:: Difference and Difference Quotient
:: by Bo Li , Yan Zhang and Xiquan Liang
::
:: Received September 29, 2006
:: Copyright (c) 2006 Association of Mizar Users

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func Shift f,h -> PartFunc of REAL , REAL means :Def1: :: DIFF_1:def 1
( dom it = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
it . x = f . (x + h) ) );
existence
ex b₁ being PartFunc of REAL , REAL st
( dom b₁ = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
b₁ . x = f . (x + h) ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of REAL , REAL st dom b₁ = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
b₁ . x = f . (x + h) ) & dom b₂ = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
b₂ . x = f . (x + h) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines Shift DIFF_1:def 1 :

for f being PartFunc of REAL , REAL
for h being real number
for b₃ being PartFunc of REAL , REAL holds
( b₃ = Shift f,h iff ( dom b₃ = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
b₃ . x = f . (x + h) ) ) );

definition

let f be Function of REAL , REAL ;
let h be real number ;
:: original: Shift
redefine func Shift f,h -> Function of REAL , REAL means :Def2: :: DIFF_1:def 2
for x being Real holds it . x = f . (x + h);
coherence
Shift f,h is Function of REAL , REAL

proof end;

compatibility
for b₁ being Function of REAL , REAL holds
( b₁ = Shift f,h iff for x being Real holds b₁ . x = f . (x + h) )

proof end;

end;

:: deftheorem Def2 defines Shift DIFF_1:def 2 :

for f being Function of REAL , REAL
for h being real number
for b₃ being Function of REAL , REAL holds
( b₃ = Shift f,h iff for x being Real holds b₃ . x = f . (x + h) );

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func fD f,h -> PartFunc of REAL , REAL equals :: DIFF_1:def 3
(Shift f,h) - f;
coherence
(Shift f,h) - f is PartFunc of REAL , REAL ;

end;

:: deftheorem defines fD DIFF_1:def 3 :

for f being PartFunc of REAL , REAL
for h being real number holds fD f,h = (Shift f,h) - f;

definition

let f be Function of REAL , REAL ;
let h be real number ;
:: original: fD
redefine func fD f,h -> Function of REAL , REAL ;
coherence
fD f,h is Function of REAL , REAL

proof end;

end;

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func bD f,h -> PartFunc of REAL , REAL equals :: DIFF_1:def 4
f - (Shift f,(- h));
coherence
f - (Shift f,(- h)) is PartFunc of REAL , REAL ;

end;

:: deftheorem defines bD DIFF_1:def 4 :

for f being PartFunc of REAL , REAL
for h being real number holds bD f,h = f - (Shift f,(- h));

definition

let f be Function of REAL , REAL ;
let h be real number ;
:: original: bD
redefine func bD f,h -> Function of REAL , REAL ;
coherence
bD f,h is Function of REAL , REAL

proof end;

end;

theorem :: DIFF_1:1

for h being Real
for f being PartFunc of REAL , REAL holds bD f,h = - (fD f,(- h)) by RFUNCT_1:31;

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func cD f,h -> PartFunc of REAL , REAL equals :: DIFF_1:def 5
(Shift f,(h / 2)) - (Shift f,(- (h / 2)));
coherence
(Shift f,(h / 2)) - (Shift f,(- (h / 2))) is PartFunc of REAL , REAL ;

end;

:: deftheorem defines cD DIFF_1:def 5 :

for f being PartFunc of REAL , REAL
for h being real number holds cD f,h = (Shift f,(h / 2)) - (Shift f,(- (h / 2)));

definition

let f be Function of REAL , REAL ;
let h be real number ;
:: original: cD
redefine func cD f,h -> Function of REAL , REAL ;
coherence
cD f,h is Function of REAL , REAL

proof end;

end;

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func forward_difference f,h -> Functional_Sequence of REAL ,REAL means :Def6: :: DIFF_1:def 6
( it . 0 = f & ( for n being Element of NAT holds it . (n + 1) = fD (it . n),h ) );
existence
ex b₁ being Functional_Sequence of REAL ,REAL st
( b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = fD (b₁ . n),h ) )

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of REAL ,REAL st b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = fD (b₁ . n),h ) & b₂ . 0 = f & ( for n being Element of NAT holds b₂ . (n + 1) = fD (b₂ . n),h ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines forward_difference DIFF_1:def 6 :

for f being PartFunc of REAL , REAL
for h being real number
for b₃ being Functional_Sequence of REAL ,REAL holds
( b₃ = forward_difference f,h iff ( b₃ . 0 = f & ( for n being Element of NAT holds b₃ . (n + 1) = fD (b₃ . n),h ) ) );

notation

let f be PartFunc of REAL , REAL ;
let h be real number ;
synonym fdif f,h for forward_difference f,h;

end;

Lm1: for h being Real
for f being Function of REAL , REAL holds fD f,h is Function of REAL , REAL
;

theorem Th2: :: DIFF_1:2

for h being Real
for f being Function of REAL , REAL
for n being Element of NAT holds (fdif f,h) . n is Function of REAL , REAL

proof end;

theorem Th3: :: DIFF_1:3

for h, x being Real
for f being Function of REAL , REAL holds (fD f,h) . x = (f . (x + h)) - (f . x)

proof end;

theorem Th4: :: DIFF_1:4

for h, x being Real
for f being Function of REAL , REAL holds (bD f,h) . x = (f . x) - (f . (x - h))

proof end;

theorem Th5: :: DIFF_1:5

for h, x being Real
for f being Function of REAL , REAL holds (cD f,h) . x = (f . (x + (h / 2))) - (f . (x - (h / 2)))

proof end;

theorem :: DIFF_1:6

for n being Element of NAT
for h being Real
for f being Function of REAL , REAL st f is constant holds
for x being Real holds ((fdif f,h) . (n + 1)) . x = 0

proof end;

theorem Th7: :: DIFF_1:7

for n being Element of NAT
for r, h, x being Real
for f being Function of REAL , REAL holds ((fdif (r (#) f),h) . (n + 1)) . x = r * (((fdif f,h) . (n + 1)) . x)

proof end;

theorem Th8: :: DIFF_1:8

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((fdif (f1 + f2),h) . (n + 1)) . x = (((fdif f1,h) . (n + 1)) . x) + (((fdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:9

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((fdif (f1 - f2),h) . (n + 1)) . x = (((fdif f1,h) . (n + 1)) . x) - (((fdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:10

for n being Element of NAT
for r1, r2, h, x being Real
for f1, f2 being Function of REAL , REAL holds ((fdif ((r1 (#) f1) + (r2 (#) f2)),h) . (n + 1)) . x = (r1 * (((fdif f1,h) . (n + 1)) . x)) + (r2 * (((fdif f2,h) . (n + 1)) . x))

proof end;

theorem :: DIFF_1:11

for h, x being Real
for f being Function of REAL , REAL holds ((fdif f,h) . 1) . x = ((Shift f,h) . x) - (f . x)

proof end;

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func backward_difference f,h -> Functional_Sequence of REAL ,REAL means :Def7: :: DIFF_1:def 7
( it . 0 = f & ( for n being Element of NAT holds it . (n + 1) = bD (it . n),h ) );
existence
ex b₁ being Functional_Sequence of REAL ,REAL st
( b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = bD (b₁ . n),h ) )

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of REAL ,REAL st b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = bD (b₁ . n),h ) & b₂ . 0 = f & ( for n being Element of NAT holds b₂ . (n + 1) = bD (b₂ . n),h ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines backward_difference DIFF_1:def 7 :

for f being PartFunc of REAL , REAL
for h being real number
for b₃ being Functional_Sequence of REAL ,REAL holds
( b₃ = backward_difference f,h iff ( b₃ . 0 = f & ( for n being Element of NAT holds b₃ . (n + 1) = bD (b₃ . n),h ) ) );

notation

let f be PartFunc of REAL , REAL ;
let h be real number ;
synonym bdif f,h for backward_difference f,h;

end;

Lm2: for h being Real
for f being Function of REAL , REAL holds bD f,h is Function of REAL , REAL
;

theorem Th12: :: DIFF_1:12

for h being Real
for f being Function of REAL , REAL
for n being Element of NAT holds (bdif f,h) . n is Function of REAL , REAL

proof end;

theorem :: DIFF_1:13

for n being Element of NAT
for h being Real
for f being Function of REAL , REAL st f is constant holds
for x being Real holds ((bdif f,h) . (n + 1)) . x = 0

proof end;

theorem Th14: :: DIFF_1:14

for n being Element of NAT
for r, h, x being Real
for f being Function of REAL , REAL holds ((bdif (r (#) f),h) . (n + 1)) . x = r * (((bdif f,h) . (n + 1)) . x)

proof end;

theorem Th15: :: DIFF_1:15

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((bdif (f1 + f2),h) . (n + 1)) . x = (((bdif f1,h) . (n + 1)) . x) + (((bdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:16

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((bdif (f1 - f2),h) . (n + 1)) . x = (((bdif f1,h) . (n + 1)) . x) - (((bdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:17

for n being Element of NAT
for r1, r2, h, x being Real
for f1, f2 being Function of REAL , REAL holds ((bdif ((r1 (#) f1) + (r2 (#) f2)),h) . (n + 1)) . x = (r1 * (((bdif f1,h) . (n + 1)) . x)) + (r2 * (((bdif f2,h) . (n + 1)) . x))

proof end;

theorem :: DIFF_1:18

for h, x being Real
for f being Function of REAL , REAL holds ((bdif f,h) . 1) . x = (f . x) - ((Shift f,(- h)) . x)

proof end;

definition

let f be PartFunc of REAL , REAL ;
let h be real number ;
func central_difference f,h -> Functional_Sequence of REAL ,REAL means :Def8: :: DIFF_1:def 8
( it . 0 = f & ( for n being Element of NAT holds it . (n + 1) = cD (it . n),h ) );
existence
ex b₁ being Functional_Sequence of REAL ,REAL st
( b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = cD (b₁ . n),h ) )

proof end;

uniqueness
for b₁, b₂ being Functional_Sequence of REAL ,REAL st b₁ . 0 = f & ( for n being Element of NAT holds b₁ . (n + 1) = cD (b₁ . n),h ) & b₂ . 0 = f & ( for n being Element of NAT holds b₂ . (n + 1) = cD (b₂ . n),h ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines central_difference DIFF_1:def 8 :

for f being PartFunc of REAL , REAL
for h being real number
for b₃ being Functional_Sequence of REAL ,REAL holds
( b₃ = central_difference f,h iff ( b₃ . 0 = f & ( for n being Element of NAT holds b₃ . (n + 1) = cD (b₃ . n),h ) ) );

notation

let f be PartFunc of REAL , REAL ;
let h be real number ;
synonym cdif f,h for central_difference f,h;

end;

Lm3: for h being Real
for f being Function of REAL , REAL holds cD f,h is Function of REAL , REAL
;

theorem Th19: :: DIFF_1:19

for h being Real
for f being Function of REAL , REAL
for n being Element of NAT holds (cdif f,h) . n is Function of REAL , REAL

proof end;

theorem :: DIFF_1:20

for n being Element of NAT
for h being Real
for f being Function of REAL , REAL st f is constant holds
for x being Real holds ((cdif f,h) . (n + 1)) . x = 0

proof end;

theorem Th21: :: DIFF_1:21

for n being Element of NAT
for r, h, x being Real
for f being Function of REAL , REAL holds ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)

proof end;

theorem Th22: :: DIFF_1:22

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((cdif (f1 + f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) + (((cdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:23

for n being Element of NAT
for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)

proof end;

theorem :: DIFF_1:24

for n being Element of NAT
for r1, r2, h, x being Real
for f1, f2 being Function of REAL , REAL holds ((cdif ((r1 (#) f1) + (r2 (#) f2)),h) . (n + 1)) . x = (r1 * (((cdif f1,h) . (n + 1)) . x)) + (r2 * (((cdif f2,h) . (n + 1)) . x))

proof end;

theorem :: DIFF_1:25

for h, x being Real
for f being Function of REAL , REAL holds ((cdif f,h) . 1) . x = ((Shift f,(h / 2)) . x) - ((Shift f,(- (h / 2))) . x)

proof end;

theorem :: DIFF_1:26

for n being Element of NAT
for h, x being Real
for f being Function of REAL , REAL holds ((fdif f,h) . n) . x = ((bdif f,h) . n) . (x + (n * h))

proof end;

theorem :: DIFF_1:27

for n being Element of NAT
for h, x being Real
for f being Function of REAL , REAL holds ((fdif f,h) . (2 * n)) . x = ((cdif f,h) . (2 * n)) . (x + (n * h))

proof end;

theorem :: DIFF_1:28

for n being Element of NAT
for h, x being Real
for f being Function of REAL , REAL holds ((fdif f,h) . ((2 * n) + 1)) . x = ((cdif f,h) . ((2 * n) + 1)) . ((x + (n * h)) + (h / 2))

proof end;

definition

let f be real-valued Function;
let x0, x1 be real number ;
func [!f,x0,x1!] -> Real equals :: DIFF_1:def 9
((f . x0) - (f . x1)) / (x0 - x1);
correctness
coherence
((f . x0) - (f . x1)) / (x0 - x1) is Real;
;

end;

:: deftheorem defines [! DIFF_1:def 9 :

for f being real-valued Function
for x0, x1 being real number holds [!f,x0,x1!] = ((f . x0) - (f . x1)) / (x0 - x1);

definition

let f be real-valued Function;
let x0, x1, x2 be real number ;
func [!f,x0,x1,x2!] -> Real equals :: DIFF_1:def 10
([!f,x0,x1!] - [!f,x1,x2!]) / (x0 - x2);
correctness
coherence
([!f,x0,x1!] - [!f,x1,x2!]) / (x0 - x2) is Real;
;

end;

:: deftheorem defines [! DIFF_1:def 10 :

for f being real-valued Function
for x0, x1, x2 being real number holds [!f,x0,x1,x2!] = ([!f,x0,x1!] - [!f,x1,x2!]) / (x0 - x2);

definition

let f be real-valued Function;
let x0, x1, x2, x3 be real number ;
func [!f,x0,x1,x2,x3!] -> Real equals :: DIFF_1:def 11
([!f,x0,x1,x2!] - [!f,x1,x2,x3!]) / (x0 - x3);
correctness
coherence
([!f,x0,x1,x2!] - [!f,x1,x2,x3!]) / (x0 - x3) is Real;
;

end;

:: deftheorem defines [! DIFF_1:def 11 :

for f being real-valued Function
for x0, x1, x2, x3 being real number holds [!f,x0,x1,x2,x3!] = ([!f,x0,x1,x2!] - [!f,x1,x2,x3!]) / (x0 - x3);

theorem :: DIFF_1:29

for x0, x1 being Real
for f being Function of REAL , REAL holds [!f,x0,x1!] = [!f,x1,x0!]

proof end;

theorem :: DIFF_1:30

for x0, x1 being Real
for f being Function of REAL , REAL st f is constant holds
[!f,x0,x1!] = 0

proof end;

theorem Th31: :: DIFF_1:31

for r, x0, x1 being Real
for f being Function of REAL , REAL holds [!(r (#) f),x0,x1!] = r * [!f,x0,x1!]

proof end;

theorem Th32: :: DIFF_1:32

for x0, x1 being Real
for f1, f2 being Function of REAL , REAL holds [!(f1 + f2),x0,x1!] = [!f1,x0,x1!] + [!f2,x0,x1!]

proof end;

theorem :: DIFF_1:33

for r1, r2, x0, x1 being Real
for f1, f2 being Function of REAL , REAL holds [!((r1 (#) f1) + (r2 (#) f2)),x0,x1!] = (r1 * [!f1,x0,x1!]) + (r2 * [!f2,x0,x1!])

proof end;

theorem Th34: :: DIFF_1:34

for x0, x1, x2 being Real
for f being Function of REAL , REAL st x0,x1,x2 are_mutually_different holds
( [!f,x0,x1,x2!] = [!f,x1,x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!] )

proof end;

theorem :: DIFF_1:35

for x0, x1, x2 being Real
for f being Function of REAL , REAL st x0,x1,x2 are_mutually_different holds
( [!f,x0,x1,x2!] = [!f,x2,x0,x1!] & [!f,x0,x1,x2!] = [!f,x1,x0,x2!] )

proof end;

theorem :: DIFF_1:36

for m, n being Element of NAT
for h, x being Real
for f being Function of REAL , REAL holds ((fdif ((fdif f,h) . m),h) . n) . x = ((fdif f,h) . (m + n)) . x

proof end;

definition

let S be Functional_Sequence of REAL ,REAL ;
attr S is Sequence-yielding means :Def12: :: DIFF_1:def 12
for n being Element of NAT holds S . n is Real_Sequence;

end;

:: deftheorem Def12 defines Sequence-yielding DIFF_1:def 12 :

for S being Functional_Sequence of REAL ,REAL holds
( S is Sequence-yielding iff for n being Element of NAT holds S . n is Real_Sequence );

Lm4: ex S being Functional_Sequence of REAL ,REAL st
for n being Element of NAT holds S . n is Real_Sequence

proof end;

registration

cluster Sequence-yielding Relation of NAT , PFuncs REAL ,REAL ;
existence
ex b₁ being Functional_Sequence of REAL ,REAL st b₁ is Sequence-yielding

proof end;

end;

definition

mode Seq_Sequence is Sequence-yielding Functional_Sequence of REAL ,REAL ;

end;

definition

let S be Seq_Sequence;
let n be Element of NAT ;
:: original: .
redefine func S . n -> Real_Sequence;
coherence
S . n is Real_Sequence by Def12;

end;

Lm5: for h, x being Real
for f1, f2 being Function of REAL , REAL holds ((fdif (f1 (#) f2),h) . 1) . x = ((f1 . x) * (((fdif f2,h) . 1) . x)) + ((((fdif f1,h) . 1) . x) * (f2 . (x + h)))

proof end;

theorem :: DIFF_1:37

for h, x being Real
for f1, f2 being Function of REAL , REAL
for S being Seq_Sequence st ( for n, i being Element of NAT st i <= n holds
(S . n) . i = ((n choose i) * (((fdif f1,h) . i) . x)) * (((fdif f2,h) . (n -' i)) . (x + (i * h))) ) holds
( ((fdif (f1 (#) f2),h) . 1) . x = Sum (S . 1),1 & ((fdif (f1 (#) f2),h) . 2) . x = Sum (S . 2),2 )

proof end;