:: RADIX_6 semantic presentation
Lemma1:
for b1 being Nat holds
not ( b1 >= 1 & not b1 + 2 > 1 )
theorem Th1: :: RADIX_6:1
theorem Th2: :: RADIX_6:2
:: deftheorem Def1 defines M0Digit RADIX_6:def 1 :
for b
1, b
2, b
3 being
Natfor b
4 being
Tuple of
(b2 + 2),
(b3 -SD ) holds
( b
1 in Seg (b2 + 2) implies ( ( b
1 >= b
2 implies
M0Digit b
4,b
1 = b
4 . b
1 ) & ( b
1 < b
2 implies
M0Digit b
4,b
1 = 0 ) ) );
definition
let c
1, c
2 be
Nat;
let c
3 be
Tuple of
(c1 + 2),
(c2 -SD );
func M0 c
3 -> Tuple of
(a1 + 2),
(a2 -SD ) means :
Def2:
:: RADIX_6:def 2
for b
1 being
Nat holds
( b
1 in Seg (a1 + 2) implies
DigA a
4,b
1 = M0Digit a
3,b
1 );
existence
ex b1 being Tuple of (c1 + 2),(c2 -SD ) st
for b2 being Nat holds
( b2 in Seg (c1 + 2) implies DigA b1,b2 = M0Digit c3,b2 )
uniqueness
for b1, b2 being Tuple of (c1 + 2),(c2 -SD ) holds
( ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b1,b3 = M0Digit c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b2,b3 = M0Digit c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines M0 RADIX_6:def 2 :
:: deftheorem Def3 defines MmaxDigit RADIX_6:def 3 :
definition
let c
1, c
2 be
Nat;
let c
3 be
Tuple of
(c1 + 2),
(c2 -SD );
func Mmax c
3 -> Tuple of
(a1 + 2),
(a2 -SD ) means :
Def4:
:: RADIX_6:def 4
for b
1 being
Nat holds
( b
1 in Seg (a1 + 2) implies
DigA a
4,b
1 = MmaxDigit a
3,b
1 );
existence
ex b1 being Tuple of (c1 + 2),(c2 -SD ) st
for b2 being Nat holds
( b2 in Seg (c1 + 2) implies DigA b1,b2 = MmaxDigit c3,b2 )
uniqueness
for b1, b2 being Tuple of (c1 + 2),(c2 -SD ) holds
( ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b1,b3 = MmaxDigit c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b2,b3 = MmaxDigit c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Mmax RADIX_6:def 4 :
:: deftheorem Def5 defines MminDigit RADIX_6:def 5 :
definition
let c
1, c
2 be
Nat;
let c
3 be
Tuple of
(c1 + 2),
(c2 -SD );
func Mmin c
3 -> Tuple of
(a1 + 2),
(a2 -SD ) means :
Def6:
:: RADIX_6:def 6
for b
1 being
Nat holds
( b
1 in Seg (a1 + 2) implies
DigA a
4,b
1 = MminDigit a
3,b
1 );
existence
ex b1 being Tuple of (c1 + 2),(c2 -SD ) st
for b2 being Nat holds
( b2 in Seg (c1 + 2) implies DigA b1,b2 = MminDigit c3,b2 )
uniqueness
for b1, b2 being Tuple of (c1 + 2),(c2 -SD ) holds
( ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b1,b3 = MminDigit c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b2,b3 = MminDigit c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines Mmin RADIX_6:def 6 :
theorem Th3: :: RADIX_6:3
theorem Th4: :: RADIX_6:4
:: deftheorem Def7 defines needs_digits_of RADIX_6:def 7 :
theorem Th5: :: RADIX_6:5
theorem Th6: :: RADIX_6:6
theorem Th7: :: RADIX_6:7
theorem Th8: :: RADIX_6:8
for b
1, b
2, b
3 being
Integer holds
not ( b
2 < b
1 + b
3 & b
3 > 0 & ( for b
4 being
Integer holds
not (
- b
3 < b
1 - (b4 * b3) & b
2 - (b4 * b3) < b
3 ) ) )
theorem Th9: :: RADIX_6:9
theorem Th10: :: RADIX_6:10
theorem Th11: :: RADIX_6:11
theorem Th12: :: RADIX_6:12
theorem Th13: :: RADIX_6:13
theorem Th14: :: RADIX_6:14
theorem Th15: :: RADIX_6:15
theorem Th16: :: RADIX_6:16
:: deftheorem Def8 defines MmaskDigit RADIX_6:def 8 :
definition
let c
1, c
2 be
Nat;
let c
3 be
Tuple of
(c1 + 2),
(c2 -SD );
func Mmask c
3 -> Tuple of
(a1 + 2),
(a2 -SD ) means :
Def9:
:: RADIX_6:def 9
for b
1 being
Nat holds
( b
1 in Seg (a1 + 2) implies
DigA a
4,b
1 = MmaskDigit a
3,b
1 );
existence
ex b1 being Tuple of (c1 + 2),(c2 -SD ) st
for b2 being Nat holds
( b2 in Seg (c1 + 2) implies DigA b1,b2 = MmaskDigit c3,b2 )
uniqueness
for b1, b2 being Tuple of (c1 + 2),(c2 -SD ) holds
( ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b1,b3 = MmaskDigit c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg (c1 + 2) implies DigA b2,b3 = MmaskDigit c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines Mmask RADIX_6:def 9 :
theorem Th17: :: RADIX_6:17
theorem Th18: :: RADIX_6:18
:: deftheorem Def10 defines FSDMinDigit RADIX_6:def 10 :
definition
let c
1, c
2, c
3 be
Nat;
func FSDMin c
1,c
2,c
3 -> Tuple of a
1,
(a3 -SD ) means :
Def11:
:: RADIX_6:def 11
for b
1 being
Nat holds
( b
1 in Seg a
1 implies
DigA a
4,b
1 = FSDMinDigit a
2,a
3,b
1 );
existence
ex b1 being Tuple of c1,(c3 -SD ) st
for b2 being Nat holds
( b2 in Seg c1 implies DigA b1,b2 = FSDMinDigit c2,c3,b2 )
uniqueness
for b1, b2 being Tuple of c1,(c3 -SD ) holds
( ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b1,b3 = FSDMinDigit c2,c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b2,b3 = FSDMinDigit c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines FSDMin RADIX_6:def 11 :
theorem Th19: :: RADIX_6:19
:: deftheorem Def12 defines is_Zero_over RADIX_6:def 12 :
theorem Th20: :: RADIX_6:20