:: RADIX_5 semantic presentation
theorem Th1: :: RADIX_5:1
theorem Th2: :: RADIX_5:2
theorem Th3: :: RADIX_5:3
theorem Th4: :: RADIX_5:4
theorem Th5: :: RADIX_5:5
theorem Th6: :: RADIX_5:6
theorem Th7: :: RADIX_5:7
theorem Th8: :: RADIX_5:8
theorem Th9: :: RADIX_5:9
theorem Th10: :: RADIX_5:10
theorem Th11: :: RADIX_5:11
Lemma10:
for b1 being Nat holds
( 2 <= b1 implies SD_Add_Carry ((Radix b1) - 1) = 1 )
Lemma11:
for b1, b2, b3 being Nat holds
( b2 >= 2 & b3 in Seg b1 implies SD_Add_Carry (DigA (DecSD 1,b1,b2),b3) = 0 )
theorem Th12: :: RADIX_5:12
theorem Th13: :: RADIX_5:13
theorem Th14: :: RADIX_5:14
for b
1 being
Nat holds
( b
1 >= 1 implies for b
2 being
Nat holds
( b
2 >= 2 implies for b
3, b
4, b
5, b
6 being
Tuple of b
1,
(b2 -SD ) holds
( ( for b
7 being
Nat holds
not ( b
7 in Seg b
1 & not (
DigA b
3,b
7 = DigA b
5,b
7 &
DigA b
4,b
7 = DigA b
6,b
7 ) & not (
DigA b
4,b
7 = DigA b
5,b
7 &
DigA b
3,b
7 = DigA b
6,b
7 ) ) ) implies
(SDDec b5) + (SDDec b6) = (SDDec b3) + (SDDec b4) ) ) )
theorem Th15: :: RADIX_5:15
for b
1, b
2 being
Nat holds
( b
1 >= 1 & b
2 >= 2 implies for b
3, b
4, b
5 being
Tuple of b
1,
(b2 -SD ) holds
( ( for b
6 being
Nat holds
not ( b
6 in Seg b
1 & not (
DigA b
3,b
6 = DigA b
5,b
6 &
DigA b
4,b
6 = 0 ) & not (
DigA b
4,b
6 = DigA b
5,b
6 &
DigA b
3,b
6 = 0 ) ) ) implies
(SDDec b5) + (SDDec (DecSD 0,b1,b2)) = (SDDec b3) + (SDDec b4) ) )
:: deftheorem Def1 defines SDMinDigit RADIX_5:def 1 :
definition
let c
1, c
2, c
3 be
Nat;
func SDMin c
1,c
2,c
3 -> Tuple of a
1,
(a3 -SD ) means :
Def2:
:: RADIX_5:def 2
for b
1 being
Nat holds
( b
1 in Seg a
1 implies
DigA a
4,b
1 = SDMinDigit 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 = SDMinDigit 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 = SDMinDigit c2,c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b2,b3 = SDMinDigit c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines SDMin RADIX_5:def 2 :
:: deftheorem Def3 defines SDMaxDigit RADIX_5:def 3 :
definition
let c
1, c
2, c
3 be
Nat;
func SDMax c
1,c
2,c
3 -> Tuple of a
1,
(a3 -SD ) means :
Def4:
:: RADIX_5:def 4
for b
1 being
Nat holds
( b
1 in Seg a
1 implies
DigA a
4,b
1 = SDMaxDigit 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 = SDMaxDigit 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 = SDMaxDigit c2,c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b2,b3 = SDMaxDigit c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines SDMax RADIX_5:def 4 :
:: deftheorem Def5 defines FminDigit RADIX_5:def 5 :
for b
1, b
2, b
3 being
Nat holds
( b
3 >= 2 implies ( ( b
1 = b
2 implies
FminDigit b
2,b
3,b
1 = 1 ) & ( not b
1 = b
2 implies
FminDigit b
2,b
3,b
1 = 0 ) ) );
definition
let c
1, c
2, c
3 be
Nat;
func Fmin c
1,c
2,c
3 -> Tuple of a
1,
(a3 -SD ) means :
Def6:
:: RADIX_5:def 6
for b
1 being
Nat holds
( b
1 in Seg a
1 implies
DigA a
4,b
1 = FminDigit 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 = FminDigit 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 = FminDigit c2,c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b2,b3 = FminDigit c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines Fmin RADIX_5:def 6 :
:: deftheorem Def7 defines FmaxDigit RADIX_5:def 7 :
definition
let c
1, c
2, c
3 be
Nat;
func Fmax c
1,c
2,c
3 -> Tuple of a
1,
(a3 -SD ) means :
Def8:
:: RADIX_5:def 8
for b
1 being
Nat holds
( b
1 in Seg a
1 implies
DigA a
4,b
1 = FmaxDigit 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 = FmaxDigit 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 = FmaxDigit c2,c3,b3 ) ) & ( for b3 being Nat holds
( b3 in Seg c1 implies DigA b2,b3 = FmaxDigit c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines Fmax RADIX_5:def 8 :
theorem Th16: :: RADIX_5:16
theorem Th17: :: RADIX_5:17
theorem Th18: :: RADIX_5:18
theorem Th19: :: RADIX_5:19