:: Improvement of Radix-$2^k$ Signed-Digit Number for High Speed Circuit
:: by Masaaki Niimura and Yasushi Fuwa
::
:: Received January 3, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

Lm1: for k being Nat st 1 <= k holds
Radix k = (Radix (k -' 1)) + (Radix (k -' 1))

proof end;

Lm2: for k being Nat st 1 <= k holds
(Radix k) - (Radix (k -' 1)) = Radix (k -' 1)

proof end;

Lm3: for k being Nat st 1 <= k holds
(- (Radix k)) + (Radix (k -' 1)) = - (Radix (k -' 1))

proof end;

Lm4: for k being Nat st 1 <= k holds
2 <= Radix k

proof end;

Lm5: for n, m, k, i being Nat st i in Seg n holds
DigA ((DecSD (m,(n + 1),k)),i) = DigA ((DecSD ((m mod ((Radix k) |^ n)),n,k)),i)

proof end;

definition

let k be Nat;

func k -SD_Sub_S -> set equals :: RADIX_3:def 1
{ e where e is Element of INT : ( e >= - (Radix (k -' 1)) & e <= (Radix (k -' 1)) - 1 ) } ;

correctness
coherence
{ e where e is Element of INT : ( e >= - (Radix (k -' 1)) & e <= (Radix (k -' 1)) - 1 ) } is set ;
;

func k -SD_Sub -> set equals :: RADIX_3:def 2
{ e where e is Element of INT : ( e >= (- (Radix (k -' 1))) - 1 & e <= Radix (k -' 1) ) } ;

correctness
coherence
{ e where e is Element of INT : ( e >= (- (Radix (k -' 1))) - 1 & e <= Radix (k -' 1) ) } is set ;
;

end;

:: deftheorem defines -SD_Sub_S RADIX_3:def 1 :
for k being Nat holds k -SD_Sub_S = { e where e is Element of INT : ( e >= - (Radix (k -' 1)) & e <= (Radix (k -' 1)) - 1 ) } ;

:: deftheorem defines -SD_Sub RADIX_3:def 2 :
for k being Nat holds k -SD_Sub = { e where e is Element of INT : ( e >= (- (Radix (k -' 1))) - 1 & e <= Radix (k -' 1) ) } ;

theorem Th1: :: RADIX_3:1

for k being Nat
for i1 being Integer st i1 in k -SD_Sub holds
( (- (Radix (k -' 1))) - 1 <= i1 & i1 <= Radix (k -' 1) )

proof end;

theorem Th2: :: RADIX_3:2

for k being Nat holds k -SD_Sub_S c= k -SD_Sub

proof end;

theorem Th3: :: RADIX_3:3

for k being Nat holds k -SD_Sub_S c= (k + 1) -SD_Sub_S

proof end;

theorem :: RADIX_3:4

for k being Nat st 2 <= k holds
k -SD_Sub c= k -SD

proof end;

theorem Th5: :: RADIX_3:5

0 in 0 -SD_Sub_S

proof end;

theorem Th6: :: RADIX_3:6

for k being Nat holds 0 in k -SD_Sub_S

proof end;

theorem Th7: :: RADIX_3:7

for k being Nat holds 0 in k -SD_Sub

proof end;

theorem Th8: :: RADIX_3:8

for k being Nat
for e being set st e in k -SD_Sub holds
e is Integer

proof end;

theorem Th9: :: RADIX_3:9

for k being Nat holds k -SD_Sub c= INT

proof end;

theorem Th10: :: RADIX_3:10

for k being Nat holds k -SD_Sub_S c= INT

proof end;

registration

let k be Nat;

cluster k -SD_Sub_S -> non empty ;

coherence
not k -SD_Sub_S is empty by Th6;

cluster k -SD_Sub -> non empty ;

coherence
not k -SD_Sub is empty by Th7;

end;

definition

let k be Nat;
:: original: -SD_Sub_S
redefine func k -SD_Sub_S -> non empty Subset of INT;
coherence
k -SD_Sub_S is non empty Subset of INT by Th10;

end;

definition

let k be Nat;
:: original: -SD_Sub
redefine func k -SD_Sub -> non empty Subset of INT;
coherence
k -SD_Sub is non empty Subset of INT by Th9;

end;

theorem Th11: :: RADIX_3:11

for i, n, k being Nat
for aSub being Tuple of n,k -SD_Sub st i in Seg n holds
aSub . i is Element of k -SD_Sub

proof end;

definition

let x be Integer;
let k be Nat;

func SDSub_Add_Carry (x,k) -> Integer equals :Def3: :: RADIX_3:def 3
1 if Radix (k -' 1) <= x
- 1 if x < - (Radix (k -' 1))
otherwise 0 ;

coherence
( ( Radix (k -' 1) <= x implies 1 is Integer ) & ( x < - (Radix (k -' 1)) implies - 1 is Integer ) & ( not Radix (k -' 1) <= x & not x < - (Radix (k -' 1)) implies 0 is Integer ) ) ;
consistency
for b₁ being Integer st Radix (k -' 1) <= x & x < - (Radix (k -' 1)) holds
( b₁ = 1 iff b₁ = - 1 ) ;

end;

:: deftheorem Def3 defines SDSub_Add_Carry RADIX_3:def 3 :
for x being Integer
for k being Nat holds
( ( Radix (k -' 1) <= x implies SDSub_Add_Carry (x,k) = 1 ) & ( x < - (Radix (k -' 1)) implies SDSub_Add_Carry (x,k) = - 1 ) & ( not Radix (k -' 1) <= x & not x < - (Radix (k -' 1)) implies SDSub_Add_Carry (x,k) = 0 ) );

definition

let x be Integer;
let k be Nat;

func SDSub_Add_Data (x,k) -> Integer equals :: RADIX_3:def 4
x - ((Radix k) * (SDSub_Add_Carry (x,k)));

coherence
x - ((Radix k) * (SDSub_Add_Carry (x,k))) is Integer ;

end;

:: deftheorem defines SDSub_Add_Data RADIX_3:def 4 :
for x being Integer
for k being Nat holds SDSub_Add_Data (x,k) = x - ((Radix k) * (SDSub_Add_Carry (x,k)));

theorem Th12: :: RADIX_3:12

for x being Integer
for k being Nat holds
( - 1 <= SDSub_Add_Carry (x,k) & SDSub_Add_Carry (x,k) <= 1 )

proof end;

theorem Th13: :: RADIX_3:13

for k being Nat
for i1 being Integer st 2 <= k & i1 in k -SD holds
( SDSub_Add_Data (i1,k) >= - (Radix (k -' 1)) & SDSub_Add_Data (i1,k) <= (Radix (k -' 1)) - 1 )

proof end;

theorem Th14: :: RADIX_3:14

for x being Integer
for k being Nat st 2 <= k holds
SDSub_Add_Carry (x,k) in k -SD_Sub_S

proof end;

theorem Th15: :: RADIX_3:15

for k being Nat
for i1, i2 being Integer st 2 <= k & i1 in k -SD holds
(SDSub_Add_Data (i1,k)) + (SDSub_Add_Carry (i2,k)) in k -SD_Sub

proof end;

theorem Th16: :: RADIX_3:16

for k being Nat holds SDSub_Add_Carry (0,k) = 0

proof end;

definition

let i, k, n be Nat;
let x be Tuple of n,k -SD_Sub ;

func DigA_SDSub (x,i) -> Integer equals :Def5: :: RADIX_3:def 5
x . i if i in Seg n
0 if i = 0
;

coherence
( ( i in Seg n implies x . i is Integer ) & ( i = 0 implies 0 is Integer ) ) by INT_1:def 2;
consistency
for b₁ being Integer st i in Seg n & i = 0 holds
( b₁ = x . i iff b₁ = 0 ) by FINSEQ_1:1;

end;

:: deftheorem Def5 defines DigA_SDSub RADIX_3:def 5 :
for i, k, n being Nat
for x being Tuple of n,k -SD_Sub holds
( ( i in Seg n implies DigA_SDSub (x,i) = x . i ) & ( i = 0 implies DigA_SDSub (x,i) = 0 ) );

definition

let i, k, n be Nat;
let x be Tuple of n,k -SD ;

func SD2SDSubDigit (x,i,k) -> Integer equals :Def6: :: RADIX_3:def 6
(SDSub_Add_Data ((DigA (x,i)),k)) + (SDSub_Add_Carry ((DigA (x,(i -' 1))),k)) if i in Seg n
SDSub_Add_Carry ((DigA (x,(i -' 1))),k) if i = n + 1
otherwise 0 ;

coherence
( ( i in Seg n implies (SDSub_Add_Data ((DigA (x,i)),k)) + (SDSub_Add_Carry ((DigA (x,(i -' 1))),k)) is Integer ) & ( i = n + 1 implies SDSub_Add_Carry ((DigA (x,(i -' 1))),k) is Integer ) & ( not i in Seg n & not i = n + 1 implies 0 is Integer ) ) ;
consistency
for b₁ being Integer st i in Seg n & i = n + 1 holds
( b₁ = (SDSub_Add_Data ((DigA (x,i)),k)) + (SDSub_Add_Carry ((DigA (x,(i -' 1))),k)) iff b₁ = SDSub_Add_Carry ((DigA (x,(i -' 1))),k) )

proof end;

end;

:: deftheorem Def6 defines SD2SDSubDigit RADIX_3:def 6 :
for i, k, n being Nat
for x being Tuple of n,k -SD holds
( ( i in Seg n implies SD2SDSubDigit (x,i,k) = (SDSub_Add_Data ((DigA (x,i)),k)) + (SDSub_Add_Carry ((DigA (x,(i -' 1))),k)) ) & ( i = n + 1 implies SD2SDSubDigit (x,i,k) = SDSub_Add_Carry ((DigA (x,(i -' 1))),k) ) & ( not i in Seg n & not i = n + 1 implies SD2SDSubDigit (x,i,k) = 0 ) );

theorem Th17: :: RADIX_3:17

for i, n, k being Nat
for a being Tuple of n,k -SD st 2 <= k & i in Seg (n + 1) holds
SD2SDSubDigit (a,i,k) is Element of k -SD_Sub

proof end;

definition

let i, k, n be Nat;
let x be Tuple of n,k -SD ;
assume A1: ( 2 <= k & i in Seg (n + 1) ) ;

func SD2SDSubDigitS (x,i,k) -> Element of k -SD_Sub equals :Def7: :: RADIX_3:def 7
SD2SDSubDigit (x,i,k);

coherence
SD2SDSubDigit (x,i,k) is Element of k -SD_Sub by A1, Th17;

end;

:: deftheorem Def7 defines SD2SDSubDigitS RADIX_3:def 7 :
for i, k, n being Nat
for x being Tuple of n,k -SD st 2 <= k & i in Seg (n + 1) holds
SD2SDSubDigitS (x,i,k) = SD2SDSubDigit (x,i,k);

definition

let n, k be Nat;
let x be Tuple of n,k -SD ;

func SD2SDSub x -> Tuple of n + 1,k -SD_Sub means :Def8: :: RADIX_3:def 8
for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (it,i) = SD2SDSubDigitS (x,i,k);

existence
ex b₁ being Tuple of n + 1,k -SD_Sub st
for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (b₁,i) = SD2SDSubDigitS (x,i,k)

proof end;

uniqueness
for b₁, b₂ being Tuple of n + 1,k -SD_Sub st ( for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (b₁,i) = SD2SDSubDigitS (x,i,k) ) & ( for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (b₂,i) = SD2SDSubDigitS (x,i,k) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines SD2SDSub RADIX_3:def 8 :
for n, k being Nat
for x being Tuple of n,k -SD
for b₄ being Tuple of n + 1,k -SD_Sub holds
( b₄ = SD2SDSub x iff for i being Nat st i in Seg (n + 1) holds
DigA_SDSub (b₄,i) = SD2SDSubDigitS (x,i,k) );

theorem :: RADIX_3:18

for i, n, k being Nat
for aSub being Tuple of n,k -SD_Sub st i in Seg n holds
DigA_SDSub (aSub,i) is Element of k -SD_Sub

proof end;

theorem :: RADIX_3:19

for k being Nat
for i1, i2 being Integer st 2 <= k & i1 in k -SD & i2 in k -SD_Sub holds
SDSub_Add_Data ((i1 + i2),k) in k -SD_Sub_S

proof end;

definition

let i, k, n be Nat;
let x be Tuple of n,k -SD_Sub ;

func DigB_SDSub (x,i) -> Element of INT equals :: RADIX_3:def 9
DigA_SDSub (x,i);

coherence
DigA_SDSub (x,i) is Element of INT by INT_1:def 2;

end;

:: deftheorem defines DigB_SDSub RADIX_3:def 9 :
for i, k, n being Nat
for x being Tuple of n,k -SD_Sub holds DigB_SDSub (x,i) = DigA_SDSub (x,i);

definition

let i, k, n be Nat;
let x be Tuple of n,k -SD_Sub ;

func SDSub2INTDigit (x,i,k) -> Element of INT equals :: RADIX_3:def 10
((Radix k) |^ (i -' 1)) * (DigB_SDSub (x,i));

coherence
((Radix k) |^ (i -' 1)) * (DigB_SDSub (x,i)) is Element of INT by INT_1:def 2;

end;

:: deftheorem defines SDSub2INTDigit RADIX_3:def 10 :
for i, k, n being Nat
for x being Tuple of n,k -SD_Sub holds SDSub2INTDigit (x,i,k) = ((Radix k) |^ (i -' 1)) * (DigB_SDSub (x,i));

definition

let n, k be Nat;
let x be Tuple of n,k -SD_Sub ;

func SDSub2INT x -> Tuple of n, INT means :Def11: :: RADIX_3:def 11
for i being Nat st i in Seg n holds
it /. i = SDSub2INTDigit (x,i,k);

existence
ex b₁ being Tuple of n, INT st
for i being Nat st i in Seg n holds
b₁ /. i = SDSub2INTDigit (x,i,k)

proof end;

uniqueness
for b₁, b₂ being Tuple of n, INT st ( for i being Nat st i in Seg n holds
b₁ /. i = SDSub2INTDigit (x,i,k) ) & ( for i being Nat st i in Seg n holds
b₂ /. i = SDSub2INTDigit (x,i,k) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines SDSub2INT RADIX_3:def 11 :
for n, k being Nat
for x being Tuple of n,k -SD_Sub
for b₄ being Tuple of n, INT holds
( b₄ = SDSub2INT x iff for i being Nat st i in Seg n holds
b₄ /. i = SDSub2INTDigit (x,i,k) );

definition

let n, k be Nat;
let x be Tuple of n,k -SD_Sub ;

func SDSub2IntOut x -> Integer equals :: RADIX_3:def 12
Sum (SDSub2INT x);

coherence
Sum (SDSub2INT x) is Integer ;

end;

:: deftheorem defines SDSub2IntOut RADIX_3:def 12 :
for n, k being Nat
for x being Tuple of n,k -SD_Sub holds SDSub2IntOut x = Sum (SDSub2INT x);

theorem Th20: :: RADIX_3:20

for n, m, k, i being Nat st i in Seg n & 2 <= k holds
DigA_SDSub ((SD2SDSub (DecSD (m,(n + 1),k))),i) = DigA_SDSub ((SD2SDSub (DecSD ((m mod ((Radix k) |^ n)),n,k))),i)

proof end;

theorem :: RADIX_3:21

for n being Nat st n >= 1 holds
for k, x being Nat st k >= 2 & x is_represented_by n,k holds
x = SDSub2IntOut (SD2SDSub (DecSD (x,n,k)))

proof end;