:: Difference and Difference Quotient
:: by Bo Li , Yan Zhang and Xiquan Liang
::
:: Received September 29, 2006
:: Copyright (c) 2006-2018 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, SUBSET_1, REAL_1, PARTFUN1, SEQFUNC, VALUED_1, RELAT_1,
ARYTM_1, MEMBER_1, FUNCT_1, ARYTM_3, TARSKI, XBOOLE_0, CARD_1, NAT_1,
VALUED_0, ZFMISC_1, SEQ_1, FUNCOP_1, XXREAL_0, CARD_3, SERIES_1, DIFF_1,
FUNCT_2, FUNCT_7, ASYMPT_1;
notations ORDINAL1, XCMPLX_0, XREAL_0, REAL_1, SERIES_1, XXREAL_0, TARSKI,
XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, FUNCT_1, PARTFUN1, NAT_1,
VALUED_0, VALUED_1, SEQ_1, FUNCOP_1, NAT_D, NEWTON, MEASURE6, FUNCT_2,
RELSET_1, SEQFUNC;
constructors REAL_1, SEQFUNC, NEWTON, SERIES_1, MEASURE6, NAT_D, PARTFUN3,
FUNCOP_1, RELSET_1, SEQ_1;
registrations XBOOLE_0, ORDINAL1, PARTFUN1, NUMBERS, XREAL_0, MEMBERED,
VALUED_0, VALUED_1, FUNCT_2, RELSET_1, NAT_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
equalities VALUED_1;
theorems FUNCT_2, PARTFUN1, FUNCT_1, XBOOLE_0, XCMPLX_1, NEWTON, FUNCOP_1,
SERIES_1, TARSKI, MEASURE6, RFUNCT_1, ZFMISC_1, VALUED_1, XREAL_1,
XREAL_0, NUMBERS, XBOOLE_1, ORDINAL1;
schemes NAT_1, RECDEF_1, SEQ_1;
begin
reserve n,m,k,i for Nat,
h,r,r1,r2,x0,x1,x2,x for Real,
S for Functional_Sequence of REAL,REAL,
y for set;
definition
let f be PartFunc of REAL,REAL;
let h be Real;
func Shift(f,h) -> PartFunc of REAL,REAL means
:Def1:
dom it = -h ++ dom f & for x st x in (-h ++ dom f) holds it.x = f.(x+h);
existence
proof
deffunc F(Real) = In(f.($1+h),REAL);
set X = -h ++ dom f;
defpred P[set] means $1 in X;
consider F being PartFunc of REAL,REAL such that
A1: (for x be Element of REAL holds x in dom F iff P[x]) & for x be
Element of REAL st x in dom F holds F.x = F(x) from SEQ_1:sch 3;
take F;
for y being object st y in X holds y in dom F by A1;
then
A2: X c= dom F by TARSKI:def 3;
for y being object st y in dom F holds y in X by A1;
then dom F c= X by TARSKI:def 3;
hence dom F = X by A2,XBOOLE_0:def 10;
for x st x in X holds F.x = f.(x+h)
proof let x;
assume x in X;
then F.x = F(x) by A1;
hence thesis;
end;
hence thesis;
end;
uniqueness
proof
let f1,f2 be PartFunc of REAL,REAL;
set X = -h ++ dom f;
assume that
A3: dom f1 = X and
A4: for x st x in X holds f1.x = f.(x+h) and
A5: dom f2 = X and
A6: for x st x in X holds f2.x = f.(x+h);
for x being Element of REAL st x in dom f1 holds f1.x = f2.x
proof
let x being Element of REAL;
assume
A7: x in dom f1;
then f1.x = f.(x+h) by A3,A4;
hence thesis by A3,A6,A7;
end;
hence f1 = f2 by A3,A5,PARTFUN1:5;
end;
end;
definition
let f be Function of REAL,REAL;
let h be Real;
redefine func Shift(f,h) -> Function of REAL,REAL means
:Def2:
for x holds it.x = f.(x+h);
coherence
proof
dom Shift(f,h) = -h ++ dom f & dom f = REAL by Def1,FUNCT_2:def 1;
then dom Shift(f,h) = REAL by MEASURE6:24;
hence thesis by FUNCT_2:def 1;
end;
compatibility
proof
for IT being Function of REAL,REAL holds IT = Shift(f,h) iff for x
holds IT.x = f.(x+h)
proof
let IT be Function of REAL,REAL;
hereby
assume
A1: IT = Shift(f,h);
let x;
dom Shift(f,h) = REAL by A1,FUNCT_2:def 1;
then x in dom Shift(f,h) by XREAL_0:def 1;
then x in (-h ++ dom f) by Def1;
hence IT.x = f.(x+h) by A1,Def1;
end;
dom Shift(f,h) = -h ++ dom f & dom f = REAL by Def1,FUNCT_2:def 1;
then dom Shift(f,h) = REAL by MEASURE6:24;
then
A2: dom IT = dom Shift(f,h) by FUNCT_2:def 1;
assume
A3: for x holds IT.x = f.(x+h);
for x being Element of REAL st x in dom IT holds IT.x = Shift(f,h).x
proof
let x be Element of REAL;
assume x in dom IT; then
A4: x in (-h ++ dom f) by A2,Def1;
IT.x = f.(x+h) by A3;
hence thesis by A4,Def1;
end;
hence thesis by A2,PARTFUN1:5;
end;
hence thesis;
end;
end;
definition
let f be PartFunc of REAL,REAL, h be Real;
func fD(f,h) -> PartFunc of REAL,REAL equals
Shift(f,h) - f;
coherence;
end;
registration
let f be Function of REAL,REAL, h be Real;
cluster fD(f,h) -> quasi_total;
coherence
proof
dom fD(f,h) = dom Shift(f,h) /\ dom f by VALUED_1:12
.= REAL /\ dom f by FUNCT_2:def 1
.= REAL /\ REAL by FUNCT_2:def 1
.= REAL;
hence thesis by FUNCT_2:def 1;
end;
end;
definition
let f be PartFunc of REAL,REAL, h be Real;
func bD(f,h) -> PartFunc of REAL,REAL equals
f - Shift(f,-h);
coherence;
end;
registration
let f be Function of REAL,REAL, h be Real;
cluster bD(f,h) -> quasi_total;
coherence
proof
dom bD(f,h) = dom Shift(f,-h) /\ dom f by VALUED_1:12
.= REAL /\ dom f by FUNCT_2:def 1
.= REAL /\ REAL by FUNCT_2:def 1
.= REAL;
hence thesis by FUNCT_2:def 1;
end;
end;
definition
let f be PartFunc of REAL,REAL, h be Real;
func cD(f,h) -> PartFunc of REAL,REAL equals
Shift(f,h/2) - Shift(f,-h/2);
coherence;
end;
registration
let f be Function of REAL,REAL, h be Real;
cluster cD(f,h) -> quasi_total;
coherence
proof
dom cD(f,h) = dom Shift(f,h/2) /\ dom Shift(f,-h/2) by VALUED_1:12
.= REAL /\ dom Shift(f,-h/2) by FUNCT_2:def 1
.= REAL /\ REAL by FUNCT_2:def 1
.= REAL;
hence thesis by FUNCT_2:def 1;
end;
end;
definition
let f be PartFunc of REAL,REAL;
let h be Real;
func forward_difference(f,h) -> Functional_Sequence of REAL,REAL means
:Def6:
it.0=f & for n being Nat holds it.(n+1)=fD(it.n,h);
existence
proof
reconsider fZ = f as Element of PFuncs(REAL,REAL) by PARTFUN1:45;
defpred R[set,set,set] means ex g being PartFunc of REAL, REAL st $2 = g &
$3 = fD(g,h);
A1: for n be Nat
for x being Element of PFuncs(REAL,REAL) ex y being Element of
PFuncs(REAL,REAL) st R[n,x,y]
proof
let n be Nat;
let x be Element of PFuncs (REAL,REAL);
reconsider x9 = x as PartFunc of REAL,REAL by PARTFUN1:46;
reconsider y = fD(x9,h) as Element of PFuncs (REAL,REAL) by PARTFUN1:45;
ex w being PartFunc of REAL, REAL st x = w & y = fD(w,h);
hence thesis;
end;
consider g be sequence of PFuncs (REAL,REAL) such that
A2: g.0 = fZ & for n being Nat holds R[n,g.n,g.(n+1)] from
RECDEF_1:sch 2(A1);
reconsider g as Functional_Sequence of REAL,REAL;
take g;
thus g.0 = f by A2;
let i be Nat;
R[i,g.i,g.(i+1)] by A2;
hence thesis;
end;
uniqueness
proof
let seq1,seq2 be Functional_Sequence of REAL,REAL;
assume that
A3: seq1.0=f and
A4: for n be Nat holds seq1.(n+1)=fD(seq1.n,h) and
A5: seq2.0=f and
A6: for n be Nat holds seq2.(n+1)=fD(seq2.n,h);
defpred P[Nat] means seq1.$1 = seq2.$1;
A7: for k st P[k] holds P[k+1]
proof
let k;
assume
A8: P[k];
thus seq1.(k+1) = fD(seq1.k,h) by A4
.= seq2.(k+1) by A6,A8;
end;
A9: P[0] by A3,A5;
for n holds P[n] from NAT_1:sch 2(A9,A7);
then for n being Element of NAT holds P[n];
hence seq1 = seq2 by FUNCT_2:63;
end;
end;
notation
let f be PartFunc of REAL,REAL;
let h be Real;
synonym fdif(f,h) for forward_difference(f,h);
end;
reserve f,f1,f2 for Function of REAL,REAL;
theorem
for f being PartFunc of REAL, REAL st x in dom f & x+h in dom f holds
fD(f,h).x = f.(x+h) - f.x
proof
let f be PartFunc of REAL, REAL;
assume
A1: x in dom f & x+h in dom f;
A2: dom Shift(f,h) = -h ++ dom f by Def1;
A3: (-h) + (x + h) in ((-h) ++ dom f) by A1,MEASURE6:46; then
A4: Shift(f,h).x = f.(x+h) by Def1;
x in (dom Shift(f,h)) /\ dom f by A3,A2,A1,XBOOLE_0:def 4; then
x in dom fD(f,h) by VALUED_1:12;
hence fD(f,h).x = f.(x+h) - f.x by A4,VALUED_1:13;
end;
theorem Th2:
for n be Nat holds fdif(f,h).n is Function of REAL,REAL
proof
defpred X[Nat] means fdif(f,h).$1 is Function of REAL,REAL;
A1: for k be Nat st X[k] holds X[k+1]
proof
let k be Nat;
assume fdif(f,h).k is Function of REAL,REAL;
then fD(fdif(f,h).k,h) is Function of REAL,REAL;
hence thesis by Def6;
end;
A2: X[0] by Def6;
for n be Nat holds X[n] from NAT_1:sch 2(A2,A1);
hence thesis;
end;
theorem Th3:
fD(f,h).x = f.(x+h) - f.x
proof
reconsider xx=x as Element of REAL by XREAL_0:def 1;
dom (Shift(f,h) - f) = REAL by FUNCT_2:def 1;
hence fD(f,h).x = Shift(f,h).xx - f.xx by VALUED_1:13
.= f.(x+h) - f.x by Def2;
end;
theorem Th4:
bD(f,h).x = f.x - f.(x-h)
proof
thus bD(f,h).x = (-fD(f,-h)).x by RFUNCT_1:19
.= - fD(f,-h).x by VALUED_1:8
.= - (f.(x+-h) - f.x) by Th3
.= f.x - f.(x-h);
end;
theorem Th5:
cD(f,h).x = f.(x+h/2) - f.(x-h/2)
proof
reconsider xx=x as Element of REAL by XREAL_0:def 1;
dom (Shift(f,h/2) - Shift(f,-h/2)) = REAL by FUNCT_2:def 1;
hence cD(f,h).x = Shift(f,h/2).xx - Shift(f,-h/2).xx by VALUED_1:13
.= f.(x+h/2) - Shift(f,-h/2).x by Def2
.= f.(x+h/2) - f.(x+-h/2) by Def2
.= f.(x+h/2) - f.(x-h/2);
end;
theorem
f is constant implies for x holds fdif(f,h).(n+1).x=0
proof
assume
A1: f is constant;
A2: for x holds f.(x+h) - f.x = 0
proof
let x;
x+h in REAL by XREAL_0:def 1; then
A3: x+h in dom f by FUNCT_2:def 1;
x in REAL by XREAL_0:def 1;
then x in dom f by FUNCT_2:def 1;
then f.x = f.(x+h) by A1,A3,FUNCT_1:def 10;
hence thesis;
end;
for x holds fdif(f,h).(n+1).x=0
proof
defpred X[Nat] means for x holds fdif(f,h).($1+1).x=0;
A4: for k st X[k] holds X[k+1]
proof
let k;
assume
A5: for x holds fdif(f,h).(k+1).x=0;
let x;
A6: fdif(f,h).(k+1).(x+h)=0 by A5;
A7: fdif(f,h).(k+1) is Function of REAL,REAL by Th2;
(fdif(f,h).(k+2)).x = (fdif(f,h).(k+1+1)).x
.= fD(fdif(f,h).(k+1),h).x by Def6
.= fdif(f,h).(k+1).(x+h) - fdif(f,h).(k+1).x by A7,Th3
.= 0 - 0 by A5,A6
.= 0;
hence thesis;
end;
A8: X[0]
proof
let x;
thus fdif(f,h).(0+1).x = fD(fdif(f,h).0,h).x by Def6
.= fD(f,h).x by Def6
.= f.(x+h) - f.x by Th3
.= 0 by A2;
end;
for n holds X[n] from NAT_1:sch 2(A8,A4);
hence thesis;
end;
hence thesis;
end;
theorem Th7:
fdif(r(#)f,h).(n+1).x = r* fdif(f,h).(n+1).x
proof
defpred X[Nat] means
for x holds fdif(r(#)f,h).($1+1).x = r* fdif(f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds fdif(r(#)f,h).(k+1).x = r* fdif(f,h).(k+1).x;
let x;
A3: fdif(r(#)f,h).(k+1).x = r* fdif(f,h).(k+1).x & fdif(r(#)f,h).(k+1).(x+
h) = r * fdif(f,h).(k+1).(x+h) by A2;
A4: fdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th2;
A5: fdif(f,h).(k+1) is Function of REAL,REAL by Th2;
fdif(r(#)f,h).(k+1+1).x = fD(fdif(r(#)f,h).(k+1),h).x by Def6
.= r * fdif(f,h).(k+1).(x+h) - r * fdif(f,h).(k+1).x by A3,A4,Th3
.= r * (fdif(f,h).(k+1).(x+h) - fdif(f,h).(k+1).x)
.= r * fD(fdif(f,h).(k+1),h).x by A5,Th3
.= r * fdif(f,h).(k+1+1).x by Def6;
hence thesis;
end;
A6: X[0]
proof
let x;
x in REAL by XREAL_0:def 1; then
A7: x in dom (r(#)f) by FUNCT_2:def 1;
x+h in REAL by XREAL_0:def 1; then
A8: x+h in dom (r(#)f) by FUNCT_2:def 1;
fdif(r(#)f,h).(0+1).x = fD(fdif(r(#)f,h).0,h).x by Def6
.= fD(r(#)f,h).x by Def6
.= (r(#)f).(x+h) - (r(#)f).x by Th3
.= r * f.(x+h) - (r(#)f).x by A8,VALUED_1:def 5
.= r * f.(x+h) - r * f.x by A7,VALUED_1:def 5
.= r * (f.(x+h) - f.x)
.= r * fD(f,h).x by Th3
.= r * fD(fdif(f,h).0,h).x by Def6
.= r * fdif(f,h).(0+1).x by Def6;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A6,A1);
hence thesis;
end;
theorem Th8:
fdif(f1+f2,h).(n+1).x = fdif(f1,h).(n+1).x + fdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds fdif(f1+f2,h).($1+1).x = fdif(f1
,h).($1+1).x + fdif(f2,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds fdif(f1+f2,h).(k+1).x = fdif(f1,h).(k+1).x + fdif(f2,h
).(k+1).x;
let x;
A3: fdif(f1+f2,h).(k+1).x = fdif(f1,h).(k+1).x + fdif(f2,h).(k+1).x & fdif
(f1+f2,h).(k+1).(x+h) = fdif(f1,h).(k+1).(x+h) + fdif(f2,h).(k+1).(x+h) by A2;
A4: fdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th2;
A5: fdif(f2,h).(k+1) is Function of REAL,REAL by Th2;
A6: fdif(f1,h).(k+1) is Function of REAL,REAL by Th2;
fdif(f1+f2,h).(k+1+1).x = fD(fdif(f1+f2,h).(k+1),h).x by Def6
.= fdif(f1+f2,h).(k+1).(x+h) - fdif(f1+f2,h).(k+1).x by A4,Th3
.= (fdif(f1,h).(k+1).(x+h) - fdif(f1,h).(k+1).x) + (fdif(f2,h).(k+1).(
x+h) - fdif(f2,h).(k+1).x) by A3
.= fD(fdif(f1,h).(k+1),h).x + (fdif(f2,h).(k+1).(x+h) - fdif(f2,h).(k+
1).x) by A6,Th3
.= fD(fdif(f1,h).(k+1),h).x + fD(fdif(f2,h).(k+1),h).x by A5,Th3
.= fdif(f1,h).(k+1+1).x + fD(fdif(f2,h).(k+1),h).x by Def6
.= fdif(f1,h).(k+1+1).x + fdif(f2,h).(k+1+1).x by Def6;
hence thesis;
end;
A7: X[0]
proof
let x;
reconsider xx=x, h as Element of REAL by XREAL_0:def 1;
fdif(f1+f2,h).(0+1).x = fD(fdif(f1+f2,h).0,h).x by Def6
.= fD(f1+f2,h).x by Def6
.= (f1+f2).(x+h) - (f1+f2).x by Th3
.= f1.(xx+h) + f2.(xx+h) - (f1+f2).xx by VALUED_1:1
.= f1.(x+h) + f2.(x+h) - (f1.x + f2.x) by VALUED_1:1
.= (f1.(x+h) - f1.x) + (f2.(x+h) - f2.x)
.= fD(f1,h).x + (f2.(x+h) - f2.x) by Th3
.= fD(f1,h).x + fD(f2,h).x by Th3
.= fD(fdif(f1,h).0,h).x + fD(f2,h).x by Def6
.= fD(fdif(f1,h).0,h).x + fD(fdif(f2,h).0,h).x by Def6
.= fdif(f1,h).(0+1).x + fD(fdif(f2,h).0,h).x by Def6
.= fdif(f1,h).(0+1).x + fdif(f2,h).(0+1).x by Def6;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A7,A1);
hence thesis;
end;
theorem
fdif(f1-f2,h).(n+1).x = fdif(f1,h).(n+1).x - fdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds fdif(f1-f2,h).($1+1).x = fdif(f1
,h).($1+1).x - fdif(f2,h).($1+1).x;
A1: X[0]
proof
let x;
x in REAL by XREAL_0:def 1;
then x in dom f1 & x in dom f2 by FUNCT_2:def 1;
then x in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A2: x in dom (f1-f2) by VALUED_1:12;
x+h in REAL by XREAL_0:def 1;
then x+h in dom f1 & x+h in dom f2 by FUNCT_2:def 1;
then x+h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A3: x+h in dom (f1-f2) by VALUED_1:12;
fdif(f1-f2,h).(0+1).x = fD(fdif(f1-f2,h).0,h).x by Def6
.= fD(f1-f2,h).x by Def6
.= (f1-f2).(x+h) - (f1-f2).x by Th3
.= f1.(x+h) - f2.(x+h) - (f1-f2).x by A3,VALUED_1:13
.= f1.(x+h) - f2.(x+h) - (f1.x - f2.x) by A2,VALUED_1:13
.= (f1.(x+h) - f1.x) - (f2.(x+h) - f2.x)
.= fD(f1,h).x - (f2.(x+h) - f2.x) by Th3
.= fD(f1,h).x - fD(f2,h).x by Th3
.= fD(fdif(f1,h).0,h).x - fD(f2,h).x by Def6
.= fD(fdif(f1,h).0,h).x - fD(fdif(f2,h).0,h).x by Def6
.= fdif(f1,h).(0+1).x - fD(fdif(f2,h).0,h).x by Def6
.= fdif(f1,h).(0+1).x - fdif(f2,h).(0+1).x by Def6;
hence thesis;
end;
A4: for k st X[k] holds X[k+1]
proof
let k;
assume
A5: for x holds fdif(f1-f2,h).(k+1).x = fdif(f1,h).(k+1).x - fdif(f2,
h).(k+1).x;
let x;
A6: fdif(f1-f2,h).(k+1).x = fdif(f1,h).(k+1).x - fdif(f2,h).(k+1).x &
fdif(f1-f2,h).(k+1).(x+h) = fdif(f1,h).(k+1).(x+h) - fdif(f2,h).(k+1).(x+h) by
A5;
A7: fdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th2;
A8: fdif(f2,h).(k+1) is Function of REAL,REAL by Th2;
A9: fdif(f1,h).(k+1) is Function of REAL,REAL by Th2;
fdif(f1-f2,h).(k+1+1).x = fD(fdif(f1-f2,h).(k+1),h).x by Def6
.= fdif(f1-f2,h).(k+1).(x+h) - fdif(f1-f2,h).(k+1).x by A7,Th3
.= (fdif(f1,h).(k+1).(x+h) - fdif(f1,h).(k+1).x) - (fdif(f2,h).(k+1).(
x+h) - fdif(f2,h).(k+1).x) by A6
.= fD(fdif(f1,h).(k+1),h).x - (fdif(f2,h).(k+1).(x+h) - fdif(f2,h).(k+
1).x) by A9,Th3
.= fD(fdif(f1,h).(k+1),h).x - fD(fdif(f2,h).(k+1),h).x by A8,Th3
.= fdif(f1,h).(k+1+1).x - fD(fdif(f2,h).(k+1),h).x by Def6
.= fdif(f1,h).(k+1+1).x - fdif(f2,h).(k+1+1).x by Def6;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A1,A4);
hence thesis;
end;
theorem
fdif(r1(#)f1+r2(#)f2,h).(n+1).x = r1* fdif(f1,h).(n+1).x + r2* fdif(f2
,h).(n+1).x
proof
set g1=r1(#)f1;
set g2=r2(#)f2;
fdif(r1(#)f1+r2(#)f2,h).(n+1).x = fdif(g1,h).(n+1).x + fdif(g2,h).(n+1).
x by Th8
.= r1* fdif(f1,h).(n+1).x + fdif(g2,h).(n+1).x by Th7
.= r1* fdif(f1,h).(n+1).x + r2* fdif(f2,h).(n+1).x by Th7;
hence thesis;
end;
theorem
(fdif(f,h).1).x = Shift(f,h).x - f.x
proof
set f1 = Shift(f,h);
(fdif(f,h).1).x = fdif(f,h).(0+1).x .= fD(fdif(f,h).0,h).x by Def6
.= fD(f,h).x by Def6
.= f.(x+h) - f.x by Th3
.= f1.x - f.x by Def2;
hence thesis;
end;
definition
let f be PartFunc of REAL,REAL;
let h be Real;
func backward_difference(f,h) -> Functional_Sequence of REAL,REAL means
:Def7: it.0=f & for n being Nat holds it.(n+1)=bD(it.n,h);
existence
proof
reconsider fZ = f as Element of PFuncs(REAL,REAL) by PARTFUN1:45;
defpred R[set,set,set] means ex g being PartFunc of REAL, REAL st $2 = g &
$3 = bD(g,h);
A1: for n being Nat
for x being Element of PFuncs(REAL,REAL) ex y being Element of
PFuncs(REAL,REAL) st R[n,x,y]
proof
let n be Nat;
let x be Element of PFuncs (REAL,REAL);
reconsider x9 = x as PartFunc of REAL,REAL by PARTFUN1:46;
reconsider y = bD(x9,h) as Element of PFuncs (REAL,REAL) by PARTFUN1:45;
ex w being PartFunc of REAL, REAL st x = w & y = bD(w,h);
hence thesis;
end;
consider g be sequence of PFuncs (REAL,REAL) such that
A2: g.0 = fZ & for n being Nat holds R[n,g.n,g.(n+1)] from
RECDEF_1:sch 2(A1);
reconsider g as Functional_Sequence of REAL,REAL;
take g;
thus g.0 = f by A2;
let i be Nat;
R[i,g.i,g.(i+1)] by A2;
hence thesis;
end;
uniqueness
proof
let seq1,seq2 be Functional_Sequence of REAL,REAL;
assume that
A3: seq1.0=f and
A4: for n be Nat holds seq1.(n+1)=bD(seq1.n,h) and
A5: seq2.0=f and
A6: for n be Nat holds seq2.(n+1)=bD(seq2.n,h);
defpred P[Nat] means seq1.$1 = seq2.$1;
A7: for k st P[k] holds P[k+1]
proof
let k;
assume
A8: P[k];
thus seq1.(k+1) = bD(seq1.k,h) by A4
.= seq2.(k+1) by A6,A8;
end;
A9: P[0] by A3,A5;
for n holds P[n] from NAT_1:sch 2(A9,A7);
then for n being Element of NAT holds P[n];
hence seq1 = seq2 by FUNCT_2:63;
end;
end;
notation
let f be PartFunc of REAL,REAL;
let h be Real;
synonym bdif(f,h) for backward_difference(f,h);
end;
theorem Th12:
for n be Nat holds bdif(f,h).n is Function of REAL,REAL
proof
defpred X[Nat] means bdif(f,h).$1 is Function of REAL,REAL;
A1: for k be Nat st X[k] holds X[k+1]
proof
let k be Nat;
assume bdif(f,h).k is Function of REAL,REAL;
then bD(bdif(f,h).k,h) is Function of REAL,REAL;
hence thesis by Def7;
end;
A2: X[0] by Def7;
for n be Nat holds X[n] from NAT_1:sch 2(A2,A1);
hence thesis;
end;
theorem
f is constant implies for x holds bdif(f,h).(n+1).x=0
proof
assume
A1: f is constant;
A2: for x holds f.x - f.(x-h) = 0
proof
let x;
x-h in REAL by XREAL_0:def 1;
then
A3: x-h in dom f by FUNCT_2:def 1;
x in REAL by XREAL_0:def 1;
then x in dom f by FUNCT_2:def 1;
then f.x = f.(x-h) by A1,A3,FUNCT_1:def 10;
hence thesis;
end;
for x holds bdif(f,h).(n+1).x=0
proof
defpred X[Nat] means for x holds bdif(f,h).($1+1).x=0;
A4: for k st X[k] holds X[k+1]
proof
let k;
assume
A5: for x holds bdif(f,h).(k+1).x=0;
let x;
A6: bdif(f,h).(k+1).(x-h)=0 by A5;
A7: bdif(f,h).(k+1) is Function of REAL,REAL by Th12;
(bdif(f,h).(k+2)).x = (bdif(f,h).(k+1+1)).x
.= bD(bdif(f,h).(k+1),h).x by Def7
.= bdif(f,h).(k+1).x - bdif(f,h).(k+1).(x-h) by A7,Th4
.= 0 by A5,A6;
hence thesis;
end;
A8: X[0]
proof
let x;
thus bdif(f,h).(0+1).x = bD(bdif(f,h).0,h).x by Def7
.= bD(f,h).x by Def7
.= f.x - f.(x-h) by Th4
.= 0 by A2;
end;
for n holds X[n] from NAT_1:sch 2(A8,A4);
hence thesis;
end;
hence thesis;
end;
theorem Th14:
bdif(r(#)f,h).(n+1).x = r* bdif(f,h).(n+1).x
proof
defpred X[Nat] means
for x holds bdif(r(#)f,h).($1+1).x = r* bdif
(f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds bdif(r(#)f,h).(k+1).x = r* bdif(f,h).(k+1).x;
let x;
A3: bdif(r(#)f,h).(k+1).x = r* bdif(f,h).(k+1).x & bdif(r(#)f,h).(k+1).(x-
h) = r * bdif(f,h).(k+1).(x-h) by A2;
A4: bdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th12;
A5: bdif(f,h).(k+1) is Function of REAL,REAL by Th12;
bdif(r(#)f,h).(k+1+1).x = bD(bdif(r(#)f,h).(k+1),h).x by Def7
.= bdif(r(#)f,h).(k+1).x - bdif(r(#)f,h).(k+1).(x-h) by A4,Th4
.= r * (bdif(f,h).(k+1).x - bdif(f,h).(k+1).(x-h)) by A3
.= r * bD(bdif(f,h).(k+1),h).x by A5,Th4
.= r * bdif(f,h).(k+1+1).x by Def7;
hence thesis;
end;
A6: X[0]
proof
let x;
x in REAL by XREAL_0:def 1;
then
A7: x in dom (r(#)f) by FUNCT_2:def 1;
x-h in REAL by XREAL_0:def 1;
then
A8: x-h in dom (r(#)f) by FUNCT_2:def 1;
bdif(r(#)f,h).(0+1).x = bD(bdif(r(#)f,h).0,h).x by Def7
.= bD(r(#)f,h).x by Def7
.= (r(#)f).x - (r(#)f).(x-h) by Th4
.= (r(#)f).x - r * f.(x-h) by A8,VALUED_1:def 5
.= r * f.x - r * f.(x-h) by A7,VALUED_1:def 5
.= r * (f.x - f.(x-h))
.= r * bD(f,h).x by Th4
.= r * bD(bdif(f,h).0,h).x by Def7
.= r * bdif(f,h).(0+1).x by Def7;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A6,A1);
hence thesis;
end;
theorem Th15:
bdif(f1+f2,h).(n+1).x = bdif(f1,h).(n+1).x + bdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds bdif(f1+f2,h).($1+1).x = bdif(f1
,h).($1+1).x + bdif(f2,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds bdif(f1+f2,h).(k+1).x = bdif(f1,h).(k+1).x + bdif(f2,h
).(k+1).x;
let x;
A3: bdif(f1+f2,h).(k+1).x = bdif(f1,h).(k+1).x + bdif(f2,h).(k+1).x & bdif
(f1+f2,h).(k+1).(x-h) = bdif(f1,h).(k+1).(x-h) + bdif(f2,h).(k+1).(x-h) by A2;
A4: bdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th12;
A5: bdif(f2,h).(k+1) is Function of REAL,REAL by Th12;
A6: bdif(f1,h).(k+1) is Function of REAL,REAL by Th12;
bdif(f1+f2,h).(k+1+1).x = bD(bdif(f1+f2,h).(k+1),h).x by Def7
.= bdif(f1+f2,h).(k+1).x - bdif(f1+f2,h).(k+1).(x-h) by A4,Th4
.= (bdif(f1,h).(k+1).x - bdif(f1,h).(k+1).(x-h)) + (bdif(f2,h).(k+1).x
- bdif(f2,h).(k+1).(x-h)) by A3
.= bD(bdif(f1,h).(k+1),h).x + (bdif(f2,h).(k+1).x - bdif(f2,h).(k+1).(
x-h)) by A6,Th4
.= bD(bdif(f1,h).(k+1),h).x + bD(bdif(f2,h).(k+1),h).x by A5,Th4
.= bdif(f1,h).(k+1+1).x + bD(bdif(f2,h).(k+1),h).x by Def7
.= bdif(f1,h).(k+1+1).x + bdif(f2,h).(k+1+1).x by Def7;
hence thesis;
end;
A7: X[0]
proof
let x;
reconsider xx=x, h as Element of REAL by XREAL_0:def 1;
bdif(f1+f2,h).(0+1).x = bD(bdif(f1+f2,h).0,h).x by Def7
.= bD(f1+f2,h).x by Def7
.= (f1+f2).x - (f1+f2).(x-h) by Th4
.= f1.xx + f2.xx - (f1+f2).(xx-h) by VALUED_1:1
.= f1.x + f2.x - (f1.(x-h) + f2.(x-h)) by VALUED_1:1
.= (f1.x - f1.(x-h)) + (f2.x - f2.(x-h))
.= bD(f1,h).x + (f2.x - f2.(x-h)) by Th4
.= bD(f1,h).x + bD(f2,h).x by Th4
.= bD(bdif(f1,h).0,h).x + bD(f2,h).x by Def7
.= bD(bdif(f1,h).0,h).x + bD(bdif(f2,h).0,h).x by Def7
.= bdif(f1,h).(0+1).x + bD(bdif(f2,h).0,h).x by Def7
.= bdif(f1,h).(0+1).x + bdif(f2,h).(0+1).x by Def7;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A7,A1);
hence thesis;
end;
theorem
bdif(f1-f2,h).(n+1).x = bdif(f1,h).(n+1).x - bdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds bdif(f1-f2,h).($1+1).x = bdif(f1
,h).($1+1).x - bdif(f2,h).($1+1).x;
A1: X[0]
proof
let x;
x in REAL by XREAL_0:def 1;
then x in dom f1 & x in dom f2 by FUNCT_2:def 1;
then x in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A2: x in dom (f1-f2) by VALUED_1:12;
x-h in REAL by XREAL_0:def 1;
then x-h in dom f1 & x-h in dom f2 by FUNCT_2:def 1;
then x-h in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A3: x-h in dom (f1-f2) by VALUED_1:12;
bdif(f1-f2,h).(0+1).x = bD(bdif(f1-f2,h).0,h).x by Def7
.= bD(f1-f2,h).x by Def7
.= (f1-f2).x - (f1-f2).(x-h) by Th4
.= f1.x - f2.x - (f1-f2).(x-h) by A2,VALUED_1:13
.= f1.x - f2.x - (f1.(x-h) - f2.(x-h)) by A3,VALUED_1:13
.= (f1.x - f1.(x-h)) - (f2.x - f2.(x-h))
.= bD(f1,h).x - (f2.x - f2.(x-h)) by Th4
.= bD(f1,h).x - bD(f2,h).x by Th4
.= bD(bdif(f1,h).0,h).x - bD(f2,h).x by Def7
.= bD(bdif(f1,h).0,h).x - bD(bdif(f2,h).0,h).x by Def7
.= bdif(f1,h).(0+1).x - bD(bdif(f2,h).0,h).x by Def7
.= bdif(f1,h).(0+1).x - bdif(f2,h).(0+1).x by Def7;
hence thesis;
end;
A4: for k st X[k] holds X[k+1]
proof
let k;
assume
A5: for x holds bdif(f1-f2,h).(k+1).x = bdif(f1,h).(k+1).x - bdif(f2,
h).(k+1).x;
let x;
A6: bdif(f1-f2,h).(k+1).x = bdif(f1,h).(k+1).x - bdif(f2,h).(k+1).x &
bdif(f1-f2,h).(k+1).(x-h) = bdif(f1,h).(k+1).(x-h) - bdif(f2,h).(k+1).(x-h) by
A5;
A7: bdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th12;
A8: bdif(f2,h).(k+1) is Function of REAL,REAL by Th12;
A9: bdif(f1,h).(k+1) is Function of REAL,REAL by Th12;
bdif(f1-f2,h).(k+1+1).x = bD(bdif(f1-f2,h).(k+1),h).x by Def7
.= bdif(f1-f2,h).(k+1).x - bdif(f1-f2,h).(k+1).(x-h) by A7,Th4
.= (bdif(f1,h).(k+1).x - bdif(f1,h).(k+1).(x-h)) - (bdif(f2,h).(k+1).x
- bdif(f2,h).(k+1).(x-h)) by A6
.= bD(bdif(f1,h).(k+1),h).x - (bdif(f2,h).(k+1).x - bdif(f2,h).(k+1).(
x-h)) by A9,Th4
.= bD(bdif(f1,h).(k+1),h).x - bD(bdif(f2,h).(k+1),h).x by A8,Th4
.= bdif(f1,h).(k+1+1).x - bD(bdif(f2,h).(k+1),h).x by Def7
.= bdif(f1,h).(k+1+1).x - bdif(f2,h).(k+1+1).x by Def7;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A1,A4);
hence thesis;
end;
theorem
bdif(r1(#)f1+r2(#)f2,h).(n+1).x = r1* bdif(f1,h).(n+1).x + r2* bdif(f2
,h).(n+1).x
proof
set g1=r1(#)f1;
set g2=r2(#)f2;
bdif(r1(#)f1+r2(#)f2,h).(n+1).x = bdif(g1,h).(n+1).x + bdif(g2,h).(n+1).
x by Th15
.= r1* bdif(f1,h).(n+1).x + bdif(g2,h).(n+1).x by Th14
.= r1* bdif(f1,h).(n+1).x + r2* bdif(f2,h).(n+1).x by Th14;
hence thesis;
end;
theorem
(bdif(f,h).1).x = f.x - Shift(f,-h).x
proof
set f2 = Shift(f,-h);
(bdif(f,h).1).x = bdif(f,h).(0+1).x .= bD(bdif(f,h).0,h).x by Def7
.= bD(f,h).x by Def7
.= f.x - f.(x-h) by Th4
.= f.x - f.(x+-h)
.= f.x - f2.x by Def2;
hence thesis;
end;
definition
let f be PartFunc of REAL,REAL;
let h be Real;
func central_difference(f,h) -> Functional_Sequence of REAL,REAL means
:Def8:
it.0=f & for n be Nat holds it.(n+1)=cD(it.n,h);
existence
proof
reconsider fZ = f as Element of PFuncs(REAL,REAL) by PARTFUN1:45;
defpred R[set,set,set] means ex g being PartFunc of REAL, REAL st $2 = g &
$3 = cD(g,h);
A1: for n being Nat
for x being Element of PFuncs(REAL,REAL) ex y being Element of
PFuncs(REAL,REAL) st R[n,x,y]
proof
let n be Nat;
let x be Element of PFuncs (REAL,REAL);
reconsider x9 = x as PartFunc of REAL,REAL by PARTFUN1:46;
reconsider y = cD(x9,h) as Element of PFuncs (REAL,REAL) by PARTFUN1:45;
ex w being PartFunc of REAL, REAL st x = w & y = cD(w,h);
hence thesis;
end;
consider g be sequence of PFuncs (REAL,REAL) such that
A2: g.0 = fZ & for n being Nat holds R[n,g.n,g.(n+1)] from
RECDEF_1:sch 2(A1);
reconsider g as Functional_Sequence of REAL,REAL;
take g;
thus g.0 = f by A2;
let i be Nat;
R[i,g.i,g.(i+1)] by A2;
hence thesis;
end;
uniqueness
proof
let seq1,seq2 be Functional_Sequence of REAL,REAL;
assume that
A3: seq1.0=f and
A4: for n be Nat holds seq1.(n+1)=cD(seq1.n,h) and
A5: seq2.0=f and
A6: for n be Nat holds seq2.(n+1)=cD(seq2.n,h);
defpred P[Nat] means seq1.$1 = seq2.$1;
A7: for k st P[k] holds P[k+1]
proof
let k;
assume
A8: P[k];
thus seq1.(k+1) = cD(seq1.k,h) by A4
.= seq2.(k+1) by A6,A8;
end;
A9: P[0] by A3,A5;
for n holds P[n] from NAT_1:sch 2(A9,A7);
then for n being Element of NAT holds P[n];
hence seq1 = seq2 by FUNCT_2:63;
end;
end;
notation
let f be PartFunc of REAL,REAL;
let h be Real;
synonym cdif(f,h) for central_difference(f,h);
end;
theorem Th19:
for n being Nat holds cdif(f,h).n is Function of REAL,REAL
proof
defpred X[Nat] means cdif(f,h).$1 is Function of REAL,REAL;
A1: for k be Nat st X[k] holds X[k+1]
proof
let k be Nat;
assume cdif(f,h).k is Function of REAL,REAL;
then cD(cdif(f,h).k,h) is Function of REAL,REAL;
hence thesis by Def8;
end;
A2: X[0] by Def8;
for n be Nat holds X[n] from NAT_1:sch 2(A2,A1);
hence thesis;
end;
theorem
f is constant implies for x holds cdif(f,h).(n+1).x=0
proof
defpred X[Nat] means for x holds cdif(f,h).($1+1).x=0;
assume
A1: f is constant;
A2: for x holds f.(x+h/2) - f.(x-h/2) = 0
proof
let x;
x-h/2 in REAL by XREAL_0:def 1;
then
A3: x-h/2 in dom f by FUNCT_2:def 1;
x+h/2 in REAL by XREAL_0:def 1;
then x+h/2 in dom f by FUNCT_2:def 1;
then f.(x+h/2) = f.(x-h/2) by A1,A3,FUNCT_1:def 10;
hence thesis;
end;
A4: X[0]
proof
let x;
thus cdif(f,h).(0+1).x = cD(cdif(f,h).0,h).x by Def8
.= cD(f,h).x by Def8
.= f.(x+h/2) - f.(x-h/2) by Th5
.= 0 by A2;
end;
A5: for k st X[k] holds X[k+1]
proof
let k;
assume
A6: for x holds cdif(f,h).(k+1).x=0;
let x;
A7: cdif(f,h).(k+1).(x-h/2)=0 by A6;
A8: cdif(f,h).(k+1) is Function of REAL,REAL by Th19;
(cdif(f,h).(k+2)).x = (cdif(f,h).(k+1+1)).x
.= cD(cdif(f,h).(k+1),h).x by Def8
.= cdif(f,h).(k+1).(x+h/2) - cdif(f,h).(k+1).(x-h/2) by A8,Th5
.= 0 by A6,A7;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A4,A5);
hence thesis;
end;
theorem Th21:
cdif(r(#)f,h).(n+1).x = r* cdif(f,h).(n+1).x
proof
defpred X[Nat] means
for x holds cdif(r(#)f,h).($1+1).x = r* cdif(f,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds cdif(r(#)f,h).(k+1).x = r* cdif(f,h).(k+1).x;
let x;
A3: cdif(r(#)f,h).(k+1).(x-h/2) = r* cdif(f,h).(k+1).(x-h/2) & cdif(r(#)f,
h).(k+ 1).(x+h/2) = r* cdif(f,h).(k+1).(x+h/2) by A2;
A4: cdif(r(#)f,h).(k+1) is Function of REAL,REAL by Th19;
A5: cdif(f,h).(k+1) is Function of REAL,REAL by Th19;
cdif(r(#)f,h).(k+1+1).x = cD(cdif(r(#)f,h).(k+1),h).x by Def8
.= cdif(r(#)f,h).(k+1).(x+h/2) - cdif(r(#)f,h).(k+1).(x-h/2) by A4,Th5
.= r * (cdif(f,h).(k+1).(x+h/2) - cdif(f,h).(k+1).(x-h/2)) by A3
.= r * cD(cdif(f,h).(k+1),h).x by A5,Th5
.= r * cdif(f,h).(k+1+1).x by Def8;
hence thesis;
end;
A6: X[0]
proof
let x;
x+h/2 in REAL by XREAL_0:def 1;
then
A7: x+h/2 in dom (r(#)f) by FUNCT_2:def 1;
x-h/2 in REAL by XREAL_0:def 1;
then
A8: x-h/2 in dom (r(#)f) by FUNCT_2:def 1;
cdif(r(#)f,h).(0+1).x = cD(cdif(r(#)f,h).0,h).x by Def8
.= cD(r(#)f,h).x by Def8
.= (r(#)f).(x+h/2) - (r(#)f).(x-h/2) by Th5
.= r * f.(x+h/2) - (r(#)f).(x-h/2) by A7,VALUED_1:def 5
.= r * f.(x+h/2) - r * f.(x-h/2) by A8,VALUED_1:def 5
.= r * (f.(x+h/2) - f.(x-h/2))
.= r * cD(f,h).x by Th5
.= r * cD(cdif(f,h).0,h).x by Def8
.= r * cdif(f,h).(0+1).x by Def8;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A6,A1);
hence thesis;
end;
theorem Th22:
cdif(f1+f2,h).(n+1).x = cdif(f1,h).(n+1).x + cdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds cdif(f1+f2,h).($1+1).x = cdif(f1
,h).($1+1).x + cdif(f2,h).($1+1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds cdif(f1+f2,h).(k+1).x = cdif(f1,h).(k+1).x + cdif(f2,h
).(k+1).x;
let x;
A3: cdif(f1+f2,h).(k+1).(x-h/2) = cdif(f1,h).(k+1).(x-h/2) + cdif(f2,h).(k
+1).(x- h/2) & cdif(f1+f2,h).(k+1).(x+h/2) = cdif(f1,h).(k+1).(x+h/2) + cdif(f2
,h).(k+ 1).(x+h/2) by A2;
A4: cdif(f1+f2,h).(k+1) is Function of REAL,REAL by Th19;
A5: cdif(f2,h).(k+1) is Function of REAL,REAL by Th19;
A6: cdif(f1,h).(k+1) is Function of REAL,REAL by Th19;
cdif(f1+f2,h).(k+1+1).x = cD(cdif(f1+f2,h).(k+1),h).x by Def8
.= cdif(f1+f2,h).(k+1).(x+h/2) - cdif(f1+f2,h).(k+1).(x-h/2) by A4,Th5
.= (cdif(f1,h).(k+1).(x+h/2) - cdif(f1,h).(k+1).(x-h/2)) + (cdif(f2,h)
.(k+1).(x+h/2) - cdif(f2,h).(k+1).(x-h/2)) by A3
.= cD(cdif(f1,h).(k+1),h).x + (cdif(f2,h).(k+1).(x+h/2) - cdif(f2,h).(
k+1).(x-h/2)) by A6,Th5
.= cD(cdif(f1,h).(k+1),h).x + cD(cdif(f2,h).(k+1),h).x by A5,Th5
.= cdif(f1,h).(k+1+1).x + cD(cdif(f2,h).(k+1),h).x by Def8
.= cdif(f1,h).(k+1+1).x + cdif(f2,h).(k+1+1).x by Def8;
hence thesis;
end;
A7: X[0]
proof
let x;
reconsider xx=x, hp = h/2 as Element of REAL by XREAL_0:def 1;
cdif(f1+f2,h).(0+1).x = cD(cdif(f1+f2,h).0,h).x by Def8
.= cD(f1+f2,h).x by Def8
.= (f1+f2).(x+h/2) - (f1+f2).(x-h/2) by Th5
.= f1.(xx+h/2) + f2.(xx+hp) - (f1+f2).(xx-hp) by VALUED_1:1
.= f1.(x+h/2) + f2.(x+hp) - (f1.(x-h/2) + f2.(x-hp)) by VALUED_1:1
.= (f1.(x+h/2) - f1.(x-h/2)) + (f2.(x+h/2) - f2.(x-h/2))
.= cD(f1,h).x + (f2.(x+h/2) - f2.(x-h/2)) by Th5
.= cD(f1,h).x + cD(f2,h).x by Th5
.= cD(cdif(f1,h).0,h).x + cD(f2,h).x by Def8
.= cD(cdif(f1,h).0,h).x + cD(cdif(f2,h).0,h).x by Def8
.= cdif(f1,h).(0+1).x + cD(cdif(f2,h).0,h).x by Def8
.= cdif(f1,h).(0+1).x + cdif(f2,h).(0+1).x by Def8;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A7,A1);
hence thesis;
end;
theorem
cdif(f1-f2,h).(n+1).x = cdif(f1,h).(n+1).x - cdif(f2,h).(n+1).x
proof
defpred X[Nat] means
for x holds cdif(f1-f2,h).($1+1).x = cdif(f1
,h).($1+1).x - cdif(f2,h).($1+1).x;
A1: X[0]
proof
let x;
x-h/2 in REAL by XREAL_0:def 1;
then x-h/2 in dom f1 & x-h/2 in dom f2 by FUNCT_2:def 1;
then x-h/2 in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A2: x-h/2 in dom (f1-f2) by VALUED_1:12;
x+h/2 in REAL by XREAL_0:def 1;
then x+h/2 in dom f1 & x+h/2 in dom f2 by FUNCT_2:def 1;
then x+h/2 in dom f1 /\ dom f2 by XBOOLE_0:def 4;
then
A3: x+h/2 in dom (f1-f2) by VALUED_1:12;
cdif(f1-f2,h).(0+1).x = cD(cdif(f1-f2,h).0,h).x by Def8
.= cD(f1-f2,h).x by Def8
.= (f1-f2).(x+h/2) - (f1-f2).(x-h/2) by Th5
.= f1.(x+h/2) - f2.(x+h/2) - (f1-f2).(x-h/2) by A3,VALUED_1:13
.= f1.(x+h/2) - f2.(x+h/2) - (f1.(x-h/2) - f2.(x-h/2)) by A2,VALUED_1:13
.= (f1.(x+h/2) - f1.(x-h/2)) - (f2.(x+h/2) - f2.(x-h/2))
.= cD(f1,h).x - (f2.(x+h/2) - f2.(x-h/2)) by Th5
.= cD(f1,h).x - cD(f2,h).x by Th5
.= cD(cdif(f1,h).0,h).x - cD(f2,h).x by Def8
.= cD(cdif(f1,h).0,h).x - cD(cdif(f2,h).0,h).x by Def8
.= cdif(f1,h).(0+1).x - cD(cdif(f2,h).0,h).x by Def8
.= cdif(f1,h).(0+1).x - cdif(f2,h).(0+1).x by Def8;
hence thesis;
end;
A4: for k st X[k] holds X[k+1]
proof
let k;
assume
A5: for x holds cdif(f1-f2,h).(k+1).x = cdif(f1,h).(k+1).x - cdif(f2,
h).(k+1).x;
let x;
A6: cdif(f1-f2,h).(k+1).(x-h/2) = cdif(f1,h).(k+1).(x-h/2) - cdif(f2,h).(
k+1).(x- h/2) & cdif(f1-f2,h).(k+1).(x+h/2) = cdif(f1,h).(k+1).(x+h/2) - cdif(
f2,h).(k+ 1).(x+h/2) by A5;
A7: cdif(f1-f2,h).(k+1) is Function of REAL,REAL by Th19;
A8: cdif(f2,h).(k+1) is Function of REAL,REAL by Th19;
A9: cdif(f1,h).(k+1) is Function of REAL,REAL by Th19;
cdif(f1-f2,h).(k+1+1).x = cD(cdif(f1-f2,h).(k+1),h).x by Def8
.= cdif(f1-f2,h).(k+1).(x+h/2) - cdif(f1-f2,h).(k+1).(x-h/2) by A7,Th5
.= (cdif(f1,h).(k+1).(x+h/2) - cdif(f1,h).(k+1).(x-h/2)) - (cdif(f2,h)
.(k+1).(x+h/2) - cdif(f2,h).(k+1).(x-h/2)) by A6
.= cD(cdif(f1,h).(k+1),h).x - (cdif(f2,h).(k+1).(x+h/2) - cdif(f2,h).(
k+1).(x-h/2)) by A9,Th5
.= cD(cdif(f1,h).(k+1),h).x - cD(cdif(f2,h).(k+1),h).x by A8,Th5
.= cdif(f1,h).(k+1+1).x - cD(cdif(f2,h).(k+1),h).x by Def8
.= cdif(f1,h).(k+1+1).x - cdif(f2,h).(k+1+1).x by Def8;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A1,A4);
hence thesis;
end;
theorem
cdif(r1(#)f1+r2(#)f2,h).(n+1).x = r1* cdif(f1,h).(n+1).x + r2* cdif(f2
,h).(n+1).x
proof
set g1=r1(#)f1;
set g2=r2(#)f2;
cdif(r1(#)f1+r2(#)f2,h).(n+1).x = cdif(g1,h).(n+1).x + cdif(g2,h).(n+1).
x by Th22
.= r1* cdif(f1,h).(n+1).x + cdif(g2,h).(n+1).x by Th21
.= r1* cdif(f1,h).(n+1).x + r2* cdif(f2,h).(n+1).x by Th21;
hence thesis;
end;
theorem
(cdif(f,h).1).x = Shift(f,h/2).x - Shift(f,-h/2).x
proof
set f2 = Shift(f,-h/2);
set f1 = Shift(f,h/2);
(cdif(f,h).1).x = cdif(f,h).(0+1).x .= cD(cdif(f,h).0,h).x by Def8
.= cD(f,h).x by Def8
.= f.(x+h/2) - f.(x-h/2) by Th5
.= f1.x - f.(x+-h/2) by Def2
.= f1.x - f2.x by Def2;
hence thesis;
end;
theorem
(fdif(f,h).n).x = (bdif(f,h).n).(x+n*h)
proof
defpred X[Nat] means for x holds (fdif(f,h).$1).x = (bdif(f,h).$1).(x+$1*h);
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds (fdif(f,h).k).x = (bdif(f,h).k).(x+k*h);
let x;
A3: (fdif(f,h).k).(x+h) = (bdif(f,h).k).(x+h+k*h) by A2;
A4: fdif(f,h).k is Function of REAL,REAL by Th2;
A5: bdif(f,h).k is Function of REAL,REAL by Th12;
(fdif(f,h).(k+1)).x = fD(fdif(f,h).k,h).x by Def6
.= (fdif(f,h).k).(x+h) - (fdif(f,h).k).x by A4,Th3
.= (bdif(f,h).k).(x+h+k*h) - (bdif(f,h).k).(x+k*h) by A2,A3
.= (bdif(f,h).k).(x+(k+1)*h) - (bdif(f,h).k).((x+(k+1)*h)-h)
.= bD(bdif(f,h).k,h).(x+(k+1)*h) by A5,Th4
.= (bdif(f,h).(k+1)).(x+(k+1)*h) by Def7;
hence thesis;
end;
A6: X[0]
proof
let x;
(fdif(f,h).0).x = f.x by Def6
.= (bdif(f,h).0).(x+0*h) by Def7;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A6,A1);
hence thesis;
end;
theorem
(fdif(f,h).(2*n)).x = (cdif(f,h).(2*n)).(x+n*h)
proof
defpred X[Nat] means
for x holds (fdif(f,h).(2*$1)).x = (cdif(f,h).(2*$1)).(x+$1*h);
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds (fdif(f,h).(2*k)).x = cdif(f,h).(2*k).(x+k*h);
let x;
A3: fdif(f,h).(2*k).(x+h+h) = cdif(f,h).(2*k).(x+h+h+k*h) by A2
.= cdif(f,h).(2*k).(x+(k+2)*h);
A4: fdif(f,h).(2*k).(x+h) = cdif(f,h).(2*k).(x+h+k*h) by A2
.= cdif(f,h).(2*k).(x+(k+1)*h);
set r3 = cdif(f,h).(2*k).(x+k*h);
set r2 = cdif(f,h).(2*k).(x+(k+1)*h);
set r1 = cdif(f,h).(2*k).(x+(k+2)*h);
A5: fdif(f,h).(2*k+1) is Function of REAL,REAL by Th2;
A6: cdif(f,h).(2*k) is Function of REAL,REAL by Th19;
A7: cdif(f,h).(2*k+1) is Function of REAL,REAL by Th19;
A8: fdif(f,h).(2*k) is Function of REAL,REAL by Th2;
A9: cdif(f,h).(2*k+1).(x+(k+1)*h-h/2) = cD(cdif(f,h).(2*k),h).(x+(k+1)*h-
h/2) by Def8
.= cdif(f,h).(2*k).(x+(k+1)*h-h/2+h/2) - cdif(f,h).(2*k).(x+(k+1)*h-h/
2-h/2) by A6,Th5
.= cdif(f,h).(2*k).(x+(k+1)*h) - cdif(f,h).(2*k).(x+k*h);
A10: cdif(f,h).(2*k+1).(x+(k+1)*h+h/2) = cD(cdif(f,h).(2*k),h).(x+(k+1)*h+
h/2) by Def8
.= cdif(f,h).(2*k).(x+(k+1)*h+h/2+h/2) - cdif(f,h).(2*k).(x+(k+1)*h+h/
2-h/2) by A6,Th5
.= cdif(f,h).(2*k).(x+(k+2)*h) - cdif(f,h).(2*k).(x+(k+1)*h);
A11: cdif(f,h).(2*(k+1)).(x+(k+1)*h) = cdif(f,h).(2*k+1+1).(x+(k+1)*h)
.= cD(cdif(f,h).(2*k+1),h).(x+(k+1)*h) by Def8
.= (r1 - r2) - (r2 - r3) by A7,A10,A9,Th5;
fdif(f,h).(2*(k+1)).x = fdif(f,h).(2*k+1+1).x
.= fD(fdif(f,h).(2*k+1),h).x by Def6
.= (fdif(f,h).(2*k+1)).(x+h) - (fdif(f,h).(2*k+1)).x by A5,Th3
.= fD(fdif(f,h).(2*k),h).(x+h) - (fdif(f,h).(2*k+1)).x by Def6
.= fD(fdif(f,h).(2*k),h).(x+h) - fD(fdif(f,h).(2*k),h).x by Def6
.= fdif(f,h).(2*k).(x+h+h) - fdif(f,h).(2*k).(x+h) - fD(fdif(f,h).(2*k
),h).x by A8,Th3
.= fdif(f,h).(2*k).(x+h+h) - fdif(f,h).(2*k).(x+h) - (fdif(f,h).(2*k).
(x+h) - fdif(f,h).(2*k).x) by A8,Th3
.= cdif(f,h).(2*k).(x+(k+2)*h) - cdif(f,h).(2*k).(x+(k+1)*h) - (cdif(f
,h).(2*k).(x+(k+1)*h) - cdif(f,h).(2*k).(x+k*h)) by A2,A3,A4;
hence thesis by A11;
end;
A12: X[0]
proof
let x;
(fdif(f,h).(2*0)).x = f.x by Def6
.= (cdif(f,h).(2*0)).(x+0*h) by Def8;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A12,A1);
hence thesis;
end;
theorem
(fdif(f,h).(2*n+1)).x = (cdif(f,h).(2*n+1)).(x+n*h+h/2)
proof
defpred X[Nat] means
for x holds (fdif(f,h).(2*$1+1)).x = (cdif(f
,h).(2*$1+1)).(x+$1*h+h/2);
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds (fdif(f,h).(2*k+1)).x = (cdif(f,h).(2*k+1)).(x+k*h+h/2);
let x;
A3: fdif(f,h).(2*k+1).(x+h+h) = cdif(f,h).(2*k+1).(x+h+h+k*h+h/2) by A2
.= cdif(f,h).(2*k+1).(x+(k+2)*h+h/2);
A4: fdif(f,h).(2*k+1).(x+h) = cdif(f,h).(2*k+1).(x+h+k*h+h/2) by A2
.= cdif(f,h).(2*k+1).(x+(k+1)*h+h/2);
set r3 = cdif(f,h).(2*k+1).(x+k*h+h/2);
set r2 = cdif(f,h).(2*k+1).(x+(k+1)*h+h/2);
set r1 = cdif(f,h).(2*k+1).(x+(k+2)*h+h/2);
A5: fdif(f,h).(2*(k+1)) is Function of REAL,REAL by Th2;
A6: cdif(f,h).(2*k+1) is Function of REAL,REAL by Th19;
A7: cdif(f,h).(2*(k+1)) is Function of REAL,REAL by Th19;
A8: fdif(f,h).(2*k+1) is Function of REAL,REAL by Th2;
A9: cdif(f,h).(2*(k+1)).(x+(k+1)*h) = cdif(f,h).(2*k+1+1).(x+(k+1)*h)
.= cD(cdif(f,h).(2*k+1),h).(x+(k+1)*h) by Def8
.= cdif(f,h).(2*k+1).(x+(k+1)*h+h/2) - cdif(f,h).(2*k+1).(x+(k+1)*h-h/
2) by A6,Th5
.= cdif(f,h).(2*k+1).(x+(k+1)*h+h/2) - cdif(f,h).(2*k+1).(x+k*h+h/2);
A10: cdif(f,h).(2*(k+1)).(x+(k+2)*h) = cdif(f,h).(2*k+1+1).(x+(k+2)*h)
.= cD(cdif(f,h).(2*k+1),h).(x+(k+2)*h) by Def8
.= cdif(f,h).(2*k+1).(x+(k+2)*h+h/2) - cdif(f,h).(2*k+1).(x+(k+2)*h-h/
2) by A6,Th5
.= cdif(f,h).(2*k+1).(x+(k+2)*h+h/2) - cdif(f,h).(2*k+1).(x+(k+1)*h+h/
2);
A11: cdif(f,h).(2*(k+1)+1).(x+(k+1)*h+h/2) = cD(cdif(f,h).(2*(k+1)),h).(x+
(k+1)*h+h/2) by Def8
.= cdif(f,h).(2*(k+1)).(x+(k+1)*h+h/2+h/2) - cdif(f,h).(2*(k+1)).(x+(k
+1)*h+h/2-h/2) by A7,Th5
.= (r1 - r2) - (r2 - r3) by A10,A9;
fdif(f,h).(2*(k+1)+1).x = fD(fdif(f,h).(2*(k+1)),h).x by Def6
.= fdif(f,h).(2*(k+1)).(x+h) - fdif(f,h).(2*(k+1)).x by A5,Th3
.= fD(fdif(f,h).(2*k+1),h).(x+h) - fdif(f,h).((2*k+1)+1).x by Def6
.= fD(fdif(f,h).(2*k+1),h).(x+h) - fD(fdif(f,h).(2*k+1),h).x by Def6
.= (fdif(f,h).(2*k+1).(x+h+h) - fdif(f,h).(2*k+1).(x+h)) - fD(fdif(f,h
).(2*k+1),h).x by A8,Th3
.= (fdif(f,h).(2*k+1).(x+h+h) - fdif(f,h).(2*k+1).(x+h)) - (fdif(f,h).
(2*k+1).(x+h) - fdif(f,h).(2*k+1).x) by A8,Th3
.= (r1 - r2) - (r2 - r3) by A2,A3,A4;
hence thesis by A11;
end;
A12: X[0]
proof
let x;
(fdif(f,h).(2*0+1)).x = fD(fdif(f,h).0,h).x by Def6
.= fD(f,h).x by Def6
.= f.(x+h) - f.x by Th3
.= f.(x+h/2+h/2) - f.(x+h/2-h/2)
.= cD(f,h).(x+h/2) by Th5
.= cD(cdif(f,h).0,h).(x+h/2) by Def8
.= (cdif(f,h).(2*0+1)).(x+0*h+h/2) by Def8;
hence thesis;
end;
for n holds X[n] from NAT_1:sch 2(A12,A1);
hence thesis;
end;
definition :: Forward difference quotient
let f be real-valued Function;
let x0,x1 be Real;
func [!f,x0,x1!] -> Real equals
(f.x0-f.x1)/(x0-x1);
correctness;
end;
definition :: Forward difference quotient of the second order
let f be real-valued Function;
let x0,x1,x2 be Real;
func [!f,x0,x1,x2!] -> Real equals
([!f,x0,x1!] - [!f,x1,x2!])/(x0-x2);
correctness;
end;
definition :: Forward difference quotient of the third order
let f be real-valued Function;
let x0,x1,x2,x3 be Real;
func [!f,x0,x1,x2,x3!] -> Real equals
([!f,x0,x1,x2!] - [!f,x1,x2,x3!])/(x0-x3);
correctness;
end;
theorem
[!f,x0,x1!] = [!f,x1,x0!]
proof
thus [!f,x0,x1!] = (-(f.x1-f.x0))/(-(x1-x0)) .= [!f,x1,x0!] by XCMPLX_1:191;
end;
theorem
f is constant implies [!f,x0,x1!] = 0
proof
x0 in REAL by XREAL_0:def 1;
then
A1: x0 in dom f by FUNCT_2:def 1;
x1 in REAL by XREAL_0:def 1;
then
A2: x1 in dom f by FUNCT_2:def 1;
assume f is constant;
then f.x0 = f.x1 by A1,A2,FUNCT_1:def 10;
hence thesis;
end;
theorem Th31:
[!(r(#)f),x0,x1!] = r*[!f,x0,x1!]
proof
x1 in REAL by XREAL_0:def 1;
then
A1: x1 in dom (r(#)f) by FUNCT_2:def 1;
x0 in REAL by XREAL_0:def 1;
then x0 in dom (r(#)f) by FUNCT_2:def 1;
then [!(r(#)f),x0,x1!] = (r*f.x0-(r(#)f).x1)/(x0-x1) by VALUED_1:def 5
.= (r*f.x0-r*f.x1)/(x0-x1) by A1,VALUED_1:def 5
.= r*(f.x0-f.x1)/(x0-x1)
.= r*[!f,x0,x1!] by XCMPLX_1:74;
hence thesis;
end;
theorem Th32:
[!(f1+f2),x0,x1!] = [!f1,x0,x1!] + [!f2,x0,x1!]
proof
reconsider xx0=x0, xx1=x1 as Element of REAL by XREAL_0:def 1;
[!(f1+f2),x0,x1!] = (f1.xx0+f2.xx0-(f1+f2).xx1)/(xx0-xx1) by VALUED_1:1
.= (f1.x0+f2.x0-(f1.x1+f2.x1))/(x0-x1) by VALUED_1:1
.= ((f1.x0-f1.x1)+(f2.x0-f2.x1))/(x0-x1)
.= [!f1,x0,x1!]+[!f2,x0,x1!] by XCMPLX_1:62;
hence thesis;
end;
theorem
[!(r1(#)f1+r2(#)f2),x0,x1!] = r1*[!f1,x0,x1!] + r2*[!f2,x0,x1!]
proof
[!(r1(#)f1+r2(#)f2),x0,x1!] = [!(r1(#)f1),x0,x1!]+[!(r2(#)f2),x0,x1!] by Th32
.= r1*[!f1,x0,x1!]+[!(r2(#)f2),x0,x1!] by Th31
.= r1*[!f1,x0,x1!]+r2*[!f2,x0,x1!] by Th31;
hence thesis;
end;
theorem Th34:
x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = [!f,x1,
x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!]
proof
set x10 = x1-x0;
set x20 = x2-x0;
set x12 = x1-x2;
assume
A1: x0,x1,x2 are_mutually_distinct;
then
A2: x1-x2<>0 by ZFMISC_1:def 5;
A3: x1-x0<>0 by A1,ZFMISC_1:def 5;
A4: [!f,x0,x1,x2!] = ((f.x0-f.x1)/(-(x1-x0)) - (f.x1-f.x2)/(x1-x2))/(-(x2-x0 ))
.= (-(f.x0-f.x1)/(x1-x0) - (f.x1-f.x2)/(x1-x2))/(-(x2-x0)) by XCMPLX_1:188
.= (-((f.x0-f.x1)/(x1-x0) + (f.x1-f.x2)/(x1-x2)))/(-(x2-x0))
.= ((f.x0-f.x1)/(x1-x0) + (f.x1-f.x2)/(x1-x2))/(x2-x0) by XCMPLX_1:191
.= (((f.x0-f.x1)*x12 + (f.x1-f.x2)*x10)/(x10*x12))/x20 by A2,A3,
XCMPLX_1:116
.= (f.x0*x12 + f.x1*x20 - f.x2*x10)/(x10*x12*x20) by XCMPLX_1:78;
A5: [!f,x2,x1,x0!] = ((-(f.x1-f.x2))/(-(x1-x2)) - (f.x1-f.x0)/(x1-x0))/(x2- x0)
.= ((f.x1-f.x2)/(x1-x2) - (f.x1-f.x0)/(x1-x0))/(x2-x0) by XCMPLX_1:191
.= (((f.x1-f.x2)*x10 - (f.x1-f.x0)*x12)/(x12*x10))/x20 by A2,A3,
XCMPLX_1:130
.= [!f,x0,x1,x2!] by A4,XCMPLX_1:78;
x2-x0<>0 by A1,ZFMISC_1:def 5;
then
[!f,x1,x2,x0!] = (((f.x1-f.x2)*x20 - (f.x2-f.x0)*x12)/(x12*x20))/x10 by A2,
XCMPLX_1:130
.= (f.x0*x12 + f.x1*x20 - f.x2*x10)/(x12*x20*x10) by XCMPLX_1:78
.= [!f,x0,x1,x2!] by A4;
hence thesis by A5;
end;
theorem
x0,x1,x2 are_mutually_distinct implies
[!f,x0,x1,x2!] = [!f,x2,x0,x1!] & [!f,x0,x1,x2!] = [!f,x1,x0,x2!]
proof
assume
A1: x0,x1,x2 are_mutually_distinct;
then
A2: x1<>x2 by ZFMISC_1:def 5;
x0<>x1 & x0<>x2 by A1,ZFMISC_1:def 5;
then
A3: x2,x0,x1 are_mutually_distinct by A2,ZFMISC_1:def 5;
then [!f,x0,x1,x2!] = [!f,x2,x0,x1!] by Th34;
hence thesis by A3,Th34;
end;
theorem
(fdif(fdif(f,h).m,h).n).x = fdif(f,h).(m+n).x
proof
defpred X[Nat] means
for x holds (fdif(fdif(f,h).m,h).$1).x =
fdif(f,h).(m+$1).x;
A1: for k st X[k] holds X[k+1]
proof
let k;
assume
A2: for x holds (fdif(fdif(f,h).m,h).k).x = fdif(f,h).(m+k).x;
let x;
A3: fdif(f,h).(m+k) is Function of REAL,REAL by Th2;
fdif(f,h).m is Function of REAL,REAL by Th2;
then
A4: fdif(fdif(f,h).m,h).k is Function of REAL,REAL by Th2;
fdif(fdif(f,h).m,h).(k+1).x = fD(fdif(fdif(f,h).m,h).k,h).x by Def6
.= fdif(fdif(f,h).m,h).k.(x+h) - fdif(fdif(f,h).m,h).k.x by A4,Th3
.= fdif(f,h).(m+k).(x+h) - fdif(fdif(f,h).m,h).k.x by A2
.= fdif(f,h).(m+k).(x+h) - fdif(f,h).(m+k).x by A2
.= fD(fdif(f,h).(m+k),h).x by A3,Th3
.= fdif(f,h).(m+k+1).x by Def6;
hence thesis;
end;
A5: X[0] by Def6;
for n holds X[n] from NAT_1:sch 2(A5,A1);
hence thesis;
end;
definition
let S;
attr S is Sequence-yielding means
:Def12:
for n holds S.n is Real_Sequence;
end;
Lm1: ex S being Functional_Sequence of REAL,REAL st
for n holds S.n is Real_Sequence
proof
set s1 = seq_const (1 qua Real);
set S = NAT --> s1;
A1: dom S = NAT by FUNCOP_1:13;
A2: Funcs(NAT,REAL) c= PFuncs(NAT,REAL) by FUNCT_2:72;
s1 in Funcs(NAT,REAL) by FUNCT_2:8;
then
A3: s1 in PFuncs(NAT,REAL) by A2;
PFuncs(NAT,REAL) c= PFuncs(REAL,REAL) by NUMBERS:19,PARTFUN1:50;
then
A4: s1 in PFuncs(REAL,REAL) by A3;
A5: {s1} c= PFuncs(REAL,REAL) by A4,ZFMISC_1:31;
rng S c= PFuncs(REAL,REAL) by A5,XBOOLE_1:1;
then reconsider S as Functional_Sequence of REAL,REAL by A1,FUNCT_2:2;
take S;
let n;
n in NAT by ORDINAL1:def 12;
then S.n = seq_const (1 qua Real) by FUNCOP_1:7;
hence S.n is Real_Sequence;
end;
registration
cluster Sequence-yielding for Functional_Sequence of REAL,REAL;
existence by Lm1,Def12;
end;
definition
mode Seq_Sequence is Sequence-yielding Functional_Sequence of REAL,REAL;
end;
definition
let S be Seq_Sequence;
let n;
redefine func S.n -> Real_Sequence;
coherence by Def12;
end;
reserve S for Seq_Sequence;
Lm2: (fdif(f1(#)f2,h).1).x = f1.x * fdif(f2,h).1.x + fdif(f1,h).1.x * f2.(x+h)
proof
(fdif(f1(#)f2,h).1).x = fdif(f1(#)f2,h).(0+1).x
.= fD(fdif(f1(#)f2,h).0,h).x by Def6
.= fD(f1(#)f2,h).x by Def6
.= (f1(#)f2).(x+h) - (f1(#)f2).x by Th3
.= f1.(x+h) * f2.(x+h) - (f1(#)f2).x by VALUED_1:5
.= f1.(x+h) * f2.(x+h) - f1.x * f2.x by VALUED_1:5
.= f1.x * (f2.(x+h) - f2.x) + (f1.(x+h) - f1.x) * f2.(x+h)
.= f1.x * fD(f2,h).x + (f1.(x+h) - f1.x) * f2.(x+h) by Th3
.= f1.x * fD(f2,h).x + fD(f1,h).x * f2.(x+h) by Th3
.= f1.x * fD(fdif(f2,h).0,h).x + fD(f1,h).x * f2.(x+h) by Def6
.= f1.x * fD(fdif(f2,h).0,h).x + fD(fdif(f1,h).0,h).x * f2.(x+h) by Def6
.= f1.x * fdif(f2,h).(0+1).x + fD(fdif(f1,h).0,h).x * f2.(x+h) by Def6
.= f1.x * fdif(f2,h).1.x + fdif(f1,h).1.x * f2.(x+h) by Def6;
hence thesis;
end;
theorem
(for n for i st i<=n holds (S.n).i=(n choose i) * fdif(f1,h).i.x *
fdif(f2,h).(n-'i).(x+i*h)) implies fdif(f1(#)f2,h).1.x = Sum(S.1, 1) & fdif(f1
(#)f2,h).2.x = Sum(S.2, 2)
proof
A1: 1-'0 = 1-0 by XREAL_1:233
.= 1;
A2: 1-'1 = 1-1 by XREAL_1:233
.= 0;
A3: fdif(f1(#)f2,h).1 is Function of REAL,REAL by Th2;
A4: 2-'1 = 2-1 by XREAL_1:233
.= 1;
A5: fdif(f2,h).1 is Function of REAL,REAL by Th2;
A6: 2-'2 = 2-2 by XREAL_1:233
.= 0;
assume
A7: for n for i st i<=n holds S.n.i=(n choose i) * fdif(f1,h).i.x * fdif
(f2,h).(n-'i).(x+i*h);
then
A8: (S.2).1 = (2 choose 1) * (fdif(f1,h).1).x * fdif(f2,h).(2-'1).(x+1*h)
.= 2 * (fdif(f1,h).1).x * fdif(f2,h).1.(x+h) by A4,NEWTON:23;
A9: (S.1).1 = (1 choose 1) * (fdif(f1,h).1).x * fdif(f2,h).(1-'1).(x+1*h) by A7
.= 1 * (fdif(f1,h).1).x * fdif(f2,h).(1-'1).(x+1*h) by NEWTON:21
.= fdif(f1,h).1.x * f2.(x+h) by A2,Def6;
A10: (S.1).0 = (1 choose 0) * (fdif(f1,h).0).x * fdif(f2,h).(1-'0).(x+0*h) by
A7
.= 1 * (fdif(f1,h).0).x * fdif(f2,h).(1-'0).(x+0*h) by NEWTON:19
.= f1.x * fdif(f2,h).1.x by A1,Def6;
A11: Sum(S.1, 1) = Partial_Sums(S.1).(0+1) by SERIES_1:def 5
.= Partial_Sums(S.1).0 + S.1.1 by SERIES_1:def 1
.= (S.1).0 + (S.1).1 by SERIES_1:def 1
.= (fdif(f1(#)f2,h).1).x by A10,A9,Lm2;
A12: fdif(f1,h).1 is Function of REAL,REAL by Th2;
A13: fdif(f1(#)f2,h).2.x = fdif(f1(#)f2,h).(1+1).x
.= fD(fdif(f1(#)f2,h).1,h).x by Def6
.= fdif(f1(#)f2,h).1.(x+h) - fdif(f1(#)f2,h).1.x by A3,Th3
.= f1.(x+h) * fdif(f2,h).1.(x+h) + fdif(f1,h).1.(x+h) * f2.(x+h+h) -
fdif(f1(#)f2,h).1.x by Lm2
.= f1.(x+h) * fdif(f2,h).1.(x+h) + fdif(f1,h).1.(x+h) * f2.(x+h+h) - (f1
.x * fdif(f2,h).1.x + fdif(f1,h).1.x * f2.(x+h)) by Lm2
.= f1.x * (fdif(f2,h).1.(x+h) - fdif(f2,h).1.x) + (f1.(x+h) - f1.x) *
fdif(f2,h).1.(x+h) + (fdif(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(
f1,h).1.x * (f2.(x+h+h) - f2.(x+h))
.= f1.x * fD(fdif(f2,h).1,h).x + (f1.(x+h) - f1.x) * fdif(f2,h).1.(x+h)
+ (fdif(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+
h+h) - f2.(x+h)) by A5,Th3
.= f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h) + (fdif
(f1,h).1.(x+h) - fdif(f1,h).1.x) * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+h+h) -
f2.(x+h)) by Th3
.= (f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * (f2.(x+h+h) - f2.(x+h)) by
A12,Th3
.= (f1.x * fD(fdif(f2,h).1,h).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Th3
.= (f1.x * fdif(f2,h).(1+1).x + fD(f1,h).x * fdif(f2,h).1.(x+h)) + fD(
fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
.= (f1.x * fdif(f2,h).(1+1).x + fD(fdif(f1,h).0,h).x * fdif(f2,h).1.(x+h
)) + fD(fdif(f1,h).1,h).x * f2.(x+h+h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by
Def6
.= (f1.x * fdif(f2,h).2.x + fD(fdif(f1,h).0,h).x * fdif(f2,h).1.(x+h)) +
fdif(f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
.= (f1.x * fdif(f2,h).2.x + fdif(f1,h).(0+1).x * fdif(f2,h).1.(x+h)) +
fdif(f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(f2,h).(x+h) by Def6
.= (f1.x * fdif(f2,h).2.x + fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) + fdif(
f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fD(fdif(f2,h).0,h).(x+h) by Def6
.= (f1.x * fdif(f2,h).2.x + fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) + fdif(
f1,h).2.x * f2.(x+2*h) + fdif(f1,h).1.x * fdif(f2,h).(0+1).(x+h) by Def6
.= f1.x * fdif(f2,h).2.x + 2 * (fdif(f1,h).1.x * fdif(f2,h).1.(x+h)) +
fdif(f1,h).2.x * f2.(x+2*h);
A14: 2-'0 = 2-0 by XREAL_1:233
.= 2;
A15: (S.2).2 = (2 choose 2) * (fdif(f1,h).2).x * fdif(f2,h).(2-'2).(x+2*h)
by A7
.= 1 * (fdif(f1,h).2).x * fdif(f2,h).(2-'2).(x+2*h) by NEWTON:21
.= fdif(f1,h).2.x * f2.(x+2*h) by A6,Def6;
A16: (S.2).0 = (2 choose 0) * (fdif(f1,h).0).x * fdif(f2,h).(2-'0).(x+0*h)
by A7
.= 1 * (fdif(f1,h).0).x * fdif(f2,h).(2-'0).(x+0*h) by NEWTON:19
.= f1.x * fdif(f2,h).2.x by A14,Def6;
Sum(S.2, 2) = Partial_Sums(S.2).(1+1) by SERIES_1:def 5
.= Partial_Sums(S.2).(0+1) + S.2.2 by SERIES_1:def 1
.= Partial_Sums(S.2).0 + S.2.1 + S.2.2 by SERIES_1:def 1
.= fdif(f1(#)f2,h).2.x by A13,A16,A8,A15,SERIES_1:def 1;
hence thesis by A11;
end;
theorem
for f being PartFunc of REAL, REAL st x in dom f & x-h in dom f holds
bD(f,h).x = f.x - f.(x-h)
proof
let f be PartFunc of REAL, REAL;
assume
A1: x in dom f & x-h in dom f;
A2: dom Shift(f,-h) = - -h ++ dom f by Def1;
A3: h + (x - - -h) in (- -h ++ dom f) by A1,MEASURE6:46; then
A4: Shift(f,-h).x = f.(x+-h) by Def1;
x in (dom Shift(f,-h)) /\ dom f by A3,A2,A1,XBOOLE_0:def 4; then
x in dom bD(f,h) by VALUED_1:12;
hence thesis by A4,VALUED_1:13;
end;
theorem
for f being PartFunc of REAL, REAL st x+h/2 in dom f & x-h/2 in dom f holds
cD(f,h).x = f.(x+h/2) - f.(x-h/2)
proof
let f be PartFunc of REAL, REAL;
assume
A1: x+h/2 in dom f & x-h/2 in dom f;
A2: dom Shift(f,-h/2) = - -(h/2) ++ dom f by Def1;
A3: dom Shift(f,h/2) = - (h/2) ++ dom f by Def1;
A4: -(h/2) + (x + (h/2)) in (-(h/2) ++ dom f) by A1,MEASURE6:46; then
A5: Shift(f,h/2).x = f.(x+h/2) by Def1;
A6: h/2 + (x - (h/2)) in (- -(h/2) ++ dom f) by A1,MEASURE6:46; then
A7: Shift(f,-h/2).x = f.(x+-h/2) by Def1;
x in (dom Shift(f,h/2)) /\ (dom Shift(f,-h/2))
by A4,A6,A3,A2,XBOOLE_0:def 4; then
x in dom cD(f,h) by VALUED_1:12;
hence thesis by A7,A5,VALUED_1:13;
end;