:: Magnitude Relation Properties of Radix-$2^k$ SD Number
:: by Masaaki Niimura and Yasushi Fuwa
::
:: Received November 7, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

theorem Th1: :: RADIX_5:1

for k being Nat st k >= 2 holds
(Radix k) - 1 in k -SD

proof end;

theorem Th2: :: RADIX_5:2

for i, n being Nat st i > 1 & i in Seg n holds
i -' 1 in Seg n

proof end;

theorem Th3: :: RADIX_5:3

for k being Nat st 2 <= k holds
4 <= Radix k

proof end;

theorem Th4: :: RADIX_5:4

for k being Nat
for tx being Tuple of 1,k -SD holds SDDec tx = DigA (tx,1)

proof end;

theorem Th5: :: RADIX_5:5

for i, k, n being Nat st i in Seg n holds
DigA ((DecSD (0,n,k)),i) = 0

proof end;

theorem :: RADIX_5:6

for n, k being Nat st n >= 1 holds
SDDec (DecSD (0,n,k)) = 0

proof end;

theorem Th7: :: RADIX_5:7

for k, n being Nat st 1 in Seg n & k >= 2 holds
DigA ((DecSD (1,n,k)),1) = 1

proof end;

theorem Th8: :: RADIX_5:8

for i, k, n being Nat st i in Seg n & i > 1 & k >= 2 holds
DigA ((DecSD (1,n,k)),i) = 0

proof end;

theorem :: RADIX_5:9

for n, k being Nat st n >= 1 & k >= 2 holds
SDDec (DecSD (1,n,k)) = 1

proof end;

theorem Th10: :: RADIX_5:10

for k being Nat st k >= 2 holds
SD_Add_Carry (Radix k) = 1

proof end;

theorem Th11: :: RADIX_5:11

for k being Nat st k >= 2 holds
SD_Add_Data ((Radix k),k) = 0

proof end;

Lm1: for k being Nat st 2 <= k holds
SD_Add_Carry ((Radix k) - 1) = 1

proof end;

Lm2: for n, k, i being Nat st k >= 2 & i in Seg n holds
SD_Add_Carry (DigA ((DecSD (1,n,k)),i)) = 0

proof end;

theorem Th12: :: RADIX_5:12

for n being Nat st n >= 1 holds
for k being Nat
for tx, ty being Tuple of n,k -SD st ( for i being Nat st i in Seg n holds
DigA (tx,i) = DigA (ty,i) ) holds
SDDec tx = SDDec ty

proof end;

theorem :: RADIX_5:13

for n being Nat st n >= 1 holds
for k being Nat
for tx, ty being Tuple of n,k -SD st ( for i being Nat st i in Seg n holds
DigA (tx,i) >= DigA (ty,i) ) holds
SDDec tx >= SDDec ty

proof end;

theorem Th14: :: RADIX_5:14

for n being Nat st n >= 1 holds
for k being Nat st k >= 2 holds
for tx, ty, tz, tw being Tuple of n,k -SD st ( for i being Nat holds
( not i in Seg n or ( DigA (tx,i) = DigA (tz,i) & DigA (ty,i) = DigA (tw,i) ) or ( DigA (ty,i) = DigA (tz,i) & DigA (tx,i) = DigA (tw,i) ) ) ) holds
(SDDec tz) + (SDDec tw) = (SDDec tx) + (SDDec ty)

proof end;

theorem :: RADIX_5:15

for n, k being Nat st n >= 1 & k >= 2 holds
for tx, ty, tz being Tuple of n,k -SD st ( for i being Nat holds
( not i in Seg n or ( DigA (tx,i) = DigA (tz,i) & DigA (ty,i) = 0 ) or ( DigA (ty,i) = DigA (tz,i) & DigA (tx,i) = 0 ) ) ) holds
(SDDec tz) + (SDDec (DecSD (0,n,k))) = (SDDec tx) + (SDDec ty)

proof end;

definition

let i, m, k be Nat;
assume A1: k >= 2 ;

func SDMinDigit (m,k,i) -> Element of k -SD equals :Def1: :: RADIX_5:def 1
(- (Radix k)) + 1 if ( 1 <= i & i < m )
otherwise 0 ;

coherence
( ( 1 <= i & i < m implies (- (Radix k)) + 1 is Element of k -SD ) & ( ( not 1 <= i or 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 SDMinDigit RADIX_5:def 1 :
for i, m, k being Nat st k >= 2 holds
( ( 1 <= i & i < m implies SDMinDigit (m,k,i) = (- (Radix k)) + 1 ) & ( ( not 1 <= i or not i < m ) implies SDMinDigit (m,k,i) = 0 ) );

definition

let n, m, k be Nat;

func SDMin (n,m,k) -> Tuple of n,k -SD means :Def2: :: RADIX_5:def 2
for i being Nat st i in Seg n holds
DigA (it,i) = SDMinDigit (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) = SDMinDigit (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) = SDMinDigit (m,k,i) ) & ( for i being Nat st i in Seg n holds
DigA (b₂,i) = SDMinDigit (m,k,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines SDMin RADIX_5:def 2 :
for n, m, k being Nat
for b₄ being Tuple of n,k -SD holds
( b₄ = SDMin (n,m,k) iff for i being Nat st i in Seg n holds
DigA (b₄,i) = SDMinDigit (m,k,i) );

definition

let i, m, k be Nat;
assume A1: k >= 2 ;

func SDMaxDigit (m,k,i) -> Element of k -SD equals :Def3: :: RADIX_5:def 3
(Radix k) - 1 if ( 1 <= i & i < m )
otherwise 0 ;

coherence
( ( 1 <= i & i < m implies (Radix k) - 1 is Element of k -SD ) & ( ( not 1 <= i or not i < m ) implies 0 is Element of k -SD ) ) by A1, Th1, RADIX_1:14;
consistency
for b₁ being Element of k -SD holds verum ;

end;

:: deftheorem Def3 defines SDMaxDigit RADIX_5:def 3 :
for i, m, k being Nat st k >= 2 holds
( ( 1 <= i & i < m implies SDMaxDigit (m,k,i) = (Radix k) - 1 ) & ( ( not 1 <= i or not i < m ) implies SDMaxDigit (m,k,i) = 0 ) );

definition

let n, m, k be Nat;

func SDMax (n,m,k) -> Tuple of n,k -SD means :Def4: :: RADIX_5:def 4
for i being Nat st i in Seg n holds
DigA (it,i) = SDMaxDigit (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) = SDMaxDigit (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) = SDMaxDigit (m,k,i) ) & ( for i being Nat st i in Seg n holds
DigA (b₂,i) = SDMaxDigit (m,k,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines SDMax RADIX_5:def 4 :
for n, m, k being Nat
for b₄ being Tuple of n,k -SD holds
( b₄ = SDMax (n,m,k) iff for i being Nat st i in Seg n holds
DigA (b₄,i) = SDMaxDigit (m,k,i) );

definition

let i, m, k be Nat;
assume A1: k >= 2 ;

func FminDigit (m,k,i) -> Element of k -SD equals :Def5: :: RADIX_5:def 5
1 if i = m
otherwise 0 ;

coherence
( ( i = m implies 1 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 Def5 defines FminDigit RADIX_5:def 5 :
for i, m, k being Nat st k >= 2 holds
( ( i = m implies FminDigit (m,k,i) = 1 ) & ( not i = m implies FminDigit (m,k,i) = 0 ) );

definition

let n, m, k be Nat;

func Fmin (n,m,k) -> Tuple of n,k -SD means :Def6: :: RADIX_5:def 6
for i being Nat st i in Seg n holds
DigA (it,i) = FminDigit (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) = FminDigit (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) = FminDigit (m,k,i) ) & ( for i being Nat st i in Seg n holds
DigA (b₂,i) = FminDigit (m,k,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines Fmin RADIX_5:def 6 :
for n, m, k being Nat
for b₄ being Tuple of n,k -SD holds
( b₄ = Fmin (n,m,k) iff for i being Nat st i in Seg n holds
DigA (b₄,i) = FminDigit (m,k,i) );

definition

let i, m, k be Nat;
assume A1: k >= 2 ;

func FmaxDigit (m,k,i) -> Element of k -SD equals :Def7: :: RADIX_5:def 7
(Radix k) - 1 if i = m
otherwise 0 ;

coherence
( ( i = m implies (Radix k) - 1 is Element of k -SD ) & ( not i = m implies 0 is Element of k -SD ) ) by A1, Th1, RADIX_1:14;
consistency
for b₁ being Element of k -SD holds verum ;

end;

:: deftheorem Def7 defines FmaxDigit RADIX_5:def 7 :
for i, m, k being Nat st k >= 2 holds
( ( i = m implies FmaxDigit (m,k,i) = (Radix k) - 1 ) & ( not i = m implies FmaxDigit (m,k,i) = 0 ) );

definition

let n, m, k be Nat;

func Fmax (n,m,k) -> Tuple of n,k -SD means :Def8: :: RADIX_5:def 8
for i being Nat st i in Seg n holds
DigA (it,i) = FmaxDigit (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) = FmaxDigit (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) = FmaxDigit (m,k,i) ) & ( for i being Nat st i in Seg n holds
DigA (b₂,i) = FmaxDigit (m,k,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines Fmax RADIX_5:def 8 :
for n, m, k being Nat
for b₄ being Tuple of n,k -SD holds
( b₄ = Fmax (n,m,k) iff for i being Nat st i in Seg n holds
DigA (b₄,i) = FmaxDigit (m,k,i) );

theorem Th16: :: RADIX_5:16

for n, m, k being Nat st k >= 2 holds
for i being Nat st i in Seg n holds
(DigA ((SDMax (n,m,k)),i)) + (DigA ((SDMin (n,m,k)),i)) = 0

proof end;

theorem :: RADIX_5:17

for n being Nat st n >= 1 holds
for m, k being Nat st k >= 2 holds
(SDDec (SDMax (n,m,k))) + (SDDec (SDMin (n,m,k))) = SDDec (DecSD (0,n,k))

proof end;

theorem :: RADIX_5:18

for n being Nat st n >= 1 holds
for m, k being Nat st m in Seg n & k >= 2 holds
SDDec (Fmin (n,m,k)) = (SDDec (SDMax (n,m,k))) + (SDDec (DecSD (1,n,k)))

proof end;

theorem :: RADIX_5:19

for n, m, k being Nat st m in Seg n & k >= 2 holds
SDDec (Fmin ((n + 1),(m + 1),k)) = (SDDec (Fmin ((n + 1),m,k))) + (SDDec (Fmax ((n + 1),m,k)))

proof end;