begin
:: deftheorem ASYMPT_0:def 1 :
canceled;
:: deftheorem ASYMPT_0:def 2 :
canceled;
:: deftheorem Def3 defines logbase ASYMPT_0:def 3 :
:: deftheorem Def4 defines eventually-nonnegative ASYMPT_0:def 4 :
:: deftheorem Def5 defines positive ASYMPT_0:def 5 :
:: deftheorem Def6 defines eventually-positive ASYMPT_0:def 6 :
:: deftheorem Def7 defines eventually-nonzero ASYMPT_0:def 7 :
:: deftheorem Def8 defines eventually-nondecreasing ASYMPT_0:def 8 :
:: deftheorem ASYMPT_0:def 9 :
canceled;
definition
let f,
g be
Real_Sequence;
func max f,
g -> Real_Sequence means :
Def10:
for
n being
Element of
NAT holds
it . n = max (f . n),
(g . n);
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = max (f . n),(g . n)
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = max (f . n),(g . n) ) & ( for n being Element of NAT holds b2 . n = max (f . n),(g . n) ) holds
b1 = b2
commutativity
for b1, f, g being Real_Sequence st ( for n being Element of NAT holds b1 . n = max (f . n),(g . n) ) holds
for n being Element of NAT holds b1 . n = max (g . n),(f . n)
;
end;
:: deftheorem Def10 defines max ASYMPT_0:def 10 :
:: deftheorem Def11 defines majorizes ASYMPT_0:def 11 :
Lm1:
for f, g being Real_Sequence
for n being Element of NAT holds (f /" g) . n = (f . n) / (g . n)
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem Th5:
begin
:: deftheorem defines Big_Oh ASYMPT_0:def 12 :
theorem Th6:
theorem
theorem
theorem
theorem Th10:
theorem Th11:
theorem Th12:
Lm2:
for f, g being eventually-nonnegative Real_Sequence holds
( ( f in Big_Oh g & g in Big_Oh f ) iff Big_Oh f = Big_Oh g )
theorem
theorem
Lm3:
for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds
f in Big_Oh g
theorem Th15:
theorem Th16:
theorem Th17:
begin
:: deftheorem defines Big_Omega ASYMPT_0:def 13 :
theorem
theorem Th19:
theorem Th20:
theorem Th21:
theorem
theorem Th23:
theorem
theorem
theorem
:: deftheorem defines Big_Theta ASYMPT_0:def 14 :
theorem Th27:
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
begin
:: deftheorem defines Big_Oh ASYMPT_0:def 15 :
:: deftheorem defines Big_Omega ASYMPT_0:def 16 :
:: deftheorem defines Big_Theta ASYMPT_0:def 17 :
theorem Th36:
:: deftheorem Def18 defines taken_every ASYMPT_0:def 18 :
:: deftheorem Def19 defines is_smooth_wrt ASYMPT_0:def 19 :
:: deftheorem Def20 defines smooth ASYMPT_0:def 20 :
theorem
theorem Th38:
theorem Th39:
theorem
begin
definition
let X be non
empty set ;
let F,
G be
FUNCTION_DOMAIN of
X,
REAL ;
func F + G -> FUNCTION_DOMAIN of
X,
REAL equals
{ t where t is Element of Funcs X,REAL : ex f, g being Element of Funcs X,REAL st
( f in F & g in G & ( for n being Element of X holds t . n = (f . n) + (g . n) ) ) } ;
coherence
{ t where t is Element of Funcs X,REAL : ex f, g being Element of Funcs X,REAL st
( f in F & g in G & ( for n being Element of X holds t . n = (f . n) + (g . n) ) ) } is FUNCTION_DOMAIN of X, REAL
end;
:: deftheorem defines + ASYMPT_0:def 21 :
definition
let X be non
empty set ;
let F,
G be
FUNCTION_DOMAIN of
X,
REAL ;
func max F,
G -> FUNCTION_DOMAIN of
X,
REAL equals
{ t where t is Element of Funcs X,REAL : ex f, g being Element of Funcs X,REAL st
( f in F & g in G & ( for n being Element of X holds t . n = max (f . n),(g . n) ) ) } ;
coherence
{ t where t is Element of Funcs X,REAL : ex f, g being Element of Funcs X,REAL st
( f in F & g in G & ( for n being Element of X holds t . n = max (f . n),(g . n) ) ) } is FUNCTION_DOMAIN of X, REAL
end;
:: deftheorem defines max ASYMPT_0:def 22 :
theorem
theorem
definition
let F,
G be
FUNCTION_DOMAIN of
NAT ,
REAL ;
func F to_power G -> FUNCTION_DOMAIN of
NAT ,
REAL equals
{ t where t is Element of Funcs NAT ,REAL : ex f, g being Element of Funcs NAT ,REAL ex N being Element of NAT st
( f in F & g in G & ( for n being Element of NAT st n >= N holds
t . n = (f . n) to_power (g . n) ) ) } ;
coherence
{ t where t is Element of Funcs NAT ,REAL : ex f, g being Element of Funcs NAT ,REAL ex N being Element of NAT st
( f in F & g in G & ( for n being Element of NAT st n >= N holds
t . n = (f . n) to_power (g . n) ) ) } is FUNCTION_DOMAIN of NAT , REAL
end;
:: deftheorem defines to_power ASYMPT_0:def 23 :