:: Some Properties of {F}ibonacci Numbers
:: by Magdalena Jastrz\c{e}bska and Adam Grabowski
::
:: Received May 10, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

theorem :: FIB_NUM2:1

for n being non zero Element of NAT holds (n -' 1) + 2 = n + 1

proof end;

theorem Th2: :: FIB_NUM2:2

for n being odd Integer holds (- 1) to_power n = - 1

proof end;

theorem Th3: :: FIB_NUM2:3

for n being even Integer holds (- 1) to_power n = 1

proof end;

theorem Th4: :: FIB_NUM2:4

for m being non zero Real
for n being Integer holds ((- 1) * m) to_power n = ((- 1) to_power n) * (m to_power n)

proof end;

theorem Th5: :: FIB_NUM2:5

for k, m being Nat
for a being Real holds a to_power (k + m) = (a to_power k) * (a to_power m)

proof end;

theorem Th6: :: FIB_NUM2:6

for n being Nat
for k being non zero Real
for m being odd Integer holds (k to_power m) to_power n = k to_power (m * n)

proof end;

theorem Th7: :: FIB_NUM2:7

for n being Nat holds ((- 1) to_power (- n)) ^2 = 1

proof end;

theorem Th8: :: FIB_NUM2:8

for k, m being Nat
for a being non zero Real holds (a to_power (- k)) * (a to_power (- m)) = a to_power ((- k) - m)

proof end;

theorem Th9: :: FIB_NUM2:9

for n being Nat holds (- 1) to_power (- (2 * n)) = 1

proof end;

theorem Th10: :: FIB_NUM2:10

for k being Nat
for a being non zero Real holds (a to_power k) * (a to_power (- k)) = 1

proof end;

registration

let n be odd Integer;

cluster - n -> odd ;

coherence
not - n is even

proof end;

end;

registration

let n be even Integer;

cluster - n -> even ;

coherence
- n is even

proof end;

end;

theorem Th11: :: FIB_NUM2:11

for n being Nat holds (- 1) to_power (- n) = (- 1) to_power n

proof end;

theorem Th12: :: FIB_NUM2:12

for n being Nat
for k, m, m1, n1 being Element of NAT st k divides m & k divides n holds
k divides (m * m1) + (n * n1)

proof end;

registration

cluster non empty finite with_non-empty_elements natural-membered for set ;

existence
ex b₁ being set st
( b₁ is finite & not b₁ is empty & b₁ is natural-membered & b₁ is with_non-empty_elements )

proof end;

end;

registration

let f be sequence of NAT;
let A be finite with_non-empty_elements natural-membered set ;

cluster f | A -> FinSubsequence-like ;

coherence
f | A is FinSubsequence-like

proof end;

end;

theorem :: FIB_NUM2:13

for p being FinSubsequence holds rng (Seq p) c= rng p by RELAT_1:26;

definition

let f be sequence of NAT;
let A be finite with_non-empty_elements natural-membered set ;

func Prefix (f,A) -> FinSequence of NAT equals :: FIB_NUM2:def 1
Seq (f | A);

coherence
Seq (f | A) is FinSequence of NAT

proof end;

end;

:: deftheorem defines Prefix FIB_NUM2:def 1 :
for f being sequence of NAT
for A being finite with_non-empty_elements natural-membered set holds Prefix (f,A) = Seq (f | A);

theorem Th14: :: FIB_NUM2:14

for n, m being Nat
for k being Element of NAT st k <> 0 & k + m <= n holds
m < n

proof end;

registration

cluster NAT -> bounded_below ;

coherence
NAT is bounded_below ;

end;

theorem Th15: :: FIB_NUM2:15

for i, j being Nat
for x, y being set st 0 < i & i < j holds
{[i,x],[j,y]} is FinSubsequence

proof end;

theorem Th16: :: FIB_NUM2:16

for i, j being Nat
for x, y being set
for q being FinSubsequence st i < j & q = {[i,x],[j,y]} holds
Seq q = <*x,y*>

proof end;

registration

let n be Nat;

cluster Seg n -> with_non-empty_elements ;

coherence
Seg n is with_non-empty_elements

proof end;

end;

registration

let A be with_non-empty_elements set ;

cluster -> with_non-empty_elements for Element of bool A;

coherence
for b₁ being Subset of A holds b₁ is with_non-empty_elements

proof end;

end;

registration

let A be with_non-empty_elements set ;
let B be set ;

cluster A /\ B -> with_non-empty_elements ;

coherence
A /\ B is with_non-empty_elements

proof end;

cluster B /\ A -> with_non-empty_elements ;

coherence
B /\ A is with_non-empty_elements ;

end;

theorem Th17: :: FIB_NUM2:17

for k being Element of NAT
for a being set st k >= 1 holds
{[k,a]} is FinSubsequence

proof end;

theorem Th18: :: FIB_NUM2:18

for i being Element of NAT
for y being set
for f being FinSubsequence st f = {[1,y]} holds
Shift (f,i) = {[(1 + i),y]}

proof end;

theorem Th19: :: FIB_NUM2:19

for q being FinSubsequence
for k, n being Element of NAT st dom q c= Seg k & n > k holds
ex p being FinSequence st
( q c= p & dom p = Seg n )

proof end;

theorem Th20: :: FIB_NUM2:20

for q being FinSubsequence ex p being FinSequence st q c= p

proof end;

scheme :: FIB_NUM2:sch 1
FibInd1{ P₁[ set ] } :

for k being non zero Nat holds P₁[k]

provided

A1: P₁[1] and
A2: P₁[2] and
A3: for k being non zero Nat st P₁[k] & P₁[k + 1] holds
P₁[k + 2]

proof end;

scheme :: FIB_NUM2:sch 2
FibInd2{ P₁[ set ] } :

for k being non trivial Nat holds P₁[k]

provided

A1: P₁[2] and
A2: P₁[3] and
A3: for k being non trivial Nat st P₁[k] & P₁[k + 1] holds
P₁[k + 2]

proof end;

theorem Th21: :: FIB_NUM2:21

Fib 2 = 1

proof end;

theorem Th22: :: FIB_NUM2:22

Fib 3 = 2

proof end;

theorem Th23: :: FIB_NUM2:23

Fib 4 = 3

proof end;

theorem Th24: :: FIB_NUM2:24

for n being Nat holds Fib (n + 2) = (Fib n) + (Fib (n + 1))

proof end;

theorem Th25: :: FIB_NUM2:25

for n being Nat holds Fib (n + 3) = (Fib (n + 2)) + (Fib (n + 1))

proof end;

theorem Th26: :: FIB_NUM2:26

for n being Nat holds Fib (n + 4) = (Fib (n + 2)) + (Fib (n + 3))

proof end;

theorem Th27: :: FIB_NUM2:27

for n being Nat holds Fib (n + 5) = (Fib (n + 3)) + (Fib (n + 4))

proof end;

Lm1: for k being Nat holds Fib ((2 * (k + 2)) + 1) = (Fib ((2 * k) + 3)) + (Fib ((2 * k) + 4))

proof end;

theorem Th28: :: FIB_NUM2:28

for n being Nat holds Fib (n + 2) = (Fib (n + 3)) - (Fib (n + 1))

proof end;

theorem Th29: :: FIB_NUM2:29

for n being Nat holds Fib (n + 1) = (Fib (n + 2)) - (Fib n)

proof end;

theorem Th30: :: FIB_NUM2:30

for n being Nat holds Fib n = (Fib (n + 2)) - (Fib (n + 1))

proof end;

theorem Th31: :: FIB_NUM2:31

for n being Nat holds ((Fib n) * (Fib (n + 2))) - ((Fib (n + 1)) ^2) = (- 1) |^ (n + 1)

proof end;

theorem :: FIB_NUM2:32

for n being non zero Element of NAT holds ((Fib (n -' 1)) * (Fib (n + 1))) - ((Fib n) ^2) = (- 1) |^ n

proof end;

theorem Th33: :: FIB_NUM2:33

tau > 0

proof end;

theorem Th34: :: FIB_NUM2:34

tau_bar = (- tau) to_power (- 1)

proof end;

theorem Th35: :: FIB_NUM2:35

for n being Nat holds (- tau) to_power ((- 1) * n) = ((- tau) to_power (- 1)) to_power n

proof end;

theorem Th36: :: FIB_NUM2:36

- (1 / tau) = tau_bar

proof end;

theorem Th37: :: FIB_NUM2:37

for r being Nat holds (((tau to_power r) ^2) - (2 * ((- 1) to_power r))) + ((tau to_power (- r)) ^2) = ((tau to_power r) - (tau_bar to_power r)) ^2

proof end;

theorem :: FIB_NUM2:38

for n, r being non zero Element of NAT st r <= n holds
((Fib n) ^2) - ((Fib (n + r)) * (Fib (n -' r))) = ((- 1) |^ (n -' r)) * ((Fib r) ^2)

proof end;

theorem :: FIB_NUM2:39

for n being Nat holds ((Fib n) ^2) + ((Fib (n + 1)) ^2) = Fib ((2 * n) + 1)

proof end;

theorem Th40: :: FIB_NUM2:40

for n being Nat
for k being non zero Element of NAT holds Fib (n + k) = ((Fib k) * (Fib (n + 1))) + ((Fib (k -' 1)) * (Fib n))

proof end;

theorem Th41: :: FIB_NUM2:41

for k being Nat
for n being non zero Element of NAT holds Fib n divides Fib (n * k)

proof end;

theorem Th42: :: FIB_NUM2:42

for n being Nat
for k being non zero Element of NAT st k divides n holds
Fib k divides Fib n

proof end;

theorem Th43: :: FIB_NUM2:43

for n being Nat holds Fib n <= Fib (n + 1)

proof end;

theorem Th44: :: FIB_NUM2:44

for n being Nat st n > 1 holds
Fib n < Fib (n + 1)

proof end;

theorem :: FIB_NUM2:45

for m, n being Nat st m >= n holds
Fib m >= Fib n

proof end;

theorem Th46: :: FIB_NUM2:46

for n, k being Nat st k > 1 & k < n holds
Fib k < Fib n

proof end;

theorem Th47: :: FIB_NUM2:47

for k being Nat holds
( Fib k = 1 iff ( k = 1 or k = 2 ) )

proof end;

theorem Th48: :: FIB_NUM2:48

for k, n being Element of NAT st n > 1 & k <> 0 & k <> 1 holds
( Fib k = Fib n iff k = n )

proof end;

theorem Th49: :: FIB_NUM2:49

for n being Element of NAT st n > 1 & n <> 4 & not n is prime holds
ex k being non zero Element of NAT st
( k <> 1 & k <> 2 & k <> n & k divides n )

proof end;

theorem :: FIB_NUM2:50

for n being Element of NAT st n > 1 & n <> 4 & Fib n is prime holds
n is prime

proof end;

definition

func FIB -> sequence of NAT means :Def2: :: FIB_NUM2:def 2
for k being Element of NAT holds it . k = Fib k;

existence
ex b₁ being sequence of NAT st
for k being Element of NAT holds b₁ . k = Fib k

proof end;

uniqueness
for b₁, b₂ being sequence of NAT st ( for k being Element of NAT holds b₁ . k = Fib k ) & ( for k being Element of NAT holds b₂ . k = Fib k ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines FIB FIB_NUM2:def 2 :
for b₁ being sequence of NAT holds
( b₁ = FIB iff for k being Element of NAT holds b₁ . k = Fib k );

definition

func EvenNAT -> Subset of NAT equals :: FIB_NUM2:def 3
{ (2 * k) where k is Nat : verum } ;

coherence
{ (2 * k) where k is Nat : verum } is Subset of NAT

proof end;

func OddNAT -> Subset of NAT equals :: FIB_NUM2:def 4
{ ((2 * k) + 1) where k is Element of NAT : verum } ;

coherence
{ ((2 * k) + 1) where k is Element of NAT : verum } is Subset of NAT

proof end;

end;

:: deftheorem defines EvenNAT FIB_NUM2:def 3 :
EvenNAT = { (2 * k) where k is Nat : verum } ;

:: deftheorem defines OddNAT FIB_NUM2:def 4 :
OddNAT = { ((2 * k) + 1) where k is Element of NAT : verum } ;

theorem Th51: :: FIB_NUM2:51

for k being Nat holds
( 2 * k in EvenNAT & not (2 * k) + 1 in EvenNAT )

proof end;

theorem Th52: :: FIB_NUM2:52

for k being Element of NAT holds
( (2 * k) + 1 in OddNAT & not 2 * k in OddNAT )

proof end;

definition

let n be Nat;

func EvenFibs n -> FinSequence of NAT equals :: FIB_NUM2:def 5
Prefix (FIB,(EvenNAT /\ (Seg n)));

coherence
Prefix (FIB,(EvenNAT /\ (Seg n))) is FinSequence of NAT ;

func OddFibs n -> FinSequence of NAT equals :: FIB_NUM2:def 6
Prefix (FIB,(OddNAT /\ (Seg n)));

coherence
Prefix (FIB,(OddNAT /\ (Seg n))) is FinSequence of NAT ;

end;

:: deftheorem defines EvenFibs FIB_NUM2:def 5 :
for n being Nat holds EvenFibs n = Prefix (FIB,(EvenNAT /\ (Seg n)));

:: deftheorem defines OddFibs FIB_NUM2:def 6 :
for n being Nat holds OddFibs n = Prefix (FIB,(OddNAT /\ (Seg n)));

theorem Th53: :: FIB_NUM2:53

EvenFibs 0 = {} ;

theorem :: FIB_NUM2:54

Seq (FIB | {2}) = <*1*>

proof end;

theorem Th55: :: FIB_NUM2:55

EvenFibs 2 = <*1*>

proof end;

theorem :: FIB_NUM2:56

EvenFibs 4 = <*1,3*>

proof end;

theorem Th57: :: FIB_NUM2:57

for k being Nat holds (EvenNAT /\ (Seg ((2 * k) + 2))) \/ {((2 * k) + 4)} = EvenNAT /\ (Seg ((2 * k) + 4))

proof end;

theorem Th58: :: FIB_NUM2:58

for k being Nat holds (FIB | (EvenNAT /\ (Seg ((2 * k) + 2)))) \/ {[((2 * k) + 4),(FIB . ((2 * k) + 4))]} = FIB | (EvenNAT /\ (Seg ((2 * k) + 4)))

proof end;

theorem Th59: :: FIB_NUM2:59

for n being Element of NAT holds EvenFibs ((2 * n) + 2) = (EvenFibs (2 * n)) ^ <*(Fib ((2 * n) + 2))*>

proof end;

theorem Th60: :: FIB_NUM2:60

OddFibs 1 = <*1*>

proof end;

theorem Th61: :: FIB_NUM2:61

OddFibs 3 = <*1,2*>

proof end;

theorem Th62: :: FIB_NUM2:62

for k being Nat holds (OddNAT /\ (Seg ((2 * k) + 3))) \/ {((2 * k) + 5)} = OddNAT /\ (Seg ((2 * k) + 5))

proof end;

theorem Th63: :: FIB_NUM2:63

for k being Nat holds (FIB | (OddNAT /\ (Seg ((2 * k) + 3)))) \/ {[((2 * k) + 5),(FIB . ((2 * k) + 5))]} = FIB | (OddNAT /\ (Seg ((2 * k) + 5)))

proof end;

theorem Th64: :: FIB_NUM2:64

for n being Nat holds OddFibs ((2 * n) + 3) = (OddFibs ((2 * n) + 1)) ^ <*(Fib ((2 * n) + 3))*>

proof end;

theorem :: FIB_NUM2:65

for n being Element of NAT holds Sum (EvenFibs ((2 * n) + 2)) = (Fib ((2 * n) + 3)) - 1

proof end;

theorem :: FIB_NUM2:66

for n being Nat holds Sum (OddFibs ((2 * n) + 1)) = Fib ((2 * n) + 2)

proof end;

theorem Th67: :: FIB_NUM2:67

for n being Element of NAT holds Fib n, Fib (n + 1) are_coprime

proof end;

theorem Th68: :: FIB_NUM2:68

for n being non zero Nat
for m being Nat st m <> 1 & m divides Fib n holds
not m divides Fib (n -' 1)

proof end;

:: Carmichael's Theorem on Prime Divisors

theorem :: FIB_NUM2:69

for m being Nat
for n being non zero Nat st m is prime & n is prime & m divides Fib n holds
for r being Nat st r < n & r <> 0 holds
not m divides Fib r

proof end;

theorem :: FIB_NUM2:70

for n being non zero Element of NAT holds {((Fib n) * (Fib (n + 3))),((2 * (Fib (n + 1))) * (Fib (n + 2))),(((Fib (n + 1)) ^2) + ((Fib (n + 2)) ^2))} is Pythagorean_triple

proof end;