:: Magnitude Relation Properties of Radix-$2^k$ SD Number
:: by Masaaki Niimura and Yasushi Fuwa
::
:: Received November 7, 2003
:: Copyright (c) 2003-2017 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, NAT_1, XXREAL_0, RADIX_1, ARYTM_1, ARYTM_3, CARD_1,
SUBSET_1, FINSEQ_1, POWER, RELAT_1, FINSEQ_2, PARTFUN1, NEWTON, FUNCT_1,
CARD_3, INT_1, TARSKI, RADIX_5;
notations TARSKI, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, NAT_D,
NEWTON, RADIX_1, POWER, FUNCT_1, PARTFUN1, FINSEQ_1, FINSEQ_2, GR_CY_1;
constructors REAL_1, NAT_D, NEWTON, POWER, GR_CY_1, RADIX_4;
registrations RELSET_1, NUMBERS, XREAL_0, NAT_1, INT_1, ORDINAL1, NEWTON,
FINSEQ_2;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions TARSKI;
theorems RADIX_1, RADIX_4, INT_1, NAT_1, FINSEQ_1, POWER, FINSEQ_2, NEWTON,
RADIX_2, NAT_2, PREPOWER, PEPIN, JORDAN5B, FUNCT_1, RVSUM_1, XREAL_1,
XXREAL_0, NAT_D, PARTFUN1, ORDINAL1, XREAL_0, CARD_1;
schemes NAT_1, FINSEQ_2;
begin :: Some Useful Theorems
theorem Th1:
for k be Nat st k >= 2 holds Radix(k) - 1 in k-SD
proof
let k be Nat;
assume k >= 2;
then Radix(k) > 2 by RADIX_4:1;
then Radix(k) > 1 by XXREAL_0:2;
then Radix(k) + Radix(k) > 1 + 1 by XREAL_1:8;
then
A1: Radix(k) - 1 > 0 + 1 - Radix(k) by XREAL_1:21;
k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1}
& Radix(k) - 1 is Element of INT by INT_1:def 2,RADIX_1:def 2;
hence thesis by A1;
end;
theorem Th2:
for i,n be Nat st i > 1 & i in Seg n holds i -' 1 in Seg n
proof
let i,n be Nat;
assume that
A1: i > 1 and
A2: i in Seg n;
i - 1 > 1 - 1 by A1,XREAL_1:9;
then i -' 1 > 0 by A1,XREAL_1:233;
then
A3: i -' 1 >= 0 + 1 by NAT_1:13;
i <= n by A2,FINSEQ_1:1;
then i -' 1 <= n by NAT_D:44;
hence thesis by A3,FINSEQ_1:1;
end;
theorem Th3:
for k be Nat st 2 <= k holds 4 <= Radix(k)
proof
defpred P[Nat] means 4 <= Radix($1);
A1: for kk be Nat st kk >= 2 & P[kk] holds P[kk + 1]
proof
let kk be Nat;
assume that
2 <= kk and
A2: 4 <= Radix(kk);
Radix(kk + 1) = 2 to_power (kk + 1) by RADIX_1:def 1
.= 2 to_power (1) * 2 to_power (kk) by POWER:27
.= 2 * 2 to_power (kk) by POWER:25
.= 2 * Radix(kk) by RADIX_1:def 1;
then Radix(kk + 1) >= Radix(kk) by XREAL_1:151;
hence thesis by A2,XXREAL_0:2;
end;
Radix(2) = 2 to_power (1+1) by RADIX_1:def 1
.= 2 to_power 1 * 2 to_power 1 by POWER:27
.= 2 * (2 to_power 1) by POWER:25
.= 2 * 2 by POWER:25;
then
A3: P[2];
for i being Nat st 2 <= i holds P[i] from NAT_1:sch 8(A3,A1);
hence thesis;
end;
theorem Th4:
for k be Nat, tx be Tuple of 1,k-SD holds SDDec(tx) = DigA(tx,1)
proof
let k be Nat, tx be Tuple of 1,k-SD;
reconsider w = DigA(tx,1) as Element of INT by INT_1:def 2;
A1: 1 in Seg 1 by FINSEQ_1:1;
then
A2: (DigitSD(tx))/.1 = SubDigit(tx,1,k) by RADIX_1:def 6
.= ( Radix(k) |^ (1-'1) )*DigB(tx,1) by RADIX_1:def 5
.= (Radix(k) |^ (1-'1))*DigA(tx,1) by RADIX_1:def 4
.= (Radix(k) |^ 0)*DigA(tx,1) by XREAL_1:232
.= 1 * DigA(tx,1) by NEWTON:4;
A3: len DigitSD(tx) = 1 by CARD_1:def 7;
then 1 in dom DigitSD(tx) by A1,FINSEQ_1:def 3;
then
A4: DigitSD(tx).1 = DigA(tx,1) by A2,PARTFUN1:def 6;
SDDec(tx) = Sum DigitSD(tx) by RADIX_1:def 7
.= Sum <*w*> by A3,A4,FINSEQ_1:40
.= DigA(tx,1) by RVSUM_1:73;
hence thesis;
end;
begin :: Properties of Primary Radix-$2^k$ SD Number
theorem Th5:
for i,k,n be Nat st i in Seg n holds DigA(DecSD(0,n,k),i) = 0
proof
let i,k,n be Nat;
assume
A1: i in Seg n;
then
A2: i >= 1 by FINSEQ_1:1;
DigA(DecSD(0,n,k),i) = DigitDC(0,i,k) by A1,RADIX_1:def 9
.= (0 mod (Radix(k) |^ i)) div (Radix(k) |^ (i-'1)) by RADIX_1:def 8
.= (0 div (Radix(k) |^ (i-'1))) mod Radix(k) by A2,RADIX_2:4,6
.= 0 mod Radix(k) by NAT_2:2;
hence thesis by NAT_D:26;
end;
theorem
for n,k be Nat st n >= 1 holds SDDec(DecSD(0,n,k)) = 0
proof
let n,k be Nat;
Radix(k) >= 0 + 1 by INT_1:7,RADIX_2:6;
then Radix(k) |^ n >= 1 by PREPOWER:11;
then
A1: 0 is_represented_by n,k by RADIX_1:def 12;
assume n >= 1;
hence thesis by A1,RADIX_1:22;
end;
theorem Th7:
for k,n be Nat st 1 in Seg n & k >= 2 holds DigA(DecSD(1,n,k),1) = 1
proof
let k,n be Nat;
assume that
A1: 1 in Seg n and
A2: k >= 2;
A3: Radix(k) > 2 by A2,RADIX_4:1;
then
A4: Radix(k) > 1 by XXREAL_0:2;
DigA(DecSD(1,n,k),1) = DigitDC(1,1,k) by A1,RADIX_1:def 9
.= (1 mod (Radix(k) |^ 1)) div (Radix(k) |^ (1-'1)) by RADIX_1:def 8
.= (1 div (Radix(k) |^ (1-'1))) mod Radix(k) by A3,RADIX_2:4
.= (1 div (Radix(k) |^ 0)) mod Radix(k) by NAT_2:8
.= (1 div 1) mod Radix(k) by NEWTON:4
.= 1 mod Radix(k) by NAT_2:4;
hence thesis by A4,NAT_D:14;
end;
theorem Th8:
for i,k,n be Nat st i in Seg n & i > 1 & k >= 2 holds DigA(DecSD(
1,n,k),i) = 0
proof
let i,k,n be Nat;
assume that
A1: i in Seg n and
A2: i > 1 and
A3: k >= 2;
i-1 > 1 - 1 by A2,XREAL_1:14;
then
A4: i-'1 > 0 by XREAL_0:def 2;
A5: Radix(k) > 2 by A3,RADIX_4:1;
then Radix(k) > 1 by XXREAL_0:2;
then
A6: 1 div (Radix(k) |^ (i-'1)) = 0 by A4,NAT_D:27,PEPIN:25;
DigA(DecSD(1,n,k),i) = DigitDC(1,i,k) by A1,RADIX_1:def 9
.=(1 mod (Radix(k) |^ i)) div (Radix(k) |^ (i-'1)) by RADIX_1:def 8
.=(1 div (Radix(k) |^ (i-'1))) mod Radix(k) by A2,A5,RADIX_2:4;
hence thesis by A6,NAT_D:26;
end;
theorem
for n,k be Nat st n >= 1 & k >= 2 holds SDDec(DecSD(1,n,k)) = 1
proof
let n,k be Nat;
assume that
A1: n >= 1 and
A2: k >= 2;
A3: Radix(k) > 2 by A2,RADIX_4:1;
Radix(k) >= 0 + 1 by INT_1:7,RADIX_2:6;
then Radix(k) |^ n >= Radix(k) by A1,PREPOWER:12;
then Radix(k) |^ n >= 2 by A3,XXREAL_0:2;
then Radix(k) |^ n > 1 by XXREAL_0:2;
then 1 is_represented_by n,k by RADIX_1:def 12;
hence thesis by A1,RADIX_1:22;
end;
theorem Th10:
for k be Nat st k >= 2 holds SD_Add_Carry(Radix(k)) = 1
proof
let k be Nat;
assume k >= 2;
then Radix(k) > 2 by RADIX_4:1;
hence thesis by RADIX_1:def 10;
end;
theorem Th11:
for k be Nat st k >= 2 holds SD_Add_Data(Radix(k),k) = 0
proof
let k be Nat;
assume k >= 2;
then
A1: SD_Add_Carry(Radix(k)) = 1 by Th10;
SD_Add_Data(Radix(k),k) = Radix(k) - SD_Add_Carry(Radix(k)) * Radix(k)
by RADIX_1:def 11
.= Radix(k) - Radix(k) by A1;
hence thesis;
end;
begin :: Primary Magnitude Relation of Radix-2^k SD Number
Lm1: for k be Nat st 2 <= k holds SD_Add_Carry(Radix(k) - 1) = 1
proof
let k be Nat;
assume k >= 2;
then Radix(k) >= 4 by Th3;
then Radix(k) - 1 >= 4 - 1 by XREAL_1:9;
then Radix(k) - 1 > 2 by XXREAL_0:2;
hence thesis by RADIX_1:def 10;
end;
Lm2: for n,k,i be Nat st k >= 2 & i in Seg n holds SD_Add_Carry(DigA(DecSD(1,n
,k),i)) = 0
proof
let n,k,i be Nat;
assume that
A1: k >= 2 and
A2: i in Seg n;
A3: i >= 1 by A2,FINSEQ_1:1;
now
per cases by A3,XXREAL_0:1;
suppose
i = 1;
then SD_Add_Carry(DigA(DecSD(1,n,k),i)) = SD_Add_Carry(1) by A1,A2,Th7;
hence thesis by RADIX_1:def 10;
end;
suppose
i > 1;
then SD_Add_Carry(DigA(DecSD(1,n,k),i)) = SD_Add_Carry(0) by A1,A2,Th8;
hence thesis by RADIX_1:def 10;
end;
end;
hence thesis;
end;
theorem Th12:
for n be Nat st n >= 1 holds for k be Nat, tx,ty be Tuple of n,k
-SD st (for i be Nat st i in Seg n holds DigA(tx,i) = DigA(ty,i)) holds SDDec(
tx) = SDDec(ty)
proof
defpred P[Nat] means for k be Nat, tx,ty be Tuple of $1,k-SD st (for i be
Nat st i in Seg $1 holds DigA(tx,i) = DigA(ty,i)) holds SDDec(tx) = SDDec(ty);
A1: for n be Nat st n >= 1 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
n >= 1 and
A2: P[n];
let k be Nat, tx,ty be Tuple of (n+1),k-SD;
assume
A3: for i be Nat st i in Seg (n+1) holds DigA(tx,i) = DigA(ty,i);
deffunc F(Nat) = DigB(tx,$1);
consider txn being FinSequence of INT such that
A4: len txn = n and
A5: for i be Nat st i in dom txn holds txn.i = F(i) from FINSEQ_2:sch
1;
A6: dom txn = Seg n by A4,FINSEQ_1:def 3;
rng txn c= k-SD
proof
let z be object;
assume z in rng txn;
then consider xx be object such that
A7: xx in dom txn and
A8: z = txn.xx by FUNCT_1:def 3;
reconsider xx as Element of NAT by A7;
dom txn = Seg n by A4,FINSEQ_1:def 3;
then xx in Seg (n+1) by A7,FINSEQ_2:8;
then
A9: DigA(tx,xx) is Element of k-SD by RADIX_1:16;
z = DigB(tx,xx) by A5,A7,A8
.= DigA(tx,xx) by RADIX_1:def 4;
hence thesis by A9;
end;
then reconsider txn as FinSequence of k-SD by FINSEQ_1:def 4;
A10: for i be Nat st i in Seg n holds txn.i = tx.i
proof
let i be Nat;
assume
A11: i in Seg n;
then
A12: i in Seg (n+1) by FINSEQ_2:8;
txn.i = DigB(tx,i) by A5,A6,A11
.= DigA(tx,i) by RADIX_1:def 4;
hence thesis by A12,RADIX_1:def 3;
end;
deffunc F(Nat) = DigB(ty,$1);
consider tyn being FinSequence of INT such that
A13: len tyn = n and
A14: for i be Nat st i in dom tyn holds tyn.i = F(i) from FINSEQ_2:sch
1;
A15: dom tyn = Seg n by A13,FINSEQ_1:def 3;
rng tyn c= k-SD
proof
let z be object;
assume z in rng tyn;
then consider yy be object such that
A16: yy in dom tyn and
A17: z = tyn.yy by FUNCT_1:def 3;
reconsider yy as Element of NAT by A16;
dom tyn = Seg n by A13,FINSEQ_1:def 3;
then yy in Seg (n+1) by A16,FINSEQ_2:8;
then
A18: DigA(ty,yy) is Element of k-SD by RADIX_1:16;
z = DigB(ty,yy) by A14,A16,A17
.= DigA(ty,yy) by RADIX_1:def 4;
hence thesis by A18;
end;
then reconsider tyn as FinSequence of k-SD by FINSEQ_1:def 4;
A19: for i be Nat st i in Seg n holds tyn.i = ty.i
proof
let i be Nat;
assume
A20: i in Seg n;
then
A21: i in Seg (n+1) by FINSEQ_2:8;
tyn.i = DigB(ty,i) by A14,A15,A20
.= DigA(ty,i) by RADIX_1:def 4;
hence thesis by A21,RADIX_1:def 3;
end;
reconsider n as Element of NAT by ORDINAL1:def 12;
tyn is Element of n-tuples_on (k-SD) by A13,FINSEQ_2:92;
then reconsider tyn as Tuple of n,k-SD;
A22: SDDec(tyn) + (Radix(k) |^ n)*DigA(ty,n+1) = SDDec(ty) by A19,RADIX_2:10;
txn is Element of n-tuples_on (k-SD) by A4,FINSEQ_2:92;
then reconsider txn as Tuple of n,k-SD;
for i be Nat st i in Seg n holds DigA(txn,i) = DigA(tyn,i)
proof
let i be Nat;
assume
A23: i in Seg n;
then
A24: DigA(tyn,i) = tyn.i by RADIX_1:def 3
.= DigB(ty,i) by A14,A15,A23
.= DigA(ty,i) by RADIX_1:def 4;
DigA(txn,i) = txn.i by A23,RADIX_1:def 3
.= DigB(tx,i) by A5,A6,A23
.= DigA(tx,i) by RADIX_1:def 4;
hence thesis by A3,A23,A24,FINSEQ_2:8;
end;
then
A25: SDDec(txn) = SDDec(tyn) by A2;
SDDec(txn) + (Radix(k) |^ n)*DigA(tx,n+1) = SDDec(tx) by A10,RADIX_2:10;
hence thesis by A3,A22,A25,FINSEQ_1:4;
end;
A26: P[1]
proof
let k be Nat, tx,ty be Tuple of 1,k-SD;
assume
A27: for i be Nat st i in Seg 1 holds DigA(tx,i) = DigA(ty,i);
A28: SDDec(ty) = DigA(ty,1) by Th4;
1 in Seg 1 & SDDec(tx) = DigA(tx,1) by Th4,FINSEQ_1:1;
hence thesis by A27,A28;
end;
for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A26,A1);
hence thesis;
end;
theorem
for n be Nat st n >= 1 holds for k be Nat, tx,ty be Tuple of n,k-SD st
(for i be Nat st i in Seg n holds DigA(tx,i) >= DigA(ty,i)) holds SDDec(tx) >=
SDDec(ty)
proof
defpred P[Nat] means for k be Nat, tx,ty be Tuple of $1,k-SD st (for i be
Nat st i in Seg $1 holds DigA(tx,i) >= DigA(ty,i)) holds SDDec(tx) >= SDDec(ty)
;
A1: for n be Nat st n >= 1 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
n >= 1 and
A2: P[n];
let k be Nat, tx,ty be Tuple of (n+1),k-SD;
assume
A3: for i be Nat st i in Seg (n+1) holds DigA(tx,i) >= DigA(ty,i);
reconsider n as Element of NAT by ORDINAL1:def 12;
deffunc F(Nat) = DigB(tx,$1);
consider txn being FinSequence of INT such that
A4: len txn = n and
A5: for i be Nat st i in dom txn holds txn.i = F(i) from FINSEQ_2:sch
1;
deffunc F(Nat) = DigB(ty,$1);
consider tyn being FinSequence of INT such that
A6: len tyn = n and
A7: for i be Nat st i in dom tyn holds tyn.i = F(i) from FINSEQ_2:sch
1;
A8: dom tyn = Seg n by A6,FINSEQ_1:def 3;
rng tyn c= k-SD
proof
let z be object;
assume z in rng tyn;
then consider yy be object such that
A9: yy in dom tyn and
A10: z = tyn.yy by FUNCT_1:def 3;
reconsider yy as Element of NAT by A9;
dom tyn = Seg n by A6,FINSEQ_1:def 3;
then yy in Seg (n+1) by A9,FINSEQ_2:8;
then
A11: DigA(ty,yy) is Element of k-SD by RADIX_1:16;
z = DigB(ty,yy) by A7,A9,A10
.= DigA(ty,yy) by RADIX_1:def 4;
hence thesis by A11;
end;
then reconsider tyn as FinSequence of k-SD by FINSEQ_1:def 4;
A12: for i be Nat st i in Seg n holds tyn.i = ty.i
proof
let i be Nat;
assume
A13: i in Seg n;
then
A14: i in Seg (n+1) by FINSEQ_2:8;
tyn.i = DigB(ty,i) by A7,A8,A13
.= DigA(ty,i) by RADIX_1:def 4;
hence thesis by A14,RADIX_1:def 3;
end;
tyn is Element of n-tuples_on (k-SD) by A6,FINSEQ_2:92;
then reconsider tyn as Tuple of n,k-SD;
A15: dom txn = Seg n by A4,FINSEQ_1:def 3;
rng txn c= k-SD
proof
let z be object;
assume z in rng txn;
then consider xx be object such that
A16: xx in dom txn and
A17: z = txn.xx by FUNCT_1:def 3;
reconsider xx as Element of NAT by A16;
dom txn = Seg n by A4,FINSEQ_1:def 3;
then xx in Seg (n+1) by A16,FINSEQ_2:8;
then
A18: DigA(tx,xx) is Element of k-SD by RADIX_1:16;
z = DigB(tx,xx) by A5,A16,A17
.= DigA(tx,xx) by RADIX_1:def 4;
hence thesis by A18;
end;
then reconsider txn as FinSequence of k-SD by FINSEQ_1:def 4;
A19: for i be Nat st i in Seg n holds txn.i = tx.i
proof
let i be Nat;
assume
A20: i in Seg n;
then
A21: i in Seg (n+1) by FINSEQ_2:8;
txn.i = DigB(tx,i) by A5,A15,A20
.= DigA(tx,i) by RADIX_1:def 4;
hence thesis by A21,RADIX_1:def 3;
end;
txn is Element of n-tuples_on (k-SD) by A4,FINSEQ_2:92;
then reconsider txn as Tuple of n,k-SD;
for i be Nat st i in Seg n holds DigA(txn,i) >= DigA(tyn,i)
proof
let i be Nat;
assume
A22: i in Seg n;
then
A23: DigA(tyn,i) = tyn.i by RADIX_1:def 3
.= DigB(ty,i) by A7,A8,A22
.= DigA(ty,i) by RADIX_1:def 4;
DigA(txn,i) = txn.i by A22,RADIX_1:def 3
.= DigB(tx,i) by A5,A15,A22
.= DigA(tx,i) by RADIX_1:def 4;
hence thesis by A3,A22,A23,FINSEQ_2:8;
end;
then
A24: SDDec(txn) >= SDDec(tyn) by A2;
(n+1) in Seg (n+1) by FINSEQ_1:4;
then
A25: (Radix(k) |^ n)*DigA(tx,n+1) >= (Radix(k) |^ n)*DigA(ty,n+1) by A3,
XREAL_1:64;
A26: SDDec(tyn) + (Radix(k) |^ n)*DigA(ty,n+1) = SDDec(ty) by A12,RADIX_2:10;
SDDec(txn) + (Radix(k) |^ n)*DigA(tx,n+1) = SDDec(tx) by A19,RADIX_2:10;
hence thesis by A26,A24,A25,XREAL_1:7;
end;
A27: P[1]
proof
let k be Nat, tx,ty be Tuple of 1,k-SD;
assume
A28: for i be Nat st i in Seg 1 holds DigA(tx,i) >= DigA(ty,i);
A29: SDDec(ty) = DigA(ty,1) by Th4;
1 in Seg 1 & SDDec(tx) = DigA(tx,1) by Th4,FINSEQ_1:1;
hence thesis by A28,A29;
end;
for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A27,A1);
hence thesis;
end;
theorem Th14:
for n be Nat st n >= 1 holds for k be Nat st k >= 2 holds for tx
,ty,tz,tw be Tuple of n,k-SD st (for i be Nat st i in Seg n holds (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
defpred P[Nat] means for k be Nat st k >= 2 holds for tx,ty,tz,tw be Tuple
of $1,k-SD st (for i be Nat st i in Seg $1 holds (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);
A1: for n be Nat st n >= 1 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
n >= 1 and
A2: P[n];
let k be Nat;
assume
A3: k >= 2;
let tx,ty,tz,tw be Tuple of (n+1),k-SD;
assume
A4: for i be Nat st i in Seg (n+1) holds 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);
deffunc F(Nat) = DigB(tx,$1);
consider txn being FinSequence of INT such that
A5: len txn = n and
A6: for i be Nat st i in dom txn holds txn.i = F(i) from FINSEQ_2:sch
1;
A7: dom txn = Seg n by A5,FINSEQ_1:def 3;
rng txn c= k-SD
proof
let z be object;
assume z in rng txn;
then consider xx be object such that
A8: xx in dom txn and
A9: z = txn.xx by FUNCT_1:def 3;
reconsider xx as Element of NAT by A8;
dom txn = Seg n by A5,FINSEQ_1:def 3;
then xx in Seg (n+1) by A8,FINSEQ_2:8;
then
A10: DigA(tx,xx) is Element of k-SD by RADIX_1:16;
z = DigB(tx,xx) by A6,A8,A9
.= DigA(tx,xx) by RADIX_1:def 4;
hence thesis by A10;
end;
then reconsider txn as FinSequence of k-SD by FINSEQ_1:def 4;
A11: for i be Nat st i in Seg n holds txn.i = tx.i
proof
let i be Nat;
assume
A12: i in Seg n;
then
A13: i in Seg (n+1) by FINSEQ_2:8;
txn.i = DigB(tx,i) by A6,A7,A12
.= DigA(tx,i) by RADIX_1:def 4;
hence thesis by A13,RADIX_1:def 3;
end;
deffunc F(Nat) = DigB(ty,$1);
consider tyn being FinSequence of INT such that
A14: len tyn = n and
A15: for i be Nat st i in dom tyn holds tyn.i = F(i) from FINSEQ_2:sch
1;
A16: dom tyn = Seg n by A14,FINSEQ_1:def 3;
rng tyn c= k-SD
proof
let z be object;
assume z in rng tyn;
then consider yy be object such that
A17: yy in dom tyn and
A18: z = tyn.yy by FUNCT_1:def 3;
reconsider yy as Element of NAT by A17;
dom tyn = Seg n by A14,FINSEQ_1:def 3;
then yy in Seg (n+1) by A17,FINSEQ_2:8;
then
A19: DigA(ty,yy) is Element of k-SD by RADIX_1:16;
z = DigB(ty,yy) by A15,A17,A18
.= DigA(ty,yy) by RADIX_1:def 4;
hence thesis by A19;
end;
then reconsider tyn as FinSequence of k-SD by FINSEQ_1:def 4;
A20: for i be Nat st i in Seg n holds tyn.i = ty.i
proof
let i be Nat;
assume
A21: i in Seg n;
then
A22: i in Seg (n+1) by FINSEQ_2:8;
tyn.i = DigB(ty,i) by A15,A16,A21
.= DigA(ty,i) by RADIX_1:def 4;
hence thesis by A22,RADIX_1:def 3;
end;
deffunc F(Nat) = DigB(tz,$1);
consider tzn being FinSequence of INT such that
A23: len tzn = n and
A24: for i be Nat st i in dom tzn holds tzn.i = F(i) from FINSEQ_2:sch
1;
A25: dom tzn = Seg n by A23,FINSEQ_1:def 3;
rng tzn c= k-SD
proof
let z be object;
assume z in rng tzn;
then consider zz be object such that
A26: zz in dom tzn and
A27: z = tzn.zz by FUNCT_1:def 3;
reconsider zz as Element of NAT by A26;
dom tzn = Seg n by A23,FINSEQ_1:def 3;
then zz in Seg (n+1) by A26,FINSEQ_2:8;
then
A28: DigA(tz,zz) is Element of k-SD by RADIX_1:16;
z = DigB(tz,zz) by A24,A26,A27
.= DigA(tz,zz) by RADIX_1:def 4;
hence thesis by A28;
end;
then reconsider tzn as FinSequence of k-SD by FINSEQ_1:def 4;
A29: for i be Nat st i in Seg n holds tzn.i = tz.i
proof
let i be Nat;
assume
A30: i in Seg n;
then
A31: i in Seg (n+1) by FINSEQ_2:8;
tzn.i = DigB(tz,i) by A24,A25,A30
.= DigA(tz,i) by RADIX_1:def 4;
hence thesis by A31,RADIX_1:def 3;
end;
deffunc F(Nat) = DigB(tw,$1);
consider twn being FinSequence of INT such that
A32: len twn = n and
A33: for i be Nat st i in dom twn holds twn.i = F(i) from FINSEQ_2:sch
1;
A34: dom twn = Seg n by A32,FINSEQ_1:def 3;
rng twn c= k-SD
proof
let w be object;
assume w in rng twn;
then consider ww be object such that
A35: ww in dom twn and
A36: w = twn.ww by FUNCT_1:def 3;
reconsider ww as Element of NAT by A35;
dom twn = Seg n by A32,FINSEQ_1:def 3;
then ww in Seg (n+1) by A35,FINSEQ_2:8;
then
A37: DigA(tw,ww) is Element of k-SD by RADIX_1:16;
w = DigB(tw,ww) by A33,A35,A36
.= DigA(tw,ww) by RADIX_1:def 4;
hence thesis by A37;
end;
then reconsider twn as FinSequence of k-SD by FINSEQ_1:def 4;
A38: for i be Nat st i in Seg n holds twn.i = tw.i
proof
let i be Nat;
assume
A39: i in Seg n;
then
A40: i in Seg (n+1) by FINSEQ_2:8;
twn.i = DigB(tw,i) by A33,A34,A39
.= DigA(tw,i) by RADIX_1:def 4;
hence thesis by A40,RADIX_1:def 3;
end;
reconsider n as Element of NAT by ORDINAL1:def 12;
twn is Element of n-tuples_on (k-SD) by A32,FINSEQ_2:92;
then reconsider twn as Tuple of n,k-SD;
tzn is Element of n-tuples_on (k-SD) by A23,FINSEQ_2:92;
then reconsider tzn as Tuple of n,k-SD;
tyn is Element of n-tuples_on (k-SD) by A14,FINSEQ_2:92;
then reconsider tyn as Tuple of n,k-SD;
A41: SDDec(tyn) + (Radix(k) |^ n)*DigA(ty,n+1) = SDDec(ty) by A20,RADIX_2:10;
n+1 in Seg (n+1) by FINSEQ_1:3;
then
A42: DigA(tx,n+1) = DigA(tz,n+1) & DigA(ty,n+1) = DigA(tw,n+1) or DigA(ty,
n+1) = DigA(tz,n+1) & DigA(tx,n+1) = DigA(tw,n+1) by A4;
A43: SDDec(twn) + (Radix(k) |^ n)*DigA(tw,n+1) = SDDec(tw) by A38,RADIX_2:10;
A44: SDDec(tzn) + (Radix(k) |^ n)*DigA(tz,n+1) = SDDec(tz) by A29,RADIX_2:10;
txn is Element of n-tuples_on (k-SD) by A5,FINSEQ_2:92;
then reconsider txn as Tuple of n,k-SD;
for i be Nat st i in Seg n holds DigA(txn,i) = DigA(tzn,i) & DigA(tyn
,i) = DigA(twn,i) or DigA(tyn,i) = DigA(tzn,i) & DigA(txn,i) = DigA(twn,i)
proof
let i be Nat;
assume
A45: i in Seg n;
then i <= n by FINSEQ_1:1;
then
A46: i <= n + 1 by NAT_1:12;
1 <= i by A45,FINSEQ_1:1;
then
A47: i in Seg (n+1) by A46,FINSEQ_1:1;
then
A48: DigA(ty,i) = ty.i by RADIX_1:def 3
.= tyn.i by A20,A45
.= DigA(tyn,i) by A45,RADIX_1:def 3;
A49: DigA(tw,i) = tw.i by A47,RADIX_1:def 3
.= twn.i by A38,A45
.= DigA(twn,i) by A45,RADIX_1:def 3;
A50: DigA(tz,i) = tz.i by A47,RADIX_1:def 3
.= tzn.i by A29,A45
.= DigA(tzn,i) by A45,RADIX_1:def 3;
DigA(tx,i) = tx.i by A47,RADIX_1:def 3
.= txn.i by A11,A45
.= DigA(txn,i) by A45,RADIX_1:def 3;
hence thesis by A4,A47,A48,A50,A49;
end;
then
A51: SDDec(tzn) + SDDec(twn) = SDDec(txn) + SDDec(tyn) by A2,A3;
SDDec(txn) + (Radix(k) |^ n)*DigA(tx,n+1) = SDDec(tx) by A11,RADIX_2:10;
hence thesis by A41,A44,A43,A51,A42;
end;
A52: P[1]
proof
let k be Nat;
assume
A53: k >= 2;
let tx,ty,tz,tw be Tuple of 1,k-SD;
A54: 1 in Seg 1 by FINSEQ_1:1;
A55: SDDec(tx '+' ty) = DigA(tx '+' ty,1) by Th4
.= Add(tx,ty,1,k) by A54,RADIX_1:def 14
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + SD_Add_Carry(DigA(tx,1-'1)+
DigA(ty,1-'1)) by A53,A54,RADIX_1:def 13
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + SD_Add_Carry(DigA(tx,0)+DigA
(ty,1-'1)) by XREAL_1:232
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + SD_Add_Carry(DigA(tx,0)+DigA
(ty,0)) by XREAL_1:232
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + SD_Add_Carry(DigA(tx,0)+0)
by RADIX_1:def 3
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + SD_Add_Carry(0+0) by
RADIX_1:def 3
.= SD_Add_Data(DigA(tx,1)+DigA(ty,1),k) + 0 by RADIX_1:def 10;
A56: SDDec(tz '+' tw) = DigA(tz '+' tw,1) by Th4
.= Add(tz,tw,1,k) by A54,RADIX_1:def 14
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + SD_Add_Carry(DigA(tz,1-'1)+
DigA(tw,1-'1)) by A53,A54,RADIX_1:def 13
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + SD_Add_Carry(DigA(tz,0)+DigA
(tw,1-'1)) by XREAL_1:232
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + SD_Add_Carry(DigA(tz,0)+DigA
(tw,0)) by XREAL_1:232
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + SD_Add_Carry(DigA(tz,0)+0)
by RADIX_1:def 3
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + SD_Add_Carry(0+0) by
RADIX_1:def 3
.= SD_Add_Data(DigA(tz,1)+DigA(tw,1),k) + 0 by RADIX_1:def 10;
A57: DigA(tx,1) = DigA(tz,1) & DigA(ty,1) = DigA(tw,1) implies SDDec(tz) +
SDDec(tw) = SDDec(tx) + SDDec(ty)
proof
assume DigA(tx,1) = DigA(tz,1) & DigA(ty,1) = DigA(tw,1);
then
SDDec(tz) + SDDec(tw) = (SDDec(tx '+' ty)) + (SD_Add_Carry(DigA(tx,
1)+DigA(ty,1))*(Radix(k) |^ 1)) by A53,A56,A55,RADIX_2:11;
hence thesis by A53,RADIX_2:11;
end;
assume
A58: for j be Nat st j in Seg 1 holds DigA(tx,j) = DigA(tz,j) & DigA(ty
,j) = DigA(tw,j) or DigA(ty,j) = DigA(tz,j) & DigA(tx,j) = DigA(tw,j);
then
A59: DigA(tx,1) = DigA(tz,1) & DigA(ty,1) = DigA(tw,1) or DigA(ty,1) = DigA
(tz,1) & DigA(tx,1) = DigA(tw,1) by A54;
DigA(ty,1) = DigA(tz,1) & DigA(tx,1) = DigA(tw,1) implies SDDec(tz) +
SDDec(tw) = SDDec(tx) + SDDec(ty)
proof
assume that
DigA(ty,1) = DigA(tz,1) and
DigA(tx,1) = DigA(tw,1);
SDDec(tz) + SDDec(tw) = (SDDec(tx '+' ty)) + SD_Add_Carry(DigA(tx,1
)+DigA(ty,1))*(Radix(k) |^ 1) by A53,A56,A55,A59,RADIX_2:11;
hence thesis by A53,RADIX_2:11;
end;
hence thesis by A58,A54,A57;
end;
for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A52,A1);
hence thesis;
end;
theorem
for n,k be Nat st n >= 1 & k >= 2 holds for tx,ty,tz be Tuple of n,k
-SD st (for i be Nat st i in Seg n holds (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
let n,k be Nat;
assume
A1: n >= 1 & k >= 2;
let tx,ty,tz be Tuple of n,k-SD;
assume
A2: for i be Nat st i in Seg n holds DigA(tx,i) = DigA(tz,i) & DigA(ty,i
) = 0 or DigA(ty,i) = DigA(tz,i) & DigA(tx,i) = 0;
for i be Nat st i in Seg n holds DigA(tx,i) = DigA(tz,i) & DigA(ty,i) =
DigA(DecSD(0,n,k),i) or DigA(ty,i) = DigA(tz,i) & DigA(tx,i) = DigA(DecSD(0,n,k
),i)
proof
let i be Nat;
assume
A3: i in Seg n;
then DigA(DecSD(0,n,k),i) = 0 by Th5;
hence thesis by A2,A3;
end;
hence thesis by A1,Th14;
end;
begin :: Definiton of Max/Min Radix-2^k SD Number in Some Digits
definition
let i,m,k be Nat;
assume
A1: k >= 2;
func SDMinDigit(m,k,i) -> Element of k-SD equals
:Def1:
-Radix(k)+1 if 1 <=
i & i < m otherwise 0;
coherence
proof
Radix(k) > 2 by A1,RADIX_4:1;
then Radix(k) > 1 by XXREAL_0:2;
then Radix(k) + Radix(k) > 1 + 1 by XREAL_1:8;
then
A2: Radix(k) - 1 > 1 - Radix(k) by XREAL_1:21;
k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1
} & - Radix(k) + 1 is Element of INT by INT_1:def 2,RADIX_1:def 2;
then -Radix(k) + 1 in k-SD by A2;
hence thesis by RADIX_1:14;
end;
consistency;
end;
definition
let n,m,k be Nat;
func SDMin(n,m,k) -> Tuple of n,k-SD means
:Def2:
for i be Nat st i in Seg n holds DigA(it,i) = SDMinDigit(m,k,i);
existence
proof
deffunc F(Nat) = SDMinDigit(m,k,$1);
consider z being FinSequence of k-SD such that
A1: len z = n and
A2: for j be Nat st j in dom z holds z.j = F(j) from FINSEQ_2:sch 1;
A3: dom z = Seg n by A1,FINSEQ_1:def 3;
z is Element of n-tuples_on (k-SD) by A1,FINSEQ_2:92;
then reconsider z as Tuple of n,(k-SD);
take z;
let i be Nat;
assume
A4: i in Seg n;
then DigA(z,i) = z.i by RADIX_1:def 3;
hence thesis by A2,A3,A4;
end;
uniqueness
proof
let k1,k2 be Tuple of n,k-SD such that
A5: for i be Nat st i in Seg n holds DigA(k1,i) = SDMinDigit(m,k,i) and
A6: for i be Nat st i in Seg n holds DigA(k2,i) = SDMinDigit(m,k,i);
A7: len k1 = n by CARD_1:def 7;
then
A8: dom k1 = Seg n by FINSEQ_1:def 3;
A9: now
let j be Nat;
assume
A10: j in dom k1;
then k1.j = DigA(k1,j) by A8,RADIX_1:def 3
.= SDMinDigit(m,k,j) by A5,A8,A10
.= DigA(k2,j) by A6,A8,A10
.= k2.j by A8,A10,RADIX_1:def 3;
hence k1.j = k2.j;
end;
len k2 = n by CARD_1:def 7;
hence thesis by A7,A9,FINSEQ_2:9;
end;
end;
definition
let i,m,k be Nat;
assume
A1: k >= 2;
func SDMaxDigit(m,k,i) -> Element of k-SD equals
:Def3:
Radix(k)-1 if 1 <= i
& i < m otherwise 0;
coherence by A1,Th1,RADIX_1:14;
consistency;
end;
definition
let n,m,k be Nat;
func SDMax(n,m,k) -> Tuple of n,k-SD means
:Def4:
for i be Nat st i in Seg n holds DigA(it,i) = SDMaxDigit(m,k,i);
existence
proof
deffunc F(Nat) = SDMaxDigit(m,k,$1);
consider z being FinSequence of k-SD such that
A1: len z = n and
A2: for j be Nat st j in dom z holds z.j = F(j) from FINSEQ_2:sch 1;
A3: dom z = Seg n by A1,FINSEQ_1:def 3;
z is Element of n-tuples_on (k-SD) by A1,FINSEQ_2:92;
then reconsider z as Tuple of n,(k-SD);
take z;
let i be Nat;
assume
A4: i in Seg n;
then DigA(z,i) = z.i by RADIX_1:def 3;
hence thesis by A2,A3,A4;
end;
uniqueness
proof
let k1,k2 be Tuple of n,k-SD such that
A5: for i be Nat st i in Seg n holds DigA(k1,i) = SDMaxDigit(m,k,i) and
A6: for i be Nat st i in Seg n holds DigA(k2,i) = SDMaxDigit(m,k,i);
A7: len k1 = n by CARD_1:def 7;
then
A8: dom k1 = Seg n by FINSEQ_1:def 3;
A9: now
let j be Nat;
assume
A10: j in dom k1;
then k1.j = DigA(k1,j) by A8,RADIX_1:def 3
.= SDMaxDigit(m,k,j) by A5,A8,A10
.= DigA(k2,j) by A6,A8,A10
.= k2.j by A8,A10,RADIX_1:def 3;
hence k1.j = k2.j;
end;
len k2 = n by CARD_1:def 7;
hence thesis by A7,A9,FINSEQ_2:9;
end;
end;
definition
let i,m,k be Nat;
assume
A1: k >= 2;
func FminDigit(m,k,i) -> Element of k-SD equals
:Def5:
1 if i = m otherwise
0;
coherence
proof
A2: Radix(k) > 2 by A1,RADIX_4:1;
then -Radix(k) < -2 by XREAL_1:24;
then
A3: -Radix(k) + 1 < -2 + 1 by XREAL_1:6;
A4: k-SD = {w where w is Element of INT:w <= Radix(k)-1 & w >= -Radix(k)+1
} & 1 is Element of INT by INT_1:def 2,RADIX_1:def 2;
Radix(k)+ -1 > 2 + -1 by A2,XREAL_1:6;
then 1 in k-SD by A3,A4;
hence thesis by RADIX_1:14;
end;
consistency;
end;
definition
let n,m,k be Nat;
func Fmin(n,m,k) -> Tuple of n,k-SD means
:Def6:
for i be Nat st i in Seg n holds DigA(it,i) = FminDigit(m,k,i);
existence
proof
deffunc F(Nat) = FminDigit(m,k,$1);
consider z being FinSequence of k-SD such that
A1: len z = n and
A2: for j be Nat st j in dom z holds z.j = F(j) from FINSEQ_2:sch 1;
A3: dom z = Seg n by A1,FINSEQ_1:def 3;
z is Element of n-tuples_on (k-SD) by A1,FINSEQ_2:92;
then reconsider z as Tuple of n,(k-SD);
take z;
let i be Nat;
assume
A4: i in Seg n;
hence DigA(z,i) = z.i by RADIX_1:def 3
.= FminDigit(m,k,i) by A2,A3,A4;
end;
uniqueness
proof
let k1,k2 be Tuple of n,k-SD such that
A5: for i be Nat st i in Seg n holds DigA(k1,i) = FminDigit(m,k,i) and
A6: for i be Nat st i in Seg n holds DigA(k2,i) = FminDigit(m,k,i);
A7: len k1 = n by CARD_1:def 7;
then
A8: dom k1 = Seg n by FINSEQ_1:def 3;
A9: now
let j be Nat;
assume
A10: j in dom k1;
then k1.j = DigA(k1,j) by A8,RADIX_1:def 3
.= FminDigit(m,k,j) by A5,A8,A10
.= DigA(k2,j) by A6,A8,A10
.= k2.j by A8,A10,RADIX_1:def 3;
hence k1.j = k2.j;
end;
len k2 = n by CARD_1:def 7;
hence thesis by A7,A9,FINSEQ_2:9;
end;
end;
definition
let i,m,k be Nat;
assume
A1: k >= 2;
func FmaxDigit(m,k,i) -> Element of k-SD equals
:Def7:
Radix(k) - 1 if i = m
otherwise 0;
coherence by A1,Th1,RADIX_1:14;
consistency;
end;
definition
let n,m,k be Nat;
func Fmax(n,m,k) -> Tuple of n,k-SD means
:Def8:
for i be Nat st i in Seg n holds DigA(it,i) = FmaxDigit(m,k,i);
existence
proof
deffunc F(Nat) = FmaxDigit(m,k,$1);
consider z being FinSequence of k-SD such that
A1: len z = n and
A2: for j be Nat st j in dom z holds z.j = F(j) from FINSEQ_2:sch 1;
A3: dom z = Seg n by A1,FINSEQ_1:def 3;
z is Element of n-tuples_on (k-SD) by A1,FINSEQ_2:92;
then reconsider z as Tuple of n,(k-SD);
take z;
let i be Nat;
assume
A4: i in Seg n;
hence DigA(z,i) = z.i by RADIX_1:def 3
.= FmaxDigit(m,k,i) by A2,A3,A4;
end;
uniqueness
proof
let k1,k2 be Tuple of n,k-SD such that
A5: for i be Nat st i in Seg n holds DigA(k1,i) = FmaxDigit(m,k,i) and
A6: for i be Nat st i in Seg n holds DigA(k2,i) = FmaxDigit(m,k,i);
A7: len k1 = n by CARD_1:def 7;
then
A8: dom k1 = Seg n by FINSEQ_1:def 3;
A9: now
let j be Nat;
assume
A10: j in dom k1;
then k1.j = DigA(k1,j) by A8,RADIX_1:def 3
.= FmaxDigit(m,k,j) by A5,A8,A10
.= DigA(k2,j) by A6,A8,A10
.= k2.j by A8,A10,RADIX_1:def 3;
hence k1.j = k2.j;
end;
len k2 = n by CARD_1:def 7;
hence thesis by A7,A9,FINSEQ_2:9;
end;
end;
begin :: Properties of Max/Min Radix-$2^k$ SD Numbers
theorem Th16:
for n,m,k be Nat st k >= 2 holds for i be Nat st i in Seg n
holds DigA(SDMax(n,m,k),i)+DigA(SDMin(n,m,k),i) = 0
proof
let n,m,k be Nat;
assume
A1: k >= 2;
let i be Nat;
reconsider a = SDMinDigit(m,k,i) as Element of INT;
reconsider b = SDMaxDigit(m,k,i) as Element of INT;
assume
A2: i in Seg n;
then
A3: i >= 1 by FINSEQ_1:1;
A4: DigA(SDMin(n,m,k),i) = SDMinDigit(m,k,i) by A2,Def2;
now
per cases;
suppose
A5: i < m;
then a + b = -Radix(k) + 1 + b by A1,A3,Def1
.= -Radix(k) + 1 + (Radix(k) - 1) by A1,A3,A5,Def3
.= 0;
hence thesis by A2,A4,Def4;
end;
suppose
A6: i >= m;
then a + b = 0 + b by A1,Def1
.= 0 + 0 by A1,A6,Def3;
hence thesis by A2,A4,Def4;
end;
end;
hence thesis;
end;
theorem
for n be Nat st n >= 1 holds for m,k be Nat st k >= 2 holds SDDec(
SDMax(n,m,k)) + SDDec(SDMin(n,m,k)) = SDDec(DecSD(0,n,k))
proof
let n be Nat;
assume
A1: n >= 1;
then
A2: n in Seg n by FINSEQ_1:1;
let m,k be Nat;
assume
A3: k >= 2;
A4: for i be Nat st i in Seg n holds DigA(DecSD(0,n,k),i) = DigA(SDMax(n,m,k
) '+' SDMin(n,m,k),i)
proof
let i be Nat;
assume
A5: i in Seg n;
then
A6: DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = Add(SDMax(n,m,k),SDMin(n,m,k),
i,k) by RADIX_1:def 14
.= SD_Add_Data(DigA(SDMax(n,m,k),i)+DigA(SDMin(n,m,k),i),k) +
SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(n,m,k),i-'1)) by A3,A5,
RADIX_1:def 13
.= SD_Add_Data(0,k) + SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(
n,m,k),i-'1)) by A3,A5,Th16
.= 0+SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(SDMin(n,m,k),i-'1)) by
RADIX_1:19;
A7: DigA(DecSD(0,n,k),i) = 0 by A5,Th5;
A8: i >= 1 by A5,FINSEQ_1:1;
now
per cases by A8,XXREAL_0:1;
suppose
A9: i = 1;
then DigA(SDMin(n,m,k),i-'1) = 0 by NAT_2:8,RADIX_1:def 3;
then DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = SD_Add_Carry(0+0) by A6,A9
,NAT_2:8,RADIX_1:def 3;
hence thesis by A7,RADIX_1:def 10;
end;
suppose
i > 1;
then i -' 1 in Seg n by A5,Th2;
then
DigA(SDMax(n,m,k) '+' SDMin(n,m,k),i) = SD_Add_Carry(0) by A3,A6,Th16;
hence thesis by A7,RADIX_1:def 10;
end;
end;
hence thesis;
end;
SDDec(SDMax(n,m,k)) + SDDec(SDMin(n,m,k)) = SDDec(SDMax(n,m,k) '+'
SDMin(n,m,k)) + SD_Add_Carry(DigA(SDMax(n,m,k),n)+DigA(SDMin(n,m,k),n)) * (
Radix(k) |^ n) by A1,A3,RADIX_2:11
.= SDDec(SDMax(n,m,k) '+' SDMin(n,m,k)) + SD_Add_Carry(0) * (Radix(k) |^
n) by A3,A2,Th16
.= SDDec(SDMax(n,m,k) '+' SDMin(n,m,k)) + 0 * (Radix(k) |^ n) by
RADIX_1:def 10;
hence thesis by A1,A4,Th12;
end;
theorem
for n be Nat st n >= 1 holds for m,k be 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
let n be Nat;
A1: 1 in Seg 1 by FINSEQ_1:1;
assume
A2: n >= 1;
then
A3: 1 in Seg n by FINSEQ_1:1;
let m,k be Nat;
assume that
A4: m in Seg n and
A5: k >= 2;
A6: n >= m by A4,FINSEQ_1:1;
A7: m >= 1 by A4,FINSEQ_1:1;
now
per cases by A2,XXREAL_0:1;
suppose
A8: n = 1;
then
A9: m = 1 by A6,A7,XXREAL_0:1;
A10: SDDec(Fmin(1,m,k)) = DigA(Fmin(1,m,k),1) by Th4
.= FminDigit(m,k,1) by A1,Def6
.= 1 by A5,A9,Def5;
A11: DigA(SDMax(1,m,k),1) = SDMaxDigit(m,k,1) by A1,Def4
.= 0 by A5,A6,A8,Def3;
SDDec(SDMax(1,m,k) '+' DecSD(1,1,k)) = DigA(SDMax(1,m,k) '+' DecSD(
1,1,k),1) by Th4
.= Add(SDMax(1,m,k),DecSD(1,1,k),1,k) by A1,RADIX_1:def 14
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(DigA(SDMax(1,m,k),1-'1)+DigA(DecSD(1,1,k),1-'1)) by A5,A1,
RADIX_1:def 13
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(DigA(SDMax(1,m,k),0)+DigA(DecSD(1,1,k),1-'1)) by XREAL_1:232
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(DigA(SDMax(1,m,k),0)+DigA(DecSD(1,1,k),0)) by XREAL_1:232
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(DigA(SDMax(1,m,k),0)+0) by RADIX_1:def 3
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) +
SD_Add_Carry(0+0) by RADIX_1:def 3
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1),k) + 0 by
RADIX_1:def 10
.= SD_Add_Data(DigA(SDMax(1,m,k),1)+1,k) by A5,A1,Th7;
then
A12: SDDec(SDMax(1,m,k) '+' DecSD(1,1,k)) = 1 - SD_Add_Carry(1) * Radix
(k) by A11,RADIX_1:def 11
.= 1 - 0 * Radix(k) by RADIX_1:def 10
.= 1;
SDDec(SDMax(1,m,k)) + SDDec(DecSD(1,1,k)) = SDDec(SDMax(1,m,k) '+'
DecSD(1,1,k)) + SD_Add_Carry(DigA(SDMax(1,m,k),1)+DigA(DecSD(1,1,k),1)) *(Radix
(k) |^ 1) by A5,RADIX_2:11
.= 1 + SD_Add_Carry(0+1) *(Radix(k) |^ 1) by A5,A1,A11,A12,Th7
.= 1 + 0 * (Radix(k) |^ 1) by RADIX_1:def 10;
hence thesis by A8,A10;
end;
suppose
A13: n > 1;
A14: n in Seg n by A2,FINSEQ_1:1;
then
A15: DigA(SDMax(n,m,k),n) = SDMaxDigit(m,k,n) by Def4
.= 0 by A5,A6,Def3;
A16: for i be Nat st i in Seg n holds DigA(Fmin(n,m,k),i) = DigA(SDMax(n
,m,k) '+' DecSD(1,n,k),i)
proof
let i be Nat;
assume
A17: i in Seg n;
then
A18: DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = Add(SDMax(n,m,k),DecSD(1,
n,k),i,k) by RADIX_1:def 14
.= SD_Add_Data(DigA(SDMax(n,m,k),i)+DigA(DecSD(1,n,k),i),k) +
SD_Add_Carry(DigA(SDMax(n,m,k),i-'1)+DigA(DecSD(1,n,k),i-'1)) by A5,A17,
RADIX_1:def 13;
A19: DigA(SDMax(n,m,k),i) = SDMaxDigit(m,k,i) by A17,Def4;
A20: DigA(Fmin(n,m,k),i) = FminDigit(m,k,i) by A17,Def6;
A21: i >= 1 by A17,FINSEQ_1:1;
now
per cases;
suppose
A22: i >= m+1;
then
A23: i > m by NAT_1:13;
then
A24: DigA(SDMax(n,m,k),i) = 0 by A5,A19,Def3;
A25: i > 1 by A7,A23,XXREAL_0:2;
then
A26: i -' 1 in Seg n & DigA(DecSD(1,n,k),i) = 0 by A5,A17,Th2,Th8;
now
per cases by A22,XXREAL_0:1;
suppose
i = m+1;
then DigA(SDMax(n,m,k),i-'1) = DigA(SDMax(n,m,k),m) by NAT_D:34
.= SDMaxDigit(m,k,m) by A4,Def4
.= 0 by A5,Def3;
then
DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(0+0,k
) + 0 by A5,A18,A24,A26,Lm2
.= 0 by RADIX_1:19;
hence thesis by A5,A20,A23,Def5;
end;
suppose
i > m+1;
then i - 1 > m + 1 - 1 by XREAL_1:14;
then
A27: i -' 1 > m by XREAL_0:def 2;
i -' 1 in Seg n by A17,A25,Th2;
then DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by Def4
.= 0 by A5,A27,Def3;
then
DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(0+0,k
) + 0 by A5,A18,A24,A26,Lm2
.= 0 by RADIX_1:19;
hence thesis by A5,A20,A23,Def5;
end;
end;
hence thesis;
end;
suppose
i < m+1;
then
A28: i <= m by NAT_1:13;
A29: i >= 1 by A17,FINSEQ_1:1;
now
per cases by A29,XXREAL_0:1;
suppose
A30: i > 1;
then
A31: m > 1 by A28,XXREAL_0:2;
then
A32: m -' 1 < m by JORDAN5B:1;
A33: m -' 1 in Seg n by A4,A31,Th2;
then
A34: m -' 1 >= 1 by FINSEQ_1:1;
now
per cases by A28,XXREAL_0:1;
suppose
A35: i = m;
then
A36: DigA(SDMax(n,m,k),i) = 0 & DigA(DecSD(1,n,k),i) = 0
by A4,A5,A19,A30,Def3,Th8;
A37: DigA(Fmin(n,m,k),i) = FminDigit(m, k,m) by A4,A35,Def6
.= 1 by A5,Def5;
A38: DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,m-'1) by A33,A35
,Def4
.= Radix(k) - 1 by A5,A34,A32,Def3;
A39: i >= 1 + 1 by A30,INT_1:7;
now
per cases by A39,XXREAL_0:1;
suppose
i = 2;
then i -' 1 = 2 - 1 by XREAL_1:233;
then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(0+0,k) +SD_Add_Carry(Radix(k) - 1 + 1) by A5,A3,A18,A36,A38,Th7
.= 0 + SD_Add_Carry(Radix(k) - 1 + 1) by RADIX_1:19
.= 1 by A5,Th10;
hence thesis by A37;
end;
suppose
A40: i > 2;
then i - 1 > 2 - 1 by XREAL_1:14;
then
A41: i -' 1 > 1 by XREAL_0:def 2;
i > 1 by A40,XXREAL_0:2;
then i -' 1 in Seg n by A17,Th2;
then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(0+0,k) + SD_Add_Carry(Radix(k) - 1 + 0) by A5,A18,A36,A38,A41,Th8
.= 0 + SD_Add_Carry(Radix(k) - 1) by RADIX_1:19
.= 1 by A5,Lm1;
hence thesis by A37;
end;
end;
hence thesis;
end;
suppose
A42: i < m;
i -' 1 < i by A30,JORDAN5B:1;
then
A43: i -' 1 < m by A42,XXREAL_0:2;
A44: DigA(DecSD(1,n,k),i) = 0 by A5,A17,A30,Th8;
A45: i -' 1 in Seg n by A17,A30,Th2;
A46: i >= 1 + 1 by A30,INT_1:7;
now
per cases by A46,XXREAL_0:1;
suppose
i = 2;
then
A47: i -' 1 = 2 - 1 by XREAL_1:233;
then
A48: DigA(DecSD(1,n,k),i-'1) = 1 by A5,A3,Th7;
DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by A45
,Def4
.= Radix(k) - 1 by A5,A43,A47,Def3;
then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(Radix(k)-1,k) +SD_Add_Carry(Radix(k) + 1 - 1) by A5,A18,A19,A21,A42
,A44,A48,Def3
.= SD_Add_Data(Radix(k)-1,k) + 1 by A5,Th10
.= (Radix(k)-1) - SD_Add_Carry(Radix(k)-1)*Radix(k
) + 1 by RADIX_1:def 11
.= (Radix(k)-1) - 1*Radix(k) + 1 by A5,Lm1
.= 0;
hence thesis by A5,A20,A42,Def5;
end;
suppose
A49: i > 2;
then i - 1 > 2 - 1 by XREAL_1:14;
then
A50: i -' 1 > 1 by XREAL_0:def 2;
i > 1 by A49,XXREAL_0:2;
then i -' 1 in Seg n by A17,Th2;
then
A51: DigA(DecSD(1,n,k),i-'1) = 0 by A5,A50,Th8;
DigA(SDMax(n,m,k),i-'1) = SDMaxDigit(m,k,i-'1) by A45
,Def4
.= Radix(k) - 1 by A5,A43,A50,Def3;
then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) =
SD_Add_Data(Radix(k)-1+0,k) +SD_Add_Carry(Radix(k) - 1 + 0) by A5,A18,A19,A21
,A42,A44,A51,Def3
.= SD_Add_Data(Radix(k)-1,k) + 1 by A5,Lm1
.= (Radix(k)-1) - SD_Add_Carry(Radix(k)-1)*Radix(k
) + 1 by RADIX_1:def 11
.= (Radix(k)-1) - 1*Radix(k) + 1 by A5,Lm1
.= 0;
hence thesis by A5,A20,A42,Def5;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
suppose
A52: i = 1;
then
A53: DigA(SDMax(n,m,k),i-'1) = 0 & DigA(DecSD(1,n,k),i-'1) = 0
by NAT_2:8,RADIX_1:def 3;
now
per cases by A28,XXREAL_0:1;
suppose
A54: i < m;
then DigA(SDMax(n,m,k),i)=Radix(k)-1 by A5,A19,A52,Def3;
then DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(
Radix(k)-1+1,k) + SD_Add_Carry(0+0) by A5,A3,A18,A52,A53,Th7
.= Radix(k) - SD_Add_Carry(Radix(k)) * Radix(k) by
RADIX_1:18,def 11
.= Radix(k) - 1 * Radix(k) by A5,Th10
.= 0;
hence thesis by A5,A20,A54,Def5;
end;
suppose
A55: i = m;
then DigA(SDMax(n,m,k),i) = 0 by A5,A19,Def3;
then
DigA(SDMax(n,m,k) '+' DecSD(1,n,k),i) = SD_Add_Data(1
+0,k)+SD_Add_Carry(0+0) by A5,A3,A18,A52,A53,Th7
.= 1 - SD_Add_Carry(1) * Radix(k) by RADIX_1:18,def 11
.= 1 - 0 * Radix(k) by RADIX_1:def 10
.= 1;
hence thesis by A5,A20,A55,Def5;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
SDDec(SDMax(n,m,k)) + SDDec(DecSD(1,n,k)) = SDDec(SDMax(n,m,k) '+'
DecSD(1,n,k)) + SD_Add_Carry(DigA(SDMax(n,m,k),n)+DigA(DecSD(1,n,k),n)) * (
Radix(k) |^ n) by A2,A5,RADIX_2:11
.= SDDec(SDMax(n,m,k) '+' DecSD(1,n,k)) + 0 * (Radix(k) |^ n) by A5,A13
,A14,A15,Th8,RADIX_1:18;
hence thesis by A2,A16,Th12;
end;
end;
hence thesis;
end;
theorem
for n,m,k be 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
let n,m,k be Nat;
assume that
A1: m in Seg n and
A2: k >= 2;
n >= m by A1,FINSEQ_1:1;
then
A3: n + 1 > m by NAT_1:13;
A4: n+1 in Seg(n+1) by FINSEQ_1:3;
then
A5: DigA(Fmin(n+1,m,k),n+1) = FminDigit(m,k,n+1) by Def6
.= 0 by A2,A3,Def5;
A6: m in Seg (n+1) by A1,FINSEQ_2:8;
A7: m >= 1 by A1,FINSEQ_1:1;
A8: for i be Nat st i in Seg (n+1) holds DigA(Fmin(n+1,m+1,k),i) = DigA(
Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i)
proof
let i be Nat;
A9: m + 1 > m by NAT_1:13;
assume
A10: i in Seg (n+1);
then
A11: DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = Add(Fmin(n+1,m,k),Fmax(n+1,
m,k),i,k) by RADIX_1:def 14
.= SD_Add_Data(DigA(Fmin(n+1,m,k),i)+DigA(Fmax(n+1,m,k),i),k) +
SD_Add_Carry(DigA(Fmin(n+1,m,k),i-'1)+DigA(Fmax(n+1,m,k),i-'1)) by A2,A10,
RADIX_1:def 13;
A12: DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,i) by A10,Def6;
A13: DigA(Fmin(n+1,m,k),i) = FminDigit(m,k,i) by A10,Def6;
A14: DigA(Fmax(n+1,m,k),i) = FmaxDigit(m,k,i) by A10,Def8;
A15: i >= 1 by A10,FINSEQ_1:1;
now
per cases;
suppose
A16: i >= m+1;
now
per cases;
suppose
A17: i = m+1;
then
A18: DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,m+1) by A10,Def6
.= 1 by A2,Def5;
A19: DigA(Fmax(n+1,m,k),i-'1) = DigA(Fmax(n+1,m,k),m) by A17,NAT_D:34
.= FmaxDigit(m,k,m) by A6,Def8
.= Radix(k) - 1 by A2,Def7;
A20: DigA(Fmax(n+1,m,k),i) = FmaxDigit(m,k,m+1) by A10,A17,Def8
.= 0 by A2,A9,Def7;
A21: DigA(Fmin(n+1,m,k),i-'1) = DigA(Fmin(n+1,m,k),m) by A17,NAT_D:34
.= FminDigit(m,k,m) by A6,Def6
.= 1 by A2,Def5;
DigA(Fmin(n+1,m,k),i) = FminDigit(m,k,m+1) by A10,A17,Def6
.= 0 by A2,A9,Def5;
then
DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = 0 + SD_Add_Carry(1+
(Radix(k)-1)) by A11,A20,A21,A19,RADIX_1:19
.= 1 by A2,Th10;
hence thesis by A18;
end;
suppose
A22: i <> m + 1;
then i > m + 1 by A16,XXREAL_0:1;
then i - 1 > m + 1 - 1 by XREAL_1:14;
then
A23: i -' 1 > m by XREAL_0:def 2;
i > m by A16,NAT_1:13;
then i > 1 by A7,XXREAL_0:2;
then
A24: i -' 1 in Seg (n+1) by A10,Th2;
then
A25: DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,i-'1) by Def6
.= 0 by A2,A23,Def5;
A26: DigA(Fmin(n+1,m+1,k),i) = 0 by A2,A12,A22,Def5;
A27: DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,i-'1) by A24,Def8
.= 0 by A2,A23,Def7;
DigA(Fmin(n+1,m,k),i) = 0 & DigA(Fmax(n+1,m,k),i) = 0 by A2,A9,A13
,A14,A16,Def5,Def7;
hence thesis by A11,A25,A27,A26,RADIX_1:18,19;
end;
end;
hence thesis;
end;
suppose
i < m+1;
then
A28: i <= m by NAT_1:13;
now
per cases by A15,XXREAL_0:1;
suppose
A29: i > 1;
then
A30: m > 1 by A28,XXREAL_0:2;
then
A31: m -' 1 < m by JORDAN5B:1;
A32: m -' 1 in Seg (n+1) by A6,A30,Th2;
now
per cases by A28,XXREAL_0:1;
suppose
A33: i = m;
then
A34: DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,m-'1) by A32,Def8
.= 0 by A2,A31,Def7;
A35: DigA(Fmax(n+1,m,k),i) = Radix(k) - 1 by A2,A14,A33,Def7;
A36: DigA(Fmin(n+1,m+1,k),i) = FminDigit(m+1,k,m) by A10,A33,Def6
.= 0 by A2,A9,Def5;
DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,m-'1) by A32,A33,Def6
.= 0 by A2,A31,Def5;
then
DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(1 +
(Radix(k) - 1),k) + 0 by A2,A11,A13,A33,A35,A34,Def5,RADIX_1:18
.= 0 by A2,Th11;
hence thesis by A36;
end;
suppose
A37: i < m;
A38: i -' 1 >= 1 by A29,NAT_D:49;
A39: i -' 1 < i by A15,JORDAN5B:1;
then i -' 1 < m by A37,XXREAL_0:2;
then i -' 1 <= n+1 by A3,XXREAL_0:2;
then
A40: i -' 1 in Seg (n+1) by A38,FINSEQ_1:1;
then
A41: DigA(Fmax(n+1,m,k),i-'1) = FmaxDigit(m,k,i-'1) by Def8
.= 0 by A2,A37,A39,Def7;
A42: DigA(Fmax(n+1,m,k),i) = 0 by A2,A14,A37,Def7;
A43: i < m + 1 by A37,NAT_1:13;
DigA(Fmin(n+1,m,k),i-'1) = FminDigit(m,k,i-'1) by A40,Def6
.= 0 by A2,A37,A39,Def5;
then
DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(0+0
,k) + SD_Add_Carry(0+0) by A2,A11,A13,A37,A42,A41,Def5
.= 0 + 0 by RADIX_1:18,19;
hence thesis by A2,A12,A43,Def5;
end;
end;
hence thesis;
end;
suppose
A44: i = 1;
then
A45: DigA(Fmax(n+1,m,k),i-'1) = 0 by NAT_2:8,RADIX_1:def 3;
now
per cases by A28,XXREAL_0:1;
suppose
A46: i < m;
then DigA(Fmin(n+1,m,k),i) = 0 & DigA(Fmax(n+1,m,k),i) = 0 by
A2,A13,A14,Def5,Def7;
then
A47: DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(0+0
,k) + SD_Add_Carry(0+0) by A11,A44,A45,NAT_2:8,RADIX_1:def 3
.= 0 + 0 by RADIX_1:18,19;
i < m + 1 by A46,NAT_1:13;
hence thesis by A2,A12,A47,Def5;
end;
suppose
A48: i = m;
then
DigA(Fmin(n+1,m,k),i) = 1 & DigA(Fmax(n+1,m,k),i) = Radix
(k) - 1 by A2,A13,A14,Def5,Def7;
then
A49: DigA(Fmin(n+1,m,k) '+' Fmax(n+1,m,k),i) = SD_Add_Data(1 +
(Radix(k) - 1),k) + 0 by A11,A44,A45,NAT_2:8,RADIX_1:18,def 3
.= 0 by A2,Th11;
i < m + 1 by A48,NAT_1:13;
hence thesis by A2,A12,A49,Def5;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
A50: DigA(Fmax(n+1,m,k),n+1) = FmaxDigit(m,k,n+1) by A4,Def8
.= 0 by A2,A3,Def7;
SDDec(Fmin(n+1,m,k)) + SDDec(Fmax(n+1,m,k)) = SDDec(Fmin(n+1,m,k) '+'
Fmax(n+1,m,k)) + SD_Add_Carry(DigA(Fmin(n+1,m,k),n+1)+DigA(Fmax(n+1,m,k),n+1))
* (Radix(k) |^ (n+1)) by A2,NAT_1:11,RADIX_2:11
.= SDDec(Fmin(n+1,m,k) '+' Fmax(n+1,m,k)) + 0 * (Radix(k) |^ (n+1)) by A5
,A50,RADIX_1:18;
hence thesis by A8,Th12,NAT_1:11;
end;