(3k+2	x(3k+2	NOUN
ejpam-6321	302	51	)	)	PUNCT
ejpam-6321	302	52	)	)	PUNCT
ejpam-6321	302	53	(	(	PUNCT
ejpam-6321	302	54	ii	ii	NOUN
ejpam-6321	302	55	)	)	PUNCT
ejpam-6321	302	56	.	.	PUNCT
ejpam-6321	303	1	if	if	SCONJ
ejpam-6321	303	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	303	3	)	)	PUNCT
ejpam-6321	303	4	,	,	PUNCT
ejpam-6321	303	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	303	6	)	)	PUNCT
ejpam-6321	303	7	,	,	PUNCT
ejpam-6321	303	8	x(3k+3	x(3k+3	NUM
ejpam-6321	303	9	)	)	PUNCT
ejpam-6321	303	10	)	)	PUNCT
ejpam-6321	303	11	is	be	AUX
ejpam-6321	303	12	a	a	DET
ejpam-6321	303	13	maximum	maximum	ADJ
ejpam-6321	303	14	term	term	NOUN
ejpam-6321	303	15	in	in	ADP
ejpam-6321	303	16	(	(	PUNCT
ejpam-6321	303	17	9	9	NUM
ejpam-6321	303	18	)	)	PUNCT
ejpam-6321	303	19	then	then	ADV
ejpam-6321	303	20	we	we	PRON
ejpam-6321	303	21	can	can	AUX
ejpam-6321	303	22	write	write	VERB
ejpam-6321	303	23	,	,	PUNCT
ejpam-6321	303	24	(	(	PUNCT
ejpam-6321	303	25	1−α4)g(x(3k+1	1−α4)g(x(3k+1	NUM
ejpam-6321	303	26	)	)	PUNCT
ejpam-6321	303	27	,	,	PUNCT
ejpam-6321	303	28	x(3k+2	x(3k+2	NOUN
ejpam-6321	303	29	)	)	PUNCT
ejpam-6321	303	30	,	,	PUNCT
ejpam-6321	303	31	x(3k+3	x(3k+3	NUM
ejpam-6321	303	32	)	)	PUNCT
ejpam-6321	303	33	)	)	PUNCT
ejpam-6321	304	1	≤	≤	NOUN
ejpam-6321	304	2	(	(	PUNCT
ejpam-6321	304	3	α1+α2	α1+α2	NOUN
ejpam-6321	304	4	+	+	NOUN
ejpam-6321	304	5	2α4)g(x3k	2α4)g(x3k	NUM
ejpam-6321	304	6	,	,	PUNCT
ejpam-6321	304	7	x(3k+1	x(3k+1	ADV
ejpam-6321	304	8	)	)	PUNCT
ejpam-6321	304	9	,	,	PUNCT
ejpam-6321	304	10	x(3k+2))+α5g(x(3k+1	x(3k+2))+α5g(x(3k+1	NUM
ejpam-6321	304	11	)	)	PUNCT
ejpam-6321	304	12	,	,	PUNCT
ejpam-6321	304	13	x(3k+2	x(3k+2	NOUN
ejpam-6321	304	14	)	)	PUNCT
ejpam-6321	304	15	,	,	PUNCT
ejpam-6321	304	16	x(3k+3	x(3k+3	NUM
ejpam-6321	304	17	)	)	PUNCT
ejpam-6321	304	18	)	)	PUNCT
ejpam-6321	305	1	(	(	PUNCT
ejpam-6321	305	2	1−	1−	NUM
ejpam-6321	305	3	α4	α4	NOUN
ejpam-6321	305	4	−	−	PROPN
ejpam-6321	305	5	α5)g(x(3k+1	α5)g(x(3k+1	PROPN
ejpam-6321	305	6	)	)	PUNCT
ejpam-6321	305	7	,	,	PUNCT
ejpam-6321	305	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	305	9	)	)	PUNCT
ejpam-6321	305	10	,	,	PUNCT
ejpam-6321	305	11	x(3k+3	x(3k+3	NUM
ejpam-6321	305	12	)	)	PUNCT
ejpam-6321	305	13	)	)	PUNCT
ejpam-6321	305	14	≤	≤	NOUN
ejpam-6321	305	15	(	(	PUNCT
ejpam-6321	305	16	α1	α1	PROPN
ejpam-6321	305	17	+	+	CCONJ
ejpam-6321	305	18	α2	α2	ADJ
ejpam-6321	305	19	+	+	CCONJ
ejpam-6321	305	20	2α4)g(x3k	2α4)g(x3k	NUM
ejpam-6321	305	21	,	,	PUNCT
ejpam-6321	305	22	x(3k+1	x(3k+1	ADV
ejpam-6321	305	23	)	)	PUNCT
ejpam-6321	305	24	,	,	PUNCT
ejpam-6321	305	25	x(3k+2	x(3k+2	NUM
ejpam-6321	305	26	)	)	PUNCT
ejpam-6321	305	27	)	)	PUNCT
ejpam-6321	306	1	m.	m.	NOUN
ejpam-6321	306	2	noorwali	noorwali	PROPN
ejpam-6321	306	3	et	et	PROPN
ejpam-6321	306	4	al	al	PROPN
ejpam-6321	306	5	.	.	PUNCT
ejpam-6321	306	6	/	/	SYM
ejpam-6321	306	7	eur	eur	PROPN
ejpam-6321	306	8	.	.	PUNCT
ejpam-6321	307	1	j.	j.	PROPN
ejpam-6321	307	2	pure	pure	PROPN
ejpam-6321	307	3	appl	appl	PROPN
ejpam-6321	307	4	.	.	PROPN
ejpam-6321	307	5	math	math	PROPN
ejpam-6321	307	6	,	,	PUNCT
ejpam-6321	307	7	18	18	NUM
ejpam-6321	307	8	(	(	PUNCT
ejpam-6321	307	9	3	3	NUM
ejpam-6321	307	10	)	)	PUNCT
ejpam-6321	307	11	(	(	PUNCT
ejpam-6321	307	12	2025	2025	NUM
ejpam-6321	307	13	)	)	PUNCT
ejpam-6321	307	14	,	,	PUNCT
ejpam-6321	307	15	6321	6321	NUM
ejpam-6321	307	16	12	12	NUM
ejpam-6321	307	17	of	of	ADP
ejpam-6321	307	18	27	27	NUM
ejpam-6321	307	19	≤	≤	NOUN
ejpam-6321	307	20	(	(	PUNCT
ejpam-6321	307	21	α1	α1	PROPN
ejpam-6321	307	22	+	+	CCONJ
ejpam-6321	307	23	α2	α2	ADJ
ejpam-6321	307	24	+	+	CCONJ
ejpam-6321	307	25	2α4	2α4	NUM
ejpam-6321	307	26	)	)	PUNCT
ejpam-6321	307	27	(	(	PUNCT
ejpam-6321	307	28	1−	1−	NUM
ejpam-6321	307	29	α4	α4	NOUN
ejpam-6321	307	30	−	−	NOUN
ejpam-6321	307	31	α5	α5	NOUN
ejpam-6321	307	32	)	)	PUNCT
ejpam-6321	307	33	g(x3k	g(x3k	NOUN
ejpam-6321	307	34	,	,	PUNCT
ejpam-6321	307	35	x(3k+1	x(3k+1	PRON
ejpam-6321	307	36	)	)	PUNCT
ejpam-6321	307	37	,	,	PUNCT
ejpam-6321	307	38	x(3k+2	x(3k+2	NOUN
ejpam-6321	307	39	)	)	PUNCT
ejpam-6321	307	40	)	)	PUNCT
ejpam-6321	307	41	,	,	PUNCT
ejpam-6321	307	42	where	where	SCONJ
ejpam-6321	307	43	γ2	γ2	NOUN
ejpam-6321	307	44	=	=	SYM
ejpam-6321	307	45	(	(	PUNCT
ejpam-6321	307	46	α1	α1	PROPN
ejpam-6321	307	47	+	+	CCONJ
ejpam-6321	307	48	α2	α2	ADJ
ejpam-6321	307	49	+	+	CCONJ
ejpam-6321	307	50	2α4	2α4	NUM
ejpam-6321	307	51	)	)	PUNCT
ejpam-6321	307	52	(	(	PUNCT
ejpam-6321	307	53	1−	1−	NUM
ejpam-6321	307	54	α4	α4	NOUN
ejpam-6321	307	55	−	−	NOUN
ejpam-6321	307	56	α5	α5	NOUN
ejpam-6321	307	57	)	)	PUNCT
ejpam-6321	307	58	<	<	X
ejpam-6321	307	59	1	1	NUM
ejpam-6321	307	60	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	307	61	)	)	PUNCT
ejpam-6321	307	62	,	,	PUNCT
ejpam-6321	307	63	x(3k+2	x(3k+2	NOUN
ejpam-6321	307	64	)	)	PUNCT
ejpam-6321	307	65	,	,	PUNCT
ejpam-6321	307	66	x(3k+3	x(3k+3	NUM
ejpam-6321	307	67	)	)	PUNCT
ejpam-6321	307	68	)	)	PUNCT
ejpam-6321	307	69	≤	≤	NUM
ejpam-6321	307	70	γ2g(x3k	γ2g(x3k	ADJ
ejpam-6321	307	71	,	,	PUNCT
ejpam-6321	307	72	x(3k+1	x(3k+1	ADV
ejpam-6321	307	73	)	)	PUNCT
ejpam-6321	307	74	,	,	PUNCT
ejpam-6321	307	75	x(3k+2	x(3k+2	NOUN
ejpam-6321	307	76	)	)	PUNCT
ejpam-6321	307	77	)	)	PUNCT
ejpam-6321	307	78	from	from	ADP
ejpam-6321	307	79	both	both	CCONJ
ejpam-6321	307	80	the	the	DET
ejpam-6321	307	81	cases	case	NOUN
ejpam-6321	307	82	we	we	PRON
ejpam-6321	307	83	get	get	VERB
ejpam-6321	307	84	,	,	PUNCT
ejpam-6321	307	85	∴	∴	NOUN
ejpam-6321	307	86	γ1	γ1	NOUN
ejpam-6321	307	87	=	=	SYM
ejpam-6321	307	88	γ2	γ2	PROPN
ejpam-6321	307	89	=	=	SYM
ejpam-6321	307	90	γ	γ	X
ejpam-6321	307	91	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	307	92	)	)	PUNCT
ejpam-6321	307	93	,	,	PUNCT
ejpam-6321	307	94	x(3k+2	x(3k+2	NOUN
ejpam-6321	307	95	)	)	PUNCT
ejpam-6321	307	96	,	,	PUNCT
ejpam-6321	307	97	x(3k+3	x(3k+3	NUM
ejpam-6321	307	98	)	)	PUNCT
ejpam-6321	307	99	)	)	PUNCT
ejpam-6321	308	1	≤	≤	NUM
ejpam-6321	308	2	γg(x3k	γg(x3k	NOUN
ejpam-6321	308	3	,	,	PUNCT
ejpam-6321	308	4	x(3k+1	x(3k+1	ADV
ejpam-6321	308	5	)	)	PUNCT
ejpam-6321	308	6	,	,	PUNCT
ejpam-6321	308	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	308	8	)	)	PUNCT
ejpam-6321	308	9	)	)	PUNCT
ejpam-6321	308	10	(	(	PUNCT
ejpam-6321	308	11	10	10	NUM
ejpam-6321	308	12	)	)	PUNCT
ejpam-6321	308	13	similarly	similarly	ADV
ejpam-6321	308	14	,	,	PUNCT
ejpam-6321	308	15	again	again	ADV
ejpam-6321	308	16	by	by	ADP
ejpam-6321	308	17	using	use	VERB
ejpam-6321	308	18	(	(	PUNCT
ejpam-6321	308	19	8)	8)	NUM
ejpam-6321	308	20	,	,	PUNCT
ejpam-6321	308	21	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	308	22	)	)	PUNCT
ejpam-6321	308	23	,	,	PUNCT
ejpam-6321	308	24	x(3k+3	x(3k+3	NUM
ejpam-6321	308	25	)	)	PUNCT
ejpam-6321	308	26	,	,	PUNCT
ejpam-6321	308	27	x(3k+4	x(3k+4	NUM
ejpam-6321	308	28	)	)	PUNCT
ejpam-6321	308	29	)	)	PUNCT
ejpam-6321	309	1	=	=	SYM
ejpam-6321	309	2	g(f1x(3k+1	g(f1x(3k+1	X
ejpam-6321	309	3	)	)	PUNCT
ejpam-6321	309	4	,	,	PUNCT
ejpam-6321	309	5	f2x(3k+2	f2x(3k+2	NUM
ejpam-6321	309	6	)	)	PUNCT
ejpam-6321	309	7	,	,	PUNCT
ejpam-6321	309	8	f3x(3k+3	f3x(3k+3	NUM
ejpam-6321	309	9	)	)	PUNCT
ejpam-6321	309	10	)	)	PUNCT
ejpam-6321	309	11	≤	≤	NUM
ejpam-6321	309	12	α1g(x(3k+1	α1g(x(3k+1	PROPN
ejpam-6321	309	13	)	)	PUNCT
ejpam-6321	309	14	,	,	PUNCT
ejpam-6321	309	15	x(3k+2	x(3k+2	NOUN
ejpam-6321	309	16	)	)	PUNCT
ejpam-6321	309	17	,	,	PUNCT
ejpam-6321	309	18	x(3k+3	x(3k+3	NUM
ejpam-6321	309	19	)	)	PUNCT
ejpam-6321	309	20	)	)	PUNCT
ejpam-6321	310	1	+	+	CCONJ
ejpam-6321	310	2	α2g(x(3k+2	α2g(x(3k+2	PROPN
ejpam-6321	310	3	)	)	PUNCT
ejpam-6321	310	4	,	,	PUNCT
ejpam-6321	310	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	310	6	)	)	PUNCT
ejpam-6321	310	7	,	,	PUNCT
ejpam-6321	310	8	f2x(3k+2	f2x(3k+2	NOUN
ejpam-6321	310	9	)	)	PUNCT
ejpam-6321	310	10	)	)	PUNCT
ejpam-6321	311	1	+	+	CCONJ
ejpam-6321	311	2	α3g(f1x(3k+1	α3g(f1x(3k+1	NUM
ejpam-6321	311	3	)	)	PUNCT
ejpam-6321	311	4	,	,	PUNCT
ejpam-6321	311	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	311	6	)	)	PUNCT
ejpam-6321	311	7	,	,	PUNCT
ejpam-6321	311	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	311	9	)	)	PUNCT
ejpam-6321	311	10	)	)	PUNCT
ejpam-6321	312	1	+	+	CCONJ
ejpam-6321	312	2	α4[g(f1x(3k+1	α4[g(f1x(3k+1	X
ejpam-6321	312	3	)	)	PUNCT
ejpam-6321	312	4	,	,	PUNCT
ejpam-6321	312	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	312	6	)	)	PUNCT
ejpam-6321	312	7	,	,	PUNCT
ejpam-6321	312	8	x(3k+3	x(3k+3	NUM
ejpam-6321	312	9	)	)	PUNCT
ejpam-6321	312	10	)	)	PUNCT
ejpam-6321	313	1	+	+	PUNCT
ejpam-6321	313	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	313	3	)	)	PUNCT
ejpam-6321	313	4	,	,	PUNCT
ejpam-6321	313	5	f2x(3k+2	f2x(3k+2	NUM
ejpam-6321	313	6	)	)	PUNCT
ejpam-6321	313	7	,	,	PUNCT
ejpam-6321	313	8	x(3k+3	x(3k+3	NUM
ejpam-6321	313	9	)	)	PUNCT
ejpam-6321	313	10	)	)	PUNCT
ejpam-6321	314	1	+	+	PUNCT
ejpam-6321	314	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	314	3	)	)	PUNCT
ejpam-6321	314	4	,	,	PUNCT
ejpam-6321	314	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	314	6	)	)	PUNCT
ejpam-6321	314	7	,	,	PUNCT
ejpam-6321	314	8	f3x(3k+3	f3x(3k+3	NUM
ejpam-6321	314	9	)	)	PUNCT
ejpam-6321	314	10	)	)	PUNCT
ejpam-6321	314	11	]	]	PUNCT
ejpam-6321	315	1	+	+	CCONJ
ejpam-6321	315	2	α5max	α5max	PROPN
ejpam-6321	315	3			VERB
ejpam-6321	315	4	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	315	5	)	)	PUNCT
ejpam-6321	315	6	,	,	PUNCT
ejpam-6321	315	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	8	)	)	PUNCT
ejpam-6321	315	9	,	,	PUNCT
ejpam-6321	315	10	f2x(3k+2	f2x(3k+2	NOUN
ejpam-6321	315	11	)	)	PUNCT
ejpam-6321	315	12	)	)	PUNCT
ejpam-6321	315	13	,	,	PUNCT
ejpam-6321	315	14	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	315	15	)	)	PUNCT
ejpam-6321	315	16	,	,	PUNCT
ejpam-6321	315	17	f1x(3k+1	f1x(3k+1	NOUN
ejpam-6321	315	18	)	)	PUNCT
ejpam-6321	315	19	,	,	PUNCT
ejpam-6321	315	20	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	21	)	)	PUNCT
ejpam-6321	315	22	)	)	PUNCT
ejpam-6321	315	23	,	,	PUNCT
ejpam-6321	315	24	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	315	25	)	)	PUNCT
ejpam-6321	315	26	,	,	PUNCT
ejpam-6321	315	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	28	)	)	PUNCT
ejpam-6321	315	29	,	,	PUNCT
ejpam-6321	315	30	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	31	)	)	PUNCT
ejpam-6321	315	32	)	)	PUNCT
ejpam-6321	315	33	,	,	PUNCT
ejpam-6321	315	34	g(f2x(3k+2	g(f2x(3k+2	NOUN
ejpam-6321	315	35	)	)	PUNCT
ejpam-6321	315	36	,	,	PUNCT
ejpam-6321	315	37	f2x(3k+2	f2x(3k+2	NUM
ejpam-6321	315	38	)	)	PUNCT
ejpam-6321	315	39	,	,	PUNCT
ejpam-6321	315	40	x(3k+3	x(3k+3	NUM
ejpam-6321	315	41	)	)	PUNCT
ejpam-6321	315	42	)	)	PUNCT
ejpam-6321	315	43	,	,	PUNCT
ejpam-6321	315	44	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	315	45	)	)	PUNCT
ejpam-6321	315	46	,	,	PUNCT
ejpam-6321	315	47	x(3k+1	x(3k+1	CCONJ
ejpam-6321	315	48	)	)	PUNCT
ejpam-6321	315	49	,	,	PUNCT
ejpam-6321	315	50	x(3k+1	x(3k+1	CCONJ
ejpam-6321	315	51	)	)	PUNCT
ejpam-6321	315	52	)	)	PUNCT
ejpam-6321	315	53	,	,	PUNCT
ejpam-6321	315	54	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	315	55	)	)	PUNCT
ejpam-6321	315	56	,	,	PUNCT
ejpam-6321	315	57	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	315	58	)	)	PUNCT
ejpam-6321	315	59	,	,	PUNCT
ejpam-6321	315	60	x(3k+1	x(3k+1	CCONJ
ejpam-6321	315	61	)	)	PUNCT
ejpam-6321	315	62	)	)	PUNCT
ejpam-6321	315	63	,	,	PUNCT
ejpam-6321	315	64	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	315	65	)	)	PUNCT
ejpam-6321	315	66	,	,	PUNCT
ejpam-6321	315	67	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	68	)	)	PUNCT
ejpam-6321	315	69	,	,	PUNCT
ejpam-6321	315	70	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	315	71	)	)	PUNCT
ejpam-6321	315	72	)	)	PUNCT
ejpam-6321	315	73			NOUN
ejpam-6321	315	74	=	=	SYM
ejpam-6321	315	75	α1g(x(3k+1	α1g(x(3k+1	PROPN
ejpam-6321	315	76	)	)	PUNCT
ejpam-6321	315	77	,	,	PUNCT
ejpam-6321	315	78	x(3k+2	x(3k+2	NOUN
ejpam-6321	315	79	)	)	PUNCT
ejpam-6321	315	80	,	,	PUNCT
ejpam-6321	315	81	x(3k+3	x(3k+3	NUM
ejpam-6321	315	82	)	)	PUNCT
ejpam-6321	315	83	)	)	PUNCT
ejpam-6321	316	1	+	+	CCONJ
ejpam-6321	317	1	α2g(x(3k+1	α2g(x(3k+1	NOUN
ejpam-6321	317	2	)	)	PUNCT
ejpam-6321	317	3	,	,	PUNCT
ejpam-6321	317	4	x(3k+2	x(3k+2	NOUN
ejpam-6321	317	5	)	)	PUNCT
ejpam-6321	317	6	,	,	PUNCT
ejpam-6321	317	7	x(3k+3	x(3k+3	NUM
ejpam-6321	317	8	)	)	PUNCT
ejpam-6321	317	9	)	)	PUNCT
ejpam-6321	318	1	+	+	CCONJ
ejpam-6321	318	2	α3g(x(3k+2	α3g(x(3k+2	X
ejpam-6321	318	3	)	)	PUNCT
ejpam-6321	318	4	,	,	PUNCT
ejpam-6321	318	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	318	6	)	)	PUNCT
ejpam-6321	318	7	,	,	PUNCT
ejpam-6321	318	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	318	9	)	)	PUNCT
ejpam-6321	318	10	)	)	PUNCT
ejpam-6321	319	1	+	+	CCONJ
ejpam-6321	320	1	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	320	2	)	)	PUNCT
ejpam-6321	320	3	,	,	PUNCT
ejpam-6321	320	4	x(3k+2	x(3k+2	NOUN
ejpam-6321	320	5	)	)	PUNCT
ejpam-6321	320	6	,	,	PUNCT
ejpam-6321	320	7	x(3k+3	x(3k+3	NUM
ejpam-6321	320	8	)	)	PUNCT
ejpam-6321	320	9	)	)	PUNCT
ejpam-6321	321	1	+	+	PUNCT
ejpam-6321	321	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	321	3	)	)	PUNCT
ejpam-6321	321	4	,	,	PUNCT
ejpam-6321	321	5	x(3k+3	x(3k+3	NUM
ejpam-6321	321	6	)	)	PUNCT
ejpam-6321	321	7	,	,	PUNCT
ejpam-6321	321	8	x(3k+4	x(3k+4	NUM
ejpam-6321	321	9	)	)	PUNCT
ejpam-6321	321	10	)	)	PUNCT
ejpam-6321	322	1	+	+	PUNCT
ejpam-6321	322	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	322	3	)	)	PUNCT
ejpam-6321	322	4	,	,	PUNCT
ejpam-6321	322	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	322	6	)	)	PUNCT
ejpam-6321	322	7	,	,	PUNCT
ejpam-6321	322	8	x(3k+3	x(3k+3	NUM
ejpam-6321	322	9	)	)	PUNCT
ejpam-6321	322	10	)	)	PUNCT
ejpam-6321	322	11	]	]	PUNCT
ejpam-6321	323	1	+	+	CCONJ
ejpam-6321	323	2	α5max	α5max	PROPN
ejpam-6321	323	3			VERB
ejpam-6321	323	4	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	323	5	)	)	PUNCT
ejpam-6321	323	6	,	,	PUNCT
ejpam-6321	323	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	8	)	)	PUNCT
ejpam-6321	323	9	,	,	PUNCT
ejpam-6321	323	10	x(3k+3	x(3k+3	NUM
ejpam-6321	323	11	)	)	PUNCT
ejpam-6321	323	12	)	)	PUNCT
ejpam-6321	323	13	,	,	PUNCT
ejpam-6321	323	14	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	323	15	)	)	PUNCT
ejpam-6321	323	16	,	,	PUNCT
ejpam-6321	323	17	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	18	)	)	PUNCT
ejpam-6321	323	19	,	,	PUNCT
ejpam-6321	323	20	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	21	)	)	PUNCT
ejpam-6321	323	22	)	)	PUNCT
ejpam-6321	323	23	,	,	PUNCT
ejpam-6321	323	24	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	323	25	)	)	PUNCT
ejpam-6321	323	26	,	,	PUNCT
ejpam-6321	323	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	28	)	)	PUNCT
ejpam-6321	323	29	,	,	PUNCT
ejpam-6321	323	30	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	31	)	)	PUNCT
ejpam-6321	323	32	)	)	PUNCT
ejpam-6321	323	33	,	,	PUNCT
ejpam-6321	323	34	g(x(3k+3	g(x(3k+3	PROPN
ejpam-6321	323	35	)	)	PUNCT
ejpam-6321	323	36	,	,	PUNCT
ejpam-6321	323	37	x(3k+3	x(3k+3	NUM
ejpam-6321	323	38	)	)	PUNCT
ejpam-6321	323	39	,	,	PUNCT
ejpam-6321	323	40	x(3k+3	x(3k+3	NUM
ejpam-6321	323	41	)	)	PUNCT
ejpam-6321	323	42	)	)	PUNCT
ejpam-6321	323	43	,	,	PUNCT
ejpam-6321	323	44	g(x(3k+4	g(x(3k+4	PROPN
ejpam-6321	323	45	)	)	PUNCT
ejpam-6321	323	46	,	,	PUNCT
ejpam-6321	323	47	x(3k+3	x(3k+3	NUM
ejpam-6321	323	48	)	)	PUNCT
ejpam-6321	323	49	,	,	PUNCT
ejpam-6321	323	50	x(3k+3	x(3k+3	NUM
ejpam-6321	323	51	)	)	PUNCT
ejpam-6321	323	52	)	)	PUNCT
ejpam-6321	323	53	,	,	PUNCT
ejpam-6321	323	54	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	323	55	)	)	PUNCT
ejpam-6321	323	56	,	,	PUNCT
ejpam-6321	323	57	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	58	)	)	PUNCT
ejpam-6321	323	59	,	,	PUNCT
ejpam-6321	323	60	x(3k+1	x(3k+1	CCONJ
ejpam-6321	323	61	)	)	PUNCT
ejpam-6321	323	62	)	)	PUNCT
ejpam-6321	323	63	,	,	PUNCT
ejpam-6321	323	64	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	323	65	)	)	PUNCT
ejpam-6321	323	66	,	,	PUNCT
ejpam-6321	323	67	x(3k+2	x(3k+2	NOUN
ejpam-6321	323	68	)	)	PUNCT
ejpam-6321	323	69	,	,	PUNCT
ejpam-6321	323	70	x(3k+3	x(3k+3	NUM
ejpam-6321	323	71	)	)	PUNCT
ejpam-6321	323	72	)	)	PUNCT
ejpam-6321	324	1			ADV
ejpam-6321	324	2	after	after	ADP
ejpam-6321	324	3	applying	apply	VERB
ejpam-6321	324	4	the	the	DET
ejpam-6321	324	5	simplification	simplification	NOUN
ejpam-6321	324	6	process	process	NOUN
ejpam-6321	324	7	utilizing	utilize	VERB
ejpam-6321	324	8	the	the	DET
ejpam-6321	324	9	definition	definition	NOUN
ejpam-6321	324	10	of	of	ADP
ejpam-6321	324	11	gm	gm	PROPN
ejpam-6321	324	12	-	-	PUNCT
ejpam-6321	324	13	space	space	NOUN
ejpam-6321	324	14	we	we	PRON
ejpam-6321	324	15	have	have	AUX
ejpam-6321	324	16	achieved	achieve	VERB
ejpam-6321	324	17	the	the	DET
ejpam-6321	324	18	following	following	ADJ
ejpam-6321	324	19	result	result	NOUN
ejpam-6321	324	20	,	,	PUNCT
ejpam-6321	324	21	≤	≤	NUM
ejpam-6321	324	22	(	(	PUNCT
ejpam-6321	324	23	α1	α1	PROPN
ejpam-6321	324	24	+	+	X
ejpam-6321	324	25	α2)g(x(3k+1	α2)g(x(3k+1	NOUN
ejpam-6321	324	26	)	)	PUNCT
ejpam-6321	324	27	,	,	PUNCT
ejpam-6321	324	28	x(3k+2	x(3k+2	NOUN
ejpam-6321	324	29	)	)	PUNCT
ejpam-6321	324	30	,	,	PUNCT
ejpam-6321	324	31	x(3k+3	x(3k+3	NUM
ejpam-6321	324	32	)	)	PUNCT
ejpam-6321	324	33	)	)	PUNCT
ejpam-6321	325	1	+	+	CCONJ
ejpam-6321	326	1	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	326	2	)	)	PUNCT
ejpam-6321	326	3	,	,	PUNCT
ejpam-6321	326	4	x(3k+2	x(3k+2	NOUN
ejpam-6321	326	5	)	)	PUNCT
ejpam-6321	326	6	,	,	PUNCT
ejpam-6321	326	7	x(3k+3	x(3k+3	NUM
ejpam-6321	326	8	)	)	PUNCT
ejpam-6321	326	9	)	)	PUNCT
ejpam-6321	327	1	+	+	PUNCT
ejpam-6321	327	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	327	3	)	)	PUNCT
ejpam-6321	327	4	,	,	PUNCT
ejpam-6321	327	5	x(3k+3	x(3k+3	NUM
ejpam-6321	327	6	)	)	PUNCT
ejpam-6321	327	7	,	,	PUNCT
ejpam-6321	327	8	x(3k+4	x(3k+4	NUM
ejpam-6321	327	9	)	)	PUNCT
ejpam-6321	327	10	)	)	PUNCT
ejpam-6321	328	1	+	+	PUNCT
ejpam-6321	328	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	328	3	)	)	PUNCT
ejpam-6321	328	4	,	,	PUNCT
ejpam-6321	328	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	328	6	)	)	PUNCT
ejpam-6321	328	7	,	,	PUNCT
ejpam-6321	328	8	x(3k+3	x(3k+3	NUM
ejpam-6321	328	9	)	)	PUNCT
ejpam-6321	328	10	)	)	PUNCT
ejpam-6321	328	11	]	]	PUNCT
ejpam-6321	329	1	+	+	CCONJ
ejpam-6321	329	2	α5max	α5max	PROPN
ejpam-6321	329	3	{	{	PUNCT
ejpam-6321	329	4	g(x(3k+1	g(x(3k+1	PROPN
ejpam-6321	329	5	)	)	PUNCT
ejpam-6321	329	6	,	,	PUNCT
ejpam-6321	329	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	329	8	)	)	PUNCT
ejpam-6321	329	9	,	,	PUNCT
ejpam-6321	329	10	x(3k+3	x(3k+3	NUM
ejpam-6321	329	11	)	)	PUNCT
ejpam-6321	329	12	)	)	PUNCT
ejpam-6321	329	13	,	,	PUNCT
ejpam-6321	329	14	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	329	15	)	)	PUNCT
ejpam-6321	329	16	,	,	PUNCT
ejpam-6321	329	17	x(3k+2	x(3k+2	NOUN
ejpam-6321	329	18	)	)	PUNCT
ejpam-6321	329	19	,	,	PUNCT
ejpam-6321	329	20	x(3k+3	x(3k+3	NUM
ejpam-6321	329	21	)	)	PUNCT
ejpam-6321	329	22	)	)	PUNCT
ejpam-6321	329	23	,	,	PUNCT
ejpam-6321	329	24	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	329	25	)	)	PUNCT
ejpam-6321	329	26	,	,	PUNCT
ejpam-6321	329	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	329	28	)	)	PUNCT
ejpam-6321	329	29	,	,	PUNCT
ejpam-6321	329	30	x(3k+3	x(3k+3	NUM
ejpam-6321	329	31	)	)	PUNCT
ejpam-6321	329	32	)	)	PUNCT
ejpam-6321	329	33	,	,	PUNCT
ejpam-6321	329	34	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	329	35	)	)	PUNCT
ejpam-6321	329	36	,	,	PUNCT
ejpam-6321	329	37	x(3k+3	x(3k+3	NUM
ejpam-6321	329	38	)	)	PUNCT
ejpam-6321	329	39	,	,	PUNCT
ejpam-6321	329	40	x(3k+4	x(3k+4	NUM
ejpam-6321	329	41	)	)	PUNCT
ejpam-6321	329	42	)	)	PUNCT
ejpam-6321	329	43	}	}	PUNCT
ejpam-6321	329	44	(	(	PUNCT
ejpam-6321	329	45	1−	1−	NUM
ejpam-6321	329	46	α4)g(x(3k+2	α4)g(x(3k+2	NUM
ejpam-6321	329	47	)	)	PUNCT
ejpam-6321	329	48	,	,	PUNCT
ejpam-6321	329	49	x(3k+3	x(3k+3	NUM
ejpam-6321	329	50	)	)	PUNCT
ejpam-6321	329	51	,	,	PUNCT
ejpam-6321	329	52	x(3k+4	x(3k+4	NUM
ejpam-6321	329	53	)	)	PUNCT
ejpam-6321	329	54	)	)	PUNCT
ejpam-6321	329	55	≤	≤	NUM
ejpam-6321	329	56	(	(	PUNCT
ejpam-6321	329	57	α1	α1	PROPN
ejpam-6321	329	58	+	+	CCONJ
ejpam-6321	329	59	α2	α2	ADJ
ejpam-6321	329	60	+	+	CCONJ
ejpam-6321	329	61	2α4)g(x(3k+1	2α4)g(x(3k+1	NUM
ejpam-6321	329	62	)	)	PUNCT
ejpam-6321	329	63	,	,	PUNCT
ejpam-6321	329	64	x(3k+2	x(3k+2	NOUN
ejpam-6321	329	65	)	)	PUNCT
ejpam-6321	329	66	,	,	PUNCT
ejpam-6321	329	67	x(3k+3	x(3k+3	NUM
ejpam-6321	329	68	)	)	PUNCT
ejpam-6321	329	69	)	)	PUNCT
ejpam-6321	330	1	+	+	CCONJ
ejpam-6321	330	2	α5max	α5max	PROPN
ejpam-6321	330	3	{	{	PUNCT
ejpam-6321	330	4	g(x(3k+1	g(x(3k+1	PROPN
ejpam-6321	330	5	)	)	PUNCT
ejpam-6321	330	6	,	,	PUNCT
ejpam-6321	330	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	330	8	)	)	PUNCT
ejpam-6321	330	9	,	,	PUNCT
ejpam-6321	330	10	x(3k+3	x(3k+3	NUM
ejpam-6321	330	11	)	)	PUNCT
ejpam-6321	330	12	)	)	PUNCT
ejpam-6321	330	13	,	,	PUNCT
ejpam-6321	330	14	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	330	15	)	)	PUNCT
ejpam-6321	330	16	,	,	PUNCT
ejpam-6321	330	17	x(3k+3	x(3k+3	NUM
ejpam-6321	330	18	)	)	PUNCT
ejpam-6321	330	19	,	,	PUNCT
ejpam-6321	330	20	x(3k+4	x(3k+4	NUM
ejpam-6321	330	21	)	)	PUNCT
ejpam-6321	330	22	)	)	PUNCT
ejpam-6321	330	23	}	}	PUNCT
ejpam-6321	330	24	(	(	PUNCT
ejpam-6321	330	25	11	11	NUM
ejpam-6321	330	26	)	)	PUNCT
ejpam-6321	330	27	now	now	ADV
ejpam-6321	330	28	,	,	PUNCT
ejpam-6321	330	29	there	there	PRON
ejpam-6321	330	30	are	be	VERB
ejpam-6321	330	31	two	two	NUM
ejpam-6321	330	32	cases	case	NOUN
ejpam-6321	330	33	:	:	PUNCT
ejpam-6321	330	34	(	(	PUNCT
ejpam-6321	330	35	i	i	NOUN
ejpam-6321	330	36	)	)	PUNCT
ejpam-6321	330	37	.	.	PUNCT
ejpam-6321	331	1	if	if	SCONJ
ejpam-6321	331	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	331	3	)	)	PUNCT
ejpam-6321	331	4	,	,	PUNCT
ejpam-6321	331	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	331	6	)	)	PUNCT
ejpam-6321	331	7	,	,	PUNCT
ejpam-6321	331	8	x(3k+3	x(3k+3	NUM
ejpam-6321	331	9	)	)	PUNCT
ejpam-6321	331	10	)	)	PUNCT
ejpam-6321	331	11	is	be	AUX
ejpam-6321	331	12	a	a	DET
ejpam-6321	331	13	maximum	maximum	ADJ
ejpam-6321	331	14	term	term	NOUN
ejpam-6321	331	15	in	in	ADP
ejpam-6321	331	16	(	(	PUNCT
ejpam-6321	331	17	11	11	NUM
ejpam-6321	331	18	)	)	PUNCT
ejpam-6321	331	19	then	then	ADV
ejpam-6321	331	20	we	we	PRON
ejpam-6321	331	21	can	can	AUX
ejpam-6321	331	22	write	write	VERB
ejpam-6321	331	23	,	,	PUNCT
ejpam-6321	331	24	(	(	PUNCT
ejpam-6321	331	25	1−α4)g(x(3k+2	1−α4)g(x(3k+2	NUM
ejpam-6321	331	26	)	)	PUNCT
ejpam-6321	331	27	,	,	PUNCT
ejpam-6321	331	28	x(3k+3	x(3k+3	NUM
ejpam-6321	331	29	)	)	PUNCT
ejpam-6321	331	30	,	,	PUNCT
ejpam-6321	331	31	x(3k+4	x(3k+4	NUM
ejpam-6321	331	32	)	)	PUNCT
ejpam-6321	331	33	)	)	PUNCT
ejpam-6321	332	1	≤	≤	NOUN
ejpam-6321	332	2	(	(	PUNCT
ejpam-6321	332	3	α1+α2	α1+α2	NOUN
ejpam-6321	332	4	+	+	NOUN
ejpam-6321	332	5	2α4)g(x(3k+1	2α4)g(x(3k+1	NUM
ejpam-6321	332	6	)	)	PUNCT
ejpam-6321	332	7	,	,	PUNCT
ejpam-6321	332	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	332	9	)	)	PUNCT
ejpam-6321	332	10	,	,	PUNCT
ejpam-6321	332	11	x(3k+3))+α5g(x(3k+1	x(3k+3))+α5g(x(3k+1	NUM
ejpam-6321	332	12	)	)	PUNCT
ejpam-6321	332	13	,	,	PUNCT
ejpam-6321	332	14	x(3k+2	x(3k+2	NOUN
ejpam-6321	332	15	)	)	PUNCT
ejpam-6321	332	16	,	,	PUNCT
ejpam-6321	332	17	x(3k+3	x(3k+3	NUM
ejpam-6321	332	18	)	)	PUNCT
ejpam-6321	332	19	)	)	PUNCT
ejpam-6321	333	1	(	(	PUNCT
ejpam-6321	333	2	α1	α1	PROPN
ejpam-6321	333	3	+	+	CCONJ
ejpam-6321	333	4	α2	α2	ADJ
ejpam-6321	334	1	+	+	CCONJ
ejpam-6321	335	1	2α4	2α4	NUM
ejpam-6321	336	1	+	+	CCONJ
ejpam-6321	336	2	α5)g(x(3k+1	α5)g(x(3k+1	NOUN
ejpam-6321	336	3	)	)	PUNCT
ejpam-6321	336	4	,	,	PUNCT
ejpam-6321	336	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	336	6	)	)	PUNCT
ejpam-6321	336	7	,	,	PUNCT
ejpam-6321	336	8	x(3k+3	x(3k+3	NUM
ejpam-6321	336	9	)	)	PUNCT
ejpam-6321	336	10	)	)	PUNCT
ejpam-6321	337	1	≤	≤	NOUN
ejpam-6321	337	2	(	(	PUNCT
ejpam-6321	337	3	α1	α1	PROPN
ejpam-6321	337	4	+	+	CCONJ
ejpam-6321	337	5	α2	α2	ADJ
ejpam-6321	338	1	+	+	CCONJ
ejpam-6321	339	1	2α4	2α4	NUM
ejpam-6321	340	1	+	+	SYM
ejpam-6321	340	2	α5	α5	NOUN
ejpam-6321	340	3	)	)	PUNCT
ejpam-6321	340	4	(	(	PUNCT
ejpam-6321	340	5	1−	1−	NUM
ejpam-6321	340	6	α4	α4	NOUN
ejpam-6321	340	7	)	)	PUNCT
ejpam-6321	340	8	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	340	9	)	)	PUNCT
ejpam-6321	340	10	,	,	PUNCT
ejpam-6321	340	11	x(3k+2	x(3k+2	NOUN
ejpam-6321	340	12	)	)	PUNCT
ejpam-6321	340	13	,	,	PUNCT
ejpam-6321	340	14	x(3k+3	x(3k+3	NUM
ejpam-6321	340	15	)	)	PUNCT
ejpam-6321	340	16	)	)	PUNCT
ejpam-6321	340	17	,	,	PUNCT
ejpam-6321	341	1	where	where	SCONJ
ejpam-6321	341	2	γ1	γ1	PROPN
ejpam-6321	341	3	=	=	SYM
ejpam-6321	341	4	(	(	PUNCT
ejpam-6321	341	5	α1	α1	PROPN
ejpam-6321	341	6	+	+	CCONJ
ejpam-6321	341	7	α2	α2	ADJ
ejpam-6321	341	8	+	+	CCONJ
ejpam-6321	341	9	2α4	2α4	NUM
ejpam-6321	341	10	+	+	SYM
ejpam-6321	341	11	α5	α5	NOUN
ejpam-6321	341	12	)	)	PUNCT
ejpam-6321	341	13	(	(	PUNCT
ejpam-6321	341	14	1−	1−	NUM
ejpam-6321	341	15	α4	α4	NOUN
ejpam-6321	341	16	)	)	PUNCT
ejpam-6321	341	17	<	<	X
ejpam-6321	341	18	1	1	NUM
ejpam-6321	341	19	m.	m.	NOUN
ejpam-6321	341	20	noorwali	noorwali	PROPN
ejpam-6321	341	21	et	et	PROPN
ejpam-6321	341	22	al	al	PROPN
ejpam-6321	341	23	.	.	PUNCT
ejpam-6321	341	24	/	/	SYM
ejpam-6321	341	25	eur	eur	PROPN
ejpam-6321	341	26	.	.	PUNCT
ejpam-6321	342	1	j.	j.	PROPN
ejpam-6321	342	2	pure	pure	PROPN
ejpam-6321	342	3	appl	appl	PROPN
ejpam-6321	342	4	.	.	PROPN
ejpam-6321	342	5	math	math	PROPN
ejpam-6321	342	6	,	,	PUNCT
ejpam-6321	342	7	18	18	NUM
ejpam-6321	342	8	(	(	PUNCT
ejpam-6321	342	9	3	3	NUM
ejpam-6321	342	10	)	)	PUNCT
ejpam-6321	342	11	(	(	PUNCT
ejpam-6321	342	12	2025	2025	NUM
ejpam-6321	342	13	)	)	PUNCT
ejpam-6321	342	14	,	,	PUNCT
ejpam-6321	342	15	6321	6321	NUM
ejpam-6321	342	16	13	13	NUM
ejpam-6321	342	17	of	of	ADP
ejpam-6321	342	18	27	27	NUM
ejpam-6321	342	19	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	342	20	)	)	PUNCT
ejpam-6321	342	21	,	,	PUNCT
ejpam-6321	342	22	x(3k+3	x(3k+3	NUM
ejpam-6321	342	23	)	)	PUNCT
ejpam-6321	342	24	,	,	PUNCT
ejpam-6321	342	25	x(3k+4	x(3k+4	NUM
ejpam-6321	342	26	)	)	PUNCT
ejpam-6321	342	27	)	)	PUNCT
ejpam-6321	343	1	≤	≤	NUM
ejpam-6321	343	2	γ1g(x(3k+1	γ1g(x(3k+1	PROPN
ejpam-6321	343	3	)	)	PUNCT
ejpam-6321	343	4	,	,	PUNCT
ejpam-6321	343	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	343	6	)	)	PUNCT
ejpam-6321	343	7	,	,	PUNCT
ejpam-6321	343	8	x(3k+3	x(3k+3	NUM
ejpam-6321	343	9	)	)	PUNCT
ejpam-6321	343	10	)	)	PUNCT
ejpam-6321	343	11	(	(	PUNCT
ejpam-6321	343	12	ii	ii	NOUN
ejpam-6321	343	13	)	)	PUNCT
ejpam-6321	343	14	.	.	PUNCT
ejpam-6321	344	1	if	if	SCONJ
ejpam-6321	344	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	344	3	)	)	PUNCT
ejpam-6321	344	4	,	,	PUNCT
ejpam-6321	344	5	x(3k+3	x(3k+3	NUM
ejpam-6321	344	6	)	)	PUNCT
ejpam-6321	344	7	,	,	PUNCT
ejpam-6321	344	8	x(3k+4	x(3k+4	NUM
ejpam-6321	344	9	)	)	PUNCT
ejpam-6321	344	10	)	)	PUNCT
ejpam-6321	344	11	is	be	AUX
ejpam-6321	344	12	a	a	DET
ejpam-6321	344	13	maximum	maximum	ADJ
ejpam-6321	344	14	term	term	NOUN
ejpam-6321	344	15	in	in	ADP
ejpam-6321	344	16	(	(	PUNCT
ejpam-6321	344	17	11	11	NUM
ejpam-6321	344	18	)	)	PUNCT
ejpam-6321	344	19	then	then	ADV
ejpam-6321	344	20	we	we	PRON
ejpam-6321	344	21	can	can	AUX
ejpam-6321	344	22	write	write	VERB
ejpam-6321	344	23	,	,	PUNCT
ejpam-6321	344	24	(	(	PUNCT
ejpam-6321	344	25	1−α4)g(x(3k+2	1−α4)g(x(3k+2	NUM
ejpam-6321	344	26	)	)	PUNCT
ejpam-6321	344	27	,	,	PUNCT
ejpam-6321	344	28	x(3k+3	x(3k+3	NUM
ejpam-6321	344	29	)	)	PUNCT
ejpam-6321	344	30	,	,	PUNCT
ejpam-6321	344	31	x(3k+4	x(3k+4	NUM
ejpam-6321	344	32	)	)	PUNCT
ejpam-6321	344	33	)	)	PUNCT
ejpam-6321	345	1	≤	≤	NOUN
ejpam-6321	345	2	(	(	PUNCT
ejpam-6321	345	3	α1+α2	α1+α2	NOUN
ejpam-6321	345	4	+	+	NOUN
ejpam-6321	345	5	2α4)g(x(3k+1	2α4)g(x(3k+1	NUM
ejpam-6321	345	6	)	)	PUNCT
ejpam-6321	345	7	,	,	PUNCT
ejpam-6321	345	8	x(3k+2	x(3k+2	NOUN
ejpam-6321	345	9	)	)	PUNCT
ejpam-6321	345	10	,	,	PUNCT
ejpam-6321	345	11	x(3k+3))+α5g(x(3k+2	x(3k+3))+α5g(x(3k+2	NUM
ejpam-6321	345	12	)	)	PUNCT
ejpam-6321	345	13	,	,	PUNCT
ejpam-6321	345	14	x(3k+3	x(3k+3	NUM
ejpam-6321	345	15	)	)	PUNCT
ejpam-6321	345	16	,	,	PUNCT
ejpam-6321	345	17	x(3k+4	x(3k+4	NUM
ejpam-6321	345	18	)	)	PUNCT
ejpam-6321	345	19	)	)	PUNCT
ejpam-6321	346	1	(	(	PUNCT
ejpam-6321	346	2	1−	1−	NUM
ejpam-6321	346	3	α4	α4	NOUN
ejpam-6321	346	4	−	−	PROPN
ejpam-6321	346	5	α5)g(x(3k+2	α5)g(x(3k+2	PROPN
ejpam-6321	346	6	)	)	PUNCT
ejpam-6321	346	7	,	,	PUNCT
ejpam-6321	346	8	x(3k+3	x(3k+3	NUM
ejpam-6321	346	9	)	)	PUNCT
ejpam-6321	346	10	,	,	PUNCT
ejpam-6321	346	11	x(3k+4	x(3k+4	NUM
ejpam-6321	346	12	)	)	PUNCT
ejpam-6321	346	13	)	)	PUNCT
ejpam-6321	347	1	≤	≤	NUM
ejpam-6321	347	2	(	(	PUNCT
ejpam-6321	347	3	α1	α1	PROPN
ejpam-6321	347	4	+	+	CCONJ
ejpam-6321	347	5	α2	α2	ADJ
ejpam-6321	347	6	+	+	CCONJ
ejpam-6321	347	7	2α4)g(x(3k+1	2α4)g(x(3k+1	NUM
ejpam-6321	347	8	)	)	PUNCT
ejpam-6321	347	9	,	,	PUNCT
ejpam-6321	347	10	x(3k+2	x(3k+2	NOUN
ejpam-6321	347	11	)	)	PUNCT
ejpam-6321	347	12	,	,	PUNCT
ejpam-6321	347	13	x(3k+3	x(3k+3	NUM
ejpam-6321	347	14	)	)	PUNCT
ejpam-6321	347	15	)	)	PUNCT
ejpam-6321	347	16	≤	≤	NOUN
ejpam-6321	347	17	(	(	PUNCT
ejpam-6321	347	18	α1	α1	PROPN
ejpam-6321	347	19	+	+	CCONJ
ejpam-6321	347	20	α2	α2	ADJ
ejpam-6321	347	21	+	+	CCONJ
ejpam-6321	347	22	2α4	2α4	NUM
ejpam-6321	347	23	)	)	PUNCT
ejpam-6321	347	24	(	(	PUNCT
ejpam-6321	347	25	1−	1−	NUM
ejpam-6321	347	26	α4	α4	NOUN
ejpam-6321	347	27	−	−	NOUN
ejpam-6321	347	28	α5	α5	NOUN
ejpam-6321	347	29	)	)	PUNCT
ejpam-6321	347	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	347	31	)	)	PUNCT
ejpam-6321	347	32	,	,	PUNCT
ejpam-6321	347	33	x(3k+2	x(3k+2	NOUN
ejpam-6321	347	34	)	)	PUNCT
ejpam-6321	347	35	,	,	PUNCT
ejpam-6321	347	36	x(3k+3	x(3k+3	NUM
ejpam-6321	347	37	)	)	PUNCT
ejpam-6321	347	38	)	)	PUNCT
ejpam-6321	347	39	,	,	PUNCT
ejpam-6321	347	40	where	where	SCONJ
ejpam-6321	347	41	γ2	γ2	NOUN
ejpam-6321	347	42	=	=	SYM
ejpam-6321	347	43	(	(	PUNCT
ejpam-6321	347	44	α1	α1	PROPN
ejpam-6321	347	45	+	+	CCONJ
ejpam-6321	347	46	α2	α2	ADJ
ejpam-6321	347	47	+	+	CCONJ
ejpam-6321	347	48	2α4	2α4	NUM
ejpam-6321	347	49	)	)	PUNCT
ejpam-6321	347	50	(	(	PUNCT
ejpam-6321	347	51	1−	1−	NUM
ejpam-6321	347	52	α4	α4	NOUN
ejpam-6321	347	53	−	−	NOUN
ejpam-6321	347	54	α5	α5	NOUN
ejpam-6321	347	55	)	)	PUNCT
ejpam-6321	347	56	<	<	X
ejpam-6321	347	57	1	1	NUM
ejpam-6321	347	58	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	347	59	)	)	PUNCT
ejpam-6321	347	60	,	,	PUNCT
ejpam-6321	347	61	x(3k+3	x(3k+3	NUM
ejpam-6321	347	62	)	)	PUNCT
ejpam-6321	347	63	,	,	PUNCT
ejpam-6321	347	64	x(3k+4	x(3k+4	NUM
ejpam-6321	347	65	)	)	PUNCT
ejpam-6321	347	66	)	)	PUNCT
ejpam-6321	347	67	≤	≤	NUM
ejpam-6321	347	68	γ2g(x(3k+1	γ2g(x(3k+1	PROPN
ejpam-6321	347	69	)	)	PUNCT
ejpam-6321	347	70	,	,	PUNCT
ejpam-6321	347	71	x(3k+2	x(3k+2	NOUN
ejpam-6321	347	72	)	)	PUNCT
ejpam-6321	347	73	,	,	PUNCT
ejpam-6321	347	74	x(3k+3	x(3k+3	NUM
ejpam-6321	347	75	)	)	PUNCT
ejpam-6321	347	76	)	)	PUNCT
ejpam-6321	347	77	from	from	ADP
ejpam-6321	347	78	both	both	CCONJ
ejpam-6321	347	79	the	the	DET
ejpam-6321	347	80	cases	case	NOUN
ejpam-6321	347	81	we	we	PRON
ejpam-6321	347	82	get	get	VERB
ejpam-6321	347	83	,	,	PUNCT
ejpam-6321	347	84	∴	∴	NOUN
ejpam-6321	347	85	γ1	γ1	NOUN
ejpam-6321	347	86	=	=	SYM
ejpam-6321	347	87	γ2	γ2	PROPN
ejpam-6321	347	88	=	=	PUNCT
ejpam-6321	347	89	γ	γ	X
ejpam-6321	347	90	g(x(3k+2	g(x(3k+2	PROPN
ejpam-6321	347	91	)	)	PUNCT
ejpam-6321	347	92	,	,	PUNCT
ejpam-6321	347	93	x(3k+3	x(3k+3	NUM
ejpam-6321	347	94	)	)	PUNCT
ejpam-6321	347	95	,	,	PUNCT
ejpam-6321	347	96	x(3k+4	x(3k+4	NUM
ejpam-6321	347	97	)	)	PUNCT
ejpam-6321	347	98	)	)	PUNCT
ejpam-6321	348	1	≤	≤	NOUN
ejpam-6321	348	2	γg(x(3k+1	γg(x(3k+1	NOUN
ejpam-6321	348	3	)	)	PUNCT
ejpam-6321	348	4	,	,	PUNCT
ejpam-6321	348	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	348	6	)	)	PUNCT
ejpam-6321	348	7	,	,	PUNCT
ejpam-6321	348	8	x(3k+3	x(3k+3	NUM
ejpam-6321	348	9	)	)	PUNCT
ejpam-6321	348	10	)	)	PUNCT
ejpam-6321	349	1	(	(	PUNCT
ejpam-6321	349	2	12	12	NUM
ejpam-6321	349	3	)	)	PUNCT
ejpam-6321	349	4	from	from	ADP
ejpam-6321	349	5	(	(	PUNCT
ejpam-6321	349	6	11	11	NUM
ejpam-6321	349	7	)	)	PUNCT
ejpam-6321	349	8	and	and	CCONJ
ejpam-6321	349	9	(	(	PUNCT
ejpam-6321	349	10	12	12	NUM
ejpam-6321	349	11	)	)	PUNCT
ejpam-6321	349	12	inductively	inductively	ADV
ejpam-6321	349	13	we	we	PRON
ejpam-6321	349	14	have	have	AUX
ejpam-6321	349	15	,	,	PUNCT
ejpam-6321	349	16	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	349	17	)	)	PUNCT
ejpam-6321	349	18	,	,	PUNCT
ejpam-6321	349	19	x(3k+2	x(3k+2	NOUN
ejpam-6321	349	20	)	)	PUNCT
ejpam-6321	349	21	,	,	PUNCT
ejpam-6321	349	22	x(3k+3	x(3k+3	NUM
ejpam-6321	349	23	)	)	PUNCT
ejpam-6321	349	24	)	)	PUNCT
ejpam-6321	350	1	≤	≤	NUM
ejpam-6321	350	2	γ3k+1g(x0	γ3k+1g(x0	NOUN
ejpam-6321	350	3	,	,	PUNCT
ejpam-6321	350	4	x1	x1	PROPN
ejpam-6321	350	5	,	,	PUNCT
ejpam-6321	350	6	x2	x2	PROPN
ejpam-6321	350	7	)	)	PUNCT
ejpam-6321	350	8	(	(	PUNCT
ejpam-6321	350	9	13	13	NUM
ejpam-6321	350	10	)	)	PUNCT
ejpam-6321	350	11	next	next	ADV
ejpam-6321	350	12	we	we	PRON
ejpam-6321	350	13	will	will	AUX
ejpam-6321	350	14	show	show	VERB
ejpam-6321	350	15	that	that	SCONJ
ejpam-6321	350	16	{	{	PUNCT
ejpam-6321	350	17	xk	xk	ADJ
ejpam-6321	350	18	}	}	PUNCT
ejpam-6321	350	19	is	be	AUX
ejpam-6321	350	20	a	a	DET
ejpam-6321	350	21	g	g	NOUN
ejpam-6321	350	22	-	-	PUNCT
ejpam-6321	350	23	cauchy	cauchy	ADJ
ejpam-6321	350	24	sequence	sequence	NOUN
ejpam-6321	350	25	in	in	ADP
ejpam-6321	350	26	x	x	NOUN
ejpam-6321	350	27	,	,	PUNCT
ejpam-6321	350	28	for	for	ADP
ejpam-6321	350	29	any	any	DET
ejpam-6321	350	30	j	j	PROPN
ejpam-6321	350	31	,	,	PUNCT
ejpam-6321	350	32	n	n	CCONJ
ejpam-6321	350	33	,	,	PUNCT
ejpam-6321	350	34	k	k	PROPN
ejpam-6321	350	35	with	with	ADP
ejpam-6321	350	36	j	j	PROPN
ejpam-6321	350	37	>	>	X
ejpam-6321	350	38	n	n	PROPN
ejpam-6321	350	39	>	>	X
ejpam-6321	350	40	k.	k.	PROPN
ejpam-6321	350	41	g(xk	g(xk	PROPN
ejpam-6321	350	42	,	,	PUNCT
ejpam-6321	350	43	xn	xn	PROPN
ejpam-6321	350	44	,	,	PUNCT
ejpam-6321	350	45	xj	xj	PROPN
ejpam-6321	350	46	)	)	PUNCT
ejpam-6321	350	47	≤	≤	NOUN
ejpam-6321	350	48	g(xk	g(xk	NOUN
ejpam-6321	350	49	,	,	PUNCT
ejpam-6321	350	50	x(k+1	x(k+1	NUM
ejpam-6321	350	51	)	)	PUNCT
ejpam-6321	350	52	,	,	PUNCT
ejpam-6321	350	53	x(k+1	x(k+1	NOUN
ejpam-6321	350	54	)	)	PUNCT
ejpam-6321	350	55	)	)	PUNCT
ejpam-6321	351	1	+	+	CCONJ
ejpam-6321	351	2	g(x(k+1	g(x(k+1	NOUN
ejpam-6321	351	3	)	)	PUNCT
ejpam-6321	351	4	,	,	PUNCT
ejpam-6321	351	5	x(k+2	x(k+2	NOUN
ejpam-6321	351	6	)	)	PUNCT
ejpam-6321	351	7	,	,	PUNCT
ejpam-6321	351	8	x(k+2	x(k+2	NOUN
ejpam-6321	351	9	)	)	PUNCT
ejpam-6321	351	10	)	)	PUNCT
ejpam-6321	352	1	+	+	CCONJ
ejpam-6321	352	2	......	......	PUNCT
ejpam-6321	352	3	+	+	ADJ
ejpam-6321	352	4	g(x(j−1	g(x(j−1	PROPN
ejpam-6321	352	5	)	)	PUNCT
ejpam-6321	352	6	,	,	PUNCT
ejpam-6321	352	7	x(j−1	x(j−1	PROPN
ejpam-6321	352	8	)	)	PUNCT
ejpam-6321	352	9	,	,	PUNCT
ejpam-6321	352	10	xj	xj	PROPN
ejpam-6321	352	11	)	)	PUNCT
ejpam-6321	352	12	≤	≤	NOUN
ejpam-6321	352	13	g(xk	g(xk	NOUN
ejpam-6321	352	14	,	,	PUNCT
ejpam-6321	352	15	x(k+1	x(k+1	NUM
ejpam-6321	352	16	)	)	PUNCT
ejpam-6321	352	17	,	,	PUNCT
ejpam-6321	352	18	x(k+2	x(k+2	NOUN
ejpam-6321	352	19	)	)	PUNCT
ejpam-6321	352	20	)	)	PUNCT
ejpam-6321	353	1	+	+	CCONJ
ejpam-6321	353	2	g(x(k+1	g(x(k+1	NOUN
ejpam-6321	353	3	)	)	PUNCT
ejpam-6321	353	4	,	,	PUNCT
ejpam-6321	353	5	x(k+2	x(k+2	NUM
ejpam-6321	353	6	)	)	PUNCT
ejpam-6321	353	7	,	,	PUNCT
ejpam-6321	353	8	x(k+3	x(k+3	NUM
ejpam-6321	353	9	)	)	PUNCT
ejpam-6321	353	10	)	)	PUNCT
ejpam-6321	354	1	+	+	CCONJ
ejpam-6321	354	2	.......	.......	PUNCT
ejpam-6321	355	1	+	+	PROPN
ejpam-6321	355	2	g(x(j−2	g(x(j−2	PROPN
ejpam-6321	355	3	)	)	PUNCT
ejpam-6321	355	4	,	,	PUNCT
ejpam-6321	355	5	x(j−1	x(j−1	PROPN
ejpam-6321	355	6	)	)	PUNCT
ejpam-6321	355	7	,	,	PUNCT
ejpam-6321	355	8	xj	xj	PROPN
ejpam-6321	355	9	)	)	PUNCT
ejpam-6321	355	10	≤	≤	PUNCT
ejpam-6321	356	1	[	[	X
ejpam-6321	356	2	yk	yk	X
ejpam-6321	356	3	+	+	X
ejpam-6321	356	4	y(k+1	y(k+1	NOUN
ejpam-6321	356	5	)	)	PUNCT
ejpam-6321	357	1	+	+	CCONJ
ejpam-6321	357	2	........	........	PUNCT
ejpam-6321	357	3	+	+	NUM
ejpam-6321	357	4	y(j−2)]g(x0	y(j−2)]g(x0	PROPN
ejpam-6321	357	5	,	,	PUNCT
ejpam-6321	357	6	x1	x1	PROPN
ejpam-6321	357	7	,	,	PUNCT
ejpam-6321	357	8	x2	x2	PROPN
ejpam-6321	357	9	)	)	PUNCT
ejpam-6321	357	10	this	this	PRON
ejpam-6321	357	11	implies	imply	VERB
ejpam-6321	357	12	that	that	SCONJ
ejpam-6321	357	13	,	,	PUNCT
ejpam-6321	357	14	g(xk	g(xk	PROPN
ejpam-6321	357	15	,	,	PUNCT
ejpam-6321	357	16	xn	xn	PROPN
ejpam-6321	357	17	,	,	PUNCT
ejpam-6321	357	18	xj	xj	PROPN
ejpam-6321	357	19	)	)	PUNCT
ejpam-6321	357	20	≤	≤	PROPN
ejpam-6321	357	21	yk	yk	PROPN
ejpam-6321	357	22	(	(	PUNCT
ejpam-6321	357	23	1−	1−	NUM
ejpam-6321	357	24	y	y	NOUN
ejpam-6321	357	25	)	)	PUNCT
ejpam-6321	357	26	g(x0	g(x0	NOUN
ejpam-6321	357	27	,	,	PUNCT
ejpam-6321	357	28	x1	x1	PROPN
ejpam-6321	357	29	,	,	PUNCT
ejpam-6321	357	30	x2	x2	PROPN
ejpam-6321	357	31	)	)	PUNCT
ejpam-6321	357	32	the	the	DET
ejpam-6321	357	33	same	same	ADJ
ejpam-6321	357	34	hold	hold	NOUN
ejpam-6321	357	35	if	if	SCONJ
ejpam-6321	357	36	j	j	PROPN
ejpam-6321	357	37	=	=	SYM
ejpam-6321	357	38	n	n	PROPN
ejpam-6321	357	39	>	>	X
ejpam-6321	357	40	k	k	NOUN
ejpam-6321	357	41	and	and	CCONJ
ejpam-6321	357	42	if	if	SCONJ
ejpam-6321	357	43	j	j	PROPN
ejpam-6321	357	44	>	>	X
ejpam-6321	357	45	n	n	PROPN
ejpam-6321	357	46	=	=	SYM
ejpam-6321	358	1	k	k	NOUN
ejpam-6321	358	2	we	we	PRON
ejpam-6321	358	3	have	have	VERB
ejpam-6321	358	4	,	,	PUNCT
ejpam-6321	358	5	g(xk	g(xk	PROPN
ejpam-6321	358	6	,	,	PUNCT
ejpam-6321	358	7	xn	xn	PROPN
ejpam-6321	358	8	,	,	PUNCT
ejpam-6321	358	9	xj	xj	PROPN
ejpam-6321	358	10	)	)	PUNCT
ejpam-6321	358	11	≤	≤	ADJ
ejpam-6321	358	12	yk−1	yk−1	PROPN
ejpam-6321	358	13	(	(	PUNCT
ejpam-6321	358	14	1−	1−	NUM
ejpam-6321	358	15	y	y	NOUN
ejpam-6321	358	16	)	)	PUNCT
ejpam-6321	358	17	g(x0	g(x0	NOUN
ejpam-6321	358	18	,	,	PUNCT
ejpam-6321	358	19	x1	x1	PROPN
ejpam-6321	358	20	,	,	PUNCT
ejpam-6321	358	21	x2	x2	PROPN
ejpam-6321	358	22	)	)	PUNCT
ejpam-6321	358	23	if	if	SCONJ
ejpam-6321	358	24	we	we	PRON
ejpam-6321	358	25	take	take	VERB
ejpam-6321	358	26	the	the	DET
ejpam-6321	358	27	limit	limit	NOUN
ejpam-6321	358	28	as	as	ADP
ejpam-6321	358	29	k	k	PROPN
ejpam-6321	358	30	,	,	PUNCT
ejpam-6321	358	31	n	n	CCONJ
ejpam-6321	358	32	,	,	PUNCT
ejpam-6321	358	33	j	j	PROPN
ejpam-6321	358	34	→∞	→∞	PROPN
ejpam-6321	358	35	we	we	PRON
ejpam-6321	358	36	getg(xk	getg(xk	VERB
ejpam-6321	358	37	,	,	PUNCT
ejpam-6321	358	38	xn	xn	PROPN
ejpam-6321	358	39	,	,	PUNCT
ejpam-6321	358	40	xj)→	xj)→	PROPN
ejpam-6321	358	41	0	0	NUM
ejpam-6321	358	42	.	.	PUNCT
ejpam-6321	359	1	hence	hence	ADV
ejpam-6321	359	2	{	{	PUNCT
ejpam-6321	359	3	xk	xk	ADJ
ejpam-6321	359	4	}	}	PUNCT
ejpam-6321	359	5	is	be	AUX
ejpam-6321	359	6	a	a	DET
ejpam-6321	359	7	g	g	NOUN
ejpam-6321	359	8	-	-	PUNCT
ejpam-6321	359	9	cauchy	cauchy	ADJ
ejpam-6321	359	10	sequence	sequence	NOUN
ejpam-6321	359	11	.	.	PUNCT
ejpam-6321	360	1	by	by	ADP
ejpam-6321	360	2	g	g	NOUN
ejpam-6321	360	3	-	-	PUNCT
ejpam-6321	360	4	completeness	completeness	NOUN
ejpam-6321	360	5	of	of	ADP
ejpam-6321	360	6	x	x	NOUN
ejpam-6321	360	7	,	,	PUNCT
ejpam-6321	360	8	there	there	PRON
ejpam-6321	360	9	exists	exist	VERB
ejpam-6321	360	10	℘	℘	PROPN
ejpam-6321	360	11	∈	∈	PROPN
ejpam-6321	360	12	x	x	PUNCT
ejpam-6321	360	13	such	such	ADJ
ejpam-6321	360	14	that	that	SCONJ
ejpam-6321	360	15	{	{	PUNCT
ejpam-6321	360	16	xk	xk	ADJ
ejpam-6321	360	17	}	}	PUNCT
ejpam-6321	360	18	converges	converge	VERB
ejpam-6321	360	19	to	to	ADP
ejpam-6321	360	20	℘	℘	PROPN
ejpam-6321	360	21	as	as	SCONJ
ejpam-6321	360	22	k	k	PROPN
ejpam-6321	360	23	→∞.	→∞.	X
ejpam-6321	360	24	we	we	PRON
ejpam-6321	360	25	have	have	VERB
ejpam-6321	360	26	to	to	PART
ejpam-6321	360	27	show	show	VERB
ejpam-6321	360	28	that	that	DET
ejpam-6321	360	29	f1℘	f1℘	NOUN
ejpam-6321	360	30	=	=	PUNCT
ejpam-6321	360	31	℘	℘	PROPN
ejpam-6321	360	32	by	by	ADP
ejpam-6321	360	33	contrary	contrary	ADJ
ejpam-6321	360	34	case	case	NOUN
ejpam-6321	360	35	let	let	VERB
ejpam-6321	360	36	f1℘	f1℘	PROPN
ejpam-6321	360	37	̸=	̸=	PROPN
ejpam-6321	360	38	℘.	℘.	PROPN
ejpam-6321	360	39	then	then	ADV
ejpam-6321	360	40	by	by	ADP
ejpam-6321	360	41	(	(	PUNCT
ejpam-6321	360	42	8)	8)	NUM
ejpam-6321	360	43	we	we	PRON
ejpam-6321	360	44	have	have	AUX
ejpam-6321	360	45	that	that	PRON
ejpam-6321	360	46	,	,	PUNCT
ejpam-6321	360	47	g(f1℘	g(f1℘	PROPN
ejpam-6321	360	48	,	,	PUNCT
ejpam-6321	360	49	x(3k+2	x(3k+2	NOUN
ejpam-6321	360	50	)	)	PUNCT
ejpam-6321	360	51	,	,	PUNCT
ejpam-6321	360	52	x(3k+3	x(3k+3	NUM
ejpam-6321	360	53	)	)	PUNCT
ejpam-6321	360	54	)	)	PUNCT
ejpam-6321	361	1	=	=	SYM
ejpam-6321	361	2	g(f1℘	g(f1℘	PROPN
ejpam-6321	361	3	,	,	PUNCT
ejpam-6321	361	4	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	361	5	)	)	PUNCT
ejpam-6321	361	6	,	,	PUNCT
ejpam-6321	361	7	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	361	8	)	)	PUNCT
ejpam-6321	361	9	)	)	PUNCT
ejpam-6321	361	10	≤	≤	NOUN
ejpam-6321	361	11	α1g(℘	α1g(℘	NOUN
ejpam-6321	361	12	,	,	PUNCT
ejpam-6321	361	13	x(3k+1	x(3k+1	PRON
ejpam-6321	361	14	)	)	PUNCT
ejpam-6321	361	15	,	,	PUNCT
ejpam-6321	361	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	361	17	)	)	PUNCT
ejpam-6321	361	18	)	)	PUNCT
ejpam-6321	362	1	+	+	CCONJ
ejpam-6321	362	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	362	3	,	,	PUNCT
ejpam-6321	362	4	x(3k+1	x(3k+1	PRON
ejpam-6321	362	5	)	)	PUNCT
ejpam-6321	362	6	,	,	PUNCT
ejpam-6321	362	7	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	362	8	)	)	PUNCT
ejpam-6321	362	9	)	)	PUNCT
ejpam-6321	363	1	+	+	CCONJ
ejpam-6321	364	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	364	2	,	,	PUNCT
ejpam-6321	364	3	x(3k+1	x(3k+1	ADV
ejpam-6321	364	4	)	)	PUNCT
ejpam-6321	364	5	,	,	PUNCT
ejpam-6321	364	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	364	7	)	)	PUNCT
ejpam-6321	364	8	)	)	PUNCT
ejpam-6321	365	1	+	+	CCONJ
ejpam-6321	365	2	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	365	3	,	,	PUNCT
ejpam-6321	365	4	x3k	x3k	NOUN
ejpam-6321	365	5	,	,	PUNCT
ejpam-6321	365	6	x3k	x3k	PUNCT
ejpam-6321	365	7	)	)	PUNCT
ejpam-6321	366	1	+	+	SYM
ejpam-6321	366	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	366	3	)	)	PUNCT
ejpam-6321	366	4	,	,	PUNCT
ejpam-6321	366	5	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	366	6	)	)	PUNCT
ejpam-6321	366	7	,	,	PUNCT
ejpam-6321	366	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	366	9	)	)	PUNCT
ejpam-6321	366	10	)	)	PUNCT
ejpam-6321	367	1	+	+	PUNCT
ejpam-6321	367	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	367	3	)	)	PUNCT
ejpam-6321	367	4	,	,	PUNCT
ejpam-6321	367	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	367	6	)	)	PUNCT
ejpam-6321	367	7	,	,	PUNCT
ejpam-6321	367	8	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	367	9	)	)	PUNCT
ejpam-6321	367	10	)	)	PUNCT
ejpam-6321	367	11	]	]	PUNCT
ejpam-6321	368	1	+	+	CCONJ
ejpam-6321	368	2	α5max	α5max	PROPN
ejpam-6321	368	3			VERB
ejpam-6321	368	4	g(℘	g(℘	NOUN
ejpam-6321	368	5	,	,	PUNCT
ejpam-6321	368	6	x(3k+1	x(3k+1	PRON
ejpam-6321	368	7	)	)	PUNCT
ejpam-6321	368	8	,	,	PUNCT
ejpam-6321	368	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	368	10	)	)	PUNCT
ejpam-6321	368	11	)	)	PUNCT
ejpam-6321	368	12	,	,	PUNCT
ejpam-6321	368	13	g(f1℘	g(f1℘	PROPN
ejpam-6321	368	14	,	,	PUNCT
ejpam-6321	368	15	f1℘	f1℘	PROPN
ejpam-6321	368	16	,	,	PUNCT
ejpam-6321	368	17	x(3k+1	x(3k+1	ADV
ejpam-6321	368	18	)	)	PUNCT
ejpam-6321	368	19	)	)	PUNCT
ejpam-6321	368	20	,	,	PUNCT
ejpam-6321	368	21	g(f1℘	g(f1℘	PROPN
ejpam-6321	368	22	,	,	PUNCT
ejpam-6321	368	23	x(3k+1	x(3k+1	ADV
ejpam-6321	368	24	)	)	PUNCT
ejpam-6321	368	25	,	,	PUNCT
ejpam-6321	368	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	368	27	)	)	PUNCT
ejpam-6321	368	28	)	)	PUNCT
ejpam-6321	368	29	,	,	PUNCT
ejpam-6321	368	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	368	31	)	)	PUNCT
ejpam-6321	368	32	,	,	PUNCT
ejpam-6321	368	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	368	34	)	)	PUNCT
ejpam-6321	368	35	,	,	PUNCT
ejpam-6321	368	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	368	37	)	)	PUNCT
ejpam-6321	368	38	)	)	PUNCT
ejpam-6321	368	39	,	,	PUNCT
ejpam-6321	368	40	g(f1℘	g(f1℘	PROPN
ejpam-6321	368	41	,	,	PUNCT
ejpam-6321	368	42	x3k	x3k	NOUN
ejpam-6321	368	43	,	,	PUNCT
ejpam-6321	368	44	x3k	x3k	NOUN
ejpam-6321	368	45	)	)	PUNCT
ejpam-6321	368	46	,	,	PUNCT
ejpam-6321	368	47	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	368	48	)	)	PUNCT
ejpam-6321	368	49	,	,	PUNCT
ejpam-6321	368	50	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	368	51	)	)	PUNCT
ejpam-6321	368	52	,	,	PUNCT
ejpam-6321	368	53	x(3k+1	x(3k+1	CCONJ
ejpam-6321	368	54	)	)	PUNCT
ejpam-6321	368	55	)	)	PUNCT
ejpam-6321	368	56	,	,	PUNCT
ejpam-6321	368	57	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	368	58	)	)	PUNCT
ejpam-6321	368	59	,	,	PUNCT
ejpam-6321	368	60	x(3k+2	x(3k+2	NOUN
ejpam-6321	368	61	)	)	PUNCT
ejpam-6321	368	62	,	,	PUNCT
ejpam-6321	368	63	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	368	64	)	)	PUNCT
ejpam-6321	368	65	)	)	PUNCT
ejpam-6321	368	66			ADP
ejpam-6321	368	67	m.	m.	NOUN
ejpam-6321	368	68	noorwali	noorwali	PROPN
ejpam-6321	368	69	et	et	PROPN
ejpam-6321	368	70	al	al	PROPN
ejpam-6321	368	71	.	.	PUNCT
ejpam-6321	368	72	/	/	SYM
ejpam-6321	368	73	eur	eur	PROPN
ejpam-6321	368	74	.	.	PUNCT
ejpam-6321	369	1	j.	j.	PROPN
ejpam-6321	369	2	pure	pure	PROPN
ejpam-6321	369	3	appl	appl	PROPN
ejpam-6321	369	4	.	.	PROPN
ejpam-6321	369	5	math	math	PROPN
ejpam-6321	369	6	,	,	PUNCT
ejpam-6321	369	7	18	18	NUM
ejpam-6321	369	8	(	(	PUNCT
ejpam-6321	369	9	3	3	NUM
ejpam-6321	369	10	)	)	PUNCT
ejpam-6321	369	11	(	(	PUNCT
ejpam-6321	369	12	2025	2025	NUM
ejpam-6321	369	13	)	)	PUNCT
ejpam-6321	369	14	,	,	PUNCT
ejpam-6321	369	15	6321	6321	NUM
ejpam-6321	369	16	14	14	NUM
ejpam-6321	369	17	of	of	ADP
ejpam-6321	369	18	27	27	NUM
ejpam-6321	369	19	=	=	SYM
ejpam-6321	369	20	α1g(℘	α1g(℘	NOUN
ejpam-6321	369	21	,	,	PUNCT
ejpam-6321	369	22	x(3k+1	x(3k+1	PRON
ejpam-6321	369	23	)	)	PUNCT
ejpam-6321	369	24	,	,	PUNCT
ejpam-6321	369	25	x(3k+2	x(3k+2	NOUN
ejpam-6321	369	26	)	)	PUNCT
ejpam-6321	369	27	)	)	PUNCT
ejpam-6321	370	1	+	+	CCONJ
ejpam-6321	370	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	370	3	,	,	PUNCT
ejpam-6321	370	4	x(3k+1	x(3k+1	PRON
ejpam-6321	370	5	)	)	PUNCT
ejpam-6321	370	6	,	,	PUNCT
ejpam-6321	370	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	370	8	)	)	PUNCT
ejpam-6321	370	9	)	)	PUNCT
ejpam-6321	371	1	+	+	CCONJ
ejpam-6321	372	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	372	2	,	,	PUNCT
ejpam-6321	372	3	x(3k+1	x(3k+1	ADV
ejpam-6321	372	4	)	)	PUNCT
ejpam-6321	372	5	,	,	PUNCT
ejpam-6321	372	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	372	7	)	)	PUNCT
ejpam-6321	372	8	)	)	PUNCT
ejpam-6321	373	1	+	+	CCONJ
ejpam-6321	373	2	α4[g(f1℘	α4[g(f1℘	ADV
ejpam-6321	373	3	,	,	PUNCT
ejpam-6321	373	4	x(3k+1	x(3k+1	ADP
ejpam-6321	373	5	)	)	PUNCT
ejpam-6321	373	6	,	,	PUNCT
ejpam-6321	373	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	373	8	)	)	PUNCT
ejpam-6321	373	9	)	)	PUNCT
ejpam-6321	374	1	+	+	ADP
ejpam-6321	374	2	g(x3k	g(x3k	NOUN
ejpam-6321	374	3	,	,	PUNCT
ejpam-6321	374	4	x(3k+2	x(3k+2	NUM
ejpam-6321	374	5	)	)	PUNCT
ejpam-6321	374	6	,	,	PUNCT
ejpam-6321	374	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	374	8	)	)	PUNCT
ejpam-6321	374	9	)	)	PUNCT
ejpam-6321	375	1	+	+	ADP
ejpam-6321	375	2	g(x3k	g(x3k	NOUN
ejpam-6321	375	3	,	,	PUNCT
ejpam-6321	375	4	x(3k+1	x(3k+1	PRON
ejpam-6321	375	5	)	)	PUNCT
ejpam-6321	375	6	,	,	PUNCT
ejpam-6321	375	7	x(3k+3	x(3k+3	NUM
ejpam-6321	375	8	)	)	PUNCT
ejpam-6321	375	9	)	)	PUNCT
ejpam-6321	375	10	]	]	PUNCT
ejpam-6321	376	1	+	+	CCONJ
ejpam-6321	376	2	α5max	α5max	PROPN
ejpam-6321	376	3			VERB
ejpam-6321	376	4	g(℘	g(℘	NOUN
ejpam-6321	376	5	,	,	PUNCT
ejpam-6321	376	6	x(3k+1	x(3k+1	PRON
ejpam-6321	376	7	)	)	PUNCT
ejpam-6321	376	8	,	,	PUNCT
ejpam-6321	376	9	x(3k+2	x(3k+2	NUM
ejpam-6321	376	10	)	)	PUNCT
ejpam-6321	376	11	)	)	PUNCT
ejpam-6321	376	12	,	,	PUNCT
ejpam-6321	376	13	g(f1℘	g(f1℘	PROPN
ejpam-6321	376	14	,	,	PUNCT
ejpam-6321	376	15	f1℘	f1℘	PROPN
ejpam-6321	376	16	,	,	PUNCT
ejpam-6321	376	17	x(3k+1	x(3k+1	ADV
ejpam-6321	376	18	)	)	PUNCT
ejpam-6321	376	19	)	)	PUNCT
ejpam-6321	376	20	,	,	PUNCT
ejpam-6321	376	21	g(℘	g(℘	PROPN
ejpam-6321	376	22	,	,	PUNCT
ejpam-6321	376	23	x(3k+1	x(3k+1	PRON
ejpam-6321	376	24	)	)	PUNCT
ejpam-6321	376	25	,	,	PUNCT
ejpam-6321	376	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	376	27	)	)	PUNCT
ejpam-6321	376	28	)	)	PUNCT
ejpam-6321	376	29	,	,	PUNCT
ejpam-6321	376	30	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	376	31	)	)	PUNCT
ejpam-6321	376	32	,	,	PUNCT
ejpam-6321	376	33	x(3k+2	x(3k+2	NOUN
ejpam-6321	376	34	)	)	PUNCT
ejpam-6321	376	35	,	,	PUNCT
ejpam-6321	376	36	x(3k+2	x(3k+2	NOUN
ejpam-6321	376	37	)	)	PUNCT
ejpam-6321	376	38	)	)	PUNCT
ejpam-6321	376	39	,	,	PUNCT
ejpam-6321	376	40	g(f1℘	g(f1℘	PROPN
ejpam-6321	376	41	,	,	PUNCT
ejpam-6321	376	42	x3k	x3k	NOUN
ejpam-6321	376	43	,	,	PUNCT
ejpam-6321	376	44	x3k	x3k	NOUN
ejpam-6321	376	45	)	)	PUNCT
ejpam-6321	376	46	,	,	PUNCT
ejpam-6321	376	47	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	376	48	)	)	PUNCT
ejpam-6321	376	49	,	,	PUNCT
ejpam-6321	376	50	x(3k+2	x(3k+2	NOUN
ejpam-6321	376	51	)	)	PUNCT
ejpam-6321	376	52	,	,	PUNCT
ejpam-6321	376	53	x(3k+1	x(3k+1	CCONJ
ejpam-6321	376	54	)	)	PUNCT
ejpam-6321	376	55	)	)	PUNCT
ejpam-6321	376	56	,	,	PUNCT
ejpam-6321	376	57	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	376	58	)	)	PUNCT
ejpam-6321	376	59	,	,	PUNCT
ejpam-6321	376	60	x(3k+2	x(3k+2	NOUN
ejpam-6321	376	61	)	)	PUNCT
ejpam-6321	376	62	,	,	PUNCT
ejpam-6321	376	63	x(3k+)3	x(3k+)3	NOUN
ejpam-6321	376	64	)	)	PUNCT
ejpam-6321	376	65			ADJ
ejpam-6321	376	66	after	after	ADP
ejpam-6321	376	67	applying	apply	VERB
ejpam-6321	376	68	the	the	DET
ejpam-6321	376	69	simplification	simplification	NOUN
ejpam-6321	376	70	process	process	NOUN
ejpam-6321	376	71	utilizing	utilize	VERB
ejpam-6321	376	72	the	the	DET
ejpam-6321	376	73	definition	definition	NOUN
ejpam-6321	376	74	of	of	ADP
ejpam-6321	376	75	gm	gm	PROPN
ejpam-6321	376	76	-	-	PUNCT
ejpam-6321	376	77	space	space	NOUN
ejpam-6321	376	78	we	we	PRON
ejpam-6321	376	79	have	have	AUX
ejpam-6321	376	80	achieved	achieve	VERB
ejpam-6321	376	81	the	the	DET
ejpam-6321	376	82	following	following	ADJ
ejpam-6321	376	83	result	result	NOUN
ejpam-6321	376	84	,	,	PUNCT
ejpam-6321	376	85	≤	≤	ADJ
ejpam-6321	376	86	α1g(℘	α1g(℘	NOUN
ejpam-6321	376	87	,	,	PUNCT
ejpam-6321	376	88	x(3k+1	x(3k+1	PRON
ejpam-6321	376	89	)	)	PUNCT
ejpam-6321	376	90	,	,	PUNCT
ejpam-6321	376	91	x(3k+2	x(3k+2	NOUN
ejpam-6321	376	92	)	)	PUNCT
ejpam-6321	376	93	)	)	PUNCT
ejpam-6321	377	1	+	+	CCONJ
ejpam-6321	377	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	377	3	,	,	PUNCT
ejpam-6321	377	4	x(3k+1	x(3k+1	ADV
ejpam-6321	377	5	)	)	PUNCT
ejpam-6321	377	6	,	,	PUNCT
ejpam-6321	377	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	377	8	)	)	PUNCT
ejpam-6321	377	9	)	)	PUNCT
ejpam-6321	378	1	+	+	CCONJ
ejpam-6321	379	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	379	2	,	,	PUNCT
ejpam-6321	379	3	x(3k+1	x(3k+1	ADV
ejpam-6321	379	4	)	)	PUNCT
ejpam-6321	379	5	,	,	PUNCT
ejpam-6321	379	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	379	7	)	)	PUNCT
ejpam-6321	379	8	)	)	PUNCT
ejpam-6321	380	1	+	+	CCONJ
ejpam-6321	381	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	381	2	,	,	PUNCT
ejpam-6321	381	3	x3k	x3k	NOUN
ejpam-6321	381	4	,	,	PUNCT
ejpam-6321	381	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	381	6	)	)	PUNCT
ejpam-6321	381	7	)	)	PUNCT
ejpam-6321	382	1	+	+	CCONJ
ejpam-6321	382	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	382	3	)	)	PUNCT
ejpam-6321	382	4	,	,	PUNCT
ejpam-6321	382	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	382	6	)	)	PUNCT
ejpam-6321	382	7	,	,	PUNCT
ejpam-6321	382	8	x(3k+3	x(3k+3	NUM
ejpam-6321	382	9	)	)	PUNCT
ejpam-6321	382	10	)	)	PUNCT
ejpam-6321	382	11	]	]	PUNCT
ejpam-6321	383	1	+	+	CCONJ
ejpam-6321	383	2	α5max	α5max	NUM
ejpam-6321	383	3	{	{	PUNCT
ejpam-6321	383	4	g(℘	g(℘	PROPN
ejpam-6321	383	5	,	,	PUNCT
ejpam-6321	383	6	x(3k+1	x(3k+1	PRON
ejpam-6321	383	7	)	)	PUNCT
ejpam-6321	383	8	,	,	PUNCT
ejpam-6321	383	9	x(3k+2	x(3k+2	NUM
ejpam-6321	383	10	)	)	PUNCT
ejpam-6321	383	11	)	)	PUNCT
ejpam-6321	383	12	,	,	PUNCT
ejpam-6321	383	13	g(f1℘	g(f1℘	PROPN
ejpam-6321	383	14	,	,	PUNCT
ejpam-6321	383	15	f1℘	f1℘	PROPN
ejpam-6321	383	16	,	,	PUNCT
ejpam-6321	383	17	x(3k+1	x(3k+1	ADV
ejpam-6321	383	18	)	)	PUNCT
ejpam-6321	383	19	)	)	PUNCT
ejpam-6321	383	20	,	,	PUNCT
ejpam-6321	383	21	g(f1℘	g(f1℘	PROPN
ejpam-6321	383	22	,	,	PUNCT
ejpam-6321	383	23	x(3k+1	x(3k+1	ADV
ejpam-6321	383	24	)	)	PUNCT
ejpam-6321	383	25	,	,	PUNCT
ejpam-6321	383	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	383	27	)	)	PUNCT
ejpam-6321	383	28	)	)	PUNCT
ejpam-6321	383	29	,	,	PUNCT
ejpam-6321	383	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	383	31	)	)	PUNCT
ejpam-6321	383	32	,	,	PUNCT
ejpam-6321	383	33	x(3k+2	x(3k+2	NOUN
ejpam-6321	383	34	)	)	PUNCT
ejpam-6321	383	35	,	,	PUNCT
ejpam-6321	383	36	x(3k+3	x(3k+3	NUM
ejpam-6321	383	37	)	)	PUNCT
ejpam-6321	383	38	)	)	PUNCT
ejpam-6321	383	39	,	,	PUNCT
ejpam-6321	383	40	g(f1℘	g(f1℘	PROPN
ejpam-6321	383	41	,	,	PUNCT
ejpam-6321	383	42	x3k	x3k	NOUN
ejpam-6321	383	43	,	,	PUNCT
ejpam-6321	383	44	x(3k+2	x(3k+2	NUM
ejpam-6321	383	45	)	)	PUNCT
ejpam-6321	383	46	)	)	PUNCT
ejpam-6321	383	47	}	}	PUNCT
ejpam-6321	383	48	(	(	PUNCT
ejpam-6321	383	49	14	14	NUM
ejpam-6321	383	50	)	)	PUNCT
ejpam-6321	383	51	now	now	ADV
ejpam-6321	383	52	,	,	PUNCT
ejpam-6321	383	53	there	there	PRON
ejpam-6321	383	54	are	be	VERB
ejpam-6321	383	55	five	five	NUM
ejpam-6321	383	56	cases	case	NOUN
ejpam-6321	383	57	:	:	PUNCT
ejpam-6321	383	58	(	(	PUNCT
ejpam-6321	383	59	i	i	NOUN
ejpam-6321	383	60	)	)	PUNCT
ejpam-6321	383	61	.	.	PUNCT
ejpam-6321	384	1	if	if	SCONJ
ejpam-6321	384	2	g(℘	g(℘	NOUN
ejpam-6321	384	3	,	,	PUNCT
ejpam-6321	384	4	x(3k+1	x(3k+1	PRON
ejpam-6321	384	5	)	)	PUNCT
ejpam-6321	384	6	,	,	PUNCT
ejpam-6321	384	7	x(3k+2	x(3k+2	NUM
ejpam-6321	384	8	)	)	PUNCT
ejpam-6321	384	9	)	)	PUNCT
ejpam-6321	384	10	is	be	AUX
ejpam-6321	384	11	a	a	DET
ejpam-6321	384	12	maximum	maximum	ADJ
ejpam-6321	384	13	term	term	NOUN
ejpam-6321	384	14	in	in	ADP
ejpam-6321	384	15	(	(	PUNCT
ejpam-6321	384	16	14	14	NUM
ejpam-6321	384	17	)	)	PUNCT
ejpam-6321	384	18	then	then	ADV
ejpam-6321	384	19	we	we	PRON
ejpam-6321	384	20	can	can	AUX
ejpam-6321	384	21	write	write	VERB
ejpam-6321	384	22	;	;	PUNCT
ejpam-6321	384	23	≤	≤	NUM
ejpam-6321	384	24	α1g(℘	α1g(℘	NOUN
ejpam-6321	384	25	,	,	PUNCT
ejpam-6321	384	26	x(3k+1	x(3k+1	PRON
ejpam-6321	384	27	)	)	PUNCT
ejpam-6321	384	28	,	,	PUNCT
ejpam-6321	384	29	x(3k+2	x(3k+2	NOUN
ejpam-6321	384	30	)	)	PUNCT
ejpam-6321	384	31	)	)	PUNCT
ejpam-6321	385	1	+	+	CCONJ
ejpam-6321	385	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	385	3	,	,	PUNCT
ejpam-6321	385	4	x(3k+1	x(3k+1	ADV
ejpam-6321	385	5	)	)	PUNCT
ejpam-6321	385	6	,	,	PUNCT
ejpam-6321	385	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	385	8	)	)	PUNCT
ejpam-6321	385	9	)	)	PUNCT
ejpam-6321	386	1	+	+	CCONJ
ejpam-6321	387	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	387	2	,	,	PUNCT
ejpam-6321	387	3	x(3k+1	x(3k+1	ADV
ejpam-6321	387	4	)	)	PUNCT
ejpam-6321	387	5	,	,	PUNCT
ejpam-6321	387	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	387	7	)	)	PUNCT
ejpam-6321	387	8	)	)	PUNCT
ejpam-6321	388	1	+	+	CCONJ
ejpam-6321	389	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	389	2	,	,	PUNCT
ejpam-6321	389	3	x3k	x3k	NOUN
ejpam-6321	389	4	,	,	PUNCT
ejpam-6321	389	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	389	6	)	)	PUNCT
ejpam-6321	389	7	)	)	PUNCT
ejpam-6321	390	1	+	+	CCONJ
ejpam-6321	390	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	390	3	)	)	PUNCT
ejpam-6321	390	4	,	,	PUNCT
ejpam-6321	390	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	390	6	)	)	PUNCT
ejpam-6321	390	7	,	,	PUNCT
ejpam-6321	390	8	x(3k+3	x(3k+3	NUM
ejpam-6321	390	9	)	)	PUNCT
ejpam-6321	390	10	)	)	PUNCT
ejpam-6321	390	11	]	]	PUNCT
ejpam-6321	391	1	+	+	NUM
ejpam-6321	391	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	391	3	,	,	PUNCT
ejpam-6321	391	4	x(3k+1	x(3k+1	PRON
ejpam-6321	391	5	)	)	PUNCT
ejpam-6321	391	6	,	,	PUNCT
ejpam-6321	391	7	x(3k+2	x(3k+2	NUM
ejpam-6321	391	8	)	)	PUNCT
ejpam-6321	391	9	)	)	PUNCT
ejpam-6321	391	10	applying	apply	VERB
ejpam-6321	391	11	limn→∞	limn→∞	PROPN
ejpam-6321	391	12	on	on	ADP
ejpam-6321	391	13	both	both	DET
ejpam-6321	391	14	sides	side	NOUN
ejpam-6321	391	15	,	,	PUNCT
ejpam-6321	391	16	we	we	PRON
ejpam-6321	391	17	get	get	VERB
ejpam-6321	391	18	,	,	PUNCT
ejpam-6321	391	19	g(f1℘	g(f1℘	PROPN
ejpam-6321	391	20	,	,	PUNCT
ejpam-6321	391	21	℘	℘	PROPN
ejpam-6321	391	22	,	,	PUNCT
ejpam-6321	391	23	℘	℘	NOUN
ejpam-6321	391	24	)	)	PUNCT
ejpam-6321	391	25	≤	≤	NOUN
ejpam-6321	391	26	α1g(℘	α1g(℘	NOUN
ejpam-6321	391	27	,	,	PUNCT
ejpam-6321	391	28	℘	℘	NOUN
ejpam-6321	391	29	,	,	PUNCT
ejpam-6321	391	30	℘	℘	NUM
ejpam-6321	391	31	)	)	PUNCT
ejpam-6321	391	32	+	+	CCONJ
ejpam-6321	391	33	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	391	34	,	,	PUNCT
ejpam-6321	391	35	℘	℘	NOUN
ejpam-6321	391	36	,	,	PUNCT
ejpam-6321	391	37	℘	℘	NUM
ejpam-6321	391	38	)	)	PUNCT
ejpam-6321	391	39	+	+	CCONJ
ejpam-6321	391	40	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	391	41	,	,	PUNCT
ejpam-6321	391	42	℘	℘	NOUN
ejpam-6321	391	43	,	,	PUNCT
ejpam-6321	391	44	℘	℘	NUM
ejpam-6321	391	45	)	)	PUNCT
ejpam-6321	391	46	+	+	NUM
ejpam-6321	391	47	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	391	48	,	,	PUNCT
ejpam-6321	391	49	℘	℘	PROPN
ejpam-6321	391	50	,	,	PUNCT
ejpam-6321	391	51	℘	℘	NUM
ejpam-6321	391	52	)	)	PUNCT
ejpam-6321	391	53	+	+	NUM
ejpam-6321	391	54	2g(℘	2g(℘	NUM
ejpam-6321	391	55	,	,	PUNCT
ejpam-6321	391	56	℘	℘	NOUN
ejpam-6321	391	57	,	,	PUNCT
ejpam-6321	391	58	℘	℘	NUM
ejpam-6321	391	59	)	)	PUNCT
ejpam-6321	391	60	]	]	PUNCT
ejpam-6321	392	1	+	+	NUM
ejpam-6321	392	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	392	3	,	,	PUNCT
ejpam-6321	392	4	℘	℘	NOUN
ejpam-6321	392	5	,	,	PUNCT
ejpam-6321	392	6	℘	℘	NOUN
ejpam-6321	392	7	)	)	PUNCT
ejpam-6321	392	8	≤	≤	NOUN
ejpam-6321	392	9	(	(	PUNCT
ejpam-6321	392	10	α2	α2	ADJ
ejpam-6321	392	11	+	+	CCONJ
ejpam-6321	392	12	α3	α3	PROPN
ejpam-6321	392	13	+	+	CCONJ
ejpam-6321	392	14	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	392	15	,	,	PUNCT
ejpam-6321	392	16	℘	℘	PROPN
ejpam-6321	392	17	,	,	PUNCT
ejpam-6321	392	18	℘	℘	NOUN
ejpam-6321	392	19	)	)	PUNCT
ejpam-6321	392	20	g(f1℘	g(f1℘	PROPN
ejpam-6321	392	21	,	,	PUNCT
ejpam-6321	392	22	℘	℘	NOUN
ejpam-6321	392	23	,	,	PUNCT
ejpam-6321	392	24	℘	℘	NUM
ejpam-6321	392	25	)	)	PUNCT
ejpam-6321	392	26	−	−	PROPN
ejpam-6321	392	27	(	(	PUNCT
ejpam-6321	392	28	α2	α2	ADJ
ejpam-6321	392	29	+	+	CCONJ
ejpam-6321	392	30	α3	α3	PROPN
ejpam-6321	392	31	+	+	CCONJ
ejpam-6321	392	32	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	392	33	,	,	PUNCT
ejpam-6321	392	34	℘	℘	PROPN
ejpam-6321	392	35	,	,	PUNCT
ejpam-6321	392	36	℘	℘	NOUN
ejpam-6321	392	37	)	)	PUNCT
ejpam-6321	392	38	≤	≤	NOUN
ejpam-6321	392	39	0	0	NUM
ejpam-6321	392	40	.	.	PUNCT
ejpam-6321	393	1	(	(	PUNCT
ejpam-6321	393	2	1	1	NUM
ejpam-6321	393	3	−	−	NOUN
ejpam-6321	393	4	α2	α2	ADJ
ejpam-6321	393	5	−	−	PROPN
ejpam-6321	393	6	α3	α3	PROPN
ejpam-6321	393	7	−	−	PROPN
ejpam-6321	393	8	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	393	9	,	,	PUNCT
ejpam-6321	393	10	℘	℘	PROPN
ejpam-6321	393	11	,	,	PUNCT
ejpam-6321	393	12	℘	℘	NOUN
ejpam-6321	393	13	)	)	PUNCT
ejpam-6321	393	14	≤	≤	NOUN
ejpam-6321	393	15	0	0	NUM
ejpam-6321	393	16	,	,	PUNCT
ejpam-6321	393	17	is	be	AUX
ejpam-6321	393	18	a	a	DET
ejpam-6321	393	19	contradiction	contradiction	NOUN
ejpam-6321	393	20	.	.	PUNCT
ejpam-6321	394	1	since	since	SCONJ
ejpam-6321	394	2	(	(	PUNCT
ejpam-6321	394	3	1−	1−	NUM
ejpam-6321	394	4	α2	α2	ADJ
ejpam-6321	394	5	−	−	PROPN
ejpam-6321	394	6	α3	α3	PROPN
ejpam-6321	394	7	−	−	PROPN
ejpam-6321	394	8	α4	α4	NOUN
ejpam-6321	394	9	)	)	PUNCT
ejpam-6321	394	10	̸=	̸=	PROPN
ejpam-6321	394	11	0	0	NUM
ejpam-6321	394	12	,	,	PUNCT
ejpam-6321	394	13	therefore	therefore	ADV
ejpam-6321	394	14	,	,	PUNCT
ejpam-6321	394	15	g(f1℘	g(f1℘	PROPN
ejpam-6321	394	16	,	,	PUNCT
ejpam-6321	394	17	℘	℘	PROPN
ejpam-6321	394	18	,	,	PUNCT
ejpam-6321	394	19	℘	℘	NUM
ejpam-6321	394	20	)	)	PUNCT
ejpam-6321	394	21	=	=	SYM
ejpam-6321	394	22	0	0	X
ejpam-6321	394	23	.	.	PUNCT
ejpam-6321	394	24	thus	thus	ADV
ejpam-6321	394	25	,	,	PUNCT
ejpam-6321	394	26	f1℘	f1℘	PROPN
ejpam-6321	394	27	=	=	SYM
ejpam-6321	394	28	℘	℘	PROPN
ejpam-6321	394	29	(	(	PUNCT
ejpam-6321	394	30	15	15	NUM
ejpam-6321	394	31	)	)	PUNCT
ejpam-6321	394	32	(	(	PUNCT
ejpam-6321	394	33	ii	ii	NOUN
ejpam-6321	394	34	)	)	PUNCT
ejpam-6321	394	35	.	.	PUNCT
ejpam-6321	395	1	if	if	SCONJ
ejpam-6321	395	2	g(f1℘	g(f1℘	PROPN
ejpam-6321	395	3	,	,	PUNCT
ejpam-6321	395	4	f1℘	f1℘	PROPN
ejpam-6321	395	5	,	,	PUNCT
ejpam-6321	395	6	x(3k+1	x(3k+1	ADV
ejpam-6321	395	7	)	)	PUNCT
ejpam-6321	395	8	)	)	PUNCT
ejpam-6321	395	9	is	be	AUX
ejpam-6321	395	10	a	a	DET
ejpam-6321	395	11	maximum	maximum	ADJ
ejpam-6321	395	12	term	term	NOUN
ejpam-6321	395	13	in	in	ADP
ejpam-6321	395	14	(	(	PUNCT
ejpam-6321	395	15	14	14	NUM
ejpam-6321	395	16	)	)	PUNCT
ejpam-6321	395	17	then	then	ADV
ejpam-6321	395	18	we	we	PRON
ejpam-6321	395	19	can	can	AUX
ejpam-6321	395	20	write	write	VERB
ejpam-6321	395	21	;	;	PUNCT
ejpam-6321	395	22	≤	≤	NUM
ejpam-6321	395	23	α1g(℘	α1g(℘	NOUN
ejpam-6321	395	24	,	,	PUNCT
ejpam-6321	395	25	x(3k+1	x(3k+1	PRON
ejpam-6321	395	26	)	)	PUNCT
ejpam-6321	395	27	,	,	PUNCT
ejpam-6321	395	28	x(3k+2	x(3k+2	NOUN
ejpam-6321	395	29	)	)	PUNCT
ejpam-6321	395	30	)	)	PUNCT
ejpam-6321	396	1	+	+	CCONJ
ejpam-6321	396	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	396	3	,	,	PUNCT
ejpam-6321	396	4	x(3k+1	x(3k+1	ADV
ejpam-6321	396	5	)	)	PUNCT
ejpam-6321	396	6	,	,	PUNCT
ejpam-6321	396	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	396	8	)	)	PUNCT
ejpam-6321	396	9	)	)	PUNCT
ejpam-6321	397	1	+	+	CCONJ
ejpam-6321	398	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	398	2	,	,	PUNCT
ejpam-6321	398	3	x(3k+1	x(3k+1	ADV
ejpam-6321	398	4	)	)	PUNCT
ejpam-6321	398	5	,	,	PUNCT
ejpam-6321	398	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	398	7	)	)	PUNCT
ejpam-6321	398	8	)	)	PUNCT
ejpam-6321	399	1	+	+	CCONJ
ejpam-6321	400	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	400	2	,	,	PUNCT
ejpam-6321	400	3	x3k	x3k	NOUN
ejpam-6321	400	4	,	,	PUNCT
ejpam-6321	400	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	400	6	)	)	PUNCT
ejpam-6321	400	7	)	)	PUNCT
ejpam-6321	401	1	+	+	CCONJ
ejpam-6321	401	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	401	3	)	)	PUNCT
ejpam-6321	401	4	,	,	PUNCT
ejpam-6321	401	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	401	6	)	)	PUNCT
ejpam-6321	401	7	,	,	PUNCT
ejpam-6321	401	8	x(3k+3	x(3k+3	NUM
ejpam-6321	401	9	)	)	PUNCT
ejpam-6321	401	10	)	)	PUNCT
ejpam-6321	401	11	]	]	PUNCT
ejpam-6321	402	1	+	+	CCONJ
ejpam-6321	402	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	402	3	,	,	PUNCT
ejpam-6321	402	4	f1℘	f1℘	PROPN
ejpam-6321	402	5	,	,	PUNCT
ejpam-6321	402	6	x(3k+1	x(3k+1	ADV
ejpam-6321	402	7	)	)	PUNCT
ejpam-6321	402	8	)	)	PUNCT
ejpam-6321	402	9	applying	apply	VERB
ejpam-6321	402	10	limn→∞	limn→∞	PROPN
ejpam-6321	402	11	on	on	ADP
ejpam-6321	402	12	both	both	DET
ejpam-6321	402	13	sides	side	NOUN
ejpam-6321	402	14	,	,	PUNCT
ejpam-6321	402	15	we	we	PRON
ejpam-6321	402	16	get	get	VERB
ejpam-6321	402	17	,	,	PUNCT
ejpam-6321	402	18	g(f1℘	g(f1℘	PROPN
ejpam-6321	402	19	,	,	PUNCT
ejpam-6321	402	20	℘	℘	PROPN
ejpam-6321	402	21	,	,	PUNCT
ejpam-6321	402	22	℘	℘	NOUN
ejpam-6321	402	23	)	)	PUNCT
ejpam-6321	402	24	≤	≤	NOUN
ejpam-6321	402	25	α1g(℘	α1g(℘	NOUN
ejpam-6321	402	26	,	,	PUNCT
ejpam-6321	402	27	℘	℘	NOUN
ejpam-6321	402	28	,	,	PUNCT
ejpam-6321	402	29	℘	℘	NUM
ejpam-6321	402	30	)	)	PUNCT
ejpam-6321	402	31	+	+	CCONJ
ejpam-6321	403	1	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	403	2	,	,	PUNCT
ejpam-6321	403	3	℘	℘	NOUN
ejpam-6321	403	4	,	,	PUNCT
ejpam-6321	403	5	℘	℘	NUM
ejpam-6321	403	6	)	)	PUNCT
ejpam-6321	404	1	+	+	CCONJ
ejpam-6321	405	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	405	2	,	,	PUNCT
ejpam-6321	405	3	℘	℘	NOUN
ejpam-6321	405	4	,	,	PUNCT
ejpam-6321	405	5	℘	℘	NUM
ejpam-6321	405	6	)	)	PUNCT
ejpam-6321	405	7	+	+	NUM
ejpam-6321	405	8	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	405	9	,	,	PUNCT
ejpam-6321	405	10	℘	℘	PROPN
ejpam-6321	405	11	,	,	PUNCT
ejpam-6321	405	12	℘	℘	NUM
ejpam-6321	405	13	)	)	PUNCT
ejpam-6321	405	14	+	+	NUM
ejpam-6321	405	15	2g(℘	2g(℘	NUM
ejpam-6321	405	16	,	,	PUNCT
ejpam-6321	405	17	℘	℘	NOUN
ejpam-6321	405	18	,	,	PUNCT
ejpam-6321	405	19	℘	℘	PROPN
ejpam-6321	405	20	)	)	PUNCT
ejpam-6321	405	21	]	]	PUNCT
ejpam-6321	406	1	+	+	CCONJ
ejpam-6321	406	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	406	3	,	,	PUNCT
ejpam-6321	406	4	f1℘	f1℘	PROPN
ejpam-6321	406	5	,	,	PUNCT
ejpam-6321	406	6	℘	℘	PROPN
ejpam-6321	406	7	)	)	PUNCT
ejpam-6321	406	8	≤	≤	NOUN
ejpam-6321	406	9	(	(	PUNCT
ejpam-6321	406	10	α2	α2	ADJ
ejpam-6321	406	11	+	+	CCONJ
ejpam-6321	406	12	α3	α3	PROPN
ejpam-6321	406	13	+	+	CCONJ
ejpam-6321	406	14	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	406	15	,	,	PUNCT
ejpam-6321	406	16	℘	℘	PROPN
ejpam-6321	406	17	,	,	PUNCT
ejpam-6321	406	18	℘	℘	NUM
ejpam-6321	406	19	)	)	PUNCT
ejpam-6321	406	20	+	+	CCONJ
ejpam-6321	406	21	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	406	22	,	,	PUNCT
ejpam-6321	406	23	f1℘	f1℘	PROPN
ejpam-6321	406	24	,	,	PUNCT
ejpam-6321	406	25	℘	℘	PROPN
ejpam-6321	406	26	)	)	PUNCT
ejpam-6321	406	27	by	by	ADP
ejpam-6321	406	28	using	use	VERB
ejpam-6321	406	29	proposition	proposition	NOUN
ejpam-6321	406	30	≤	≤	NOUN
ejpam-6321	406	31	(	(	PUNCT
ejpam-6321	406	32	α2	α2	ADJ
ejpam-6321	406	33	+	+	CCONJ
ejpam-6321	406	34	α3	α3	PROPN
ejpam-6321	406	35	+	+	CCONJ
ejpam-6321	406	36	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	406	37	,	,	PUNCT
ejpam-6321	406	38	℘	℘	PROPN
ejpam-6321	406	39	,	,	PUNCT
ejpam-6321	406	40	℘	℘	NUM
ejpam-6321	406	41	)	)	PUNCT
ejpam-6321	407	1	+	+	CCONJ
ejpam-6321	407	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	407	3	,	,	PUNCT
ejpam-6321	407	4	℘	℘	NOUN
ejpam-6321	407	5	,	,	PUNCT
ejpam-6321	407	6	℘	℘	NOUN
ejpam-6321	407	7	)	)	PUNCT
ejpam-6321	407	8	g(f1℘	g(f1℘	PROPN
ejpam-6321	407	9	,	,	PUNCT
ejpam-6321	407	10	℘	℘	PROPN
ejpam-6321	407	11	,	,	PUNCT
ejpam-6321	407	12	℘)−	℘)−	NOUN
ejpam-6321	407	13	(	(	PUNCT
ejpam-6321	407	14	α2	α2	ADJ
ejpam-6321	407	15	+	+	CCONJ
ejpam-6321	407	16	α3	α3	ADJ
ejpam-6321	407	17	+	+	CCONJ
ejpam-6321	407	18	α4	α4	NOUN
ejpam-6321	407	19	+	+	CCONJ
ejpam-6321	407	20	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	407	21	,	,	PUNCT
ejpam-6321	407	22	℘	℘	PROPN
ejpam-6321	407	23	,	,	PUNCT
ejpam-6321	407	24	℘	℘	NOUN
ejpam-6321	407	25	)	)	PUNCT
ejpam-6321	407	26	≤	≤	NOUN
ejpam-6321	407	27	0	0	NUM
ejpam-6321	407	28	.	.	PUNCT
ejpam-6321	408	1	(	(	PUNCT
ejpam-6321	408	2	1−	1−	NUM
ejpam-6321	408	3	α2	α2	ADJ
ejpam-6321	408	4	−	−	PROPN
ejpam-6321	408	5	α3	α3	PROPN
ejpam-6321	408	6	−	−	PROPN
ejpam-6321	408	7	α4	α4	NOUN
ejpam-6321	408	8	−	−	NOUN
ejpam-6321	408	9	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	408	10	,	,	PUNCT
ejpam-6321	408	11	℘	℘	PROPN
ejpam-6321	408	12	,	,	PUNCT
ejpam-6321	408	13	℘	℘	NOUN
ejpam-6321	408	14	)	)	PUNCT
ejpam-6321	408	15	≤	≤	NOUN
ejpam-6321	408	16	0	0	NUM
ejpam-6321	408	17	,	,	PUNCT
ejpam-6321	408	18	is	be	AUX
ejpam-6321	408	19	a	a	DET
ejpam-6321	408	20	contradiction	contradiction	NOUN
ejpam-6321	408	21	.	.	PUNCT
ejpam-6321	409	1	since	since	SCONJ
ejpam-6321	409	2	(	(	PUNCT
ejpam-6321	409	3	1−	1−	NUM
ejpam-6321	409	4	α2	α2	ADJ
ejpam-6321	409	5	−	−	PROPN
ejpam-6321	409	6	α3	α3	PROPN
ejpam-6321	409	7	−	−	PROPN
ejpam-6321	409	8	α4	α4	NOUN
ejpam-6321	409	9	−	−	NOUN
ejpam-6321	409	10	α5	α5	NOUN
ejpam-6321	409	11	)	)	PUNCT
ejpam-6321	409	12	̸=	̸=	PROPN
ejpam-6321	409	13	0	0	NUM
ejpam-6321	409	14	,	,	PUNCT
ejpam-6321	409	15	therefore	therefore	ADV
ejpam-6321	409	16	,	,	PUNCT
ejpam-6321	409	17	g(f1℘	g(f1℘	PROPN
ejpam-6321	409	18	,	,	PUNCT
ejpam-6321	409	19	℘	℘	PROPN
ejpam-6321	409	20	,	,	PUNCT
ejpam-6321	409	21	℘	℘	NUM
ejpam-6321	409	22	)	)	PUNCT
ejpam-6321	409	23	=	=	SYM
ejpam-6321	409	24	0	0	X
ejpam-6321	409	25	.	.	PUNCT
ejpam-6321	409	26	thus	thus	ADV
ejpam-6321	409	27	,	,	PUNCT
ejpam-6321	409	28	f1℘	f1℘	PROPN
ejpam-6321	409	29	=	=	SYM
ejpam-6321	409	30	℘	℘	PROPN
ejpam-6321	409	31	(	(	PUNCT
ejpam-6321	409	32	16	16	NUM
ejpam-6321	409	33	)	)	PUNCT
ejpam-6321	409	34	m.	m.	NOUN
ejpam-6321	409	35	noorwali	noorwali	PROPN
ejpam-6321	409	36	et	et	PROPN
ejpam-6321	409	37	al	al	PROPN
ejpam-6321	409	38	.	.	PUNCT
ejpam-6321	409	39	/	/	SYM
ejpam-6321	409	40	eur	eur	PROPN
ejpam-6321	409	41	.	.	PUNCT
ejpam-6321	410	1	j.	j.	PROPN
ejpam-6321	410	2	pure	pure	PROPN
ejpam-6321	410	3	appl	appl	PROPN
ejpam-6321	410	4	.	.	PROPN
ejpam-6321	410	5	math	math	PROPN
ejpam-6321	410	6	,	,	PUNCT
ejpam-6321	410	7	18	18	NUM
ejpam-6321	410	8	(	(	PUNCT
ejpam-6321	410	9	3	3	NUM
ejpam-6321	410	10	)	)	PUNCT
ejpam-6321	410	11	(	(	PUNCT
ejpam-6321	410	12	2025	2025	NUM
ejpam-6321	410	13	)	)	PUNCT
ejpam-6321	410	14	,	,	PUNCT
ejpam-6321	410	15	6321	6321	NUM
ejpam-6321	410	16	15	15	NUM
ejpam-6321	410	17	of	of	ADP
ejpam-6321	410	18	27	27	NUM
ejpam-6321	410	19	(	(	PUNCT
ejpam-6321	410	20	iii	iii	NOUN
ejpam-6321	410	21	)	)	PUNCT
ejpam-6321	410	22	.	.	PUNCT
ejpam-6321	411	1	if	if	SCONJ
ejpam-6321	411	2	g(f1℘	g(f1℘	PROPN
ejpam-6321	411	3	,	,	PUNCT
ejpam-6321	411	4	x(3k+1	x(3k+1	ADV
ejpam-6321	411	5	)	)	PUNCT
ejpam-6321	411	6	,	,	PUNCT
ejpam-6321	411	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	411	8	)	)	PUNCT
ejpam-6321	411	9	)	)	PUNCT
ejpam-6321	411	10	is	be	AUX
ejpam-6321	411	11	a	a	DET
ejpam-6321	411	12	maximum	maximum	ADJ
ejpam-6321	411	13	term	term	NOUN
ejpam-6321	411	14	in	in	ADP
ejpam-6321	411	15	(	(	PUNCT
ejpam-6321	411	16	14	14	NUM
ejpam-6321	411	17	)	)	PUNCT
ejpam-6321	411	18	then	then	ADV
ejpam-6321	411	19	we	we	PRON
ejpam-6321	411	20	can	can	AUX
ejpam-6321	411	21	write	write	VERB
ejpam-6321	411	22	;	;	PUNCT
ejpam-6321	411	23	≤	≤	NUM
ejpam-6321	411	24	α1g(℘	α1g(℘	NOUN
ejpam-6321	411	25	,	,	PUNCT
ejpam-6321	411	26	x(3k+1	x(3k+1	PRON
ejpam-6321	411	27	)	)	PUNCT
ejpam-6321	411	28	,	,	PUNCT
ejpam-6321	411	29	x(3k+2	x(3k+2	NOUN
ejpam-6321	411	30	)	)	PUNCT
ejpam-6321	411	31	)	)	PUNCT
ejpam-6321	412	1	+	+	CCONJ
ejpam-6321	412	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	412	3	,	,	PUNCT
ejpam-6321	412	4	x(3k+1	x(3k+1	ADV
ejpam-6321	412	5	)	)	PUNCT
ejpam-6321	412	6	,	,	PUNCT
ejpam-6321	412	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	412	8	)	)	PUNCT
ejpam-6321	412	9	)	)	PUNCT
ejpam-6321	413	1	+	+	CCONJ
ejpam-6321	414	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	414	2	,	,	PUNCT
ejpam-6321	414	3	x(3k+1	x(3k+1	ADV
ejpam-6321	414	4	)	)	PUNCT
ejpam-6321	414	5	,	,	PUNCT
ejpam-6321	414	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	414	7	)	)	PUNCT
ejpam-6321	414	8	)	)	PUNCT
ejpam-6321	415	1	+	+	CCONJ
ejpam-6321	416	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	416	2	,	,	PUNCT
ejpam-6321	416	3	x3k	x3k	NOUN
ejpam-6321	416	4	,	,	PUNCT
ejpam-6321	416	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	416	6	)	)	PUNCT
ejpam-6321	416	7	)	)	PUNCT
ejpam-6321	417	1	+	+	CCONJ
ejpam-6321	417	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	417	3	)	)	PUNCT
ejpam-6321	417	4	,	,	PUNCT
ejpam-6321	417	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	417	6	)	)	PUNCT
ejpam-6321	417	7	,	,	PUNCT
ejpam-6321	417	8	x(3k+3	x(3k+3	NUM
ejpam-6321	417	9	)	)	PUNCT
ejpam-6321	417	10	)	)	PUNCT
ejpam-6321	417	11	]	]	PUNCT
ejpam-6321	418	1	+	+	CCONJ
ejpam-6321	418	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	418	3	,	,	PUNCT
ejpam-6321	418	4	x(3k+1	x(3k+1	ADV
ejpam-6321	418	5	)	)	PUNCT
ejpam-6321	418	6	,	,	PUNCT
ejpam-6321	418	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	418	8	)	)	PUNCT
ejpam-6321	418	9	)	)	PUNCT
ejpam-6321	418	10	applying	apply	VERB
ejpam-6321	418	11	limn→∞	limn→∞	PROPN
ejpam-6321	418	12	on	on	ADP
ejpam-6321	418	13	both	both	DET
ejpam-6321	418	14	sides	side	NOUN
ejpam-6321	418	15	,	,	PUNCT
ejpam-6321	418	16	we	we	PRON
ejpam-6321	418	17	get	get	VERB
ejpam-6321	418	18	,	,	PUNCT
ejpam-6321	418	19	g(f1℘	g(f1℘	PROPN
ejpam-6321	418	20	,	,	PUNCT
ejpam-6321	418	21	℘	℘	PROPN
ejpam-6321	418	22	,	,	PUNCT
ejpam-6321	418	23	℘	℘	NOUN
ejpam-6321	418	24	)	)	PUNCT
ejpam-6321	418	25	≤	≤	NOUN
ejpam-6321	418	26	α1g(℘	α1g(℘	NOUN
ejpam-6321	418	27	,	,	PUNCT
ejpam-6321	418	28	℘	℘	NOUN
ejpam-6321	418	29	,	,	PUNCT
ejpam-6321	418	30	℘	℘	NUM
ejpam-6321	418	31	)	)	PUNCT
ejpam-6321	418	32	+	+	CCONJ
ejpam-6321	419	1	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	419	2	,	,	PUNCT
ejpam-6321	419	3	℘	℘	NOUN
ejpam-6321	419	4	,	,	PUNCT
ejpam-6321	419	5	℘	℘	NUM
ejpam-6321	419	6	)	)	PUNCT
ejpam-6321	420	1	+	+	CCONJ
ejpam-6321	421	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	421	2	,	,	PUNCT
ejpam-6321	421	3	℘	℘	NOUN
ejpam-6321	421	4	,	,	PUNCT
ejpam-6321	421	5	℘	℘	NUM
ejpam-6321	421	6	)	)	PUNCT
ejpam-6321	421	7	+	+	NUM
ejpam-6321	421	8	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	421	9	,	,	PUNCT
ejpam-6321	421	10	℘	℘	PROPN
ejpam-6321	421	11	,	,	PUNCT
ejpam-6321	421	12	℘	℘	NUM
ejpam-6321	421	13	)	)	PUNCT
ejpam-6321	421	14	+	+	NUM
ejpam-6321	421	15	2g(℘	2g(℘	NUM
ejpam-6321	421	16	,	,	PUNCT
ejpam-6321	421	17	℘	℘	NOUN
ejpam-6321	421	18	,	,	PUNCT
ejpam-6321	421	19	℘	℘	PROPN
ejpam-6321	421	20	)	)	PUNCT
ejpam-6321	421	21	]	]	PUNCT
ejpam-6321	422	1	+	+	CCONJ
ejpam-6321	422	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	422	3	,	,	PUNCT
ejpam-6321	422	4	℘	℘	NOUN
ejpam-6321	422	5	,	,	PUNCT
ejpam-6321	422	6	℘	℘	NOUN
ejpam-6321	422	7	)	)	PUNCT
ejpam-6321	422	8	≤	≤	NOUN
ejpam-6321	422	9	(	(	PUNCT
ejpam-6321	422	10	α2	α2	ADJ
ejpam-6321	422	11	+	+	CCONJ
ejpam-6321	422	12	α3	α3	ADJ
ejpam-6321	422	13	+	+	CCONJ
ejpam-6321	422	14	α4	α4	NOUN
ejpam-6321	422	15	+	+	CCONJ
ejpam-6321	422	16	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	422	17	,	,	PUNCT
ejpam-6321	422	18	℘	℘	PROPN
ejpam-6321	422	19	,	,	PUNCT
ejpam-6321	422	20	℘	℘	NOUN
ejpam-6321	422	21	)	)	PUNCT
ejpam-6321	422	22	g(f1℘	g(f1℘	PROPN
ejpam-6321	422	23	,	,	PUNCT
ejpam-6321	422	24	℘	℘	PROPN
ejpam-6321	422	25	,	,	PUNCT
ejpam-6321	422	26	℘)−	℘)−	NOUN
ejpam-6321	422	27	(	(	PUNCT
ejpam-6321	422	28	α2	α2	ADJ
ejpam-6321	422	29	+	+	CCONJ
ejpam-6321	422	30	α3	α3	ADJ
ejpam-6321	422	31	+	+	CCONJ
ejpam-6321	422	32	α4	α4	NOUN
ejpam-6321	422	33	+	+	CCONJ
ejpam-6321	422	34	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	422	35	,	,	PUNCT
ejpam-6321	422	36	℘	℘	PROPN
ejpam-6321	422	37	,	,	PUNCT
ejpam-6321	422	38	℘	℘	NOUN
ejpam-6321	422	39	)	)	PUNCT
ejpam-6321	422	40	≤	≤	NOUN
ejpam-6321	422	41	0	0	NUM
ejpam-6321	422	42	.	.	PUNCT
ejpam-6321	423	1	(	(	PUNCT
ejpam-6321	423	2	1−	1−	NUM
ejpam-6321	423	3	α2	α2	ADJ
ejpam-6321	423	4	−	−	PROPN
ejpam-6321	423	5	α3	α3	PROPN
ejpam-6321	423	6	−	−	PROPN
ejpam-6321	423	7	α4	α4	NOUN
ejpam-6321	423	8	−	−	NOUN
ejpam-6321	423	9	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	423	10	,	,	PUNCT
ejpam-6321	423	11	℘	℘	PROPN
ejpam-6321	423	12	,	,	PUNCT
ejpam-6321	423	13	℘	℘	NOUN
ejpam-6321	423	14	)	)	PUNCT
ejpam-6321	423	15	≤	≤	NOUN
ejpam-6321	423	16	0	0	NUM
ejpam-6321	423	17	,	,	PUNCT
ejpam-6321	423	18	is	be	AUX
ejpam-6321	423	19	a	a	DET
ejpam-6321	423	20	contradiction	contradiction	NOUN
ejpam-6321	423	21	.	.	PUNCT
ejpam-6321	424	1	since	since	SCONJ
ejpam-6321	424	2	(	(	PUNCT
ejpam-6321	424	3	1−	1−	NUM
ejpam-6321	424	4	α2	α2	ADJ
ejpam-6321	424	5	−	−	PROPN
ejpam-6321	424	6	α3	α3	PROPN
ejpam-6321	424	7	−	−	PROPN
ejpam-6321	424	8	α4	α4	NOUN
ejpam-6321	424	9	−	−	NOUN
ejpam-6321	424	10	α5	α5	NOUN
ejpam-6321	424	11	)	)	PUNCT
ejpam-6321	424	12	̸=	̸=	PROPN
ejpam-6321	424	13	0	0	NUM
ejpam-6321	424	14	,	,	PUNCT
ejpam-6321	424	15	therefore	therefore	ADV
ejpam-6321	424	16	,	,	PUNCT
ejpam-6321	424	17	g(f1℘	g(f1℘	PROPN
ejpam-6321	424	18	,	,	PUNCT
ejpam-6321	424	19	℘	℘	PROPN
ejpam-6321	424	20	,	,	PUNCT
ejpam-6321	424	21	℘	℘	NUM
ejpam-6321	424	22	)	)	PUNCT
ejpam-6321	424	23	=	=	SYM
ejpam-6321	424	24	0	0	X
ejpam-6321	424	25	.	.	PUNCT
ejpam-6321	424	26	thus	thus	ADV
ejpam-6321	424	27	,	,	PUNCT
ejpam-6321	424	28	f1℘	f1℘	PROPN
ejpam-6321	424	29	=	=	SYM
ejpam-6321	424	30	℘	℘	PROPN
ejpam-6321	424	31	(	(	PUNCT
ejpam-6321	424	32	17	17	NUM
ejpam-6321	424	33	)	)	PUNCT
ejpam-6321	424	34	(	(	PUNCT
ejpam-6321	424	35	iv	iv	X
ejpam-6321	424	36	)	)	PUNCT
ejpam-6321	424	37	.	.	PUNCT
ejpam-6321	425	1	if	if	SCONJ
ejpam-6321	425	2	g(f1℘	g(f1℘	PROPN
ejpam-6321	425	3	,	,	PUNCT
ejpam-6321	425	4	x3k	x3k	NOUN
ejpam-6321	425	5	,	,	PUNCT
ejpam-6321	425	6	x(3k+2	x(3k+2	NUM
ejpam-6321	425	7	)	)	PUNCT
ejpam-6321	425	8	)	)	PUNCT
ejpam-6321	425	9	is	be	AUX
ejpam-6321	425	10	a	a	DET
ejpam-6321	425	11	maximum	maximum	ADJ
ejpam-6321	425	12	term	term	NOUN
ejpam-6321	425	13	in	in	ADP
ejpam-6321	425	14	(	(	PUNCT
ejpam-6321	425	15	14	14	NUM
ejpam-6321	425	16	)	)	PUNCT
ejpam-6321	425	17	then	then	ADV
ejpam-6321	425	18	we	we	PRON
ejpam-6321	425	19	can	can	AUX
ejpam-6321	425	20	write	write	VERB
ejpam-6321	425	21	;	;	PUNCT
ejpam-6321	425	22	≤	≤	NUM
ejpam-6321	425	23	α1g(℘	α1g(℘	NOUN
ejpam-6321	425	24	,	,	PUNCT
ejpam-6321	425	25	x(3k+1	x(3k+1	PRON
ejpam-6321	425	26	)	)	PUNCT
ejpam-6321	425	27	,	,	PUNCT
ejpam-6321	425	28	x(3k+2	x(3k+2	NOUN
ejpam-6321	425	29	)	)	PUNCT
ejpam-6321	425	30	)	)	PUNCT
ejpam-6321	426	1	+	+	CCONJ
ejpam-6321	426	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	426	3	,	,	PUNCT
ejpam-6321	426	4	x(3k+1	x(3k+1	ADV
ejpam-6321	426	5	)	)	PUNCT
ejpam-6321	426	6	,	,	PUNCT
ejpam-6321	426	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	426	8	)	)	PUNCT
ejpam-6321	426	9	)	)	PUNCT
ejpam-6321	427	1	+	+	CCONJ
ejpam-6321	428	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	428	2	,	,	PUNCT
ejpam-6321	428	3	x(3k+1	x(3k+1	ADV
ejpam-6321	428	4	)	)	PUNCT
ejpam-6321	428	5	,	,	PUNCT
ejpam-6321	428	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	428	7	)	)	PUNCT
ejpam-6321	428	8	)	)	PUNCT
ejpam-6321	429	1	+	+	CCONJ
ejpam-6321	430	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	430	2	,	,	PUNCT
ejpam-6321	430	3	x3k	x3k	NOUN
ejpam-6321	430	4	,	,	PUNCT
ejpam-6321	430	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	430	6	)	)	PUNCT
ejpam-6321	430	7	)	)	PUNCT
ejpam-6321	431	1	+	+	CCONJ
ejpam-6321	431	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	431	3	)	)	PUNCT
ejpam-6321	431	4	,	,	PUNCT
ejpam-6321	431	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	431	6	)	)	PUNCT
ejpam-6321	431	7	,	,	PUNCT
ejpam-6321	431	8	x(3k+3	x(3k+3	NUM
ejpam-6321	431	9	)	)	PUNCT
ejpam-6321	431	10	)	)	PUNCT
ejpam-6321	431	11	]	]	PUNCT
ejpam-6321	432	1	+	+	CCONJ
ejpam-6321	432	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	432	3	,	,	PUNCT
ejpam-6321	432	4	x3k	x3k	NOUN
ejpam-6321	432	5	,	,	PUNCT
ejpam-6321	432	6	x(3k+2	x(3k+2	NUM
ejpam-6321	432	7	)	)	PUNCT
ejpam-6321	432	8	)	)	PUNCT
ejpam-6321	432	9	applying	apply	VERB
ejpam-6321	432	10	limn→∞	limn→∞	PROPN
ejpam-6321	432	11	on	on	ADP
ejpam-6321	432	12	both	both	DET
ejpam-6321	432	13	sides	side	NOUN
ejpam-6321	432	14	,	,	PUNCT
ejpam-6321	432	15	we	we	PRON
ejpam-6321	432	16	get	get	VERB
ejpam-6321	432	17	,	,	PUNCT
ejpam-6321	432	18	g(f1℘	g(f1℘	PROPN
ejpam-6321	432	19	,	,	PUNCT
ejpam-6321	432	20	℘	℘	PROPN
ejpam-6321	432	21	,	,	PUNCT
ejpam-6321	432	22	℘	℘	NOUN
ejpam-6321	432	23	)	)	PUNCT
ejpam-6321	432	24	≤	≤	NOUN
ejpam-6321	432	25	α1g(℘	α1g(℘	NOUN
ejpam-6321	432	26	,	,	PUNCT
ejpam-6321	432	27	℘	℘	NOUN
ejpam-6321	432	28	,	,	PUNCT
ejpam-6321	432	29	℘	℘	NUM
ejpam-6321	432	30	)	)	PUNCT
ejpam-6321	432	31	+	+	CCONJ
ejpam-6321	433	1	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	433	2	,	,	PUNCT
ejpam-6321	433	3	℘	℘	NOUN
ejpam-6321	433	4	,	,	PUNCT
ejpam-6321	433	5	℘	℘	NUM
ejpam-6321	433	6	)	)	PUNCT
ejpam-6321	434	1	+	+	CCONJ
ejpam-6321	435	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	435	2	,	,	PUNCT
ejpam-6321	435	3	℘	℘	NOUN
ejpam-6321	435	4	,	,	PUNCT
ejpam-6321	435	5	℘	℘	NUM
ejpam-6321	435	6	)	)	PUNCT
ejpam-6321	435	7	+	+	NUM
ejpam-6321	435	8	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	435	9	,	,	PUNCT
ejpam-6321	435	10	℘	℘	PROPN
ejpam-6321	435	11	,	,	PUNCT
ejpam-6321	435	12	℘	℘	NUM
ejpam-6321	435	13	)	)	PUNCT
ejpam-6321	435	14	+	+	NUM
ejpam-6321	435	15	2g(℘	2g(℘	NUM
ejpam-6321	435	16	,	,	PUNCT
ejpam-6321	435	17	℘	℘	NOUN
ejpam-6321	435	18	,	,	PUNCT
ejpam-6321	435	19	℘	℘	PROPN
ejpam-6321	435	20	)	)	PUNCT
ejpam-6321	435	21	]	]	PUNCT
ejpam-6321	436	1	+	+	CCONJ
ejpam-6321	436	2	α5g(f1℘	α5g(f1℘	PROPN
ejpam-6321	436	3	,	,	PUNCT
ejpam-6321	436	4	℘	℘	NOUN
ejpam-6321	436	5	,	,	PUNCT
ejpam-6321	436	6	℘	℘	NOUN
ejpam-6321	436	7	)	)	PUNCT
ejpam-6321	436	8	≤	≤	NOUN
ejpam-6321	436	9	(	(	PUNCT
ejpam-6321	436	10	α2	α2	ADJ
ejpam-6321	436	11	+	+	CCONJ
ejpam-6321	436	12	α3	α3	ADJ
ejpam-6321	436	13	+	+	CCONJ
ejpam-6321	436	14	α4	α4	NOUN
ejpam-6321	436	15	+	+	CCONJ
ejpam-6321	436	16	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	436	17	,	,	PUNCT
ejpam-6321	436	18	℘	℘	PROPN
ejpam-6321	436	19	,	,	PUNCT
ejpam-6321	436	20	℘	℘	NOUN
ejpam-6321	436	21	)	)	PUNCT
ejpam-6321	436	22	g(f1℘	g(f1℘	PROPN
ejpam-6321	436	23	,	,	PUNCT
ejpam-6321	436	24	℘	℘	PROPN
ejpam-6321	436	25	,	,	PUNCT
ejpam-6321	436	26	℘)−	℘)−	NOUN
ejpam-6321	436	27	(	(	PUNCT
ejpam-6321	436	28	α2	α2	ADJ
ejpam-6321	436	29	+	+	CCONJ
ejpam-6321	436	30	α3	α3	ADJ
ejpam-6321	436	31	+	+	CCONJ
ejpam-6321	436	32	α4	α4	NOUN
ejpam-6321	436	33	+	+	CCONJ
ejpam-6321	436	34	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	436	35	,	,	PUNCT
ejpam-6321	436	36	℘	℘	PROPN
ejpam-6321	436	37	,	,	PUNCT
ejpam-6321	436	38	℘	℘	NOUN
ejpam-6321	436	39	)	)	PUNCT
ejpam-6321	436	40	≤	≤	NOUN
ejpam-6321	436	41	0	0	NUM
ejpam-6321	436	42	.	.	PUNCT
ejpam-6321	437	1	(	(	PUNCT
ejpam-6321	437	2	1−	1−	NUM
ejpam-6321	437	3	α2	α2	ADJ
ejpam-6321	437	4	−	−	PROPN
ejpam-6321	437	5	α3	α3	PROPN
ejpam-6321	437	6	−	−	PROPN
ejpam-6321	437	7	α4	α4	NOUN
ejpam-6321	437	8	−	−	NOUN
ejpam-6321	437	9	α5)g(f1℘	α5)g(f1℘	NOUN
ejpam-6321	437	10	,	,	PUNCT
ejpam-6321	437	11	℘	℘	PROPN
ejpam-6321	437	12	,	,	PUNCT
ejpam-6321	437	13	℘	℘	NOUN
ejpam-6321	437	14	)	)	PUNCT
ejpam-6321	437	15	≤	≤	NOUN
ejpam-6321	437	16	0	0	NUM
ejpam-6321	437	17	,	,	PUNCT
ejpam-6321	437	18	is	be	AUX
ejpam-6321	437	19	a	a	DET
ejpam-6321	437	20	contradiction	contradiction	NOUN
ejpam-6321	437	21	.	.	PUNCT
ejpam-6321	438	1	since	since	SCONJ
ejpam-6321	438	2	(	(	PUNCT
ejpam-6321	438	3	1−	1−	NUM
ejpam-6321	438	4	α2	α2	ADJ
ejpam-6321	438	5	−	−	PROPN
ejpam-6321	438	6	α3	α3	PROPN
ejpam-6321	438	7	−	−	PROPN
ejpam-6321	438	8	α4	α4	NOUN
ejpam-6321	438	9	−	−	NOUN
ejpam-6321	438	10	α5	α5	NOUN
ejpam-6321	438	11	)	)	PUNCT
ejpam-6321	438	12	̸=	̸=	PROPN
ejpam-6321	438	13	0	0	NUM
ejpam-6321	438	14	,	,	PUNCT
ejpam-6321	438	15	therefore	therefore	ADV
ejpam-6321	438	16	,	,	PUNCT
ejpam-6321	438	17	g(f1℘	g(f1℘	PROPN
ejpam-6321	438	18	,	,	PUNCT
ejpam-6321	438	19	℘	℘	PROPN
ejpam-6321	438	20	,	,	PUNCT
ejpam-6321	438	21	℘	℘	NUM
ejpam-6321	438	22	)	)	PUNCT
ejpam-6321	438	23	=	=	SYM
ejpam-6321	438	24	0	0	X
ejpam-6321	438	25	.	.	PUNCT
ejpam-6321	438	26	thus	thus	ADV
ejpam-6321	438	27	,	,	PUNCT
ejpam-6321	438	28	f1℘	f1℘	PROPN
ejpam-6321	438	29	=	=	SYM
ejpam-6321	438	30	℘	℘	PROPN
ejpam-6321	438	31	(	(	PUNCT
ejpam-6321	438	32	18	18	NUM
ejpam-6321	438	33	)	)	PUNCT
ejpam-6321	438	34	(	(	PUNCT
ejpam-6321	438	35	v	v	NOUN
ejpam-6321	438	36	)	)	PUNCT
ejpam-6321	438	37	.	.	PUNCT
ejpam-6321	439	1	if	if	SCONJ
ejpam-6321	439	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	439	3	)	)	PUNCT
ejpam-6321	439	4	,	,	PUNCT
ejpam-6321	439	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	439	6	)	)	PUNCT
ejpam-6321	439	7	,	,	PUNCT
ejpam-6321	439	8	x(3k+3	x(3k+3	NUM
ejpam-6321	439	9	)	)	PUNCT
ejpam-6321	439	10	)	)	PUNCT
ejpam-6321	439	11	is	be	AUX
ejpam-6321	439	12	a	a	DET
ejpam-6321	439	13	maximum	maximum	ADJ
ejpam-6321	439	14	term	term	NOUN
ejpam-6321	439	15	in	in	ADP
ejpam-6321	439	16	(	(	PUNCT
ejpam-6321	439	17	14	14	NUM
ejpam-6321	439	18	)	)	PUNCT
ejpam-6321	439	19	then	then	ADV
ejpam-6321	439	20	we	we	PRON
ejpam-6321	439	21	can	can	AUX
ejpam-6321	439	22	write	write	VERB
ejpam-6321	439	23	;	;	PUNCT
ejpam-6321	439	24	≤	≤	NUM
ejpam-6321	439	25	α1g(℘	α1g(℘	NOUN
ejpam-6321	439	26	,	,	PUNCT
ejpam-6321	439	27	x(3k+1	x(3k+1	PRON
ejpam-6321	439	28	)	)	PUNCT
ejpam-6321	439	29	,	,	PUNCT
ejpam-6321	439	30	x(3k+2	x(3k+2	NOUN
ejpam-6321	439	31	)	)	PUNCT
ejpam-6321	439	32	)	)	PUNCT
ejpam-6321	440	1	+	+	CCONJ
ejpam-6321	440	2	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	440	3	,	,	PUNCT
ejpam-6321	440	4	x(3k+1	x(3k+1	ADV
ejpam-6321	440	5	)	)	PUNCT
ejpam-6321	440	6	,	,	PUNCT
ejpam-6321	440	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	440	8	)	)	PUNCT
ejpam-6321	440	9	)	)	PUNCT
ejpam-6321	441	1	+	+	CCONJ
ejpam-6321	442	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	442	2	,	,	PUNCT
ejpam-6321	442	3	x(3k+1	x(3k+1	ADV
ejpam-6321	442	4	)	)	PUNCT
ejpam-6321	442	5	,	,	PUNCT
ejpam-6321	442	6	x(3k+1	x(3k+1	CCONJ
ejpam-6321	442	7	)	)	PUNCT
ejpam-6321	442	8	)	)	PUNCT
ejpam-6321	443	1	+	+	CCONJ
ejpam-6321	444	1	α4[g(f1℘	α4[g(f1℘	NOUN
ejpam-6321	444	2	,	,	PUNCT
ejpam-6321	444	3	x3k	x3k	NOUN
ejpam-6321	444	4	,	,	PUNCT
ejpam-6321	444	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	444	6	)	)	PUNCT
ejpam-6321	444	7	)	)	PUNCT
ejpam-6321	445	1	+	+	CCONJ
ejpam-6321	445	2	2g(x(3k+1	2g(x(3k+1	NUM
ejpam-6321	445	3	)	)	PUNCT
ejpam-6321	445	4	,	,	PUNCT
ejpam-6321	445	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	445	6	)	)	PUNCT
ejpam-6321	445	7	,	,	PUNCT
ejpam-6321	445	8	x(3k+3	x(3k+3	NUM
ejpam-6321	445	9	)	)	PUNCT
ejpam-6321	445	10	)	)	PUNCT
ejpam-6321	445	11	]	]	PUNCT
ejpam-6321	446	1	+	+	CCONJ
ejpam-6321	446	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	446	3	)	)	PUNCT
ejpam-6321	446	4	,	,	PUNCT
ejpam-6321	446	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	446	6	)	)	PUNCT
ejpam-6321	446	7	,	,	PUNCT
ejpam-6321	446	8	x(3k+3	x(3k+3	NUM
ejpam-6321	446	9	)	)	PUNCT
ejpam-6321	446	10	)	)	PUNCT
ejpam-6321	447	1	applying	apply	VERB
ejpam-6321	447	2	limn→∞	limn→∞	PROPN
ejpam-6321	447	3	on	on	ADP
ejpam-6321	447	4	both	both	DET
ejpam-6321	447	5	sides	side	NOUN
ejpam-6321	447	6	,	,	PUNCT
ejpam-6321	447	7	we	we	PRON
ejpam-6321	447	8	get	get	VERB
ejpam-6321	447	9	,	,	PUNCT
ejpam-6321	447	10	g(f1℘	g(f1℘	PROPN
ejpam-6321	447	11	,	,	PUNCT
ejpam-6321	447	12	℘	℘	PROPN
ejpam-6321	447	13	,	,	PUNCT
ejpam-6321	447	14	℘	℘	NOUN
ejpam-6321	447	15	)	)	PUNCT
ejpam-6321	447	16	≤	≤	NOUN
ejpam-6321	447	17	α1g(℘	α1g(℘	NOUN
ejpam-6321	447	18	,	,	PUNCT
ejpam-6321	447	19	℘	℘	NOUN
ejpam-6321	447	20	,	,	PUNCT
ejpam-6321	447	21	℘	℘	NUM
ejpam-6321	447	22	)	)	PUNCT
ejpam-6321	447	23	+	+	CCONJ
ejpam-6321	448	1	α2g(f1℘	α2g(f1℘	NOUN
ejpam-6321	448	2	,	,	PUNCT
ejpam-6321	448	3	℘	℘	NOUN
ejpam-6321	448	4	,	,	PUNCT
ejpam-6321	448	5	℘	℘	NUM
ejpam-6321	448	6	)	)	PUNCT
ejpam-6321	449	1	+	+	CCONJ
ejpam-6321	450	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	450	2	,	,	PUNCT
ejpam-6321	450	3	℘	℘	NOUN
ejpam-6321	450	4	,	,	PUNCT
ejpam-6321	450	5	℘	℘	NUM
ejpam-6321	450	6	)	)	PUNCT
ejpam-6321	450	7	+	+	NUM
ejpam-6321	450	8	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	450	9	,	,	PUNCT
ejpam-6321	450	10	℘	℘	PROPN
ejpam-6321	450	11	,	,	PUNCT
ejpam-6321	450	12	℘	℘	NUM
ejpam-6321	450	13	)	)	PUNCT
ejpam-6321	450	14	+	+	NUM
ejpam-6321	450	15	2g(℘	2g(℘	NUM
ejpam-6321	450	16	,	,	PUNCT
ejpam-6321	450	17	℘	℘	NOUN
ejpam-6321	450	18	,	,	PUNCT
ejpam-6321	450	19	℘	℘	NUM
ejpam-6321	450	20	)	)	PUNCT
ejpam-6321	450	21	]	]	PUNCT
ejpam-6321	451	1	+	+	NUM
ejpam-6321	451	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	451	3	,	,	PUNCT
ejpam-6321	451	4	℘	℘	NOUN
ejpam-6321	451	5	,	,	PUNCT
ejpam-6321	451	6	℘	℘	NOUN
ejpam-6321	451	7	)	)	PUNCT
ejpam-6321	451	8	≤	≤	NOUN
ejpam-6321	451	9	(	(	PUNCT
ejpam-6321	451	10	α2	α2	ADJ
ejpam-6321	451	11	+	+	CCONJ
ejpam-6321	451	12	α3	α3	PROPN
ejpam-6321	451	13	+	+	CCONJ
ejpam-6321	451	14	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	451	15	,	,	PUNCT
ejpam-6321	451	16	℘	℘	PROPN
ejpam-6321	451	17	,	,	PUNCT
ejpam-6321	451	18	℘	℘	NOUN
ejpam-6321	451	19	)	)	PUNCT
ejpam-6321	451	20	g(f1℘	g(f1℘	PROPN
ejpam-6321	451	21	,	,	PUNCT
ejpam-6321	451	22	℘	℘	NOUN
ejpam-6321	451	23	,	,	PUNCT
ejpam-6321	451	24	℘	℘	NUM
ejpam-6321	451	25	)	)	PUNCT
ejpam-6321	451	26	−	−	PROPN
ejpam-6321	451	27	(	(	PUNCT
ejpam-6321	451	28	α2	α2	ADJ
ejpam-6321	451	29	+	+	CCONJ
ejpam-6321	451	30	α3	α3	PROPN
ejpam-6321	451	31	+	+	CCONJ
ejpam-6321	451	32	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	451	33	,	,	PUNCT
ejpam-6321	451	34	℘	℘	PROPN
ejpam-6321	451	35	,	,	PUNCT
ejpam-6321	451	36	℘	℘	NOUN
ejpam-6321	451	37	)	)	PUNCT
ejpam-6321	451	38	≤	≤	NOUN
ejpam-6321	451	39	0	0	NUM
ejpam-6321	451	40	.	.	PUNCT
ejpam-6321	452	1	(	(	PUNCT
ejpam-6321	452	2	1	1	NUM
ejpam-6321	452	3	−	−	NOUN
ejpam-6321	452	4	α2	α2	ADJ
ejpam-6321	452	5	−	−	PROPN
ejpam-6321	452	6	α3	α3	PROPN
ejpam-6321	452	7	−	−	PROPN
ejpam-6321	452	8	α4)g(f1℘	α4)g(f1℘	NOUN
ejpam-6321	452	9	,	,	PUNCT
ejpam-6321	452	10	℘	℘	PROPN
ejpam-6321	452	11	,	,	PUNCT
ejpam-6321	452	12	℘	℘	NOUN
ejpam-6321	452	13	)	)	PUNCT
ejpam-6321	452	14	≤	≤	NOUN
ejpam-6321	452	15	0	0	NUM
ejpam-6321	452	16	,	,	PUNCT
ejpam-6321	452	17	is	be	AUX
ejpam-6321	452	18	a	a	DET
ejpam-6321	452	19	contradiction	contradiction	NOUN
ejpam-6321	452	20	.	.	PUNCT
ejpam-6321	453	1	since	since	SCONJ
ejpam-6321	453	2	(	(	PUNCT
ejpam-6321	453	3	1−	1−	NUM
ejpam-6321	453	4	α2	α2	ADJ
ejpam-6321	453	5	−	−	PROPN
ejpam-6321	453	6	α3	α3	PROPN
ejpam-6321	453	7	−	−	PROPN
ejpam-6321	453	8	α4	α4	NOUN
ejpam-6321	453	9	)	)	PUNCT
ejpam-6321	453	10	̸=	̸=	PROPN
ejpam-6321	453	11	0	0	NUM
ejpam-6321	453	12	,	,	PUNCT
ejpam-6321	453	13	therefore	therefore	ADV
ejpam-6321	453	14	,	,	PUNCT
ejpam-6321	453	15	g(f1℘	g(f1℘	PROPN
ejpam-6321	453	16	,	,	PUNCT
ejpam-6321	453	17	℘	℘	PROPN
ejpam-6321	453	18	,	,	PUNCT
ejpam-6321	453	19	℘	℘	NUM
ejpam-6321	453	20	)	)	PUNCT
ejpam-6321	453	21	=	=	SYM
ejpam-6321	453	22	0	0	X
ejpam-6321	453	23	.	.	PUNCT
ejpam-6321	453	24	thus	thus	ADV
ejpam-6321	453	25	,	,	PUNCT
ejpam-6321	453	26	f1℘	f1℘	PROPN
ejpam-6321	453	27	=	=	SYM
ejpam-6321	453	28	℘	℘	PROPN
ejpam-6321	453	29	(	(	PUNCT
ejpam-6321	453	30	19	19	NUM
ejpam-6321	453	31	)	)	PUNCT
ejpam-6321	453	32	by	by	ADP
ejpam-6321	453	33	the	the	DET
ejpam-6321	453	34	view	view	NOUN
ejpam-6321	453	35	of	of	ADP
ejpam-6321	453	36	(	(	PUNCT
ejpam-6321	453	37	15)−	15)−	NUM
ejpam-6321	453	38	(	(	PUNCT
ejpam-6321	453	39	19	19	NUM
ejpam-6321	453	40	)	)	PUNCT
ejpam-6321	453	41	,	,	PUNCT
ejpam-6321	453	42	we	we	PRON
ejpam-6321	453	43	obtain	obtain	VERB
ejpam-6321	453	44	that	that	SCONJ
ejpam-6321	453	45	℘	℘	PROPN
ejpam-6321	453	46	is	be	AUX
ejpam-6321	453	47	a	a	DET
ejpam-6321	453	48	fp	fp	X
ejpam-6321	453	49	of	of	ADP
ejpam-6321	453	50	f1	f1	NOUN
ejpam-6321	453	51	,	,	PUNCT
ejpam-6321	453	52	f2	f2	PROPN
ejpam-6321	453	53	and	and	CCONJ
ejpam-6321	453	54	f3	f3	PROPN
ejpam-6321	453	55	,	,	PUNCT
ejpam-6321	453	56	i.e	i.e	PROPN
ejpam-6321	453	57	,	,	PUNCT
ejpam-6321	454	1	f1℘	f1℘	PROPN
ejpam-6321	454	2	=	=	SYM
ejpam-6321	454	3	℘	℘	PROPN
ejpam-6321	454	4	(	(	PUNCT
ejpam-6321	454	5	20	20	NUM
ejpam-6321	454	6	)	)	PUNCT
ejpam-6321	454	7	m.	m.	NOUN
ejpam-6321	454	8	noorwali	noorwali	PROPN
ejpam-6321	454	9	et	et	PROPN
ejpam-6321	454	10	al	al	PROPN
ejpam-6321	454	11	.	.	PUNCT
ejpam-6321	454	12	/	/	SYM
ejpam-6321	454	13	eur	eur	PROPN
ejpam-6321	454	14	.	.	PUNCT
ejpam-6321	455	1	j.	j.	PROPN
ejpam-6321	455	2	pure	pure	PROPN
ejpam-6321	455	3	appl	appl	PROPN
ejpam-6321	455	4	.	.	PROPN
ejpam-6321	455	5	math	math	PROPN
ejpam-6321	455	6	,	,	PUNCT
ejpam-6321	455	7	18	18	NUM
ejpam-6321	455	8	(	(	PUNCT
ejpam-6321	455	9	3	3	NUM
ejpam-6321	455	10	)	)	PUNCT
ejpam-6321	455	11	(	(	PUNCT
ejpam-6321	455	12	2025	2025	NUM
ejpam-6321	455	13	)	)	PUNCT
ejpam-6321	455	14	,	,	PUNCT
ejpam-6321	455	15	6321	6321	NUM
ejpam-6321	455	16	16	16	NUM
ejpam-6321	455	17	of	of	ADP
ejpam-6321	455	18	27	27	NUM
ejpam-6321	455	19	next	next	ADV
ejpam-6321	455	20	,	,	PUNCT
ejpam-6321	455	21	we	we	PRON
ejpam-6321	455	22	have	have	VERB
ejpam-6321	455	23	to	to	PART
ejpam-6321	455	24	show	show	VERB
ejpam-6321	455	25	that	that	SCONJ
ejpam-6321	455	26	f2℘	f2℘	NOUN
ejpam-6321	455	27	=	=	PUNCT
ejpam-6321	455	28	℘	℘	PROPN
ejpam-6321	455	29	by	by	ADP
ejpam-6321	455	30	contrary	contrary	ADJ
ejpam-6321	455	31	case	case	NOUN
ejpam-6321	455	32	let	let	VERB
ejpam-6321	455	33	f2℘	f2℘	NOUN
ejpam-6321	455	34	̸=	̸=	PROPN
ejpam-6321	455	35	℘.	℘.	PROPN
ejpam-6321	455	36	then	then	ADV
ejpam-6321	455	37	from	from	ADP
ejpam-6321	455	38	(	(	PUNCT
ejpam-6321	455	39	8)	8)	NUM
ejpam-6321	455	40	we	we	PRON
ejpam-6321	455	41	have	have	VERB
ejpam-6321	455	42	that	that	PRON
ejpam-6321	455	43	,	,	PUNCT
ejpam-6321	455	44	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	455	45	)	)	PUNCT
ejpam-6321	455	46	,	,	PUNCT
ejpam-6321	455	47	f2℘	f2℘	NOUN
ejpam-6321	455	48	,	,	PUNCT
ejpam-6321	455	49	x(3k+3	x(3k+3	NUM
ejpam-6321	455	50	)	)	PUNCT
ejpam-6321	455	51	)	)	PUNCT
ejpam-6321	456	1	=	=	PUNCT
ejpam-6321	456	2	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	456	3	,	,	PUNCT
ejpam-6321	456	4	f2℘	f2℘	NOUN
ejpam-6321	456	5	,	,	PUNCT
ejpam-6321	456	6	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	456	7	)	)	PUNCT
ejpam-6321	456	8	)	)	PUNCT
ejpam-6321	457	1	≤	≤	NUM
ejpam-6321	457	2	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	457	3	,	,	PUNCT
ejpam-6321	457	4	℘	℘	PROPN
ejpam-6321	457	5	,	,	PUNCT
ejpam-6321	457	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	457	7	)	)	PUNCT
ejpam-6321	457	8	)	)	PUNCT
ejpam-6321	458	1	+	+	CCONJ
ejpam-6321	458	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	458	3	,	,	PUNCT
ejpam-6321	458	4	℘	℘	PROPN
ejpam-6321	458	5	,	,	PUNCT
ejpam-6321	458	6	f2℘	f2℘	NOUN
ejpam-6321	458	7	)	)	PUNCT
ejpam-6321	458	8	+	+	CCONJ
ejpam-6321	458	9	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	458	10	,	,	PUNCT
ejpam-6321	458	11	℘	℘	NOUN
ejpam-6321	458	12	,	,	PUNCT
ejpam-6321	458	13	℘	℘	NUM
ejpam-6321	458	14	)	)	PUNCT
ejpam-6321	458	15	+	+	SYM
ejpam-6321	458	16	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	458	17	,	,	PUNCT
ejpam-6321	458	18	℘	℘	PROPN
ejpam-6321	458	19	,	,	PUNCT
ejpam-6321	458	20	x(3k+2	x(3k+2	NOUN
ejpam-6321	458	21	)	)	PUNCT
ejpam-6321	458	22	)	)	PUNCT
ejpam-6321	459	1	+	+	ADP
ejpam-6321	459	2	g(x3k	g(x3k	NOUN
ejpam-6321	459	3	,	,	PUNCT
ejpam-6321	459	4	f2℘	f2℘	NOUN
ejpam-6321	459	5	,	,	PUNCT
ejpam-6321	459	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	459	7	)	)	PUNCT
ejpam-6321	459	8	)	)	PUNCT
ejpam-6321	460	1	+	+	ADP
ejpam-6321	460	2	g(x3k	g(x3k	NOUN
ejpam-6321	460	3	,	,	PUNCT
ejpam-6321	460	4	℘	℘	NOUN
ejpam-6321	460	5	,	,	PUNCT
ejpam-6321	460	6	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	460	7	)	)	PUNCT
ejpam-6321	460	8	)	)	PUNCT
ejpam-6321	460	9	]	]	PUNCT
ejpam-6321	461	1	+	+	CCONJ
ejpam-6321	461	2	α5max	α5max	PROPN
ejpam-6321	461	3			PROPN
ejpam-6321	461	4	g(x3k	g(x3k	NOUN
ejpam-6321	461	5	,	,	PUNCT
ejpam-6321	461	6	℘	℘	NOUN
ejpam-6321	461	7	,	,	PUNCT
ejpam-6321	461	8	f2℘	f2℘	NOUN
ejpam-6321	461	9	)	)	PUNCT
ejpam-6321	461	10	,	,	PUNCT
ejpam-6321	461	11	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	461	12	,	,	PUNCT
ejpam-6321	461	13	f1x3k	f1x3k	NUM
ejpam-6321	461	14	,	,	PUNCT
ejpam-6321	461	15	℘	℘	PROPN
ejpam-6321	461	16	)	)	PUNCT
ejpam-6321	461	17	,	,	PUNCT
ejpam-6321	461	18	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	461	19	,	,	PUNCT
ejpam-6321	461	20	℘	℘	PROPN
ejpam-6321	461	21	,	,	PUNCT
ejpam-6321	461	22	℘	℘	PROPN
ejpam-6321	461	23	)	)	PUNCT
ejpam-6321	461	24	,	,	PUNCT
ejpam-6321	461	25	g(f2℘	g(f2℘	PROPN
ejpam-6321	461	26	,	,	PUNCT
ejpam-6321	461	27	f2℘	f2℘	NOUN
ejpam-6321	461	28	,	,	PUNCT
ejpam-6321	461	29	℘	℘	PROPN
ejpam-6321	461	30	)	)	PUNCT
ejpam-6321	461	31	,	,	PUNCT
ejpam-6321	461	32	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	461	33	,	,	PUNCT
ejpam-6321	461	34	x3k	x3k	NOUN
ejpam-6321	461	35	,	,	PUNCT
ejpam-6321	461	36	x3k	x3k	NOUN
ejpam-6321	461	37	)	)	PUNCT
ejpam-6321	461	38	,	,	PUNCT
ejpam-6321	461	39	g(℘	g(℘	PROPN
ejpam-6321	461	40	,	,	PUNCT
ejpam-6321	461	41	f2℘	f2℘	NOUN
ejpam-6321	461	42	,	,	PUNCT
ejpam-6321	461	43	℘	℘	PROPN
ejpam-6321	461	44	)	)	PUNCT
ejpam-6321	461	45	,	,	PUNCT
ejpam-6321	461	46	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	461	47	)	)	PUNCT
ejpam-6321	461	48	)	)	PUNCT
ejpam-6321	461	49	,	,	PUNCT
ejpam-6321	461	50	x(3k+2	x(3k+2	NOUN
ejpam-6321	461	51	)	)	PUNCT
ejpam-6321	461	52	)	)	PUNCT
ejpam-6321	461	53	,	,	PUNCT
ejpam-6321	461	54	f3x(3k+2	f3x(3k+2	NOUN
ejpam-6321	461	55	)	)	PUNCT
ejpam-6321	461	56	)	)	PUNCT
ejpam-6321	462	1			NOUN
ejpam-6321	462	2	=	=	SYM
ejpam-6321	462	3	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	462	4	,	,	PUNCT
ejpam-6321	462	5	℘	℘	PROPN
ejpam-6321	462	6	,	,	PUNCT
ejpam-6321	462	7	x(3k+2	x(3k+2	NOUN
ejpam-6321	462	8	)	)	PUNCT
ejpam-6321	462	9	)	)	PUNCT
ejpam-6321	463	1	+	+	CCONJ
ejpam-6321	463	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	463	3	,	,	PUNCT
ejpam-6321	463	4	℘	℘	PROPN
ejpam-6321	463	5	,	,	PUNCT
ejpam-6321	463	6	f2℘	f2℘	NOUN
ejpam-6321	463	7	)	)	PUNCT
ejpam-6321	463	8	+	+	NUM
ejpam-6321	463	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	463	10	)	)	PUNCT
ejpam-6321	463	11	,	,	PUNCT
ejpam-6321	463	12	℘	℘	PROPN
ejpam-6321	463	13	,	,	PUNCT
ejpam-6321	463	14	℘	℘	NUM
ejpam-6321	463	15	)	)	PUNCT
ejpam-6321	463	16	+	+	NUM
ejpam-6321	463	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	463	18	)	)	PUNCT
ejpam-6321	463	19	,	,	PUNCT
ejpam-6321	463	20	℘	℘	PROPN
ejpam-6321	463	21	,	,	PUNCT
ejpam-6321	463	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	463	23	)	)	PUNCT
ejpam-6321	463	24	)	)	PUNCT
ejpam-6321	464	1	+	+	ADP
ejpam-6321	464	2	g(x3k	g(x3k	NOUN
ejpam-6321	464	3	,	,	PUNCT
ejpam-6321	464	4	f2℘	f2℘	NOUN
ejpam-6321	464	5	,	,	PUNCT
ejpam-6321	464	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	464	7	)	)	PUNCT
ejpam-6321	464	8	)	)	PUNCT
ejpam-6321	465	1	+	+	ADP
ejpam-6321	465	2	g(x3k	g(x3k	NOUN
ejpam-6321	465	3	,	,	PUNCT
ejpam-6321	465	4	℘	℘	PROPN
ejpam-6321	465	5	,	,	PUNCT
ejpam-6321	465	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	465	7	)	)	PUNCT
ejpam-6321	465	8	)	)	PUNCT
ejpam-6321	465	9	]	]	PUNCT
ejpam-6321	466	1	+	+	CCONJ
ejpam-6321	466	2	α5max	α5max	PROPN
ejpam-6321	466	3			PART
ejpam-6321	466	4	g(x3k	g(x3k	NOUN
ejpam-6321	466	5	,	,	PUNCT
ejpam-6321	466	6	℘	℘	NOUN
ejpam-6321	466	7	,	,	PUNCT
ejpam-6321	466	8	f2℘	f2℘	NOUN
ejpam-6321	466	9	)	)	PUNCT
ejpam-6321	466	10	,	,	PUNCT
ejpam-6321	466	11	g(x3k	g(x3k	NOUN
ejpam-6321	466	12	,	,	PUNCT
ejpam-6321	466	13	x3k	x3k	NOUN
ejpam-6321	466	14	,	,	PUNCT
ejpam-6321	466	15	℘	℘	PROPN
ejpam-6321	466	16	)	)	PUNCT
ejpam-6321	466	17	,	,	PUNCT
ejpam-6321	466	18	g(x3k	g(x3k	NOUN
ejpam-6321	466	19	,	,	PUNCT
ejpam-6321	466	20	℘	℘	PROPN
ejpam-6321	466	21	,	,	PUNCT
ejpam-6321	466	22	℘	℘	NUM
ejpam-6321	466	23	)	)	PUNCT
ejpam-6321	466	24	,	,	PUNCT
ejpam-6321	466	25	g(f2℘	g(f2℘	PROPN
ejpam-6321	466	26	,	,	PUNCT
ejpam-6321	466	27	f2℘	f2℘	NOUN
ejpam-6321	466	28	,	,	PUNCT
ejpam-6321	466	29	x(3k+2	x(3k+2	NOUN
ejpam-6321	466	30	)	)	PUNCT
ejpam-6321	466	31	)	)	PUNCT
ejpam-6321	466	32	,	,	PUNCT
ejpam-6321	466	33	g(x3k	g(x3k	NOUN
ejpam-6321	466	34	,	,	PUNCT
ejpam-6321	466	35	x3k	x3k	NOUN
ejpam-6321	466	36	,	,	PUNCT
ejpam-6321	466	37	x3k	x3k	NOUN
ejpam-6321	466	38	)	)	PUNCT
ejpam-6321	466	39	,	,	PUNCT
ejpam-6321	466	40	g(℘	g(℘	PROPN
ejpam-6321	466	41	,	,	PUNCT
ejpam-6321	466	42	f2℘	f2℘	NOUN
ejpam-6321	466	43	,	,	PUNCT
ejpam-6321	466	44	℘	℘	PROPN
ejpam-6321	466	45	)	)	PUNCT
ejpam-6321	466	46	,	,	PUNCT
ejpam-6321	466	47	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	466	48	)	)	PUNCT
ejpam-6321	466	49	)	)	PUNCT
ejpam-6321	466	50	,	,	PUNCT
ejpam-6321	466	51	x(3k+2	x(3k+2	NOUN
ejpam-6321	466	52	)	)	PUNCT
ejpam-6321	466	53	)	)	PUNCT
ejpam-6321	466	54	,	,	PUNCT
ejpam-6321	466	55	x(3k+2	x(3k+2	NUM
ejpam-6321	466	56	)	)	PUNCT
ejpam-6321	466	57	)	)	PUNCT
ejpam-6321	466	58			ADV
ejpam-6321	466	59	(	(	PUNCT
ejpam-6321	466	60	21	21	NUM
ejpam-6321	466	61	)	)	PUNCT
ejpam-6321	466	62	now	now	ADV
ejpam-6321	466	63	there	there	PRON
ejpam-6321	466	64	are	be	VERB
ejpam-6321	466	65	seven	seven	NUM
ejpam-6321	466	66	cases	case	NOUN
ejpam-6321	466	67	:	:	PUNCT
ejpam-6321	466	68	(	(	PUNCT
ejpam-6321	466	69	i	i	NOUN
ejpam-6321	466	70	)	)	PUNCT
ejpam-6321	466	71	.	.	PUNCT
ejpam-6321	467	1	if	if	SCONJ
ejpam-6321	467	2	g(x3k	g(x3k	NOUN
ejpam-6321	467	3	,	,	PUNCT
ejpam-6321	467	4	℘	℘	NOUN
ejpam-6321	467	5	,	,	PUNCT
ejpam-6321	467	6	f2℘	f2℘	NOUN
ejpam-6321	467	7	)	)	PUNCT
ejpam-6321	467	8	is	be	AUX
ejpam-6321	467	9	a	a	DET
ejpam-6321	467	10	maximum	maximum	ADJ
ejpam-6321	467	11	term	term	NOUN
ejpam-6321	467	12	in	in	ADP
ejpam-6321	467	13	(	(	PUNCT
ejpam-6321	467	14	21	21	NUM
ejpam-6321	467	15	)	)	PUNCT
ejpam-6321	467	16	then	then	ADV
ejpam-6321	467	17	we	we	PRON
ejpam-6321	467	18	can	can	AUX
ejpam-6321	467	19	write	write	VERB
ejpam-6321	467	20	;	;	PUNCT
ejpam-6321	467	21	≤	≤	NUM
ejpam-6321	467	22	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	467	23	,	,	PUNCT
ejpam-6321	467	24	℘	℘	PROPN
ejpam-6321	467	25	,	,	PUNCT
ejpam-6321	467	26	x(3k+2	x(3k+2	NOUN
ejpam-6321	467	27	)	)	PUNCT
ejpam-6321	467	28	)	)	PUNCT
ejpam-6321	468	1	+	+	CCONJ
ejpam-6321	468	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	468	3	,	,	PUNCT
ejpam-6321	468	4	℘	℘	PROPN
ejpam-6321	468	5	,	,	PUNCT
ejpam-6321	468	6	f2℘	f2℘	NOUN
ejpam-6321	468	7	)	)	PUNCT
ejpam-6321	468	8	+	+	NUM
ejpam-6321	468	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	468	10	)	)	PUNCT
ejpam-6321	468	11	,	,	PUNCT
ejpam-6321	468	12	℘	℘	PROPN
ejpam-6321	468	13	,	,	PUNCT
ejpam-6321	468	14	℘	℘	NUM
ejpam-6321	468	15	)	)	PUNCT
ejpam-6321	468	16	+	+	NUM
ejpam-6321	468	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	468	18	)	)	PUNCT
ejpam-6321	468	19	,	,	PUNCT
ejpam-6321	468	20	℘	℘	PROPN
ejpam-6321	468	21	,	,	PUNCT
ejpam-6321	468	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	468	23	)	)	PUNCT
ejpam-6321	468	24	)	)	PUNCT
ejpam-6321	469	1	+	+	ADP
ejpam-6321	469	2	g(x3k	g(x3k	NOUN
ejpam-6321	469	3	,	,	PUNCT
ejpam-6321	469	4	f2℘	f2℘	NOUN
ejpam-6321	469	5	,	,	PUNCT
ejpam-6321	469	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	469	7	)	)	PUNCT
ejpam-6321	469	8	)	)	PUNCT
ejpam-6321	470	1	+	+	ADP
ejpam-6321	470	2	g(x3k	g(x3k	NOUN
ejpam-6321	470	3	,	,	PUNCT
ejpam-6321	470	4	℘	℘	PROPN
ejpam-6321	470	5	,	,	PUNCT
ejpam-6321	470	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	470	7	)	)	PUNCT
ejpam-6321	470	8	)	)	PUNCT
ejpam-6321	470	9	]	]	PUNCT
ejpam-6321	471	1	+	+	CCONJ
ejpam-6321	471	2	α5g(x3k	α5g(x3k	NOUN
ejpam-6321	471	3	,	,	PUNCT
ejpam-6321	471	4	℘	℘	PROPN
ejpam-6321	471	5	,	,	PUNCT
ejpam-6321	471	6	f2℘	f2℘	NOUN
ejpam-6321	471	7	)	)	PUNCT
ejpam-6321	471	8	applying	apply	VERB
ejpam-6321	471	9	limn→∞	limn→∞	PROPN
ejpam-6321	471	10	on	on	ADP
ejpam-6321	471	11	both	both	DET
ejpam-6321	471	12	sides	side	NOUN
ejpam-6321	471	13	,	,	PUNCT
ejpam-6321	471	14	we	we	PRON
ejpam-6321	471	15	get	get	VERB
ejpam-6321	471	16	,	,	PUNCT
ejpam-6321	471	17	g(℘	g(℘	NOUN
ejpam-6321	471	18	,	,	PUNCT
ejpam-6321	471	19	f2℘	f2℘	NOUN
ejpam-6321	471	20	,	,	PUNCT
ejpam-6321	471	21	℘	℘	PROPN
ejpam-6321	471	22	)	)	PUNCT
ejpam-6321	471	23	≤	≤	NOUN
ejpam-6321	471	24	α1g(℘	α1g(℘	NOUN
ejpam-6321	471	25	,	,	PUNCT
ejpam-6321	471	26	℘	℘	NOUN
ejpam-6321	471	27	,	,	PUNCT
ejpam-6321	471	28	℘	℘	NUM
ejpam-6321	471	29	)	)	PUNCT
ejpam-6321	471	30	+	+	NUM
ejpam-6321	471	31	α2g(℘	α2g(℘	NOUN
ejpam-6321	471	32	,	,	PUNCT
ejpam-6321	471	33	℘	℘	NOUN
ejpam-6321	471	34	,	,	PUNCT
ejpam-6321	471	35	f2℘	f2℘	NOUN
ejpam-6321	471	36	)	)	PUNCT
ejpam-6321	471	37	+	+	CCONJ
ejpam-6321	471	38	α3g(℘	α3g(℘	NUM
ejpam-6321	471	39	,	,	PUNCT
ejpam-6321	471	40	℘	℘	NOUN
ejpam-6321	471	41	,	,	PUNCT
ejpam-6321	471	42	℘)+	℘)+	NOUN
ejpam-6321	471	43	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	471	44	,	,	PUNCT
ejpam-6321	471	45	℘	℘	NOUN
ejpam-6321	471	46	,	,	PUNCT
ejpam-6321	471	47	℘	℘	NUM
ejpam-6321	471	48	)	)	PUNCT
ejpam-6321	471	49	+	+	NOUN
ejpam-6321	471	50	g(℘	g(℘	NOUN
ejpam-6321	471	51	,	,	PUNCT
ejpam-6321	471	52	f2℘	f2℘	NOUN
ejpam-6321	471	53	,	,	PUNCT
ejpam-6321	471	54	℘	℘	PROPN
ejpam-6321	471	55	)	)	PUNCT
ejpam-6321	471	56	+	+	NOUN
ejpam-6321	471	57	g(℘	g(℘	NOUN
ejpam-6321	471	58	,	,	PUNCT
ejpam-6321	471	59	℘	℘	NOUN
ejpam-6321	471	60	,	,	PUNCT
ejpam-6321	471	61	℘	℘	NUM
ejpam-6321	471	62	)	)	PUNCT
ejpam-6321	471	63	]	]	PUNCT
ejpam-6321	472	1	+	+	NUM
ejpam-6321	472	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	472	3	,	,	PUNCT
ejpam-6321	472	4	℘	℘	NOUN
ejpam-6321	472	5	,	,	PUNCT
ejpam-6321	472	6	f2℘	f2℘	NOUN
ejpam-6321	472	7	)	)	PUNCT
ejpam-6321	472	8	≤	≤	NUM
ejpam-6321	472	9	α2	α2	ADJ
ejpam-6321	472	10	+	+	CCONJ
ejpam-6321	472	11	α4	α4	NOUN
ejpam-6321	472	12	+	+	CCONJ
ejpam-6321	472	13	α5)g(℘	α5)g(℘	NUM
ejpam-6321	472	14	,	,	PUNCT
ejpam-6321	472	15	℘	℘	NOUN
ejpam-6321	472	16	,	,	PUNCT
ejpam-6321	472	17	f2℘	f2℘	NOUN
ejpam-6321	472	18	)	)	PUNCT
ejpam-6321	472	19	g(℘	g(℘	PROPN
ejpam-6321	472	20	,	,	PUNCT
ejpam-6321	472	21	f2℘	f2℘	NOUN
ejpam-6321	472	22	,	,	PUNCT
ejpam-6321	472	23	℘	℘	PROPN
ejpam-6321	472	24	)	)	PUNCT
ejpam-6321	472	25	−	−	PROPN
ejpam-6321	473	1	(	(	PUNCT
ejpam-6321	473	2	α2	α2	ADJ
ejpam-6321	473	3	+	+	CCONJ
ejpam-6321	473	4	α4	α4	NOUN
ejpam-6321	473	5	+	+	CCONJ
ejpam-6321	473	6	α5)g(℘	α5)g(℘	NUM
ejpam-6321	473	7	,	,	PUNCT
ejpam-6321	473	8	℘	℘	NOUN
ejpam-6321	473	9	,	,	PUNCT
ejpam-6321	473	10	f2℘	f2℘	NOUN
ejpam-6321	473	11	)	)	PUNCT
ejpam-6321	473	12	≤	≤	NOUN
ejpam-6321	473	13	0	0	NUM
ejpam-6321	473	14	.	.	PUNCT
ejpam-6321	474	1	(	(	PUNCT
ejpam-6321	474	2	1	1	NUM
ejpam-6321	474	3	−	−	ADV
ejpam-6321	474	4	α2	α2	ADJ
ejpam-6321	474	5	−	−	NOUN
ejpam-6321	474	6	α4	α4	NOUN
ejpam-6321	474	7	−	−	ADP
ejpam-6321	474	8	α5)g(℘	α5)g(℘	NUM
ejpam-6321	474	9	,	,	PUNCT
ejpam-6321	474	10	℘	℘	PROPN
ejpam-6321	474	11	,	,	PUNCT
ejpam-6321	474	12	f2℘	f2℘	NOUN
ejpam-6321	474	13	)	)	PUNCT
ejpam-6321	474	14	≤	≤	NOUN
ejpam-6321	474	15	0	0	NUM
ejpam-6321	474	16	is	be	AUX
ejpam-6321	474	17	a	a	DET
ejpam-6321	474	18	contradiction	contradiction	NOUN
ejpam-6321	474	19	.	.	PUNCT
ejpam-6321	475	1	since	since	SCONJ
ejpam-6321	475	2	(	(	PUNCT
ejpam-6321	475	3	1−	1−	NUM
ejpam-6321	475	4	α2	α2	ADJ
ejpam-6321	475	5	−	−	PROPN
ejpam-6321	475	6	α4	α4	NOUN
ejpam-6321	475	7	−	−	NOUN
ejpam-6321	475	8	α5	α5	NOUN
ejpam-6321	475	9	)	)	PUNCT
ejpam-6321	475	10	̸=	̸=	PROPN
ejpam-6321	475	11	0	0	NUM
ejpam-6321	475	12	,	,	PUNCT
ejpam-6321	475	13	therefore	therefore	ADV
ejpam-6321	475	14	,	,	PUNCT
ejpam-6321	475	15	g(f2℘	g(f2℘	PROPN
ejpam-6321	475	16	,	,	PUNCT
ejpam-6321	475	17	℘	℘	PROPN
ejpam-6321	475	18	,	,	PUNCT
ejpam-6321	475	19	℘	℘	NUM
ejpam-6321	475	20	)	)	PUNCT
ejpam-6321	475	21	=	=	SYM
ejpam-6321	475	22	0	0	X
ejpam-6321	475	23	.	.	PUNCT
ejpam-6321	475	24	thus	thus	ADV
ejpam-6321	475	25	,	,	PUNCT
ejpam-6321	475	26	f2℘	f2℘	NOUN
ejpam-6321	475	27	=	=	SYM
ejpam-6321	475	28	℘	℘	PROPN
ejpam-6321	475	29	(	(	PUNCT
ejpam-6321	475	30	22	22	NUM
ejpam-6321	475	31	)	)	PUNCT
ejpam-6321	475	32	(	(	PUNCT
ejpam-6321	475	33	ii	ii	NOUN
ejpam-6321	475	34	)	)	PUNCT
ejpam-6321	475	35	.	.	PUNCT
ejpam-6321	476	1	if	if	SCONJ
ejpam-6321	476	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	476	3	)	)	PUNCT
ejpam-6321	476	4	,	,	PUNCT
ejpam-6321	476	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	476	6	)	)	PUNCT
ejpam-6321	476	7	,	,	PUNCT
ejpam-6321	476	8	℘	℘	PROPN
ejpam-6321	476	9	)	)	PUNCT
ejpam-6321	476	10	is	be	AUX
ejpam-6321	476	11	a	a	DET
ejpam-6321	476	12	maximum	maximum	ADJ
ejpam-6321	476	13	term	term	NOUN
ejpam-6321	476	14	in	in	ADP
ejpam-6321	476	15	(	(	PUNCT
ejpam-6321	476	16	21	21	NUM
ejpam-6321	476	17	)	)	PUNCT
ejpam-6321	476	18	then	then	ADV
ejpam-6321	476	19	we	we	PRON
ejpam-6321	476	20	can	can	AUX
ejpam-6321	476	21	write	write	VERB
ejpam-6321	476	22	;	;	PUNCT
ejpam-6321	476	23	≤	≤	NUM
ejpam-6321	476	24	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	476	25	,	,	PUNCT
ejpam-6321	476	26	℘	℘	PROPN
ejpam-6321	476	27	,	,	PUNCT
ejpam-6321	476	28	x(3k+2	x(3k+2	NOUN
ejpam-6321	476	29	)	)	PUNCT
ejpam-6321	476	30	)	)	PUNCT
ejpam-6321	477	1	+	+	CCONJ
ejpam-6321	477	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	477	3	,	,	PUNCT
ejpam-6321	477	4	℘	℘	PROPN
ejpam-6321	477	5	,	,	PUNCT
ejpam-6321	477	6	f2℘	f2℘	NOUN
ejpam-6321	477	7	)	)	PUNCT
ejpam-6321	477	8	+	+	NUM
ejpam-6321	477	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	477	10	)	)	PUNCT
ejpam-6321	477	11	,	,	PUNCT
ejpam-6321	477	12	℘	℘	PROPN
ejpam-6321	477	13	,	,	PUNCT
ejpam-6321	477	14	℘	℘	NUM
ejpam-6321	477	15	)	)	PUNCT
ejpam-6321	477	16	+	+	NUM
ejpam-6321	477	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	477	18	)	)	PUNCT
ejpam-6321	477	19	,	,	PUNCT
ejpam-6321	477	20	℘	℘	PROPN
ejpam-6321	477	21	,	,	PUNCT
ejpam-6321	477	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	477	23	)	)	PUNCT
ejpam-6321	477	24	)	)	PUNCT
ejpam-6321	478	1	+	+	ADP
ejpam-6321	478	2	g(x3k	g(x3k	NOUN
ejpam-6321	478	3	,	,	PUNCT
ejpam-6321	478	4	f2℘	f2℘	NOUN
ejpam-6321	478	5	,	,	PUNCT
ejpam-6321	478	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	478	7	)	)	PUNCT
ejpam-6321	478	8	)	)	PUNCT
ejpam-6321	479	1	+	+	ADP
ejpam-6321	479	2	g(x3k	g(x3k	NOUN
ejpam-6321	479	3	,	,	PUNCT
ejpam-6321	479	4	℘	℘	PROPN
ejpam-6321	479	5	,	,	PUNCT
ejpam-6321	479	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	479	7	)	)	PUNCT
ejpam-6321	479	8	)	)	PUNCT
ejpam-6321	479	9	]	]	PUNCT
ejpam-6321	480	1	+	+	CCONJ
ejpam-6321	480	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	480	3	)	)	PUNCT
ejpam-6321	480	4	,	,	PUNCT
ejpam-6321	480	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	480	6	)	)	PUNCT
ejpam-6321	480	7	,	,	PUNCT
ejpam-6321	480	8	℘	℘	PROPN
ejpam-6321	480	9	)	)	PUNCT
ejpam-6321	480	10	applying	apply	VERB
ejpam-6321	480	11	limn→∞	limn→∞	PROPN
ejpam-6321	480	12	on	on	ADP
ejpam-6321	480	13	both	both	DET
ejpam-6321	480	14	sides	side	NOUN
ejpam-6321	480	15	,	,	PUNCT
ejpam-6321	480	16	we	we	PRON
ejpam-6321	480	17	get	get	VERB
ejpam-6321	480	18	,	,	PUNCT
ejpam-6321	480	19	g(℘	g(℘	NOUN
ejpam-6321	480	20	,	,	PUNCT
ejpam-6321	480	21	f2℘	f2℘	NOUN
ejpam-6321	480	22	,	,	PUNCT
ejpam-6321	480	23	℘	℘	PROPN
ejpam-6321	480	24	)	)	PUNCT
ejpam-6321	480	25	≤	≤	NOUN
ejpam-6321	480	26	α1g(℘	α1g(℘	NOUN
ejpam-6321	480	27	,	,	PUNCT
ejpam-6321	480	28	℘	℘	NOUN
ejpam-6321	480	29	,	,	PUNCT
ejpam-6321	480	30	℘	℘	NUM
ejpam-6321	480	31	)	)	PUNCT
ejpam-6321	480	32	+	+	NUM
ejpam-6321	480	33	α2g(℘	α2g(℘	NOUN
ejpam-6321	480	34	,	,	PUNCT
ejpam-6321	480	35	℘	℘	NOUN
ejpam-6321	480	36	,	,	PUNCT
ejpam-6321	480	37	f2℘	f2℘	NOUN
ejpam-6321	480	38	)	)	PUNCT
ejpam-6321	480	39	+	+	CCONJ
ejpam-6321	480	40	α3g(℘	α3g(℘	NUM
ejpam-6321	480	41	,	,	PUNCT
ejpam-6321	480	42	℘	℘	NOUN
ejpam-6321	480	43	,	,	PUNCT
ejpam-6321	480	44	℘)+	℘)+	NOUN
ejpam-6321	480	45	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	480	46	,	,	PUNCT
ejpam-6321	480	47	℘	℘	NOUN
ejpam-6321	480	48	,	,	PUNCT
ejpam-6321	480	49	℘	℘	NUM
ejpam-6321	480	50	)	)	PUNCT
ejpam-6321	481	1	+	+	NOUN
ejpam-6321	481	2	g(℘	g(℘	NOUN
ejpam-6321	481	3	,	,	PUNCT
ejpam-6321	481	4	f2℘	f2℘	NOUN
ejpam-6321	481	5	,	,	PUNCT
ejpam-6321	481	6	℘	℘	PROPN
ejpam-6321	481	7	)	)	PUNCT
ejpam-6321	481	8	+	+	NOUN
ejpam-6321	481	9	g(℘	g(℘	NOUN
ejpam-6321	481	10	,	,	PUNCT
ejpam-6321	481	11	℘	℘	NOUN
ejpam-6321	481	12	,	,	PUNCT
ejpam-6321	481	13	℘	℘	NUM
ejpam-6321	481	14	)	)	PUNCT
ejpam-6321	481	15	]	]	PUNCT
ejpam-6321	482	1	+	+	NUM
ejpam-6321	482	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	482	3	,	,	PUNCT
ejpam-6321	482	4	℘	℘	NOUN
ejpam-6321	482	5	,	,	PUNCT
ejpam-6321	482	6	℘	℘	NUM
ejpam-6321	482	7	)	)	PUNCT
ejpam-6321	482	8	≤	≤	NOUN
ejpam-6321	482	9	α2	α2	ADJ
ejpam-6321	482	10	+	+	CCONJ
ejpam-6321	483	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	483	2	,	,	PUNCT
ejpam-6321	483	3	℘	℘	PROPN
ejpam-6321	483	4	,	,	PUNCT
ejpam-6321	483	5	f2℘	f2℘	NOUN
ejpam-6321	483	6	)	)	PUNCT
ejpam-6321	483	7	g(℘	g(℘	PROPN
ejpam-6321	483	8	,	,	PUNCT
ejpam-6321	483	9	f2℘	f2℘	NOUN
ejpam-6321	483	10	,	,	PUNCT
ejpam-6321	483	11	℘	℘	PROPN
ejpam-6321	483	12	)	)	PUNCT
ejpam-6321	483	13	−	−	PROPN
ejpam-6321	483	14	(	(	PUNCT
ejpam-6321	483	15	α2	α2	PROPN
ejpam-6321	483	16	+	+	CCONJ
ejpam-6321	483	17	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	483	18	,	,	PUNCT
ejpam-6321	483	19	℘	℘	NOUN
ejpam-6321	483	20	,	,	PUNCT
ejpam-6321	483	21	f2℘	f2℘	NOUN
ejpam-6321	483	22	)	)	PUNCT
ejpam-6321	483	23	≤	≤	NOUN
ejpam-6321	483	24	0	0	NUM
ejpam-6321	483	25	.	.	PUNCT
ejpam-6321	484	1	(	(	PUNCT
ejpam-6321	484	2	1	1	NUM
ejpam-6321	484	3	−	−	NOUN
ejpam-6321	484	4	α2	α2	ADJ
ejpam-6321	484	5	−	−	PROPN
ejpam-6321	485	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	485	2	,	,	PUNCT
ejpam-6321	485	3	℘	℘	PROPN
ejpam-6321	485	4	,	,	PUNCT
ejpam-6321	485	5	f2℘	f2℘	NOUN
ejpam-6321	485	6	)	)	PUNCT
ejpam-6321	485	7	≤	≤	NOUN
ejpam-6321	485	8	0	0	NUM
ejpam-6321	485	9	is	be	AUX
ejpam-6321	485	10	a	a	DET
ejpam-6321	485	11	contradiction	contradiction	NOUN
ejpam-6321	485	12	.	.	PUNCT
ejpam-6321	486	1	since	since	SCONJ
ejpam-6321	486	2	(	(	PUNCT
ejpam-6321	486	3	1−	1−	NUM
ejpam-6321	486	4	α2	α2	ADJ
ejpam-6321	486	5	−	−	PROPN
ejpam-6321	486	6	α4	α4	NOUN
ejpam-6321	486	7	)	)	PUNCT
ejpam-6321	486	8	̸=	̸=	PROPN
ejpam-6321	486	9	0	0	NUM
ejpam-6321	486	10	,	,	PUNCT
ejpam-6321	486	11	therefore	therefore	ADV
ejpam-6321	486	12	,	,	PUNCT
ejpam-6321	486	13	g(f2℘	g(f2℘	PROPN
ejpam-6321	486	14	,	,	PUNCT
ejpam-6321	486	15	℘	℘	PROPN
ejpam-6321	486	16	,	,	PUNCT
ejpam-6321	486	17	℘	℘	NUM
ejpam-6321	486	18	)	)	PUNCT
ejpam-6321	486	19	=	=	SYM
ejpam-6321	486	20	0	0	X
ejpam-6321	486	21	.	.	PUNCT
ejpam-6321	486	22	thus	thus	ADV
ejpam-6321	486	23	,	,	PUNCT
ejpam-6321	486	24	f2℘	f2℘	NOUN
ejpam-6321	486	25	=	=	SYM
ejpam-6321	486	26	℘	℘	PROPN
ejpam-6321	486	27	(	(	PUNCT
ejpam-6321	486	28	23	23	NUM
ejpam-6321	486	29	)	)	PUNCT
ejpam-6321	486	30	m.	m.	NOUN
ejpam-6321	486	31	noorwali	noorwali	PROPN
ejpam-6321	486	32	et	et	PROPN
ejpam-6321	486	33	al	al	PROPN
ejpam-6321	486	34	.	.	PUNCT
ejpam-6321	486	35	/	/	SYM
ejpam-6321	486	36	eur	eur	PROPN
ejpam-6321	486	37	.	.	PUNCT
ejpam-6321	487	1	j.	j.	PROPN
ejpam-6321	487	2	pure	pure	PROPN
ejpam-6321	487	3	appl	appl	PROPN
ejpam-6321	487	4	.	.	PROPN
ejpam-6321	487	5	math	math	PROPN
ejpam-6321	487	6	,	,	PUNCT
ejpam-6321	487	7	18	18	NUM
ejpam-6321	487	8	(	(	PUNCT
ejpam-6321	487	9	3	3	NUM
ejpam-6321	487	10	)	)	PUNCT
ejpam-6321	487	11	(	(	PUNCT
ejpam-6321	487	12	2025	2025	NUM
ejpam-6321	487	13	)	)	PUNCT
ejpam-6321	487	14	,	,	PUNCT
ejpam-6321	487	15	6321	6321	NUM
ejpam-6321	487	16	17	17	NUM
ejpam-6321	487	17	of	of	ADP
ejpam-6321	487	18	27	27	NUM
ejpam-6321	487	19	(	(	PUNCT
ejpam-6321	487	20	iii	iii	NOUN
ejpam-6321	487	21	)	)	PUNCT
ejpam-6321	487	22	.	.	PUNCT
ejpam-6321	488	1	if	if	SCONJ
ejpam-6321	488	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	488	3	)	)	PUNCT
ejpam-6321	488	4	,	,	PUNCT
ejpam-6321	488	5	℘	℘	PROPN
ejpam-6321	488	6	,	,	PUNCT
ejpam-6321	488	7	℘	℘	NUM
ejpam-6321	488	8	)	)	PUNCT
ejpam-6321	488	9	is	be	AUX
ejpam-6321	488	10	a	a	DET
ejpam-6321	488	11	maximum	maximum	ADJ
ejpam-6321	488	12	term	term	NOUN
ejpam-6321	488	13	in	in	ADP
ejpam-6321	488	14	(	(	PUNCT
ejpam-6321	488	15	21	21	NUM
ejpam-6321	488	16	)	)	PUNCT
ejpam-6321	488	17	then	then	ADV
ejpam-6321	488	18	we	we	PRON
ejpam-6321	488	19	can	can	AUX
ejpam-6321	488	20	write	write	VERB
ejpam-6321	488	21	;	;	PUNCT
ejpam-6321	488	22	≤	≤	NUM
ejpam-6321	488	23	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	488	24	,	,	PUNCT
ejpam-6321	488	25	℘	℘	PROPN
ejpam-6321	488	26	,	,	PUNCT
ejpam-6321	488	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	488	28	)	)	PUNCT
ejpam-6321	488	29	)	)	PUNCT
ejpam-6321	489	1	+	+	CCONJ
ejpam-6321	489	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	489	3	,	,	PUNCT
ejpam-6321	489	4	℘	℘	PROPN
ejpam-6321	489	5	,	,	PUNCT
ejpam-6321	489	6	f2℘	f2℘	NOUN
ejpam-6321	489	7	)	)	PUNCT
ejpam-6321	489	8	+	+	NUM
ejpam-6321	489	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	489	10	)	)	PUNCT
ejpam-6321	489	11	,	,	PUNCT
ejpam-6321	489	12	℘	℘	PROPN
ejpam-6321	489	13	,	,	PUNCT
ejpam-6321	489	14	℘	℘	NUM
ejpam-6321	489	15	)	)	PUNCT
ejpam-6321	489	16	+	+	NUM
ejpam-6321	489	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	489	18	)	)	PUNCT
ejpam-6321	489	19	,	,	PUNCT
ejpam-6321	489	20	℘	℘	PROPN
ejpam-6321	489	21	,	,	PUNCT
ejpam-6321	489	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	489	23	)	)	PUNCT
ejpam-6321	489	24	)	)	PUNCT
ejpam-6321	490	1	+	+	ADP
ejpam-6321	490	2	g(x3k	g(x3k	NOUN
ejpam-6321	490	3	,	,	PUNCT
ejpam-6321	490	4	f2℘	f2℘	NOUN
ejpam-6321	490	5	,	,	PUNCT
ejpam-6321	490	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	490	7	)	)	PUNCT
ejpam-6321	490	8	)	)	PUNCT
ejpam-6321	491	1	+	+	ADP
ejpam-6321	491	2	g(x3k	g(x3k	NOUN
ejpam-6321	491	3	,	,	PUNCT
ejpam-6321	491	4	℘	℘	PROPN
ejpam-6321	491	5	,	,	PUNCT
ejpam-6321	491	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	491	7	)	)	PUNCT
ejpam-6321	491	8	)	)	PUNCT
ejpam-6321	491	9	]	]	PUNCT
ejpam-6321	492	1	+	+	CCONJ
ejpam-6321	492	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	492	3	)	)	PUNCT
ejpam-6321	492	4	,	,	PUNCT
ejpam-6321	492	5	℘	℘	PROPN
ejpam-6321	492	6	,	,	PUNCT
ejpam-6321	492	7	℘	℘	NOUN
ejpam-6321	492	8	)	)	PUNCT
ejpam-6321	492	9	applying	apply	VERB
ejpam-6321	492	10	limn→∞	limn→∞	PROPN
ejpam-6321	492	11	on	on	ADP
ejpam-6321	492	12	both	both	DET
ejpam-6321	492	13	sides	side	NOUN
ejpam-6321	492	14	,	,	PUNCT
ejpam-6321	492	15	we	we	PRON
ejpam-6321	492	16	get	get	VERB
ejpam-6321	492	17	,	,	PUNCT
ejpam-6321	492	18	g(℘	g(℘	NOUN
ejpam-6321	492	19	,	,	PUNCT
ejpam-6321	492	20	f2℘	f2℘	NOUN
ejpam-6321	492	21	,	,	PUNCT
ejpam-6321	492	22	℘	℘	PROPN
ejpam-6321	492	23	)	)	PUNCT
ejpam-6321	492	24	≤	≤	NOUN
ejpam-6321	492	25	α1g(℘	α1g(℘	NOUN
ejpam-6321	492	26	,	,	PUNCT
ejpam-6321	492	27	℘	℘	NOUN
ejpam-6321	492	28	,	,	PUNCT
ejpam-6321	492	29	℘	℘	NUM
ejpam-6321	492	30	)	)	PUNCT
ejpam-6321	492	31	+	+	NUM
ejpam-6321	492	32	α2g(℘	α2g(℘	NOUN
ejpam-6321	492	33	,	,	PUNCT
ejpam-6321	492	34	℘	℘	NOUN
ejpam-6321	492	35	,	,	PUNCT
ejpam-6321	492	36	f2℘	f2℘	NOUN
ejpam-6321	492	37	)	)	PUNCT
ejpam-6321	493	1	+	+	CCONJ
ejpam-6321	493	2	α3g(℘	α3g(℘	NUM
ejpam-6321	493	3	,	,	PUNCT
ejpam-6321	493	4	℘	℘	NOUN
ejpam-6321	493	5	,	,	PUNCT
ejpam-6321	493	6	℘)+	℘)+	NOUN
ejpam-6321	493	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	493	8	,	,	PUNCT
ejpam-6321	493	9	℘	℘	NOUN
ejpam-6321	493	10	,	,	PUNCT
ejpam-6321	493	11	℘	℘	NUM
ejpam-6321	493	12	)	)	PUNCT
ejpam-6321	493	13	+	+	NOUN
ejpam-6321	493	14	g(℘	g(℘	NOUN
ejpam-6321	493	15	,	,	PUNCT
ejpam-6321	493	16	f2℘	f2℘	NOUN
ejpam-6321	493	17	,	,	PUNCT
ejpam-6321	493	18	℘	℘	PROPN
ejpam-6321	493	19	)	)	PUNCT
ejpam-6321	493	20	+	+	NOUN
ejpam-6321	493	21	g(℘	g(℘	NOUN
ejpam-6321	493	22	,	,	PUNCT
ejpam-6321	493	23	℘	℘	NOUN
ejpam-6321	493	24	,	,	PUNCT
ejpam-6321	493	25	℘	℘	NUM
ejpam-6321	493	26	)	)	PUNCT
ejpam-6321	493	27	]	]	PUNCT
ejpam-6321	494	1	+	+	NUM
ejpam-6321	494	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	494	3	,	,	PUNCT
ejpam-6321	494	4	℘	℘	NOUN
ejpam-6321	494	5	,	,	PUNCT
ejpam-6321	494	6	℘	℘	NUM
ejpam-6321	494	7	)	)	PUNCT
ejpam-6321	494	8	≤	≤	NOUN
ejpam-6321	494	9	α2	α2	ADJ
ejpam-6321	494	10	+	+	CCONJ
ejpam-6321	495	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	495	2	,	,	PUNCT
ejpam-6321	495	3	℘	℘	PROPN
ejpam-6321	495	4	,	,	PUNCT
ejpam-6321	495	5	f2℘	f2℘	NOUN
ejpam-6321	495	6	)	)	PUNCT
ejpam-6321	495	7	g(℘	g(℘	PROPN
ejpam-6321	495	8	,	,	PUNCT
ejpam-6321	495	9	f2℘	f2℘	NOUN
ejpam-6321	495	10	,	,	PUNCT
ejpam-6321	495	11	℘	℘	PROPN
ejpam-6321	495	12	)	)	PUNCT
ejpam-6321	495	13	−	−	PROPN
ejpam-6321	495	14	(	(	PUNCT
ejpam-6321	495	15	α2	α2	PROPN
ejpam-6321	495	16	+	+	CCONJ
ejpam-6321	495	17	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	495	18	,	,	PUNCT
ejpam-6321	495	19	℘	℘	NOUN
ejpam-6321	495	20	,	,	PUNCT
ejpam-6321	495	21	f2℘	f2℘	NOUN
ejpam-6321	495	22	)	)	PUNCT
ejpam-6321	495	23	≤	≤	NOUN
ejpam-6321	495	24	0	0	NUM
ejpam-6321	495	25	.	.	PUNCT
ejpam-6321	496	1	(	(	PUNCT
ejpam-6321	496	2	1	1	NUM
ejpam-6321	496	3	−	−	NOUN
ejpam-6321	496	4	α2	α2	ADJ
ejpam-6321	496	5	−	−	PROPN
ejpam-6321	497	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	497	2	,	,	PUNCT
ejpam-6321	497	3	℘	℘	PROPN
ejpam-6321	497	4	,	,	PUNCT
ejpam-6321	497	5	f2℘	f2℘	NOUN
ejpam-6321	497	6	)	)	PUNCT
ejpam-6321	497	7	≤	≤	NOUN
ejpam-6321	497	8	0	0	NUM
ejpam-6321	497	9	is	be	AUX
ejpam-6321	497	10	a	a	DET
ejpam-6321	497	11	contradiction	contradiction	NOUN
ejpam-6321	497	12	.	.	PUNCT
ejpam-6321	498	1	since	since	SCONJ
ejpam-6321	498	2	(	(	PUNCT
ejpam-6321	498	3	1−	1−	NUM
ejpam-6321	498	4	α2	α2	ADJ
ejpam-6321	498	5	−	−	PROPN
ejpam-6321	498	6	α4	α4	NOUN
ejpam-6321	498	7	)	)	PUNCT
ejpam-6321	498	8	̸=	̸=	PROPN
ejpam-6321	498	9	0	0	NUM
ejpam-6321	498	10	,	,	PUNCT
ejpam-6321	498	11	therefore	therefore	ADV
ejpam-6321	498	12	,	,	PUNCT
ejpam-6321	498	13	g(f2℘	g(f2℘	PROPN
ejpam-6321	498	14	,	,	PUNCT
ejpam-6321	498	15	℘	℘	PROPN
ejpam-6321	498	16	,	,	PUNCT
ejpam-6321	498	17	℘	℘	NUM
ejpam-6321	498	18	)	)	PUNCT
ejpam-6321	498	19	=	=	SYM
ejpam-6321	498	20	0	0	X
ejpam-6321	498	21	.	.	PUNCT
ejpam-6321	498	22	thus	thus	ADV
ejpam-6321	498	23	,	,	PUNCT
ejpam-6321	498	24	f2℘	f2℘	NOUN
ejpam-6321	498	25	=	=	PUNCT
ejpam-6321	498	26	℘	℘	PROPN
ejpam-6321	498	27	(	(	PUNCT
ejpam-6321	498	28	24	24	NUM
ejpam-6321	498	29	)	)	PUNCT
ejpam-6321	498	30	(	(	PUNCT
ejpam-6321	498	31	iv	iv	X
ejpam-6321	498	32	)	)	PUNCT
ejpam-6321	498	33	.	.	PUNCT
ejpam-6321	499	1	if	if	SCONJ
ejpam-6321	499	2	g(f2℘	g(f2℘	PROPN
ejpam-6321	499	3	,	,	PUNCT
ejpam-6321	499	4	f2℘	f2℘	NOUN
ejpam-6321	499	5	,	,	PUNCT
ejpam-6321	499	6	℘	℘	PROPN
ejpam-6321	499	7	)	)	PUNCT
ejpam-6321	499	8	is	be	AUX
ejpam-6321	499	9	a	a	DET
ejpam-6321	499	10	maximum	maximum	ADJ
ejpam-6321	499	11	term	term	NOUN
ejpam-6321	499	12	in	in	ADP
ejpam-6321	499	13	(	(	PUNCT
ejpam-6321	499	14	21	21	NUM
ejpam-6321	499	15	)	)	PUNCT
ejpam-6321	499	16	then	then	ADV
ejpam-6321	499	17	we	we	PRON
ejpam-6321	499	18	can	can	AUX
ejpam-6321	499	19	write	write	VERB
ejpam-6321	499	20	;	;	PUNCT
ejpam-6321	499	21	≤	≤	NUM
ejpam-6321	499	22	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	499	23	,	,	PUNCT
ejpam-6321	499	24	℘	℘	PROPN
ejpam-6321	499	25	,	,	PUNCT
ejpam-6321	499	26	x(3k+2	x(3k+2	NOUN
ejpam-6321	499	27	)	)	PUNCT
ejpam-6321	499	28	)	)	PUNCT
ejpam-6321	500	1	+	+	CCONJ
ejpam-6321	500	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	500	3	,	,	PUNCT
ejpam-6321	500	4	℘	℘	PROPN
ejpam-6321	500	5	,	,	PUNCT
ejpam-6321	500	6	f2℘	f2℘	NOUN
ejpam-6321	500	7	)	)	PUNCT
ejpam-6321	500	8	+	+	NUM
ejpam-6321	500	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	500	10	)	)	PUNCT
ejpam-6321	500	11	,	,	PUNCT
ejpam-6321	500	12	℘	℘	PROPN
ejpam-6321	500	13	,	,	PUNCT
ejpam-6321	500	14	℘	℘	NUM
ejpam-6321	500	15	)	)	PUNCT
ejpam-6321	500	16	+	+	NUM
ejpam-6321	500	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	500	18	)	)	PUNCT
ejpam-6321	500	19	,	,	PUNCT
ejpam-6321	500	20	℘	℘	PROPN
ejpam-6321	500	21	,	,	PUNCT
ejpam-6321	500	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	500	23	)	)	PUNCT
ejpam-6321	500	24	)	)	PUNCT
ejpam-6321	501	1	+	+	ADP
ejpam-6321	501	2	g(x3k	g(x3k	NOUN
ejpam-6321	501	3	,	,	PUNCT
ejpam-6321	501	4	f2℘	f2℘	NOUN
ejpam-6321	501	5	,	,	PUNCT
ejpam-6321	501	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	501	7	)	)	PUNCT
ejpam-6321	501	8	)	)	PUNCT
ejpam-6321	502	1	+	+	ADP
ejpam-6321	502	2	g(x3k	g(x3k	NOUN
ejpam-6321	502	3	,	,	PUNCT
ejpam-6321	502	4	℘	℘	PROPN
ejpam-6321	502	5	,	,	PUNCT
ejpam-6321	502	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	502	7	)	)	PUNCT
ejpam-6321	502	8	)	)	PUNCT
ejpam-6321	502	9	]	]	PUNCT
ejpam-6321	503	1	+	+	CCONJ
ejpam-6321	503	2	α5g(f2℘	α5g(f2℘	PROPN
ejpam-6321	503	3	,	,	PUNCT
ejpam-6321	503	4	f2℘	f2℘	NOUN
ejpam-6321	503	5	,	,	PUNCT
ejpam-6321	503	6	℘	℘	PROPN
ejpam-6321	503	7	)	)	PUNCT
ejpam-6321	503	8	applying	apply	VERB
ejpam-6321	503	9	limn→∞	limn→∞	PROPN
ejpam-6321	503	10	on	on	ADP
ejpam-6321	503	11	both	both	DET
ejpam-6321	503	12	sides	side	NOUN
ejpam-6321	503	13	,	,	PUNCT
ejpam-6321	503	14	we	we	PRON
ejpam-6321	503	15	get	get	VERB
ejpam-6321	503	16	,	,	PUNCT
ejpam-6321	503	17	g(℘	g(℘	NOUN
ejpam-6321	503	18	,	,	PUNCT
ejpam-6321	503	19	f2℘	f2℘	NOUN
ejpam-6321	503	20	,	,	PUNCT
ejpam-6321	503	21	℘	℘	PROPN
ejpam-6321	503	22	)	)	PUNCT
ejpam-6321	503	23	≤	≤	NOUN
ejpam-6321	503	24	α1g(℘	α1g(℘	NOUN
ejpam-6321	503	25	,	,	PUNCT
ejpam-6321	503	26	℘	℘	NOUN
ejpam-6321	503	27	,	,	PUNCT
ejpam-6321	503	28	℘	℘	NUM
ejpam-6321	503	29	)	)	PUNCT
ejpam-6321	503	30	+	+	NUM
ejpam-6321	503	31	α2g(℘	α2g(℘	NOUN
ejpam-6321	503	32	,	,	PUNCT
ejpam-6321	503	33	℘	℘	NOUN
ejpam-6321	503	34	,	,	PUNCT
ejpam-6321	503	35	f2℘	f2℘	NOUN
ejpam-6321	503	36	)	)	PUNCT
ejpam-6321	504	1	+	+	CCONJ
ejpam-6321	504	2	α3g(℘	α3g(℘	NUM
ejpam-6321	504	3	,	,	PUNCT
ejpam-6321	504	4	℘	℘	NOUN
ejpam-6321	504	5	,	,	PUNCT
ejpam-6321	504	6	℘)+	℘)+	NOUN
ejpam-6321	504	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	504	8	,	,	PUNCT
ejpam-6321	504	9	℘	℘	NOUN
ejpam-6321	504	10	,	,	PUNCT
ejpam-6321	504	11	℘	℘	NUM
ejpam-6321	504	12	)	)	PUNCT
ejpam-6321	504	13	+	+	NOUN
ejpam-6321	504	14	g(℘	g(℘	NOUN
ejpam-6321	504	15	,	,	PUNCT
ejpam-6321	504	16	f2℘	f2℘	NOUN
ejpam-6321	504	17	,	,	PUNCT
ejpam-6321	504	18	℘	℘	PROPN
ejpam-6321	504	19	)	)	PUNCT
ejpam-6321	504	20	+	+	NOUN
ejpam-6321	504	21	g(℘	g(℘	NOUN
ejpam-6321	504	22	,	,	PUNCT
ejpam-6321	504	23	℘	℘	NOUN
ejpam-6321	504	24	,	,	PUNCT
ejpam-6321	504	25	℘	℘	NUM
ejpam-6321	504	26	)	)	PUNCT
ejpam-6321	504	27	]	]	PUNCT
ejpam-6321	505	1	+	+	CCONJ
ejpam-6321	505	2	α5g(f2℘	α5g(f2℘	PROPN
ejpam-6321	505	3	,	,	PUNCT
ejpam-6321	505	4	f2℘	f2℘	NOUN
ejpam-6321	505	5	,	,	PUNCT
ejpam-6321	505	6	℘	℘	PROPN
ejpam-6321	505	7	)	)	PUNCT
ejpam-6321	505	8	≤	≤	NOUN
ejpam-6321	505	9	α2	α2	ADJ
ejpam-6321	505	10	+	+	CCONJ
ejpam-6321	506	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	506	2	,	,	PUNCT
ejpam-6321	506	3	℘	℘	NOUN
ejpam-6321	506	4	,	,	PUNCT
ejpam-6321	506	5	f2℘	f2℘	NOUN
ejpam-6321	506	6	)	)	PUNCT
ejpam-6321	507	1	+	+	CCONJ
ejpam-6321	507	2	α5g(f2℘	α5g(f2℘	PROPN
ejpam-6321	507	3	,	,	PUNCT
ejpam-6321	507	4	f2℘	f2℘	NOUN
ejpam-6321	507	5	,	,	PUNCT
ejpam-6321	507	6	℘	℘	PROPN
ejpam-6321	507	7	)	)	PUNCT
ejpam-6321	507	8	by	by	ADP
ejpam-6321	507	9	using	use	VERB
ejpam-6321	507	10	proposition	proposition	NOUN
ejpam-6321	507	11	≤	≤	PUNCT
ejpam-6321	507	12	α2	α2	ADJ
ejpam-6321	507	13	+	+	CCONJ
ejpam-6321	507	14	α4	α4	NOUN
ejpam-6321	507	15	+	+	CCONJ
ejpam-6321	507	16	α5)g(℘	α5)g(℘	NUM
ejpam-6321	507	17	,	,	PUNCT
ejpam-6321	507	18	℘	℘	NOUN
ejpam-6321	507	19	,	,	PUNCT
ejpam-6321	507	20	f2℘	f2℘	NOUN
ejpam-6321	507	21	)	)	PUNCT
ejpam-6321	507	22	g(℘	g(℘	PROPN
ejpam-6321	507	23	,	,	PUNCT
ejpam-6321	507	24	f2℘	f2℘	NOUN
ejpam-6321	507	25	,	,	PUNCT
ejpam-6321	507	26	℘	℘	PROPN
ejpam-6321	507	27	)	)	PUNCT
ejpam-6321	507	28	−	−	PROPN
ejpam-6321	507	29	(	(	PUNCT
ejpam-6321	507	30	α2	α2	ADJ
ejpam-6321	507	31	+	+	CCONJ
ejpam-6321	507	32	α4	α4	NOUN
ejpam-6321	507	33	+	+	CCONJ
ejpam-6321	507	34	α5)g(℘	α5)g(℘	NUM
ejpam-6321	507	35	,	,	PUNCT
ejpam-6321	507	36	℘	℘	NOUN
ejpam-6321	507	37	,	,	PUNCT
ejpam-6321	507	38	f2℘	f2℘	NOUN
ejpam-6321	507	39	)	)	PUNCT
ejpam-6321	507	40	≤	≤	NOUN
ejpam-6321	507	41	0	0	NUM
ejpam-6321	507	42	.	.	PUNCT
ejpam-6321	508	1	(	(	PUNCT
ejpam-6321	508	2	1	1	NUM
ejpam-6321	508	3	−	−	ADV
ejpam-6321	508	4	α2	α2	ADJ
ejpam-6321	508	5	−	−	NOUN
ejpam-6321	508	6	α4	α4	NOUN
ejpam-6321	508	7	−	−	ADP
ejpam-6321	508	8	α5)g(℘	α5)g(℘	NUM
ejpam-6321	508	9	,	,	PUNCT
ejpam-6321	508	10	℘	℘	PROPN
ejpam-6321	508	11	,	,	PUNCT
ejpam-6321	508	12	f2℘	f2℘	NOUN
ejpam-6321	508	13	)	)	PUNCT
ejpam-6321	508	14	≤	≤	NOUN
ejpam-6321	508	15	0	0	NUM
ejpam-6321	508	16	is	be	AUX
ejpam-6321	508	17	a	a	DET
ejpam-6321	508	18	contradiction	contradiction	NOUN
ejpam-6321	508	19	.	.	PUNCT
ejpam-6321	509	1	since	since	SCONJ
ejpam-6321	509	2	(	(	PUNCT
ejpam-6321	509	3	1−	1−	NUM
ejpam-6321	509	4	α2	α2	ADJ
ejpam-6321	509	5	−	−	PROPN
ejpam-6321	509	6	α4	α4	NOUN
ejpam-6321	509	7	−	−	NOUN
ejpam-6321	509	8	α5	α5	NOUN
ejpam-6321	509	9	)	)	PUNCT
ejpam-6321	509	10	̸=	̸=	PROPN
ejpam-6321	509	11	0	0	NUM
ejpam-6321	509	12	,	,	PUNCT
ejpam-6321	509	13	therefore	therefore	ADV
ejpam-6321	509	14	,	,	PUNCT
ejpam-6321	509	15	g(f2℘	g(f2℘	PROPN
ejpam-6321	509	16	,	,	PUNCT
ejpam-6321	509	17	℘	℘	PROPN
ejpam-6321	509	18	,	,	PUNCT
ejpam-6321	509	19	℘	℘	NUM
ejpam-6321	509	20	)	)	PUNCT
ejpam-6321	509	21	=	=	SYM
ejpam-6321	509	22	0	0	X
ejpam-6321	509	23	.	.	PUNCT
ejpam-6321	509	24	thus	thus	ADV
ejpam-6321	509	25	,	,	PUNCT
ejpam-6321	509	26	f2℘	f2℘	NOUN
ejpam-6321	509	27	=	=	PUNCT
ejpam-6321	509	28	℘	℘	PROPN
ejpam-6321	509	29	(	(	PUNCT
ejpam-6321	509	30	25	25	NUM
ejpam-6321	509	31	)	)	PUNCT
ejpam-6321	509	32	(	(	PUNCT
ejpam-6321	509	33	v	v	NOUN
ejpam-6321	509	34	)	)	PUNCT
ejpam-6321	509	35	.	.	PUNCT
ejpam-6321	510	1	if	if	SCONJ
ejpam-6321	510	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	510	3	)	)	PUNCT
ejpam-6321	510	4	,	,	PUNCT
ejpam-6321	510	5	x3k	x3k	NOUN
ejpam-6321	510	6	,	,	PUNCT
ejpam-6321	510	7	x3k	x3k	PRON
ejpam-6321	510	8	)	)	PUNCT
ejpam-6321	510	9	is	be	AUX
ejpam-6321	510	10	a	a	DET
ejpam-6321	510	11	maximum	maximum	ADJ
ejpam-6321	510	12	term	term	NOUN
ejpam-6321	510	13	in	in	ADP
ejpam-6321	510	14	(	(	PUNCT
ejpam-6321	510	15	21	21	NUM
ejpam-6321	510	16	)	)	PUNCT
ejpam-6321	510	17	then	then	ADV
ejpam-6321	510	18	we	we	PRON
ejpam-6321	510	19	can	can	AUX
ejpam-6321	510	20	write	write	VERB
ejpam-6321	510	21	;	;	PUNCT
ejpam-6321	510	22	≤	≤	NUM
ejpam-6321	510	23	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	510	24	,	,	PUNCT
ejpam-6321	510	25	℘	℘	PROPN
ejpam-6321	510	26	,	,	PUNCT
ejpam-6321	510	27	x(3k+2	x(3k+2	NOUN
ejpam-6321	510	28	)	)	PUNCT
ejpam-6321	510	29	)	)	PUNCT
ejpam-6321	511	1	+	+	CCONJ
ejpam-6321	511	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	511	3	,	,	PUNCT
ejpam-6321	511	4	℘	℘	PROPN
ejpam-6321	511	5	,	,	PUNCT
ejpam-6321	511	6	f2℘	f2℘	NOUN
ejpam-6321	511	7	)	)	PUNCT
ejpam-6321	511	8	+	+	NUM
ejpam-6321	511	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	511	10	)	)	PUNCT
ejpam-6321	511	11	,	,	PUNCT
ejpam-6321	511	12	℘	℘	PROPN
ejpam-6321	511	13	,	,	PUNCT
ejpam-6321	511	14	℘	℘	NUM
ejpam-6321	511	15	)	)	PUNCT
ejpam-6321	511	16	+	+	NUM
ejpam-6321	511	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	511	18	)	)	PUNCT
ejpam-6321	511	19	,	,	PUNCT
ejpam-6321	511	20	℘	℘	PROPN
ejpam-6321	511	21	,	,	PUNCT
ejpam-6321	511	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	511	23	)	)	PUNCT
ejpam-6321	511	24	)	)	PUNCT
ejpam-6321	512	1	+	+	ADP
ejpam-6321	512	2	g(x3k	g(x3k	NOUN
ejpam-6321	512	3	,	,	PUNCT
ejpam-6321	512	4	f2℘	f2℘	NOUN
ejpam-6321	512	5	,	,	PUNCT
ejpam-6321	512	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	512	7	)	)	PUNCT
ejpam-6321	512	8	)	)	PUNCT
ejpam-6321	513	1	+	+	ADP
ejpam-6321	513	2	g(x3k	g(x3k	NOUN
ejpam-6321	513	3	,	,	PUNCT
ejpam-6321	513	4	℘	℘	PROPN
ejpam-6321	513	5	,	,	PUNCT
ejpam-6321	513	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	513	7	)	)	PUNCT
ejpam-6321	513	8	)	)	PUNCT
ejpam-6321	513	9	]	]	PUNCT
ejpam-6321	514	1	+	+	CCONJ
ejpam-6321	514	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	514	3	)	)	PUNCT
ejpam-6321	514	4	,	,	PUNCT
ejpam-6321	514	5	x3k	x3k	NOUN
ejpam-6321	514	6	,	,	PUNCT
ejpam-6321	514	7	x3k	x3k	NOUN
ejpam-6321	514	8	)	)	PUNCT
ejpam-6321	514	9	applying	apply	VERB
ejpam-6321	514	10	limn→∞	limn→∞	PROPN
ejpam-6321	514	11	on	on	ADP
ejpam-6321	514	12	both	both	DET
ejpam-6321	514	13	sides	side	NOUN
ejpam-6321	514	14	,	,	PUNCT
ejpam-6321	514	15	we	we	PRON
ejpam-6321	514	16	get	get	VERB
ejpam-6321	514	17	,	,	PUNCT
ejpam-6321	514	18	g(℘	g(℘	NOUN
ejpam-6321	514	19	,	,	PUNCT
ejpam-6321	514	20	f2℘	f2℘	NOUN
ejpam-6321	514	21	,	,	PUNCT
ejpam-6321	514	22	℘	℘	PROPN
ejpam-6321	514	23	)	)	PUNCT
ejpam-6321	514	24	≤	≤	NOUN
ejpam-6321	514	25	α1g(℘	α1g(℘	NOUN
ejpam-6321	514	26	,	,	PUNCT
ejpam-6321	514	27	℘	℘	NOUN
ejpam-6321	514	28	,	,	PUNCT
ejpam-6321	514	29	℘	℘	NUM
ejpam-6321	514	30	)	)	PUNCT
ejpam-6321	514	31	+	+	NUM
ejpam-6321	514	32	α2g(℘	α2g(℘	NOUN
ejpam-6321	514	33	,	,	PUNCT
ejpam-6321	514	34	℘	℘	NOUN
ejpam-6321	514	35	,	,	PUNCT
ejpam-6321	514	36	f2℘	f2℘	NOUN
ejpam-6321	514	37	)	)	PUNCT
ejpam-6321	515	1	+	+	CCONJ
ejpam-6321	515	2	α3g(℘	α3g(℘	NUM
ejpam-6321	515	3	,	,	PUNCT
ejpam-6321	515	4	℘	℘	NOUN
ejpam-6321	515	5	,	,	PUNCT
ejpam-6321	515	6	℘)+	℘)+	NOUN
ejpam-6321	515	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	515	8	,	,	PUNCT
ejpam-6321	515	9	℘	℘	NOUN
ejpam-6321	515	10	,	,	PUNCT
ejpam-6321	515	11	℘	℘	NUM
ejpam-6321	515	12	)	)	PUNCT
ejpam-6321	515	13	+	+	NOUN
ejpam-6321	515	14	g(℘	g(℘	NOUN
ejpam-6321	515	15	,	,	PUNCT
ejpam-6321	515	16	f2℘	f2℘	NOUN
ejpam-6321	515	17	,	,	PUNCT
ejpam-6321	515	18	℘	℘	PROPN
ejpam-6321	515	19	)	)	PUNCT
ejpam-6321	515	20	+	+	NOUN
ejpam-6321	515	21	g(℘	g(℘	NOUN
ejpam-6321	515	22	,	,	PUNCT
ejpam-6321	515	23	℘	℘	NOUN
ejpam-6321	515	24	,	,	PUNCT
ejpam-6321	515	25	℘	℘	NUM
ejpam-6321	515	26	)	)	PUNCT
ejpam-6321	515	27	]	]	PUNCT
ejpam-6321	516	1	+	+	NUM
ejpam-6321	516	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	516	3	,	,	PUNCT
ejpam-6321	516	4	℘	℘	NOUN
ejpam-6321	516	5	,	,	PUNCT
ejpam-6321	516	6	℘	℘	NUM
ejpam-6321	516	7	)	)	PUNCT
ejpam-6321	516	8	≤	≤	NOUN
ejpam-6321	516	9	α2	α2	ADJ
ejpam-6321	516	10	+	+	CCONJ
ejpam-6321	517	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	517	2	,	,	PUNCT
ejpam-6321	517	3	℘	℘	PROPN
ejpam-6321	517	4	,	,	PUNCT
ejpam-6321	517	5	f2℘	f2℘	NOUN
ejpam-6321	517	6	)	)	PUNCT
ejpam-6321	517	7	g(℘	g(℘	PROPN
ejpam-6321	517	8	,	,	PUNCT
ejpam-6321	517	9	f2℘	f2℘	NOUN
ejpam-6321	517	10	,	,	PUNCT
ejpam-6321	517	11	℘	℘	PROPN
ejpam-6321	517	12	)	)	PUNCT
ejpam-6321	517	13	−	−	PROPN
ejpam-6321	517	14	(	(	PUNCT
ejpam-6321	517	15	α2	α2	PROPN
ejpam-6321	517	16	+	+	CCONJ
ejpam-6321	517	17	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	517	18	,	,	PUNCT
ejpam-6321	517	19	℘	℘	NOUN
ejpam-6321	517	20	,	,	PUNCT
ejpam-6321	517	21	f2℘	f2℘	NOUN
ejpam-6321	517	22	)	)	PUNCT
ejpam-6321	517	23	≤	≤	NOUN
ejpam-6321	517	24	0	0	NUM
ejpam-6321	517	25	.	.	PUNCT
ejpam-6321	518	1	(	(	PUNCT
ejpam-6321	518	2	1	1	NUM
ejpam-6321	518	3	−	−	NOUN
ejpam-6321	518	4	α2	α2	ADJ
ejpam-6321	518	5	−	−	PROPN
ejpam-6321	519	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	519	2	,	,	PUNCT
ejpam-6321	519	3	℘	℘	PROPN
ejpam-6321	519	4	,	,	PUNCT
ejpam-6321	519	5	f2℘	f2℘	NOUN
ejpam-6321	519	6	)	)	PUNCT
ejpam-6321	519	7	≤	≤	NOUN
ejpam-6321	519	8	0	0	NUM
ejpam-6321	519	9	is	be	AUX
ejpam-6321	519	10	a	a	DET
ejpam-6321	519	11	contradiction	contradiction	NOUN
ejpam-6321	519	12	.	.	PUNCT
ejpam-6321	520	1	since	since	SCONJ
ejpam-6321	520	2	(	(	PUNCT
ejpam-6321	520	3	1−	1−	NUM
ejpam-6321	520	4	α2	α2	ADJ
ejpam-6321	520	5	−	−	PROPN
ejpam-6321	520	6	α4	α4	NOUN
ejpam-6321	520	7	)	)	PUNCT
ejpam-6321	520	8	̸=	̸=	PROPN
ejpam-6321	520	9	0	0	NUM
ejpam-6321	520	10	,	,	PUNCT
ejpam-6321	520	11	therefore	therefore	ADV
ejpam-6321	520	12	,	,	PUNCT
ejpam-6321	520	13	g(f2℘	g(f2℘	PROPN
ejpam-6321	520	14	,	,	PUNCT
ejpam-6321	520	15	℘	℘	PROPN
ejpam-6321	520	16	,	,	PUNCT
ejpam-6321	520	17	℘	℘	NUM
ejpam-6321	520	18	)	)	PUNCT
ejpam-6321	520	19	=	=	SYM
ejpam-6321	520	20	0	0	X
ejpam-6321	520	21	.	.	PUNCT
ejpam-6321	520	22	thus	thus	ADV
ejpam-6321	520	23	,	,	PUNCT
ejpam-6321	520	24	f2℘	f2℘	NOUN
ejpam-6321	520	25	=	=	PUNCT
ejpam-6321	520	26	℘	℘	PROPN
ejpam-6321	520	27	(	(	PUNCT
ejpam-6321	520	28	26	26	NUM
ejpam-6321	520	29	)	)	PUNCT
ejpam-6321	520	30	m.	m.	NOUN
ejpam-6321	520	31	noorwali	noorwali	PROPN
ejpam-6321	520	32	et	et	PROPN
ejpam-6321	520	33	al	al	PROPN
ejpam-6321	520	34	.	.	PUNCT
ejpam-6321	520	35	/	/	SYM
ejpam-6321	520	36	eur	eur	PROPN
ejpam-6321	520	37	.	.	PUNCT
ejpam-6321	521	1	j.	j.	PROPN
ejpam-6321	521	2	pure	pure	PROPN
ejpam-6321	521	3	appl	appl	PROPN
ejpam-6321	521	4	.	.	PROPN
ejpam-6321	521	5	math	math	PROPN
ejpam-6321	521	6	,	,	PUNCT
ejpam-6321	521	7	18	18	NUM
ejpam-6321	521	8	(	(	PUNCT
ejpam-6321	521	9	3	3	NUM
ejpam-6321	521	10	)	)	PUNCT
ejpam-6321	521	11	(	(	PUNCT
ejpam-6321	521	12	2025	2025	NUM
ejpam-6321	521	13	)	)	PUNCT
ejpam-6321	521	14	,	,	PUNCT
ejpam-6321	521	15	6321	6321	NUM
ejpam-6321	521	16	18	18	NUM
ejpam-6321	521	17	of	of	ADP
ejpam-6321	521	18	27	27	NUM
ejpam-6321	521	19	(	(	PUNCT
ejpam-6321	521	20	vi	vi	NOUN
ejpam-6321	521	21	)	)	PUNCT
ejpam-6321	521	22	.	.	PUNCT
ejpam-6321	522	1	if	if	SCONJ
ejpam-6321	522	2	g(℘	g(℘	NOUN
ejpam-6321	522	3	,	,	PUNCT
ejpam-6321	522	4	f2℘	f2℘	NOUN
ejpam-6321	522	5	,	,	PUNCT
ejpam-6321	522	6	℘	℘	PROPN
ejpam-6321	522	7	)	)	PUNCT
ejpam-6321	522	8	is	be	AUX
ejpam-6321	522	9	a	a	DET
ejpam-6321	522	10	maximum	maximum	ADJ
ejpam-6321	522	11	term	term	NOUN
ejpam-6321	522	12	in	in	ADP
ejpam-6321	522	13	(	(	PUNCT
ejpam-6321	522	14	21	21	NUM
ejpam-6321	522	15	)	)	PUNCT
ejpam-6321	522	16	then	then	ADV
ejpam-6321	522	17	we	we	PRON
ejpam-6321	522	18	can	can	AUX
ejpam-6321	522	19	write	write	VERB
ejpam-6321	522	20	;	;	PUNCT
ejpam-6321	522	21	≤	≤	NUM
ejpam-6321	522	22	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	522	23	,	,	PUNCT
ejpam-6321	522	24	℘	℘	PROPN
ejpam-6321	522	25	,	,	PUNCT
ejpam-6321	522	26	x(3k+2	x(3k+2	NOUN
ejpam-6321	522	27	)	)	PUNCT
ejpam-6321	522	28	)	)	PUNCT
ejpam-6321	523	1	+	+	CCONJ
ejpam-6321	523	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	523	3	,	,	PUNCT
ejpam-6321	523	4	℘	℘	PROPN
ejpam-6321	523	5	,	,	PUNCT
ejpam-6321	523	6	f2℘	f2℘	NOUN
ejpam-6321	523	7	)	)	PUNCT
ejpam-6321	523	8	+	+	NUM
ejpam-6321	523	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	523	10	)	)	PUNCT
ejpam-6321	523	11	,	,	PUNCT
ejpam-6321	523	12	℘	℘	PROPN
ejpam-6321	523	13	,	,	PUNCT
ejpam-6321	523	14	℘	℘	NUM
ejpam-6321	523	15	)	)	PUNCT
ejpam-6321	523	16	+	+	NUM
ejpam-6321	523	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	523	18	)	)	PUNCT
ejpam-6321	523	19	,	,	PUNCT
ejpam-6321	523	20	℘	℘	PROPN
ejpam-6321	523	21	,	,	PUNCT
ejpam-6321	523	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	523	23	)	)	PUNCT
ejpam-6321	523	24	)	)	PUNCT
ejpam-6321	524	1	+	+	ADP
ejpam-6321	524	2	g(x3k	g(x3k	NOUN
ejpam-6321	524	3	,	,	PUNCT
ejpam-6321	524	4	f2℘	f2℘	NOUN
ejpam-6321	524	5	,	,	PUNCT
ejpam-6321	524	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	524	7	)	)	PUNCT
ejpam-6321	524	8	)	)	PUNCT
ejpam-6321	525	1	+	+	ADP
ejpam-6321	525	2	g(x3k	g(x3k	NOUN
ejpam-6321	525	3	,	,	PUNCT
ejpam-6321	525	4	℘	℘	PROPN
ejpam-6321	525	5	,	,	PUNCT
ejpam-6321	525	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	525	7	)	)	PUNCT
ejpam-6321	525	8	)	)	PUNCT
ejpam-6321	525	9	]	]	PUNCT
ejpam-6321	526	1	+	+	CCONJ
ejpam-6321	526	2	α5g(f2℘	α5g(f2℘	PROPN
ejpam-6321	526	3	,	,	PUNCT
ejpam-6321	526	4	f2℘	f2℘	NOUN
ejpam-6321	526	5	,	,	PUNCT
ejpam-6321	526	6	℘	℘	PROPN
ejpam-6321	526	7	)	)	PUNCT
ejpam-6321	526	8	applying	apply	VERB
ejpam-6321	526	9	limn→∞	limn→∞	PROPN
ejpam-6321	526	10	on	on	ADP
ejpam-6321	526	11	both	both	DET
ejpam-6321	526	12	sides	side	NOUN
ejpam-6321	526	13	,	,	PUNCT
ejpam-6321	526	14	we	we	PRON
ejpam-6321	526	15	get	get	VERB
ejpam-6321	526	16	,	,	PUNCT
ejpam-6321	526	17	g(℘	g(℘	NOUN
ejpam-6321	526	18	,	,	PUNCT
ejpam-6321	526	19	f2℘	f2℘	NOUN
ejpam-6321	526	20	,	,	PUNCT
ejpam-6321	526	21	℘	℘	PROPN
ejpam-6321	526	22	)	)	PUNCT
ejpam-6321	526	23	≤	≤	NOUN
ejpam-6321	526	24	α1g(℘	α1g(℘	NOUN
ejpam-6321	526	25	,	,	PUNCT
ejpam-6321	526	26	℘	℘	NOUN
ejpam-6321	526	27	,	,	PUNCT
ejpam-6321	526	28	℘	℘	NUM
ejpam-6321	526	29	)	)	PUNCT
ejpam-6321	526	30	+	+	NUM
ejpam-6321	526	31	α2g(℘	α2g(℘	NOUN
ejpam-6321	526	32	,	,	PUNCT
ejpam-6321	526	33	℘	℘	NOUN
ejpam-6321	526	34	,	,	PUNCT
ejpam-6321	526	35	f2℘	f2℘	NOUN
ejpam-6321	526	36	)	)	PUNCT
ejpam-6321	527	1	+	+	CCONJ
ejpam-6321	527	2	α3g(℘	α3g(℘	NUM
ejpam-6321	527	3	,	,	PUNCT
ejpam-6321	527	4	℘	℘	NOUN
ejpam-6321	527	5	,	,	PUNCT
ejpam-6321	527	6	℘)+	℘)+	NOUN
ejpam-6321	527	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	527	8	,	,	PUNCT
ejpam-6321	527	9	℘	℘	NOUN
ejpam-6321	527	10	,	,	PUNCT
ejpam-6321	527	11	℘	℘	NUM
ejpam-6321	527	12	)	)	PUNCT
ejpam-6321	527	13	+	+	NOUN
ejpam-6321	527	14	g(℘	g(℘	NOUN
ejpam-6321	527	15	,	,	PUNCT
ejpam-6321	527	16	f2℘	f2℘	NOUN
ejpam-6321	527	17	,	,	PUNCT
ejpam-6321	527	18	℘	℘	PROPN
ejpam-6321	527	19	)	)	PUNCT
ejpam-6321	527	20	+	+	NOUN
ejpam-6321	527	21	g(℘	g(℘	NOUN
ejpam-6321	527	22	,	,	PUNCT
ejpam-6321	527	23	℘	℘	NOUN
ejpam-6321	527	24	,	,	PUNCT
ejpam-6321	527	25	℘	℘	NUM
ejpam-6321	527	26	)	)	PUNCT
ejpam-6321	527	27	]	]	PUNCT
ejpam-6321	528	1	+	+	NUM
ejpam-6321	528	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	528	3	,	,	PUNCT
ejpam-6321	528	4	f2℘	f2℘	NOUN
ejpam-6321	528	5	,	,	PUNCT
ejpam-6321	528	6	℘	℘	PROPN
ejpam-6321	528	7	)	)	PUNCT
ejpam-6321	528	8	≤	≤	NOUN
ejpam-6321	528	9	α2	α2	ADJ
ejpam-6321	528	10	+	+	CCONJ
ejpam-6321	528	11	α4	α4	NOUN
ejpam-6321	528	12	+	+	CCONJ
ejpam-6321	528	13	α5)g(℘	α5)g(℘	NUM
ejpam-6321	528	14	,	,	PUNCT
ejpam-6321	528	15	℘	℘	NOUN
ejpam-6321	528	16	,	,	PUNCT
ejpam-6321	528	17	f2℘	f2℘	NOUN
ejpam-6321	528	18	)	)	PUNCT
ejpam-6321	528	19	g(℘	g(℘	PROPN
ejpam-6321	528	20	,	,	PUNCT
ejpam-6321	528	21	f2℘	f2℘	NOUN
ejpam-6321	528	22	,	,	PUNCT
ejpam-6321	528	23	℘	℘	PROPN
ejpam-6321	528	24	)	)	PUNCT
ejpam-6321	528	25	−	−	PROPN
ejpam-6321	529	1	(	(	PUNCT
ejpam-6321	529	2	α2	α2	ADJ
ejpam-6321	529	3	+	+	CCONJ
ejpam-6321	529	4	α4	α4	NOUN
ejpam-6321	529	5	+	+	CCONJ
ejpam-6321	529	6	α5)g(℘	α5)g(℘	NUM
ejpam-6321	529	7	,	,	PUNCT
ejpam-6321	529	8	℘	℘	NOUN
ejpam-6321	529	9	,	,	PUNCT
ejpam-6321	529	10	f2℘	f2℘	NOUN
ejpam-6321	529	11	)	)	PUNCT
ejpam-6321	529	12	≤	≤	NOUN
ejpam-6321	529	13	0	0	NUM
ejpam-6321	529	14	.	.	PUNCT
ejpam-6321	530	1	(	(	PUNCT
ejpam-6321	530	2	1	1	NUM
ejpam-6321	530	3	−	−	ADV
ejpam-6321	530	4	α2	α2	ADJ
ejpam-6321	530	5	−	−	NOUN
ejpam-6321	530	6	α4	α4	NOUN
ejpam-6321	530	7	−	−	ADP
ejpam-6321	530	8	α5)g(℘	α5)g(℘	NUM
ejpam-6321	530	9	,	,	PUNCT
ejpam-6321	530	10	℘	℘	PROPN
ejpam-6321	530	11	,	,	PUNCT
ejpam-6321	530	12	f2℘	f2℘	NOUN
ejpam-6321	530	13	)	)	PUNCT
ejpam-6321	530	14	≤	≤	NOUN
ejpam-6321	530	15	0	0	NUM
ejpam-6321	530	16	is	be	AUX
ejpam-6321	530	17	a	a	DET
ejpam-6321	530	18	contradiction	contradiction	NOUN
ejpam-6321	530	19	.	.	PUNCT
ejpam-6321	531	1	since	since	SCONJ
ejpam-6321	531	2	(	(	PUNCT
ejpam-6321	531	3	1−	1−	NUM
ejpam-6321	531	4	α2	α2	ADJ
ejpam-6321	531	5	−	−	PROPN
ejpam-6321	531	6	α4	α4	NOUN
ejpam-6321	531	7	−	−	NOUN
ejpam-6321	531	8	α5	α5	NOUN
ejpam-6321	531	9	)	)	PUNCT
ejpam-6321	531	10	̸=	̸=	PROPN
ejpam-6321	531	11	0	0	NUM
ejpam-6321	531	12	,	,	PUNCT
ejpam-6321	531	13	therefore	therefore	ADV
ejpam-6321	531	14	,	,	PUNCT
ejpam-6321	531	15	g(f2℘	g(f2℘	PROPN
ejpam-6321	531	16	,	,	PUNCT
ejpam-6321	531	17	℘	℘	PROPN
ejpam-6321	531	18	,	,	PUNCT
ejpam-6321	531	19	℘	℘	NUM
ejpam-6321	531	20	)	)	PUNCT
ejpam-6321	531	21	=	=	SYM
ejpam-6321	531	22	0	0	X
ejpam-6321	531	23	.	.	PUNCT
ejpam-6321	531	24	thus	thus	ADV
ejpam-6321	531	25	,	,	PUNCT
ejpam-6321	531	26	f2℘	f2℘	NOUN
ejpam-6321	531	27	=	=	SYM
ejpam-6321	531	28	℘	℘	PROPN
ejpam-6321	531	29	(	(	PUNCT
ejpam-6321	531	30	27	27	NUM
ejpam-6321	531	31	)	)	PUNCT
ejpam-6321	531	32	(	(	PUNCT
ejpam-6321	531	33	vii	vii	PROPN
ejpam-6321	531	34	)	)	PUNCT
ejpam-6321	531	35	.	.	PUNCT
ejpam-6321	532	1	if	if	SCONJ
ejpam-6321	532	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	532	3	)	)	PUNCT
ejpam-6321	532	4	,	,	PUNCT
ejpam-6321	532	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	532	6	)	)	PUNCT
ejpam-6321	532	7	,	,	PUNCT
ejpam-6321	532	8	x(3k+3	x(3k+3	NUM
ejpam-6321	532	9	)	)	PUNCT
ejpam-6321	532	10	)	)	PUNCT
ejpam-6321	532	11	is	be	AUX
ejpam-6321	532	12	a	a	DET
ejpam-6321	532	13	maximum	maximum	ADJ
ejpam-6321	532	14	term	term	NOUN
ejpam-6321	532	15	in	in	ADP
ejpam-6321	532	16	(	(	PUNCT
ejpam-6321	532	17	21	21	NUM
ejpam-6321	532	18	)	)	PUNCT
ejpam-6321	532	19	then	then	ADV
ejpam-6321	532	20	we	we	PRON
ejpam-6321	532	21	can	can	AUX
ejpam-6321	532	22	write	write	VERB
ejpam-6321	532	23	;	;	PUNCT
ejpam-6321	532	24	≤	≤	NUM
ejpam-6321	532	25	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	532	26	,	,	PUNCT
ejpam-6321	532	27	℘	℘	PROPN
ejpam-6321	532	28	,	,	PUNCT
ejpam-6321	532	29	x(3k+2	x(3k+2	NOUN
ejpam-6321	532	30	)	)	PUNCT
ejpam-6321	532	31	)	)	PUNCT
ejpam-6321	533	1	+	+	CCONJ
ejpam-6321	533	2	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	533	3	,	,	PUNCT
ejpam-6321	533	4	℘	℘	PROPN
ejpam-6321	533	5	,	,	PUNCT
ejpam-6321	533	6	f2℘	f2℘	NOUN
ejpam-6321	533	7	)	)	PUNCT
ejpam-6321	533	8	+	+	NUM
ejpam-6321	533	9	α3g(x(3k+1	α3g(x(3k+1	NOUN
ejpam-6321	533	10	)	)	PUNCT
ejpam-6321	533	11	,	,	PUNCT
ejpam-6321	533	12	℘	℘	PROPN
ejpam-6321	533	13	,	,	PUNCT
ejpam-6321	533	14	℘	℘	NUM
ejpam-6321	533	15	)	)	PUNCT
ejpam-6321	533	16	+	+	NUM
ejpam-6321	533	17	α4[g(x(3k+1	α4[g(x(3k+1	NOUN
ejpam-6321	533	18	)	)	PUNCT
ejpam-6321	533	19	,	,	PUNCT
ejpam-6321	533	20	℘	℘	PROPN
ejpam-6321	533	21	,	,	PUNCT
ejpam-6321	533	22	x(3k+2	x(3k+2	NOUN
ejpam-6321	533	23	)	)	PUNCT
ejpam-6321	533	24	)	)	PUNCT
ejpam-6321	534	1	+	+	ADP
ejpam-6321	534	2	g(x3k	g(x3k	NOUN
ejpam-6321	534	3	,	,	PUNCT
ejpam-6321	534	4	f2℘	f2℘	NOUN
ejpam-6321	534	5	,	,	PUNCT
ejpam-6321	534	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	534	7	)	)	PUNCT
ejpam-6321	534	8	)	)	PUNCT
ejpam-6321	535	1	+	+	ADP
ejpam-6321	535	2	g(x3k	g(x3k	NOUN
ejpam-6321	535	3	,	,	PUNCT
ejpam-6321	535	4	℘	℘	PROPN
ejpam-6321	535	5	,	,	PUNCT
ejpam-6321	535	6	x(3k+2	x(3k+2	NOUN
ejpam-6321	535	7	)	)	PUNCT
ejpam-6321	535	8	)	)	PUNCT
ejpam-6321	535	9	]	]	PUNCT
ejpam-6321	536	1	+	+	PUNCT
ejpam-6321	536	2	α5g(x(3k+2	α5g(x(3k+2	NOUN
ejpam-6321	536	3	)	)	PUNCT
ejpam-6321	536	4	,	,	PUNCT
ejpam-6321	536	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	536	6	)	)	PUNCT
ejpam-6321	536	7	,	,	PUNCT
ejpam-6321	536	8	x(3k+3	x(3k+3	NUM
ejpam-6321	536	9	)	)	PUNCT
ejpam-6321	536	10	)	)	PUNCT
ejpam-6321	536	11	applying	apply	VERB
ejpam-6321	536	12	limn→∞	limn→∞	PROPN
ejpam-6321	536	13	on	on	ADP
ejpam-6321	536	14	both	both	DET
ejpam-6321	536	15	sides	side	NOUN
ejpam-6321	536	16	,	,	PUNCT
ejpam-6321	536	17	we	we	PRON
ejpam-6321	536	18	get	get	VERB
ejpam-6321	536	19	,	,	PUNCT
ejpam-6321	536	20	g(℘	g(℘	NOUN
ejpam-6321	536	21	,	,	PUNCT
ejpam-6321	536	22	f2℘	f2℘	NOUN
ejpam-6321	536	23	,	,	PUNCT
ejpam-6321	536	24	℘	℘	PROPN
ejpam-6321	536	25	)	)	PUNCT
ejpam-6321	536	26	≤	≤	NOUN
ejpam-6321	536	27	α1g(℘	α1g(℘	NOUN
ejpam-6321	536	28	,	,	PUNCT
ejpam-6321	536	29	℘	℘	NOUN
ejpam-6321	536	30	,	,	PUNCT
ejpam-6321	536	31	℘	℘	NUM
ejpam-6321	536	32	)	)	PUNCT
ejpam-6321	536	33	+	+	NUM
ejpam-6321	536	34	α2g(℘	α2g(℘	NOUN
ejpam-6321	536	35	,	,	PUNCT
ejpam-6321	536	36	℘	℘	NOUN
ejpam-6321	536	37	,	,	PUNCT
ejpam-6321	536	38	f2℘	f2℘	NOUN
ejpam-6321	536	39	)	)	PUNCT
ejpam-6321	537	1	+	+	CCONJ
ejpam-6321	537	2	α3g(℘	α3g(℘	NUM
ejpam-6321	537	3	,	,	PUNCT
ejpam-6321	537	4	℘	℘	NOUN
ejpam-6321	537	5	,	,	PUNCT
ejpam-6321	537	6	℘)+	℘)+	NOUN
ejpam-6321	537	7	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	537	8	,	,	PUNCT
ejpam-6321	537	9	℘	℘	NOUN
ejpam-6321	537	10	,	,	PUNCT
ejpam-6321	537	11	℘	℘	NUM
ejpam-6321	537	12	)	)	PUNCT
ejpam-6321	537	13	+	+	NOUN
ejpam-6321	537	14	g(℘	g(℘	NOUN
ejpam-6321	537	15	,	,	PUNCT
ejpam-6321	537	16	f2℘	f2℘	NOUN
ejpam-6321	537	17	,	,	PUNCT
ejpam-6321	537	18	℘	℘	PROPN
ejpam-6321	537	19	)	)	PUNCT
ejpam-6321	537	20	+	+	NOUN
ejpam-6321	537	21	g(℘	g(℘	NOUN
ejpam-6321	537	22	,	,	PUNCT
ejpam-6321	537	23	℘	℘	NOUN
ejpam-6321	537	24	,	,	PUNCT
ejpam-6321	537	25	℘	℘	NUM
ejpam-6321	537	26	)	)	PUNCT
ejpam-6321	537	27	]	]	PUNCT
ejpam-6321	538	1	+	+	NUM
ejpam-6321	538	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	538	3	,	,	PUNCT
ejpam-6321	538	4	℘	℘	NOUN
ejpam-6321	538	5	,	,	PUNCT
ejpam-6321	538	6	℘	℘	NUM
ejpam-6321	538	7	)	)	PUNCT
ejpam-6321	538	8	≤	≤	NOUN
ejpam-6321	538	9	α2	α2	ADJ
ejpam-6321	538	10	+	+	CCONJ
ejpam-6321	539	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	539	2	,	,	PUNCT
ejpam-6321	539	3	℘	℘	PROPN
ejpam-6321	539	4	,	,	PUNCT
ejpam-6321	539	5	f2℘	f2℘	NOUN
ejpam-6321	539	6	)	)	PUNCT
ejpam-6321	539	7	g(℘	g(℘	PROPN
ejpam-6321	539	8	,	,	PUNCT
ejpam-6321	539	9	f2℘	f2℘	NOUN
ejpam-6321	539	10	,	,	PUNCT
ejpam-6321	539	11	℘	℘	PROPN
ejpam-6321	539	12	)	)	PUNCT
ejpam-6321	539	13	−	−	PROPN
ejpam-6321	539	14	(	(	PUNCT
ejpam-6321	539	15	α2	α2	PROPN
ejpam-6321	539	16	+	+	CCONJ
ejpam-6321	539	17	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	539	18	,	,	PUNCT
ejpam-6321	539	19	℘	℘	NOUN
ejpam-6321	539	20	,	,	PUNCT
ejpam-6321	539	21	f2℘	f2℘	NOUN
ejpam-6321	539	22	)	)	PUNCT
ejpam-6321	539	23	≤	≤	NOUN
ejpam-6321	539	24	0	0	NUM
ejpam-6321	539	25	.	.	PUNCT
ejpam-6321	540	1	(	(	PUNCT
ejpam-6321	540	2	1	1	NUM
ejpam-6321	540	3	−	−	NOUN
ejpam-6321	540	4	α2	α2	ADJ
ejpam-6321	540	5	−	−	PROPN
ejpam-6321	541	1	α4)g(℘	α4)g(℘	PROPN
ejpam-6321	541	2	,	,	PUNCT
ejpam-6321	541	3	℘	℘	PROPN
ejpam-6321	541	4	,	,	PUNCT
ejpam-6321	541	5	f2℘	f2℘	NOUN
ejpam-6321	541	6	)	)	PUNCT
ejpam-6321	541	7	≤	≤	NOUN
ejpam-6321	541	8	0	0	NUM
ejpam-6321	541	9	is	be	AUX
ejpam-6321	541	10	a	a	DET
ejpam-6321	541	11	contradiction	contradiction	NOUN
ejpam-6321	541	12	.	.	PUNCT
ejpam-6321	542	1	since	since	SCONJ
ejpam-6321	542	2	(	(	PUNCT
ejpam-6321	542	3	1−	1−	NUM
ejpam-6321	542	4	α2	α2	ADJ
ejpam-6321	542	5	−	−	PROPN
ejpam-6321	542	6	α4	α4	NOUN
ejpam-6321	542	7	)	)	PUNCT
ejpam-6321	542	8	̸=	̸=	PROPN
ejpam-6321	542	9	0	0	NUM
ejpam-6321	542	10	,	,	PUNCT
ejpam-6321	542	11	therefore	therefore	ADV
ejpam-6321	542	12	,	,	PUNCT
ejpam-6321	542	13	g(f2℘	g(f2℘	PROPN
ejpam-6321	542	14	,	,	PUNCT
ejpam-6321	542	15	℘	℘	PROPN
ejpam-6321	542	16	,	,	PUNCT
ejpam-6321	542	17	℘	℘	NUM
ejpam-6321	542	18	)	)	PUNCT
ejpam-6321	542	19	=	=	SYM
ejpam-6321	542	20	0	0	X
ejpam-6321	542	21	.	.	PUNCT
ejpam-6321	542	22	thus	thus	ADV
ejpam-6321	542	23	,	,	PUNCT
ejpam-6321	542	24	f2℘	f2℘	NOUN
ejpam-6321	542	25	=	=	SYM
ejpam-6321	542	26	℘	℘	PROPN
ejpam-6321	542	27	(	(	PUNCT
ejpam-6321	542	28	28	28	NUM
ejpam-6321	542	29	)	)	PUNCT
ejpam-6321	542	30	by	by	ADP
ejpam-6321	542	31	the	the	DET
ejpam-6321	542	32	view	view	NOUN
ejpam-6321	542	33	of	of	ADP
ejpam-6321	542	34	(	(	PUNCT
ejpam-6321	542	35	22)−	22)−	NUM
ejpam-6321	542	36	(	(	PUNCT
ejpam-6321	542	37	28	28	NUM
ejpam-6321	542	38	)	)	PUNCT
ejpam-6321	542	39	,	,	PUNCT
ejpam-6321	542	40	we	we	PRON
ejpam-6321	542	41	obtain	obtain	VERB
ejpam-6321	542	42	that	that	SCONJ
ejpam-6321	542	43	℘	℘	PROPN
ejpam-6321	542	44	is	be	AUX
ejpam-6321	542	45	a	a	DET
ejpam-6321	542	46	fp	fp	X
ejpam-6321	542	47	of	of	ADP
ejpam-6321	542	48	f1	f1	NOUN
ejpam-6321	542	49	,	,	PUNCT
ejpam-6321	542	50	f2	f2	PROPN
ejpam-6321	542	51	and	and	CCONJ
ejpam-6321	542	52	f3	f3	PROPN
ejpam-6321	542	53	,	,	PUNCT
ejpam-6321	542	54	i.e	i.e	NOUN
ejpam-6321	542	55	,	,	PUNCT
ejpam-6321	542	56	f2℘	f2℘	NOUN
ejpam-6321	542	57	=	=	PUNCT
ejpam-6321	542	58	℘	℘	PROPN
ejpam-6321	542	59	(	(	PUNCT
ejpam-6321	542	60	29	29	NUM
ejpam-6321	542	61	)	)	PUNCT
ejpam-6321	542	62	next	next	ADV
ejpam-6321	542	63	,	,	PUNCT
ejpam-6321	542	64	we	we	PRON
ejpam-6321	542	65	have	have	VERB
ejpam-6321	542	66	to	to	PART
ejpam-6321	542	67	show	show	VERB
ejpam-6321	542	68	that	that	SCONJ
ejpam-6321	542	69	f3℘	f3℘	NOUN
ejpam-6321	542	70	=	=	SYM
ejpam-6321	542	71	℘	℘	PROPN
ejpam-6321	542	72	by	by	ADP
ejpam-6321	542	73	contrary	contrary	ADJ
ejpam-6321	542	74	case	case	NOUN
ejpam-6321	542	75	let	let	VERB
ejpam-6321	542	76	f3℘	f3℘	PROPN
ejpam-6321	542	77	̸=	̸=	PROPN
ejpam-6321	542	78	℘.	℘.	PROPN
ejpam-6321	542	79	then	then	ADV
ejpam-6321	542	80	from	from	ADP
ejpam-6321	542	81	(	(	PUNCT
ejpam-6321	542	82	8)	8)	NUM
ejpam-6321	542	83	we	we	PRON
ejpam-6321	542	84	have	have	VERB
ejpam-6321	542	85	that	that	PRON
ejpam-6321	542	86	,	,	PUNCT
ejpam-6321	542	87	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	542	88	)	)	PUNCT
ejpam-6321	542	89	,	,	PUNCT
ejpam-6321	542	90	x(3k+2	x(3k+2	NOUN
ejpam-6321	542	91	)	)	PUNCT
ejpam-6321	542	92	,	,	PUNCT
ejpam-6321	542	93	f3℘	f3℘	X
ejpam-6321	542	94	)	)	PUNCT
ejpam-6321	542	95	=	=	SYM
ejpam-6321	542	96	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	542	97	,	,	PUNCT
ejpam-6321	542	98	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	542	99	)	)	PUNCT
ejpam-6321	542	100	,	,	PUNCT
ejpam-6321	542	101	f3℘	f3℘	NOUN
ejpam-6321	542	102	)	)	PUNCT
ejpam-6321	542	103	≤	≤	NUM
ejpam-6321	542	104	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	542	105	,	,	PUNCT
ejpam-6321	542	106	x(3k+1	x(3k+1	PRON
ejpam-6321	542	107	)	)	PUNCT
ejpam-6321	542	108	,	,	PUNCT
ejpam-6321	542	109	℘	℘	PROPN
ejpam-6321	542	110	)	)	PUNCT
ejpam-6321	542	111	+	+	NUM
ejpam-6321	542	112	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	542	113	,	,	PUNCT
ejpam-6321	542	114	x(3k+1	x(3k+1	PRON
ejpam-6321	542	115	)	)	PUNCT
ejpam-6321	542	116	,	,	PUNCT
ejpam-6321	542	117	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	542	118	)	)	PUNCT
ejpam-6321	542	119	)	)	PUNCT
ejpam-6321	543	1	+	+	CCONJ
ejpam-6321	543	2	α3g(f1x3k	α3g(f1x3k	NUM
ejpam-6321	543	3	,	,	PUNCT
ejpam-6321	543	4	x(3k+1	x(3k+1	ADV
ejpam-6321	543	5	)	)	PUNCT
ejpam-6321	543	6	,	,	PUNCT
ejpam-6321	543	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	543	8	)	)	PUNCT
ejpam-6321	543	9	)	)	PUNCT
ejpam-6321	544	1	+	+	CCONJ
ejpam-6321	544	2	α4[g(f1x3k	α4[g(f1x3k	PROPN
ejpam-6321	544	3	,	,	PUNCT
ejpam-6321	544	4	x(3k+1	x(3k+1	ADV
ejpam-6321	544	5	)	)	PUNCT
ejpam-6321	544	6	,	,	PUNCT
ejpam-6321	544	7	℘	℘	PROPN
ejpam-6321	544	8	)	)	PUNCT
ejpam-6321	544	9	+	+	NOUN
ejpam-6321	544	10	g(x3k	g(x3k	NOUN
ejpam-6321	544	11	,	,	PUNCT
ejpam-6321	544	12	f2x(3k+1	f2x(3k+1	NUM
ejpam-6321	544	13	)	)	PUNCT
ejpam-6321	544	14	,	,	PUNCT
ejpam-6321	544	15	℘	℘	PROPN
ejpam-6321	544	16	)	)	PUNCT
ejpam-6321	545	1	+	+	NOUN
ejpam-6321	545	2	g(x3k	g(x3k	NOUN
ejpam-6321	545	3	,	,	PUNCT
ejpam-6321	545	4	x(3k+1	x(3k+1	PRON
ejpam-6321	545	5	)	)	PUNCT
ejpam-6321	545	6	,	,	PUNCT
ejpam-6321	545	7	℘	℘	PROPN
ejpam-6321	545	8	)	)	PUNCT
ejpam-6321	545	9	]	]	PUNCT
ejpam-6321	546	1	+	+	CCONJ
ejpam-6321	546	2	α5max	α5max	PROPN
ejpam-6321	546	3			PART
ejpam-6321	546	4	g(x3k	g(x3k	NOUN
ejpam-6321	546	5	,	,	PUNCT
ejpam-6321	546	6	x(3k+1	x(3k+1	PRON
ejpam-6321	546	7	)	)	PUNCT
ejpam-6321	546	8	,	,	PUNCT
ejpam-6321	546	9	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	546	10	)	)	PUNCT
ejpam-6321	546	11	)	)	PUNCT
ejpam-6321	546	12	,	,	PUNCT
ejpam-6321	546	13	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	546	14	,	,	PUNCT
ejpam-6321	546	15	f1x3k	f1x3k	NUM
ejpam-6321	546	16	,	,	PUNCT
ejpam-6321	546	17	x(3k+1	x(3k+1	NUM
ejpam-6321	546	18	)	)	PUNCT
ejpam-6321	546	19	)	)	PUNCT
ejpam-6321	546	20	,	,	PUNCT
ejpam-6321	546	21	g(f1x3k	g(f1x3k	NOUN
ejpam-6321	546	22	,	,	PUNCT
ejpam-6321	546	23	x(3k+1	x(3k+1	ADV
ejpam-6321	546	24	)	)	PUNCT
ejpam-6321	546	25	,	,	PUNCT
ejpam-6321	546	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	546	27	)	)	PUNCT
ejpam-6321	546	28	)	)	PUNCT
ejpam-6321	546	29	,	,	PUNCT
ejpam-6321	546	30	g(f2x(3k+1	g(f2x(3k+1	NOUN
ejpam-6321	546	31	)	)	PUNCT
ejpam-6321	546	32	,	,	PUNCT
ejpam-6321	546	33	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	546	34	)	)	PUNCT
ejpam-6321	546	35	,	,	PUNCT
ejpam-6321	546	36	℘	℘	PROPN
ejpam-6321	546	37	)	)	PUNCT
ejpam-6321	546	38	,	,	PUNCT
ejpam-6321	546	39	g(f1x(3k+1	g(f1x(3k+1	NOUN
ejpam-6321	546	40	)	)	PUNCT
ejpam-6321	546	41	,	,	PUNCT
ejpam-6321	546	42	x3k	x3k	NOUN
ejpam-6321	546	43	,	,	PUNCT
ejpam-6321	546	44	x3k	x3k	NOUN
ejpam-6321	546	45	)	)	PUNCT
ejpam-6321	546	46	,	,	PUNCT
ejpam-6321	546	47	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	546	48	)	)	PUNCT
ejpam-6321	546	49	,	,	PUNCT
ejpam-6321	546	50	f2x(3k+1	f2x(3k+1	NOUN
ejpam-6321	546	51	)	)	PUNCT
ejpam-6321	546	52	,	,	PUNCT
ejpam-6321	546	53	x(3k+1	x(3k+1	CCONJ
ejpam-6321	546	54	)	)	PUNCT
ejpam-6321	546	55	)	)	PUNCT
ejpam-6321	546	56	,	,	PUNCT
ejpam-6321	546	57	g(℘	g(℘	PROPN
ejpam-6321	546	58	,	,	PUNCT
ejpam-6321	546	59	℘	℘	NOUN
ejpam-6321	546	60	,	,	PUNCT
ejpam-6321	546	61	f3℘	f3℘	NOUN
ejpam-6321	546	62	)	)	PUNCT
ejpam-6321	546	63			ADJ
ejpam-6321	546	64	=	=	SYM
ejpam-6321	546	65	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	546	66	,	,	PUNCT
ejpam-6321	546	67	x(3k+1	x(3k+1	PRON
ejpam-6321	546	68	)	)	PUNCT
ejpam-6321	546	69	,	,	PUNCT
ejpam-6321	546	70	℘	℘	PROPN
ejpam-6321	546	71	)	)	PUNCT
ejpam-6321	546	72	+	+	NUM
ejpam-6321	546	73	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	546	74	,	,	PUNCT
ejpam-6321	546	75	x(3k+1	x(3k+1	CCONJ
ejpam-6321	546	76	)	)	PUNCT
ejpam-6321	546	77	,	,	PUNCT
ejpam-6321	546	78	x(3k+1))+	x(3k+1))+	PROPN
ejpam-6321	546	79	α3g(x3k	α3g(x3k	PROPN
ejpam-6321	546	80	,	,	PUNCT
ejpam-6321	546	81	x(3k+1	x(3k+1	ADV
ejpam-6321	546	82	)	)	PUNCT
ejpam-6321	546	83	,	,	PUNCT
ejpam-6321	546	84	x(3k+1	x(3k+1	CCONJ
ejpam-6321	546	85	)	)	PUNCT
ejpam-6321	546	86	)	)	PUNCT
ejpam-6321	547	1	+	+	CCONJ
ejpam-6321	547	2	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	547	3	,	,	PUNCT
ejpam-6321	547	4	x3k	x3k	NOUN
ejpam-6321	547	5	,	,	PUNCT
ejpam-6321	547	6	x3k	x3k	PUNCT
ejpam-6321	547	7	)	)	PUNCT
ejpam-6321	547	8	+	+	SYM
ejpam-6321	547	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	547	10	)	)	PUNCT
ejpam-6321	547	11	,	,	PUNCT
ejpam-6321	547	12	x(3k+1	x(3k+1	CCONJ
ejpam-6321	547	13	)	)	PUNCT
ejpam-6321	547	14	,	,	PUNCT
ejpam-6321	547	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	547	16	)	)	PUNCT
ejpam-6321	547	17	)	)	PUNCT
ejpam-6321	548	1	+	+	PUNCT
ejpam-6321	548	2	g(℘	g(℘	NOUN
ejpam-6321	548	3	,	,	PUNCT
ejpam-6321	548	4	℘	℘	NOUN
ejpam-6321	548	5	,	,	PUNCT
ejpam-6321	548	6	f3℘	f3℘	NOUN
ejpam-6321	548	7	)	)	PUNCT
ejpam-6321	548	8	]	]	PUNCT
ejpam-6321	549	1	+	+	CCONJ
ejpam-6321	549	2	α5max	α5max	PROPN
ejpam-6321	549	3			PART
ejpam-6321	549	4	g(x3k	g(x3k	NOUN
ejpam-6321	549	5	,	,	PUNCT
ejpam-6321	549	6	x(3k+1	x(3k+1	PRON
ejpam-6321	549	7	)	)	PUNCT
ejpam-6321	549	8	,	,	PUNCT
ejpam-6321	549	9	x(3k+1	x(3k+1	CCONJ
ejpam-6321	549	10	)	)	PUNCT
ejpam-6321	549	11	)	)	PUNCT
ejpam-6321	549	12	,	,	PUNCT
ejpam-6321	549	13	g(x3k	g(x3k	NOUN
ejpam-6321	549	14	,	,	PUNCT
ejpam-6321	549	15	x3k	x3k	NOUN
ejpam-6321	549	16	,	,	PUNCT
ejpam-6321	549	17	x(3k+1	x(3k+1	ADV
ejpam-6321	549	18	)	)	PUNCT
ejpam-6321	549	19	)	)	PUNCT
ejpam-6321	549	20	,	,	PUNCT
ejpam-6321	549	21	g(x3k	g(x3k	NOUN
ejpam-6321	549	22	,	,	PUNCT
ejpam-6321	549	23	x(3k+1	x(3k+1	PRON
ejpam-6321	549	24	)	)	PUNCT
ejpam-6321	549	25	,	,	PUNCT
ejpam-6321	549	26	x(3k+1	x(3k+1	CCONJ
ejpam-6321	549	27	)	)	PUNCT
ejpam-6321	549	28	)	)	PUNCT
ejpam-6321	549	29	,	,	PUNCT
ejpam-6321	549	30	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	549	31	)	)	PUNCT
ejpam-6321	549	32	,	,	PUNCT
ejpam-6321	549	33	x(3k+1	x(3k+1	ADP
ejpam-6321	549	34	)	)	PUNCT
ejpam-6321	549	35	,	,	PUNCT
ejpam-6321	549	36	x(3k	x(3k	X
ejpam-6321	549	37	+	+	ADJ
ejpam-6321	549	38	2	2	NUM
ejpam-6321	549	39	)	)	PUNCT
ejpam-6321	549	40	)	)	PUNCT
ejpam-6321	549	41	,	,	PUNCT
ejpam-6321	549	42	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	549	43	)	)	PUNCT
ejpam-6321	549	44	,	,	PUNCT
ejpam-6321	549	45	x3k	x3k	NOUN
ejpam-6321	549	46	,	,	PUNCT
ejpam-6321	549	47	x3k	x3k	NOUN
ejpam-6321	549	48	)	)	PUNCT
ejpam-6321	549	49	,	,	PUNCT
ejpam-6321	549	50	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	549	51	)	)	PUNCT
ejpam-6321	549	52	,	,	PUNCT
ejpam-6321	549	53	x(3k+1	x(3k+1	CCONJ
ejpam-6321	549	54	)	)	PUNCT
ejpam-6321	549	55	,	,	PUNCT
ejpam-6321	549	56	x(3k+1	x(3k+1	CCONJ
ejpam-6321	549	57	)	)	PUNCT
ejpam-6321	549	58	)	)	PUNCT
ejpam-6321	549	59	,	,	PUNCT
ejpam-6321	549	60	g(℘	g(℘	PROPN
ejpam-6321	549	61	,	,	PUNCT
ejpam-6321	549	62	℘	℘	NOUN
ejpam-6321	549	63	,	,	PUNCT
ejpam-6321	549	64	f3℘	f3℘	NOUN
ejpam-6321	549	65	)	)	PUNCT
ejpam-6321	549	66			ADJ
ejpam-6321	549	67	m.	m.	NOUN
ejpam-6321	549	68	noorwali	noorwali	PROPN
ejpam-6321	549	69	et	et	PROPN
ejpam-6321	549	70	al	al	PROPN
ejpam-6321	549	71	.	.	PUNCT
ejpam-6321	549	72	/	/	SYM
ejpam-6321	549	73	eur	eur	PROPN
ejpam-6321	549	74	.	.	PUNCT
ejpam-6321	550	1	j.	j.	PROPN
ejpam-6321	550	2	pure	pure	PROPN
ejpam-6321	550	3	appl	appl	PROPN
ejpam-6321	550	4	.	.	PROPN
ejpam-6321	550	5	math	math	PROPN
ejpam-6321	550	6	,	,	PUNCT
ejpam-6321	550	7	18	18	NUM
ejpam-6321	550	8	(	(	PUNCT
ejpam-6321	550	9	3	3	NUM
ejpam-6321	550	10	)	)	PUNCT
ejpam-6321	550	11	(	(	PUNCT
ejpam-6321	550	12	2025	2025	NUM
ejpam-6321	550	13	)	)	PUNCT
ejpam-6321	550	14	,	,	PUNCT
ejpam-6321	550	15	6321	6321	NUM
ejpam-6321	550	16	19	19	NUM
ejpam-6321	550	17	of	of	ADP
ejpam-6321	550	18	27	27	NUM
ejpam-6321	550	19	after	after	ADP
ejpam-6321	550	20	applying	apply	VERB
ejpam-6321	550	21	the	the	DET
ejpam-6321	550	22	simplification	simplification	NOUN
ejpam-6321	550	23	process	process	NOUN
ejpam-6321	550	24	utilizing	utilize	VERB
ejpam-6321	550	25	the	the	DET
ejpam-6321	550	26	definition	definition	NOUN
ejpam-6321	550	27	of	of	ADP
ejpam-6321	550	28	gm	gm	PROPN
ejpam-6321	550	29	-	-	PUNCT
ejpam-6321	550	30	space	space	NOUN
ejpam-6321	550	31	we	we	PRON
ejpam-6321	550	32	have	have	AUX
ejpam-6321	550	33	achieved	achieve	VERB
ejpam-6321	550	34	the	the	DET
ejpam-6321	550	35	following	follow	VERB
ejpam-6321	550	36	result	result	NOUN
ejpam-6321	550	37	,	,	PUNCT
ejpam-6321	550	38	=	=	SYM
ejpam-6321	550	39	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	550	40	,	,	PUNCT
ejpam-6321	550	41	x(3k+1	x(3k+1	PRON
ejpam-6321	550	42	)	)	PUNCT
ejpam-6321	550	43	,	,	PUNCT
ejpam-6321	550	44	℘	℘	PROPN
ejpam-6321	550	45	)	)	PUNCT
ejpam-6321	550	46	+	+	NUM
ejpam-6321	550	47	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	550	48	,	,	PUNCT
ejpam-6321	550	49	x(3k+1	x(3k+1	CCONJ
ejpam-6321	550	50	)	)	PUNCT
ejpam-6321	550	51	,	,	PUNCT
ejpam-6321	550	52	x(3k+1	x(3k+1	CCONJ
ejpam-6321	550	53	)	)	PUNCT
ejpam-6321	550	54	)	)	PUNCT
ejpam-6321	551	1	+	+	CCONJ
ejpam-6321	551	2	α4[g(x3k	α4[g(x3k	NOUN
ejpam-6321	551	3	,	,	PUNCT
ejpam-6321	551	4	x3k	x3k	NOUN
ejpam-6321	551	5	,	,	PUNCT
ejpam-6321	551	6	x3k	x3k	PUNCT
ejpam-6321	551	7	)	)	PUNCT
ejpam-6321	551	8	+	+	SYM
ejpam-6321	551	9	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	551	10	)	)	PUNCT
ejpam-6321	551	11	,	,	PUNCT
ejpam-6321	551	12	x(3k+1	x(3k+1	CCONJ
ejpam-6321	551	13	)	)	PUNCT
ejpam-6321	551	14	,	,	PUNCT
ejpam-6321	551	15	x(3k+1	x(3k+1	CCONJ
ejpam-6321	551	16	)	)	PUNCT
ejpam-6321	551	17	)	)	PUNCT
ejpam-6321	552	1	+	+	PUNCT
ejpam-6321	552	2	g(℘	g(℘	NOUN
ejpam-6321	552	3	,	,	PUNCT
ejpam-6321	552	4	℘	℘	NOUN
ejpam-6321	552	5	,	,	PUNCT
ejpam-6321	552	6	f3℘	f3℘	NOUN
ejpam-6321	552	7	)	)	PUNCT
ejpam-6321	552	8	]	]	PUNCT
ejpam-6321	553	1	+	+	CCONJ
ejpam-6321	553	2	α5max	α5max	PROPN
ejpam-6321	553	3	{	{	PUNCT
ejpam-6321	553	4	g(x3k	g(x3k	NOUN
ejpam-6321	553	5	,	,	PUNCT
ejpam-6321	553	6	x(3k+1	x(3k+1	PRON
ejpam-6321	553	7	)	)	PUNCT
ejpam-6321	553	8	,	,	PUNCT
ejpam-6321	553	9	x(3k+1	x(3k+1	CCONJ
ejpam-6321	553	10	)	)	PUNCT
ejpam-6321	553	11	)	)	PUNCT
ejpam-6321	553	12	,	,	PUNCT
ejpam-6321	553	13	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	553	14	)	)	PUNCT
ejpam-6321	553	15	,	,	PUNCT
ejpam-6321	553	16	x(3k+2	x(3k+2	NOUN
ejpam-6321	553	17	)	)	PUNCT
ejpam-6321	553	18	,	,	PUNCT
ejpam-6321	553	19	℘	℘	PROPN
ejpam-6321	553	20	)	)	PUNCT
ejpam-6321	553	21	,	,	PUNCT
ejpam-6321	553	22	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	553	23	)	)	PUNCT
ejpam-6321	553	24	,	,	PUNCT
ejpam-6321	553	25	x(3k+2	x(3k+2	NOUN
ejpam-6321	553	26	)	)	PUNCT
ejpam-6321	553	27	,	,	PUNCT
ejpam-6321	553	28	x(3k+3	x(3k+3	NUM
ejpam-6321	553	29	)	)	PUNCT
ejpam-6321	553	30	)	)	PUNCT
ejpam-6321	553	31	,	,	PUNCT
ejpam-6321	553	32	g(℘	g(℘	PROPN
ejpam-6321	553	33	,	,	PUNCT
ejpam-6321	553	34	℘	℘	NOUN
ejpam-6321	553	35	,	,	PUNCT
ejpam-6321	553	36	f3℘	f3℘	NOUN
ejpam-6321	553	37	)	)	PUNCT
ejpam-6321	553	38	}	}	PUNCT
ejpam-6321	553	39	(	(	PUNCT
ejpam-6321	553	40	30	30	NUM
ejpam-6321	553	41	)	)	PUNCT
ejpam-6321	553	42	now	now	ADV
ejpam-6321	553	43	there	there	PRON
ejpam-6321	553	44	are	be	VERB
ejpam-6321	553	45	four	four	NUM
ejpam-6321	553	46	cases	case	NOUN
ejpam-6321	553	47	:	:	PUNCT
ejpam-6321	553	48	(	(	PUNCT
ejpam-6321	553	49	i	i	NOUN
ejpam-6321	553	50	)	)	PUNCT
ejpam-6321	553	51	.	.	PUNCT
ejpam-6321	554	1	if	if	SCONJ
ejpam-6321	554	2	g(x3k	g(x3k	NOUN
ejpam-6321	554	3	,	,	PUNCT
ejpam-6321	554	4	x(3k+1	x(3k+1	PRON
ejpam-6321	554	5	)	)	PUNCT
ejpam-6321	554	6	,	,	PUNCT
ejpam-6321	554	7	x(3k+1	x(3k+1	CCONJ
ejpam-6321	554	8	)	)	PUNCT
ejpam-6321	554	9	)	)	PUNCT
ejpam-6321	555	1	is	be	AUX
ejpam-6321	555	2	a	a	DET
ejpam-6321	555	3	maximum	maximum	ADJ
ejpam-6321	555	4	term	term	NOUN
ejpam-6321	555	5	in	in	ADP
ejpam-6321	555	6	(	(	PUNCT
ejpam-6321	555	7	30	30	NUM
ejpam-6321	555	8	)	)	PUNCT
ejpam-6321	555	9	then	then	ADV
ejpam-6321	555	10	we	we	PRON
ejpam-6321	555	11	can	can	AUX
ejpam-6321	555	12	write	write	VERB
ejpam-6321	555	13	;	;	PUNCT
ejpam-6321	555	14	=	=	SYM
ejpam-6321	555	15	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	555	16	,	,	PUNCT
ejpam-6321	555	17	x(3k+1	x(3k+1	PRON
ejpam-6321	555	18	)	)	PUNCT
ejpam-6321	555	19	,	,	PUNCT
ejpam-6321	555	20	℘	℘	PROPN
ejpam-6321	555	21	)	)	PUNCT
ejpam-6321	555	22	+	+	NUM
ejpam-6321	555	23	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	555	24	,	,	PUNCT
ejpam-6321	555	25	x(3k+1	x(3k+1	CCONJ
ejpam-6321	555	26	)	)	PUNCT
ejpam-6321	555	27	,	,	PUNCT
ejpam-6321	555	28	x(3k+1))+	x(3k+1))+	PROPN
ejpam-6321	555	29	α4[g(x3k	α4[g(x3k	PROPN
ejpam-6321	555	30	,	,	PUNCT
ejpam-6321	555	31	x3k	x3k	NOUN
ejpam-6321	555	32	,	,	PUNCT
ejpam-6321	555	33	x3k	x3k	PUNCT
ejpam-6321	555	34	)	)	PUNCT
ejpam-6321	556	1	+	+	SYM
ejpam-6321	556	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	556	3	)	)	PUNCT
ejpam-6321	556	4	,	,	PUNCT
ejpam-6321	556	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	556	6	)	)	PUNCT
ejpam-6321	556	7	,	,	PUNCT
ejpam-6321	556	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	556	9	)	)	PUNCT
ejpam-6321	556	10	)	)	PUNCT
ejpam-6321	557	1	+	+	PUNCT
ejpam-6321	557	2	g(℘	g(℘	NOUN
ejpam-6321	557	3	,	,	PUNCT
ejpam-6321	557	4	℘	℘	NOUN
ejpam-6321	557	5	,	,	PUNCT
ejpam-6321	557	6	f3℘	f3℘	NOUN
ejpam-6321	557	7	)	)	PUNCT
ejpam-6321	557	8	]	]	PUNCT
ejpam-6321	558	1	+	+	CCONJ
ejpam-6321	558	2	α5g(x3k	α5g(x3k	NOUN
ejpam-6321	558	3	,	,	PUNCT
ejpam-6321	558	4	x(3k+1	x(3k+1	PRON
ejpam-6321	558	5	)	)	PUNCT
ejpam-6321	558	6	,	,	PUNCT
ejpam-6321	558	7	x(3k+2	x(3k+2	NUM
ejpam-6321	558	8	)	)	PUNCT
ejpam-6321	558	9	)	)	PUNCT
ejpam-6321	558	10	applying	apply	VERB
ejpam-6321	558	11	limn→∞	limn→∞	PROPN
ejpam-6321	558	12	on	on	ADP
ejpam-6321	558	13	both	both	DET
ejpam-6321	558	14	sides	side	NOUN
ejpam-6321	558	15	,	,	PUNCT
ejpam-6321	558	16	we	we	PRON
ejpam-6321	558	17	get	get	VERB
ejpam-6321	558	18	,	,	PUNCT
ejpam-6321	558	19	g(℘	g(℘	PROPN
ejpam-6321	558	20	,	,	PUNCT
ejpam-6321	558	21	℘	℘	NOUN
ejpam-6321	558	22	,	,	PUNCT
ejpam-6321	558	23	f3℘	f3℘	NOUN
ejpam-6321	558	24	)	)	PUNCT
ejpam-6321	558	25	≤	≤	NOUN
ejpam-6321	558	26	α1g(℘	α1g(℘	NOUN
ejpam-6321	558	27	,	,	PUNCT
ejpam-6321	558	28	℘	℘	NOUN
ejpam-6321	558	29	,	,	PUNCT
ejpam-6321	558	30	℘	℘	NUM
ejpam-6321	558	31	)	)	PUNCT
ejpam-6321	558	32	+	+	NUM
ejpam-6321	558	33	α2g(℘	α2g(℘	NOUN
ejpam-6321	558	34	,	,	PUNCT
ejpam-6321	558	35	℘	℘	NOUN
ejpam-6321	558	36	,	,	PUNCT
ejpam-6321	558	37	℘)+	℘)+	PROPN
ejpam-6321	558	38	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	558	39	,	,	PUNCT
ejpam-6321	558	40	℘	℘	NOUN
ejpam-6321	558	41	,	,	PUNCT
ejpam-6321	558	42	℘	℘	NUM
ejpam-6321	558	43	)	)	PUNCT
ejpam-6321	559	1	+	+	NOUN
ejpam-6321	559	2	g(℘	g(℘	NOUN
ejpam-6321	559	3	,	,	PUNCT
ejpam-6321	559	4	℘	℘	NOUN
ejpam-6321	559	5	,	,	PUNCT
ejpam-6321	559	6	℘	℘	NUM
ejpam-6321	559	7	)	)	PUNCT
ejpam-6321	559	8	+	+	NOUN
ejpam-6321	559	9	g(℘	g(℘	NOUN
ejpam-6321	559	10	,	,	PUNCT
ejpam-6321	559	11	℘	℘	NOUN
ejpam-6321	559	12	,	,	PUNCT
ejpam-6321	559	13	f3℘	f3℘	NOUN
ejpam-6321	559	14	)	)	PUNCT
ejpam-6321	559	15	]	]	PUNCT
ejpam-6321	560	1	+	+	NUM
ejpam-6321	560	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	560	3	,	,	PUNCT
ejpam-6321	560	4	℘	℘	NOUN
ejpam-6321	560	5	,	,	PUNCT
ejpam-6321	560	6	℘	℘	NUM
ejpam-6321	560	7	)	)	PUNCT
ejpam-6321	560	8	we	we	PRON
ejpam-6321	560	9	get	get	VERB
ejpam-6321	560	10	that	that	PRON
ejpam-6321	560	11	,	,	PUNCT
ejpam-6321	560	12	g(℘	g(℘	PROPN
ejpam-6321	560	13	,	,	PUNCT
ejpam-6321	560	14	℘	℘	NOUN
ejpam-6321	560	15	,	,	PUNCT
ejpam-6321	560	16	f3℘	f3℘	NOUN
ejpam-6321	560	17	)	)	PUNCT
ejpam-6321	560	18	=	=	SYM
ejpam-6321	561	1	0	0	X
ejpam-6321	561	2	.	.	PUNCT
ejpam-6321	562	1	thus	thus	ADV
ejpam-6321	562	2	,	,	PUNCT
ejpam-6321	562	3	f3℘	f3℘	NOUN
ejpam-6321	562	4	=	=	SYM
ejpam-6321	562	5	℘	℘	PROPN
ejpam-6321	562	6	(	(	PUNCT
ejpam-6321	562	7	31	31	NUM
ejpam-6321	562	8	)	)	PUNCT
ejpam-6321	562	9	(	(	PUNCT
ejpam-6321	562	10	ii	ii	NOUN
ejpam-6321	562	11	)	)	PUNCT
ejpam-6321	562	12	.	.	PUNCT
ejpam-6321	563	1	if	if	SCONJ
ejpam-6321	563	2	g(x(3k+2	g(x(3k+2	NOUN
ejpam-6321	563	3	)	)	PUNCT
ejpam-6321	563	4	,	,	PUNCT
ejpam-6321	563	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	563	6	)	)	PUNCT
ejpam-6321	563	7	,	,	PUNCT
ejpam-6321	563	8	℘	℘	PROPN
ejpam-6321	563	9	)	)	PUNCT
ejpam-6321	563	10	is	be	AUX
ejpam-6321	563	11	a	a	DET
ejpam-6321	563	12	maximum	maximum	ADJ
ejpam-6321	563	13	term	term	NOUN
ejpam-6321	563	14	in	in	ADP
ejpam-6321	563	15	(	(	PUNCT
ejpam-6321	563	16	30	30	NUM
ejpam-6321	563	17	)	)	PUNCT
ejpam-6321	563	18	then	then	ADV
ejpam-6321	563	19	we	we	PRON
ejpam-6321	563	20	can	can	AUX
ejpam-6321	563	21	write	write	VERB
ejpam-6321	563	22	;	;	PUNCT
ejpam-6321	563	23	=	=	SYM
ejpam-6321	563	24	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	563	25	,	,	PUNCT
ejpam-6321	563	26	x(3k+1	x(3k+1	PRON
ejpam-6321	563	27	)	)	PUNCT
ejpam-6321	563	28	,	,	PUNCT
ejpam-6321	563	29	℘	℘	PROPN
ejpam-6321	563	30	)	)	PUNCT
ejpam-6321	563	31	+	+	NUM
ejpam-6321	563	32	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	563	33	,	,	PUNCT
ejpam-6321	563	34	x(3k+1	x(3k+1	CCONJ
ejpam-6321	563	35	)	)	PUNCT
ejpam-6321	563	36	,	,	PUNCT
ejpam-6321	563	37	x(3k+1))+	x(3k+1))+	PROPN
ejpam-6321	563	38	α4[g(x3k	α4[g(x3k	PROPN
ejpam-6321	563	39	,	,	PUNCT
ejpam-6321	563	40	x3k	x3k	NOUN
ejpam-6321	563	41	,	,	PUNCT
ejpam-6321	563	42	x3k	x3k	PUNCT
ejpam-6321	563	43	)	)	PUNCT
ejpam-6321	564	1	+	+	SYM
ejpam-6321	564	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	564	3	)	)	PUNCT
ejpam-6321	564	4	,	,	PUNCT
ejpam-6321	564	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	564	6	)	)	PUNCT
ejpam-6321	564	7	,	,	PUNCT
ejpam-6321	564	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	564	9	)	)	PUNCT
ejpam-6321	564	10	)	)	PUNCT
ejpam-6321	565	1	+	+	PUNCT
ejpam-6321	565	2	g(℘	g(℘	NOUN
ejpam-6321	565	3	,	,	PUNCT
ejpam-6321	565	4	℘	℘	NOUN
ejpam-6321	565	5	,	,	PUNCT
ejpam-6321	565	6	f3℘	f3℘	NOUN
ejpam-6321	565	7	)	)	PUNCT
ejpam-6321	565	8	]	]	PUNCT
ejpam-6321	566	1	+	+	CCONJ
ejpam-6321	566	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	566	3	)	)	PUNCT
ejpam-6321	566	4	,	,	PUNCT
ejpam-6321	566	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	566	6	)	)	PUNCT
ejpam-6321	566	7	,	,	PUNCT
ejpam-6321	566	8	℘	℘	PROPN
ejpam-6321	566	9	)	)	PUNCT
ejpam-6321	566	10	applying	apply	VERB
ejpam-6321	566	11	limn→∞	limn→∞	PROPN
ejpam-6321	566	12	on	on	ADP
ejpam-6321	566	13	both	both	DET
ejpam-6321	566	14	sides	side	NOUN
ejpam-6321	566	15	,	,	PUNCT
ejpam-6321	566	16	we	we	PRON
ejpam-6321	566	17	get	get	VERB
ejpam-6321	566	18	,	,	PUNCT
ejpam-6321	566	19	g(℘	g(℘	PROPN
ejpam-6321	566	20	,	,	PUNCT
ejpam-6321	566	21	℘	℘	NOUN
ejpam-6321	566	22	,	,	PUNCT
ejpam-6321	566	23	f3℘	f3℘	NOUN
ejpam-6321	566	24	)	)	PUNCT
ejpam-6321	566	25	≤	≤	NOUN
ejpam-6321	566	26	α1g(℘	α1g(℘	NOUN
ejpam-6321	566	27	,	,	PUNCT
ejpam-6321	566	28	℘	℘	NOUN
ejpam-6321	566	29	,	,	PUNCT
ejpam-6321	566	30	℘	℘	NUM
ejpam-6321	566	31	)	)	PUNCT
ejpam-6321	566	32	+	+	NUM
ejpam-6321	566	33	α2g(℘	α2g(℘	NOUN
ejpam-6321	566	34	,	,	PUNCT
ejpam-6321	566	35	℘	℘	NOUN
ejpam-6321	566	36	,	,	PUNCT
ejpam-6321	566	37	℘)+	℘)+	PROPN
ejpam-6321	566	38	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	566	39	,	,	PUNCT
ejpam-6321	566	40	℘	℘	NOUN
ejpam-6321	566	41	,	,	PUNCT
ejpam-6321	566	42	℘	℘	NUM
ejpam-6321	566	43	)	)	PUNCT
ejpam-6321	567	1	+	+	NOUN
ejpam-6321	567	2	g(℘	g(℘	NOUN
ejpam-6321	567	3	,	,	PUNCT
ejpam-6321	567	4	℘	℘	NOUN
ejpam-6321	567	5	,	,	PUNCT
ejpam-6321	567	6	℘	℘	NUM
ejpam-6321	567	7	)	)	PUNCT
ejpam-6321	567	8	+	+	NOUN
ejpam-6321	567	9	g(℘	g(℘	NOUN
ejpam-6321	567	10	,	,	PUNCT
ejpam-6321	567	11	℘	℘	NOUN
ejpam-6321	567	12	,	,	PUNCT
ejpam-6321	567	13	f3℘	f3℘	NOUN
ejpam-6321	567	14	)	)	PUNCT
ejpam-6321	567	15	]	]	PUNCT
ejpam-6321	568	1	+	+	NUM
ejpam-6321	568	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	568	3	,	,	PUNCT
ejpam-6321	568	4	℘	℘	NOUN
ejpam-6321	568	5	,	,	PUNCT
ejpam-6321	568	6	℘	℘	NUM
ejpam-6321	568	7	)	)	PUNCT
ejpam-6321	568	8	we	we	PRON
ejpam-6321	568	9	get	get	VERB
ejpam-6321	568	10	that	that	PRON
ejpam-6321	568	11	,	,	PUNCT
ejpam-6321	568	12	g(℘	g(℘	PROPN
ejpam-6321	568	13	,	,	PUNCT
ejpam-6321	568	14	℘	℘	NOUN
ejpam-6321	568	15	,	,	PUNCT
ejpam-6321	568	16	f3℘	f3℘	NOUN
ejpam-6321	568	17	)	)	PUNCT
ejpam-6321	568	18	=	=	SYM
ejpam-6321	569	1	0	0	X
ejpam-6321	569	2	.	.	PUNCT
ejpam-6321	570	1	thus	thus	ADV
ejpam-6321	570	2	,	,	PUNCT
ejpam-6321	570	3	f3℘	f3℘	NOUN
ejpam-6321	570	4	=	=	SYM
ejpam-6321	570	5	℘	℘	PROPN
ejpam-6321	570	6	(	(	PUNCT
ejpam-6321	570	7	32	32	NUM
ejpam-6321	570	8	)	)	PUNCT
ejpam-6321	570	9	(	(	PUNCT
ejpam-6321	570	10	iii	iii	NOUN
ejpam-6321	570	11	)	)	PUNCT
ejpam-6321	570	12	.	.	PUNCT
ejpam-6321	571	1	if	if	SCONJ
ejpam-6321	571	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	571	3	)	)	PUNCT
ejpam-6321	571	4	,	,	PUNCT
ejpam-6321	571	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	571	6	)	)	PUNCT
ejpam-6321	571	7	,	,	PUNCT
ejpam-6321	571	8	x(3k+3	x(3k+3	NUM
ejpam-6321	571	9	)	)	PUNCT
ejpam-6321	571	10	)	)	PUNCT
ejpam-6321	571	11	is	be	AUX
ejpam-6321	571	12	a	a	DET
ejpam-6321	571	13	maximum	maximum	ADJ
ejpam-6321	571	14	term	term	NOUN
ejpam-6321	571	15	in	in	ADP
ejpam-6321	571	16	(	(	PUNCT
ejpam-6321	571	17	30	30	NUM
ejpam-6321	571	18	)	)	PUNCT
ejpam-6321	571	19	then	then	ADV
ejpam-6321	571	20	we	we	PRON
ejpam-6321	571	21	can	can	AUX
ejpam-6321	571	22	write	write	VERB
ejpam-6321	571	23	;	;	PUNCT
ejpam-6321	571	24	=	=	SYM
ejpam-6321	571	25	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	571	26	,	,	PUNCT
ejpam-6321	571	27	x(3k+1	x(3k+1	PRON
ejpam-6321	571	28	)	)	PUNCT
ejpam-6321	571	29	,	,	PUNCT
ejpam-6321	571	30	℘	℘	PROPN
ejpam-6321	571	31	)	)	PUNCT
ejpam-6321	571	32	+	+	NUM
ejpam-6321	571	33	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	571	34	,	,	PUNCT
ejpam-6321	571	35	x(3k+1	x(3k+1	CCONJ
ejpam-6321	571	36	)	)	PUNCT
ejpam-6321	571	37	,	,	PUNCT
ejpam-6321	571	38	x(3k+1))+	x(3k+1))+	PROPN
ejpam-6321	571	39	α4[g(x3k	α4[g(x3k	PROPN
ejpam-6321	571	40	,	,	PUNCT
ejpam-6321	571	41	x3k	x3k	NOUN
ejpam-6321	571	42	,	,	PUNCT
ejpam-6321	571	43	x3k	x3k	PUNCT
ejpam-6321	571	44	)	)	PUNCT
ejpam-6321	572	1	+	+	SYM
ejpam-6321	572	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	572	3	)	)	PUNCT
ejpam-6321	572	4	,	,	PUNCT
ejpam-6321	572	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	572	6	)	)	PUNCT
ejpam-6321	572	7	,	,	PUNCT
ejpam-6321	572	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	572	9	)	)	PUNCT
ejpam-6321	572	10	)	)	PUNCT
ejpam-6321	573	1	+	+	PUNCT
ejpam-6321	573	2	g(℘	g(℘	NOUN
ejpam-6321	573	3	,	,	PUNCT
ejpam-6321	573	4	℘	℘	NOUN
ejpam-6321	573	5	,	,	PUNCT
ejpam-6321	573	6	f3℘	f3℘	NOUN
ejpam-6321	573	7	)	)	PUNCT
ejpam-6321	573	8	]	]	PUNCT
ejpam-6321	574	1	+	+	CCONJ
ejpam-6321	574	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	574	3	)	)	PUNCT
ejpam-6321	574	4	,	,	PUNCT
ejpam-6321	574	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	574	6	)	)	PUNCT
ejpam-6321	574	7	,	,	PUNCT
ejpam-6321	574	8	x(3k+3	x(3k+3	NUM
ejpam-6321	574	9	)	)	PUNCT
ejpam-6321	574	10	)	)	PUNCT
ejpam-6321	575	1	applying	apply	VERB
ejpam-6321	575	2	limn→∞	limn→∞	PROPN
ejpam-6321	575	3	on	on	ADP
ejpam-6321	575	4	both	both	DET
ejpam-6321	575	5	sides	side	NOUN
ejpam-6321	575	6	,	,	PUNCT
ejpam-6321	575	7	we	we	PRON
ejpam-6321	575	8	get	get	VERB
ejpam-6321	575	9	,	,	PUNCT
ejpam-6321	575	10	g(℘	g(℘	PROPN
ejpam-6321	575	11	,	,	PUNCT
ejpam-6321	575	12	℘	℘	NOUN
ejpam-6321	575	13	,	,	PUNCT
ejpam-6321	575	14	f3℘	f3℘	NOUN
ejpam-6321	575	15	)	)	PUNCT
ejpam-6321	575	16	≤	≤	NOUN
ejpam-6321	575	17	α1g(℘	α1g(℘	NOUN
ejpam-6321	575	18	,	,	PUNCT
ejpam-6321	575	19	℘	℘	NOUN
ejpam-6321	575	20	,	,	PUNCT
ejpam-6321	575	21	℘	℘	NUM
ejpam-6321	575	22	)	)	PUNCT
ejpam-6321	575	23	+	+	NUM
ejpam-6321	575	24	α2g(℘	α2g(℘	NOUN
ejpam-6321	575	25	,	,	PUNCT
ejpam-6321	575	26	℘	℘	NOUN
ejpam-6321	575	27	,	,	PUNCT
ejpam-6321	575	28	℘)+	℘)+	PROPN
ejpam-6321	575	29	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	575	30	,	,	PUNCT
ejpam-6321	575	31	℘	℘	NOUN
ejpam-6321	575	32	,	,	PUNCT
ejpam-6321	575	33	℘	℘	NUM
ejpam-6321	575	34	)	)	PUNCT
ejpam-6321	576	1	+	+	NOUN
ejpam-6321	576	2	g(℘	g(℘	NOUN
ejpam-6321	576	3	,	,	PUNCT
ejpam-6321	576	4	℘	℘	NOUN
ejpam-6321	576	5	,	,	PUNCT
ejpam-6321	576	6	℘	℘	NUM
ejpam-6321	576	7	)	)	PUNCT
ejpam-6321	576	8	+	+	NOUN
ejpam-6321	576	9	g(℘	g(℘	NOUN
ejpam-6321	576	10	,	,	PUNCT
ejpam-6321	576	11	℘	℘	NOUN
ejpam-6321	576	12	,	,	PUNCT
ejpam-6321	576	13	f3℘	f3℘	NOUN
ejpam-6321	576	14	)	)	PUNCT
ejpam-6321	576	15	]	]	PUNCT
ejpam-6321	577	1	+	+	NUM
ejpam-6321	577	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	577	3	,	,	PUNCT
ejpam-6321	577	4	℘	℘	NOUN
ejpam-6321	577	5	,	,	PUNCT
ejpam-6321	577	6	℘	℘	NUM
ejpam-6321	577	7	)	)	PUNCT
ejpam-6321	577	8	we	we	PRON
ejpam-6321	577	9	get	get	VERB
ejpam-6321	577	10	that	that	PRON
ejpam-6321	577	11	,	,	PUNCT
ejpam-6321	577	12	g(℘	g(℘	PROPN
ejpam-6321	577	13	,	,	PUNCT
ejpam-6321	577	14	℘	℘	NOUN
ejpam-6321	577	15	,	,	PUNCT
ejpam-6321	577	16	f3℘	f3℘	NOUN
ejpam-6321	577	17	)	)	PUNCT
ejpam-6321	577	18	=	=	SYM
ejpam-6321	578	1	0	0	X
ejpam-6321	578	2	.	.	PUNCT
ejpam-6321	579	1	thus	thus	ADV
ejpam-6321	579	2	,	,	PUNCT
ejpam-6321	579	3	f3℘	f3℘	NOUN
ejpam-6321	579	4	=	=	SYM
ejpam-6321	579	5	℘	℘	PROPN
ejpam-6321	579	6	(	(	PUNCT
ejpam-6321	579	7	33	33	NUM
ejpam-6321	579	8	)	)	PUNCT
ejpam-6321	579	9	(	(	PUNCT
ejpam-6321	579	10	iv	iv	X
ejpam-6321	579	11	)	)	PUNCT
ejpam-6321	579	12	.	.	PUNCT
ejpam-6321	580	1	if	if	SCONJ
ejpam-6321	580	2	g(℘	g(℘	NOUN
ejpam-6321	580	3	,	,	PUNCT
ejpam-6321	580	4	℘	℘	NOUN
ejpam-6321	580	5	,	,	PUNCT
ejpam-6321	580	6	f3℘	f3℘	NOUN
ejpam-6321	580	7	)	)	PUNCT
ejpam-6321	580	8	is	be	AUX
ejpam-6321	580	9	a	a	DET
ejpam-6321	580	10	maximum	maximum	ADJ
ejpam-6321	580	11	term	term	NOUN
ejpam-6321	580	12	in	in	ADP
ejpam-6321	580	13	(	(	PUNCT
ejpam-6321	580	14	30	30	NUM
ejpam-6321	580	15	)	)	PUNCT
ejpam-6321	580	16	then	then	ADV
ejpam-6321	580	17	we	we	PRON
ejpam-6321	580	18	can	can	AUX
ejpam-6321	580	19	write	write	VERB
ejpam-6321	580	20	;	;	PUNCT
ejpam-6321	580	21	=	=	SYM
ejpam-6321	580	22	α1g(x3k	α1g(x3k	NOUN
ejpam-6321	580	23	,	,	PUNCT
ejpam-6321	580	24	x(3k+1	x(3k+1	PRON
ejpam-6321	580	25	)	)	PUNCT
ejpam-6321	580	26	,	,	PUNCT
ejpam-6321	580	27	℘	℘	PROPN
ejpam-6321	580	28	)	)	PUNCT
ejpam-6321	580	29	+	+	NUM
ejpam-6321	580	30	α2g(x3k	α2g(x3k	NOUN
ejpam-6321	580	31	,	,	PUNCT
ejpam-6321	580	32	x(3k+1	x(3k+1	CCONJ
ejpam-6321	580	33	)	)	PUNCT
ejpam-6321	580	34	,	,	PUNCT
ejpam-6321	580	35	x(3k+1))+	x(3k+1))+	PROPN
ejpam-6321	580	36	α4[g(x3k	α4[g(x3k	PROPN
ejpam-6321	580	37	,	,	PUNCT
ejpam-6321	580	38	x3k	x3k	NOUN
ejpam-6321	580	39	,	,	PUNCT
ejpam-6321	580	40	x3k	x3k	PUNCT
ejpam-6321	580	41	)	)	PUNCT
ejpam-6321	581	1	+	+	SYM
ejpam-6321	581	2	g(x(3k+1	g(x(3k+1	NOUN
ejpam-6321	581	3	)	)	PUNCT
ejpam-6321	581	4	,	,	PUNCT
ejpam-6321	581	5	x(3k+1	x(3k+1	CCONJ
ejpam-6321	581	6	)	)	PUNCT
ejpam-6321	581	7	,	,	PUNCT
ejpam-6321	581	8	x(3k+1	x(3k+1	CCONJ
ejpam-6321	581	9	)	)	PUNCT
ejpam-6321	581	10	)	)	PUNCT
ejpam-6321	582	1	+	+	PUNCT
ejpam-6321	582	2	g(℘	g(℘	NOUN
ejpam-6321	582	3	,	,	PUNCT
ejpam-6321	582	4	℘	℘	NOUN
ejpam-6321	582	5	,	,	PUNCT
ejpam-6321	582	6	f3℘	f3℘	NOUN
ejpam-6321	582	7	)	)	PUNCT
ejpam-6321	582	8	]	]	PUNCT
ejpam-6321	583	1	+	+	CCONJ
ejpam-6321	583	2	α5g(x(3k+1	α5g(x(3k+1	NOUN
ejpam-6321	583	3	)	)	PUNCT
ejpam-6321	583	4	,	,	PUNCT
ejpam-6321	583	5	x(3k+2	x(3k+2	NOUN
ejpam-6321	583	6	)	)	PUNCT
ejpam-6321	583	7	,	,	PUNCT
ejpam-6321	583	8	x(3k+3	x(3k+3	NUM
ejpam-6321	583	9	)	)	PUNCT
ejpam-6321	583	10	)	)	PUNCT
ejpam-6321	584	1	m.	m.	NOUN
ejpam-6321	584	2	noorwali	noorwali	PROPN
ejpam-6321	584	3	et	et	PROPN
ejpam-6321	584	4	al	al	PROPN
ejpam-6321	584	5	.	.	PUNCT
ejpam-6321	584	6	/	/	SYM
ejpam-6321	584	7	eur	eur	PROPN
ejpam-6321	584	8	.	.	PUNCT
ejpam-6321	585	1	j.	j.	PROPN
ejpam-6321	585	2	pure	pure	PROPN
ejpam-6321	585	3	appl	appl	PROPN
ejpam-6321	585	4	.	.	PROPN
ejpam-6321	585	5	math	math	PROPN
ejpam-6321	585	6	,	,	PUNCT
ejpam-6321	585	7	18	18	NUM
ejpam-6321	585	8	(	(	PUNCT
ejpam-6321	585	9	3	3	NUM
ejpam-6321	585	10	)	)	PUNCT
ejpam-6321	585	11	(	(	PUNCT
ejpam-6321	585	12	2025	2025	NUM
ejpam-6321	585	13	)	)	PUNCT
ejpam-6321	585	14	,	,	PUNCT
ejpam-6321	585	15	6321	6321	NUM
ejpam-6321	585	16	20	20	NUM
ejpam-6321	585	17	of	of	ADP
ejpam-6321	585	18	27	27	NUM
ejpam-6321	585	19	applying	apply	VERB
ejpam-6321	585	20	limn→∞	limn→∞	PROPN
ejpam-6321	585	21	on	on	ADP
ejpam-6321	585	22	both	both	DET
ejpam-6321	585	23	sides	side	NOUN
ejpam-6321	585	24	,	,	PUNCT
ejpam-6321	585	25	we	we	PRON
ejpam-6321	585	26	get	get	VERB
ejpam-6321	585	27	,	,	PUNCT
ejpam-6321	585	28	g(℘	g(℘	PROPN
ejpam-6321	585	29	,	,	PUNCT
ejpam-6321	585	30	℘	℘	NOUN
ejpam-6321	585	31	,	,	PUNCT
ejpam-6321	585	32	f3℘	f3℘	NOUN
ejpam-6321	585	33	)	)	PUNCT
ejpam-6321	585	34	≤	≤	NOUN
ejpam-6321	585	35	α1g(℘	α1g(℘	NOUN
ejpam-6321	585	36	,	,	PUNCT
ejpam-6321	585	37	℘	℘	NOUN
ejpam-6321	585	38	,	,	PUNCT
ejpam-6321	585	39	℘	℘	NUM
ejpam-6321	585	40	)	)	PUNCT
ejpam-6321	585	41	+	+	NUM
ejpam-6321	585	42	α2g(℘	α2g(℘	NOUN
ejpam-6321	585	43	,	,	PUNCT
ejpam-6321	585	44	℘	℘	NOUN
ejpam-6321	585	45	,	,	PUNCT
ejpam-6321	585	46	℘)+	℘)+	PROPN
ejpam-6321	585	47	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	585	48	,	,	PUNCT
ejpam-6321	585	49	℘	℘	NOUN
ejpam-6321	585	50	,	,	PUNCT
ejpam-6321	585	51	℘	℘	NUM
ejpam-6321	585	52	)	)	PUNCT
ejpam-6321	586	1	+	+	NOUN
ejpam-6321	586	2	g(℘	g(℘	NOUN
ejpam-6321	586	3	,	,	PUNCT
ejpam-6321	586	4	℘	℘	NOUN
ejpam-6321	586	5	,	,	PUNCT
ejpam-6321	586	6	℘	℘	NUM
ejpam-6321	586	7	)	)	PUNCT
ejpam-6321	586	8	+	+	NOUN
ejpam-6321	586	9	g(℘	g(℘	NOUN
ejpam-6321	586	10	,	,	PUNCT
ejpam-6321	586	11	℘	℘	NOUN
ejpam-6321	586	12	,	,	PUNCT
ejpam-6321	586	13	f3℘	f3℘	NOUN
ejpam-6321	586	14	)	)	PUNCT
ejpam-6321	586	15	]	]	PUNCT
ejpam-6321	587	1	+	+	NUM
ejpam-6321	587	2	α5g(℘	α5g(℘	NOUN
ejpam-6321	587	3	,	,	PUNCT
ejpam-6321	587	4	℘	℘	NOUN
ejpam-6321	587	5	,	,	PUNCT
ejpam-6321	587	6	f3℘	f3℘	NOUN
ejpam-6321	587	7	)	)	PUNCT
ejpam-6321	587	8	g(℘	g(℘	PROPN
ejpam-6321	587	9	,	,	PUNCT
ejpam-6321	587	10	℘	℘	NOUN
ejpam-6321	587	11	,	,	PUNCT
ejpam-6321	587	12	f3℘)−α5g(℘	f3℘)−α5g(℘	NOUN
ejpam-6321	587	13	,	,	PUNCT
ejpam-6321	587	14	℘	℘	PROPN
ejpam-6321	587	15	,	,	PUNCT
ejpam-6321	587	16	f3℘	f3℘	NOUN
ejpam-6321	587	17	)	)	PUNCT
ejpam-6321	587	18	≤	≤	NOUN
ejpam-6321	587	19	0	0	NUM
ejpam-6321	587	20	.	.	PUNCT
ejpam-6321	588	1	(	(	PUNCT
ejpam-6321	588	2	1−α5)g(℘	1−α5)g(℘	NUM
ejpam-6321	588	3	,	,	PUNCT
ejpam-6321	588	4	℘	℘	PROPN
ejpam-6321	588	5	,	,	PUNCT
ejpam-6321	588	6	f3℘	f3℘	NOUN
ejpam-6321	588	7	)	)	PUNCT
ejpam-6321	588	8	≤	≤	NOUN
ejpam-6321	588	9	0	0	NUM
ejpam-6321	588	10	is	be	AUX
ejpam-6321	588	11	a	a	DET
ejpam-6321	588	12	contradiction	contradiction	NOUN
ejpam-6321	588	13	.	.	PUNCT
ejpam-6321	589	1	since	since	SCONJ
ejpam-6321	589	2	(	(	PUNCT
ejpam-6321	589	3	1−α5	1−α5	NUM
ejpam-6321	589	4	)	)	PUNCT
ejpam-6321	589	5	̸=	̸=	PROPN
ejpam-6321	589	6	0	0	NUM
ejpam-6321	589	7	,	,	PUNCT
ejpam-6321	589	8	therefore	therefore	ADV
ejpam-6321	589	9	,	,	PUNCT
ejpam-6321	589	10	g(f2℘	g(f2℘	PROPN
ejpam-6321	589	11	,	,	PUNCT
ejpam-6321	589	12	℘	℘	PROPN
ejpam-6321	589	13	,	,	PUNCT
ejpam-6321	589	14	℘	℘	NUM
ejpam-6321	589	15	)	)	PUNCT
ejpam-6321	589	16	=	=	SYM
ejpam-6321	589	17	0	0	X
ejpam-6321	589	18	.	.	PUNCT
ejpam-6321	590	1	thus	thus	ADV
ejpam-6321	590	2	,	,	PUNCT
ejpam-6321	590	3	f3℘	f3℘	NOUN
ejpam-6321	590	4	=	=	SYM
ejpam-6321	590	5	℘	℘	PROPN
ejpam-6321	590	6	(	(	PUNCT
ejpam-6321	590	7	34	34	NUM
ejpam-6321	590	8	)	)	PUNCT
ejpam-6321	590	9	by	by	ADP
ejpam-6321	590	10	the	the	DET
ejpam-6321	590	11	view	view	NOUN
ejpam-6321	590	12	of	of	ADP
ejpam-6321	590	13	(	(	PUNCT
ejpam-6321	590	14	31)−	31)−	NUM
ejpam-6321	590	15	(	(	PUNCT
ejpam-6321	590	16	34	34	NUM
ejpam-6321	590	17	)	)	PUNCT
ejpam-6321	590	18	,	,	PUNCT
ejpam-6321	590	19	we	we	PRON
ejpam-6321	590	20	obtain	obtain	VERB
ejpam-6321	590	21	that	that	SCONJ
ejpam-6321	590	22	℘	℘	PROPN
ejpam-6321	590	23	is	be	AUX
ejpam-6321	590	24	a	a	DET
ejpam-6321	590	25	fp	fp	X
ejpam-6321	590	26	of	of	ADP
ejpam-6321	590	27	f1	f1	NOUN
ejpam-6321	590	28	,	,	PUNCT
ejpam-6321	590	29	f2	f2	PROPN
ejpam-6321	590	30	and	and	CCONJ
ejpam-6321	590	31	f3	f3	PROPN
ejpam-6321	590	32	,	,	PUNCT
ejpam-6321	590	33	i.e	i.e	PROPN
ejpam-6321	590	34	,	,	PUNCT
ejpam-6321	590	35	f3℘	f3℘	NOUN
ejpam-6321	590	36	=	=	SYM
ejpam-6321	590	37	℘	℘	PROPN
ejpam-6321	590	38	(	(	PUNCT
ejpam-6321	590	39	35	35	NUM
ejpam-6321	590	40	)	)	PUNCT
ejpam-6321	590	41	thus	thus	ADV
ejpam-6321	590	42	from	from	ADP
ejpam-6321	590	43	(	(	PUNCT
ejpam-6321	590	44	15	15	NUM
ejpam-6321	590	45	)	)	PUNCT
ejpam-6321	590	46	,	,	PUNCT
ejpam-6321	590	47	(	(	PUNCT
ejpam-6321	590	48	34	34	NUM
ejpam-6321	590	49	)	)	PUNCT
ejpam-6321	590	50	and	and	CCONJ
ejpam-6321	590	51	(	(	PUNCT
ejpam-6321	590	52	35	35	NUM
ejpam-6321	590	53	)	)	PUNCT
ejpam-6321	590	54	,	,	PUNCT
ejpam-6321	590	55	it	it	PRON
ejpam-6321	590	56	is	be	AUX
ejpam-6321	590	57	proved	prove	VERB
ejpam-6321	590	58	that	that	SCONJ
ejpam-6321	590	59	℘	℘	PROPN
ejpam-6321	590	60	is	be	AUX
ejpam-6321	590	61	a	a	DET
ejpam-6321	590	62	cfp	cfp	NOUN
ejpam-6321	590	63	of	of	ADP
ejpam-6321	590	64	f1	f1	NOUN
ejpam-6321	590	65	,	,	PUNCT
ejpam-6321	590	66	f2	f2	PROPN
ejpam-6321	590	67	and	and	CCONJ
ejpam-6321	590	68	f3	f3	ADJ
ejpam-6321	590	69	,	,	PUNCT
ejpam-6321	590	70	such	such	ADJ
ejpam-6321	590	71	that	that	DET
ejpam-6321	590	72	f1℘	f1℘	PROPN
ejpam-6321	590	73	=	=	X
ejpam-6321	590	74	f2℘	f2℘	NOUN
ejpam-6321	590	75	=	=	SYM
ejpam-6321	590	76	f3℘	f3℘	NOUN
ejpam-6321	590	77	=	=	SYM
ejpam-6321	590	78	℘	℘	PROPN
ejpam-6321	590	79	uniqueness	uniqueness	NOUN
ejpam-6321	590	80	:	:	PUNCT
ejpam-6321	590	81	suppose	suppose	VERB
ejpam-6321	590	82	that	that	SCONJ
ejpam-6321	590	83	℘⋄	℘⋄	ADJ
ejpam-6321	590	84	∈	∈	NOUN
ejpam-6321	590	85	x	x	PUNCT
ejpam-6321	590	86	be	be	AUX
ejpam-6321	590	87	the	the	DET
ejpam-6321	590	88	other	other	ADJ
ejpam-6321	590	89	cfp	cfp	PROPN
ejpam-6321	590	90	of	of	ADP
ejpam-6321	590	91	f1	f1	NOUN
ejpam-6321	590	92	,	,	PUNCT
ejpam-6321	590	93	f2	f2	PROPN
ejpam-6321	590	94	and	and	CCONJ
ejpam-6321	590	95	f3	f3	ADJ
ejpam-6321	590	96	,	,	PUNCT
ejpam-6321	590	97	so	so	SCONJ
ejpam-6321	590	98	that	that	SCONJ
ejpam-6321	590	99	f1℘	f1℘	PROPN
ejpam-6321	590	100	⋄	⋄	NOUN
ejpam-6321	590	101	=	=	PUNCT
ejpam-6321	590	102	f2℘	f2℘	PROPN
ejpam-6321	590	103	⋄	⋄	NOUN
ejpam-6321	590	104	=	=	PUNCT
ejpam-6321	590	105	f3℘	f3℘	X
ejpam-6321	590	106	⋄	⋄	NOUN
ejpam-6321	590	107	=	=	SYM
ejpam-6321	590	108	℘⋄	℘⋄	X
ejpam-6321	590	109	then	then	ADV
ejpam-6321	590	110	from	from	ADP
ejpam-6321	590	111	(	(	PUNCT
ejpam-6321	590	112	8)	8)	NUM
ejpam-6321	590	113	we	we	PRON
ejpam-6321	590	114	have	have	VERB
ejpam-6321	590	115	that	that	PRON
ejpam-6321	590	116	,	,	PUNCT
ejpam-6321	590	117	g(℘	g(℘	PROPN
ejpam-6321	590	118	,	,	PUNCT
ejpam-6321	590	119	℘⋄	℘⋄	NUM
ejpam-6321	590	120	,	,	PUNCT
ejpam-6321	590	121	℘⋄	℘⋄	NUM
ejpam-6321	590	122	)	)	PUNCT
ejpam-6321	590	123	=	=	SYM
ejpam-6321	590	124	g(f1℘	g(f1℘	NOUN
ejpam-6321	590	125	,	,	PUNCT
ejpam-6321	590	126	f2℘	f2℘	PROPN
ejpam-6321	590	127	⋄	⋄	PROPN
ejpam-6321	590	128	,	,	PUNCT
ejpam-6321	590	129	f3℘	f3℘	PROPN
ejpam-6321	590	130	⋄	⋄	NOUN
ejpam-6321	590	131	)	)	PUNCT
ejpam-6321	590	132	≤	≤	NOUN
ejpam-6321	590	133	α1g(℘	α1g(℘	NOUN
ejpam-6321	590	134	,	,	PUNCT
ejpam-6321	590	135	℘⋄	℘⋄	X
ejpam-6321	590	136	,	,	PUNCT
ejpam-6321	590	137	℘⋄	℘⋄	NUM
ejpam-6321	590	138	)	)	PUNCT
ejpam-6321	590	139	+	+	NUM
ejpam-6321	590	140	α2g(℘	α2g(℘	NOUN
ejpam-6321	590	141	,	,	PUNCT
ejpam-6321	590	142	℘⋄	℘⋄	NUM
ejpam-6321	590	143	,	,	PUNCT
ejpam-6321	590	144	f2℘	f2℘	PROPN
ejpam-6321	590	145	⋄	⋄	PROPN
ejpam-6321	590	146	)	)	PUNCT
ejpam-6321	590	147	+	+	CCONJ
ejpam-6321	591	1	α3g(f1℘	α3g(f1℘	PROPN
ejpam-6321	591	2	,	,	PUNCT
ejpam-6321	591	3	℘	℘	PROPN
ejpam-6321	591	4	⋄	⋄	PROPN
ejpam-6321	591	5	,	,	PUNCT
ejpam-6321	591	6	℘⋄	℘⋄	NUM
ejpam-6321	591	7	)	)	PUNCT
ejpam-6321	592	1	+	+	NUM
ejpam-6321	592	2	α4[g(f1℘	α4[g(f1℘	PROPN
ejpam-6321	592	3	,	,	PUNCT
ejpam-6321	592	4	℘	℘	PROPN
ejpam-6321	592	5	,	,	PUNCT
ejpam-6321	592	6	℘	℘	NUM
ejpam-6321	592	7	)	)	PUNCT
ejpam-6321	592	8	+	+	NOUN
ejpam-6321	592	9	g(℘⋄	g(℘⋄	NUM
ejpam-6321	592	10	,	,	PUNCT
ejpam-6321	592	11	f2℘	f2℘	PROPN
ejpam-6321	592	12	⋄	⋄	PROPN
ejpam-6321	592	13	,	,	PUNCT
ejpam-6321	592	14	℘⋄	℘⋄	NUM
ejpam-6321	592	15	)	)	PUNCT
ejpam-6321	593	1	+	+	NOUN
ejpam-6321	593	2	g(℘⋄	g(℘⋄	NUM
ejpam-6321	593	3	,	,	PUNCT
ejpam-6321	593	4	℘⋄	℘⋄	NUM
ejpam-6321	593	5	,	,	PUNCT
ejpam-6321	593	6	f3℘	f3℘	PROPN
ejpam-6321	593	7	⋄	⋄	NOUN
ejpam-6321	593	8	)	)	PUNCT
ejpam-6321	593	9	]	]	PUNCT
ejpam-6321	594	1	+	+	CCONJ
ejpam-6321	594	2	α5max	α5max	PROPN
ejpam-6321	594	3			PROPN
ejpam-6321	594	4	g(℘	g(℘	PROPN
ejpam-6321	594	5	,	,	PUNCT
ejpam-6321	594	6	℘⋄	℘⋄	NUM
ejpam-6321	594	7	,	,	PUNCT
ejpam-6321	594	8	f2℘	f2℘	PROPN
ejpam-6321	594	9	⋄	⋄	PROPN
ejpam-6321	594	10	)	)	PUNCT
ejpam-6321	594	11	,	,	PUNCT
ejpam-6321	594	12	g(f1℘	g(f1℘	PROPN
ejpam-6321	594	13	,	,	PUNCT
ejpam-6321	594	14	f1℘	f1℘	PROPN
ejpam-6321	594	15	,	,	PUNCT
ejpam-6321	594	16	℘	℘	PROPN
ejpam-6321	594	17	⋄	⋄	PROPN
ejpam-6321	594	18	)	)	PUNCT
ejpam-6321	594	19	,	,	PUNCT
ejpam-6321	594	20	g(f1℘	g(f1℘	PROPN
ejpam-6321	594	21	,	,	PUNCT
ejpam-6321	594	22	℘	℘	PROPN
ejpam-6321	594	23	⋄	⋄	PROPN
ejpam-6321	594	24	,	,	PUNCT
ejpam-6321	594	25	℘⋄	℘⋄	NUM
ejpam-6321	594	26	)	)	PUNCT
ejpam-6321	594	27	,	,	PUNCT
ejpam-6321	594	28	g(f2℘	g(f2℘	PROPN
ejpam-6321	594	29	⋄	⋄	PROPN
ejpam-6321	594	30	,	,	PUNCT
ejpam-6321	594	31	f2℘	f2℘	PROPN
ejpam-6321	594	32	⋄	⋄	PROPN
ejpam-6321	594	33	,	,	PUNCT
ejpam-6321	594	34	℘⋄	℘⋄	NUM
ejpam-6321	594	35	)	)	PUNCT
ejpam-6321	594	36	,	,	PUNCT
ejpam-6321	594	37	g(f1℘	g(f1℘	PROPN
ejpam-6321	594	38	,	,	PUNCT
ejpam-6321	594	39	℘	℘	PROPN
ejpam-6321	594	40	,	,	PUNCT
ejpam-6321	594	41	℘	℘	PROPN
ejpam-6321	594	42	)	)	PUNCT
ejpam-6321	594	43	,	,	PUNCT
ejpam-6321	594	44	g(f2℘	g(f2℘	PROPN
ejpam-6321	594	45	⋄	⋄	PROPN
ejpam-6321	594	46	,	,	PUNCT
ejpam-6321	594	47	f2℘	f2℘	PROPN
ejpam-6321	594	48	⋄	⋄	PROPN
ejpam-6321	594	49	,	,	PUNCT
ejpam-6321	594	50	℘⋄	℘⋄	NUM
ejpam-6321	594	51	)	)	PUNCT
ejpam-6321	594	52	,	,	PUNCT
ejpam-6321	594	53	g(℘⋄	g(℘⋄	NUM
ejpam-6321	594	54	,	,	PUNCT
ejpam-6321	594	55	℘⋄	℘⋄	NUM
ejpam-6321	594	56	,	,	PUNCT
ejpam-6321	594	57	f3℘	f3℘	PROPN
ejpam-6321	594	58	⋄	⋄	NOUN
ejpam-6321	594	59	)	)	PUNCT
ejpam-6321	595	1			PROPN
ejpam-6321	595	2	≤	≤	NUM
ejpam-6321	595	3	α1g(℘	α1g(℘	NOUN
ejpam-6321	595	4	,	,	PUNCT
ejpam-6321	595	5	℘⋄	℘⋄	X
ejpam-6321	595	6	,	,	PUNCT
ejpam-6321	595	7	℘⋄	℘⋄	NUM
ejpam-6321	595	8	)	)	PUNCT
ejpam-6321	596	1	+	+	NUM
ejpam-6321	596	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	596	3	,	,	PUNCT
ejpam-6321	596	4	℘⋄	℘⋄	X
ejpam-6321	596	5	,	,	PUNCT
ejpam-6321	596	6	℘⋄	℘⋄	NUM
ejpam-6321	596	7	)	)	PUNCT
ejpam-6321	596	8	+	+	CCONJ
ejpam-6321	596	9	α3g(℘	α3g(℘	NUM
ejpam-6321	596	10	,	,	PUNCT
ejpam-6321	596	11	℘⋄	℘⋄	X
ejpam-6321	596	12	,	,	PUNCT
ejpam-6321	596	13	℘⋄	℘⋄	NUM
ejpam-6321	596	14	)	)	PUNCT
ejpam-6321	597	1	+	+	CCONJ
ejpam-6321	597	2	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	597	3	,	,	PUNCT
ejpam-6321	597	4	℘	℘	NOUN
ejpam-6321	597	5	,	,	PUNCT
ejpam-6321	597	6	℘	℘	NUM
ejpam-6321	597	7	)	)	PUNCT
ejpam-6321	597	8	+	+	NOUN
ejpam-6321	597	9	g(℘⋄	g(℘⋄	NUM
ejpam-6321	597	10	,	,	PUNCT
ejpam-6321	597	11	℘⋄	℘⋄	NUM
ejpam-6321	597	12	,	,	PUNCT
ejpam-6321	597	13	℘⋄	℘⋄	NUM
ejpam-6321	597	14	)	)	PUNCT
ejpam-6321	598	1	+	+	NOUN
ejpam-6321	598	2	g(℘⋄	g(℘⋄	NUM
ejpam-6321	598	3	,	,	PUNCT
ejpam-6321	598	4	℘⋄	℘⋄	NUM
ejpam-6321	598	5	,	,	PUNCT
ejpam-6321	598	6	℘⋄	℘⋄	NUM
ejpam-6321	598	7	)	)	PUNCT
ejpam-6321	598	8	]	]	PUNCT
ejpam-6321	599	1	+	+	CCONJ
ejpam-6321	599	2	α5max	α5max	PROPN
ejpam-6321	599	3			PROPN
ejpam-6321	599	4	g(℘	g(℘	PROPN
ejpam-6321	599	5	,	,	PUNCT
ejpam-6321	599	6	℘⋄	℘⋄	NUM
ejpam-6321	599	7	,	,	PUNCT
ejpam-6321	599	8	℘⋄	℘⋄	NUM
ejpam-6321	599	9	)	)	PUNCT
ejpam-6321	599	10	,	,	PUNCT
ejpam-6321	599	11	g(℘	g(℘	PROPN
ejpam-6321	599	12	,	,	PUNCT
ejpam-6321	599	13	℘	℘	NOUN
ejpam-6321	599	14	,	,	PUNCT
ejpam-6321	599	15	℘⋄	℘⋄	NUM
ejpam-6321	599	16	)	)	PUNCT
ejpam-6321	599	17	,	,	PUNCT
ejpam-6321	599	18	g(℘	g(℘	PROPN
ejpam-6321	599	19	,	,	PUNCT
ejpam-6321	599	20	℘⋄	℘⋄	NUM
ejpam-6321	599	21	,	,	PUNCT
ejpam-6321	599	22	℘⋄	℘⋄	NUM
ejpam-6321	599	23	)	)	PUNCT
ejpam-6321	599	24	,	,	PUNCT
ejpam-6321	599	25	g(℘⋄	g(℘⋄	NUM
ejpam-6321	599	26	,	,	PUNCT
ejpam-6321	599	27	℘⋄	℘⋄	X
ejpam-6321	599	28	,	,	PUNCT
ejpam-6321	599	29	℘⋄	℘⋄	NUM
ejpam-6321	599	30	)	)	PUNCT
ejpam-6321	599	31	,	,	PUNCT
ejpam-6321	599	32	g(℘	g(℘	PROPN
ejpam-6321	599	33	,	,	PUNCT
ejpam-6321	599	34	℘	℘	NOUN
ejpam-6321	599	35	,	,	PUNCT
ejpam-6321	599	36	℘	℘	PROPN
ejpam-6321	599	37	)	)	PUNCT
ejpam-6321	599	38	,	,	PUNCT
ejpam-6321	599	39	g(f2℘	g(f2℘	PROPN
ejpam-6321	599	40	⋄	⋄	PROPN
ejpam-6321	599	41	,	,	PUNCT
ejpam-6321	599	42	℘⋄	℘⋄	X
ejpam-6321	599	43	,	,	PUNCT
ejpam-6321	599	44	℘⋄	℘⋄	NUM
ejpam-6321	599	45	)	)	PUNCT
ejpam-6321	599	46	,	,	PUNCT
ejpam-6321	599	47	g(℘⋄	g(℘⋄	NUM
ejpam-6321	599	48	,	,	PUNCT
ejpam-6321	599	49	℘⋄	℘⋄	X
ejpam-6321	599	50	,	,	PUNCT
ejpam-6321	599	51	℘⋄	℘⋄	NUM
ejpam-6321	599	52	)	)	PUNCT
ejpam-6321	599	53			NOUN
ejpam-6321	599	54	after	after	ADP
ejpam-6321	599	55	applying	apply	VERB
ejpam-6321	599	56	the	the	DET
ejpam-6321	599	57	simplification	simplification	NOUN
ejpam-6321	599	58	process	process	NOUN
ejpam-6321	599	59	utilizing	utilize	VERB
ejpam-6321	599	60	the	the	DET
ejpam-6321	599	61	definition	definition	NOUN
ejpam-6321	599	62	of	of	ADP
ejpam-6321	599	63	gm	gm	PROPN
ejpam-6321	599	64	-	-	PUNCT
ejpam-6321	599	65	space	space	NOUN
ejpam-6321	599	66	we	we	PRON
ejpam-6321	599	67	have	have	AUX
ejpam-6321	599	68	achieved	achieve	VERB
ejpam-6321	599	69	the	the	DET
ejpam-6321	599	70	following	following	ADJ
ejpam-6321	599	71	result	result	NOUN
ejpam-6321	599	72	,	,	PUNCT
ejpam-6321	599	73	≤	≤	ADJ
ejpam-6321	599	74	α1g(℘	α1g(℘	NOUN
ejpam-6321	599	75	,	,	PUNCT
ejpam-6321	599	76	℘⋄	℘⋄	X
ejpam-6321	599	77	,	,	PUNCT
ejpam-6321	599	78	℘⋄	℘⋄	NUM
ejpam-6321	599	79	)	)	PUNCT
ejpam-6321	600	1	+	+	NUM
ejpam-6321	600	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	600	3	,	,	PUNCT
ejpam-6321	600	4	℘⋄	℘⋄	X
ejpam-6321	600	5	,	,	PUNCT
ejpam-6321	600	6	℘⋄	℘⋄	NUM
ejpam-6321	600	7	)	)	PUNCT
ejpam-6321	600	8	+	+	CCONJ
ejpam-6321	600	9	α3g(℘	α3g(℘	NUM
ejpam-6321	600	10	,	,	PUNCT
ejpam-6321	600	11	℘⋄	℘⋄	X
ejpam-6321	600	12	,	,	PUNCT
ejpam-6321	600	13	℘⋄	℘⋄	NUM
ejpam-6321	600	14	)	)	PUNCT
ejpam-6321	601	1	+	+	CCONJ
ejpam-6321	601	2	α4[g(℘	α4[g(℘	NOUN
ejpam-6321	601	3	,	,	PUNCT
ejpam-6321	601	4	℘	℘	NOUN
ejpam-6321	601	5	,	,	PUNCT
ejpam-6321	601	6	℘	℘	NUM
ejpam-6321	601	7	)	)	PUNCT
ejpam-6321	601	8	+	+	NOUN
ejpam-6321	601	9	g(℘⋄	g(℘⋄	NUM
ejpam-6321	601	10	,	,	PUNCT
ejpam-6321	601	11	℘⋄	℘⋄	NUM
ejpam-6321	601	12	,	,	PUNCT
ejpam-6321	601	13	℘⋄	℘⋄	NUM
ejpam-6321	601	14	)	)	PUNCT
ejpam-6321	602	1	+	+	NOUN
ejpam-6321	602	2	g(℘⋄	g(℘⋄	NUM
ejpam-6321	602	3	,	,	PUNCT
ejpam-6321	602	4	℘⋄	℘⋄	NUM
ejpam-6321	602	5	,	,	PUNCT
ejpam-6321	602	6	℘⋄	℘⋄	NUM
ejpam-6321	602	7	)	)	PUNCT
ejpam-6321	602	8	]	]	PUNCT
ejpam-6321	603	1	+	+	CCONJ
ejpam-6321	603	2	α5max	α5max	PROPN
ejpam-6321	603	3			PROPN
ejpam-6321	603	4	g(℘	g(℘	PROPN
ejpam-6321	603	5	,	,	PUNCT
ejpam-6321	603	6	℘⋄	℘⋄	NUM
ejpam-6321	603	7	,	,	PUNCT
ejpam-6321	603	8	℘⋄	℘⋄	NUM
ejpam-6321	603	9	)	)	PUNCT
ejpam-6321	603	10	,	,	PUNCT
ejpam-6321	603	11	g(℘	g(℘	PROPN
ejpam-6321	603	12	,	,	PUNCT
ejpam-6321	603	13	℘	℘	NOUN
ejpam-6321	603	14	,	,	PUNCT
ejpam-6321	603	15	℘⋄	℘⋄	NUM
ejpam-6321	603	16	)	)	PUNCT
ejpam-6321	603	17	,	,	PUNCT
ejpam-6321	603	18	g(℘	g(℘	PROPN
ejpam-6321	603	19	,	,	PUNCT
ejpam-6321	603	20	℘⋄	℘⋄	NUM
ejpam-6321	603	21	,	,	PUNCT
ejpam-6321	603	22	℘⋄	℘⋄	NUM
ejpam-6321	603	23	)	)	PUNCT
ejpam-6321	603	24	,	,	PUNCT
ejpam-6321	603	25	g(℘⋄	g(℘⋄	NUM
ejpam-6321	603	26	,	,	PUNCT
ejpam-6321	603	27	℘⋄	℘⋄	X
ejpam-6321	603	28	,	,	PUNCT
ejpam-6321	603	29	℘⋄	℘⋄	NUM
ejpam-6321	603	30	)	)	PUNCT
ejpam-6321	603	31	,	,	PUNCT
ejpam-6321	603	32	g(℘	g(℘	PROPN
ejpam-6321	603	33	,	,	PUNCT
ejpam-6321	603	34	℘	℘	NOUN
ejpam-6321	603	35	,	,	PUNCT
ejpam-6321	603	36	℘	℘	PROPN
ejpam-6321	603	37	)	)	PUNCT
ejpam-6321	603	38	,	,	PUNCT
ejpam-6321	603	39	g(f2℘	g(f2℘	PROPN
ejpam-6321	603	40	⋄	⋄	PROPN
ejpam-6321	603	41	,	,	PUNCT
ejpam-6321	603	42	℘⋄	℘⋄	X
ejpam-6321	603	43	,	,	PUNCT
ejpam-6321	603	44	℘⋄	℘⋄	NUM
ejpam-6321	603	45	)	)	PUNCT
ejpam-6321	603	46	,	,	PUNCT
ejpam-6321	603	47	g(℘⋄	g(℘⋄	NUM
ejpam-6321	603	48	,	,	PUNCT
ejpam-6321	603	49	℘⋄	℘⋄	X
ejpam-6321	603	50	,	,	PUNCT
ejpam-6321	603	51	℘⋄	℘⋄	NUM
ejpam-6321	603	52	)	)	PUNCT
ejpam-6321	603	53			NOUN
ejpam-6321	603	54	≤	≤	NUM
ejpam-6321	603	55	α1g(℘	α1g(℘	NOUN
ejpam-6321	603	56	,	,	PUNCT
ejpam-6321	603	57	℘⋄	℘⋄	X
ejpam-6321	603	58	,	,	PUNCT
ejpam-6321	603	59	℘⋄	℘⋄	NUM
ejpam-6321	603	60	)	)	PUNCT
ejpam-6321	604	1	+	+	NUM
ejpam-6321	604	2	α2g(℘	α2g(℘	NOUN
ejpam-6321	604	3	,	,	PUNCT
ejpam-6321	604	4	℘⋄	℘⋄	X
ejpam-6321	604	5	,	,	PUNCT
ejpam-6321	604	6	℘⋄	℘⋄	NUM
ejpam-6321	604	7	)	)	PUNCT
ejpam-6321	604	8	+	+	CCONJ
ejpam-6321	604	9	α3g(℘	α3g(℘	NUM
ejpam-6321	604	10	,	,	PUNCT
ejpam-6321	604	11	℘⋄	℘⋄	X
ejpam-6321	604	12	,	,	PUNCT
ejpam-6321	604	13	℘⋄	℘⋄	NUM
ejpam-6321	604	14	)	)	PUNCT
ejpam-6321	604	15	+	+	NUM
ejpam-6321	604	16	α5g(℘	α5g(℘	NOUN
ejpam-6321	604	17	,	,	PUNCT
ejpam-6321	604	18	℘⋄	℘⋄	X
ejpam-6321	604	19	,	,	PUNCT
ejpam-6321	604	20	℘⋄	℘⋄	NUM
ejpam-6321	604	21	)	)	PUNCT
ejpam-6321	604	22	(	(	PUNCT
ejpam-6321	604	23	α1	α1	PROPN
ejpam-6321	604	24	+	+	CCONJ
ejpam-6321	604	25	α2	α2	ADJ
ejpam-6321	604	26	+	+	CCONJ
ejpam-6321	604	27	α3	α3	NOUN
ejpam-6321	604	28	+	+	NOUN
ejpam-6321	604	29	α5)g(℘	α5)g(℘	NUM
ejpam-6321	604	30	,	,	PUNCT
ejpam-6321	604	31	℘⋄	℘⋄	NUM
ejpam-6321	604	32	,	,	PUNCT
ejpam-6321	604	33	℘⋄)(1−	℘⋄)(1−	PROPN
ejpam-6321	604	34	α1	α1	PROPN
ejpam-6321	604	35	−	−	PROPN
ejpam-6321	604	36	α2	α2	ADJ
ejpam-6321	604	37	−	−	PROPN
ejpam-6321	604	38	α3	α3	NOUN
ejpam-6321	604	39	−	−	PROPN
ejpam-6321	604	40	α5)g(℘	α5)g(℘	PROPN
ejpam-6321	604	41	,	,	PUNCT
ejpam-6321	604	42	℘⋄	℘⋄	X
ejpam-6321	604	43	,	,	PUNCT
ejpam-6321	604	44	℘⋄	℘⋄	NUM
ejpam-6321	604	45	)	)	PUNCT
ejpam-6321	604	46	≤	≤	NOUN
ejpam-6321	604	47	0	0	NUM
ejpam-6321	605	1	,	,	PUNCT
ejpam-6321	605	2	m.	m.	NOUN
ejpam-6321	605	3	noorwali	noorwali	PROPN
ejpam-6321	605	4	et	et	PROPN
ejpam-6321	605	5	al	al	PROPN
ejpam-6321	605	6	.	.	PUNCT
ejpam-6321	605	7	/	/	SYM
ejpam-6321	605	8	eur	eur	PROPN
ejpam-6321	605	9	.	.	PUNCT
ejpam-6321	606	1	j.	j.	PROPN
ejpam-6321	606	2	pure	pure	PROPN
ejpam-6321	606	3	appl	appl	PROPN
ejpam-6321	606	4	.	.	PROPN
ejpam-6321	606	5	math	math	PROPN
ejpam-6321	606	6	,	,	PUNCT
ejpam-6321	606	7	18	18	NUM
ejpam-6321	606	8	(	(	PUNCT
ejpam-6321	606	9	3	3	NUM
ejpam-6321	606	10	)	)	PUNCT
ejpam-6321	606	11	(	(	PUNCT
ejpam-6321	606	12	2025	2025	NUM
ejpam-6321	606	13	)	)	PUNCT
ejpam-6321	606	14	,	,	PUNCT
ejpam-6321	606	15	6321	6321	NUM
ejpam-6321	606	16	21	21	NUM
ejpam-6321	606	17	of	of	ADP
ejpam-6321	606	18	27	27	NUM
ejpam-6321	606	19	is	be	AUX
ejpam-6321	606	20	a	a	DET
ejpam-6321	606	21	contradiction	contradiction	NOUN
ejpam-6321	606	22	.	.	PUNCT
ejpam-6321	607	1	since	since	SCONJ
ejpam-6321	607	2	,	,	PUNCT
ejpam-6321	607	3	(	(	PUNCT
ejpam-6321	607	4	1−	1−	NUM
ejpam-6321	607	5	α1	α1	PROPN
ejpam-6321	607	6	−	−	PROPN
ejpam-6321	607	7	α2	α2	ADJ
ejpam-6321	607	8	−	−	PROPN
ejpam-6321	607	9	α3	α3	NOUN
ejpam-6321	607	10	−	−	PROPN
ejpam-6321	607	11	α5	α5	NOUN
ejpam-6321	607	12	)	)	PUNCT
ejpam-6321	607	13	̸=	̸=	PROPN
ejpam-6321	607	14	0	0	NUM
ejpam-6321	607	15	,	,	PUNCT
ejpam-6321	607	16	therefore	therefore	ADV
ejpam-6321	607	17	g(℘	g(℘	NOUN
ejpam-6321	607	18	,	,	PUNCT
ejpam-6321	607	19	℘⋄	℘⋄	NUM
ejpam-6321	607	20	,	,	PUNCT
ejpam-6321	607	21	℘⋄	℘⋄	NUM
ejpam-6321	607	22	)	)	PUNCT
ejpam-6321	607	23	=	=	SYM
ejpam-6321	607	24	0	0	X
ejpam-6321	607	25	.	.	PUNCT
ejpam-6321	607	26	thus	thus	ADV
ejpam-6321	607	27	,	,	PUNCT
ejpam-6321	607	28	℘	℘	PROPN
ejpam-6321	607	29	=	=	SYM
ejpam-6321	607	30	℘⋄	℘⋄	NOUN
ejpam-6321	607	31	it	it	PRON
ejpam-6321	607	32	is	be	AUX
ejpam-6321	607	33	proved	prove	VERB
ejpam-6321	607	34	that	that	SCONJ
ejpam-6321	607	35	the	the	DET
ejpam-6321	607	36	three	three	NUM
ejpam-6321	607	37	individual	individual	ADJ
ejpam-6321	607	38	self	self	NOUN
ejpam-6321	607	39	-	-	PUNCT
ejpam-6321	607	40	mappings	mapping	NOUN
ejpam-6321	607	41	referred	refer	VERB
ejpam-6321	607	42	as	as	ADP
ejpam-6321	607	43	f1	f1	NOUN
ejpam-6321	607	44	,	,	PUNCT
ejpam-6321	607	45	f2	f2	PROPN
ejpam-6321	607	46	and	and	CCONJ
ejpam-6321	607	47	f3	f3	PROPN
ejpam-6321	607	48	possess	possess	VERB
ejpam-6321	607	49	a	a	DET
ejpam-6321	607	50	ucfp	ucfp	NOUN
ejpam-6321	607	51	in	in	ADP
ejpam-6321	607	52	x.	x.	NOUN
ejpam-6321	607	53	the	the	DET
ejpam-6321	607	54	theorem	theorem	ADJ
ejpam-6321	607	55	2	2	NUM
ejpam-6321	607	56	provides	provide	VERB
ejpam-6321	607	57	conditions	condition	NOUN
ejpam-6321	607	58	under	under	ADP
ejpam-6321	607	59	which	which	PRON
ejpam-6321	607	60	three	three	NUM
ejpam-6321	607	61	self	self	NOUN
ejpam-6321	607	62	-	-	PUNCT
ejpam-6321	607	63	mappings	mapping	NOUN
ejpam-6321	607	64	f1	f1	NOUN
ejpam-6321	607	65	,	,	PUNCT
ejpam-6321	607	66	f2	f2	PROPN
ejpam-6321	607	67	,	,	PUNCT
ejpam-6321	607	68	f3	f3	PROPN
ejpam-6321	607	69	on	on	ADP
ejpam-6321	607	70	a	a	DET
ejpam-6321	607	71	g	g	NOUN
ejpam-6321	607	72	-	-	PUNCT
ejpam-6321	607	73	metric	metric	ADJ
ejpam-6321	607	74	space	space	NOUN
ejpam-6321	607	75	(	(	PUNCT
ejpam-6321	607	76	x	x	NOUN
ejpam-6321	607	77	,	,	PUNCT
ejpam-6321	607	78	g	g	NOUN
ejpam-6321	607	79	)	)	PUNCT
ejpam-6321	607	80	have	have	VERB
ejpam-6321	607	81	a	a	DET
ejpam-6321	607	82	unique	unique	ADJ
ejpam-6321	607	83	common	common	ADJ
ejpam-6321	607	84	fixed	fix	VERB
ejpam-6321	607	85	point	point	NOUN
ejpam-6321	607	86	.	.	PUNCT
ejpam-6321	608	1	below	below	ADV
ejpam-6321	608	2	is	be	AUX
ejpam-6321	608	3	an	an	DET
ejpam-6321	608	4	algorithm	algorithm	NOUN
ejpam-6321	608	5	3	3	NUM
ejpam-6321	608	6	representation	representation	NOUN
ejpam-6321	608	7	of	of	ADP
ejpam-6321	608	8	the	the	DET
ejpam-6321	608	9	fixed	fix	VERB
ejpam-6321	608	10	-	-	PUNCT
ejpam-6321	608	11	point	point	NOUN
ejpam-6321	608	12	iteration	iteration	NOUN
ejpam-6321	608	13	process	process	NOUN
ejpam-6321	608	14	:	:	PUNCT
ejpam-6321	608	15	algorithm	algorithm	NOUN
ejpam-6321	608	16	3	3	NUM
ejpam-6321	608	17	common	common	ADJ
ejpam-6321	608	18	fixed	fix	VERB
ejpam-6321	608	19	-	-	PUNCT
ejpam-6321	608	20	point	point	NOUN
ejpam-6321	608	21	iteration	iteration	NOUN
ejpam-6321	608	22	for	for	ADP
ejpam-6321	608	23	three	three	NUM
ejpam-6321	608	24	mappings	mapping	NOUN
ejpam-6321	608	25	1	1	NUM
ejpam-6321	608	26	:	:	PUNCT
ejpam-6321	608	27	input	input	NOUN
ejpam-6321	608	28	:	:	PUNCT
ejpam-6321	608	29	•	•	ADP
ejpam-6321	608	30	a	a	DET
ejpam-6321	608	31	g	g	NOUN
ejpam-6321	608	32	-	-	PUNCT
ejpam-6321	608	33	metric	metric	ADJ
ejpam-6321	608	34	space	space	NOUN
ejpam-6321	608	35	(	(	PUNCT
ejpam-6321	608	36	x	x	NOUN
ejpam-6321	608	37	,	,	PUNCT
ejpam-6321	608	38	g	g	NOUN
ejpam-6321	608	39	)	)	PUNCT
ejpam-6321	608	40	•	•	NOUN
ejpam-6321	608	41	three	three	NUM
ejpam-6321	608	42	self	self	NOUN
ejpam-6321	608	43	-	-	PUNCT
ejpam-6321	608	44	mappings	mapping	NOUN
ejpam-6321	608	45	f1	f1	NOUN
ejpam-6321	608	46	,	,	PUNCT
ejpam-6321	608	47	f2	f2	PROPN
ejpam-6321	608	48	,	,	PUNCT
ejpam-6321	608	49	f3	f3	PROPN
ejpam-6321	608	50	:	:	PUNCT
ejpam-6321	608	51	x	x	X
ejpam-6321	608	52	×x	×x	VERB
ejpam-6321	608	53	×x	×x	X
ejpam-6321	608	54	→	→	SYM
ejpam-6321	608	55	x	x	SYM
ejpam-6321	608	56	•	•	NOUN
ejpam-6321	608	57	an	an	DET
ejpam-6321	608	58	initial	initial	ADJ
ejpam-6321	608	59	point	point	NOUN
ejpam-6321	608	60	x0	x0	PROPN
ejpam-6321	609	1	∈	∈	PROPN
ejpam-6321	609	2	x	x	SYM
ejpam-6321	609	3	•	•	NUM
ejpam-6321	609	4	coefficients	coefficient	NOUN
ejpam-6321	609	5	α1	α1	PROPN
ejpam-6321	609	6	,	,	PUNCT
ejpam-6321	609	7	α2	α2	ADJ
ejpam-6321	609	8	,	,	PUNCT
ejpam-6321	609	9	α3	α3	NOUN
ejpam-6321	609	10	,	,	PUNCT
ejpam-6321	609	11	α4	α4	NOUN
ejpam-6321	609	12	,	,	PUNCT
ejpam-6321	609	13	α5	α5	NOUN
ejpam-6321	609	14	satisfying	satisfying	NOUN
ejpam-6321	609	15	:	:	PUNCT
ejpam-6321	609	16	α1	α1	PROPN
ejpam-6321	609	17	+	+	CCONJ
ejpam-6321	609	18	α2	α2	ADJ
ejpam-6321	609	19	+	+	CCONJ
ejpam-6321	609	20	α3	α3	ADJ
ejpam-6321	609	21	+	+	CCONJ
ejpam-6321	609	22	α4	α4	NOUN
ejpam-6321	609	23	+	+	CCONJ
ejpam-6321	609	24	α5	α5	X
ejpam-6321	609	25	<	<	X
ejpam-6321	609	26	1	1	NUM
ejpam-6321	609	27	2	2	NUM
ejpam-6321	609	28	:	:	PUNCT
ejpam-6321	609	29	initialize	initialize	VERB
ejpam-6321	609	30	:	:	PUNCT
ejpam-6321	609	31	•	•	NUM
ejpam-6321	609	32	set	set	VERB
ejpam-6321	609	33	k	k	PROPN
ejpam-6321	609	34	=	=	SYM
ejpam-6321	609	35	0	0	NUM
ejpam-6321	609	36	•	•	NUM
ejpam-6321	609	37	define	define	VERB
ejpam-6321	609	38	iterative	iterative	ADJ
ejpam-6321	609	39	sequences	sequence	NOUN
ejpam-6321	609	40	:	:	PUNCT
ejpam-6321	609	41	x3k+1	x3k+1	X
ejpam-6321	609	42	=	=	SYM
ejpam-6321	609	43	f1x3k	f1x3k	X
ejpam-6321	609	44	,	,	PUNCT
ejpam-6321	609	45	x3k+2	x3k+2	PUNCT
ejpam-6321	610	1	=	=	SYM
ejpam-6321	610	2	f2x3k+1	f2x3k+1	PROPN
ejpam-6321	610	3	,	,	PUNCT
ejpam-6321	610	4	x3k+3	x3k+3	PUNCT
ejpam-6321	611	1	=	=	NOUN
ejpam-6321	611	2	f3x3k+2	f3x3k+2	PROPN
ejpam-6321	611	3	3	3	NUM
ejpam-6321	611	4	:	:	PUNCT
ejpam-6321	611	5	iterate	iterate	NOUN
ejpam-6321	611	6	:	:	PUNCT
ejpam-6321	611	7	•	•	NUM
ejpam-6321	611	8	compute	compute	NOUN
ejpam-6321	611	9	xk+1	xk+1	NUM
ejpam-6321	611	10	using	use	VERB
ejpam-6321	611	11	the	the	DET
ejpam-6321	611	12	above	above	ADJ
ejpam-6321	611	13	definitions	definition	NOUN
ejpam-6321	611	14	•	•	NOUN
ejpam-6321	611	15	check	check	VERB
ejpam-6321	611	16	contraction	contraction	NOUN
ejpam-6321	611	17	condition	condition	NOUN
ejpam-6321	611	18	:	:	PUNCT
ejpam-6321	611	19	g(xk+1	g(xk+1	NOUN
ejpam-6321	611	20	,	,	PUNCT
ejpam-6321	611	21	xk+2	xk+2	NUM
ejpam-6321	611	22	,	,	PUNCT
ejpam-6321	611	23	xk+3	xk+3	NOUN
ejpam-6321	611	24	)	)	PUNCT
ejpam-6321	611	25	≤	≤	NOUN
ejpam-6321	611	26	γg(xk	γg(xk	PROPN
ejpam-6321	611	27	,	,	PUNCT
ejpam-6321	611	28	xk+1	xk+1	PROPN
ejpam-6321	611	29	,	,	PUNCT
ejpam-6321	611	30	xk+2	xk+2	NUM
ejpam-6321	611	31	)	)	PUNCT
ejpam-6321	611	32	where	where	SCONJ
ejpam-6321	611	33	γ	γ	PROPN
ejpam-6321	611	34	=	=	SYM
ejpam-6321	611	35	max	max	PROPN
ejpam-6321	611	36	(	(	PUNCT
ejpam-6321	611	37	α1	α1	PROPN
ejpam-6321	611	38	+	+	CCONJ
ejpam-6321	611	39	α2	α2	ADJ
ejpam-6321	611	40	+	+	CCONJ
ejpam-6321	611	41	2α4	2α4	NUM
ejpam-6321	611	42	+	+	SYM
ejpam-6321	611	43	α5	α5	NOUN
ejpam-6321	611	44	1−	1−	NUM
ejpam-6321	611	45	α4	α4	NOUN
ejpam-6321	611	46	,	,	PUNCT
ejpam-6321	611	47	α1	α1	PROPN
ejpam-6321	611	48	+	+	CCONJ
ejpam-6321	611	49	α2	α2	ADJ
ejpam-6321	611	50	+	+	CCONJ
ejpam-6321	611	51	2α4	2α4	NUM
ejpam-6321	611	52	1−	1−	NUM
ejpam-6321	611	53	α4	α4	NOUN
ejpam-6321	611	54	−	−	PROPN
ejpam-6321	611	55	α5	α5	NOUN
ejpam-6321	611	56	)	)	PUNCT
ejpam-6321	611	57	<	<	X
ejpam-6321	611	58	1	1	NUM
ejpam-6321	611	59	•	•	NOUN
ejpam-6321	611	60	repeat	repeat	NOUN
ejpam-6321	611	61	until	until	ADP
ejpam-6321	611	62	g(xk	g(xk	NOUN
ejpam-6321	611	63	,	,	PUNCT
ejpam-6321	611	64	xk+1	xk+1	NUM
ejpam-6321	611	65	,	,	PUNCT
ejpam-6321	611	66	xk+2	xk+2	NUM
ejpam-6321	611	67	)	)	PUNCT
ejpam-6321	611	68	<	<	X
ejpam-6321	612	1	ϵ	ϵ	X
ejpam-6321	612	2	for	for	ADP
ejpam-6321	612	3	tolerance	tolerance	NOUN
ejpam-6321	612	4	ϵ	ϵ	X
ejpam-6321	612	5	>	>	X
ejpam-6321	612	6	0	0	NUM
ejpam-6321	612	7	4	4	NUM
ejpam-6321	612	8	:	:	PUNCT
ejpam-6321	612	9	output	output	NOUN
ejpam-6321	612	10	:	:	PUNCT
ejpam-6321	612	11	the	the	DET
ejpam-6321	612	12	limit	limit	NOUN
ejpam-6321	612	13	℘	℘	NOUN
ejpam-6321	612	14	=	=	SYM
ejpam-6321	613	1	limk→∞	limk→∞	ADJ
ejpam-6321	613	2	xk	xk	PROPN
ejpam-6321	613	3	,	,	PUNCT
ejpam-6321	613	4	the	the	DET
ejpam-6321	613	5	unique	unique	ADJ
ejpam-6321	613	6	common	common	ADJ
ejpam-6321	613	7	fixed	fix	VERB
ejpam-6321	613	8	point	point	NOUN
ejpam-6321	613	9	of	of	ADP
ejpam-6321	613	10	f1	f1	NOUN
ejpam-6321	613	11	,	,	PUNCT
ejpam-6321	613	12	f2	f2	PROPN
ejpam-6321	613	13	,	,	PUNCT
ejpam-6321	613	14	f3	f3	PROPN
ejpam-6321	613	15	m.	m.	PROPN
ejpam-6321	613	16	noorwali	noorwali	PROPN
ejpam-6321	613	17	et	et	PROPN
ejpam-6321	613	18	al	al	PROPN
ejpam-6321	613	19	.	.	PUNCT
ejpam-6321	613	20	/	/	SYM
ejpam-6321	613	21	eur	eur	PROPN
ejpam-6321	613	22	.	.	PUNCT
ejpam-6321	614	1	j.	j.	PROPN
ejpam-6321	614	2	pure	pure	PROPN
ejpam-6321	614	3	appl	appl	PROPN
ejpam-6321	614	4	.	.	PROPN
ejpam-6321	614	5	math	math	PROPN
ejpam-6321	614	6	,	,	PUNCT
ejpam-6321	614	7	18	18	NUM
ejpam-6321	614	8	(	(	PUNCT
ejpam-6321	614	9	3	3	NUM
ejpam-6321	614	10	)	)	PUNCT
ejpam-6321	614	11	(	(	PUNCT
ejpam-6321	614	12	2025	2025	NUM
ejpam-6321	614	13	)	)	PUNCT
ejpam-6321	614	14	,	,	PUNCT
ejpam-6321	614	15	6321	6321	NUM
ejpam-6321	614	16	22	22	NUM
ejpam-6321	614	17	of	of	ADP
ejpam-6321	614	18	27	27	NUM
ejpam-6321	614	19	figure	figure	NOUN
ejpam-6321	614	20	3	3	NUM
ejpam-6321	614	21	:	:	PUNCT
ejpam-6321	614	22	convergence	convergence	NOUN
ejpam-6321	614	23	of	of	ADP
ejpam-6321	614	24	fixed	fix	VERB
ejpam-6321	614	25	-	-	PUNCT
ejpam-6321	614	26	point	point	NOUN
ejpam-6321	614	27	iteration	iteration	NOUN
ejpam-6321	614	28	in	in	ADP
ejpam-6321	614	29	g	g	NOUN
ejpam-6321	614	30	-	-	PUNCT
ejpam-6321	614	31	metric	metric	ADJ
ejpam-6321	614	32	space	space	NOUN
ejpam-6321	614	33	with	with	ADP
ejpam-6321	614	34	applications	application	NOUN
ejpam-6321	614	35	in	in	ADP
ejpam-6321	614	36	ai	ai	NOUN
ejpam-6321	614	37	and	and	CCONJ
ejpam-6321	614	38	cryptography	cryptography	NOUN
ejpam-6321	614	39	figure	figure	NOUN
ejpam-6321	614	40	3	3	NUM
ejpam-6321	614	41	depicts	depict	VERB
ejpam-6321	614	42	the	the	DET
ejpam-6321	614	43	convergence	convergence	NOUN
ejpam-6321	614	44	behavior	behavior	NOUN
ejpam-6321	614	45	of	of	ADP
ejpam-6321	614	46	a	a	DET
ejpam-6321	614	47	fixed	fix	VERB
ejpam-6321	614	48	-	-	PUNCT
ejpam-6321	614	49	point	point	NOUN
ejpam-6321	614	50	iteration	iteration	NOUN
ejpam-6321	614	51	scheme	scheme	NOUN
ejpam-6321	614	52	governed	govern	VERB
ejpam-6321	614	53	by	by	ADP
ejpam-6321	614	54	theorem	theorem	NOUN
ejpam-6321	614	55	2	2	NUM
ejpam-6321	614	56	in	in	ADP
ejpam-6321	614	57	a	a	DET
ejpam-6321	614	58	g	g	NOUN
ejpam-6321	614	59	-	-	PUNCT
ejpam-6321	614	60	metric	metric	ADJ
ejpam-6321	614	61	space	space	NOUN
ejpam-6321	614	62	.	.	PUNCT
ejpam-6321	615	1	this	this	DET
ejpam-6321	615	2	visual	visual	ADJ
ejpam-6321	615	3	representation	representation	NOUN
ejpam-6321	615	4	not	not	PART
ejpam-6321	615	5	only	only	ADV
ejpam-6321	615	6	affirms	affirm	VERB
ejpam-6321	615	7	the	the	DET
ejpam-6321	615	8	theoretical	theoretical	ADJ
ejpam-6321	615	9	convergence	convergence	NOUN
ejpam-6321	615	10	conditions	condition	NOUN
ejpam-6321	615	11	but	but	CCONJ
ejpam-6321	615	12	also	also	ADV
ejpam-6321	615	13	demonstrates	demonstrate	VERB
ejpam-6321	615	14	the	the	DET
ejpam-6321	615	15	models	model	NOUN
ejpam-6321	615	16	strength	strength	NOUN
ejpam-6321	615	17	through	through	ADP
ejpam-6321	615	18	real	real	ADJ
ejpam-6321	615	19	-	-	PUNCT
ejpam-6321	615	20	world	world	NOUN
ejpam-6321	615	21	simulations	simulation	NOUN
ejpam-6321	615	22	in	in	ADP
ejpam-6321	615	23	ai	ai	ADJ
ejpam-6321	615	24	consensus	consensus	NOUN
ejpam-6321	615	25	modeling	modeling	NOUN
ejpam-6321	615	26	,	,	PUNCT
ejpam-6321	615	27	cryptographic	cryptographic	ADJ
ejpam-6321	615	28	secure	secure	ADJ
ejpam-6321	615	29	aggregation	aggregation	NOUN
ejpam-6321	615	30	,	,	PUNCT
ejpam-6321	615	31	and	and	CCONJ
ejpam-6321	615	32	fixed	fix	VERB
ejpam-6321	615	33	-	-	PUNCT
ejpam-6321	615	34	point	point	NOUN
ejpam-6321	615	35	learning	learning	NOUN
ejpam-6321	615	36	in	in	ADP
ejpam-6321	615	37	neural	neural	ADJ
ejpam-6321	615	38	networkshighlighting	networkshighlighte	VERB
ejpam-6321	615	39	both	both	CCONJ
ejpam-6321	615	40	the	the	DET
ejpam-6321	615	41	validity	validity	NOUN
ejpam-6321	615	42	and	and	CCONJ
ejpam-6321	615	43	superiority	superiority	NOUN
ejpam-6321	615	44	of	of	ADP
ejpam-6321	615	45	the	the	DET
ejpam-6321	615	46	proposed	propose	VERB
ejpam-6321	615	47	method	method	NOUN
ejpam-6321	615	48	.	.	PUNCT
ejpam-6321	616	1	1	1	X
ejpam-6321	616	2	.	.	X
ejpam-6321	616	3	application	application	NOUN
ejpam-6321	616	4	in	in	ADP
ejpam-6321	616	5	ai	ai	NOUN
ejpam-6321	616	6	:	:	PUNCT
ejpam-6321	616	7	multi	multi	ADJ
ejpam-6321	616	8	-	-	ADJ
ejpam-6321	616	9	agent	agent	ADJ
ejpam-6321	616	10	systems	system	NOUN
ejpam-6321	616	11	•	•	NOUN
ejpam-6321	616	12	scenario	scenario	NOUN
ejpam-6321	616	13	:	:	PUNCT
ejpam-6321	616	14	three	three	NUM
ejpam-6321	616	15	ai	ai	PROPN
ejpam-6321	616	16	agents	agent	NOUN
ejpam-6321	616	17	f1	f1	NOUN
ejpam-6321	616	18	,	,	PUNCT
ejpam-6321	616	19	f2	f2	PROPN
ejpam-6321	616	20	,	,	PUNCT
ejpam-6321	616	21	f3	f3	PROPN
ejpam-6321	616	22	collaboratively	collaboratively	ADV
ejpam-6321	616	23	optimize	optimize	VERB
ejpam-6321	616	24	a	a	DET
ejpam-6321	616	25	shared	share	VERB
ejpam-6321	616	26	objective	objective	NOUN
ejpam-6321	616	27	(	(	PUNCT
ejpam-6321	616	28	e.g.	e.g.	ADV
ejpam-6321	616	29	,	,	PUNCT
ejpam-6321	616	30	in	in	ADP
ejpam-6321	616	31	reinforcement	reinforcement	NOUN
ejpam-6321	616	32	learning	learning	NOUN
ejpam-6321	616	33	)	)	PUNCT
ejpam-6321	616	34	.	.	PUNCT
ejpam-6321	617	1	each	each	DET
ejpam-6321	617	2	agent	agent	NOUN
ejpam-6321	617	3	updates	update	VERB
ejpam-6321	617	4	its	its	PRON
ejpam-6321	617	5	policy	policy	NOUN
ejpam-6321	617	6	based	base	VERB
ejpam-6321	617	7	on	on	ADP
ejpam-6321	617	8	others	other	NOUN
ejpam-6321	617	9	’	’	PART
ejpam-6321	617	10	outputs	output	NOUN
ejpam-6321	617	11	.	.	PUNCT
ejpam-6321	618	1	•	•	NUM
ejpam-6321	618	2	fixed	fix	VERB
ejpam-6321	618	3	-	-	PUNCT
ejpam-6321	618	4	point	point	NOUN
ejpam-6321	618	5	interpretation	interpretation	NOUN
ejpam-6321	618	6	:	:	PUNCT
ejpam-6321	618	7	the	the	DET
ejpam-6321	618	8	theorem	theorem	NOUN
ejpam-6321	618	9	ensures	ensure	VERB
ejpam-6321	618	10	iterative	iterative	NOUN
ejpam-6321	618	11	updates	update	NOUN
ejpam-6321	618	12	converge	converge	VERB
ejpam-6321	618	13	to	to	ADP
ejpam-6321	618	14	a	a	DET
ejpam-6321	618	15	unique	unique	ADJ
ejpam-6321	618	16	shared	share	VERB
ejpam-6321	618	17	solution	solution	NOUN
ejpam-6321	618	18	under	under	ADP
ejpam-6321	618	19	contraction	contraction	NOUN
ejpam-6321	618	20	conditions	condition	NOUN
ejpam-6321	618	21	.	.	PUNCT
ejpam-6321	619	1	•	•	NUM
ejpam-6321	619	2	example	example	NOUN
ejpam-6321	619	3	:	:	PUNCT
ejpam-6321	619	4	in	in	ADP
ejpam-6321	619	5	federated	federated	ADJ
ejpam-6321	619	6	learning	learning	NOUN
ejpam-6321	619	7	,	,	PUNCT
ejpam-6321	619	8	three	three	NUM
ejpam-6321	619	9	models	model	NOUN
ejpam-6321	619	10	f1	f1	NOUN
ejpam-6321	619	11	,	,	PUNCT
ejpam-6321	619	12	f2	f2	PROPN
ejpam-6321	619	13	,	,	PUNCT
ejpam-6321	619	14	f3	f3	PROPN
ejpam-6321	619	15	represent	represent	VERB
ejpam-6321	619	16	local	local	ADJ
ejpam-6321	619	17	updates	update	NOUN
ejpam-6321	619	18	on	on	ADP
ejpam-6321	619	19	different	different	ADJ
ejpam-6321	619	20	devices	device	NOUN
ejpam-6321	619	21	.	.	PUNCT
ejpam-6321	620	1	the	the	DET
ejpam-6321	620	2	theorem	theorem	ADJ
ejpam-6321	620	3	guarantees	guarantee	NOUN
ejpam-6321	620	4	convergence	convergence	NOUN
ejpam-6321	620	5	to	to	ADP
ejpam-6321	620	6	a	a	DET
ejpam-6321	620	7	consensus	consensus	NOUN
ejpam-6321	620	8	model	model	NOUN
ejpam-6321	620	9	.	.	PUNCT
ejpam-6321	621	1	2	2	X
ejpam-6321	621	2	.	.	X
ejpam-6321	621	3	application	application	NOUN
ejpam-6321	621	4	in	in	ADP
ejpam-6321	621	5	cryptography	cryptography	NOUN
ejpam-6321	621	6	:	:	PUNCT
ejpam-6321	621	7	secure	secure	VERB
ejpam-6321	621	8	multi	multi	ADJ
ejpam-6321	621	9	-	-	ADJ
ejpam-6321	621	10	party	party	ADJ
ejpam-6321	621	11	computation	computation	NOUN
ejpam-6321	621	12	•	•	NOUN
ejpam-6321	621	13	scenario	scenario	NOUN
ejpam-6321	621	14	:	:	PUNCT
ejpam-6321	621	15	three	three	NUM
ejpam-6321	621	16	parties	party	NOUN
ejpam-6321	621	17	f1	f1	NOUN
ejpam-6321	621	18	,	,	PUNCT
ejpam-6321	621	19	f2	f2	PROPN
ejpam-6321	621	20	,	,	PUNCT
ejpam-6321	621	21	f3	f3	PROPN
ejpam-6321	621	22	compute	compute	VERB
ejpam-6321	621	23	a	a	DET
ejpam-6321	621	24	function	function	NOUN
ejpam-6321	621	25	f(x	f(x	PROPN
ejpam-6321	621	26	)	)	PUNCT
ejpam-6321	621	27	collaboratively	collaboratively	ADV
ejpam-6321	621	28	without	without	ADP
ejpam-6321	621	29	revealing	reveal	VERB
ejpam-6321	621	30	private	private	ADJ
ejpam-6321	621	31	inputs	input	NOUN
ejpam-6321	621	32	,	,	PUNCT
ejpam-6321	621	33	using	use	VERB
ejpam-6321	621	34	iterative	iterative	NOUN
ejpam-6321	621	35	encrypted	encrypt	VERB
ejpam-6321	621	36	updates	update	NOUN
ejpam-6321	621	37	.	.	PUNCT
ejpam-6321	622	1	•	•	NUM
ejpam-6321	622	2	fixed	fix	VERB
ejpam-6321	622	3	-	-	PUNCT
ejpam-6321	622	4	point	point	NOUN
ejpam-6321	622	5	interpretation	interpretation	NOUN
ejpam-6321	622	6	:	:	PUNCT
ejpam-6321	622	7	the	the	DET
ejpam-6321	622	8	protocol	protocol	NOUN
ejpam-6321	622	9	converges	converge	VERB
ejpam-6321	622	10	to	to	ADP
ejpam-6321	622	11	a	a	DET
ejpam-6321	622	12	unique	unique	ADJ
ejpam-6321	622	13	fixed	fix	VERB
ejpam-6321	622	14	point	point	NOUN
ejpam-6321	622	15	(	(	PUNCT
ejpam-6321	622	16	correct	correct	ADJ
ejpam-6321	622	17	output	output	NOUN
ejpam-6321	622	18	of	of	ADP
ejpam-6321	622	19	f(x	f(x	PROPN
ejpam-6321	622	20	)	)	PUNCT
ejpam-6321	622	21	)	)	PUNCT
ejpam-6321	622	22	even	even	ADV
ejpam-6321	622	23	with	with	ADP
ejpam-6321	622	24	noisy	noisy	ADJ
ejpam-6321	622	25	updates	update	NOUN
ejpam-6321	622	26	,	,	PUNCT
ejpam-6321	622	27	provided	provide	VERB
ejpam-6321	622	28	contraction	contraction	NOUN
ejpam-6321	622	29	conditions	condition	NOUN
ejpam-6321	622	30	hold	hold	VERB
ejpam-6321	622	31	.	.	PUNCT
ejpam-6321	623	1	•	•	NUM
ejpam-6321	623	2	example	example	NOUN
ejpam-6321	623	3	:	:	PUNCT
ejpam-6321	623	4	in	in	ADP
ejpam-6321	623	5	blockchain	blockchain	PROPN
ejpam-6321	623	6	smart	smart	ADJ
ejpam-6321	623	7	contracts	contract	NOUN
ejpam-6321	623	8	,	,	PUNCT
ejpam-6321	623	9	three	three	NUM
ejpam-6321	623	10	oracles	oracle	NOUN
ejpam-6321	623	11	f1	f1	NOUN
ejpam-6321	623	12	,	,	PUNCT
ejpam-6321	623	13	f2	f2	PROPN
ejpam-6321	623	14	,	,	PUNCT
ejpam-6321	623	15	f3	f3	ADJ
ejpam-6321	623	16	aggregate	aggregate	ADJ
ejpam-6321	623	17	data	datum	NOUN
ejpam-6321	623	18	.	.	PUNCT
ejpam-6321	624	1	the	the	DET
ejpam-6321	624	2	theorem	theorem	NOUN
ejpam-6321	624	3	prevents	prevent	VERB
ejpam-6321	624	4	divergence	divergence	ADJ
ejpam-6321	624	5	/	/	SYM
ejpam-6321	624	6	manipulation	manipulation	NOUN
ejpam-6321	624	7	by	by	ADP
ejpam-6321	624	8	ensuring	ensure	VERB
ejpam-6321	624	9	unique	unique	ADJ
ejpam-6321	624	10	convergence	convergence	NOUN
ejpam-6321	624	11	.	.	PUNCT
ejpam-6321	625	1	3	3	X
ejpam-6321	625	2	.	.	X
ejpam-6321	625	3	application	application	NOUN
ejpam-6321	625	4	in	in	ADP
ejpam-6321	625	5	ai	ai	NOUN
ejpam-6321	625	6	:	:	PUNCT
ejpam-6321	625	7	neural	neural	ADJ
ejpam-6321	625	8	network	network	NOUN
ejpam-6321	625	9	fixed	fix	VERB
ejpam-6321	625	10	-	-	PUNCT
ejpam-6321	625	11	point	point	NOUN
ejpam-6321	625	12	learning	learn	VERB
ejpam-6321	625	13	•	•	NUM
ejpam-6321	625	14	scenario	scenario	NOUN
ejpam-6321	625	15	:	:	PUNCT
ejpam-6321	625	16	a	a	DET
ejpam-6321	625	17	neural	neural	ADJ
ejpam-6321	625	18	network	network	NOUN
ejpam-6321	625	19	with	with	ADP
ejpam-6321	625	20	three	three	NUM
ejpam-6321	625	21	sub	sub	ADJ
ejpam-6321	625	22	-	-	ADJ
ejpam-6321	625	23	modules	module	NOUN
ejpam-6321	625	24	f1	f1	NOUN
ejpam-6321	625	25	(	(	PUNCT
ejpam-6321	625	26	encoder	encoder	NOUN
ejpam-6321	625	27	)	)	PUNCT
ejpam-6321	625	28	,	,	PUNCT
ejpam-6321	625	29	f2	f2	PROPN
ejpam-6321	625	30	(	(	PUNCT
ejpam-6321	625	31	decoder	decoder	NOUN
ejpam-6321	625	32	)	)	PUNCT
ejpam-6321	625	33	,	,	PUNCT
ejpam-6321	625	34	and	and	CCONJ
ejpam-6321	625	35	f3	f3	ADJ
ejpam-6321	625	36	(	(	PUNCT
ejpam-6321	625	37	attention	attention	NOUN
ejpam-6321	625	38	mechanism	mechanism	NOUN
ejpam-6321	625	39	)	)	PUNCT
ejpam-6321	625	40	learns	learn	VERB
ejpam-6321	625	41	stable	stable	ADJ
ejpam-6321	625	42	representations	representation	NOUN
ejpam-6321	625	43	.	.	PUNCT
ejpam-6321	626	1	•	•	NUM
ejpam-6321	626	2	fixed	fix	VERB
ejpam-6321	626	3	-	-	PUNCT
ejpam-6321	626	4	point	point	NOUN
ejpam-6321	626	5	interpretation	interpretation	NOUN
ejpam-6321	626	6	:	:	PUNCT
ejpam-6321	626	7	iterative	iterative	NOUN
ejpam-6321	626	8	application	application	NOUN
ejpam-6321	626	9	of	of	ADP
ejpam-6321	626	10	modules	module	NOUN
ejpam-6321	626	11	converges	converge	VERB
ejpam-6321	626	12	to	to	ADP
ejpam-6321	626	13	unique	unique	ADJ
ejpam-6321	626	14	representations	representation	NOUN
ejpam-6321	626	15	under	under	ADP
ejpam-6321	626	16	theorem	theorem	ADJ
ejpam-6321	626	17	conditions	condition	NOUN
ejpam-6321	626	18	,	,	PUNCT
ejpam-6321	626	19	ensuring	ensure	VERB
ejpam-6321	626	20	stability	stability	NOUN
ejpam-6321	626	21	in	in	ADP
ejpam-6321	626	22	deep	deep	ADJ
ejpam-6321	626	23	learning	learning	NOUN
ejpam-6321	626	24	.	.	PUNCT
ejpam-6321	627	1	m.	m.	PROPN
ejpam-6321	627	2	noorwali	noorwali	PROPN
ejpam-6321	627	3	et	et	PROPN
ejpam-6321	627	4	al	al	PROPN
ejpam-6321	627	5	.	.	PUNCT
ejpam-6321	627	6	/	/	SYM
ejpam-6321	627	7	eur	eur	PROPN
ejpam-6321	627	8	.	.	PUNCT
ejpam-6321	628	1	j.	j.	PROPN
ejpam-6321	628	2	pure	pure	PROPN
ejpam-6321	628	3	appl	appl	PROPN
ejpam-6321	628	4	.	.	PROPN
ejpam-6321	628	5	math	math	PROPN
ejpam-6321	628	6	,	,	PUNCT
ejpam-6321	628	7	18	18	NUM
ejpam-6321	628	8	(	(	PUNCT
ejpam-6321	628	9	3	3	NUM
ejpam-6321	628	10	)	)	PUNCT
ejpam-6321	628	11	(	(	PUNCT
ejpam-6321	628	12	2025	2025	NUM
ejpam-6321	628	13	)	)	PUNCT
ejpam-6321	628	14	,	,	PUNCT
ejpam-6321	628	15	6321	6321	NUM
ejpam-6321	628	16	23	23	NUM
ejpam-6321	628	17	of	of	ADP
ejpam-6321	628	18	27	27	NUM
ejpam-6321	628	19	example	example	NOUN
ejpam-6321	628	20	2	2	NUM
ejpam-6321	628	21	.	.	X
ejpam-6321	629	1	let	let	AUX
ejpam-6321	629	2	(	(	PUNCT
ejpam-6321	629	3	x	x	NOUN
ejpam-6321	629	4	,	,	PUNCT
ejpam-6321	629	5	g	g	NOUN
ejpam-6321	629	6	)	)	PUNCT
ejpam-6321	629	7	be	be	AUX
ejpam-6321	629	8	a	a	DET
ejpam-6321	629	9	g	g	NOUN
ejpam-6321	629	10	-	-	PUNCT
ejpam-6321	629	11	metric	metric	ADJ
ejpam-6321	629	12	space	space	NOUN
ejpam-6321	629	13	,	,	PUNCT
ejpam-6321	629	14	where	where	SCONJ
ejpam-6321	629	15	x	x	X
ejpam-6321	629	16	∈	∈	PROPN
ejpam-6321	630	1	[	[	X
ejpam-6321	630	2	0	0	NUM
ejpam-6321	630	3	,	,	PUNCT
ejpam-6321	630	4	1	1	NUM
ejpam-6321	630	5	]	]	PUNCT
ejpam-6321	630	6	and	and	CCONJ
ejpam-6321	630	7	g	g	NOUN
ejpam-6321	630	8	:	:	PUNCT
ejpam-6321	630	9	x	x	X
ejpam-6321	630	10	×x	×x	X
ejpam-6321	630	11	×x	×x	X
ejpam-6321	630	12	→	→	SYM
ejpam-6321	630	13	r	r	NOUN
ejpam-6321	630	14	,	,	PUNCT
ejpam-6321	630	15	with	with	ADP
ejpam-6321	630	16	g(x1	g(x1	NOUN
ejpam-6321	630	17	,	,	PUNCT
ejpam-6321	630	18	x2	x2	PROPN
ejpam-6321	630	19	,	,	PUNCT
ejpam-6321	630	20	x3	x3	ADJ
ejpam-6321	630	21	)	)	PUNCT
ejpam-6321	630	22	=	=	PUNCT
ejpam-6321	630	23	max{|x1	max{|x1	NUM
ejpam-6321	630	24	−	−	PROPN
ejpam-6321	630	25	x2|	x2|	PROPN
ejpam-6321	630	26	,	,	PUNCT
ejpam-6321	630	27	|x2	|x2	ADV
ejpam-6321	630	28	−	−	PROPN
ejpam-6321	630	29	x3|	x3|	PROPN
ejpam-6321	630	30	,	,	PUNCT
ejpam-6321	630	31	|x3	|x3	X
ejpam-6321	630	32	−	−	PROPN
ejpam-6321	631	1	x1|	x1|	X
ejpam-6321	631	2	}	}	PUNCT
ejpam-6321	631	3	,	,	PUNCT
ejpam-6321	631	4	for	for	ADP
ejpam-6321	631	5	all	all	DET
ejpam-6321	631	6	x1	x1	PROPN
ejpam-6321	631	7	,	,	PUNCT
ejpam-6321	631	8	x2	x2	PROPN
ejpam-6321	631	9	,	,	PUNCT
ejpam-6321	631	10	x3	x3	PROPN
ejpam-6321	631	11	∈	∈	PROPN
ejpam-6321	631	12	x.	x.	NOUN
ejpam-6321	631	13	now	now	ADV
ejpam-6321	631	14	we	we	PRON
ejpam-6321	631	15	define	define	VERB
ejpam-6321	631	16	the	the	DET
ejpam-6321	631	17	three	three	NUM
ejpam-6321	631	18	self	self	NOUN
ejpam-6321	631	19	-	-	PUNCT
ejpam-6321	631	20	mappings	mapping	NOUN
ejpam-6321	631	21	,	,	PUNCT
ejpam-6321	631	22	f1	f1	NOUN
ejpam-6321	631	23	,	,	PUNCT
ejpam-6321	631	24	f2	f2	PROPN
ejpam-6321	631	25	,	,	PUNCT
ejpam-6321	631	26	f3	f3	ADJ
ejpam-6321	631	27	:	:	PUNCT
ejpam-6321	631	28	x	x	X
ejpam-6321	631	29	→	→	SYM
ejpam-6321	631	30	xby	xby	PROPN
ejpam-6321	632	1	f1x	f1x	PROPN
ejpam-6321	632	2	=	=	SYM
ejpam-6321	632	3	f2x	f2x	VERB
ejpam-6321	632	4	=	=	SYM
ejpam-6321	632	5	f3x	f3x	NOUN
ejpam-6321	632	6	=	=	PUNCT
ejpam-6321	632	7	{	{	PUNCT
ejpam-6321	632	8	3x	3x	NUM
ejpam-6321	632	9	5	5	NUM
ejpam-6321	632	10	+	+	CCONJ
ejpam-6321	632	11	2	2	NUM
ejpam-6321	632	12	5	5	NUM
ejpam-6321	632	13	forall	forall	NOUN
ejpam-6321	632	14	x	x	SYM
ejpam-6321	632	15	∈	∈	PROPN
ejpam-6321	633	1	[	[	X
ejpam-6321	633	2	0	0	NUM
ejpam-6321	633	3	,	,	PUNCT
ejpam-6321	633	4	1	1	NUM
ejpam-6321	633	5	]	]	SYM
ejpam-6321	633	6	6x+2	6x+2	PROPN
ejpam-6321	633	7	7	7	NUM
ejpam-6321	633	8	for	for	ADP
ejpam-6321	633	9	xin[1,∞	xin[1,∞	PROPN
ejpam-6321	633	10	)	)	PUNCT
ejpam-6321	633	11	(	(	PUNCT
ejpam-6321	633	12	36	36	NUM
ejpam-6321	633	13	)	)	PUNCT
ejpam-6321	633	14	then	then	ADV
ejpam-6321	633	15	,	,	PUNCT
ejpam-6321	633	16	we	we	PRON
ejpam-6321	633	17	get	get	VERB
ejpam-6321	633	18	that	that	PRON
ejpam-6321	633	19	,	,	PUNCT
ejpam-6321	633	20	g(f1x	g(f1x	ADJ
ejpam-6321	633	21	,	,	PUNCT
ejpam-6321	633	22	f2x	f2x	X
ejpam-6321	633	23	,	,	PUNCT
ejpam-6321	633	24	f3x	f3x	NOUN
ejpam-6321	633	25	)	)	PUNCT
ejpam-6321	633	26	=	=	SYM
ejpam-6321	633	27	3	3	NUM
ejpam-6321	633	28	5	5	NUM
ejpam-6321	633	29	max{|x1	max{|x1	NUM
ejpam-6321	633	30	−	−	PROPN
ejpam-6321	633	31	x2|	x2|	PROPN
ejpam-6321	633	32	,	,	PUNCT
ejpam-6321	633	33	|x2	|x2	ADV
ejpam-6321	633	34	−	−	PROPN
ejpam-6321	633	35	x3|	x3|	PROPN
ejpam-6321	633	36	,	,	PUNCT
ejpam-6321	633	37	|x3	|x3	PART
ejpam-6321	633	38	−	−	PUNCT
ejpam-6321	634	1	x1|	x1|	NUM
ejpam-6321	634	2	}	}	PUNCT
ejpam-6321	634	3	g(f1x	g(f1x	PROPN
ejpam-6321	634	4	,	,	PUNCT
ejpam-6321	634	5	f2x	f2x	X
ejpam-6321	634	6	,	,	PUNCT
ejpam-6321	634	7	f3x	f3x	NOUN
ejpam-6321	634	8	)	)	PUNCT
ejpam-6321	634	9	=	=	SYM
ejpam-6321	634	10	3	3	NUM
ejpam-6321	634	11	5	5	NUM
ejpam-6321	634	12	g(x1	g(x1	NOUN
ejpam-6321	634	13	,	,	PUNCT
ejpam-6321	634	14	x2	x2	PROPN
ejpam-6321	634	15	,	,	PUNCT
ejpam-6321	634	16	x3	x3	ADJ
ejpam-6321	634	17	)	)	PUNCT
ejpam-6321	634	18	(	(	PUNCT
ejpam-6321	634	19	37	37	NUM
ejpam-6321	634	20	)	)	PUNCT
ejpam-6321	634	21	now	now	ADV
ejpam-6321	634	22	from	from	ADP
ejpam-6321	634	23	(	(	PUNCT
ejpam-6321	634	24	36	36	NUM
ejpam-6321	634	25	)	)	PUNCT
ejpam-6321	634	26	we	we	PRON
ejpam-6321	634	27	have	have	VERB
ejpam-6321	634	28	that	that	PRON
ejpam-6321	634	29	:	:	PUNCT
ejpam-6321	634	30	g(x1	g(x1	NOUN
ejpam-6321	634	31	,	,	PUNCT
ejpam-6321	634	32	x2	x2	PROPN
ejpam-6321	634	33	,	,	PUNCT
ejpam-6321	634	34	f2x2	f2x2	NOUN
ejpam-6321	634	35	)	)	PUNCT
ejpam-6321	634	36	=	=	SYM
ejpam-6321	634	37	1	1	NUM
ejpam-6321	634	38	5	5	NUM
ejpam-6321	634	39	max{5|x1	max{5|x1	NOUN
ejpam-6321	634	40	−	−	PROPN
ejpam-6321	634	41	x2|	x2|	PROPN
ejpam-6321	634	42	,	,	PUNCT
ejpam-6321	634	43	2|x2	2|x2	NUM
ejpam-6321	635	1	−	−	NOUN
ejpam-6321	635	2	1|	1|	NUM
ejpam-6321	635	3	,	,	PUNCT
ejpam-6321	635	4	|3x2	|3x2	ADV
ejpam-6321	635	5	+	+	ADJ
ejpam-6321	635	6	2−	2−	NUM
ejpam-6321	635	7	5x1|	5x1|	NUM
ejpam-6321	635	8	}	}	PUNCT
ejpam-6321	635	9	,	,	PUNCT
ejpam-6321	635	10	g(f1x1	g(f1x1	PROPN
ejpam-6321	635	11	,	,	PUNCT
ejpam-6321	635	12	x2	x2	PROPN
ejpam-6321	635	13	,	,	PUNCT
ejpam-6321	635	14	x2	x2	PROPN
ejpam-6321	635	15	)	)	PUNCT
ejpam-6321	635	16	=	=	SYM
ejpam-6321	635	17	|f1x1	|f1x1	NOUN
ejpam-6321	635	18	−	−	PROPN
ejpam-6321	635	19	x2|	x2|	PUNCT
ejpam-6321	636	1	=	=	NOUN
ejpam-6321	636	2	1	1	NUM
ejpam-6321	636	3	5	5	NUM
ejpam-6321	636	4	|3x1	|3x1	NOUN
ejpam-6321	636	5	+	+	CCONJ
ejpam-6321	636	6	2−	2−	NUM
ejpam-6321	636	7	5x2|	5x2|	NOUN
ejpam-6321	636	8	,	,	PUNCT
ejpam-6321	636	9	g(f1x1	g(f1x1	PROPN
ejpam-6321	636	10	,	,	PUNCT
ejpam-6321	636	11	x1	x1	PROPN
ejpam-6321	636	12	,	,	PUNCT
ejpam-6321	636	13	x1	x1	PROPN
ejpam-6321	636	14	)	)	PUNCT
ejpam-6321	637	1	=	=	SYM
ejpam-6321	637	2	|x1	|x1	NUM
ejpam-6321	637	3	−	−	PROPN
ejpam-6321	637	4	f1x1|	f1x1|	NOUN
ejpam-6321	637	5	=	=	NOUN
ejpam-6321	637	6	2	2	NUM
ejpam-6321	637	7	5	5	NUM
ejpam-6321	637	8	|x1	|x1	ADP
ejpam-6321	637	9	−	−	PROPN
ejpam-6321	637	10	1|	1|	NUM
ejpam-6321	637	11	,	,	PUNCT
ejpam-6321	637	12	g(x2	g(x2	NOUN
ejpam-6321	637	13	,	,	PUNCT
ejpam-6321	637	14	f2x2	f2x2	X
ejpam-6321	637	15	,	,	PUNCT
ejpam-6321	637	16	x2	x2	PROPN
ejpam-6321	637	17	)	)	PUNCT
ejpam-6321	637	18	=	=	SYM
ejpam-6321	638	1	|x2	|x2	ADJ
ejpam-6321	638	2	−	−	PROPN
ejpam-6321	638	3	f2x2|	f2x2|	PROPN
ejpam-6321	638	4	=	=	NOUN
ejpam-6321	638	5	2	2	NUM
ejpam-6321	638	6	5	5	NUM
ejpam-6321	638	7	|x2	|x2	NOUN
ejpam-6321	638	8	−	−	PROPN
ejpam-6321	638	9	1|	1|	NUM
ejpam-6321	638	10	,	,	PUNCT
ejpam-6321	638	11	g(x3	g(x3	NUM
ejpam-6321	638	12	,	,	PUNCT
ejpam-6321	638	13	x3	x3	ADJ
ejpam-6321	638	14	,	,	PUNCT
ejpam-6321	638	15	f3x3	f3x3	X
ejpam-6321	638	16	)	)	PUNCT
ejpam-6321	638	17	=	=	SYM
ejpam-6321	638	18	|x3	|x3	NUM
ejpam-6321	638	19	−	−	ADP
ejpam-6321	639	1	f3x3|	f3x3|	PROPN
ejpam-6321	639	2	=	=	SYM
ejpam-6321	639	3	2	2	NUM
ejpam-6321	639	4	5	5	NUM
ejpam-6321	639	5	|x3	|x3	NUM
ejpam-6321	639	6	−	−	PROPN
ejpam-6321	639	7	1|	1|	NUM
ejpam-6321	639	8	,	,	PUNCT
ejpam-6321	639	9	g(f1x1	g(f1x1	PROPN
ejpam-6321	639	10	,	,	PUNCT
ejpam-6321	639	11	f1x1	f1x1	NOUN
ejpam-6321	639	12	,	,	PUNCT
ejpam-6321	639	13	x2	x2	PROPN
ejpam-6321	639	14	)	)	PUNCT
ejpam-6321	639	15	=	=	SYM
ejpam-6321	639	16	g(f1x1	g(f1x1	NOUN
ejpam-6321	639	17	,	,	PUNCT
ejpam-6321	639	18	x2	x2	PROPN
ejpam-6321	639	19	,	,	PUNCT
ejpam-6321	639	20	x2	x2	PROPN
ejpam-6321	639	21	)	)	PUNCT
ejpam-6321	639	22	=	=	SYM
ejpam-6321	639	23	|f1x1	|f1x1	NOUN
ejpam-6321	639	24	−	−	PROPN
ejpam-6321	639	25	x2|	x2|	PUNCT
ejpam-6321	640	1	=	=	NOUN
ejpam-6321	640	2	1	1	NUM
ejpam-6321	640	3	5	5	NUM
ejpam-6321	640	4	|3x1	|3x1	NOUN
ejpam-6321	640	5	+	+	CCONJ
ejpam-6321	640	6	2−	2−	NUM
ejpam-6321	640	7	5x2|	5x2|	NOUN
ejpam-6321	640	8	,	,	PUNCT
ejpam-6321	640	9	g(f2x2	g(f2x2	NOUN
ejpam-6321	640	10	,	,	PUNCT
ejpam-6321	640	11	f2x2	f2x2	CCONJ
ejpam-6321	640	12	,	,	PUNCT
ejpam-6321	640	13	x3	x3	ADJ
ejpam-6321	640	14	)	)	PUNCT
ejpam-6321	640	15	=	=	PUNCT
ejpam-6321	640	16	g(f2x2	g(f2x2	NOUN
ejpam-6321	640	17	,	,	PUNCT
ejpam-6321	640	18	x3	x3	ADJ
ejpam-6321	640	19	,	,	PUNCT
ejpam-6321	640	20	x3	x3	ADJ
ejpam-6321	640	21	)	)	PUNCT
ejpam-6321	640	22	=	=	SYM
ejpam-6321	640	23	|f2x2	|f2x2	NOUN
ejpam-6321	640	24	−	−	PROPN
ejpam-6321	640	25	x3|	x3|	PUNCT
ejpam-6321	641	1	=	=	SYM
ejpam-6321	641	2	1	1	NUM
ejpam-6321	641	3	5	5	NUM
ejpam-6321	641	4	|3x2	|3x2	ADV
ejpam-6321	641	5	+	+	CCONJ
ejpam-6321	641	6	2−	2−	NUM
ejpam-6321	641	7	5x3|	5x3|	NUM
ejpam-6321	641	8	.	.	PUNCT
ejpam-6321	642	1	now	now	ADV
ejpam-6321	642	2	from	from	ADP
ejpam-6321	642	3	(	(	PUNCT
ejpam-6321	642	4	8)	8)	NUM
ejpam-6321	642	5	and	and	CCONJ
ejpam-6321	642	6	(	(	PUNCT
ejpam-6321	642	7	37	37	NUM
ejpam-6321	642	8	)	)	PUNCT
ejpam-6321	642	9	we	we	PRON
ejpam-6321	642	10	have	have	VERB
ejpam-6321	642	11	that	that	DET
ejpam-6321	642	12	g(f1x1	g(f1x1	NOUN
ejpam-6321	642	13	,	,	PUNCT
ejpam-6321	642	14	f2x2	f2x2	NOUN
ejpam-6321	642	15	,	,	PUNCT
ejpam-6321	642	16	f3x3	f3x3	X
ejpam-6321	642	17	)	)	PUNCT
ejpam-6321	642	18	=	=	SYM
ejpam-6321	642	19	3	3	NUM
ejpam-6321	642	20	5	5	NUM
ejpam-6321	642	21	g(x1	g(x1	NOUN
ejpam-6321	642	22	,	,	PUNCT
ejpam-6321	642	23	x2	x2	PROPN
ejpam-6321	642	24	,	,	PUNCT
ejpam-6321	642	25	x3	x3	ADJ
ejpam-6321	642	26	)	)	PUNCT
ejpam-6321	642	27	≤	≤	NUM
ejpam-6321	642	28	1	1	NUM
ejpam-6321	642	29	5	5	NUM
ejpam-6321	642	30	(	(	PUNCT
ejpam-6321	642	31	3	3	NUM
ejpam-6321	642	32	5	5	NUM
ejpam-6321	642	33	g(x1	g(x1	NOUN
ejpam-6321	642	34	,	,	PUNCT
ejpam-6321	642	35	x2	x2	PROPN
ejpam-6321	642	36	,	,	PUNCT
ejpam-6321	642	37	x3	x3	ADJ
ejpam-6321	642	38	)	)	PUNCT
ejpam-6321	642	39	)	)	PUNCT
ejpam-6321	643	1	+	+	CCONJ
ejpam-6321	643	2	1	1	NUM
ejpam-6321	643	3	8	8	NUM
ejpam-6321	643	4	(	(	PUNCT
ejpam-6321	643	5	1	1	NUM
ejpam-6321	643	6	5	5	NUM
ejpam-6321	643	7	max{5|x1	max{5|x1	NOUN
ejpam-6321	643	8	−	−	PROPN
ejpam-6321	643	9	x2|	x2|	PROPN
ejpam-6321	643	10	,	,	PUNCT
ejpam-6321	643	11	2|x2	2|x2	NUM
ejpam-6321	643	12	−	−	NOUN
ejpam-6321	644	1	1|	1|	NUM
ejpam-6321	644	2	,	,	PUNCT
ejpam-6321	644	3	|3x2	|3x2	ADV
ejpam-6321	644	4	+	+	ADJ
ejpam-6321	644	5	2−	2−	NUM
ejpam-6321	644	6	5x1|	5x1|	NUM
ejpam-6321	644	7	}	}	PUNCT
ejpam-6321	644	8	)	)	PUNCT
ejpam-6321	645	1	+	+	CCONJ
ejpam-6321	645	2	1	1	NUM
ejpam-6321	645	3	10	10	NUM
ejpam-6321	645	4	(	(	PUNCT
ejpam-6321	645	5	1	1	NUM
ejpam-6321	645	6	5	5	NUM
ejpam-6321	645	7	|3x1	|3x1	NOUN
ejpam-6321	645	8	+	+	CCONJ
ejpam-6321	645	9	2−	2−	NUM
ejpam-6321	645	10	5x2|	5x2|	NUM
ejpam-6321	645	11	)	)	PUNCT
ejpam-6321	646	1	+	+	CCONJ
ejpam-6321	646	2	1	1	NUM
ejpam-6321	646	3	30	30	NUM
ejpam-6321	646	4	(	(	PUNCT
ejpam-6321	646	5	2	2	NUM
ejpam-6321	646	6	5	5	NUM
ejpam-6321	647	1	[	[	X
ejpam-6321	647	2	|x1	|x1	NUM
ejpam-6321	647	3	−	−	PROPN
ejpam-6321	647	4	1|+	1|+	NUM
ejpam-6321	647	5	|x2	|x2	NOUN
ejpam-6321	647	6	−	−	PROPN
ejpam-6321	647	7	1|+	1|+	NUM
ejpam-6321	647	8	|x3	|x3	NUM
ejpam-6321	647	9	−	−	PROPN
ejpam-6321	647	10	1|	1|	NUM
ejpam-6321	647	11	]	]	PUNCT
ejpam-6321	647	12	)	)	PUNCT
ejpam-6321	648	1	+	+	CCONJ
ejpam-6321	648	2	1	1	NUM
ejpam-6321	648	3	40	40	NUM
ejpam-6321	648	4	max	max	NOUN
ejpam-6321	648	5			PROPN
ejpam-6321	648	6	(	(	PUNCT
ejpam-6321	648	7	1	1	NUM
ejpam-6321	648	8	5	5	NUM
ejpam-6321	648	9	max{5|x1	max{5|x1	NOUN
ejpam-6321	648	10	−	−	PROPN
ejpam-6321	648	11	x2|	x2|	PROPN
ejpam-6321	648	12	,	,	PUNCT
ejpam-6321	648	13	2|x2	2|x2	NUM
ejpam-6321	648	14	−	−	NOUN
ejpam-6321	649	1	1|	1|	NUM
ejpam-6321	649	2	,	,	PUNCT
ejpam-6321	649	3	|3x2	|3x2	ADV
ejpam-6321	649	4	+	+	ADJ
ejpam-6321	649	5	2−	2−	NUM
ejpam-6321	649	6	5x1|	5x1|	NUM
ejpam-6321	649	7	}	}	PUNCT
ejpam-6321	649	8	,	,	PUNCT
ejpam-6321	649	9	1	1	NUM
ejpam-6321	649	10	5	5	NUM
ejpam-6321	649	11	|3x1	|3x1	NOUN
ejpam-6321	649	12	+	+	CCONJ
ejpam-6321	649	13	2−	2−	NUM
ejpam-6321	649	14	5x2|	5x2|	NOUN
ejpam-6321	649	15	,	,	PUNCT
ejpam-6321	649	16	1	1	NUM
ejpam-6321	649	17	5	5	NUM
ejpam-6321	649	18	|3x1	|3x1	NOUN
ejpam-6321	649	19	+	+	CCONJ
ejpam-6321	649	20	2−	2−	NUM
ejpam-6321	649	21	5x2|	5x2|	NOUN
ejpam-6321	649	22	,	,	PUNCT
ejpam-6321	649	23	1	1	NUM
ejpam-6321	649	24	5	5	NUM
ejpam-6321	649	25	|3x2	|3x2	ADV
ejpam-6321	649	26	+	+	CCONJ
ejpam-6321	649	27	2−	2−	NUM
ejpam-6321	649	28	5x3|	5x3|	NUM
ejpam-6321	649	29	,	,	PUNCT
ejpam-6321	649	30	2	2	NUM
ejpam-6321	649	31	5	5	NUM
ejpam-6321	649	32	|x1	|x1	ADP
ejpam-6321	649	33	−	−	PROPN
ejpam-6321	649	34	1|	1|	NUM
ejpam-6321	649	35	,	,	PUNCT
ejpam-6321	649	36	2	2	NUM
ejpam-6321	649	37	5	5	NUM
ejpam-6321	649	38	|x2	|x2	NOUN
ejpam-6321	649	39	−	−	PROPN
ejpam-6321	649	40	1|	1|	NUM
ejpam-6321	649	41	,	,	PUNCT
ejpam-6321	649	42	2	2	NUM
ejpam-6321	649	43	5	5	NUM
ejpam-6321	649	44	|x3	|x3	NUM
ejpam-6321	649	45	−	−	PROPN
ejpam-6321	649	46	1|	1|	NUM
ejpam-6321	649	47	)	)	PUNCT
ejpam-6321	649	48			VERB
ejpam-6321	649	49	1	1	NUM
ejpam-6321	649	50	5	5	NUM
ejpam-6321	649	51	g(x1	g(x1	NOUN
ejpam-6321	649	52	,	,	PUNCT
ejpam-6321	649	53	x2	x2	PROPN
ejpam-6321	649	54	,	,	PUNCT
ejpam-6321	649	55	x3	x3	ADJ
ejpam-6321	649	56	)	)	PUNCT
ejpam-6321	649	57	+	+	CCONJ
ejpam-6321	649	58	1	1	NUM
ejpam-6321	649	59	8	8	NUM
ejpam-6321	649	60	g(x1	g(x1	NOUN
ejpam-6321	649	61	,	,	PUNCT
ejpam-6321	649	62	x2	x2	PROPN
ejpam-6321	649	63	,	,	PUNCT
ejpam-6321	649	64	f2x2	f2x2	X
ejpam-6321	649	65	)	)	PUNCT
ejpam-6321	649	66	+	+	CCONJ
ejpam-6321	649	67	1	1	NUM
ejpam-6321	649	68	10	10	NUM
ejpam-6321	649	69	g(f1x1	g(f1x1	NOUN
ejpam-6321	649	70	,	,	PUNCT
ejpam-6321	649	71	x2	x2	PROPN
ejpam-6321	649	72	,	,	PUNCT
ejpam-6321	649	73	x2)+	x2)+	PROPN
ejpam-6321	649	74	1	1	NUM
ejpam-6321	649	75	30	30	NUM
ejpam-6321	649	76	[	[	X
ejpam-6321	649	77	g(f1x1	g(f1x1	PROPN
ejpam-6321	649	78	,	,	PUNCT
ejpam-6321	649	79	x1	x1	PROPN
ejpam-6321	649	80	,	,	PUNCT
ejpam-6321	649	81	x1	x1	PROPN
ejpam-6321	649	82	)	)	PUNCT
ejpam-6321	650	1	+	+	NUM
ejpam-6321	650	2	g(x2	g(x2	NOUN
ejpam-6321	650	3	,	,	PUNCT
ejpam-6321	650	4	f2x2	f2x2	NOUN
ejpam-6321	650	5	,	,	PUNCT
ejpam-6321	650	6	x2	x2	PROPN
ejpam-6321	650	7	)	)	PUNCT
ejpam-6321	651	1	+	+	ADJ
ejpam-6321	651	2	g(x3	g(x3	ADJ
ejpam-6321	651	3	,	,	PUNCT
ejpam-6321	651	4	x3	x3	ADJ
ejpam-6321	651	5	,	,	PUNCT
ejpam-6321	651	6	f3x3	f3x3	X
ejpam-6321	651	7	)	)	PUNCT
ejpam-6321	651	8	]	]	PUNCT
ejpam-6321	652	1	+	+	CCONJ
ejpam-6321	652	2	1	1	NUM
ejpam-6321	652	3	40	40	NUM
ejpam-6321	652	4	max	max	PROPN
ejpam-6321	652	5	{	{	PUNCT
ejpam-6321	652	6	g(x1	g(x1	PROPN
ejpam-6321	652	7	,	,	PUNCT
ejpam-6321	652	8	x2	x2	PROPN
ejpam-6321	652	9	,	,	PUNCT
ejpam-6321	652	10	f2x2	f2x2	NOUN
ejpam-6321	652	11	)	)	PUNCT
ejpam-6321	652	12	,	,	PUNCT
ejpam-6321	652	13	g(f1x1	g(f1x1	PROPN
ejpam-6321	652	14	,	,	PUNCT
ejpam-6321	652	15	f1x1	f1x1	NOUN
ejpam-6321	652	16	,	,	PUNCT
ejpam-6321	652	17	x2	x2	PROPN
ejpam-6321	652	18	)	)	PUNCT
ejpam-6321	652	19	,	,	PUNCT
ejpam-6321	652	20	g(f1x1	g(f1x1	PROPN
ejpam-6321	652	21	,	,	PUNCT
ejpam-6321	652	22	x2	x2	PROPN
ejpam-6321	652	23	,	,	PUNCT
ejpam-6321	652	24	x2	x2	PROPN
ejpam-6321	652	25	)	)	PUNCT
ejpam-6321	652	26	,	,	PUNCT
ejpam-6321	652	27	g(f2x2	g(f2x2	NOUN
ejpam-6321	652	28	,	,	PUNCT
ejpam-6321	652	29	f2x2	f2x2	NOUN
ejpam-6321	652	30	,	,	PUNCT
ejpam-6321	652	31	x3	x3	ADJ
ejpam-6321	652	32	)	)	PUNCT
ejpam-6321	652	33	,	,	PUNCT
ejpam-6321	652	34	g(f1x1	g(f1x1	PROPN
ejpam-6321	652	35	,	,	PUNCT
ejpam-6321	652	36	x1	x1	PROPN
ejpam-6321	652	37	,	,	PUNCT
ejpam-6321	652	38	x1	x1	PROPN
ejpam-6321	652	39	)	)	PUNCT
ejpam-6321	652	40	,	,	PUNCT
ejpam-6321	652	41	g(x2	g(x2	NOUN
ejpam-6321	652	42	,	,	PUNCT
ejpam-6321	652	43	f2x2	f2x2	X
ejpam-6321	652	44	,	,	PUNCT
ejpam-6321	652	45	x2	x2	PROPN
ejpam-6321	652	46	)	)	PUNCT
ejpam-6321	652	47	,	,	PUNCT
ejpam-6321	652	48	g(x3	g(x3	NOUN
ejpam-6321	652	49	,	,	PUNCT
ejpam-6321	652	50	x3	x3	ADJ
ejpam-6321	652	51	,	,	PUNCT
ejpam-6321	652	52	f3x3	f3x3	X
ejpam-6321	652	53	)	)	PUNCT
ejpam-6321	652	54	}	}	PUNCT
ejpam-6321	652	55	hence	hence	ADV
ejpam-6321	652	56	,	,	PUNCT
ejpam-6321	652	57	it	it	PRON
ejpam-6321	652	58	satisfied	satisfy	VERB
ejpam-6321	652	59	all	all	DET
ejpam-6321	652	60	the	the	DET
ejpam-6321	652	61	conditions	condition	NOUN
ejpam-6321	652	62	of	of	ADP
ejpam-6321	652	63	theorem	theorem	NOUN
ejpam-6321	652	64	2	2	NUM
ejpam-6321	652	65	with	with	ADP
ejpam-6321	652	66	α1	α1	PROPN
ejpam-6321	652	67	=	=	SYM
ejpam-6321	652	68	1	1	NUM
ejpam-6321	652	69	5	5	NUM
ejpam-6321	652	70	,	,	PUNCT
ejpam-6321	652	71	α2	α2	NOUN
ejpam-6321	652	72	=	=	SYM
ejpam-6321	652	73	1	1	NUM
ejpam-6321	652	74	8	8	NUM
ejpam-6321	652	75	,	,	PUNCT
ejpam-6321	652	76	α3	α3	NOUN
ejpam-6321	652	77	=	=	SYM
ejpam-6321	652	78	2	2	NUM
ejpam-6321	652	79	5	5	NUM
ejpam-6321	652	80	,	,	PUNCT
ejpam-6321	652	81	α4	α4	NOUN
ejpam-6321	652	82	=	=	SYM
ejpam-6321	652	83	1	1	NUM
ejpam-6321	652	84	20	20	NUM
ejpam-6321	652	85	,	,	PUNCT
ejpam-6321	652	86	α5	α5	NOUN
ejpam-6321	652	87	=	=	SYM
ejpam-6321	652	88	1	1	NUM
ejpam-6321	652	89	40	40	NUM
ejpam-6321	652	90	.	.	PUNCT
ejpam-6321	653	1	i.e.	i.e.	X
ejpam-6321	653	2	α1	α1	PROPN
ejpam-6321	653	3	,	,	PUNCT
ejpam-6321	653	4	α2	α2	ADJ
ejpam-6321	653	5	,	,	PUNCT
ejpam-6321	653	6	α3	α3	NOUN
ejpam-6321	653	7	,	,	PUNCT
ejpam-6321	653	8	α4	α4	NOUN
ejpam-6321	653	9	,	,	PUNCT
ejpam-6321	653	10	α5	α5	NOUN
ejpam-6321	653	11	=	=	SYM
ejpam-6321	653	12	20	20	NUM
ejpam-6321	653	13	40	40	NUM
ejpam-6321	653	14	=	=	SYM
ejpam-6321	653	15	0.5	0.5	NUM
ejpam-6321	653	16	<	<	SYM
ejpam-6321	653	17	1	1	NUM
ejpam-6321	653	18	and	and	CCONJ
ejpam-6321	653	19	α1	α1	PROPN
ejpam-6321	653	20	,	,	PUNCT
ejpam-6321	653	21	α2	α2	ADJ
ejpam-6321	653	22	,	,	PUNCT
ejpam-6321	653	23	α3	α3	NOUN
ejpam-6321	653	24	,	,	PUNCT
ejpam-6321	653	25	α4	α4	NOUN
ejpam-6321	653	26	,	,	PUNCT
ejpam-6321	653	27	α5	α5	NOUN
ejpam-6321	653	28	=	=	SYM
ejpam-6321	653	29	30	30	NUM
ejpam-6321	653	30	40	40	NUM
ejpam-6321	653	31	=	=	SYM
ejpam-6321	653	32	0.75	0.75	NUM
ejpam-6321	653	33	<	<	X
ejpam-6321	653	34	1	1	NUM
ejpam-6321	653	35	,	,	PUNCT
ejpam-6321	653	36	under	under	ADP
ejpam-6321	653	37	these	these	DET
ejpam-6321	653	38	conditions	condition	NOUN
ejpam-6321	653	39	,	,	PUNCT
ejpam-6321	653	40	the	the	DET
ejpam-6321	653	41	three	three	NUM
ejpam-6321	653	42	individual	individual	ADJ
ejpam-6321	653	43	self	self	NOUN
ejpam-6321	653	44	-	-	PUNCT
ejpam-6321	653	45	mappings	mapping	NOUN
ejpam-6321	653	46	referred	refer	VERB
ejpam-6321	653	47	as	as	ADP
ejpam-6321	653	48	f1	f1	NOUN
ejpam-6321	653	49	,	,	PUNCT
ejpam-6321	653	50	f2	f2	PROPN
ejpam-6321	653	51	and	and	CCONJ
ejpam-6321	653	52	f3	f3	PROPN
ejpam-6321	653	53	possess	possess	VERB
ejpam-6321	653	54	a	a	DET
ejpam-6321	653	55	unique	unique	ADJ
ejpam-6321	653	56	cfp	cfp	NOUN
ejpam-6321	653	57	within	within	ADP
ejpam-6321	653	58	f	f	PROPN
ejpam-6321	653	59	that	that	PRON
ejpam-6321	653	60	is	be	AUX
ejpam-6321	653	61	2	2	NUM
ejpam-6321	653	62	∈	∈	NOUN
ejpam-6321	654	1	[	[	X
ejpam-6321	654	2	0,∞	0,∞	NOUN
ejpam-6321	654	3	)	)	PUNCT
ejpam-6321	654	4	.	.	PUNCT
ejpam-6321	655	1	m.	m.	PROPN
ejpam-6321	655	2	noorwali	noorwali	PROPN
ejpam-6321	655	3	et	et	PROPN
ejpam-6321	655	4	al	al	PROPN
ejpam-6321	655	5	.	.	PUNCT
ejpam-6321	655	6	/	/	SYM
ejpam-6321	655	7	eur	eur	PROPN
ejpam-6321	655	8	.	.	PUNCT
ejpam-6321	656	1	j.	j.	PROPN
ejpam-6321	656	2	pure	pure	PROPN
ejpam-6321	656	3	appl	appl	PROPN
ejpam-6321	656	4	.	.	PROPN
ejpam-6321	656	5	math	math	PROPN
ejpam-6321	656	6	,	,	PUNCT
ejpam-6321	656	7	18	18	NUM
ejpam-6321	656	8	(	(	PUNCT
ejpam-6321	656	9	3	3	NUM
ejpam-6321	656	10	)	)	PUNCT
ejpam-6321	656	11	(	(	PUNCT
ejpam-6321	656	12	2025	2025	NUM
ejpam-6321	656	13	)	)	PUNCT
ejpam-6321	656	14	,	,	PUNCT
ejpam-6321	656	15	6321	6321	NUM
ejpam-6321	656	16	24	24	NUM
ejpam-6321	656	17	of	of	ADP
ejpam-6321	656	18	27	27	NUM
ejpam-6321	656	19	example	example	NOUN
ejpam-6321	656	20	3	3	NUM
ejpam-6321	656	21	.	.	X
ejpam-6321	657	1	let	let	AUX
ejpam-6321	657	2	(	(	PUNCT
ejpam-6321	657	3	x	x	NOUN
ejpam-6321	657	4	,	,	PUNCT
ejpam-6321	657	5	g	g	NOUN
ejpam-6321	657	6	)	)	PUNCT
ejpam-6321	657	7	be	be	AUX
ejpam-6321	657	8	a	a	DET
ejpam-6321	657	9	g	g	NOUN
ejpam-6321	657	10	-	-	PUNCT
ejpam-6321	657	11	metric	metric	ADJ
ejpam-6321	657	12	space	space	NOUN
ejpam-6321	657	13	,	,	PUNCT
ejpam-6321	657	14	where	where	SCONJ
ejpam-6321	657	15	x	x	X
ejpam-6321	657	16	∈	∈	PROPN
ejpam-6321	658	1	[	[	X
ejpam-6321	658	2	0	0	NUM
ejpam-6321	658	3	,	,	PUNCT
ejpam-6321	658	4	2	2	NUM
ejpam-6321	658	5	]	]	PUNCT
ejpam-6321	658	6	represents	represent	VERB
ejpam-6321	658	7	either	either	CCONJ
ejpam-6321	658	8	normalized	normalize	VERB
ejpam-6321	658	9	neural	neural	ADJ
ejpam-6321	658	10	activations	activation	NOUN
ejpam-6321	658	11	in	in	ADP
ejpam-6321	658	12	an	an	DET
ejpam-6321	658	13	ai	ai	NOUN
ejpam-6321	658	14	interpretation	interpretation	NOUN
ejpam-6321	658	15	or	or	CCONJ
ejpam-6321	658	16	scaled	scale	VERB
ejpam-6321	658	17	cryptographic	cryptographic	ADJ
ejpam-6321	658	18	message	message	NOUN
ejpam-6321	658	19	values	value	NOUN
ejpam-6321	658	20	in	in	ADP
ejpam-6321	658	21	a	a	DET
ejpam-6321	658	22	cryptographic	cryptographic	ADJ
ejpam-6321	658	23	interpretation	interpretation	NOUN
ejpam-6321	658	24	.	.	PUNCT
ejpam-6321	659	1	the	the	DET
ejpam-6321	659	2	g	g	NOUN
ejpam-6321	659	3	-	-	PUNCT
ejpam-6321	659	4	metric	metric	ADJ
ejpam-6321	659	5	is	be	AUX
ejpam-6321	659	6	defined	define	VERB
ejpam-6321	659	7	as	as	ADP
ejpam-6321	659	8	:	:	PUNCT
ejpam-6321	659	9	g(x1	g(x1	NOUN
ejpam-6321	659	10	,	,	PUNCT
ejpam-6321	659	11	x2	x2	PROPN
ejpam-6321	659	12	,	,	PUNCT
ejpam-6321	659	13	x3	x3	ADJ
ejpam-6321	659	14	)	)	PUNCT
ejpam-6321	660	1	=	=	PUNCT
ejpam-6321	660	2	max{|x1	max{|x1	NUM
ejpam-6321	660	3	−	−	PROPN
ejpam-6321	660	4	x2|	x2|	PROPN
ejpam-6321	660	5	,	,	PUNCT
ejpam-6321	660	6	|x2	|x2	ADV
ejpam-6321	660	7	−	−	PROPN
ejpam-6321	660	8	x3|	x3|	PROPN
ejpam-6321	660	9	,	,	PUNCT
ejpam-6321	660	10	|x3	|x3	X
ejpam-6321	660	11	−	−	PROPN
ejpam-6321	661	1	x1|	x1|	PRON
ejpam-6321	661	2	}	}	PUNCT
ejpam-6321	661	3	define	define	VERB
ejpam-6321	661	4	three	three	NUM
ejpam-6321	661	5	contractive	contractive	ADJ
ejpam-6321	661	6	mappings	mapping	NOUN
ejpam-6321	661	7	with	with	ADP
ejpam-6321	661	8	modified	modified	ADJ
ejpam-6321	661	9	parameters	parameter	NOUN
ejpam-6321	661	10	:	:	PUNCT
ejpam-6321	661	11	fix	fix	NOUN
ejpam-6321	661	12	=	=	SYM
ejpam-6321	661	13	{	{	PUNCT
ejpam-6321	661	14	2x	2x	NUM
ejpam-6321	661	15	7	7	NUM
ejpam-6321	661	16	+	+	CCONJ
ejpam-6321	661	17	3	3	NUM
ejpam-6321	661	18	7	7	NUM
ejpam-6321	661	19	for	for	ADP
ejpam-6321	661	20	x	x	PROPN
ejpam-6321	661	21	∈	∈	PROPN
ejpam-6321	662	1	[	[	X
ejpam-6321	662	2	0	0	NUM
ejpam-6321	662	3	,	,	PUNCT
ejpam-6321	662	4	2	2	NUM
ejpam-6321	662	5	]	]	SYM
ejpam-6321	662	6	5x+1	5x+1	NUM
ejpam-6321	662	7	8	8	NUM
ejpam-6321	662	8	for	for	ADP
ejpam-6321	662	9	x	x	PROPN
ejpam-6321	662	10	∈	∈	PROPN
ejpam-6321	662	11	(	(	PUNCT
ejpam-6321	662	12	2,∞	2,∞	NUM
ejpam-6321	662	13	)	)	PUNCT
ejpam-6321	662	14	for	for	ADP
ejpam-6321	662	15	i	i	PROPN
ejpam-6321	662	16	=	=	SYM
ejpam-6321	662	17	1	1	NUM
ejpam-6321	662	18	,	,	PUNCT
ejpam-6321	662	19	2	2	NUM
ejpam-6321	662	20	,	,	PUNCT
ejpam-6321	662	21	3	3	NUM
ejpam-6321	662	22	for	for	ADP
ejpam-6321	662	23	x1	x1	PROPN
ejpam-6321	662	24	,	,	PUNCT
ejpam-6321	662	25	x2	x2	PROPN
ejpam-6321	662	26	,	,	PUNCT
ejpam-6321	662	27	x3	x3	PROPN
ejpam-6321	662	28	∈	∈	PROPN
ejpam-6321	663	1	[	[	X
ejpam-6321	663	2	0	0	NUM
ejpam-6321	663	3	,	,	PUNCT
ejpam-6321	663	4	2	2	NUM
ejpam-6321	663	5	]	]	PUNCT
ejpam-6321	663	6	,	,	PUNCT
ejpam-6321	663	7	we	we	PRON
ejpam-6321	663	8	verify	verify	VERB
ejpam-6321	663	9	:	:	PUNCT
ejpam-6321	663	10	g(f1x	g(f1x	ADJ
ejpam-6321	663	11	,	,	PUNCT
ejpam-6321	663	12	f2x	f2x	X
ejpam-6321	663	13	,	,	PUNCT
ejpam-6321	663	14	f3x	f3x	NOUN
ejpam-6321	663	15	)	)	PUNCT
ejpam-6321	663	16	=	=	SYM
ejpam-6321	663	17	2	2	NUM
ejpam-6321	663	18	7	7	NUM
ejpam-6321	663	19	max{|x1	max{|x1	NUM
ejpam-6321	663	20	−	−	PROPN
ejpam-6321	663	21	x2|	x2|	PROPN
ejpam-6321	663	22	,	,	PUNCT
ejpam-6321	663	23	|x2	|x2	ADV
ejpam-6321	663	24	−	−	PROPN
ejpam-6321	663	25	x3|	x3|	PROPN
ejpam-6321	663	26	,	,	PUNCT
ejpam-6321	663	27	|x3	|x3	PART
ejpam-6321	663	28	−	−	PUNCT
ejpam-6321	664	1	x1|	x1|	PRON
ejpam-6321	664	2	}	}	PUNCT
ejpam-6321	664	3	=	=	SYM
ejpam-6321	664	4	2	2	NUM
ejpam-6321	664	5	7	7	NUM
ejpam-6321	664	6	g(x1	g(x1	NOUN
ejpam-6321	664	7	,	,	PUNCT
ejpam-6321	664	8	x2	x2	PROPN
ejpam-6321	664	9	,	,	PUNCT
ejpam-6321	664	10	x3	x3	ADJ
ejpam-6321	664	11	)	)	PUNCT
ejpam-6321	664	12	(	(	PUNCT
ejpam-6321	664	13	improved	improve	VERB
ejpam-6321	664	14	contraction	contraction	NOUN
ejpam-6321	664	15	)	)	PUNCT
ejpam-6321	664	16	for	for	ADP
ejpam-6321	664	17	cryptographic	cryptographic	ADJ
ejpam-6321	664	18	protocols	protocol	NOUN
ejpam-6321	664	19	:	:	PUNCT
ejpam-6321	664	20	g(x1	g(x1	NOUN
ejpam-6321	664	21	,	,	PUNCT
ejpam-6321	664	22	x2	x2	PROPN
ejpam-6321	664	23	,	,	PUNCT
ejpam-6321	664	24	f2x2	f2x2	NOUN
ejpam-6321	664	25	)	)	PUNCT
ejpam-6321	664	26	=	=	SYM
ejpam-6321	664	27	1	1	NUM
ejpam-6321	664	28	7	7	NUM
ejpam-6321	664	29	max{7|x1	max{7|x1	NOUN
ejpam-6321	664	30	−	−	PROPN
ejpam-6321	664	31	x2|	x2|	PROPN
ejpam-6321	664	32	,	,	PUNCT
ejpam-6321	664	33	3|x2	3|x2	NUM
ejpam-6321	664	34	−	−	PROPN
ejpam-6321	664	35	1|	1|	NUM
ejpam-6321	664	36	,	,	PUNCT
ejpam-6321	664	37	|2x2	|2x2	NOUN
ejpam-6321	664	38	+	+	CCONJ
ejpam-6321	664	39	3−	3−	NUM
ejpam-6321	664	40	7x1|	7x1|	NOUN
ejpam-6321	664	41	}	}	PUNCT
ejpam-6321	664	42	g(f1x1	g(f1x1	NOUN
ejpam-6321	664	43	,	,	PUNCT
ejpam-6321	664	44	x2	x2	PROPN
ejpam-6321	664	45	,	,	PUNCT
ejpam-6321	664	46	x2	x2	PROPN
ejpam-6321	664	47	)	)	PUNCT
ejpam-6321	664	48	=	=	SYM
ejpam-6321	664	49	1	1	NUM
ejpam-6321	664	50	7	7	NUM
ejpam-6321	664	51	|2x1	|2x1	NOUN
ejpam-6321	664	52	+	+	CCONJ
ejpam-6321	664	53	3−	3−	NUM
ejpam-6321	664	54	7x2|	7x2|	NOUN
ejpam-6321	664	55	(	(	PUNCT
ejpam-6321	664	56	tighter	tight	ADJ
ejpam-6321	664	57	bound	bind	VERB
ejpam-6321	664	58	)	)	PUNCT
ejpam-6321	664	59	the	the	DET
ejpam-6321	664	60	modified	modify	VERB
ejpam-6321	664	61	coefficients	coefficient	NOUN
ejpam-6321	664	62	satisfy	satisfy	VERB
ejpam-6321	664	63	:	:	PUNCT
ejpam-6321	664	64	1	1	NUM
ejpam-6321	664	65	7	7	NUM
ejpam-6321	664	66	g(x1	g(x1	NOUN
ejpam-6321	664	67	,	,	PUNCT
ejpam-6321	664	68	x2	x2	PROPN
ejpam-6321	664	69	,	,	PUNCT
ejpam-6321	664	70	x3	x3	ADJ
ejpam-6321	664	71	)	)	PUNCT
ejpam-6321	664	72	+	+	CCONJ
ejpam-6321	664	73	1	1	NUM
ejpam-6321	664	74	10	10	NUM
ejpam-6321	664	75	g(x1	g(x1	NOUN
ejpam-6321	664	76	,	,	PUNCT
ejpam-6321	664	77	x2	x2	PROPN
ejpam-6321	664	78	,	,	PUNCT
ejpam-6321	664	79	f2x2	f2x2	X
ejpam-6321	664	80	)	)	PUNCT
ejpam-6321	664	81	+	+	CCONJ
ejpam-6321	664	82	1	1	NUM
ejpam-6321	664	83	14	14	NUM
ejpam-6321	664	84	g(f1x1	g(f1x1	NOUN
ejpam-6321	664	85	,	,	PUNCT
ejpam-6321	664	86	x2	x2	PROPN
ejpam-6321	664	87	,	,	PUNCT
ejpam-6321	664	88	x2	x2	PROPN
ejpam-6321	664	89	)	)	PUNCT
ejpam-6321	665	1	+	+	CCONJ
ejpam-6321	665	2	1	1	NUM
ejpam-6321	665	3	35	35	NUM
ejpam-6321	665	4	[	[	X
ejpam-6321	665	5	g(f1x1	g(f1x1	PROPN
ejpam-6321	665	6	,	,	PUNCT
ejpam-6321	665	7	x1	x1	PROPN
ejpam-6321	665	8	,	,	PUNCT
ejpam-6321	665	9	x1	x1	PROPN
ejpam-6321	665	10	)	)	PUNCT
ejpam-6321	666	1	+	+	NUM
ejpam-6321	666	2	g(x2	g(x2	NOUN
ejpam-6321	666	3	,	,	PUNCT
ejpam-6321	666	4	f2x2	f2x2	NOUN
ejpam-6321	666	5	,	,	PUNCT
ejpam-6321	666	6	x2	x2	PROPN
ejpam-6321	666	7	)	)	PUNCT
ejpam-6321	667	1	+	+	ADJ
ejpam-6321	667	2	g(x3	g(x3	ADJ
ejpam-6321	667	3	,	,	PUNCT
ejpam-6321	667	4	x3	x3	ADJ
ejpam-6321	667	5	,	,	PUNCT
ejpam-6321	667	6	f3x3	f3x3	X
ejpam-6321	667	7	)	)	PUNCT
ejpam-6321	667	8	]	]	PUNCT
ejpam-6321	668	1	+	+	CCONJ
ejpam-6321	668	2	1	1	NUM
ejpam-6321	668	3	50	50	NUM
ejpam-6321	668	4	max	max	NOUN
ejpam-6321	668	5			PROPN
ejpam-6321	668	6	g(x1	g(x1	NOUN
ejpam-6321	668	7	,	,	PUNCT
ejpam-6321	668	8	x2	x2	PROPN
ejpam-6321	668	9	,	,	PUNCT
ejpam-6321	668	10	f2x2	f2x2	NOUN
ejpam-6321	668	11	)	)	PUNCT
ejpam-6321	668	12	,	,	PUNCT
ejpam-6321	668	13	g(f1x1	g(f1x1	PROPN
ejpam-6321	668	14	,	,	PUNCT
ejpam-6321	668	15	f1x1	f1x1	NOUN
ejpam-6321	668	16	,	,	PUNCT
ejpam-6321	668	17	x2	x2	PROPN
ejpam-6321	668	18	)	)	PUNCT
ejpam-6321	668	19	,	,	PUNCT
ejpam-6321	668	20	g(f1x1	g(f1x1	PROPN
ejpam-6321	668	21	,	,	PUNCT
ejpam-6321	668	22	x2	x2	PROPN
ejpam-6321	668	23	,	,	PUNCT
ejpam-6321	668	24	x2	x2	PROPN
ejpam-6321	668	25	)	)	PUNCT
ejpam-6321	668	26	,	,	PUNCT
ejpam-6321	668	27	g(f2x2	g(f2x2	NOUN
ejpam-6321	668	28	,	,	PUNCT
ejpam-6321	668	29	f2x2	f2x2	NOUN
ejpam-6321	668	30	,	,	PUNCT
ejpam-6321	668	31	x3	x3	ADJ
ejpam-6321	668	32	)	)	PUNCT
ejpam-6321	668	33	,	,	PUNCT
ejpam-6321	668	34	g(f1x1	g(f1x1	PROPN
ejpam-6321	668	35	,	,	PUNCT
ejpam-6321	668	36	x1	x1	PROPN
ejpam-6321	668	37	,	,	PUNCT
ejpam-6321	668	38	x1	x1	PROPN
ejpam-6321	668	39	)	)	PUNCT
ejpam-6321	668	40	,	,	PUNCT
ejpam-6321	668	41	g(x2	g(x2	NOUN
ejpam-6321	668	42	,	,	PUNCT
ejpam-6321	668	43	f2x2	f2x2	X
ejpam-6321	668	44	,	,	PUNCT
ejpam-6321	668	45	x2	x2	PROPN
ejpam-6321	668	46	)	)	PUNCT
ejpam-6321	668	47	,	,	PUNCT
ejpam-6321	668	48	g(x3	g(x3	NOUN
ejpam-6321	668	49	,	,	PUNCT
ejpam-6321	668	50	x3	x3	ADJ
ejpam-6321	668	51	,	,	PUNCT
ejpam-6321	668	52	f3x3	f3x3	X
ejpam-6321	668	53	)	)	PUNCT
ejpam-6321	668	54			X
ejpam-6321	668	55	<	<	X
ejpam-6321	668	56	1	1	NUM
ejpam-6321	668	57	the	the	DET
ejpam-6321	668	58	new	new	ADJ
ejpam-6321	668	59	parameters	parameter	NOUN
ejpam-6321	668	60	satisfy	satisfy	VERB
ejpam-6321	668	61	theorem	theorem	VERB
ejpam-6321	668	62	2	2	NUM
ejpam-6321	668	63	with	with	ADP
ejpam-6321	668	64	:	:	PUNCT
ejpam-6321	668	65	α1	α1	PROPN
ejpam-6321	668	66	=	=	SYM
ejpam-6321	668	67	1	1	NUM
ejpam-6321	668	68	7	7	NUM
ejpam-6321	668	69	,	,	PUNCT
ejpam-6321	668	70	α2	α2	NOUN
ejpam-6321	668	71	=	=	SYM
ejpam-6321	668	72	1	1	NUM
ejpam-6321	668	73	10	10	NUM
ejpam-6321	668	74	,	,	PUNCT
ejpam-6321	668	75	α3	α3	NOUN
ejpam-6321	668	76	=	=	SYM
ejpam-6321	668	77	2	2	NUM
ejpam-6321	668	78	7	7	NUM
ejpam-6321	668	79	,	,	PUNCT
ejpam-6321	668	80	α4	α4	NOUN
ejpam-6321	668	81	=	=	SYM
ejpam-6321	668	82	1	1	NUM
ejpam-6321	668	83	35	35	NUM
ejpam-6321	668	84	,	,	PUNCT
ejpam-6321	668	85	α5	α5	NOUN
ejpam-6321	668	86	=	=	SYM
ejpam-6321	668	87	1	1	NUM
ejpam-6321	668	88	50	50	NUM
ejpam-6321	668	89	the	the	DET
ejpam-6321	668	90	unique	unique	ADJ
ejpam-6321	668	91	common	common	ADJ
ejpam-6321	668	92	fixed	fix	VERB
ejpam-6321	668	93	point	point	NOUN
ejpam-6321	668	94	℘	℘	NOUN
ejpam-6321	668	95	=	=	SYM
ejpam-6321	668	96	1	1	NUM
ejpam-6321	668	97	(	(	PUNCT
ejpam-6321	668	98	solved	solve	VERB
ejpam-6321	668	99	by	by	ADP
ejpam-6321	668	100	setting	set	VERB
ejpam-6321	668	101	x	x	PUNCT
ejpam-6321	668	102	=	=	NOUN
ejpam-6321	668	103	2x	2x	NUM
ejpam-6321	668	104	7	7	NUM
ejpam-6321	668	105	+	+	CCONJ
ejpam-6321	668	106	3	3	NUM
ejpam-6321	668	107	7	7	NUM
ejpam-6321	668	108	)	)	PUNCT
ejpam-6321	668	109	represents	represent	VERB
ejpam-6321	668	110	:	:	PUNCT
ejpam-6321	668	111	the	the	DET
ejpam-6321	668	112	fixedpoint	fixedpoint	NOUN
ejpam-6321	668	113	result	result	NOUN
ejpam-6321	668	114	yields	yield	VERB
ejpam-6321	668	115	a	a	DET
ejpam-6321	668	116	stable	stable	ADJ
ejpam-6321	668	117	activation	activation	NOUN
ejpam-6321	668	118	value	value	NOUN
ejpam-6321	668	119	of	of	ADP
ejpam-6321	668	120	1.0	1.0	NUM
ejpam-6321	668	121	in	in	ADP
ejpam-6321	668	122	neural	neural	ADJ
ejpam-6321	668	123	networks	network	NOUN
ejpam-6321	668	124	and	and	CCONJ
ejpam-6321	668	125	a	a	DET
ejpam-6321	668	126	secure	secure	ADJ
ejpam-6321	668	127	consensus	consensus	NOUN
ejpam-6321	668	128	value	value	NOUN
ejpam-6321	668	129	in	in	ADP
ejpam-6321	668	130	cryptographic	cryptographic	ADJ
ejpam-6321	668	131	protocols	protocol	NOUN
ejpam-6321	668	132	.	.	PUNCT
ejpam-6321	669	1	algorithm	algorithm	NOUN
ejpam-6321	669	2	4	4	NUM
ejpam-6321	669	3	fixed	fix	VERB
ejpam-6321	669	4	-	-	PUNCT
ejpam-6321	669	5	point	point	NOUN
ejpam-6321	669	6	computation	computation	NOUN
ejpam-6321	669	7	for	for	ADP
ejpam-6321	669	8	ai	ai	NOUN
ejpam-6321	669	9	/	/	SYM
ejpam-6321	669	10	crypto	crypto	NOUN
ejpam-6321	669	11	systems	system	NOUN
ejpam-6321	669	12	1	1	NUM
ejpam-6321	669	13	:	:	PUNCT
ejpam-6321	669	14	initialize	initialize	VERB
ejpam-6321	669	15	x0	x0	PROPN
ejpam-6321	669	16	∈	∈	PROPN
ejpam-6321	670	1	[	[	X
ejpam-6321	670	2	0	0	NUM
ejpam-6321	670	3	,	,	PUNCT
ejpam-6321	670	4	1	1	NUM
ejpam-6321	670	5	]	]	PUNCT
ejpam-6321	670	6	(	(	PUNCT
ejpam-6321	670	7	input	input	NOUN
ejpam-6321	670	8	data	datum	NOUN
ejpam-6321	670	9	/	/	SYM
ejpam-6321	670	10	encrypted	encrypt	VERB
ejpam-6321	670	11	message	message	NOUN
ejpam-6321	670	12	)	)	PUNCT
ejpam-6321	670	13	2	2	NUM
ejpam-6321	670	14	:	:	PUNCT
ejpam-6321	670	15	for	for	ADP
ejpam-6321	670	16	k	k	PROPN
ejpam-6321	670	17	=	=	SYM
ejpam-6321	670	18	0	0	PROPN
ejpam-6321	670	19	to	to	ADP
ejpam-6321	670	20	max	max	PROPN
ejpam-6321	670	21	iter	iter	NOUN
ejpam-6321	670	22	do	do	VERB
ejpam-6321	670	23	3	3	NUM
ejpam-6321	670	24	:	:	PUNCT
ejpam-6321	670	25	xk+1	xk+1	PROPN
ejpam-6321	670	26	←	←	PROPN
ejpam-6321	670	27	f3(f2(f1(xk	f3(f2(f1(xk	PROPN
ejpam-6321	670	28	)	)	PUNCT
ejpam-6321	670	29	)	)	PUNCT
ejpam-6321	670	30	)	)	PUNCT
ejpam-6321	671	1	▷	▷	ADV
ejpam-6321	671	2	ai	ai	VERB
ejpam-6321	671	3	:	:	PUNCT
ejpam-6321	671	4	layer	layer	NOUN
ejpam-6321	671	5	composition	composition	NOUN
ejpam-6321	671	6	/	/	SYM
ejpam-6321	671	7	crypto	crypto	NOUN
ejpam-6321	671	8	:	:	PUNCT
ejpam-6321	671	9	protocol	protocol	NOUN
ejpam-6321	671	10	round	round	ADV
ejpam-6321	671	11	4	4	NUM
ejpam-6321	671	12	:	:	PUNCT
ejpam-6321	671	13	if	if	SCONJ
ejpam-6321	671	14	g(xk	g(xk	NOUN
ejpam-6321	671	15	,	,	PUNCT
ejpam-6321	671	16	xk+1	xk+1	NUM
ejpam-6321	671	17	,	,	PUNCT
ejpam-6321	671	18	xk+2	xk+2	NUM
ejpam-6321	671	19	)	)	PUNCT
ejpam-6321	671	20	<	<	X
ejpam-6321	672	1	ϵ	ϵ	X
ejpam-6321	672	2	then	then	ADV
ejpam-6321	672	3	5	5	NUM
ejpam-6321	672	4	:	:	PUNCT
ejpam-6321	672	5	return	return	NOUN
ejpam-6321	672	6	xk+1	xk+1	PROPN
ejpam-6321	673	1	▷	▷	ADV
ejpam-6321	673	2	fixed	fix	VERB
ejpam-6321	673	3	point	point	NOUN
ejpam-6321	673	4	reached	reach	VERB
ejpam-6321	673	5	6	6	NUM
ejpam-6321	673	6	:	:	PUNCT
ejpam-6321	673	7	end	end	VERB
ejpam-6321	673	8	if	if	SCONJ
ejpam-6321	673	9	7	7	NUM
ejpam-6321	673	10	:	:	PUNCT
ejpam-6321	673	11	end	end	NOUN
ejpam-6321	673	12	for	for	ADP
ejpam-6321	673	13	example	example	NOUN
ejpam-6321	673	14	3	3	NUM
ejpam-6321	673	15	and	and	CCONJ
ejpam-6321	673	16	algorithm	algorithm	PROPN
ejpam-6321	673	17	3	3	NUM
ejpam-6321	673	18	together	together	ADV
ejpam-6321	673	19	demonstrate	demonstrate	VERB
ejpam-6321	673	20	the	the	DET
ejpam-6321	673	21	convergence	convergence	NOUN
ejpam-6321	673	22	of	of	ADP
ejpam-6321	673	23	fixed	fix	VERB
ejpam-6321	673	24	-	-	PUNCT
ejpam-6321	673	25	point	point	NOUN
ejpam-6321	673	26	iterations	iteration	NOUN
ejpam-6321	673	27	in	in	ADP
ejpam-6321	673	28	a	a	DET
ejpam-6321	673	29	g	g	NOUN
ejpam-6321	673	30	-	-	PUNCT
ejpam-6321	673	31	metric	metric	ADJ
ejpam-6321	673	32	space	space	NOUN
ejpam-6321	673	33	,	,	PUNCT
ejpam-6321	673	34	representing	represent	VERB
ejpam-6321	673	35	either	either	CCONJ
ejpam-6321	673	36	stable	stable	ADJ
ejpam-6321	673	37	neural	neural	ADJ
ejpam-6321	673	38	activation	activation	NOUN
ejpam-6321	673	39	in	in	ADP
ejpam-6321	673	40	ai	ai	PROPN
ejpam-6321	673	41	systems	system	NOUN
ejpam-6321	673	42	or	or	CCONJ
ejpam-6321	673	43	secure	secure	VERB
ejpam-6321	673	44	consensus	consensus	NOUN
ejpam-6321	673	45	values	value	NOUN
ejpam-6321	673	46	in	in	ADP
ejpam-6321	673	47	cryptographic	cryptographic	ADJ
ejpam-6321	673	48	protocols	protocol	NOUN
ejpam-6321	673	49	.	.	PUNCT
ejpam-6321	674	1	m.	m.	NOUN
ejpam-6321	674	2	noorwali	noorwali	PROPN
ejpam-6321	674	3	et	et	PROPN
ejpam-6321	674	4	al	al	PROPN
ejpam-6321	674	5	.	.	PUNCT
ejpam-6321	674	6	/	/	SYM
ejpam-6321	674	7	eur	eur	PROPN
ejpam-6321	674	8	.	.	PUNCT
ejpam-6321	675	1	j.	j.	PROPN
ejpam-6321	675	2	pure	pure	PROPN
ejpam-6321	675	3	appl	appl	PROPN
ejpam-6321	675	4	.	.	PROPN
ejpam-6321	675	5	math	math	PROPN
ejpam-6321	675	6	,	,	PUNCT
ejpam-6321	675	7	18	18	NUM
ejpam-6321	675	8	(	(	PUNCT
ejpam-6321	675	9	3	3	NUM
ejpam-6321	675	10	)	)	PUNCT
ejpam-6321	675	11	(	(	PUNCT
ejpam-6321	675	12	2025	2025	NUM
ejpam-6321	675	13	)	)	PUNCT
ejpam-6321	675	14	,	,	PUNCT
ejpam-6321	675	15	6321	6321	NUM
ejpam-6321	675	16	25	25	NUM
ejpam-6321	675	17	of	of	ADP
ejpam-6321	675	18	27	27	NUM
ejpam-6321	675	19	figure	figure	NOUN
ejpam-6321	675	20	4	4	NUM
ejpam-6321	675	21	:	:	PUNCT
ejpam-6321	675	22	fixed	fix	VERB
ejpam-6321	675	23	-	-	PUNCT
ejpam-6321	675	24	point	point	NOUN
ejpam-6321	675	25	convergence	convergence	NOUN
ejpam-6321	675	26	in	in	ADP
ejpam-6321	675	27	g	g	NOUN
ejpam-6321	675	28	-	-	PUNCT
ejpam-6321	675	29	metric	metric	ADJ
ejpam-6321	675	30	space	space	NOUN
ejpam-6321	675	31	using	use	VERB
ejpam-6321	675	32	the	the	DET
ejpam-6321	675	33	mapping	mapping	NOUN
ejpam-6321	675	34	.	.	PUNCT
ejpam-6321	676	1	figure	figure	NOUN
ejpam-6321	676	2	4	4	NUM
ejpam-6321	676	3	demonstrates	demonstrate	VERB
ejpam-6321	676	4	the	the	DET
ejpam-6321	676	5	convergence	convergence	NOUN
ejpam-6321	676	6	of	of	ADP
ejpam-6321	676	7	the	the	DET
ejpam-6321	676	8	fixed	fix	VERB
ejpam-6321	676	9	-	-	PUNCT
ejpam-6321	676	10	point	point	NOUN
ejpam-6321	676	11	iteration	iteration	NOUN
ejpam-6321	676	12	governed	govern	VERB
ejpam-6321	676	13	by	by	ADP
ejpam-6321	676	14	the	the	DET
ejpam-6321	676	15	contraction	contraction	NOUN
ejpam-6321	676	16	condition	condition	NOUN
ejpam-6321	676	17	in	in	ADP
ejpam-6321	676	18	a	a	DET
ejpam-6321	676	19	g	g	NOUN
ejpam-6321	676	20	-	-	PUNCT
ejpam-6321	676	21	metric	metric	ADJ
ejpam-6321	676	22	space	space	NOUN
ejpam-6321	676	23	,	,	PUNCT
ejpam-6321	676	24	validating	validate	VERB
ejpam-6321	676	25	the	the	DET
ejpam-6321	676	26	theoretical	theoretical	ADJ
ejpam-6321	676	27	result	result	NOUN
ejpam-6321	676	28	numerically	numerically	ADV
ejpam-6321	676	29	.	.	PUNCT
ejpam-6321	677	1	4	4	X
ejpam-6321	677	2	.	.	X
ejpam-6321	677	3	conclusion	conclusion	VERB
ejpam-6321	677	4	the	the	DET
ejpam-6321	677	5	field	field	NOUN
ejpam-6321	677	6	of	of	ADP
ejpam-6321	677	7	fixed	fix	VERB
ejpam-6321	677	8	-	-	PUNCT
ejpam-6321	677	9	point	point	NOUN
ejpam-6321	677	10	theory	theory	NOUN
ejpam-6321	677	11	is	be	AUX
ejpam-6321	677	12	a	a	DET
ejpam-6321	677	13	well	well	ADV
ejpam-6321	677	14	-	-	PUNCT
ejpam-6321	677	15	established	establish	VERB
ejpam-6321	677	16	and	and	CCONJ
ejpam-6321	677	17	substantial	substantial	ADJ
ejpam-6321	677	18	area	area	NOUN
ejpam-6321	677	19	of	of	ADP
ejpam-6321	677	20	analysis	analysis	NOUN
ejpam-6321	677	21	with	with	ADP
ejpam-6321	677	22	numerous	numerous	ADJ
ejpam-6321	677	23	applications	application	NOUN
ejpam-6321	677	24	across	across	ADP
ejpam-6321	677	25	various	various	ADJ
ejpam-6321	677	26	disciplines	discipline	NOUN
ejpam-6321	677	27	.	.	PUNCT
ejpam-6321	678	1	within	within	ADP
ejpam-6321	678	2	this	this	DET
ejpam-6321	678	3	field	field	NOUN
ejpam-6321	678	4	,	,	PUNCT
ejpam-6321	678	5	particular	particular	ADJ
ejpam-6321	678	6	attention	attention	NOUN
ejpam-6321	678	7	has	have	AUX
ejpam-6321	678	8	been	be	AUX
ejpam-6321	678	9	given	give	VERB
ejpam-6321	678	10	to	to	ADP
ejpam-6321	678	11	the	the	DET
ejpam-6321	678	12	study	study	NOUN
ejpam-6321	678	13	of	of	ADP
ejpam-6321	678	14	contractive	contractive	ADJ
ejpam-6321	678	15	-	-	PUNCT
ejpam-6321	678	16	type	type	NOUN
ejpam-6321	678	17	inequalities	inequality	NOUN
ejpam-6321	678	18	.	.	PUNCT
ejpam-6321	679	1	in	in	ADP
ejpam-6321	679	2	this	this	DET
ejpam-6321	679	3	work	work	NOUN
ejpam-6321	679	4	,	,	PUNCT
ejpam-6321	679	5	we	we	PRON
ejpam-6321	679	6	establish	establish	VERB
ejpam-6321	679	7	conditions	condition	NOUN
ejpam-6321	679	8	under	under	ADP
ejpam-6321	679	9	which	which	PRON
ejpam-6321	679	10	three	three	NUM
ejpam-6321	679	11	self	self	NOUN
ejpam-6321	679	12	-	-	PUNCT
ejpam-6321	679	13	mappings	mapping	NOUN
ejpam-6321	679	14	in	in	ADP
ejpam-6321	679	15	generalized	generalized	ADJ
ejpam-6321	679	16	metric	metric	ADJ
ejpam-6321	679	17	spaces	space	NOUN
ejpam-6321	679	18	(	(	PUNCT
ejpam-6321	679	19	gm	gm	NOUN
ejpam-6321	679	20	-	-	PUNCT
ejpam-6321	679	21	spaces	space	NOUN
ejpam-6321	679	22	)	)	PUNCT
ejpam-6321	679	23	possess	possess	VERB
ejpam-6321	679	24	common	common	ADJ
ejpam-6321	679	25	fixed	fix	VERB
ejpam-6321	679	26	points	point	NOUN
ejpam-6321	679	27	(	(	PUNCT
ejpam-6321	679	28	cfp	cfp	NOUN
ejpam-6321	679	29	)	)	PUNCT
ejpam-6321	679	30	and	and	CCONJ
ejpam-6321	679	31	unique	unique	ADJ
ejpam-6321	679	32	common	common	ADJ
ejpam-6321	679	33	fixed	fix	VERB
ejpam-6321	679	34	points	point	NOUN
ejpam-6321	679	35	(	(	PUNCT
ejpam-6321	679	36	ucfp	ucfp	NOUN
ejpam-6321	679	37	)	)	PUNCT
ejpam-6321	679	38	.	.	PUNCT
ejpam-6321	680	1	these	these	DET
ejpam-6321	680	2	results	result	NOUN
ejpam-6321	680	3	present	present	VERB
ejpam-6321	680	4	key	key	ADJ
ejpam-6321	680	5	inequalities	inequality	NOUN
ejpam-6321	680	6	involving	involve	VERB
ejpam-6321	680	7	specific	specific	ADJ
ejpam-6321	680	8	constants	constant	NOUN
ejpam-6321	680	9	that	that	PRON
ejpam-6321	680	10	guarantee	guarantee	VERB
ejpam-6321	680	11	the	the	DET
ejpam-6321	680	12	existence	existence	NOUN
ejpam-6321	680	13	of	of	ADP
ejpam-6321	680	14	cfp	cfp	PROPN
ejpam-6321	680	15	and	and	CCONJ
ejpam-6321	680	16	ucfp	ucfp	NOUN
ejpam-6321	680	17	under	under	ADP
ejpam-6321	680	18	certain	certain	ADJ
ejpam-6321	680	19	assumptions	assumption	NOUN
ejpam-6321	680	20	.	.	PUNCT
ejpam-6321	681	1	through	through	ADP
ejpam-6321	681	2	an	an	DET
ejpam-6321	681	3	illustrative	illustrative	ADJ
ejpam-6321	681	4	example	example	NOUN
ejpam-6321	681	5	,	,	PUNCT
ejpam-6321	681	6	we	we	PRON
ejpam-6321	681	7	demonstrate	demonstrate	VERB
ejpam-6321	681	8	the	the	DET
ejpam-6321	681	9	applicability	applicability	NOUN
ejpam-6321	681	10	of	of	ADP
ejpam-6321	681	11	these	these	DET
ejpam-6321	681	12	conditions	condition	NOUN
ejpam-6321	681	13	to	to	ADP
ejpam-6321	681	14	a	a	DET
ejpam-6321	681	15	gm	gm	ADV
ejpam-6321	681	16	-	-	PUNCT
ejpam-6321	681	17	metric	metric	NOUN
ejpam-6321	681	18	defined	define	VERB
ejpam-6321	681	19	on	on	ADP
ejpam-6321	681	20	the	the	DET
ejpam-6321	681	21	interval	interval	NOUN
ejpam-6321	681	22	[	[	X
ejpam-6321	681	23	0	0	NUM
ejpam-6321	681	24	,	,	PUNCT
ejpam-6321	681	25	1	1	NUM
ejpam-6321	681	26	]	]	PUNCT
ejpam-6321	681	27	,	,	PUNCT
ejpam-6321	681	28	thereby	thereby	ADV
ejpam-6321	681	29	confirming	confirm	VERB
ejpam-6321	681	30	the	the	DET
ejpam-6321	681	31	existence	existence	NOUN
ejpam-6321	681	32	of	of	ADP
ejpam-6321	681	33	a	a	DET
ejpam-6321	681	34	ucfp	ucfp	NOUN
ejpam-6321	681	35	for	for	ADP
ejpam-6321	681	36	the	the	DET
ejpam-6321	681	37	given	give	VERB
ejpam-6321	681	38	mappings	mapping	NOUN
ejpam-6321	681	39	.	.	PUNCT
ejpam-6321	682	1	this	this	DET
ejpam-6321	682	2	study	study	NOUN
ejpam-6321	682	3	contributes	contribute	VERB
ejpam-6321	682	4	to	to	ADP
ejpam-6321	682	5	the	the	DET
ejpam-6321	682	6	advancement	advancement	NOUN
ejpam-6321	682	7	of	of	ADP
ejpam-6321	682	8	fixed	fix	VERB
ejpam-6321	682	9	-	-	PUNCT
ejpam-6321	682	10	point	point	NOUN
ejpam-6321	682	11	theory	theory	NOUN
ejpam-6321	682	12	in	in	ADP
ejpam-6321	682	13	gm	gm	PROPN
ejpam-6321	682	14	-	-	PUNCT
ejpam-6321	682	15	spaces	space	NOUN
ejpam-6321	682	16	and	and	CCONJ
ejpam-6321	682	17	highlights	highlight	NOUN
ejpam-6321	682	18	the	the	DET
ejpam-6321	682	19	importance	importance	NOUN
ejpam-6321	682	20	of	of	ADP
ejpam-6321	682	21	metric	metric	ADJ
ejpam-6321	682	22	-	-	PUNCT
ejpam-6321	682	23	related	relate	VERB
ejpam-6321	682	24	conditions	condition	NOUN
ejpam-6321	682	25	in	in	ADP
ejpam-6321	682	26	determining	determine	VERB
ejpam-6321	682	27	the	the	DET
ejpam-6321	682	28	existence	existence	NOUN
ejpam-6321	682	29	of	of	ADP
ejpam-6321	682	30	fixed	fix	VERB
ejpam-6321	682	31	points	point	NOUN
ejpam-6321	682	32	in	in	ADP
ejpam-6321	682	33	such	such	ADJ
ejpam-6321	682	34	spaces	space	NOUN
ejpam-6321	682	35	.	.	PUNCT
ejpam-6321	683	1	the	the	DET
ejpam-6321	683	2	findings	finding	NOUN
ejpam-6321	683	3	also	also	ADV
ejpam-6321	683	4	build	build	VERB
ejpam-6321	683	5	upon	upon	SCONJ
ejpam-6321	683	6	and	and	CCONJ
ejpam-6321	683	7	expand	expand	VERB
ejpam-6321	683	8	recent	recent	ADJ
ejpam-6321	683	9	developments	development	NOUN
ejpam-6321	683	10	in	in	ADP
ejpam-6321	683	11	the	the	DET
ejpam-6321	683	12	existing	exist	VERB
ejpam-6321	683	13	literature	literature	NOUN
ejpam-6321	683	14	.	.	PUNCT
ejpam-6321	684	1	moreover	moreover	ADV
ejpam-6321	684	2	,	,	PUNCT
ejpam-6321	684	3	the	the	DET
ejpam-6321	684	4	implications	implication	NOUN
ejpam-6321	684	5	of	of	ADP
ejpam-6321	684	6	these	these	DET
ejpam-6321	684	7	results	result	NOUN
ejpam-6321	684	8	extend	extend	VERB
ejpam-6321	684	9	beyond	beyond	ADP
ejpam-6321	684	10	pure	pure	ADJ
ejpam-6321	684	11	mathematics	mathematic	NOUN
ejpam-6321	684	12	:	:	PUNCT
ejpam-6321	684	13	in	in	ADP
ejpam-6321	684	14	artificial	artificial	ADJ
ejpam-6321	684	15	intelligence	intelligence	NOUN
ejpam-6321	684	16	,	,	PUNCT
ejpam-6321	684	17	fixed	fix	VERB
ejpam-6321	684	18	-	-	PUNCT
ejpam-6321	684	19	point	point	NOUN
ejpam-6321	684	20	iterations	iteration	NOUN
ejpam-6321	684	21	in	in	ADP
ejpam-6321	684	22	gm	gm	PROPN
ejpam-6321	684	23	-	-	PUNCT
ejpam-6321	684	24	spaces	space	NOUN
ejpam-6321	684	25	can	can	AUX
ejpam-6321	684	26	model	model	VERB
ejpam-6321	684	27	convergence	convergence	NOUN
ejpam-6321	684	28	behavior	behavior	NOUN
ejpam-6321	684	29	in	in	ADP
ejpam-6321	684	30	deep	deep	ADJ
ejpam-6321	684	31	learning	learning	NOUN
ejpam-6321	684	32	layers	layer	NOUN
ejpam-6321	684	33	,	,	PUNCT
ejpam-6321	684	34	iterative	iterative	ADJ
ejpam-6321	684	35	reasoning	reasoning	NOUN
ejpam-6321	684	36	in	in	ADP
ejpam-6321	684	37	agents	agent	NOUN
ejpam-6321	684	38	,	,	PUNCT
ejpam-6321	684	39	and	and	CCONJ
ejpam-6321	684	40	equilibrium	equilibrium	NOUN
ejpam-6321	684	41	states	state	NOUN
ejpam-6321	684	42	in	in	ADP
ejpam-6321	684	43	feedback	feedback	NOUN
ejpam-6321	684	44	-	-	PUNCT
ejpam-6321	684	45	driven	drive	VERB
ejpam-6321	684	46	models	model	NOUN
ejpam-6321	684	47	.	.	PUNCT
ejpam-6321	685	1	similarly	similarly	ADV
ejpam-6321	685	2	,	,	PUNCT
ejpam-6321	685	3	in	in	ADP
ejpam-6321	685	4	cryptography	cryptography	NOUN
ejpam-6321	685	5	,	,	PUNCT
ejpam-6321	685	6	such	such	ADJ
ejpam-6321	685	7	fixed	fix	VERB
ejpam-6321	685	8	points	point	NOUN
ejpam-6321	685	9	can	can	AUX
ejpam-6321	685	10	represent	represent	VERB
ejpam-6321	685	11	secure	secure	ADJ
ejpam-6321	685	12	consensus	consensus	NOUN
ejpam-6321	685	13	states	state	NOUN
ejpam-6321	685	14	in	in	ADP
ejpam-6321	685	15	distributed	distribute	VERB
ejpam-6321	685	16	protocols	protocol	NOUN
ejpam-6321	685	17	,	,	PUNCT
ejpam-6321	685	18	iterative	iterative	NOUN
ejpam-6321	685	19	agreement	agreement	NOUN
ejpam-6321	685	20	in	in	ADP
ejpam-6321	685	21	key	key	ADJ
ejpam-6321	685	22	-	-	PUNCT
ejpam-6321	685	23	exchange	exchange	NOUN
ejpam-6321	685	24	mechanisms	mechanism	NOUN
ejpam-6321	685	25	,	,	PUNCT
ejpam-6321	685	26	or	or	CCONJ
ejpam-6321	685	27	stability	stability	NOUN
ejpam-6321	685	28	in	in	ADP
ejpam-6321	685	29	homomorphic	homomorphic	ADJ
ejpam-6321	685	30	encryption	encryption	NOUN
ejpam-6321	685	31	loops	loop	NOUN
ejpam-6321	685	32	.	.	PUNCT
ejpam-6321	686	1	future	future	ADJ
ejpam-6321	686	2	research	research	NOUN
ejpam-6321	686	3	in	in	ADP
ejpam-6321	686	4	fixed	fix	VERB
ejpam-6321	686	5	-	-	PUNCT
ejpam-6321	686	6	point	point	NOUN
ejpam-6321	686	7	theory	theory	NOUN
ejpam-6321	686	8	,	,	PUNCT
ejpam-6321	686	9	particularly	particularly	ADV
ejpam-6321	686	10	within	within	ADP
ejpam-6321	686	11	the	the	DET
ejpam-6321	686	12	framework	framework	NOUN
ejpam-6321	686	13	of	of	ADP
ejpam-6321	686	14	gm	gm	PROPN
ejpam-6321	686	15	-	-	PUNCT
ejpam-6321	686	16	spaces	space	NOUN
ejpam-6321	686	17	,	,	PUNCT
ejpam-6321	686	18	could	could	AUX
ejpam-6321	686	19	explore	explore	VERB
ejpam-6321	686	20	several	several	ADJ
ejpam-6321	686	21	promising	promising	ADJ
ejpam-6321	686	22	directions	direction	NOUN
ejpam-6321	686	23	.	.	PUNCT
ejpam-6321	687	1	one	one	NUM
ejpam-6321	687	2	avenue	avenue	NOUN
ejpam-6321	687	3	involves	involve	VERB
ejpam-6321	687	4	extending	extend	VERB
ejpam-6321	687	5	contractive	contractive	ADJ
ejpam-6321	687	6	-	-	PUNCT
ejpam-6321	687	7	type	type	NOUN
ejpam-6321	687	8	inequalities	inequality	NOUN
ejpam-6321	687	9	to	to	ADP
ejpam-6321	687	10	more	more	ADJ
ejpam-6321	687	11	general	general	ADJ
ejpam-6321	687	12	settings	setting	NOUN
ejpam-6321	687	13	,	,	PUNCT
ejpam-6321	687	14	such	such	ADJ
ejpam-6321	687	15	as	as	ADP
ejpam-6321	687	16	non	non	ADJ
ejpam-6321	687	17	-	-	ADJ
ejpam-6321	687	18	metric	metric	ADJ
ejpam-6321	687	19	spaces	space	NOUN
ejpam-6321	687	20	or	or	CCONJ
ejpam-6321	687	21	spaces	space	NOUN
ejpam-6321	687	22	with	with	ADP
ejpam-6321	687	23	weaker	weak	ADJ
ejpam-6321	687	24	topological	topological	ADJ
ejpam-6321	687	25	structures	structure	NOUN
ejpam-6321	687	26	.	.	PUNCT
ejpam-6321	688	1	investigating	investigate	VERB
ejpam-6321	688	2	the	the	DET
ejpam-6321	688	3	relationships	relationship	NOUN
ejpam-6321	688	4	between	between	ADP
ejpam-6321	688	5	various	various	ADJ
ejpam-6321	688	6	types	type	NOUN
ejpam-6321	688	7	of	of	ADP
ejpam-6321	688	8	inequalities	inequality	NOUN
ejpam-6321	688	9	,	,	PUNCT
ejpam-6321	688	10	including	include	VERB
ejpam-6321	688	11	newly	newly	ADV
ejpam-6321	688	12	formulated	formulate	VERB
ejpam-6321	688	13	generalized	generalized	ADJ
ejpam-6321	688	14	contractive	contractive	ADJ
ejpam-6321	688	15	conditions	condition	NOUN
ejpam-6321	688	16	,	,	PUNCT
ejpam-6321	688	17	may	may	AUX
ejpam-6321	688	18	deepen	deepen	VERB
ejpam-6321	688	19	the	the	DET
ejpam-6321	688	20	theoretical	theoretical	ADJ
ejpam-6321	688	21	understanding	understanding	NOUN
ejpam-6321	688	22	of	of	ADP
ejpam-6321	688	23	fixed	fix	VERB
ejpam-6321	688	24	-	-	PUNCT
ejpam-6321	688	25	point	point	NOUN
ejpam-6321	688	26	phenomena	phenomenon	NOUN
ejpam-6321	688	27	across	across	ADP
ejpam-6321	688	28	broader	broad	ADJ
ejpam-6321	688	29	contexts	context	NOUN
ejpam-6321	688	30	.	.	PUNCT
ejpam-6321	689	1	another	another	DET
ejpam-6321	689	2	potential	potential	ADJ
ejpam-6321	689	3	direction	direction	NOUN
ejpam-6321	689	4	is	be	AUX
ejpam-6321	689	5	the	the	DET
ejpam-6321	689	6	application	application	NOUN
ejpam-6321	689	7	of	of	ADP
ejpam-6321	689	8	these	these	DET
ejpam-6321	689	9	fixed	fix	VERB
ejpam-6321	689	10	-	-	PUNCT
ejpam-6321	689	11	point	point	NOUN
ejpam-6321	689	12	results	result	NOUN
ejpam-6321	689	13	to	to	ADP
ejpam-6321	689	14	practical	practical	ADJ
ejpam-6321	689	15	problems	problem	NOUN
ejpam-6321	689	16	in	in	ADP
ejpam-6321	689	17	fields	field	NOUN
ejpam-6321	689	18	such	such	ADJ
ejpam-6321	689	19	as	as	ADP
ejpam-6321	689	20	optimization	optimization	NOUN
ejpam-6321	689	21	,	,	PUNCT
ejpam-6321	689	22	machine	machine	NOUN
ejpam-6321	689	23	learning	learning	NOUN
ejpam-6321	689	24	,	,	PUNCT
ejpam-6321	689	25	and	and	CCONJ
ejpam-6321	689	26	secure	secure	ADJ
ejpam-6321	689	27	computation	computation	NOUN
ejpam-6321	689	28	,	,	PUNCT
ejpam-6321	689	29	where	where	SCONJ
ejpam-6321	689	30	fixed	fix	VERB
ejpam-6321	689	31	points	point	NOUN
ejpam-6321	689	32	are	be	AUX
ejpam-6321	689	33	fundamental	fundamental	ADJ
ejpam-6321	689	34	to	to	ADP
ejpam-6321	689	35	algorithm	algorithm	NOUN
ejpam-6321	689	36	design	design	NOUN
ejpam-6321	689	37	and	and	CCONJ
ejpam-6321	689	38	decision	decision	NOUN
ejpam-6321	689	39	-	-	PUNCT
ejpam-6321	689	40	making	making	NOUN
ejpam-6321	689	41	.	.	PUNCT
ejpam-6321	690	1	moreover	moreover	ADV
ejpam-6321	690	2	,	,	PUNCT
ejpam-6321	690	3	generalizing	generalize	VERB
ejpam-6321	690	4	the	the	DET
ejpam-6321	690	5	current	current	ADJ
ejpam-6321	690	6	results	result	NOUN
ejpam-6321	690	7	to	to	PART
ejpam-6321	690	8	accommodate	accommodate	VERB
ejpam-6321	690	9	more	more	ADJ
ejpam-6321	690	10	complex	complex	ADJ
ejpam-6321	690	11	classes	class	NOUN
ejpam-6321	690	12	of	of	ADP
ejpam-6321	690	13	mappings	mapping	NOUN
ejpam-6321	690	14	,	,	PUNCT
ejpam-6321	690	15	such	such	ADJ
ejpam-6321	690	16	as	as	ADP
ejpam-6321	690	17	non	non	ADJ
ejpam-6321	690	18	-	-	ADJ
ejpam-6321	690	19	linear	linear	ADJ
ejpam-6321	690	20	or	or	CCONJ
ejpam-6321	690	21	non	non	ADJ
ejpam-6321	690	22	-	-	ADJ
ejpam-6321	690	23	continuous	continuous	ADJ
ejpam-6321	690	24	mappings	mapping	NOUN
ejpam-6321	690	25	,	,	PUNCT
ejpam-6321	690	26	could	could	AUX
ejpam-6321	690	27	yield	yield	VERB
ejpam-6321	690	28	valuable	valuable	ADJ
ejpam-6321	690	29	tools	tool	NOUN
ejpam-6321	690	30	and	and	CCONJ
ejpam-6321	690	31	insights	insight	NOUN
ejpam-6321	690	32	m.	m.	NOUN
ejpam-6321	690	33	noorwali	noorwali	PROPN
ejpam-6321	690	34	et	et	PROPN
ejpam-6321	690	35	al	al	PROPN
ejpam-6321	690	36	.	.	PUNCT
ejpam-6321	690	37	/	/	SYM
ejpam-6321	690	38	eur	eur	PROPN
ejpam-6321	690	39	.	.	PUNCT
ejpam-6321	691	1	j.	j.	PROPN
ejpam-6321	691	2	pure	pure	PROPN
ejpam-6321	691	3	appl	appl	PROPN
ejpam-6321	691	4	.	.	PROPN
ejpam-6321	691	5	math	math	PROPN
ejpam-6321	691	6	,	,	PUNCT
ejpam-6321	691	7	18	18	NUM
ejpam-6321	691	8	(	(	PUNCT
ejpam-6321	691	9	3	3	NUM
ejpam-6321	691	10	)	)	PUNCT
ejpam-6321	691	11	(	(	PUNCT
ejpam-6321	691	12	2025	2025	NUM
ejpam-6321	691	13	)	)	PUNCT
ejpam-6321	691	14	,	,	PUNCT
ejpam-6321	691	15	6321	6321	NUM
ejpam-6321	691	16	26	26	NUM
ejpam-6321	691	17	of	of	ADP
ejpam-6321	691	18	27	27	NUM
ejpam-6321	691	19	for	for	ADP
ejpam-6321	691	20	researchers	researcher	NOUN
ejpam-6321	691	21	.	.	PUNCT
ejpam-6321	692	1	finally	finally	ADV
ejpam-6321	692	2	,	,	PUNCT
ejpam-6321	692	3	examining	examine	VERB
ejpam-6321	692	4	the	the	DET
ejpam-6321	692	5	robustness	robustness	NOUN
ejpam-6321	692	6	of	of	ADP
ejpam-6321	692	7	fixed	fix	VERB
ejpam-6321	692	8	-	-	PUNCT
ejpam-6321	692	9	point	point	NOUN
ejpam-6321	692	10	results	result	NOUN
ejpam-6321	692	11	in	in	ADP
ejpam-6321	692	12	gm	gm	PROPN
ejpam-6321	692	13	-	-	PUNCT
ejpam-6321	692	14	spaces	space	NOUN
ejpam-6321	692	15	under	under	ADP
ejpam-6321	692	16	perturbations	perturbation	NOUN
ejpam-6321	692	17	or	or	CCONJ
ejpam-6321	692	18	uncertainty	uncertainty	NOUN
ejpam-6321	692	19	could	could	AUX
ejpam-6321	692	20	enhance	enhance	VERB
ejpam-6321	692	21	their	their	PRON
ejpam-6321	692	22	practical	practical	ADJ
ejpam-6321	692	23	relevance	relevance	NOUN
ejpam-6321	692	24	,	,	PUNCT
ejpam-6321	692	25	particularly	particularly	ADV
ejpam-6321	692	26	in	in	ADP
ejpam-6321	692	27	real	real	ADJ
ejpam-6321	692	28	-	-	PUNCT
ejpam-6321	692	29	world	world	NOUN
ejpam-6321	692	30	scenarios	scenario	NOUN
ejpam-6321	692	31	where	where	SCONJ
ejpam-6321	692	32	ideal	ideal	ADJ
ejpam-6321	692	33	conditions	condition	NOUN
ejpam-6321	692	34	may	may	AUX
ejpam-6321	692	35	not	not	PART
ejpam-6321	692	36	be	be	AUX
ejpam-6321	692	37	met	meet	VERB
ejpam-6321	692	38	.	.	PUNCT
ejpam-6321	693	1	by	by	ADP
ejpam-6321	693	2	expanding	expand	VERB
ejpam-6321	693	3	the	the	DET
ejpam-6321	693	4	scope	scope	NOUN
ejpam-6321	693	5	of	of	ADP
ejpam-6321	693	6	gm	gm	PROPN
ejpam-6321	693	7	-	-	PUNCT
ejpam-6321	693	8	spaces	space	NOUN
ejpam-6321	693	9	and	and	CCONJ
ejpam-6321	693	10	refining	refine	VERB
ejpam-6321	693	11	the	the	DET
ejpam-6321	693	12	criteria	criterion	NOUN
ejpam-6321	693	13	for	for	ADP
ejpam-6321	693	14	the	the	DET
ejpam-6321	693	15	existence	existence	NOUN
ejpam-6321	693	16	of	of	ADP
ejpam-6321	693	17	cfp	cfp	PROPN
ejpam-6321	693	18	and	and	CCONJ
ejpam-6321	693	19	ucfp	ucfp	NOUN
ejpam-6321	693	20	,	,	PUNCT
ejpam-6321	693	21	this	this	DET
ejpam-6321	693	22	line	line	NOUN
ejpam-6321	693	23	of	of	ADP
ejpam-6321	693	24	research	research	NOUN
ejpam-6321	693	25	holds	hold	VERB
ejpam-6321	693	26	considerable	considerable	ADJ
ejpam-6321	693	27	potential	potential	NOUN
ejpam-6321	693	28	to	to	PART
ejpam-6321	693	29	contribute	contribute	VERB
ejpam-6321	693	30	significantly	significantly	ADV
ejpam-6321	693	31	to	to	ADP
ejpam-6321	693	32	both	both	CCONJ
ejpam-6321	693	33	theoretical	theoretical	ADJ
ejpam-6321	693	34	and	and	CCONJ
ejpam-6321	693	35	applied	applied	ADJ
ejpam-6321	693	36	mathematics	mathematic	NOUN
ejpam-6321	693	37	—	—	PUNCT
ejpam-6321	693	38	including	include	VERB
ejpam-6321	693	39	emerging	emerge	VERB
ejpam-6321	693	40	domains	domain	NOUN
ejpam-6321	693	41	like	like	ADP
ejpam-6321	693	42	adaptive	adaptive	ADJ
ejpam-6321	693	43	ai	ai	VERB
ejpam-6321	693	44	control	control	NOUN
ejpam-6321	693	45	and	and	CCONJ
ejpam-6321	693	46	privacy	privacy	NOUN
ejpam-6321	693	47	-	-	PUNCT
ejpam-6321	693	48	preserving	preserve	VERB
ejpam-6321	693	49	cryptographic	cryptographic	ADJ
ejpam-6321	693	50	infrastructure	infrastructure	NOUN
ejpam-6321	693	51	.	.	PUNCT
ejpam-6321	694	1	acknowledgements	acknowledgement	NOUN
ejpam-6321	694	2	(	(	PUNCT
ejpam-6321	694	3	i	i	NOUN
ejpam-6321	694	4	)	)	PUNCT
ejpam-6321	694	5	the	the	DET
ejpam-6321	694	6	project	project	NOUN
ejpam-6321	694	7	was	be	AUX
ejpam-6321	694	8	funded	fund	VERB
ejpam-6321	694	9	by	by	ADP
ejpam-6321	694	10	the	the	DET
ejpam-6321	694	11	kau	kau	PROPN
ejpam-6321	694	12	endowment	endowment	PROPN
ejpam-6321	694	13	(	(	PUNCT
ejpam-6321	694	14	waqf	waqf	NOUN
ejpam-6321	694	15	)	)	PUNCT
ejpam-6321	694	16	at	at	ADP
ejpam-6321	694	17	king	king	PROPN
ejpam-6321	694	18	abdulaziz	abdulaziz	PROPN
ejpam-6321	694	19	university	university	PROPN
ejpam-6321	694	20	,	,	PUNCT
ejpam-6321	694	21	jeddah	jeddah	PROPN
ejpam-6321	694	22	,	,	PUNCT
ejpam-6321	694	23	saudi	saudi	PROPN
ejpam-6321	694	24	arabia	arabia	PROPN
ejpam-6321	694	25	.	.	PUNCT
ejpam-6321	695	1	the	the	DET
ejpam-6321	695	2	authors	author	NOUN
ejpam-6321	695	3	,	,	PUNCT
ejpam-6321	695	4	therefore	therefore	ADV
ejpam-6321	695	5	,	,	PUNCT
ejpam-6321	695	6	acknowledge	acknowledge	VERB
ejpam-6321	695	7	waqf	waqf	NOUN
ejpam-6321	695	8	and	and	CCONJ
ejpam-6321	695	9	the	the	DET
ejpam-6321	695	10	deanship	deanship	NOUN
ejpam-6321	695	11	of	of	ADP
ejpam-6321	695	12	scientific	scientific	ADJ
ejpam-6321	695	13	research	research	NOUN
ejpam-6321	695	14	(	(	PUNCT
ejpam-6321	695	15	dsr	dsr	PROPN
ejpam-6321	695	16	)	)	PUNCT
ejpam-6321	695	17	for	for	ADP
ejpam-6321	695	18	technical	technical	ADJ
ejpam-6321	695	19	and	and	CCONJ
ejpam-6321	695	20	financial	financial	ADJ
ejpam-6321	695	21	support	support	NOUN
ejpam-6321	695	22	.	.	PUNCT
ejpam-6321	696	1	(	(	PUNCT
ejpam-6321	696	2	ii	ii	X
ejpam-6321	696	3	)	)	PUNCT
ejpam-6321	696	4	we	we	PRON
ejpam-6321	696	5	acknowledge	acknowledge	VERB
ejpam-6321	696	6	ho	ho	PROPN
ejpam-6321	696	7	chi	chi	PROPN
ejpam-6321	696	8	minh	minh	PROPN
ejpam-6321	696	9	city	city	PROPN
ejpam-6321	696	10	university	university	PROPN
ejpam-6321	696	11	of	of	ADP
ejpam-6321	696	12	technology	technology	NOUN
ejpam-6321	696	13	(	(	PUNCT
ejpam-6321	696	14	hcmut	hcmut	NOUN
ejpam-6321	696	15	)	)	PUNCT
ejpam-6321	696	16	,	,	PUNCT
ejpam-6321	696	17	vnu	vnu	PROPN
ejpam-6321	696	18	-	-	PUNCT
ejpam-6321	696	19	hcm	hcm	PROPN
ejpam-6321	696	20	for	for	ADP
ejpam-6321	696	21	supporting	support	VERB
ejpam-6321	696	22	this	this	DET
ejpam-6321	696	23	study	study	NOUN
ejpam-6321	696	24	.	.	PUNCT
ejpam-6321	697	1	(	(	PUNCT
ejpam-6321	697	2	iii	iii	X
ejpam-6321	697	3	)	)	PUNCT
ejpam-6321	697	4	the	the	DET
ejpam-6321	697	5	authors	author	NOUN
ejpam-6321	697	6	extend	extend	VERB
ejpam-6321	697	7	their	their	PRON
ejpam-6321	697	8	appreciation	appreciation	NOUN
ejpam-6321	697	9	to	to	ADP
ejpam-6321	697	10	the	the	DET
ejpam-6321	697	11	deanship	deanship	NOUN
ejpam-6321	697	12	of	of	ADP
ejpam-6321	697	13	scientific	scientific	ADJ
ejpam-6321	697	14	research	research	NOUN
ejpam-6321	697	15	at	at	ADP
ejpam-6321	697	16	northern	northern	ADJ
ejpam-6321	697	17	border	border	NOUN
ejpam-6321	697	18	university	university	PROPN
ejpam-6321	697	19	,	,	PUNCT
ejpam-6321	697	20	arar	arar	PROPN
ejpam-6321	697	21	,	,	PUNCT
ejpam-6321	697	22	ksa	ksa	PROPN
ejpam-6321	697	23	for	for	ADP
ejpam-6321	697	24	funding	fund	VERB
ejpam-6321	697	25	this	this	DET
ejpam-6321	697	26	research	research	NOUN
ejpam-6321	697	27	work	work	NOUN
ejpam-6321	697	28	through	through	ADP
ejpam-6321	697	29	the	the	DET
ejpam-6321	697	30	project	project	NOUN
ejpam-6321	697	31	number	number	NOUN
ejpam-6321	697	32	“	"	PUNCT
ejpam-6321	697	33	nbu	nbu	NOUN
ejpam-6321	697	34	-	-	PUNCT
ejpam-6321	697	35	ffr-2025	ffr-2025	NOUN
ejpam-6321	697	36	-	-	PUNCT
ejpam-6321	697	37	2727	2727	NUM
ejpam-6321	697	38	-	-	SYM
ejpam-6321	697	39	07	07	NUM
ejpam-6321	697	40	”	"	PUNCT
ejpam-6321	697	41	.	.	PUNCT
ejpam-6321	698	1	author	author	NOUN
ejpam-6321	698	2	contributions	contribution	NOUN
ejpam-6321	698	3	:	:	PUNCT
ejpam-6321	698	4	all	all	DET
ejpam-6321	698	5	authors	author	NOUN
ejpam-6321	698	6	equally	equally	ADV
ejpam-6321	698	7	contributed	contribute	VERB
ejpam-6321	698	8	.	.	PUNCT
ejpam-6321	699	1	conflicts	conflict	NOUN
ejpam-6321	699	2	of	of	ADP
ejpam-6321	699	3	interest	interest	NOUN
ejpam-6321	699	4	:	:	PUNCT
ejpam-6321	699	5	the	the	DET
ejpam-6321	699	6	authors	author	NOUN
ejpam-6321	699	7	declare	declare	VERB
ejpam-6321	699	8	no	no	DET
ejpam-6321	699	9	conflict	conflict	NOUN
ejpam-6321	699	10	of	of	ADP
ejpam-6321	699	11	interest	interest	NOUN
ejpam-6321	699	12	.	.	PUNCT
ejpam-6321	700	1	availability	availability	NOUN
ejpam-6321	700	2	of	of	ADP
ejpam-6321	700	3	data	datum	NOUN
ejpam-6321	700	4	and	and	CCONJ
ejpam-6321	700	5	materials	material	NOUN
ejpam-6321	700	6	:	:	PUNCT
ejpam-6321	700	7	all	all	DET
ejpam-6321	700	8	the	the	DET
ejpam-6321	700	9	data	datum	NOUN
ejpam-6321	700	10	is	be	AUX
ejpam-6321	700	11	provided	provide	VERB
ejpam-6321	700	12	in	in	ADP
ejpam-6321	700	13	the	the	DET
ejpam-6321	700	14	manuscript	manuscript	NOUN
ejpam-6321	700	15	.	.	PUNCT
ejpam-6321	701	1	references	reference	NOUN
ejpam-6321	701	2	[	[	X
ejpam-6321	701	3	1	1	NUM
ejpam-6321	701	4	]	]	PUNCT
ejpam-6321	701	5	s.	s.	PROPN
ejpam-6321	701	6	kakutani	kakutani	PROPN
ejpam-6321	701	7	.	.	PUNCT
ejpam-6321	702	1	a	a	DET
ejpam-6321	702	2	generalization	generalization	NOUN
ejpam-6321	702	3	of	of	ADP
ejpam-6321	702	4	brouwer	brouwer	PROPN
ejpam-6321	702	5	’s	’s	PART
ejpam-6321	702	6	fixed	fix	VERB
ejpam-6321	702	7	point	point	NOUN
ejpam-6321	702	8	theorem	theorem	VERB
ejpam-6321	702	9	.	.	PROPN
ejpam-6321	703	1	duke	duke	PROPN
ejpam-6321	703	2	mathematical	mathematical	PROPN
ejpam-6321	703	3	journal	journal	PROPN
ejpam-6321	703	4	,	,	PUNCT
ejpam-6321	703	5	8(3):457–459	8(3):457–459	NUM
ejpam-6321	703	6	,	,	PUNCT
ejpam-6321	703	7	1941	1941	NUM
ejpam-6321	703	8	.	.	PUNCT
ejpam-6321	704	1	[	[	X
ejpam-6321	704	2	2	2	X
ejpam-6321	704	3	]	]	PUNCT
ejpam-6321	704	4	g.	g.	PROPN
ejpam-6321	704	5	jungck	jungck	PROPN
ejpam-6321	704	6	and	and	CCONJ
ejpam-6321	704	7	b.	b.	PROPN
ejpam-6321	704	8	e.	e.	PROPN
ejpam-6321	704	9	rhoades	rhoades	PROPN
ejpam-6321	704	10	.	.	PUNCT
ejpam-6321	705	1	some	some	DET
ejpam-6321	705	2	fixed	fix	VERB
ejpam-6321	705	3	point	point	NOUN
ejpam-6321	705	4	theorems	theorem	NOUN
ejpam-6321	705	5	for	for	ADP
ejpam-6321	705	6	compatible	compatible	ADJ
ejpam-6321	705	7	maps	map	NOUN
ejpam-6321	705	8	.	.	PUNCT
ejpam-6321	706	1	international	international	ADJ
ejpam-6321	706	2	journal	journal	PROPN
ejpam-6321	706	3	of	of	ADP
ejpam-6321	706	4	mathematics	mathematics	PROPN
ejpam-6321	706	5	and	and	CCONJ
ejpam-6321	706	6	mathematical	mathematical	ADJ
ejpam-6321	706	7	sciences	science	NOUN
ejpam-6321	706	8	,	,	PUNCT
ejpam-6321	706	9	16(3):417–428	16(3):417–428	NUM
ejpam-6321	706	10	,	,	PUNCT
ejpam-6321	706	11	1993	1993	NUM
ejpam-6321	706	12	.	.	PUNCT
ejpam-6321	707	1	[	[	X
ejpam-6321	707	2	3	3	X
ejpam-6321	707	3	]	]	X
ejpam-6321	707	4	g.	g.	NOUN
ejpam-6321	707	5	minak	minak	PROPN
ejpam-6321	707	6	,	,	PUNCT
ejpam-6321	707	7	m.	m.	NOUN
ejpam-6321	707	8	olgun	olgun	NOUN
ejpam-6321	707	9	,	,	PUNCT
ejpam-6321	707	10	and	and	CCONJ
ejpam-6321	707	11	i.	i.	NOUN
ejpam-6321	707	12	altun	altun	PROPN
ejpam-6321	707	13	.	.	PUNCT
ejpam-6321	708	1	a	a	DET
ejpam-6321	708	2	new	new	ADJ
ejpam-6321	708	3	approach	approach	NOUN
ejpam-6321	708	4	to	to	ADP
ejpam-6321	708	5	fixed	fix	VERB
ejpam-6321	708	6	point	point	NOUN
ejpam-6321	708	7	theorems	theorem	NOUN
ejpam-6321	708	8	for	for	ADP
ejpam-6321	708	9	multivalued	multivalued	ADJ
ejpam-6321	708	10	contractive	contractive	ADJ
ejpam-6321	708	11	maps	map	NOUN
ejpam-6321	708	12	.	.	PUNCT
ejpam-6321	709	1	carpathian	carpathian	ADJ
ejpam-6321	709	2	journal	journal	PROPN
ejpam-6321	709	3	of	of	ADP
ejpam-6321	709	4	mathematics	mathematic	NOUN
ejpam-6321	709	5	,	,	PUNCT
ejpam-6321	709	6	pages	page	NOUN
ejpam-6321	709	7	241–248	241–248	NUM
ejpam-6321	709	8	,	,	PUNCT
ejpam-6321	709	9	2015	2015	NUM
ejpam-6321	709	10	.	.	PUNCT
ejpam-6321	710	1	[	[	X
ejpam-6321	710	2	4	4	X
ejpam-6321	710	3	]	]	X
ejpam-6321	710	4	f.	f.	PROPN
ejpam-6321	710	5	sullivan	sullivan	PROPN
ejpam-6321	710	6	.	.	PUNCT
ejpam-6321	711	1	a	a	DET
ejpam-6321	711	2	characterization	characterization	NOUN
ejpam-6321	711	3	of	of	ADP
ejpam-6321	711	4	complete	complete	ADJ
ejpam-6321	711	5	metric	metric	ADJ
ejpam-6321	711	6	spaces	space	NOUN
ejpam-6321	711	7	.	.	PUNCT
ejpam-6321	712	1	proceedings	proceeding	NOUN
ejpam-6321	712	2	of	of	ADP
ejpam-6321	712	3	the	the	DET
ejpam-6321	712	4	american	american	PROPN
ejpam-6321	712	5	mathematical	mathematical	PROPN
ejpam-6321	712	6	society	society	NOUN
ejpam-6321	712	7	,	,	PUNCT
ejpam-6321	712	8	83(2):345–346	83(2):345–346	NUM
ejpam-6321	712	9	,	,	PUNCT
ejpam-6321	712	10	1981	1981	NUM
ejpam-6321	712	11	.	.	PUNCT
ejpam-6321	713	1	[	[	X
ejpam-6321	713	2	5	5	NUM
ejpam-6321	713	3	]	]	PUNCT
ejpam-6321	713	4	a.	a.	NOUN
ejpam-6321	713	5	george	george	PROPN
ejpam-6321	713	6	and	and	CCONJ
ejpam-6321	713	7	p.	p.	PROPN
ejpam-6321	713	8	veeramani	veeramani	PROPN
ejpam-6321	713	9	.	.	PUNCT
ejpam-6321	714	1	on	on	ADP
ejpam-6321	714	2	some	some	DET
ejpam-6321	714	3	results	result	NOUN
ejpam-6321	714	4	in	in	ADP
ejpam-6321	714	5	fuzzy	fuzzy	ADJ
ejpam-6321	714	6	metric	metric	ADJ
ejpam-6321	714	7	spaces	space	NOUN
ejpam-6321	714	8	.	.	PUNCT
ejpam-6321	715	1	fuzzy	fuzzy	ADJ
ejpam-6321	715	2	sets	set	NOUN
ejpam-6321	715	3	and	and	CCONJ
ejpam-6321	715	4	systems	system	NOUN
ejpam-6321	715	5	,	,	PUNCT
ejpam-6321	715	6	64(3):395–399	64(3):395–399	PROPN
ejpam-6321	715	7	,	,	PUNCT
ejpam-6321	715	8	1994	1994	NUM
ejpam-6321	715	9	.	.	PUNCT
ejpam-6321	716	1	[	[	X
ejpam-6321	716	2	6	6	NUM
ejpam-6321	716	3	]	]	PUNCT
ejpam-6321	716	4	t.	t.	NOUN
ejpam-6321	716	5	g.	g.	PROPN
ejpam-6321	716	6	bhaskar	bhaskar	PROPN
ejpam-6321	716	7	and	and	CCONJ
ejpam-6321	716	8	v.	v.	ADP
ejpam-6321	716	9	lakshmikantham	lakshmikantham	NOUN
ejpam-6321	716	10	.	.	PUNCT
ejpam-6321	717	1	fixed	fix	VERB
ejpam-6321	717	2	point	point	NOUN
ejpam-6321	717	3	theorems	theorem	NOUN
ejpam-6321	717	4	in	in	ADP
ejpam-6321	717	5	partially	partially	ADV
ejpam-6321	717	6	ordered	order	VERB
ejpam-6321	717	7	metric	metric	ADJ
ejpam-6321	717	8	spaces	space	NOUN
ejpam-6321	717	9	and	and	CCONJ
ejpam-6321	717	10	applications	application	NOUN
ejpam-6321	717	11	.	.	PUNCT
ejpam-6321	718	1	nonlinear	nonlinear	ADJ
ejpam-6321	718	2	analysis	analysis	NOUN
ejpam-6321	718	3	:	:	PUNCT
ejpam-6321	718	4	theory	theory	NOUN
ejpam-6321	718	5	,	,	PUNCT
ejpam-6321	718	6	methods	method	NOUN
ejpam-6321	718	7	&	&	CCONJ
ejpam-6321	718	8	applications	application	NOUN
ejpam-6321	718	9	,	,	PUNCT
ejpam-6321	718	10	65(7):1379	65(7):1379	NUM
ejpam-6321	718	11	–	–	PUNCT
ejpam-6321	718	12	1393	1393	NUM
ejpam-6321	718	13	,	,	PUNCT
ejpam-6321	718	14	2006	2006	NUM
ejpam-6321	718	15	.	.	PUNCT
ejpam-6321	719	1	[	[	X
ejpam-6321	719	2	7	7	X
ejpam-6321	719	3	]	]	X
ejpam-6321	719	4	m.	m.	NOUN
ejpam-6321	719	5	a.	a.	NOUN
ejpam-6321	719	6	kutbi	kutbi	PROPN
ejpam-6321	719	7	,	,	PUNCT
ejpam-6321	719	8	j.	j.	PROPN
ejpam-6321	719	9	ahmad	ahmad	PROPN
ejpam-6321	719	10	,	,	PUNCT
ejpam-6321	719	11	n.	n.	PROPN
ejpam-6321	719	12	hussain	hussain	PROPN
ejpam-6321	719	13	,	,	PUNCT
ejpam-6321	719	14	and	and	CCONJ
ejpam-6321	719	15	m.	m.	PROPN
ejpam-6321	719	16	arshad	arshad	PROPN
ejpam-6321	719	17	.	.	PUNCT
ejpam-6321	720	1	common	common	ADJ
ejpam-6321	720	2	fixed	fix	VERB
ejpam-6321	720	3	point	point	NOUN
ejpam-6321	720	4	results	result	NOUN
ejpam-6321	720	5	for	for	ADP
ejpam-6321	720	6	mappings	mapping	NOUN
ejpam-6321	720	7	with	with	ADP
ejpam-6321	720	8	rational	rational	ADJ
ejpam-6321	720	9	expressions	expression	NOUN
ejpam-6321	720	10	.	.	PUNCT
ejpam-6321	721	1	abstract	abstract	ADJ
ejpam-6321	721	2	and	and	CCONJ
ejpam-6321	721	3	applied	apply	VERB
ejpam-6321	721	4	analysis	analysis	NOUN
ejpam-6321	721	5	,	,	PUNCT
ejpam-6321	721	6	2013	2013	NUM
ejpam-6321	721	7	,	,	PUNCT
ejpam-6321	721	8	2013	2013	NUM
ejpam-6321	721	9	.	.	PUNCT
ejpam-6321	722	1	[	[	X
ejpam-6321	722	2	8	8	NUM
ejpam-6321	722	3	]	]	PUNCT
ejpam-6321	722	4	m.	m.	NOUN
ejpam-6321	722	5	t.	t.	PROPN
ejpam-6321	722	6	yaseen	yaseen	PROPN
ejpam-6321	722	7	,	,	PUNCT
ejpam-6321	722	8	a.	a.	PROPN
ejpam-6321	722	9	h.	h.	PROPN
ejpam-6321	722	10	ali	ali	PROPN
ejpam-6321	722	11	,	,	PUNCT
ejpam-6321	722	12	a.	a.	PROPN
ejpam-6321	722	13	a.	a.	PROPN
ejpam-6321	722	14	al	al	PROPN
ejpam-6321	722	15	-	-	PUNCT
ejpam-6321	722	16	moneef	moneef	PROPN
ejpam-6321	722	17	,	,	PUNCT
ejpam-6321	722	18	o.	o.	PROPN
ejpam-6321	722	19	bazighifan	bazighifan	PROPN
ejpam-6321	722	20	,	,	PUNCT
ejpam-6321	722	21	t.	t.	PROPN
ejpam-6321	722	22	a.	a.	NOUN
ejpam-6321	722	23	nofal	nofal	PROPN
ejpam-6321	722	24	,	,	PUNCT
ejpam-6321	722	25	and	and	CCONJ
ejpam-6321	722	26	f.	f.	PROPN
ejpam-6321	722	27	ghanim	ghanim	PROPN
ejpam-6321	722	28	.	.	PUNCT
ejpam-6321	723	1	new	new	ADJ
ejpam-6321	723	2	results	result	NOUN
ejpam-6321	723	3	of	of	ADP
ejpam-6321	723	4	fixed	fix	VERB
ejpam-6321	723	5	-	-	PUNCT
ejpam-6321	723	6	point	point	NOUN
ejpam-6321	723	7	theorems	theorem	NOUN
ejpam-6321	723	8	in	in	ADP
ejpam-6321	723	9	complete	complete	ADJ
ejpam-6321	723	10	metric	metric	ADJ
ejpam-6321	723	11	spaces	space	NOUN
ejpam-6321	723	12	.	.	PUNCT
ejpam-6321	724	1	mathematical	mathematical	ADJ
ejpam-6321	724	2	problems	problem	NOUN
ejpam-6321	724	3	in	in	ADP
ejpam-6321	724	4	engineering	engineering	NOUN
ejpam-6321	724	5	,	,	PUNCT
ejpam-6321	724	6	2022	2022	NUM
ejpam-6321	724	7	,	,	PUNCT
ejpam-6321	724	8	2022	2022	NUM
ejpam-6321	724	9	.	.	PUNCT
ejpam-6321	725	1	[	[	X
ejpam-6321	725	2	9	9	NUM
ejpam-6321	725	3	]	]	X
ejpam-6321	725	4	b.	b.	PROPN
ejpam-6321	725	5	e.	e.	PROPN
ejpam-6321	725	6	rhoades	rhoades	PROPN
ejpam-6321	725	7	.	.	PUNCT
ejpam-6321	726	1	a	a	DET
ejpam-6321	726	2	fixed	fix	VERB
ejpam-6321	726	3	point	point	NOUN
ejpam-6321	726	4	theorem	theorem	NOUN
ejpam-6321	726	5	for	for	ADP
ejpam-6321	726	6	generalized	generalized	ADJ
ejpam-6321	726	7	metric	metric	ADJ
ejpam-6321	726	8	spaces	space	NOUN
ejpam-6321	726	9	.	.	PUNCT
ejpam-6321	727	1	international	international	ADJ
ejpam-6321	727	2	journal	journal	PROPN
ejpam-6321	727	3	of	of	ADP
ejpam-6321	727	4	mathematics	mathematics	PROPN
ejpam-6321	727	5	and	and	CCONJ
ejpam-6321	727	6	mathematical	mathematical	ADJ
ejpam-6321	727	7	sciences	science	NOUN
ejpam-6321	727	8	,	,	PUNCT
ejpam-6321	727	9	19:457–460	19:457–460	NUM
ejpam-6321	727	10	,	,	PUNCT
ejpam-6321	727	11	1996	1996	NUM
ejpam-6321	727	12	.	.	PUNCT
ejpam-6321	728	1	[	[	X
ejpam-6321	728	2	10	10	NUM
ejpam-6321	728	3	]	]	PUNCT
ejpam-6321	728	4	z.	z.	PROPN
ejpam-6321	728	5	mustafa	mustafa	PROPN
ejpam-6321	728	6	and	and	CCONJ
ejpam-6321	728	7	b.	b.	PROPN
ejpam-6321	728	8	sims	sim	NOUN
ejpam-6321	728	9	.	.	PUNCT
ejpam-6321	729	1	a	a	DET
ejpam-6321	729	2	new	new	ADJ
ejpam-6321	729	3	approach	approach	NOUN
ejpam-6321	729	4	to	to	ADP
ejpam-6321	729	5	generalized	generalize	VERB
ejpam-6321	729	6	metric	metric	ADJ
ejpam-6321	729	7	spaces	space	NOUN
ejpam-6321	729	8	.	.	PUNCT
ejpam-6321	730	1	journal	journal	PROPN
ejpam-6321	730	2	of	of	ADP
ejpam-6321	730	3	nonlinear	nonlinear	ADJ
ejpam-6321	730	4	and	and	CCONJ
ejpam-6321	730	5	convex	convex	ADJ
ejpam-6321	730	6	analysis	analysis	NOUN
ejpam-6321	730	7	,	,	PUNCT
ejpam-6321	730	8	7(2):289	7(2):289	NUM
ejpam-6321	730	9	,	,	PUNCT
ejpam-6321	730	10	2006	2006	NUM
ejpam-6321	730	11	.	.	PUNCT
ejpam-6321	731	1	[	[	X
ejpam-6321	731	2	11	11	NUM
ejpam-6321	731	3	]	]	X
ejpam-6321	731	4	a.	a.	NOUN
ejpam-6321	731	5	azam	azam	PROPN
ejpam-6321	731	6	and	and	CCONJ
ejpam-6321	731	7	m.	m.	PROPN
ejpam-6321	731	8	arshad	arshad	PROPN
ejpam-6321	731	9	.	.	PUNCT
ejpam-6321	732	1	kannan	kannan	PROPN
ejpam-6321	732	2	fixed	fix	VERB
ejpam-6321	732	3	point	point	NOUN
ejpam-6321	732	4	theorem	theorem	VERB
ejpam-6321	732	5	on	on	ADP
ejpam-6321	732	6	generalized	generalized	ADJ
ejpam-6321	732	7	metric	metric	ADJ
ejpam-6321	732	8	spaces	space	NOUN
ejpam-6321	732	9	.	.	PUNCT
ejpam-6321	733	1	the	the	DET
ejpam-6321	733	2	journal	journal	PROPN
ejpam-6321	733	3	of	of	ADP
ejpam-6321	733	4	nonlinear	nonlinear	PROPN
ejpam-6321	733	5	sciences	sciences	PROPN
ejpam-6321	733	6	and	and	CCONJ
ejpam-6321	733	7	its	its	PRON
ejpam-6321	733	8	applications	application	NOUN
ejpam-6321	733	9	,	,	PUNCT
ejpam-6321	733	10	1(1):45–48	1(1):45–48	NUM
ejpam-6321	733	11	,	,	PUNCT
ejpam-6321	733	12	2008	2008	NUM
ejpam-6321	733	13	.	.	PUNCT
ejpam-6321	734	1	m.	m.	PROPN
ejpam-6321	734	2	noorwali	noorwali	PROPN
ejpam-6321	734	3	et	et	PROPN
ejpam-6321	734	4	al	al	PROPN
ejpam-6321	734	5	.	.	PUNCT
ejpam-6321	734	6	/	/	SYM
ejpam-6321	734	7	eur	eur	PROPN
ejpam-6321	734	8	.	.	PUNCT
ejpam-6321	735	1	j.	j.	PROPN
ejpam-6321	735	2	pure	pure	PROPN
ejpam-6321	735	3	appl	appl	PROPN
ejpam-6321	735	4	.	.	PROPN
ejpam-6321	735	5	math	math	PROPN
ejpam-6321	735	6	,	,	PUNCT
ejpam-6321	735	7	18	18	NUM
ejpam-6321	735	8	(	(	PUNCT
ejpam-6321	735	9	3	3	NUM
ejpam-6321	735	10	)	)	PUNCT
ejpam-6321	735	11	(	(	PUNCT
ejpam-6321	735	12	2025	2025	NUM
ejpam-6321	735	13	)	)	PUNCT
ejpam-6321	735	14	,	,	PUNCT
ejpam-6321	735	15	6321	6321	NUM
ejpam-6321	735	16	27	27	NUM
ejpam-6321	735	17	of	of	ADP
ejpam-6321	735	18	27	27	NUM
ejpam-6321	735	19	[	[	X
ejpam-6321	735	20	12	12	NUM
ejpam-6321	735	21	]	]	X
ejpam-6321	735	22	i.	i.	PROPN
ejpam-6321	735	23	r.	r.	PROPN
ejpam-6321	735	24	sarma	sarma	PROPN
ejpam-6321	735	25	,	,	PUNCT
ejpam-6321	735	26	j.	j.	PROPN
ejpam-6321	735	27	m.	m.	PROPN
ejpam-6321	735	28	rao	rao	PROPN
ejpam-6321	735	29	,	,	PUNCT
ejpam-6321	735	30	and	and	CCONJ
ejpam-6321	735	31	s.	s.	PROPN
ejpam-6321	735	32	s.	s.	PROPN
ejpam-6321	735	33	rao	rao	PROPN
ejpam-6321	735	34	.	.	PUNCT
ejpam-6321	736	1	contractions	contraction	NOUN
ejpam-6321	736	2	over	over	ADP
ejpam-6321	736	3	generalized	generalized	ADJ
ejpam-6321	736	4	metric	metric	ADJ
ejpam-6321	736	5	spaces	space	NOUN
ejpam-6321	736	6	.	.	PUNCT
ejpam-6321	737	1	the	the	DET
ejpam-6321	737	2	journal	journal	PROPN
ejpam-6321	737	3	of	of	ADP
ejpam-6321	737	4	nonlinear	nonlinear	PROPN
ejpam-6321	737	5	sciences	sciences	PROPN
ejpam-6321	737	6	and	and	CCONJ
ejpam-6321	737	7	its	its	PRON
ejpam-6321	737	8	applications	application	NOUN
ejpam-6321	737	9	,	,	PUNCT
ejpam-6321	737	10	2(3):180–182	2(3):180–182	NUM
ejpam-6321	737	11	,	,	PUNCT
ejpam-6321	737	12	2009	2009	NUM
ejpam-6321	737	13	.	.	PUNCT
ejpam-6321	738	1	[	[	X
ejpam-6321	738	2	13	13	NUM
ejpam-6321	738	3	]	]	X
ejpam-6321	738	4	d.	d.	PROPN
ejpam-6321	738	5	mihet	mihet	PROPN
ejpam-6321	738	6	.	.	PUNCT
ejpam-6321	739	1	on	on	ADP
ejpam-6321	739	2	kannan	kannan	PROPN
ejpam-6321	739	3	fixed	fix	VERB
ejpam-6321	739	4	point	point	PROPN
ejpam-6321	739	5	principle	principle	NOUN
ejpam-6321	739	6	in	in	ADP
ejpam-6321	739	7	generalized	generalized	ADJ
ejpam-6321	739	8	metric	metric	ADJ
ejpam-6321	739	9	spaces	space	NOUN
ejpam-6321	739	10	.	.	PUNCT
ejpam-6321	740	1	the	the	DET
ejpam-6321	740	2	journal	journal	PROPN
ejpam-6321	740	3	of	of	ADP
ejpam-6321	740	4	nonlinear	nonlinear	PROPN
ejpam-6321	740	5	sciences	sciences	PROPN
ejpam-6321	740	6	and	and	CCONJ
ejpam-6321	740	7	its	its	PRON
ejpam-6321	740	8	applications	application	NOUN
ejpam-6321	740	9	,	,	PUNCT
ejpam-6321	740	10	2(2):92–96	2(2):92–96	NUM
ejpam-6321	740	11	,	,	PUNCT
ejpam-6321	740	12	2009	2009	NUM
ejpam-6321	740	13	.	.	PUNCT
ejpam-6321	741	1	[	[	X
ejpam-6321	741	2	14	14	NUM
ejpam-6321	741	3	]	]	PUNCT
ejpam-6321	741	4	p.	p.	NOUN
ejpam-6321	741	5	das	das	PROPN
ejpam-6321	741	6	and	and	CCONJ
ejpam-6321	741	7	l.	l.	PROPN
ejpam-6321	741	8	k.	k.	PROPN
ejpam-6321	741	9	dey	dey	PROPN
ejpam-6321	741	10	.	.	PROPN
ejpam-6321	741	11	fixed	fix	VERB
ejpam-6321	741	12	point	point	NOUN
ejpam-6321	741	13	of	of	ADP
ejpam-6321	741	14	contractive	contractive	ADJ
ejpam-6321	741	15	mappings	mapping	NOUN
ejpam-6321	741	16	in	in	ADP
ejpam-6321	741	17	generalized	generalized	ADJ
ejpam-6321	741	18	metric	metric	ADJ
ejpam-6321	741	19	spaces	space	NOUN
ejpam-6321	741	20	.	.	PUNCT
ejpam-6321	742	1	mathematica	mathematica	PROPN
ejpam-6321	742	2	slovaca	slovaca	PROPN
ejpam-6321	742	3	,	,	PUNCT
ejpam-6321	742	4	59(4):499–504	59(4):499–504	PROPN
ejpam-6321	742	5	,	,	PUNCT
ejpam-6321	742	6	2009	2009	NUM
ejpam-6321	742	7	.	.	PUNCT
ejpam-6321	743	1	[	[	X
ejpam-6321	743	2	15	15	NUM
ejpam-6321	743	3	]	]	X
ejpam-6321	743	4	w.	w.	PROPN
ejpam-6321	743	5	shatanawi	shatanawi	PROPN
ejpam-6321	743	6	.	.	PUNCT
ejpam-6321	744	1	coupled	couple	VERB
ejpam-6321	744	2	fixed	fix	VERB
ejpam-6321	744	3	point	point	NOUN
ejpam-6321	744	4	theorems	theorem	NOUN
ejpam-6321	744	5	in	in	ADP
ejpam-6321	744	6	generalized	generalized	ADJ
ejpam-6321	744	7	metric	metric	ADJ
ejpam-6321	744	8	spaces	space	NOUN
ejpam-6321	744	9	.	.	PUNCT
ejpam-6321	745	1	hacettepe	hacettepe	PROPN
ejpam-6321	745	2	journal	journal	PROPN
ejpam-6321	745	3	of	of	ADP
ejpam-6321	745	4	mathematics	mathematic	NOUN
ejpam-6321	745	5	and	and	CCONJ
ejpam-6321	745	6	statistics	statistic	NOUN
ejpam-6321	745	7	,	,	PUNCT
ejpam-6321	745	8	40(3):441–447	40(3):441–447	NOUN
ejpam-6321	745	9	,	,	PUNCT
ejpam-6321	745	10	2011	2011	NUM
ejpam-6321	745	11	.	.	PUNCT
ejpam-6321	746	1	[	[	X
ejpam-6321	746	2	16	16	NUM
ejpam-6321	746	3	]	]	X
ejpam-6321	746	4	c.	c.	PROPN
ejpam-6321	746	5	di	di	PROPN
ejpam-6321	746	6	bari	bari	NOUN
ejpam-6321	746	7	and	and	CCONJ
ejpam-6321	746	8	p.	p.	NOUN
ejpam-6321	746	9	vetro	vetro	PROPN
ejpam-6321	746	10	.	.	PUNCT
ejpam-6321	747	1	common	common	ADJ
ejpam-6321	747	2	fixed	fix	VERB
ejpam-6321	747	3	points	point	NOUN
ejpam-6321	747	4	in	in	ADP
ejpam-6321	747	5	generalized	generalized	ADJ
ejpam-6321	747	6	metric	metric	ADJ
ejpam-6321	747	7	spaces	space	NOUN
ejpam-6321	747	8	.	.	PUNCT
ejpam-6321	748	1	applied	apply	VERB
ejpam-6321	748	2	mathematics	mathematic	NOUN
ejpam-6321	748	3	and	and	CCONJ
ejpam-6321	748	4	computation	computation	NOUN
ejpam-6321	748	5	,	,	PUNCT
ejpam-6321	748	6	218(13):7322–7325	218(13):7322–7325	NUM
ejpam-6321	748	7	,	,	PUNCT
ejpam-6321	748	8	2012	2012	NUM
ejpam-6321	748	9	.	.	PUNCT
ejpam-6321	749	1	[	[	X
ejpam-6321	749	2	17	17	NUM
ejpam-6321	749	3	]	]	X
ejpam-6321	749	4	s.	s.	PROPN
ejpam-6321	749	5	romaguera	romaguera	PROPN
ejpam-6321	749	6	.	.	PUNCT
ejpam-6321	750	1	fixed	fix	VERB
ejpam-6321	750	2	point	point	NOUN
ejpam-6321	750	3	theorems	theorem	NOUN
ejpam-6321	750	4	for	for	ADP
ejpam-6321	750	5	generalized	generalized	ADJ
ejpam-6321	750	6	contractions	contraction	NOUN
ejpam-6321	750	7	on	on	ADP
ejpam-6321	750	8	partial	partial	ADJ
ejpam-6321	750	9	metric	metric	ADJ
ejpam-6321	750	10	spaces	space	NOUN
ejpam-6321	750	11	.	.	PUNCT
ejpam-6321	751	1	topology	topology	NOUN
ejpam-6321	751	2	and	and	CCONJ
ejpam-6321	751	3	its	its	PRON
ejpam-6321	751	4	applications	application	NOUN
ejpam-6321	751	5	,	,	PUNCT
ejpam-6321	751	6	159(1):194–199	159(1):194–199	NUM
ejpam-6321	751	7	,	,	PUNCT
ejpam-6321	751	8	2012	2012	NUM
ejpam-6321	751	9	.	.	PUNCT
ejpam-6321	752	1	[	[	X
ejpam-6321	752	2	18	18	NUM
ejpam-6321	752	3	]	]	PUNCT
ejpam-6321	752	4	s.	s.	PROPN
ejpam-6321	752	5	k.	k.	PROPN
ejpam-6321	752	6	mohanta	mohanta	PROPN
ejpam-6321	752	7	and	and	CCONJ
ejpam-6321	752	8	s.	s.	PROPN
ejpam-6321	752	9	mohanta	mohanta	PROPN
ejpam-6321	752	10	.	.	PUNCT
ejpam-6321	753	1	a	a	DET
ejpam-6321	753	2	common	common	ADJ
ejpam-6321	753	3	fixed	fix	VERB
ejpam-6321	753	4	point	point	NOUN
ejpam-6321	753	5	theorem	theorem	VERB
ejpam-6321	753	6	in	in	ADP
ejpam-6321	753	7	g	g	NOUN
ejpam-6321	753	8	-	-	PUNCT
ejpam-6321	753	9	metric	metric	ADJ
ejpam-6321	753	10	spaces	space	NOUN
ejpam-6321	753	11	.	.	PUNCT
ejpam-6321	754	1	cubo	cubo	NOUN
ejpam-6321	754	2	(	(	PUNCT
ejpam-6321	754	3	temuco	temuco	PROPN
ejpam-6321	754	4	)	)	PUNCT
ejpam-6321	754	5	,	,	PUNCT
ejpam-6321	754	6	14(3):85–101	14(3):85–101	NUM
ejpam-6321	754	7	,	,	PUNCT
ejpam-6321	754	8	2012	2012	NUM
ejpam-6321	754	9	.	.	PUNCT
ejpam-6321	755	1	[	[	X
ejpam-6321	755	2	19	19	NUM
ejpam-6321	755	3	]	]	PUNCT
ejpam-6321	755	4	m.	m.	NOUN
ejpam-6321	755	5	gugnani	gugnani	PROPN
ejpam-6321	755	6	,	,	PUNCT
ejpam-6321	755	7	m.	m.	NOUN
ejpam-6321	755	8	aggarwal	aggarwal	NOUN
ejpam-6321	755	9	,	,	PUNCT
ejpam-6321	755	10	and	and	CCONJ
ejpam-6321	755	11	r.	r.	PROPN
ejpam-6321	755	12	chugh	chugh	NOUN
ejpam-6321	755	13	.	.	PUNCT
ejpam-6321	756	1	common	common	ADJ
ejpam-6321	756	2	fixed	fix	VERB
ejpam-6321	756	3	point	point	NOUN
ejpam-6321	756	4	results	result	NOUN
ejpam-6321	756	5	in	in	ADP
ejpam-6321	756	6	g	g	NOUN
ejpam-6321	756	7	-	-	PUNCT
ejpam-6321	756	8	metric	metric	ADJ
ejpam-6321	756	9	spaces	space	NOUN
ejpam-6321	756	10	and	and	CCONJ
ejpam-6321	756	11	applications	application	NOUN
ejpam-6321	756	12	.	.	PUNCT
ejpam-6321	757	1	international	international	ADJ
ejpam-6321	757	2	journal	journal	PROPN
ejpam-6321	757	3	of	of	ADP
ejpam-6321	757	4	computer	computer	NOUN
ejpam-6321	757	5	applications	application	NOUN
ejpam-6321	757	6	,	,	PUNCT
ejpam-6321	757	7	43(11):38–42	43(11):38–42	NOUN
ejpam-6321	757	8	,	,	PUNCT
ejpam-6321	757	9	2012	2012	NUM
ejpam-6321	757	10	.	.	PUNCT
ejpam-6321	758	1	[	[	X
ejpam-6321	758	2	20	20	NUM
ejpam-6321	758	3	]	]	X
ejpam-6321	758	4	h.	h.	PROPN
ejpam-6321	758	5	aydi	aydi	VERB
ejpam-6321	758	6	.	.	PUNCT
ejpam-6321	759	1	a	a	DET
ejpam-6321	759	2	common	common	ADJ
ejpam-6321	759	3	fixed	fix	VERB
ejpam-6321	759	4	point	point	NOUN
ejpam-6321	759	5	of	of	ADP
ejpam-6321	759	6	integral	integral	ADJ
ejpam-6321	759	7	type	type	NOUN
ejpam-6321	759	8	contraction	contraction	NOUN
ejpam-6321	759	9	in	in	ADP
ejpam-6321	759	10	generalized	generalized	ADJ
ejpam-6321	759	11	metric	metric	ADJ
ejpam-6321	759	12	spaces	space	NOUN
ejpam-6321	759	13	.	.	PUNCT
ejpam-6321	760	1	journal	journal	NOUN
ejpam-6321	760	2	of	of	ADP
ejpam-6321	760	3	advanced	advanced	ADJ
ejpam-6321	760	4	mathematical	mathematical	ADJ
ejpam-6321	760	5	studies	study	NOUN
ejpam-6321	760	6	,	,	PUNCT
ejpam-6321	760	7	5(1):111–118	5(1):111–118	NUM
ejpam-6321	760	8	,	,	PUNCT
ejpam-6321	760	9	2012	2012	NUM
ejpam-6321	760	10	.	.	PUNCT
ejpam-6321	761	1	[	[	X
ejpam-6321	761	2	21	21	NUM
ejpam-6321	761	3	]	]	PUNCT
ejpam-6321	761	4	z.	z.	PROPN
ejpam-6321	761	5	kadelburg	kadelburg	PROPN
ejpam-6321	761	6	and	and	CCONJ
ejpam-6321	761	7	s.	s.	PROPN
ejpam-6321	761	8	radenovic	radenovic	PROPN
ejpam-6321	761	9	.	.	PUNCT
ejpam-6321	762	1	on	on	ADP
ejpam-6321	762	2	generalized	generalized	ADJ
ejpam-6321	762	3	metric	metric	ADJ
ejpam-6321	762	4	spaces	space	NOUN
ejpam-6321	762	5	:	:	PUNCT
ejpam-6321	762	6	a	a	DET
ejpam-6321	762	7	survey	survey	NOUN
ejpam-6321	762	8	.	.	PUNCT
ejpam-6321	763	1	twms	twms	PROPN
ejpam-6321	763	2	journal	journal	PROPN
ejpam-6321	763	3	of	of	ADP
ejpam-6321	763	4	pure	pure	ADJ
ejpam-6321	763	5	and	and	CCONJ
ejpam-6321	763	6	applied	applied	ADJ
ejpam-6321	763	7	mathematics	mathematic	NOUN
ejpam-6321	763	8	,	,	PUNCT
ejpam-6321	763	9	5(1):3–13	5(1):3–13	NUM
ejpam-6321	763	10	,	,	PUNCT
ejpam-6321	763	11	2014	2014	NUM
ejpam-6321	763	12	.	.	PUNCT
ejpam-6321	764	1	[	[	X
ejpam-6321	764	2	22	22	NUM
ejpam-6321	764	3	]	]	PUNCT
ejpam-6321	764	4	m.	m.	NOUN
ejpam-6321	764	5	jleli	jleli	PROPN
ejpam-6321	764	6	and	and	CCONJ
ejpam-6321	764	7	b.	b.	PROPN
ejpam-6321	764	8	samet	samet	PROPN
ejpam-6321	764	9	.	.	PUNCT
ejpam-6321	765	1	a	a	DET
ejpam-6321	765	2	generalized	generalize	VERB
ejpam-6321	765	3	metric	metric	ADJ
ejpam-6321	765	4	space	space	NOUN
ejpam-6321	765	5	and	and	CCONJ
ejpam-6321	765	6	related	relate	VERB
ejpam-6321	765	7	fixed	fix	VERB
ejpam-6321	765	8	point	point	NOUN
ejpam-6321	765	9	theorems	theorem	NOUN
ejpam-6321	765	10	.	.	PUNCT
ejpam-6321	766	1	fixed	fix	VERB
ejpam-6321	766	2	point	point	NOUN
ejpam-6321	766	3	theory	theory	NOUN
ejpam-6321	766	4	and	and	CCONJ
ejpam-6321	766	5	applications	application	NOUN
ejpam-6321	766	6	,	,	PUNCT
ejpam-6321	766	7	pages	page	NOUN
ejpam-6321	766	8	1–14	1–14	PROPN
ejpam-6321	766	9	,	,	PUNCT
ejpam-6321	766	10	2015	2015	NUM
ejpam-6321	766	11	.	.	PUNCT
ejpam-6321	767	1	[	[	X
ejpam-6321	767	2	23	23	NUM
ejpam-6321	767	3	]	]	PUNCT
ejpam-6321	767	4	s.	s.	PROPN
ejpam-6321	767	5	s.	s.	PROPN
ejpam-6321	767	6	abed	abe	VERB
ejpam-6321	767	7	and	and	CCONJ
ejpam-6321	767	8	h.	h.	PROPN
ejpam-6321	767	9	h.	h.	PROPN
ejpam-6321	767	10	luaibi	luaibi	PROPN
ejpam-6321	767	11	.	.	PUNCT
ejpam-6321	768	1	two	two	NUM
ejpam-6321	768	2	fixed	fix	VERB
ejpam-6321	768	3	point	point	NOUN
ejpam-6321	768	4	theorems	theorem	NOUN
ejpam-6321	768	5	in	in	ADP
ejpam-6321	768	6	generalized	generalized	ADJ
ejpam-6321	768	7	metric	metric	ADJ
ejpam-6321	768	8	spaces	space	NOUN
ejpam-6321	768	9	.	.	PUNCT
ejpam-6321	769	1	international	international	ADJ
ejpam-6321	769	2	journal	journal	NOUN
ejpam-6321	769	3	of	of	ADP
ejpam-6321	769	4	advanced	advanced	ADJ
ejpam-6321	769	5	statistics	statistic	NOUN
ejpam-6321	769	6	and	and	CCONJ
ejpam-6321	769	7	probability	probability	NOUN
ejpam-6321	769	8	,	,	PUNCT
ejpam-6321	769	9	4(1):16–19	4(1):16–19	NUM
ejpam-6321	769	10	,	,	PUNCT
ejpam-6321	769	11	2016	2016	NUM
ejpam-6321	769	12	.	.	PUNCT
ejpam-6321	770	1	[	[	X
ejpam-6321	770	2	24	24	NUM
ejpam-6321	770	3	]	]	X
ejpam-6321	770	4	s.	s.	PROPN
ejpam-6321	770	5	lin	lin	PROPN
ejpam-6321	770	6	and	and	CCONJ
ejpam-6321	770	7	z.	z.	PROPN
ejpam-6321	770	8	yun	yun	PROPN
ejpam-6321	770	9	.	.	PUNCT
ejpam-6321	771	1	generalized	generalize	VERB
ejpam-6321	771	2	metric	metric	ADJ
ejpam-6321	771	3	spaces	space	NOUN
ejpam-6321	771	4	and	and	CCONJ
ejpam-6321	771	5	mappings	mapping	NOUN
ejpam-6321	771	6	.	.	PUNCT
ejpam-6321	772	1	springer	springer	NOUN
ejpam-6321	772	2	,	,	PUNCT
ejpam-6321	772	3	2016	2016	NUM
ejpam-6321	772	4	.	.	PUNCT
ejpam-6321	773	1	[	[	X
ejpam-6321	773	2	25	25	NUM
ejpam-6321	773	3	]	]	PUNCT
ejpam-6321	773	4	l.	l.	PROPN
ejpam-6321	773	5	x.	x.	PROPN
ejpam-6321	773	6	peng	peng	PROPN
ejpam-6321	773	7	and	and	CCONJ
ejpam-6321	773	8	y.	y.	PROPN
ejpam-6321	773	9	sun	sun	PROPN
ejpam-6321	773	10	.	.	PUNCT
ejpam-6321	774	1	a	a	DET
ejpam-6321	774	2	study	study	NOUN
ejpam-6321	774	3	on	on	ADP
ejpam-6321	774	4	symmetric	symmetric	ADJ
ejpam-6321	774	5	products	product	NOUN
ejpam-6321	774	6	of	of	ADP
ejpam-6321	774	7	generalized	generalized	ADJ
ejpam-6321	774	8	metric	metric	ADJ
ejpam-6321	774	9	spaces	space	NOUN
ejpam-6321	774	10	.	.	PUNCT
ejpam-6321	775	1	topology	topology	NOUN
ejpam-6321	775	2	and	and	CCONJ
ejpam-6321	775	3	its	its	PRON
ejpam-6321	775	4	applications	application	NOUN
ejpam-6321	775	5	,	,	PUNCT
ejpam-6321	775	6	231:411–429	231:411–429	NUM
ejpam-6321	775	7	,	,	PUNCT
ejpam-6321	775	8	2017	2017	NUM
ejpam-6321	775	9	.	.	PUNCT