:: Basic Properties of Rational Numbers
:: by Andrzej Kondracki
::
:: Copyright (c) 1990-2012 Association of Mizar Users

begin

Lm1: omega c= ( { [c,d] where c, d is Element of omega : ( c,d are_relative_prime & d <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega
by XBOOLE_1:7;

Lm2: for i1, j1 being Element of NAT holds quotient (i1,j1) = i1 / j1
proof end;

;

then reconsider 09 = 0 as Element of REAL+ by ARYTM_2:2;

Lm3: for a being real number
for x9 being Element of REAL+ st x9 = a holds
09 - x9 = - a

proof end;

Lm4: for x being set st x in RAT holds
ex m, n being Integer st x = m / n

proof end;

Lm5: for x being set
for l being Element of NAT
for m being Integer st x = m / l holds
x in RAT

proof end;

Lm6: for x being set
for m, n being Integer st x = m / n holds
x in RAT

proof end;

definition
redefine func RAT means :Def1: :: RAT_1:def 1
for x being set holds
( x in it iff ex m, n being Integer st x = m / n );
compatibility
for b1 being set holds
( b1 = RAT iff for x being set holds
( x in b1 iff ex m, n being Integer st x = m / n ) )
proof end;
end;

:: deftheorem Def1 defines RAT RAT_1:def 1 :
for b1 being set holds
( b1 = RAT iff for x being set holds
( x in b1 iff ex m, n being Integer st x = m / n ) );

definition
let r be number ;
attr r is rational means :Def2: :: RAT_1:def 2
r in RAT ;
end;

:: deftheorem Def2 defines rational RAT_1:def 2 :
for r being number holds
( r is rational iff r in RAT );

registration
existence
ex b1 being Real st b1 is rational
proof end;
end;

registration
existence
ex b1 being number st b1 is rational
proof end;
end;

definition end;

theorem Th1: :: RAT_1:1
for x being set st x in RAT holds
ex m, n being Integer st
( n <> 0 & x = m / n )
proof end;

theorem Th2: :: RAT_1:2
for x being set st x is Rational holds
ex m, n being Integer st
( n <> 0 & x = m / n )
proof end;

registration
cluster rational -> real for set ;
coherence
for b1 being number st b1 is rational holds
b1 is real
proof end;
end;

theorem Th3: :: RAT_1:3
for m, n being Integer holds m / n is rational
proof end;

registration
coherence
for b1 being number st b1 is integer holds
b1 is rational
proof end;
end;

:: Now we prove that fractions of integer denominator and natural numerator
:: or of natural denominator and numerator, etc., are all rational numbers.
registration
let p, q be Rational;
cluster p * q -> rational ;
coherence
p * q is rational
proof end;
cluster p + q -> rational ;
coherence
p + q is rational
proof end;
cluster p - q -> rational ;
coherence
p - q is rational
proof end;
cluster p / q -> rational ;
coherence
p / q is rational
proof end;
end;

registration
let p be Rational;
cluster - p -> rational ;
coherence
- p is rational
proof end;
cluster p " -> rational ;
coherence
p " is rational
proof end;
end;

theorem :: RAT_1:4
canceled;

theorem :: RAT_1:5
canceled;

theorem :: RAT_1:6
canceled;

:: RAT is dense, that is there exists at least one rational number between
:: any two distinct real numbers.
theorem :: RAT_1:7
for a, b being real number st a < b holds
ex p being Rational st
( a < p & p < b )
proof end;

theorem Th8: :: RAT_1:8
for p being Rational ex m being Integer ex k being Element of NAT st
( k <> 0 & p = m / k )
proof end;

:: Each rational number can be uniquely expressed as a ratio of two
:: relatively prime numbers, the first is integer and the latter is natural
:: (but not equal to zero). Function denominator(p) is defined as the least
:: natural denominator of all denominators of fractions integer/natural=p.
:: Function numerator(p) is defined as a product of p and denominator(p).
theorem Th9: :: RAT_1:9
for p being Rational ex m being Integer ex k being Element of NAT st
( k <> 0 & p = m / k & ( for n being Integer
for l being Element of NAT st l <> 0 & p = n / l holds
k <= l ) )
proof end;

definition
let p be Rational;
func denominator p -> Element of NAT means :Def3: :: RAT_1:def 3
( it <> 0 & ex m being Integer st p = m / it & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
it <= k ) );
existence
ex b1 being Element of NAT st
( b1 <> 0 & ex m being Integer st p = m / b1 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b1 <= k ) )
proof end;
uniqueness
for b1, b2 being Element of NAT st b1 <> 0 & ex m being Integer st p = m / b1 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b1 <= k ) & b2 <> 0 & ex m being Integer st p = m / b2 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b2 <= k ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines denominator RAT_1:def 3 :
for p being Rational
for b2 being Element of NAT holds
( b2 = denominator p iff ( b2 <> 0 & ex m being Integer st p = m / b2 & ( for n being Integer
for k being Element of NAT st k <> 0 & p = n / k holds
b2 <= k ) ) );

definition
let p be Rational;
func numerator p -> Integer equals :: RAT_1:def 4
() * p;
coherence
() * p is Integer
proof end;
end;

:: deftheorem defines numerator RAT_1:def 4 :
for p being Rational holds numerator p = () * p;

:: Some basic theorems concerning p, numerator(p) and denominator(p).
theorem Th10: :: RAT_1:10
for p being Rational holds 0 < denominator p
proof end;

theorem Th11: :: RAT_1:11
for p being Rational holds 1 <= denominator p
proof end;

theorem Th12: :: RAT_1:12
for p being Rational holds 0 < () "
proof end;

theorem Th13: :: RAT_1:13
for p being Rational holds 1 >= () "
proof end;

theorem Th14: :: RAT_1:14
for p being Rational holds
( numerator p = 0 iff p = 0 )
proof end;

theorem Th15: :: RAT_1:15
for p being Rational holds
( p = () / () & p = () * (() ") & p = (() ") * () )
proof end;

theorem :: RAT_1:16
for p being Rational st p <> 0 holds
denominator p = () / p
proof end;

theorem Th17: :: RAT_1:17
for p being Rational st p is Integer holds
( denominator p = 1 & numerator p = p )
proof end;

theorem Th18: :: RAT_1:18
for p being Rational st ( numerator p = p or denominator p = 1 ) holds
p is Integer
proof end;

theorem :: RAT_1:19
for p being Rational holds
( numerator p = p iff denominator p = 1 ) by Th17;

theorem :: RAT_1:20
for p being Rational st ( numerator p = p or denominator p = 1 ) & 0 <= p holds
p is Element of NAT
proof end;

theorem :: RAT_1:21
for p being Rational holds
( 1 < denominator p iff not p is integer )
proof end;

Lm7: 1 " = 1
;

theorem :: RAT_1:22
for p being Rational holds
( 1 > () " iff not p is integer )
proof end;

theorem Th23: :: RAT_1:23
for p being Rational holds
( numerator p = denominator p iff p = 1 )
proof end;

theorem Th24: :: RAT_1:24
for p being Rational holds
( numerator p = - () iff p = - 1 )
proof end;

theorem :: RAT_1:25
for p being Rational holds
( - () = denominator p iff p = - 1 )
proof end;

:: We can multiple the numerator and the denominator of any rational number
:: by any integer (natural) number which is not equal to zero.
theorem Th26: :: RAT_1:26
for m being Integer
for p being Rational st m <> 0 holds
p = (() * m) / (() * m)
proof end;

theorem Th27: :: RAT_1:27
for k being Element of NAT
for m being Integer
for p being Rational st k <> 0 & p = m / k holds
ex l being Element of NAT st
( m = () * l & k = () * l )
proof end;

theorem :: RAT_1:28
for m, n being Integer
for p being Rational st p = m / n & n <> 0 holds
ex m1 being Integer st
( m = () * m1 & n = () * m1 )
proof end;

:: Fraction numerator(p)/denominator(p) is irreducible.
theorem Th29: :: RAT_1:29
for p being Rational
for l being Element of NAT holds
( not 1 < l or for m being Integer
for k being Element of NAT holds
( not numerator p = m * l or not denominator p = k * l ) )
proof end;

theorem Th30: :: RAT_1:30
for k being Element of NAT
for m being Integer
for p being Rational st p = m / k & k <> 0 & ( for l being Element of NAT holds
( not 1 < l or for m1 being Integer
for k1 being Element of NAT holds
( not m = m1 * l or not k = k1 * l ) ) ) holds
( k = denominator p & m = numerator p )
proof end;

theorem Th31: :: RAT_1:31
for p being Rational holds
( p < - 1 iff numerator p < - () )
proof end;

theorem Th32: :: RAT_1:32
for p being Rational holds
( p <= - 1 iff numerator p <= - () )
proof end;

theorem :: RAT_1:33
for p being Rational holds
( p < - 1 iff denominator p < - () )
proof end;

theorem :: RAT_1:34
for p being Rational holds
( p <= - 1 iff denominator p <= - () )
proof end;

theorem Th35: :: RAT_1:35
for p being Rational holds
( p < 1 iff numerator p < denominator p )
proof end;

theorem :: RAT_1:36
for p being Rational holds
( p <= 1 iff numerator p <= denominator p )
proof end;

theorem :: RAT_1:37
for p being Rational holds
( p < 0 iff numerator p < 0 )
proof end;

theorem Th38: :: RAT_1:38
for p being Rational holds
( p <= 0 iff numerator p <= 0 )
proof end;

theorem Th39: :: RAT_1:39
for a being real number
for p being Rational holds
( a < p iff a * () < numerator p )
proof end;

theorem :: RAT_1:40
for a being real number
for p being Rational holds
( a <= p iff a * () <= numerator p )
proof end;

theorem :: RAT_1:41
for p, q being Rational st denominator p = denominator q & numerator p = numerator q holds
p = q
proof end;

theorem :: RAT_1:42
for p, q being Rational holds
( p < q iff () * () < () * () )
proof end;

theorem Th43: :: RAT_1:43
for p being Rational holds
( denominator (- p) = denominator p & numerator (- p) = - () )
proof end;

theorem Th44: :: RAT_1:44
for p, q being Rational holds
( 0 < p & q = 1 / p iff ( numerator q = denominator p & denominator q = numerator p ) )
proof end;

theorem :: RAT_1:45
for p, q being Rational holds
( p < 0 & q = 1 / p iff ( numerator q = - () & denominator q = - () ) )
proof end;