:: Asymptotic notation. Part I: Theory
:: by Richard Krueger , Piotr Rudnicki and Paul Shelley
::
:: Received November 4, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

scheme :: ASYMPT_0:sch 1
FinSegRng1{ F₁() -> Nat, F₂() -> Nat, F₃() -> non empty set , F₄( set ) -> Element of F₃() } :

{ F₄(i) where i is Element of NAT : ( F₁() <= i & i <= F₂() ) } is non empty finite Subset of F₃()

provided

A1: F₁() <= F₂()

proof end;

scheme :: ASYMPT_0:sch 2
FinImInit1{ F₁() -> Nat, F₂() -> non empty set , F₃( set ) -> Element of F₂() } :

{ F₃(n) where n is Element of NAT : n <= F₁() } is non empty finite Subset of F₂()

proof end;

scheme :: ASYMPT_0:sch 3
FinImInit2{ F₁() -> Nat, F₂() -> non empty set , F₃( set ) -> Element of F₂() } :

{ F₃(n) where n is Element of NAT : n < F₁() } is non empty finite Subset of F₂()

provided

A1: F₁() > 0

proof end;

definition

let c be Real;

attr c is logbase means :: ASYMPT_0:def 1
( c > 0 & c <> 1 );

end;

:: deftheorem defines logbase ASYMPT_0:def 1 :
for c being Real holds
( c is logbase iff ( c > 0 & c <> 1 ) );

reconsider jj = 1, dwa = 2 as Element of REAL by XREAL_0:def 1;

registration

cluster V19() ext-real positive V38() for object ;

existence
ex b₁ being Real st b₁ is positive

proof end;

cluster V19() ext-real negative V38() for object ;

existence
ex b₁ being Real st b₁ is negative

proof end;

cluster V19() ext-real V38() logbase for object ;

existence
ex b₁ being Real st b₁ is logbase

proof end;

cluster V19() ext-real non negative V38() for object ;

existence
not for b₁ being Real holds b₁ is negative by XXREAL_0:def 7;

cluster V19() ext-real non positive V38() for object ;

existence
not for b₁ being Real holds b₁ is positive by XXREAL_0:def 6;

cluster V19() ext-real V38() non logbase for object ;

existence
not for b₁ being Real holds b₁ is logbase

proof end;

end;

definition

let f be Real_Sequence;

attr f is eventually-nonnegative means :Def2: :: ASYMPT_0:def 2
ex N being Nat st
for n being Nat st n >= N holds
f . n >= 0 ;

attr f is positive means :Def3: :: ASYMPT_0:def 3
for n being Nat holds f . n > 0 ;

attr f is eventually-positive means :Def4: :: ASYMPT_0:def 4
ex N being Nat st
for n being Nat st n >= N holds
f . n > 0 ;

attr f is eventually-nonzero means :Def5: :: ASYMPT_0:def 5
ex N being Nat st
for n being Nat st n >= N holds
f . n <> 0 ;

attr f is eventually-nondecreasing means :Def6: :: ASYMPT_0:def 6
ex N being Nat st
for n being Nat st n >= N holds
f . n <= f . (n + 1);

end;

:: deftheorem Def2 defines eventually-nonnegative ASYMPT_0:def 2 :
for f being Real_Sequence holds
( f is eventually-nonnegative iff ex N being Nat st
for n being Nat st n >= N holds
f . n >= 0 );

:: deftheorem Def3 defines positive ASYMPT_0:def 3 :
for f being Real_Sequence holds
( f is positive iff for n being Nat holds f . n > 0 );

:: deftheorem Def4 defines eventually-positive ASYMPT_0:def 4 :
for f being Real_Sequence holds
( f is eventually-positive iff ex N being Nat st
for n being Nat st n >= N holds
f . n > 0 );

:: deftheorem Def5 defines eventually-nonzero ASYMPT_0:def 5 :
for f being Real_Sequence holds
( f is eventually-nonzero iff ex N being Nat st
for n being Nat st n >= N holds
f . n <> 0 );

:: deftheorem Def6 defines eventually-nondecreasing ASYMPT_0:def 6 :
for f being Real_Sequence holds
( f is eventually-nondecreasing iff ex N being Nat st
for n being Nat st n >= N holds
f . n <= f . (n + 1) );

registration

cluster V1() V4( NAT ) V5( REAL ) non empty Function-like V28( NAT ) quasi_total V39() V40() V41() eventually-nonnegative positive eventually-positive eventually-nonzero eventually-nondecreasing for Element of K16(K17(NAT,REAL));

existence
ex b₁ being Real_Sequence st
( b₁ is eventually-nonnegative & b₁ is eventually-nonzero & b₁ is positive & b₁ is eventually-positive & b₁ is eventually-nondecreasing )

proof end;

end;

registration

cluster Function-like quasi_total positive -> eventually-positive for Element of K16(K17(NAT,REAL));

coherence
for b₁ being Real_Sequence st b₁ is positive holds
b₁ is eventually-positive

proof end;

cluster Function-like quasi_total eventually-positive -> eventually-nonnegative eventually-nonzero for Element of K16(K17(NAT,REAL));

coherence
for b₁ being Real_Sequence st b₁ is eventually-positive holds
( b₁ is eventually-nonnegative & b₁ is eventually-nonzero )

proof end;

cluster Function-like quasi_total eventually-nonnegative eventually-nonzero -> eventually-positive for Element of K16(K17(NAT,REAL));

coherence
for b₁ being Real_Sequence st b₁ is eventually-nonnegative & b₁ is eventually-nonzero holds
b₁ is eventually-positive

proof end;

end;

definition

let f, g be eventually-nonnegative Real_Sequence;
:: original: +
redefine func f + g -> eventually-nonnegative Real_Sequence;
coherence
f + g is eventually-nonnegative Real_Sequence

proof end;

end;

definition

let f be eventually-nonnegative Real_Sequence;
let c be positive Real;
:: original: (#)
redefine func c (#) f -> eventually-nonnegative Real_Sequence;
coherence
c (#) f is eventually-nonnegative Real_Sequence

proof end;

end;

definition

let f be eventually-nonnegative Real_Sequence;
let c be non negative Real;
:: original: +
redefine func c + f -> eventually-nonnegative Real_Sequence;
coherence
c + f is eventually-nonnegative Real_Sequence

proof end;

end;

definition

let f be eventually-nonnegative Real_Sequence;
let c be positive Real;
:: original: +
redefine func c + f -> eventually-positive Real_Sequence;
coherence
c + f is eventually-positive Real_Sequence

proof end;

end;

definition

let f, g be Real_Sequence;

func max (f,g) -> Real_Sequence means :Def7: :: ASYMPT_0:def 7
for n being Nat holds it . n = max ((f . n),(g . n));

existence
ex b₁ being Real_Sequence st
for n being Nat holds b₁ . n = max ((f . n),(g . n))

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Nat holds b₁ . n = max ((f . n),(g . n)) ) & ( for n being Nat holds b₂ . n = max ((f . n),(g . n)) ) holds
b₁ = b₂

proof end;

commutativity
for b₁, f, g being Real_Sequence st ( for n being Nat holds b₁ . n = max ((f . n),(g . n)) ) holds
for n being Nat holds b₁ . n = max ((g . n),(f . n)) ;

end;

:: deftheorem Def7 defines max ASYMPT_0:def 7 :
for f, g, b₃ being Real_Sequence holds
( b₃ = max (f,g) iff for n being Nat holds b₃ . n = max ((f . n),(g . n)) );

registration

let f be Real_Sequence;
let g be eventually-nonnegative Real_Sequence;

cluster max (f,g) -> eventually-nonnegative ;

coherence
max (f,g) is eventually-nonnegative

proof end;

end;

registration

let f be Real_Sequence;
let g be eventually-positive Real_Sequence;

cluster max (f,g) -> eventually-positive ;

coherence
max (f,g) is eventually-positive

proof end;

end;

definition

let f, g be Real_Sequence;

pred g majorizes f means :: ASYMPT_0:def 8
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
f . n <= g . n;

end;

:: deftheorem defines majorizes ASYMPT_0:def 8 :
for f, g being Real_Sequence holds
( g majorizes f iff ex N being Element of NAT st
for n being Element of NAT st n >= N holds
f . n <= g . n );

Lm1: for f, g being Real_Sequence
for n being Nat holds (f /" g) . n = (f . n) / (g . n)

proof end;

theorem Th1: :: ASYMPT_0:1

for f being Real_Sequence
for N being Nat st ( for n being Nat st n >= N holds
f . n <= f . (n + 1) ) holds
for n, m being Nat st N <= n & n <= m holds
f . n <= f . m

proof end;

theorem Th2: :: ASYMPT_0:2

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) <> 0 holds
( g /" f is convergent & lim (g /" f) = (lim (f /" g)) " )

proof end;

theorem Th3: :: ASYMPT_0:3

for f being eventually-nonnegative Real_Sequence st f is convergent holds
0 <= lim f

proof end;

theorem Th4: :: ASYMPT_0:4

for f, g being Real_Sequence st f is convergent & g is convergent & g majorizes f holds
lim f <= lim g

proof end;

theorem Th5: :: ASYMPT_0:5

for f being Real_Sequence
for g being eventually-nonzero Real_Sequence st f /" g is divergent_to+infty holds
( g /" f is convergent & lim (g /" f) = 0 )

proof end;

definition

let f be eventually-nonnegative Real_Sequence;

func Big_Oh f -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 9
{ t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;

coherence
{ t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Oh ASYMPT_0:def 9 :
for f being eventually-nonnegative Real_Sequence holds Big_Oh f = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;

theorem Th6: :: ASYMPT_0:6

for x being set
for f being eventually-nonnegative Real_Sequence st x in Big_Oh f holds
x is eventually-nonnegative Real_Sequence

proof end;

theorem :: ASYMPT_0:7

for f being positive Real_Sequence
for t being eventually-nonnegative Real_Sequence holds
( t in Big_Oh f iff ex c being Real st
( c > 0 & ( for n being Element of NAT holds t . n <= c * (f . n) ) ) )

proof end;

theorem :: ASYMPT_0:8

for f being eventually-positive Real_Sequence
for t being eventually-nonnegative Real_Sequence
for N being Element of NAT st t in Big_Oh f & ( for n being Element of NAT st n >= N holds
f . n > 0 ) holds
ex c being Real st
( c > 0 & ( for n being Element of NAT st n >= N holds
t . n <= c * (f . n) ) )

proof end;

theorem :: ASYMPT_0:9

for f, g being eventually-nonnegative Real_Sequence holds Big_Oh (f + g) = Big_Oh (max (f,g))

proof end;

theorem Th10: :: ASYMPT_0:10

for f being eventually-nonnegative Real_Sequence holds f in Big_Oh f

proof end;

theorem Th11: :: ASYMPT_0:11

for f, g being eventually-nonnegative Real_Sequence st f in Big_Oh g holds
Big_Oh f c= Big_Oh g

proof end;

theorem Th12: :: ASYMPT_0:12

for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Oh g & g in Big_Oh h holds
f in Big_Oh h

proof end;

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 )

proof end;

theorem :: ASYMPT_0:13

for f being eventually-nonnegative Real_Sequence
for c being positive Real holds Big_Oh f = Big_Oh (c (#) f)

proof end;

theorem :: ASYMPT_0:14

for c being non negative Real
for x, f being eventually-nonnegative Real_Sequence st x in Big_Oh f holds
x in Big_Oh (c + f)

proof end;

Lm3: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds
f in Big_Oh g

proof end;

theorem Th15: :: ASYMPT_0:15

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds
Big_Oh f = Big_Oh g

proof end;

theorem Th16: :: ASYMPT_0:16

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds
( f in Big_Oh g & not g in Big_Oh f )

proof end;

theorem Th17: :: ASYMPT_0:17

for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds
( not f in Big_Oh g & g in Big_Oh f )

proof end;

definition

let f be eventually-nonnegative Real_Sequence;

func Big_Omega f -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 10
{ t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;

coherence
{ t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Omega ASYMPT_0:def 10 :
for f being eventually-nonnegative Real_Sequence holds Big_Omega f = { t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;

theorem :: ASYMPT_0:18

for x being set
for f being eventually-nonnegative Real_Sequence st x in Big_Omega f holds
x is eventually-nonnegative Real_Sequence

proof end;

theorem Th19: :: ASYMPT_0:19

for f, g being eventually-nonnegative Real_Sequence holds
( f in Big_Omega g iff g in Big_Oh f )

proof end;

theorem Th20: :: ASYMPT_0:20

for f being eventually-nonnegative Real_Sequence holds f in Big_Omega f

proof end;

theorem Th21: :: ASYMPT_0:21

for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Omega g & g in Big_Omega h holds
f in Big_Omega h

proof end;

theorem :: ASYMPT_0:22

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds
Big_Omega f = Big_Omega g

proof end;

theorem Th23: :: ASYMPT_0:23

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds
( g in Big_Omega f & not f in Big_Omega g )

proof end;

theorem :: ASYMPT_0:24

for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds
( not g in Big_Omega f & f in Big_Omega g )

proof end;

theorem :: ASYMPT_0:25

for f, t being positive Real_Sequence holds
( t in Big_Omega f iff ex d being Real st
( d > 0 & ( for n being Element of NAT holds d * (f . n) <= t . n ) ) )

proof end;

theorem :: ASYMPT_0:26

for f, g being eventually-nonnegative Real_Sequence holds Big_Omega (f + g) = Big_Omega (max (f,g))

proof end;

definition

let f be eventually-nonnegative Real_Sequence;

func Big_Theta f -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 11
(Big_Oh f) /\ (Big_Omega f);

coherence
(Big_Oh f) /\ (Big_Omega f) is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Theta ASYMPT_0:def 11 :
for f being eventually-nonnegative Real_Sequence holds Big_Theta f = (Big_Oh f) /\ (Big_Omega f);

theorem Th27: :: ASYMPT_0:27

for f being eventually-nonnegative Real_Sequence holds Big_Theta f = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) }

proof end;

theorem :: ASYMPT_0:28

for f being eventually-nonnegative Real_Sequence holds f in Big_Theta f

proof end;

theorem :: ASYMPT_0:29

for f, g being eventually-nonnegative Real_Sequence st f in Big_Theta g holds
g in Big_Theta f

proof end;

theorem :: ASYMPT_0:30

for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Theta g & g in Big_Theta h holds
f in Big_Theta h

proof end;

theorem :: ASYMPT_0:31

for f, t being positive Real_Sequence holds
( t in Big_Theta f iff ex c, d being Real st
( c > 0 & d > 0 & ( for n being Element of NAT holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) )

proof end;

theorem :: ASYMPT_0:32

for f, g being eventually-nonnegative Real_Sequence holds Big_Theta (f + g) = Big_Theta (max (f,g))

proof end;

theorem :: ASYMPT_0:33

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds
f in Big_Theta g

proof end;

theorem :: ASYMPT_0:34

for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds
( f in Big_Oh g & not f in Big_Theta g )

proof end;

theorem :: ASYMPT_0:35

for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds
( f in Big_Omega g & not f in Big_Theta g )

proof end;

definition

let f be eventually-nonnegative Real_Sequence;
let X be set ;

func Big_Oh (f,X) -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 12
{ t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;

coherence
{ t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Oh ASYMPT_0:def 12 :
for f being eventually-nonnegative Real_Sequence
for X being set holds Big_Oh (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;

definition

let f be eventually-nonnegative Real_Sequence;
let X be set ;

func Big_Omega (f,X) -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 13
{ t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;

coherence
{ t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Omega ASYMPT_0:def 13 :
for f being eventually-nonnegative Real_Sequence
for X being set holds Big_Omega (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;

definition

let f be eventually-nonnegative Real_Sequence;
let X be set ;

func Big_Theta (f,X) -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 14
{ t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ;

coherence
{ t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } is FUNCTION_DOMAIN of NAT , REAL

proof end;

end;

:: deftheorem defines Big_Theta ASYMPT_0:def 14 :
for f being eventually-nonnegative Real_Sequence
for X being set holds Big_Theta (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ;

theorem Th36: :: ASYMPT_0:36

for f being eventually-nonnegative Real_Sequence
for X being set holds Big_Theta (f,X) = (Big_Oh (f,X)) /\ (Big_Omega (f,X))

proof end;

definition

let f be Real_Sequence;
let b be Nat;

func f taken_every b -> Real_Sequence means :Def15: :: ASYMPT_0:def 15
for n being Element of NAT holds it . n = f . (b * n);

existence
ex b₁ being Real_Sequence st
for n being Element of NAT holds b₁ . n = f . (b * n)

proof end;

uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Element of NAT holds b₁ . n = f . (b * n) ) & ( for n being Element of NAT holds b₂ . n = f . (b * n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def15 defines taken_every ASYMPT_0:def 15 :
for f being Real_Sequence
for b being Nat
for b₃ being Real_Sequence holds
( b₃ = f taken_every b iff for n being Element of NAT holds b₃ . n = f . (b * n) );

definition

let f be eventually-nonnegative Real_Sequence;
let b be Nat;

pred f is_smooth_wrt b means :: ASYMPT_0:def 16
( f is eventually-nondecreasing & f taken_every b in Big_Oh f );

end;

:: deftheorem defines is_smooth_wrt ASYMPT_0:def 16 :
for f being eventually-nonnegative Real_Sequence
for b being Nat holds
( f is_smooth_wrt b iff ( f is eventually-nondecreasing & f taken_every b in Big_Oh f ) );

definition

let f be eventually-nonnegative Real_Sequence;

attr f is smooth means :: ASYMPT_0:def 17
for b being Element of NAT st b >= 2 holds
f is_smooth_wrt b;

end;

:: deftheorem defines smooth ASYMPT_0:def 17 :
for f being eventually-nonnegative Real_Sequence holds
( f is smooth iff for b being Element of NAT st b >= 2 holds
f is_smooth_wrt b );

theorem :: ASYMPT_0:37

for f being eventually-nonnegative Real_Sequence st ex b being Element of NAT st
( b >= 2 & f is_smooth_wrt b ) holds
f is smooth

proof end;

theorem Th38: :: ASYMPT_0:38

for f being eventually-nonnegative Real_Sequence
for t being eventually-nonnegative eventually-nondecreasing Real_Sequence
for b being Nat st f is smooth & b >= 2 & t in Big_Oh (f, { (b |^ n) where n is Element of NAT : verum } ) holds
t in Big_Oh f

proof end;

theorem Th39: :: ASYMPT_0:39

for f being eventually-nonnegative Real_Sequence
for t being eventually-nonnegative eventually-nondecreasing Real_Sequence
for b being Element of NAT st f is smooth & b >= 2 & t in Big_Omega (f, { (b |^ n) where n is Element of NAT : verum } ) holds
t in Big_Omega f

proof end;

theorem :: ASYMPT_0:40

for f being eventually-nonnegative Real_Sequence
for t being eventually-nonnegative eventually-nondecreasing Real_Sequence
for b being Element of NAT st f is smooth & b >= 2 & t in Big_Theta (f, { (b |^ n) where n is Element of NAT : verum } ) holds
t in Big_Theta f

proof end;

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 :: ASYMPT_0:def 18
{ 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

proof end;

end;

:: deftheorem defines + ASYMPT_0:def 18 :
for X being non empty set
for F, G being FUNCTION_DOMAIN of X, REAL holds F + G = { 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) ) ) } ;

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 :: ASYMPT_0:def 19
{ 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

proof end;

end;

:: deftheorem defines max ASYMPT_0:def 19 :
for X being non empty set
for F, G being FUNCTION_DOMAIN of X, REAL holds max (F,G) = { 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)) ) ) } ;

theorem :: ASYMPT_0:41

for f, g being eventually-nonnegative Real_Sequence holds (Big_Oh f) + (Big_Oh g) = Big_Oh (f + g)

proof end;

theorem :: ASYMPT_0:42

for f, g being eventually-nonnegative Real_Sequence holds max ((Big_Oh f),(Big_Oh g)) = Big_Oh (max (f,g))

proof end;

definition

let F, G be FUNCTION_DOMAIN of NAT , REAL ;

func F to_power G -> FUNCTION_DOMAIN of NAT , REAL equals :: ASYMPT_0:def 20
{ 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

proof end;

end;

:: deftheorem defines to_power ASYMPT_0:def 20 :
for F, G being FUNCTION_DOMAIN of NAT , REAL holds F to_power G = { 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) ) ) } ;