:: Euler's Function
:: by Yoshinori Fujisawa and Yasushi Fuwa
::
:: Received December 10, 1997
:: Copyright (c) 1997-2011 Association of Mizar Users

begin

Lm1: for k being Nat
for j being Integer st k <> 0 & [\(j / k)/] + 1 >= j / k holds
k >= j - ([\(j / k)/] * k)

proof end;

Lm2: not 1 is prime
by INT_2:def 5;

theorem :: EULER_1:1

canceled;

theorem Th2: :: EULER_1:2

for n being Nat holds
( n,n are_relative_prime iff n = 1 )

proof end;

theorem Th3: :: EULER_1:3

for k, n being Nat st k <> 0 & k < n & n is prime holds
k,n are_relative_prime

proof end;

theorem Th4: :: EULER_1:4

for n, k being Nat holds
( n is prime & k in { kk where kk is Element of NAT : ( n,kk are_relative_prime & kk >= 1 & kk <= n ) } iff ( n is prime & k in n & not k in {0} ) )

proof end;

theorem Th5: :: EULER_1:5

for A being finite set
for x being set st x in A holds
card (A \ {x}) = (card A) - (card {x})

proof end;

theorem Th6: :: EULER_1:6

for a, b being Nat st a gcd b = 1 holds
for c being Nat holds (a * c) gcd (b * c) = c

proof end;

theorem Th7: :: EULER_1:7

for a, c, b being Nat st a <> 0 & c <> 0 & (a * c) gcd (b * c) = c holds
a,b are_relative_prime

proof end;

theorem Th8: :: EULER_1:8

for a, b being Nat st a gcd b = 1 holds
(a + b) gcd b = 1

proof end;

theorem Th9: :: EULER_1:9

for a, b, c being Nat holds (a + (b * c)) gcd b = a gcd b

proof end;

theorem Th10: :: EULER_1:10

for m, n being Nat st m,n are_relative_prime holds
ex k being Nat st
( ex i0, j0 being Integer st
( k = (i0 * m) + (j0 * n) & k > 0 ) & ( for l being Nat st ex i, j being Integer st
( l = (i * m) + (j * n) & l > 0 ) holds
k <= l ) )

proof end;

theorem Th11: :: EULER_1:11

for m, n being Nat st m,n are_relative_prime holds
for k being Nat ex i, j being Integer st (i * m) + (j * n) = k

proof end;

theorem Th12: :: EULER_1:12

for A being set
for B being non empty set
for f being Function of A,B st f is bijective holds
card A = card B

proof end;

theorem Th13: :: EULER_1:13

for i, k, n being Integer holds (i + (k * n)) mod n = i mod n

proof end;

theorem Th14: :: EULER_1:14

for c, a, b being Nat st c divides a * b & a,c are_relative_prime holds
c divides b

proof end;

theorem Th15: :: EULER_1:15

for a, c, b being Nat st a <> 0 & c <> 0 & a,c are_relative_prime & b,c are_relative_prime holds
a * b,c are_relative_prime

proof end;

theorem Th16: :: EULER_1:16

for x, i, y being Integer st x <> 0 & i > 0 holds
(i * x) gcd (i * y) = i * (x gcd y)

proof end;

theorem Th17: :: EULER_1:17

for a, b being Nat
for x being Integer st a <> 0 holds
(a + (x * b)) gcd b = a gcd b

proof end;

begin

definition

let n be Nat;
func Euler n -> Element of NAT equals :: EULER_1:def 1
card { k where k is Element of NAT : ( n,k are_relative_prime & k >= 1 & k <= n ) } ;
coherence
card { k where k is Element of NAT : ( n,k are_relative_prime & k >= 1 & k <= n ) } is Element of NAT

proof end;

end;

:: deftheorem defines Euler EULER_1:def 1 :
for n being Nat holds Euler n = card { k where k is Element of NAT : ( n,k are_relative_prime & k >= 1 & k <= n ) } ;

set X = { k where k is Element of NAT : ( 1,k are_relative_prime & k >= 1 & k <= 1 ) } ;

for x being set holds
( x in { k where k is Element of NAT : ( 1,k are_relative_prime & k >= 1 & k <= 1 ) } iff x = 1 )

proof end;

then Lm3: card {1} = Euler 1
by TARSKI:def 1;

theorem :: EULER_1:18

Euler 1 = 1 by Lm3, CARD_1:50;

set X = { k where k is Element of NAT : ( 2,k are_relative_prime & k >= 1 & k <= 2 ) } ;

for x being set holds
( x in { k where k is Element of NAT : ( 2,k are_relative_prime & k >= 1 & k <= 2 ) } iff x = 1 )

proof end;

then Lm4: card {1} = Euler 2
by TARSKI:def 1;

theorem :: EULER_1:19

Euler 2 = 1 by Lm4, CARD_1:50;

theorem Th20: :: EULER_1:20

for n being Nat st n > 1 holds
Euler n <= n - 1

proof end;

theorem :: EULER_1:21

for n being Nat st n is prime holds
Euler n = n - 1

proof end;

theorem :: EULER_1:22

for m, n being Nat st m > 1 & n > 1 & m,n are_relative_prime holds
Euler (m * n) = (Euler m) * (Euler n)

proof end;