:: Euler's Function
:: by Yoshinori Fujisawa and Yasushi Fuwa
::
:: Received December 10, 1997
:: Copyright (c) 1997 Association of Mizar Users
Lm1:
for k being Nat
for j being Integer st k <> 0 & [\(j / k)/] + 1 >= j / k holds
k >= j - ([\(j / k)/] * k)
Lm2:
not 1 is prime
by INT_2:def 5;
theorem :: EULER_1:1
canceled;
theorem Th2: :: EULER_1:2
theorem Th3: :: EULER_1:3
theorem Th4: :: EULER_1:4
theorem Th5: :: EULER_1:5
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
theorem Th7: :: EULER_1:7
theorem Th8: :: EULER_1:8
theorem Th9: :: EULER_1:9
theorem Th10: :: EULER_1:10
theorem Th11: :: EULER_1:11
theorem Th12: :: EULER_1:12
theorem Th13: :: EULER_1:13
theorem Th14: :: EULER_1:14
theorem Th15: :: EULER_1:15
theorem Th16: :: EULER_1:16
theorem Th17: :: EULER_1:17
:: deftheorem defines Euler EULER_1:def 1 :
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 )
then Lm3:
card {1} = Euler 1
by TARSKI:def 1;
theorem :: EULER_1:18
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 )
then Lm4:
card {1} = Euler 2
by TARSKI:def 1;
theorem :: EULER_1:19
theorem Th20: :: EULER_1:20
theorem :: EULER_1:21
theorem :: EULER_1:22