:: Irrationality of e
:: by Freek Wiedijk
::
:: Received July 2, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

notation

let x be Real;
antonym irrational x for rational ;

end;

notation

let x, y be Real;
synonym x ^ y for x to_power y;

end;

:: The Irrationality of the Square Root of 2

theorem Th1: :: IRRAT_1:1

for p being Nat st p is prime holds
sqrt p is irrational

proof end;

theorem :: IRRAT_1:2

ex x, y being Real st
( x is irrational & y is irrational & x ^ y is rational )

proof end;

scheme :: IRRAT_1:sch 1
LambdaRealSeq{ F₁( set ) -> Real } :

( ex seq being Real_Sequence st
for n being Nat holds seq . n = F₁(n) & ( for seq1, seq2 being Real_Sequence st ( for n being Nat holds seq1 . n = F₁(n) ) & ( for n being Nat holds seq2 . n = F₁(n) ) holds
seq1 = seq2 ) )

proof end;

definition

let k be Nat;

func aseq k -> Real_Sequence means :Def1: :: IRRAT_1:def 1
for n being Nat holds it . n = (n - k) / n;

correctness
existence
ex b₁ being Real_Sequence st
for n being Nat holds b₁ . n = (n - k) / n;
uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Nat holds b₁ . n = (n - k) / n ) & ( for n being Nat holds b₂ . n = (n - k) / n ) holds
b₁ = b₂;

proof end;

func bseq k -> Real_Sequence means :Def2: :: IRRAT_1:def 2
for n being Nat holds it . n = (n choose k) * (n ^ (- k));

correctness
existence
ex b₁ being Real_Sequence st
for n being Nat holds b₁ . n = (n choose k) * (n ^ (- k));
uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Nat holds b₁ . n = (n choose k) * (n ^ (- k)) ) & ( for n being Nat holds b₂ . n = (n choose k) * (n ^ (- k)) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def1 defines aseq IRRAT_1:def 1 :
for k being Nat
for b₂ being Real_Sequence holds
( b₂ = aseq k iff for n being Nat holds b₂ . n = (n - k) / n );

:: deftheorem Def2 defines bseq IRRAT_1:def 2 :
for k being Nat
for b₂ being Real_Sequence holds
( b₂ = bseq k iff for n being Nat holds b₂ . n = (n choose k) * (n ^ (- k)) );

definition

let n be Nat;

func cseq n -> Real_Sequence means :Def3: :: IRRAT_1:def 3
for k being Nat holds it . k = (n choose k) * (n ^ (- k));

correctness
existence
ex b₁ being Real_Sequence st
for k being Nat holds b₁ . k = (n choose k) * (n ^ (- k));
uniqueness
for b₁, b₂ being Real_Sequence st ( for k being Nat holds b₁ . k = (n choose k) * (n ^ (- k)) ) & ( for k being Nat holds b₂ . k = (n choose k) * (n ^ (- k)) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def3 defines cseq IRRAT_1:def 3 :
for n being Nat
for b₂ being Real_Sequence holds
( b₂ = cseq n iff for k being Nat holds b₂ . k = (n choose k) * (n ^ (- k)) );

theorem Th3: :: IRRAT_1:3

for k, n being Nat holds (cseq n) . k = (bseq k) . n

proof end;

definition

func dseq -> Real_Sequence means :Def4: :: IRRAT_1:def 4
for n being Nat holds it . n = (1 + (1 / n)) ^ n;

correctness
existence
ex b₁ being Real_Sequence st
for n being Nat holds b₁ . n = (1 + (1 / n)) ^ n;
uniqueness
for b₁, b₂ being Real_Sequence st ( for n being Nat holds b₁ . n = (1 + (1 / n)) ^ n ) & ( for n being Nat holds b₂ . n = (1 + (1 / n)) ^ n ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def4 defines dseq IRRAT_1:def 4 :
for b₁ being Real_Sequence holds
( b₁ = dseq iff for n being Nat holds b₁ . n = (1 + (1 / n)) ^ n );

definition

func eseq -> Real_Sequence means :Def5: :: IRRAT_1:def 5
for k being Nat holds it . k = 1 / (k !);

correctness
existence
ex b₁ being Real_Sequence st
for k being Nat holds b₁ . k = 1 / (k !);
uniqueness
for b₁, b₂ being Real_Sequence st ( for k being Nat holds b₁ . k = 1 / (k !) ) & ( for k being Nat holds b₂ . k = 1 / (k !) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def5 defines eseq IRRAT_1:def 5 :
for b₁ being Real_Sequence holds
( b₁ = eseq iff for k being Nat holds b₁ . k = 1 / (k !) );

:: lim(n:(n choose k)*(n ^ (-k))) = 1/(k!)

theorem Th4: :: IRRAT_1:4

for k, n being Nat st n > 0 holds
n ^ (- (k + 1)) = (n ^ (- k)) / n

proof end;

theorem Th5: :: IRRAT_1:5

for k, n being Nat holds n choose (k + 1) = ((n - k) / (k + 1)) * (n choose k)

proof end;

theorem Th6: :: IRRAT_1:6

for k, n being Nat st n > 0 holds
(bseq (k + 1)) . n = ((1 / (k + 1)) * ((bseq k) . n)) * ((aseq k) . n)

proof end;

theorem Th7: :: IRRAT_1:7

for k, n being Nat st n > 0 holds
(aseq k) . n = 1 - (k / n)

proof end;

theorem Th8: :: IRRAT_1:8

for k being Nat holds
( aseq k is convergent & lim (aseq k) = 1 )

proof end;

theorem Th9: :: IRRAT_1:9

for x being Real
for seq being Real_Sequence st ( for n being Nat holds seq . n = x ) holds
( seq is convergent & lim seq = x )

proof end;

theorem Th10: :: IRRAT_1:10

for n being Nat holds (bseq 0) . n = 1

proof end;

theorem Th11: :: IRRAT_1:11

for k being Nat holds (1 / (k + 1)) * (1 / (k !)) = 1 / ((k + 1) !)

proof end;

theorem Th12: :: IRRAT_1:12

for k being Nat holds
( bseq k is convergent & lim (bseq k) = 1 / (k !) & lim (bseq k) = eseq . k )

proof end;

:: 0 <= (n choose k)*(n ^ (-k))) <= 1/(k!)

theorem Th13: :: IRRAT_1:13

for k, n being Nat st k < n holds
( 0 < (aseq k) . n & (aseq k) . n <= 1 )

proof end;

theorem Th14: :: IRRAT_1:14

for k, n being Nat st n > 0 holds
( 0 <= (bseq k) . n & (bseq k) . n <= 1 / (k !) & (bseq k) . n <= eseq . k & 0 <= (cseq n) . k & (cseq n) . k <= 1 / (k !) & (cseq n) . k <= eseq . k )

proof end;

:: n>0 implies (1+(1/n)) ^ n = Sum (k:(n choose k)*(n ^ (-k)))

theorem Th15: :: IRRAT_1:15

for seq being Real_Sequence st seq ^\ 1 is summable holds
( seq is summable & Sum seq = (seq . 0) + (Sum (seq ^\ 1)) )

proof end;

theorem Th16: :: IRRAT_1:16

for k being Nat
for D being non empty set
for sq being FinSequence of D st 1 <= k & k < len sq holds
(sq /^ 1) . k = sq . (k + 1)

proof end;

theorem Th17: :: IRRAT_1:17

for sq being FinSequence of REAL st len sq > 0 holds
Sum sq = (sq . 1) + (Sum (sq /^ 1))

proof end;

theorem Th18: :: IRRAT_1:18

for n being Nat
for seq being Real_Sequence
for sq being FinSequence of REAL st len sq = n & ( for k being Nat st k < n holds
seq . k = sq . (k + 1) ) & ( for k being Nat st k >= n holds
seq . k = 0 ) holds
( seq is summable & Sum seq = Sum sq )

proof end;

theorem Th19: :: IRRAT_1:19

for k, n being Nat
for x, y being Real st k <= n holds
((x,y) In_Power n) . (k + 1) = ((n choose k) * (x ^ (n - k))) * (y ^ k)

proof end;

theorem Th20: :: IRRAT_1:20

for k, n being Nat st n > 0 & k <= n holds
(cseq n) . k = ((1,(1 / n)) In_Power n) . (k + 1)

proof end;

theorem Th21: :: IRRAT_1:21

for n being Nat st n > 0 holds
( cseq n is summable & Sum (cseq n) = (1 + (1 / n)) ^ n & Sum (cseq n) = dseq . n )

proof end;

:: number_e = lim(n:(1+(1/n)) ^ n)

theorem Th22: :: IRRAT_1:22

( dseq is convergent & lim dseq = number_e )

proof end;

:: exp(1) = Sum (k:1/(k!))

theorem Th23: :: IRRAT_1:23

( eseq is summable & Sum eseq = exp_R 1 )

proof end;

:: lim(n:(1+(1/n)) ^ n) = Sum (k:1/(k!))

theorem Th24: :: IRRAT_1:24

for K being Nat
for dseqK being Real_Sequence st ( for n being Nat holds dseqK . n = (Partial_Sums (cseq n)) . K ) holds
( dseqK is convergent & lim dseqK = (Partial_Sums eseq) . K )

proof end;

theorem Th25: :: IRRAT_1:25

for x being Real
for seq being Real_Sequence st seq is convergent & lim seq = x holds
for eps being Real st eps > 0 holds
ex N being Nat st
for n being Nat st n >= N holds
seq . n > x - eps

proof end;

theorem Th26: :: IRRAT_1:26

for x being Real
for seq being Real_Sequence st ( for eps being Real st eps > 0 holds
ex N being Nat st
for n being Nat st n >= N holds
seq . n > x - eps ) & ex N being Nat st
for n being Nat st n >= N holds
seq . n <= x holds
( seq is convergent & lim seq = x )

proof end;

theorem Th27: :: IRRAT_1:27

for seq being Real_Sequence st seq is summable holds
for eps being Real st eps > 0 holds
ex K being Nat st (Partial_Sums seq) . K > (Sum seq) - eps

proof end;

theorem Th28: :: IRRAT_1:28

for n being Nat st n >= 1 holds
dseq . n <= Sum eseq

proof end;

theorem Th29: :: IRRAT_1:29

for K being Nat
for seq being Real_Sequence st seq is summable & ( for k being Nat holds seq . k >= 0 ) holds
Sum seq >= (Partial_Sums seq) . K

proof end;

theorem Th30: :: IRRAT_1:30

( dseq is convergent & lim dseq = Sum eseq )

proof end;

:: number_e = exp(1)

definition

redefine func number_e equals :: IRRAT_1:def 6
Sum eseq;

compatibility
for b₁ being object holds
( b₁ = number_e iff b₁ = Sum eseq ) by Th22, Th30;

end;

:: deftheorem defines number_e IRRAT_1:def 6 :
number_e = Sum eseq;

definition

redefine func number_e equals :: IRRAT_1:def 7
exp_R 1;

compatibility
for b₁ being object holds
( b₁ = number_e iff b₁ = exp_R 1 ) by Th23;

end;

:: deftheorem defines number_e IRRAT_1:def 7 :
number_e = exp_R 1;

theorem Th31: :: IRRAT_1:31

for x being Real st x is rational holds
ex n being Nat st
( n >= 2 & (n !) * x is integer )

proof end;

theorem Th32: :: IRRAT_1:32

for k, n being Nat holds (n !) * (eseq . k) = (n !) / (k !)

proof end;

theorem :: IRRAT_1:33

for k, n being Nat holds (n !) / (k !) > 0 by XREAL_1:139;

theorem Th34: :: IRRAT_1:34

for seq being Real_Sequence st seq is summable & ( for n being Nat holds seq . n > 0 ) holds
Sum seq > 0

proof end;

theorem Th35: :: IRRAT_1:35

for n being Nat holds (n !) * (Sum (eseq ^\ (n + 1))) > 0

proof end;

theorem Th36: :: IRRAT_1:36

for k, n being Nat st k <= n holds
(n !) / (k !) is integer

proof end;

theorem Th37: :: IRRAT_1:37

for n being Nat holds (n !) * ((Partial_Sums eseq) . n) is integer

proof end;

theorem Th38: :: IRRAT_1:38

for k, n being Nat
for x being Real st x = 1 / (n + 1) holds
(n !) / (((n + k) + 1) !) <= x ^ (k + 1)

proof end;

theorem Th39: :: IRRAT_1:39

for n being Nat
for x being Real st n > 0 & x = 1 / (n + 1) holds
(n !) * (Sum (eseq ^\ (n + 1))) <= x / (1 - x)

proof end;

theorem Th40: :: IRRAT_1:40

for x, n being Real st n >= 2 & x = 1 / (n + 1) holds
x / (1 - x) < 1

proof end;

:: Irrationality of e

theorem :: IRRAT_1:41

number_e is irrational

proof end;