:: High Speed Adder Algorithm with Radix-$2^k$ SD_Sub Number
:: by Masaaki Niimura and Yasushi Fuwa
::
:: Received January 3, 2003
:: Copyright (c) 2003 Association of Mizar Users

begin

theorem Th1: :: RADIX_4:1

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

proof end;

Lm1: for k being Nat
for i1 being Integer st i1 in k -SD_Sub_S holds
( i1 >= - (Radix (k -' 1)) & i1 <= (Radix (k -' 1)) - 1 )

proof end;

Lm2: for n, m, k, i being Nat st i in Seg n holds
DigA (DecSD m,(n + 1),k),i = DigA (DecSD m,n,k),i

proof end;

Lm3: for k, x, n being Nat st n >= 1 holds
DigA (DecSD (x mod ((Radix k) |^ n)),n,k),n = DigA (DecSD x,n,k),n

proof end;

begin

theorem Th2: :: RADIX_4:2

for x, y being Integer
for k being Nat st 3 <= k holds
SDSub_Add_Carry ((SDSub_Add_Carry x,k) + (SDSub_Add_Carry y,k)),k = 0

proof end;

theorem Th3: :: RADIX_4:3

for k, m, n being Nat st 2 <= k holds
DigA_SDSub (SD2SDSub (DecSD m,n,k)),(n + 1) = SDSub_Add_Carry (DigA (DecSD m,n,k),n),k

proof end;

theorem Th4: :: RADIX_4:4

for k, m being Nat st 2 <= k & m is_represented_by 1,k holds
DigA_SDSub (SD2SDSub (DecSD m,1,k)),(1 + 1) = SDSub_Add_Carry m,k

proof end;

theorem Th5: :: RADIX_4:5

for k, x, n being Nat st n >= 1 & k >= 3 & x is_represented_by n + 1,k holds
DigA_SDSub (SD2SDSub (DecSD (x mod ((Radix k) |^ n)),n,k)),(n + 1) = SDSub_Add_Carry (DigA (DecSD x,n,k),n),k

proof end;

theorem Th6: :: RADIX_4:6

for k, m being Nat st 2 <= k & m is_represented_by 1,k holds
DigA_SDSub (SD2SDSub (DecSD m,1,k)),1 = m - ((SDSub_Add_Carry m,k) * (Radix k))

proof end;

theorem Th7: :: RADIX_4:7

for k, x, n being Nat st k >= 2 holds
((Radix k) |^ n) * (DigA_SDSub (SD2SDSub (DecSD x,(n + 1),k)),(n + 1)) = ((((Radix k) |^ n) * (DigA (DecSD x,(n + 1),k),(n + 1))) - (((Radix k) |^ (n + 1)) * (SDSub_Add_Carry (DigA (DecSD x,(n + 1),k),(n + 1)),k))) + (((Radix k) |^ n) * (SDSub_Add_Carry (DigA (DecSD x,(n + 1),k),n),k))

proof end;

begin

definition

let i, n, k be Nat;
let x, y be Tuple of n,k -SD_Sub ;
assume that
A1: i in Seg n and
A2: k >= 2 ;
func SDSubAddDigit x,y,i,k -> Element of k -SD_Sub equals :Def1: :: RADIX_4:def 1
(SDSub_Add_Data ((DigA_SDSub x,i) + (DigA_SDSub y,i)),k) + (SDSub_Add_Carry ((DigA_SDSub x,(i -' 1)) + (DigA_SDSub y,(i -' 1))),k);
coherence
(SDSub_Add_Data ((DigA_SDSub x,i) + (DigA_SDSub y,i)),k) + (SDSub_Add_Carry ((DigA_SDSub x,(i -' 1)) + (DigA_SDSub y,(i -' 1))),k) is Element of k -SD_Sub

proof end;

end;

:: deftheorem Def1 defines SDSubAddDigit RADIX_4:def 1 :

for i, n, k being Nat
for x, y being Tuple of n,k -SD_Sub st i in Seg n & k >= 2 holds
SDSubAddDigit x,y,i,k = (SDSub_Add_Data ((DigA_SDSub x,i) + (DigA_SDSub y,i)),k) + (SDSub_Add_Carry ((DigA_SDSub x,(i -' 1)) + (DigA_SDSub y,(i -' 1))),k);

definition

let n, k be Nat;
let x, y be Tuple of n,k -SD_Sub ;
func x '+' y -> Tuple of n,k -SD_Sub means :Def2: :: RADIX_4:def 2
for i being Nat st i in Seg n holds
DigA_SDSub it,i = SDSubAddDigit x,y,i,k;
existence
ex b₁ being Tuple of n,k -SD_Sub st
for i being Nat st i in Seg n holds
DigA_SDSub b₁,i = SDSubAddDigit x,y,i,k

proof end;

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

proof end;

end;

:: deftheorem Def2 defines '+' RADIX_4:def 2 :

for n, k being Nat
for x, y, b₅ being Tuple of n,k -SD_Sub holds
( b₅ = x '+' y iff for i being Nat st i in Seg n holds
DigA_SDSub b₅,i = SDSubAddDigit x,y,i,k );

theorem Th8: :: RADIX_4:8

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

proof end;

theorem :: RADIX_4:9

for n being Nat st n >= 1 holds
for k, x, y being Nat st k >= 3 & x is_represented_by n,k & y is_represented_by n,k holds
x + y = SDSub2IntOut ((SD2SDSub (DecSD x,n,k)) '+' (SD2SDSub (DecSD y,n,k)))

proof end;