### Definitions of Radix-\$2^k\$ Signed-Digit Number and its Adder Algorithm

by
Yoshinori Fujisawa, and
Yasushi Fuwa

Received September 7, 1999

Copyright (c) 1999 Association of Mizar Users

[ MML identifier index ]

```environ

vocabulary INT_1, NAT_1, ARYTM_1, ARYTM_3, POWER, MIDSP_3, FINSEQ_1, FUNCT_1,
RELAT_1, RLVECT_1, RADIX_1, FINSEQ_4, GROUP_1;
notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, INT_1, NAT_1, FUNCT_1,
POWER, FINSEQ_1, FINSEQ_4, BINARITH, TREES_4, WSIERP_1, MIDSP_3;
constructors REAL_1, POWER, BINARITH, FINSEQ_4, EULER_2, WSIERP_1, MEMBERED,
INT_1;
clusters INT_1, RELSET_1, GROUP_2, NAT_1, XREAL_0, MEMBERED;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions TARSKI;
theorems REAL_2, REAL_1, NAT_1, NAT_2, INT_1, FINSEQ_1, AXIOMS, GOBOARD9,
GROUP_4, RLVECT_1, GR_CY_1, EULER_1, BINARITH, PREPOWER, POWER, JORDAN4,
RVSUM_1, FINSEQ_2, FINSEQ_4, TARSKI, FUNCT_1, GR_CY_2, PEPIN, FINSEQ_5,
SCMFSA_7, JORDAN3, NEWTON, SCMFSA9A, SQUARE_1, XCMPLX_0, XCMPLX_1;
schemes NAT_1, FINSEQ_2, INT_2;

begin :: Some Useful Theorems

reserve i,k,m,n,p,x,y for Nat,
i1,i2,i3 for Integer,
e for set;

canceled;

theorem Th2:
n mod k = k - 1 implies (n+1) mod k = 0
proof
per cases;
suppose A1: k <> 0;
assume A2: n mod k = k-1;
k >= 1 by A1,RLVECT_1:99;
then k - 1 >= 0 by SQUARE_1:12;
then reconsider K = k - 1 as Nat by INT_1:16;
K + 1 = k - (1 - 1) by XCMPLX_1:37
.= k - 0;
then (n+1) mod k = k mod k by A2,GR_CY_1:2;
hence thesis by GR_CY_1:5;
suppose k=0;
hence thesis by NAT_1:def 2;
end;

theorem Th3:
k <> 0 & n mod k < k - 1 implies (n+1) mod k = (n mod k) + 1
proof
assume k <> 0 & n mod k < k - 1;
then A1:(n mod k) + 1 < k by REAL_1:86;
(n+1) mod k = ((n mod k)+1) mod k by GR_CY_1:2;
hence thesis by A1,GR_CY_1:4;
end;

theorem Th4:
m <> 0 implies (k mod (m*n)) mod n = k mod n
proof
assume A1:m <> 0;
per cases;
suppose A2: n <> 0;
reconsider k' = k, m' = m, n' = n as Integer;
A3:m'*n' <> 0 by A1,A2,XCMPLX_1:6;
k mod (m*n) = k' mod (m'*n') by SCMFSA9A:5;
hence (k mod (m*n)) mod n = (k' mod (m'*n')) mod n' by SCMFSA9A:5
.= (k' - (k' div (m'*n'))*(m'*n')) mod n' by A3,INT_1:def 8
.= (k' - ((k' div m'*n')*m')*n') mod n' by XCMPLX_1:4
.= (k' + (-(m'*(k' div m'*n'))*n')) mod n' by XCMPLX_0:def 8
.= (k' + (-(m'*(k' div m'*n')))*n') mod n' by XCMPLX_1:175
.= k' mod n' by EULER_1:13
.= k mod n by SCMFSA9A:5;
suppose A4: n = 0;
hence (k mod (m*n)) mod n = 0 by NAT_1:def 2
.= k mod n by A4,NAT_1:def 2;
end;

theorem
k <> 0 implies (n+1) mod k = 0 or (n+1) mod k = (n mod k) + 1
proof
assume A1:k <> 0;
then k >= 1 by RLVECT_1:99;
then k - 1 >= 0 by SQUARE_1:12;
then reconsider K = k - 1 as Nat by INT_1:16;
k > 0 by A1,NAT_1:19;
then n mod k < k by NAT_1:46;
then n mod k < k + 1 - 1 by XCMPLX_1:26;
then n mod k < k - 1 + 1 by XCMPLX_1:29;
then A2:n mod k <= K by NAT_1:38;
per cases by A2,AXIOMS:21;
suppose n mod k < k - 1;
hence thesis by A1,Th3;
suppose n mod k = k - 1;
hence thesis by Th2;
end;

theorem Th6:
i <> 0 & k <> 0 implies (n mod (i |^ k)) div (i |^ (k -'1)) < i
proof
assume A1:i <> 0 & k <> 0;
then A2:i > 0 by NAT_1:19;
then i |^ k > 0 by PREPOWER:13;
then A3:n mod (i |^ k) < (i |^ k) by NAT_1:46;
set I1 = i |^ (k -'1);
set T = n mod (i |^ k);
A4:I1 > 0 by A2,PREPOWER:13;
A5:I1 <> 0 by A2,PREPOWER:13;
i |^ k = i*(i |^ (k -'1)) by A1,PEPIN:27;
then T mod I1 = n mod I1 by A1,Th4
.= n - I1*(n div I1) by A4,GR_CY_2:1;
then T = I1*(T div I1) + (n - I1*(n div I1)) by A4,NAT_1:47;
then I1*(T div I1) + (n - I1*(n div I1)) < i*I1 by A1,A3,PEPIN:27;
then I1*(T div I1) + n - I1*(n div I1) < i*I1 by XCMPLX_1:29;
then (I1*(T div I1) + n - I1*(n div I1))/I1 < i*I1/I1 by A4,REAL_1:73;
then I1*(T div I1)/I1 + n/I1 - I1*(n div I1)/I1 < i*I1/I1
by XCMPLX_1:125;
then (T div I1) + n/I1 - I1*(n div I1)/I1 < i*I1/I1 by A5,XCMPLX_1:90
;
then (T div I1) + n/I1 - (n div I1) < i*I1/I1 by A5,XCMPLX_1:90;
then A6:(T div I1) + n/I1 - (n div I1) < i by A5,XCMPLX_1:90;
reconsider n' = n , I'1 = I1 as Integer;
A7:n div I1 = n' div I'1 by SCMFSA9A:5
.= [\n/I1/] by INT_1:def 7;
[\n/I1/] <= n/I1 by INT_1:def 4;
then (T div I1) + [\n/I1/] <= (T div I1) + n/I1 by AXIOMS:24;
then (T div I1) + [\n/I1/] - [\n/I1/] <= (T div I1) + n/I1 - (n div I1)
by A7,REAL_1:49;
then (T div I1) + ([\n/I1/] - [\n/I1/]) <= (T div I1) + n/I1 - (n div I1)
by XCMPLX_1:29;
then (T div I1) + 0 <= (T div I1) + n/I1 - (n div I1) by XCMPLX_1:14;
hence thesis by A6,AXIOMS:22;
end;

theorem Th7:
k <= n implies m |^ k divides m |^ n
proof
assume k <= n;
then consider t being Nat such that
A1:n = k + t by NAT_1:28;
m |^ n = (m |^ k)*(m |^ t) by A1,NEWTON:13;
hence thesis by NAT_1:def 3;
end;

theorem
i2 > 0 & i3 >= 0 implies (i1 mod (i2*i3)) mod i3 = i1 mod i3
proof
assume A1:i2 > 0 & i3 >= 0;
per cases by A1;
suppose i3 > 0;
then i2*i3 <> 0 by A1,XCMPLX_1:6;
then (i1 mod (i2*i3)) mod i3
= (i1 - (i1 div (i2*i3))*(i2*i3)) mod i3 by INT_1:def 8
.= (i1 - ((i1 div i2*i3)*i2)*i3) mod i3 by XCMPLX_1:4
.= (i1 + (-(i2*(i1 div i2*i3))*i3)) mod i3 by XCMPLX_0:def 8
.= (i1 + (-(i2*(i1 div i2*i3)))*i3) mod i3 by XCMPLX_1:175;
hence thesis by EULER_1:13;
suppose A2:i3 = 0;
then (i1 mod (i2*i3)) mod i3 = 0 by INT_1:def 8;
hence thesis by A2,INT_1:def 8;
end;

begin :: Definition for Radix-2^k, k-SD

definition let n;
func Radix(n) -> Nat equals
:Def1:
2 to_power n;
correctness
proof
defpred P[Nat] means 2 to_power \$1 is Nat;
A1: P[0] by POWER:29;
A2:for m be Nat st P[m]  holds P[m+1]
proof
let m be Nat;
assume 2 to_power m is Nat;
then reconsider k = 2 to_power m as Nat;
2 to_power (m+1) = 2 to_power m * 2 to_power 1 by POWER:32
.= k * 2 by POWER:30;
hence thesis;
end;
for m be Nat holds P[m] from Ind(A1,A2);
hence 2 to_power n is Nat;
end;
end;

definition let k;
func k-SD -> set equals
:Def2:
{e where e is Element of INT:e <= Radix(k)-1 & e >= -Radix(k)+1};
correctness;
end;

theorem Th9:
proof
Radix(n) = 2 to_power n by Def1;
hence thesis by POWER:39;
end;

Lm1:
for k be Nat st k >= 2 holds Radix(k) >= 2 + 2
proof
defpred P[Nat] means Radix(\$1) >= 2+2;
Radix(2) = 2 to_power (1+1) by Def1
.= 2 to_power 1 * 2 to_power 1 by POWER:32
.= 2 * 2 to_power 1 by POWER:30
.= 2 * 2 by POWER:30;
then A1:P[2];
A2:for kk be Nat st kk >= 2 & P[kk]
holds P[kk+1]
proof
let kk be Nat;
assume kk >= 2;
assume A3:Radix(kk) >= 2 + 2;
Radix(kk) <> 0 by Th9;
then A4:Radix(kk) > 0 by NAT_1:19;
Radix(kk+1) = 2 to_power (kk+1) by Def1
.= 2 to_power kk * 2 to_power 1 by POWER:32
.= 2 to_power kk * 2 by POWER:30
.= Radix(kk) * 2 by Def1;
hence thesis by A3,AXIOMS:22;
end;
for k be Nat st k >= 2 holds P[k] from Ind1(A1,A2);
hence thesis;
end;

theorem Th10:
for e holds e in 0-SD iff e = 0
proof
let e be set;
Radix(0) = 2 to_power(0) by Def1
.= 1 by POWER:29;
then A1:0-SD = {w where w is Element of INT:w <= 1-1 & w >= -1+1} by Def2
.= {w where w is Element of INT:w <= 0 & w >= 0};
hereby assume e in 0-SD;
then consider b be Element of INT such that
A2:e = b and
A3:b <= 0 & b >= 0 by A1;
thus e = 0 by A2,A3;
end;
assume
A4:  e = 0;
then e is Element of INT by INT_1:def 2;
hence thesis by A1,A4;
end;

theorem
0-SD = {0} by Th10,TARSKI:def 1;

theorem Th12:
k-SD c= (k+1)-SD
proof
let e be set;
assume A1:e in k-SD;
A2:(k+1)-SD
= {w where w is Element of INT:w <= Radix(k+1)-1 & w >= -Radix(k+1)+1}
by Def2;
k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1}
by Def2;
then consider g being Element of INT such that
A3:e = g and
A4:g <= Radix(k)-1 and
A5:g >= -Radix(k)+1 by A1;
A6:2 to_power k = Radix(k) by Def1;
k < k+1 by NAT_1:38;
then 2 to_power k < 2 to_power (k+1) by POWER:44;
then A8:g <= Radix(k+1)-1 by A4,AXIOMS:22;
then g >= -Radix(k+1)+1 by A5,AXIOMS:22;
hence thesis by A2,A3,A8;
end;

theorem Th13:
e in k-SD implies e is Integer
proof
assume A1:e in k-SD;
k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1}
by Def2;
then consider t be Element of INT such that
A2:e = t and
t <= Radix(k)-1 & t >= -Radix(k)+1 by A1;
thus thesis by A2;
end;

theorem Th14:
k-SD c= INT
proof
let e be set;
assume e in k-SD;
then e is Integer by Th13;
hence thesis by INT_1:12;
end;

theorem Th15:
i1 in k-SD implies i1 <= Radix(k)-1 & i1 >= -Radix(k)+1
proof
assume A1:i1 in k-SD;
k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1}
by Def2;
then consider i be Element of INT such that
A2:i = i1 and
A3:i <= Radix(k)-1 & i >= -Radix(k)+1 by A1;
thus thesis by A2,A3;
end;

theorem Th16:
0 in k-SD
proof
defpred P[Nat] means 0 in \$1-SD;
A1: P[0] by Th10;
A2:for k being Nat st P[k] holds P[k+1]
proof
let kk be Nat;
assume A3:0 in kk-SD;
kk-SD c= (kk+1)-SD by Th12;
hence thesis by A3;
end;
for k being Nat holds P[k] from Ind(A1,A2);
hence thesis;
end;

definition let k;
cluster k-SD -> non empty;
coherence by Th16;
end;

definition let k;
redefine func k-SD -> non empty Subset of INT;
coherence by Th14;
end;

begin :: Functions for generating Radix-2^k SD numbers from natural numbers
:: and natural numbers from Radix-2^k SD numbers

reserve a for Tuple of n,(k-SD);

canceled;

theorem Th18:
i in Seg n implies a.i is Element of k-SD
proof
assume A1:i in Seg n;
len a = n by FINSEQ_2:109;
then i in dom a by A1,FINSEQ_1:def 3;
then a.i in rng a & rng a c= k-SD by FINSEQ_1:def 4,FUNCT_1:def 5;
hence thesis;
end;

definition let i,k,n be Nat, x be Tuple of n,(k-SD);
func DigA(x,i) -> Integer equals
:Def3:
x.i if i in Seg n,
0 if i = 0;
coherence
proof
i in Seg n implies x.i is Integer
proof
assume i in Seg n;
then x.i is Element of k-SD by Th18;
hence thesis by INT_1:def 2;
end;
hence thesis;
end;
consistency by FINSEQ_1:3;
end;

definition let i,k,n be Nat, x be Tuple of n,(k-SD);
func DigB(x,i) -> Element of INT equals
:Def4:
DigA(x,i);
coherence by INT_1:def 2;
end;

theorem Th19:
i in Seg n implies DigA(a,i) is Element of k-SD
proof
assume A1:i in Seg n;
then a.i = DigA(a,i) by Def3;
hence thesis by A1,Th18;
end;

theorem Th20:
for x be Tuple of 1,INT st x/.1 = m holds x = <*m*>
proof
let x be Tuple of 1,INT;
assume A1:x/.1 = m;
A2:len x = 1 by FINSEQ_2:109;
then x/.1 = x.1 by FINSEQ_4:24;
hence thesis by A1,A2,FINSEQ_1:57;
end;

definition let i,k,n be Nat, x be Tuple of n,(k-SD);
func SubDigit(x,i,k) -> Element of INT equals
:Def5:
(Radix(k) |^ (i -'1))*DigB(x,i);
coherence by INT_1:12;
end;

definition let n,k be Nat, x be Tuple of n,(k-SD);
func DigitSD(x) -> Tuple of n,INT means
:Def6:
for i be Nat st i in Seg n holds it/.i = SubDigit(x,i,k);
existence
proof
deffunc F(Nat)=SubDigit(x,\$1,k);
consider z being FinSequence of INT such that
A1:len z = n and
A2:for j be Nat st j in Seg n holds z.j = F(j)
reconsider z as Tuple of n,INT by A1,FINSEQ_2:110;
take z;
let i;
assume A3:i in Seg n;
then i in dom z by A1,FINSEQ_1:def 3;
hence z/.i = z.i by FINSEQ_4:def 4
.= SubDigit(x,i,k) by A2,A3;
end;
uniqueness
proof
let k1,k2 be Tuple of n,INT such that
A4:for i be Nat st i in Seg n holds k1/.i = SubDigit(x,i,k) and
A5:for i be Nat st i in Seg n holds k2/.i = SubDigit(x,i,k);
A6:len k1 = n & len k2 = n by FINSEQ_2:109;
now let j be Nat;
assume A7:j in Seg n;
then A8:j in dom k1 & j in dom k2 by A6,FINSEQ_1:def 3;
then k1.j = k1/.j by FINSEQ_4:def 4
.= SubDigit(x,j,k) by A4,A7
.= k2/.j by A5,A7
.= k2.j by A8,FINSEQ_4:def 4;
hence k1.j = k2.j;
end;
hence k1 = k2 by A6,FINSEQ_2:10;
end;
end;

definition let n,k be Nat, x be Tuple of n,(k-SD);
func SDDec(x) -> Integer equals
:Def7:
Sum DigitSD(x);
coherence;
end;

definition let i,k,x be Nat;
func DigitDC(x,i,k) -> Element of k-SD equals
:Def8:
(x mod (Radix(k) |^ i)) div (Radix(k) |^ (i -'1));
coherence
proof
set T = Radix(k) |^ i;
set S = Radix(k) |^ (i -'1);
A1:Radix(k) <> 0 by Th9;
then A2:Radix(k) >= 1 by RLVECT_1:99;
then A3:T >= 1 by PREPOWER:19;
T <> 0 by A2,PREPOWER:19;
then A4:T > 0 by NAT_1:19;
S <> 0 by A2,PREPOWER:19;
then A5:S > 0 by NAT_1:19;
then A6:0 div S = 0 by JORDAN4:5;
defpred P[Nat] means (\$1 mod T) div S is Element of k-SD;
0 in k-SD by Th16;
then A7: P[0] by A4,A6,GR_CY_1:4;
A8:for x being Nat st P[x]
holds P[(x+1)]
proof
let xx be Nat;
assume (xx mod T) div S is Element of k-SD;
now per cases by A3,AXIOMS:21;
suppose A9:T = 1;
then (xx+1) mod T = (xx+1) - T*((xx+1) div 1) by GR_CY_2:1
.= (xx+1) - (xx+1) by A9,NAT_2:6
.= 0 by XCMPLX_1:14;
then ((xx+1) mod T) div S = 0 by A5,JORDAN4:5;
hence thesis by Th16;
suppose T > 1;
now
set X = ((xx+1) mod T) div S;
A10:k-SD =
{e where e is Element of INT:e <= Radix(k)-1 & e >=
Radix(k)-1 >= 0 by A2,SQUARE_1:12;
then reconsider R = Radix(k)-1 as Nat by INT_1:16;
now per cases by NAT_1:18;
suppose A11:i = 0;
Radix(k) |^ i = Radix(k) to_power i by A1,POWER:46
.= 1 by A11,POWER:29;
then ((xx+1) mod T) = (((xx+1)*1) mod 1)
.= 0 by GROUP_4:101;
then X = 0 by A5,JORDAN4:5;
hence thesis by Th16;
suppose i > 0;
then X < Radix(k) by A1,Th6;
then X < Radix(k) + 1 - 1 by XCMPLX_1:26;
then X < Radix(k) - 1 + 1 by XCMPLX_1:29;
then A12:X <= R by NAT_1:38;
A13:  X is Element of INT by INT_1:def 2;
A14:X >= 0 by NAT_1:18;
-1 >= -Radix(k) by A2,REAL_1:50;
then 0 >= -Radix(k)+1 by REAL_2:109;
then X in k-SD by A10,A12,A13,A14;
hence thesis;
end;
hence thesis;
end;
hence thesis;
end;
hence thesis;
end;
for x being Nat holds P[x] from Ind(A7,A8);
hence thesis;
end;
end;

definition let k,n,x be Nat;
func DecSD(x,n,k) -> Tuple of n,(k-SD) means
:Def9:
for i be Nat st i in Seg n holds DigA(it,i) = DigitDC(x,i,k);
existence
proof
deffunc F(Nat)=DigitDC(x,\$1,k);
consider z being FinSequence of k-SD such that
A1: len z = n and
A2: for j be Nat st j in Seg n holds z.j = F(j)
reconsider z as Tuple of n,(k-SD) by A1,FINSEQ_2:110;
take z;
let i;
assume A3:i in Seg n;
hence DigA(z,i) = z.i by Def3
.= DigitDC(x,i,k) by A2,A3;
end;
uniqueness
proof
let k1,k2 be Tuple of n,(k-SD) such that
A4:for i be Nat st i in Seg n holds DigA(k1,i) = DigitDC(x,i,k) and
A5:for i be Nat st i in Seg n holds DigA(k2,i) = DigitDC(x,i,k);
A6:len k1 = n & len k2 = n by FINSEQ_2:109;
now let j be Nat;
assume A7:j in Seg n;
then k1.j = DigA(k1,j) by Def3
.= DigitDC(x,j,k) by A4,A7
.= DigA(k2,j) by A5,A7
.= k2.j by A7,Def3;
hence k1.j = k2.j;
end;
hence k1 = k2 by A6,FINSEQ_2:10;
end;
end;

begin :: Definition for carry & data components of Addition

definition let x be Integer;
func SD_Add_Carry(x) -> Integer equals
:Def10:
1 if x > 2,
-1 if x < -2
otherwise 0;
coherence;
consistency by AXIOMS:22;
end;

theorem Th21:
SD_Add_Carry(0) = 0 by Def10;

Lm2:
for x be Integer holds -1 <= SD_Add_Carry(x) & SD_Add_Carry(x) <= 1
proof
let x be Integer;
per cases;
suppose A1:x < -2;
-1 <= -1 & -1 <= 1;
hence thesis by A1,Def10;
suppose -2 <= x & x <= 2;
then SD_Add_Carry(x) = 0 by Def10;
hence thesis;
suppose A2:x > 2;
-1 <= 1 & 1 <= 1;
hence thesis by A2,Def10;
end;

definition let x be Integer, k be Nat;
func SD_Add_Data(x,k) -> Integer equals
:Def11:
coherence;
end;

theorem
proof
SD_Add_Data(0,k) = 0 - 0*Radix(k) by Def11,Th21
.= 0;
hence thesis;
end;

theorem Th23:
k >= 2 & i1 in k-SD & i2 in k-SD implies
proof
assume A1:k >= 2 & i1 in k-SD & i2 in k-SD;
then A2:-Radix(k)+1 <= i1 & i1 <= Radix(k)-1 &
-Radix(k)+1 <= i2 & i2 <= Radix(k)-1 by Th15;
set z = i1+i2;
A3: -Radix(k)+1 + (-Radix(k)+1) <= z &
z <= Radix(k)-1 + (Radix(k)-1) by A2,REAL_1:55;
per cases;
suppose A4:z < -2;
then SD_Add_Carry(z) = -1 by Def10;
then A5:SD_Add_Data(z,k) = z - (-1)*Radix(k) by Def11
.= z - (-Radix(k)) by XCMPLX_1:180
.= z + Radix(k) by XCMPLX_1:151;
-Radix(k)+1 + (1 + -Radix(k)) <= z by A2,REAL_1:55;
then (-Radix(k)+1+1)+ -Radix(k) <= z by XCMPLX_1:1;
then (-Radix(k)+(1+1))+ -Radix(k) <= z by XCMPLX_1:1;
then A6: (-Radix(k)+(1+1))-Radix(k) <= z by XCMPLX_0:def 8;
z + Radix(k) < -2 + Radix(k) by A4,REAL_1:53;
hence thesis by A5,A6,REAL_1:86,XCMPLX_0:def 8;
suppose A7:-2 <= z & z <= 2;
then SD_Add_Carry(z) = 0 by Def10;
then A8:SD_Add_Data(z,k) = z - 0 * Radix(k) by Def11
.= z;
A9: Radix(k) >= 2+2 by A1,Lm1;
then A10:Radix(k)-2 >= 2 by REAL_1:84;
-Radix(k) <= -(2+2) by A9,REAL_1:50;
then -Radix(k) <= -2 + -2;
then -Radix(k) - -2 <= -2 by REAL_1:86;
then -Radix(k) + 2 <= -2 by XCMPLX_1:151;
hence thesis by A7,A8,A10,AXIOMS:22;
suppose A11:z > 2;
then SD_Add_Carry(z) = 1 by Def10;
then A12:SD_Add_Data(z,k) = z - 1*Radix(k) by Def11
.= z - Radix(k);
z <= Radix(k)-1 + Radix(k) - 1 by A3,XCMPLX_1:29;
then z <= Radix(k)-1 + Radix(k) + -1 by XCMPLX_0:def 8;
then z <= Radix(k)-1+ -1 + Radix(k) by XCMPLX_1:1;
then z <= Radix(k)-1 -1 + Radix(k) by XCMPLX_0:def 8;
then A13: z <= Radix(k)-(1 + 1) + Radix(k) by XCMPLX_1:36;
z - Radix(k) > 2 - Radix(k) by A11,REAL_1:54;
hence thesis by A12,A13,REAL_1:86,XCMPLX_0:def 8;
end;

begin :: Definition for checking whether or not a natural number can be
:: expressed as n digits Radix-2^k SD number

definition let n,x,k be Nat;
pred x is_represented_by n,k means
:Def12:
x < (Radix(k) |^ n);
end;

theorem Th24:
m is_represented_by 1,k implies DigA(DecSD(m,1,k),1) = m
proof
assume m is_represented_by 1,k;
then A1:m < Radix(k) |^ 1 by Def12;
1 in Seg 1 by FINSEQ_1:3;
hence DigA(DecSD(m,1,k),1)
= DigitDC(m,1,k) by Def9
.= (m mod (Radix(k) |^ 1)) div (Radix(k) |^ (1 -'1)) by Def8
.= (m mod (Radix(k) |^ 1)) div (Radix(k) |^ 0) by GOBOARD9:1
.= (m mod (Radix(k) |^ 1)) div 1 by NEWTON:9
.= m mod (Radix(k) |^ 1) by NAT_2:6
.= m by A1,GR_CY_1:4;
end;

theorem
for n st n >= 1 holds
for m st m is_represented_by n,k holds m = SDDec(DecSD(m,n,k))
proof
defpred P[Nat] means
for m st m is_represented_by \$1,k holds m = SDDec(DecSD(m,\$1,k));
A1:P[1]
proof
let m;
assume A2:m is_represented_by 1,k;
A3:1 in Seg 1 by FINSEQ_1:3;
SubDigit(DecSD(m,1,k),1,k) = (Radix(k) |^ (1 -'1))*DigB(DecSD(m,1,k),1)
by Def5
.= (Radix(k) |^ 0)*DigB(DecSD(m,1,k),1)
by GOBOARD9:1
.= 1*DigB(DecSD(m,1,k),1) by NEWTON:9
.= DigA(DecSD(m,1,k),1) by Def4
.= m by A2,Th24;
then A4:(DigitSD(DecSD(m,1,k)))/.1 = m by A3,Def6;
m is Element of INT by INT_1:def 2;
then reconsider M = <*m*> as FinSequence of INT by SCMFSA_7:22;
thus SDDec(DecSD(m,1,k)) = Sum (DigitSD(DecSD(m,1,k))) by Def7
.= Sum M by A4,Th20
.= m by RVSUM_1:103;
end;
A5:for u be Nat st u >= 1 & P[u] holds P[u+1]
proof
let u be Nat;
assume A6:u >= 1 & P[u];
p is_represented_by (u+1),k implies p = SDDec(DecSD(p,(u+1),k))
proof
assume p is_represented_by (u+1),k;
then A7:p < Radix(k) |^ (u+1) by Def12;
set R = Radix(k) |^ u;
Radix(k) <> 0 by Th9;
then R <> 0 by PREPOWER:12;
then A8:R > 0 by NAT_1:19;
then A9:p = R*(p div R) + (p mod R) by NAT_1:47;
set M = p mod R;
set N = R*(p div R);
M < R by A8,NAT_1:46;
then M is_represented_by u,k by Def12;
then A10:p = SDDec(DecSD(M,u,k)) + N by A6,A9;
N is Element of INT by INT_1:def 2;
then reconsider NN = <*N*> as FinSequence of INT by SCMFSA_7:22;
reconsider DD = DigitSD(DecSD(M,u,k)) as FinSequence of INT;
A11:p = Sum DigitSD(DecSD(M,u,k)) + N by A10,Def7
.= Sum (DD^NN) by RVSUM_1:104;
set I = u + 1;
A12:I -' 1 = u by BINARITH:39;
then A13:p div R
= (p mod (Radix(k) |^ I)) div (Radix(k) |^ (I -'1))
by A7,GR_CY_1:4
.= DigitDC(p,(u+1),k) by Def8;
A14:1 <= u+1 & u+1 <= u+1 by NAT_1:37;
then A15:u+1 in Seg (u+1) by FINSEQ_1:3;
A16:u+1 <= len (DigitSD(DecSD(p,(u+1),k))) by FINSEQ_2:109;
A17:N = (Radix(k) |^ (I -'1))*DigA(DecSD(p,I,k),I) by A12,A13,A15,Def9
.= (Radix(k) |^ (I -'1))*DigB(DecSD(p,I,k),I) by Def4
.= SubDigit(DecSD(p,(u+1),k),(u+1),k) by Def5
.= (DigitSD(DecSD(p,(u+1),k)))/.(u+1) by A15,Def6
.= DigitSD(DecSD(p,(u+1),k)).(u+1) by A14,A16,FINSEQ_4:24;

DigitSD(DecSD(M,u,k))^<*N*> = DigitSD(DecSD(p,(u+1),k))
proof
A18:N is Element of INT by INT_1:def 2;
set z0 = DigitSD(DecSD(M,u,k));
reconsider z1 = <*N*> as FinSequence of INT by A18,SCMFSA_7:22;
reconsider DD = DigitSD(DecSD(p,(u+1),k)) as FinSequence of INT;
reconsider z = z0^z1 as FinSequence of INT;
A19:len z = len (DigitSD(DecSD(M,u,k))) + len <*N*> by FINSEQ_1:35
.= u + len <*N*> by FINSEQ_2:109
.= u + 1 by FINSEQ_1:56;
A20:len DD = u+1 by FINSEQ_2:109;
for i be Nat st 1 <= i & i <= len z holds z/.i = DD/.i
proof
let i be Nat;
assume 1 <= i & i <= len z;
then A21:i in Seg (u+1) by A19,FINSEQ_1:3;
per cases by A21,FINSEQ_2:8;
suppose A22:i in Seg u;
A23:M mod (Radix(k) |^ i) = p mod (Radix(k) |^ i)
proof
i <= u by A22,FINSEQ_1:3;
then Radix(k) |^ i divides Radix(k) |^ u by Th7;
then consider t being Nat such that
A24:Radix(k) |^ u = (Radix(k) |^ i)*t by NAT_1:def 3;
Radix(k) <> 0 by Th9;
then t <> 0 by A24,PREPOWER:12;
hence thesis by A24,Th4;
end;
len z0 = u by FINSEQ_2:109;
then A25:i in dom z0 by A22,FINSEQ_1:def 3;
then A26:(z0^z1).i
= DigitSD(DecSD(M,u,k)).i by FINSEQ_1:def 7
.= (DigitSD(DecSD(M,u,k)))/.i by A25,FINSEQ_4:def 4
.= SubDigit(DecSD(M,u,k),i,k) by A22,Def6
.= (Radix(k) |^ (i -'1))*DigB(DecSD(M,u,k),i) by Def5
.= (Radix(k) |^ (i -'1))*DigA(DecSD(M,u,k),i) by Def4
.= (Radix(k) |^ (i -'1))*DigitDC(M,i,k) by A22,Def9
.= (Radix(k) |^ (i -'1))*((p mod (Radix(k) |^ i))
div (Radix(k) |^ (i -'1))) by A23,Def8;
len DD = u+1 by FINSEQ_2:109;
then A27:i in dom DD by A21,FINSEQ_1:def 3;
then A28:DD.i = (DigitSD(DecSD(p,(u+1),k)))/.i by FINSEQ_4:def 4
.= SubDigit(DecSD(p,(u+1),k),i,k) by A21,Def6
.= (Radix(k) |^ (i -'1))*DigB(DecSD(p,(u+1),k),i) by Def5
.= (Radix(k) |^ (i -'1))*DigA(DecSD(p,(u+1),k),i) by Def4
.= (Radix(k) |^ (i -'1))*DigitDC(p,i,k) by A21,Def9
.= (Radix(k) |^ (i -'1))*((p mod (Radix(k) |^ i))
div (Radix(k) |^ (i -'1))) by Def8;
A29:DD.i = DD/.i by A27,FINSEQ_4:def 4;
i in dom (z0^z1) by A19,A21,FINSEQ_1:def 3;
hence thesis by A26,A28,A29,FINSEQ_4:def 4;
suppose A30:i = u+1;
hence z/.i = z.(u+1) by A14,A19,FINSEQ_4:24
.= (z0^z1).(len z0 + 1) by FINSEQ_2:109
.= DigitSD(DecSD(p,(u+1),k)).(u+1) by A17,FINSEQ_1:59
.= DD/.i by A14,A16,A30,FINSEQ_4:24;
end;
hence thesis by A19,A20,FINSEQ_5:14;
end;
hence thesis by A11,Def7;
end;
hence thesis;
end;
for u be Nat st u >= 1 holds P[u] from Ind1(A1,A5);
hence thesis;
end;

theorem Th26:
k >= 2 & m is_represented_by 1,k & n is_represented_by 1,k implies
proof
assume A1:k >= 2 & m is_represented_by 1,k & n is_represented_by 1,k;
= SD_Add_Carry(m + DigA(DecSD(n,1,k),1)) by Th24
.= SD_Add_Carry(m + n) by A1,Th24;
hence thesis;
end;

Lm3:
for i st i in Seg n holds
DigA(DecSD(m,n+1,k),i)=DigA(DecSD((m mod (Radix(k) |^ n)),n,k),i)
proof
let i;
assume A1:i in Seg n;
then A2:1 <= i & i <= n by FINSEQ_1:3;
then i <= n+1 by NAT_1:37;
then A3:i in Seg (n+1) by A2,FINSEQ_1:3;
Radix(k) |^ i divides Radix(k) |^ n by A2,Th7;
then consider t be Nat such that
A4:Radix(k) |^ n = (Radix(k) |^ i)*t by NAT_1:def 3;
Radix(k) <> 0 by Th9;
then t <> 0 by A4,PREPOWER:12;
then A5:(m mod (Radix(k) |^ n)) mod (Radix(k) |^ i)
= m mod (Radix(k) |^ i) by A4,Th4;
DigA(DecSD((m mod (Radix(k) |^ n)),n,k),i)
= DigitDC((m mod (Radix(k) |^ n)),i,k) by A1,Def9
.= (m mod (Radix(k) |^ i)) div (Radix(k) |^ (i -'1)) by A5,Def8
.= DigitDC(m,i,k) by Def8
.= DigA(DecSD(m,n+1,k),i) by A3,Def9;
hence thesis;
end;

theorem Th27:
m is_represented_by (n+1),k implies
DigA(DecSD(m,(n+1),k),n+1) = m div (Radix(k) |^ n)
proof
assume A1:m is_represented_by (n+1),k;
A2: n+1 in Seg (n+1) by FINSEQ_1:5;
A3:m < Radix(k) |^ (n+1) by A1,Def12;
DigA(DecSD(m,(n+1),k),n+1)
= DigitDC(m,n+1,k) by A2,Def9
.= (m mod (Radix(k) |^ (n+1))) div (Radix(k) |^ ((n+1) -'1)) by Def8
.= m div (Radix(k) |^ ((n+1) -'1)) by A3,GR_CY_1:4
.= m div (Radix(k) |^ n) by BINARITH:39;
hence thesis;
end;

begin :: Definition for Addition operation for a high-speed adder algorithm
:: on Radix-2^k SD number

definition let k,i,n be Nat, x,y be Tuple of n,(k-SD);
assume A1:i in Seg n & k >= 2;
func Add(x,y,i,k) -> Element of k-SD equals
:Def13:
+ SD_Add_Carry(DigA(x,i -'1)+DigA(y,i -'1));
coherence
proof
A2:k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >=
set SDD = SD_Add_Data(DigA(x,i)+DigA(y,i),k);
set SDC = SD_Add_Carry(DigA(x,i -'1)+DigA(y,i -'1));
A3:DigA(x,i) is Element of k-SD by A1,Th19;
DigA(y,i) is Element of k-SD by A1,Th19;
then A4:-Radix(k)+2 <= SDD & SDD <= Radix(k)-2 by A1,A3,Th23;
.= Radix(k)-1-1+1 by XCMPLX_1:36
.= Radix(k)-1-(1-1) by XCMPLX_1:37
A6: -Radix(k)+2 +(-1) = -Radix(k)+(1+1)-1 by XCMPLX_0:def 8
.= -Radix(k)+1+1-1 by XCMPLX_1:1
.= -Radix(k)+1 by XCMPLX_1:26;
A7: SDD + SDC is Element of INT by INT_1:def 2;
-1 <= SDC & SDC <= 1 by Lm2;
then -Radix(k)+2 +(-1) <= SDD + SDC & SDD + SDC <= Radix(k)-2 + 1
by A4,REAL_1:55;
then SDD + SDC in k-SD by A2,A5,A6,A7;
hence thesis;
end;
end;

definition let n,k be Nat,x,y be Tuple of n,(k-SD);
func x '+' y -> Tuple of n,(k-SD) means
:Def14:
for i st i in Seg n holds DigA(it,i) = Add(x,y,i,k);
existence
proof
consider z being FinSequence of (k-SD) such that
A1: len z = n and
A2: for j be Nat st j in Seg n holds z.j = F(j) from SeqLambdaD;
reconsider z as Tuple of n,(k-SD) by A1,FINSEQ_2:110;
take z;
let i;
assume A3:i in Seg n;
hence DigA(z,i) = z.i by Def3
.= Add(x,y,i,k) by A2,A3;
end;
uniqueness
proof
let k1,k2 be Tuple of n,(k-SD) such that
A4:for i be Nat st i in Seg n holds DigA(k1,i) = Add(x,y,i,k) and
A5:for i be Nat st i in Seg n holds DigA(k2,i) = Add(x,y,i,k);
A6:len k1 = n & len k2 = n by FINSEQ_2:109;
now let j be Nat;
assume A7:j in Seg n;
then k1.j = DigA(k1,j) by Def3
.= Add(x,y,j,k) by A4,A7
.= DigA(k2,j) by A5,A7
.= k2.j by A7,Def3;
hence k1.j = k2.j;
end;
hence k1 = k2 by A6,FINSEQ_2:10;
end;
end;

theorem Th28:
k >= 2 & m is_represented_by 1,k & n is_represented_by 1,k implies
proof
assume A1:k >= 2 & m is_represented_by 1,k & n is_represented_by 1,k;
set M = DecSD(m,1,k);
set N = DecSD(n,1,k);

A2:1 in Seg 1 by FINSEQ_1:3;
then A3:DigA((M '+' N),1)
= Add(M,N,1,k) by Def14
+ SD_Add_Carry(DigA(M,1 -'1)+DigA(N,1 -'1)) by A1,A2,Def13
+ SD_Add_Carry(DigA(M,0)+DigA(N,1 -'1)) by GOBOARD9:1
by GOBOARD9:1
.= SD_Add_Data(DigA(M,1)+DigA(N,1),k) + 0 by Def3,Th21
.= SD_Add_Data(m+DigA(N,1),k) by A1,Th24
.= SD_Add_Data(m+n,k) by A1,Th24;
A4:(DigitSD(M '+' N))/.1 = SubDigit((M '+' N),1,k) by A2,Def6
.= (Radix(k) |^ (1 -'1))*DigB((M '+' N),1) by Def5
.= (Radix(k) |^ (1 -'1))*DigA((M '+' N),1) by Def4
.= (Radix(k) |^ 0)*DigA((M '+' N),1) by GOBOARD9:1
.= 1*DigA((M '+' N),1) by NEWTON:9
.= SD_Add_Data(m+n,k) by A3;
A5:len (DigitSD(M '+' N)) = 1 by FINSEQ_2:109;
1 in Seg 1 by FINSEQ_1:3;
then 1 in dom DigitSD(M '+' N) by A5,FINSEQ_1:def 3;
then A6:DigitSD(M '+' N).1 = SD_Add_Data(m+n,k) by A4,FINSEQ_4:def 4;
reconsider w = SD_Add_Data(m+n,k) as Element of INT by INT_1:def 2;
SDDec(M '+' N) = Sum DigitSD(M '+' N) by Def7
.= Sum <*w*> by A5,A6,FINSEQ_1:57
.= SD_Add_Data(m+n,k) by RVSUM_1:103;
hence thesis;
end;

theorem
for n st n >= 1 holds
for k,x,y st
k >= 2 & x is_represented_by n,k & y is_represented_by n,k holds
x + y = (SDDec(DecSD(x,n,k) '+' DecSD(y,n,k)))
+ (Radix(k) |^ n)*
proof
defpred P[Nat] means
for k,x,y be Nat st
k >= 2 & x is_represented_by \$1,k & y is_represented_by \$1,k holds
x + y = (SDDec(DecSD(x,\$1,k) '+' DecSD(y,\$1,k)))+(Radix(k) |^ \$1)*

A1:P[1]
proof
let k,x,y be Nat;
assume A2:k >= 2 & x is_represented_by 1,k & y is_represented_by 1,k;
then A3:SDDec(DecSD(x,1,k)'+'DecSD(y,1,k))
= SD_Add_Data(x + y,k) by Th28;
= SD_Add_Carry(x+y) by A2,Th26;
SD_Add_Data(x + y,k) = x + y - SD_Add_Carry(x+y)*Radix(k) by Def11;
by XCMPLX_1:37
.= x + y - 0 by XCMPLX_1:14;
hence thesis by A3,A4,NEWTON:10;
end;

A5:for n be Nat st n >= 1 & P[n] holds P[n+1]
proof
let n be Nat;
assume A6:n >= 1 & P[n];
then n <> 0;
then A7: n in Seg n by FINSEQ_1:5;
A8: n+1 in Seg (n+1) by FINSEQ_1:5;
let k,x,y be Nat;
assume A9:k >= 2 &
x is_represented_by (n+1),k & y is_represented_by (n+1),k;
set x1 = x div (Radix(k) |^ n);
set xn = x mod (Radix(k) |^ n);
set y1 = y div (Radix(k) |^ n);
set yn = y mod (Radix(k) |^ n);
set z = DecSD(x,(n+1),k) '+' DecSD(y,(n+1),k);
set zn = DecSD(xn,n,k) '+' DecSD(yn,n,k);

A10:DigA(DecSD(y,(n+1),k),n+1) = y1 by A9,Th27;

Radix(k) <> 0 by Th9;
then Radix(k) > 0 by NAT_1:19;
then A11:(Radix(k) |^ n) > 0 by PREPOWER:13;
then A12:x = (x div (Radix(k) |^ n))*(Radix(k) |^ n)
+ (x mod (Radix(k) |^ n)) by NAT_1:47;

xn < Radix(k) |^ n & yn < Radix(k) |^ n by A11,NAT_1:46;
then A13:xn is_represented_by n,k & yn is_represented_by n,k by Def12;

.= x1 + y1 - 0 by XCMPLX_1:14;
A15:DigitSD(z) = DigitSD(zn)^<*SubDigit(z,n+1,k)*>
proof
A16:len DigitSD(z) = n+1 by FINSEQ_2:109;
A17:len DigitSD(zn) = n by FINSEQ_2:109;
A18:len <*SubDigit(z,n+1,k)*> = 1 by FINSEQ_1:56;
len (DigitSD(zn)^<*SubDigit(z,n+1,k)*>)
= len DigitSD(zn) + len <*SubDigit(z,n+1,k)*> by FINSEQ_1:35
.= n+1 by A18,FINSEQ_2:109;
then A19:len DigitSD(z) = len (DigitSD(zn)^<*SubDigit(z,n+1,k)*>)
by FINSEQ_2:109;

for i st 1 <= i & i <= len DigitSD(z) holds
(DigitSD(z)).i = ((DigitSD(zn))^<*SubDigit(z,n+1,k)*>).i
proof
let i;
assume A20:1 <= i & i <= len DigitSD(z);
then A21:i <= n+1 by FINSEQ_2:109;
then A22:i in Seg (n+1) by A20,FINSEQ_1:3;
then A23:      i in dom DigitSD(z) by A16,FINSEQ_1:def 3;
A24:i -'1 = i - 1 by A20,SCMFSA_7:3;
per cases by A22,FINSEQ_2:8;
suppose A25:i in Seg n;
then A26:i in dom DigitSD(zn) by A17,FINSEQ_1:def 3;
then A27:((DigitSD(zn))^<*SubDigit(z,n+1,k)*>).i
= DigitSD(zn).i by FINSEQ_1:def 7
.= (DigitSD(zn))/.i by A26,FINSEQ_4:def 4
.= SubDigit(zn,i,k) by A25,Def6
.= (Radix(k) |^ (i -'1))*DigB(zn,i) by Def5
.= (Radix(k) |^ (i -'1))*DigA(zn,i) by Def4
.= (Radix(k) |^ (i -'1))
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(yn,n,k),i -'1))) by A9,A25,Def13;
A28:DigitSD(z).i
= (DigitSD(z))/.i by A23,FINSEQ_4:def 4
.= SubDigit(z,i,k) by A22,Def6
.= (Radix(k) |^ (i -'1))*DigB(z,i) by Def5
.= (Radix(k) |^ (i -'1))*DigA(z,i) by Def4
.= (Radix(k) |^ (i -'1))
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)
+ DigA(DecSD(y,(n+1),k),i -'1))) by A9,A22,Def13;
now per cases by A20,AXIOMS:21;
suppose A29:i = 1;
then A30:((DigitSD(zn))^<*SubDigit(z,n+1,k)*>).i
= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(yn,n,k),0))) by A27,GOBOARD9:1
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(yn,n,k),0))) by A29,GOBOARD9:1
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ 0)) by Def3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)+0) by Def3,Th21;
DigitSD(z).i
= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)
+ DigA(DecSD(y,(n+1),k),0))) by A28,A29,GOBOARD9:1
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)
+ DigA(DecSD(y,(n+1),k),0))) by A29,GOBOARD9:1
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)
+ 0)) by Def3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)+0) by Def3,Th21
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)) by A25,Lm3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)) by A25,Lm3;
hence thesis by A30;
suppose i > 1;
then A31:i -'1 >= 1 by JORDAN3:12;
i - 1 <= n by A21,REAL_1:86;
then A32:i -'1 in Seg n by A24,A31,FINSEQ_1:3;
DigitSD(z).i
= (Radix(k) |^ (i -'1))
+ DigA(DecSD(y,(n+1),k),i),k)
+ DigA(DecSD(y,(n+1),k),i -'1))) by A25,A28,Lm3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(y,(n+1),k),i -'1))) by A25,Lm3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(y,(n+1),k),i -'1))) by A32,Lm3
.= (Radix(k) |^ (i -'1))
+ DigA(DecSD(yn,n,k),i),k)
+ DigA(DecSD(yn,n,k),i -'1))) by A32,Lm3;
hence thesis by A27;
end;
hence thesis;
suppose A33:i = n+1;
then ((DigitSD(zn))^<*SubDigit(z,n+1,k)*>).i
= ((DigitSD(zn))^<*SubDigit(z,n+1,k)*>).((len (DigitSD(zn)))+1)
by FINSEQ_2:109
.= SubDigit(z,n+1,k) by FINSEQ_1:59
.= (DigitSD(z))/.(n+1) by A8,Def6
.= DigitSD(z).(n+1) by A23,A33,FINSEQ_4:def 4;
hence thesis by A33;
end;
hence thesis by A19,FINSEQ_1:18;
end;

A34:DigitSD(zn) is FinSequence of INT;
A35:SubDigit(z,n+1,k) = (Radix(k) |^ ((n+1)-'1))*DigB(z,(n+1)) by Def5
.= (Radix(k) |^ n)*DigB(z,(n+1)) by BINARITH:39;

SDDec(z)
= Sum(DigitSD(zn)^<*SubDigit(z,n+1,k)*>) by A15,Def7
.= (Radix(k) |^ n)*DigB(z,(n+1)) + Sum DigitSD(zn) by A34,A35,RVSUM_1:104
.= DigA(z,(n+1))*(Radix(k) |^ n) + Sum DigitSD(zn) by Def4
+ Sum DigitSD(zn) by A8,Def14
+ DigA(DecSD(y,(n+1),k),(n+1)),k)
+ DigA(DecSD(y,(n+1),k),(n+1)-'1)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by A8,A9,Def13
+ DigA(DecSD(y,(n+1),k),(n+1)),k)
+ DigA(DecSD(y,(n+1),k),n)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by BINARITH:39
+ DigA(DecSD(y,(n+1),k),(n+1)),k)
+ DigA(DecSD(y,(n+1),k),n)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by BINARITH:39
+ DigA(DecSD(y,(n+1),k),(n+1)),k)
+ DigA(DecSD(y,(n+1),k),n)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by A7,Lm3
+ DigA(DecSD(y,(n+1),k),(n+1)),k)
+ DigA(DecSD(yn,n,k),n)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by A7,Lm3
+ DigA(DecSD(yn,n,k),n)))*(Radix(k) |^ n)
+ Sum DigitSD(zn) by A9,A10,Th27
+ DigA(DecSD(yn,n,k),n)))*(Radix(k) |^ n)
+ SDDec(zn) by Def7
+ DigA(DecSD(yn,n,k),n))*(Radix(k) |^ n)
+ (SDDec(DecSD(xn,n,k) '+' DecSD(yn,n,k))) by XCMPLX_1:8
*(Radix(k) |^ n)+(SDDec(DecSD(xn,n,k)'+'DecSD(yn,n,k)))) by XCMPLX_1:1
.= SD_Add_Data(x1+y1,k)*(Radix(k) |^ n) + (xn + yn) by A6,A9,A13
.= SD_Add_Data(x1+y1,k)*(Radix(k) |^ n) + xn + yn by XCMPLX_1:1;
= SD_Add_Data(x1+y1,k)*(Radix(k) |^ n) + (xn + yn)
+ SD_Add_Carry(x1+y1)*(Radix(k) |^ (n+1))+ (xn + yn)
by XCMPLX_1:1
+ SD_Add_Carry(x1+y1)*(Radix(k) |^ (n+1))+ xn + yn
by XCMPLX_1:1
by NEWTON:11
by XCMPLX_1:4
.= (x1+y1)*(Radix(k) |^ n)+ xn + yn by A14,XCMPLX_1:8
.= x1*(Radix(k) |^ n) + y1*(Radix(k) |^ n) + xn + yn by XCMPLX_1:8
.= (x1*(Radix(k) |^ n) + xn) + y1*(Radix(k) |^ n) + yn by XCMPLX_1:1
.= (x1*(Radix(k) |^ n) + xn) + (y1*(Radix(k) |^ n) + yn) by XCMPLX_1:1
.= x + y by A11,A12,NAT_1:47;
hence thesis by A9,A10,Th27;
end;
for n be Nat st n >= 1 holds P[n] from Ind1(A1,A5);
hence thesis;
end;

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