:: High Speed Modulo Calculation Algorithm with Radix-$2^k$ SD Number
:: by Masaaki Niimura and Yasushi Fuwa
::
:: Received November 7, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users


Lm1: for m being Nat st m >= 1 holds
m + 2 > 1
proof end;

theorem Th1: :: RADIX_6:1
for n being Nat st n >= 1 holds
for m, k being Nat st m >= 1 & k >= 2 holds
SDDec (Fmin ((m + n),m,k)) = SDDec (Fmin (m,m,k))
proof end;

theorem :: RADIX_6:2
for m, k being Nat st m >= 1 & k >= 2 holds
SDDec (Fmin (m,m,k)) > 0
proof end;

definition
let i, m, k be Nat;
let r be Tuple of m + 2,k -SD ;
assume A1: i in Seg (m + 2) ;
func M0Digit (r,i) -> Element of k -SD equals :Def1: :: RADIX_6:def 1
r . i if i >= m
otherwise 0 ;
coherence
( ( i >= m implies r . i is Element of k -SD ) & ( not i >= m implies 0 is Element of k -SD ) )
proof end;
consistency
for b1 being Element of k -SD holds verum ;
end;

:: deftheorem Def1 defines M0Digit RADIX_6:def 1 :
for i, m, k being Nat
for r being Tuple of m + 2,k -SD st i in Seg (m + 2) holds
( ( i >= m implies M0Digit (r,i) = r . i ) & ( not i >= m implies M0Digit (r,i) = 0 ) );

definition
let m, k be Nat;
let r be Tuple of m + 2,k -SD ;
func M0 r -> Tuple of m + 2,k -SD means :Def2: :: RADIX_6:def 2
 for i being Nat st i in Seg (m + 2) holds
DigA (it,i) = M0Digit (r,i);
existence
ex b1 being Tuple of m + 2,k -SD st
for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = M0Digit (r,i)
proof end;
uniqueness
for b1, b2 being Tuple of m + 2,k -SD st ( for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = M0Digit (r,i) ) & ( for i being Nat st i in Seg (m + 2) holds
DigA (b2,i) = M0Digit (r,i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines M0 RADIX_6:def 2 :
for m, k being Nat
for r, b4 being Tuple of m + 2,k -SD holds
( b4 = M0 r iff for i being Nat st i in Seg (m + 2) holds
DigA (b4,i) = M0Digit (r,i) );

definition
let i, m, k be Nat;
let r be Tuple of m + 2,k -SD ;
assume that
A1: k >= 2 and
A2: i in Seg (m + 2) ;
func MmaxDigit (r,i) -> Element of k -SD equals :Def3: :: RADIX_6:def 3
r . i if i >= m
otherwise (Radix k) - 1;
coherence
( ( i >= m implies r . i is Element of k -SD ) & ( not i >= m implies (Radix k) - 1 is Element of k -SD ) )
proof end;
consistency
for b1 being Element of k -SD holds verum ;
end;

:: deftheorem Def3 defines MmaxDigit RADIX_6:def 3 :
for i, m, k being Nat
for r being Tuple of m + 2,k -SD st k >= 2 & i in Seg (m + 2) holds
( ( i >= m implies MmaxDigit (r,i) = r . i ) & ( not i >= m implies MmaxDigit (r,i) = (Radix k) - 1 ) );

definition
let m, k be Nat;
let r be Tuple of m + 2,k -SD ;
func Mmax r -> Tuple of m + 2,k -SD means :Def4: :: RADIX_6:def 4
 for i being Nat st i in Seg (m + 2) holds
DigA (it,i) = MmaxDigit (r,i);
existence
ex b1 being Tuple of m + 2,k -SD st
for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MmaxDigit (r,i)
proof end;
uniqueness
for b1, b2 being Tuple of m + 2,k -SD st ( for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MmaxDigit (r,i) ) & ( for i being Nat st i in Seg (m + 2) holds
DigA (b2,i) = MmaxDigit (r,i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines Mmax RADIX_6:def 4 :
for m, k being Nat
for r, b4 being Tuple of m + 2,k -SD holds
( b4 = Mmax r iff for i being Nat st i in Seg (m + 2) holds
DigA (b4,i) = MmaxDigit (r,i) );

definition
let i, m, k be Nat;
let r be Tuple of m + 2,k -SD ;
assume that
A1: k >= 2 and
A2: i in Seg (m + 2) ;
func MminDigit (r,i) -> Element of k -SD equals :Def5: :: RADIX_6:def 5
r . i if i >= m
otherwise (- (Radix k)) + 1;
coherence
( ( i >= m implies r . i is Element of k -SD ) & ( not i >= m implies (- (Radix k)) + 1 is Element of k -SD ) )
proof end;
consistency
for b1 being Element of k -SD holds verum ;
end;

:: deftheorem Def5 defines MminDigit RADIX_6:def 5 :
for i, m, k being Nat
for r being Tuple of m + 2,k -SD st k >= 2 & i in Seg (m + 2) holds
( ( i >= m implies MminDigit (r,i) = r . i ) & ( not i >= m implies MminDigit (r,i) = (- (Radix k)) + 1 ) );

definition
let m, k be Nat;
let r be Tuple of m + 2,k -SD ;
func Mmin r -> Tuple of m + 2,k -SD means :Def6: :: RADIX_6:def 6
 for i being Nat st i in Seg (m + 2) holds
DigA (it,i) = MminDigit (r,i);
existence
ex b1 being Tuple of m + 2,k -SD st
for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MminDigit (r,i)
proof end;
uniqueness
for b1, b2 being Tuple of m + 2,k -SD st ( for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MminDigit (r,i) ) & ( for i being Nat st i in Seg (m + 2) holds
DigA (b2,i) = MminDigit (r,i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines Mmin RADIX_6:def 6 :
for m, k being Nat
for r, b4 being Tuple of m + 2,k -SD holds
( b4 = Mmin r iff for i being Nat st i in Seg (m + 2) holds
DigA (b4,i) = MminDigit (r,i) );

theorem Th3: :: RADIX_6:3
for m, k being Nat st k >= 2 holds
for r being Tuple of m + 2,k -SD holds SDDec (Mmax r) >= SDDec r
proof end;

theorem Th4: :: RADIX_6:4
for m, k being Nat st k >= 2 holds
for r being Tuple of m + 2,k -SD holds SDDec r >= SDDec (Mmin r)
proof end;

definition
let n, k be Nat;
let x be Integer;
pred x needs_digits_of n,k means :: RADIX_6:def 7
( x < (Radix k) |^ n & x >= (Radix k) |^ (n -' 1) );
end;

:: deftheorem defines needs_digits_of RADIX_6:def 7 :
for n, k being Nat
for x being Integer holds
( x needs_digits_of n,k iff ( x < (Radix k) |^ n & x >= (Radix k) |^ (n -' 1) ) );

theorem Th5: :: RADIX_6:5
for x, n, k, i being Nat st i in Seg n holds
DigA ((DecSD (x,n,k)),i) >= 0
proof end;

theorem Th6: :: RADIX_6:6
for n, k, x being Nat st n >= 1 & x needs_digits_of n,k holds
DigA ((DecSD (x,n,k)),n) > 0
proof end;

theorem Th7: :: RADIX_6:7
for f, m, k being Nat st m >= 1 & k >= 2 & f needs_digits_of m,k holds
f >= SDDec (Fmin ((m + 2),m,k))
proof end;

theorem Th8: :: RADIX_6:8
for mlow, mhigh, f being Integer st mhigh < mlow + f & f > 0 holds
ex s being Integer st
( - f < mlow - (s * f) & mhigh - (s * f) < f )
proof end;

theorem Th9: :: RADIX_6:9
for m, k being Nat st k >= 2 holds
for r being Tuple of m + 2,k -SD holds (SDDec (Mmax r)) + (SDDec (DecSD (0,(m + 2),k))) = (SDDec (M0 r)) + (SDDec (SDMax ((m + 2),m,k)))
proof end;

theorem Th10: :: RADIX_6:10
for m, k being Nat st m >= 1 & k >= 2 holds
for r being Tuple of m + 2,k -SD holds SDDec (Mmax r) < (SDDec (M0 r)) + (SDDec (Fmin ((m + 2),m,k)))
proof end;

theorem Th11: :: RADIX_6:11
for m, k being Nat st k >= 2 holds
for r being Tuple of m + 2,k -SD holds (SDDec (Mmin r)) + (SDDec (DecSD (0,(m + 2),k))) = (SDDec (M0 r)) + (SDDec (SDMin ((m + 2),m,k)))
proof end;

theorem Th12: :: RADIX_6:12
for m, k being Nat
for r being Tuple of m + 2,k -SD st m >= 1 & k >= 2 holds
(SDDec (M0 r)) + (SDDec (DecSD (0,(m + 2),k))) = (SDDec (Mmin r)) + (SDDec (SDMax ((m + 2),m,k)))
proof end;

theorem Th13: :: RADIX_6:13
for m, k being Nat st m >= 1 & k >= 2 holds
for r being Tuple of m + 2,k -SD holds SDDec (M0 r) < (SDDec (Mmin r)) + (SDDec (Fmin ((m + 2),m,k)))
proof end;

theorem :: RADIX_6:14
for m, k, f being Nat
for r being Tuple of m + 2,k -SD st m >= 1 & k >= 2 & f needs_digits_of m,k holds
ex s being Integer st
( - f < (SDDec (M0 r)) - (s * f) & (SDDec (Mmax r)) - (s * f) < f )
proof end;

theorem :: RADIX_6:15
for m, k, f being Nat
for r being Tuple of m + 2,k -SD st m >= 1 & k >= 2 & f needs_digits_of m,k holds
ex s being Integer st
( - f < (SDDec (Mmin r)) - (s * f) & (SDDec (M0 r)) - (s * f) < f )
proof end;

theorem :: RADIX_6:16
for m, k being Nat
for r being Tuple of m + 2,k -SD holds
( not k >= 2 or ( SDDec (M0 r) <= SDDec r & SDDec r <= SDDec (Mmax r) ) or ( SDDec (Mmin r) <= SDDec r & SDDec r < SDDec (M0 r) ) )
proof end;

definition
let i, m, k be Nat;
let r be Tuple of m + 2,k -SD ;
assume A1: i in Seg (m + 2) ;
func MmaskDigit (r,i) -> Element of k -SD equals :Def8: :: RADIX_6:def 8
r . i if i < m
otherwise 0 ;
coherence
( ( i < m implies r . i is Element of k -SD ) & ( not i < m implies 0 is Element of k -SD ) )
proof end;
consistency
for b1 being Element of k -SD holds verum ;
end;

:: deftheorem Def8 defines MmaskDigit RADIX_6:def 8 :
for i, m, k being Nat
for r being Tuple of m + 2,k -SD st i in Seg (m + 2) holds
( ( i < m implies MmaskDigit (r,i) = r . i ) & ( not i < m implies MmaskDigit (r,i) = 0 ) );

definition
let m, k be Nat;
let r be Tuple of m + 2,k -SD ;
func Mmask r -> Tuple of m + 2,k -SD means :Def9: :: RADIX_6:def 9
 for i being Nat st i in Seg (m + 2) holds
DigA (it,i) = MmaskDigit (r,i);
existence
ex b1 being Tuple of m + 2,k -SD st
for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MmaskDigit (r,i)
proof end;
uniqueness
for b1, b2 being Tuple of m + 2,k -SD st ( for i being Nat st i in Seg (m + 2) holds
DigA (b1,i) = MmaskDigit (r,i) ) & ( for i being Nat st i in Seg (m + 2) holds
DigA (b2,i) = MmaskDigit (r,i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines Mmask RADIX_6:def 9 :
for m, k being Nat
for r, b4 being Tuple of m + 2,k -SD holds
( b4 = Mmask r iff for i being Nat st i in Seg (m + 2) holds
DigA (b4,i) = MmaskDigit (r,i) );

theorem Th17: :: RADIX_6:17
for m, k being Nat
for r being Tuple of m + 2,k -SD st k >= 2 holds
(SDDec (M0 r)) + (SDDec (Mmask r)) = (SDDec r) + (SDDec (DecSD (0,(m + 2),k)))
proof end;

theorem :: RADIX_6:18
for m, k being Nat
for r being Tuple of m + 2,k -SD st m >= 1 & k >= 2 & SDDec (Mmask r) > 0 holds
SDDec r > SDDec (M0 r)
proof end;

definition
let i, m, k be Nat;
assume A1: k >= 2 ;
func FSDMinDigit (m,k,i) -> Element of k -SD equals :Def10: :: RADIX_6:def 10
 0 if i > m
1 if i = m
otherwise (- (Radix k)) + 1;
coherence
( ( i > m implies 0 is Element of k -SD ) & ( i = m implies 1 is Element of k -SD ) & ( not i > m & not i = m implies (- (Radix k)) + 1 is Element of k -SD ) )
proof end;
consistency
for b1 being Element of k -SD st i > m & i = m holds
( b1 = 0 iff b1 = 1 ) ;
end;

:: deftheorem Def10 defines FSDMinDigit RADIX_6:def 10 :
for i, m, k being Nat st k >= 2 holds
( ( i > m implies FSDMinDigit (m,k,i) = 0 ) & ( i = m implies FSDMinDigit (m,k,i) = 1 ) & ( not i > m & not i = m implies FSDMinDigit (m,k,i) = (- (Radix k)) + 1 ) );

definition
let n, m, k be Nat;
func FSDMin (n,m,k) -> Tuple of n,k -SD means :Def11: :: RADIX_6:def 11
 for i being Nat st i in Seg n holds
DigA (it,i) = FSDMinDigit (m,k,i);
existence
ex b1 being Tuple of n,k -SD st
for i being Nat st i in Seg n holds
DigA (b1,i) = FSDMinDigit (m,k,i)
proof end;
uniqueness
for b1, b2 being Tuple of n,k -SD st ( for i being Nat st i in Seg n holds
DigA (b1,i) = FSDMinDigit (m,k,i) ) & ( for i being Nat st i in Seg n holds
DigA (b2,i) = FSDMinDigit (m,k,i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines FSDMin RADIX_6:def 11 :
for n, m, k being Nat
for b4 being Tuple of n,k -SD holds
( b4 = FSDMin (n,m,k) iff for i being Nat st i in Seg n holds
DigA (b4,i) = FSDMinDigit (m,k,i) );

theorem Th19: :: RADIX_6:19
for n being Nat st n >= 1 holds
for m, k being Nat st m in Seg n & k >= 2 holds
SDDec (FSDMin (n,m,k)) = 1
proof end;

definition
let n, m, k be Nat;
let r be Tuple of m + 2,k -SD ;
pred r is_Zero_over n means :: RADIX_6:def 12
 for i being Nat st i > n holds
DigA (r,i) = 0 ;
end;

:: deftheorem defines is_Zero_over RADIX_6:def 12 :
for n, m, k being Nat
for r being Tuple of m + 2,k -SD holds
( r is_Zero_over n iff for i being Nat st i > n holds
DigA (r,i) = 0 );

theorem :: RADIX_6:20
for m being Nat st m >= 1 holds
for n, k being Nat
for r being Tuple of m + 2,k -SD st k >= 2 & n in Seg (m + 2) & Mmask r is_Zero_over n & DigA ((Mmask r),n) > 0 holds
SDDec (Mmask r) > 0
proof end;