Article RADIX_2, MML version 4.99.1005
:: RADIX_2:th 1
theorem
for b1 being natural set holds
b1 mod 1 = 0;
:: RADIX_2:th 2
theorem
for b1, b2 being integer set
for b3 being natural set
st 0 < b3
holds ((b1 mod b3) + (b2 mod b3)) mod b3 = (b1 + (b2 mod b3)) mod b3;
:: RADIX_2:th 3
theorem
for b1, b2 being integer set
for b3 being natural set
st 0 < b3
holds (b1 * b2) mod b3 = (b1 * (b2 mod b3)) mod b3;
:: RADIX_2:th 4
theorem
for b1, b2, b3 being natural set
st 1 <= b3 & 0 < b2
holds (b1 mod (b2 |^ b3)) div (b2 |^ (b3 -' 1)) = (b1 div (b2 |^ (b3 -' 1))) mod b2;
:: RADIX_2:th 5
theorem
for b1, b2 being natural set
st b1 in Seg b2
holds b1 + 1 in Seg (b2 + 1);
:: RADIX_2:th 6
theorem
for b1 being natural set holds
0 < Radix b1;
:: RADIX_2:th 7
theorem
for b1 being natural set
for b2 being Element of 1 -tuples_on (b1 -SD) holds
SDDec b2 = DigA(b2,1);
:: RADIX_2:th 8
theorem
for b1 being natural set
for b2 being integer set holds
(SD_Add_Data(b2,b1)) + ((SD_Add_Carry b2) * Radix b1) = b2;
:: RADIX_2:th 9
theorem
for b1, b2 being natural set
for b3 being Element of (b2 + 1) -tuples_on (b1 -SD)
for b4 being Element of b2 -tuples_on (b1 -SD)
st for b5 being natural set
st b5 in Seg b2
holds b3 . b5 = b4 . b5
holds Sum DigitSD b3 = Sum ((DigitSD b4) ^ <*SubDigit(b3,b2 + 1,b1)*>);
:: RADIX_2:th 10
theorem
for b1, b2 being natural set
for b3 being Element of (b2 + 1) -tuples_on (b1 -SD)
for b4 being Element of b2 -tuples_on (b1 -SD)
st for b5 being natural set
st b5 in Seg b2
holds b3 . b5 = b4 . b5
holds (SDDec b4) + (((Radix b1) |^ b2) * DigA(b3,b2 + 1)) = SDDec b3;
:: RADIX_2:th 11
theorem
for b1, b2 being natural set
st 1 <= b2
for b3, b4 being Element of b2 -tuples_on (b1 -SD)
st 2 <= b1
holds (SDDec (b3 '+' b4)) + ((SD_Add_Carry ((DigA(b3,b2)) + DigA(b4,b2))) * ((Radix b1) |^ b2)) = (SDDec b3) + SDDec b4;
:: RADIX_2:funcnot 1 => RADIX_2:func 1
definition
let a1, a2, a3 be natural set;
let a4 be Element of a3 -tuples_on NAT;
func SubDigit2(A4,A1,A2) -> Element of NAT equals
((Radix a2) |^ (a1 -' 1)) * (a4 . a1);
end;
:: RADIX_2:def 1
theorem
for b1, b2, b3 being natural set
for b4 being Element of b3 -tuples_on NAT holds
SubDigit2(b4,b1,b2) = ((Radix b2) |^ (b1 -' 1)) * (b4 . b1);
:: RADIX_2:funcnot 2 => RADIX_2:func 2
definition
let a1, a2 be natural set;
let a3 be Element of a1 -tuples_on NAT;
func DigitSD2(A3,A2) -> Element of a1 -tuples_on NAT means
for b1 being natural set
st b1 in Seg a1
holds it /. b1 = SubDigit2(a3,b1,a2);
end;
:: RADIX_2:def 2
theorem
for b1, b2 being natural set
for b3, b4 being Element of b1 -tuples_on NAT holds
b4 = DigitSD2(b3,b2)
iff
for b5 being natural set
st b5 in Seg b1
holds b4 /. b5 = SubDigit2(b3,b5,b2);
:: RADIX_2:funcnot 3 => RADIX_2:func 3
definition
let a1, a2 be natural set;
let a3 be Element of a1 -tuples_on NAT;
func SDDec2(A3,A2) -> Element of NAT equals
Sum DigitSD2(a3,a2);
end;
:: RADIX_2:def 3
theorem
for b1, b2 being natural set
for b3 being Element of b1 -tuples_on NAT holds
SDDec2(b3,b2) = Sum DigitSD2(b3,b2);
:: RADIX_2:funcnot 4 => RADIX_2:func 4
definition
let a1, a2, a3 be natural set;
func DigitDC2(A3,A1,A2) -> Element of NAT equals
(a3 mod ((Radix a2) |^ a1)) div ((Radix a2) |^ (a1 -' 1));
end;
:: RADIX_2:def 4
theorem
for b1, b2, b3 being natural set holds
DigitDC2(b3,b1,b2) = (b3 mod ((Radix b2) |^ b1)) div ((Radix b2) |^ (b1 -' 1));
:: RADIX_2:funcnot 5 => RADIX_2:func 5
definition
let a1, a2, a3 be natural set;
func DecSD2(A3,A2,A1) -> Element of a2 -tuples_on NAT means
for b1 being natural set
st b1 in Seg a2
holds it . b1 = DigitDC2(a3,b1,a1);
end;
:: RADIX_2:def 5
theorem
for b1, b2, b3 being natural set
for b4 being Element of b2 -tuples_on NAT holds
b4 = DecSD2(b3,b2,b1)
iff
for b5 being natural set
st b5 in Seg b2
holds b4 . b5 = DigitDC2(b3,b5,b1);
:: RADIX_2:th 12
theorem
for b1, b2 being natural set
for b3 being Element of b1 -tuples_on NAT
for b4 being Element of b1 -tuples_on (b2 -SD)
st b3 = b4
holds DigitSD2(b3,b2) = DigitSD b4;
:: RADIX_2:th 13
theorem
for b1, b2 being natural set
for b3 being Element of b1 -tuples_on NAT
for b4 being Element of b1 -tuples_on (b2 -SD)
st b3 = b4
holds SDDec2(b3,b2) = SDDec b4;
:: RADIX_2:th 14
theorem
for b1, b2, b3 being natural set holds
DecSD2(b1,b2,b3) = DecSD(b1,b2,b3);
:: RADIX_2:th 15
theorem
for b1 being natural set
st 1 <= b1
for b2, b3 being natural set
st b2 is_represented_by b1,b3
holds b2 = SDDec2(DecSD2(b2,b1,b3),b3);
:: RADIX_2:funcnot 6 => RADIX_2:func 6
definition
let a1 be integer set;
let a2, a3, a4, a5 be natural set;
let a6 be Element of a5 -tuples_on (a4 -SD);
func Table1(A1,A6,A2,A3) -> integer set equals
(a1 * DigA(a6,a3)) mod a2;
end;
:: RADIX_2:def 6
theorem
for b1 being integer set
for b2, b3, b4, b5 being natural set
for b6 being Element of b5 -tuples_on (b4 -SD) holds
Table1(b1,b6,b2,b3) = (b1 * DigA(b6,b3)) mod b2;
:: RADIX_2:funcnot 7 => RADIX_2:func 7
definition
let a1 be integer set;
let a2, a3, a4 be natural set;
let a5 be Element of a4 -tuples_on (a2 -SD);
assume 1 <= a4;
func Mul_mod(A1,A5,A3,A2) -> Element of a4 -tuples_on INT means
it . 1 = Table1(a1,a5,a3,a4) &
(for b1 being natural set
st 1 <= b1 & b1 <= a4 - 1
holds ex b2, b3 being integer set st
b2 = it . b1 &
b3 = it . (b1 + 1) &
b3 = (((Radix a2) * b2) + Table1(a1,a5,a3,a4 -' b1)) mod a3);
end;
:: RADIX_2:def 7
theorem
for b1 being integer set
for b2, b3, b4 being natural set
for b5 being Element of b4 -tuples_on (b2 -SD)
st 1 <= b4
for b6 being Element of b4 -tuples_on INT holds
b6 = Mul_mod(b1,b5,b3,b2)
iff
b6 . 1 = Table1(b1,b5,b3,b4) &
(for b7 being natural set
st 1 <= b7 & b7 <= b4 - 1
holds ex b8, b9 being integer set st
b8 = b6 . b7 &
b9 = b6 . (b7 + 1) &
b9 = (((Radix b2) * b8) + Table1(b1,b5,b3,b4 -' b7)) mod b3);
:: RADIX_2:th 16
theorem
for b1 being natural set
st 1 <= b1
for b2 being integer set
for b3, b4, b5 being natural set
st b3 is_represented_by b1,b5 & 0 < b4
for b6 being Element of b1 -tuples_on (b5 -SD)
st b6 = DecSD(b3,b1,b5)
holds (Mul_mod(b2,b6,b4,b5)) . b1 = (b2 * b3) mod b4;
:: RADIX_2:funcnot 8 => RADIX_2:func 8
definition
let a1, a2, a3, a4 be natural set;
let a5 be Element of a1 -tuples_on NAT;
func Table2(A4,A5,A2,A3) -> Element of NAT equals
(a4 |^ (a5 /. a3)) mod a2;
end;
:: RADIX_2:def 8
theorem
for b1, b2, b3, b4 being natural set
for b5 being Element of b1 -tuples_on NAT holds
Table2(b4,b5,b2,b3) = (b4 |^ (b5 /. b3)) mod b2;
:: RADIX_2:funcnot 9 => RADIX_2:func 9
definition
let a1, a2, a3, a4 be natural set;
let a5 be Element of a4 -tuples_on NAT;
assume 1 <= a4;
func Pow_mod(A3,A5,A2,A1) -> Element of a4 -tuples_on NAT means
it . 1 = Table2(a3,a5,a2,a4) &
(for b1 being natural set
st 1 <= b1 & b1 <= a4 - 1
holds ex b2, b3 being natural set st
b2 = it . b1 &
b3 = it . (b1 + 1) &
b3 = (((b2 |^ Radix a1) mod a2) * Table2(a3,a5,a2,a4 -' b1)) mod a2);
end;
:: RADIX_2:def 9
theorem
for b1, b2, b3, b4 being natural set
for b5 being Element of b4 -tuples_on NAT
st 1 <= b4
for b6 being Element of b4 -tuples_on NAT holds
b6 = Pow_mod(b3,b5,b2,b1)
iff
b6 . 1 = Table2(b3,b5,b2,b4) &
(for b7 being natural set
st 1 <= b7 & b7 <= b4 - 1
holds ex b8, b9 being natural set st
b8 = b6 . b7 &
b9 = b6 . (b7 + 1) &
b9 = (((b8 |^ Radix b1) mod b2) * Table2(b3,b5,b2,b4 -' b7)) mod b2);
:: RADIX_2:th 17
theorem
for b1 being natural set
st 1 <= b1
for b2, b3, b4, b5 being natural set
st b5 is_represented_by b1,b3 & 0 < b4
for b6 being Element of b1 -tuples_on NAT
st b6 = DecSD2(b5,b1,b3)
holds (Pow_mod(b2,b6,b4,b3)) . b1 = (b2 |^ b5) mod b4;