FUNCT_4: The Modification of a Function by a Functionand the Iteration of the Composition of a Function

:: The Modification of a Function by a Functionand the Iteration of the Composition of a Function
:: by Czes{\l}aw Byli\'nski
::
:: Received March 1, 1990
:: Copyright (c) 1990 Association of Mizar Users

Lm1: for x, y, Z being set st [x,y] in Z holds
( x in union (union Z) & y in union (union Z) )

proof end;

Lm2: for x, x', y, y', x1, x1', y1, y1' being set st [[x,x'],[y,y']] = [[x1,x1'],[y1,y1']] holds
( x = x1 & y = y1 & x' = x1' & y' = y1' )

proof end;

theorem Th1: :: FUNCT_4:1

for Z being set st ( for z being set st z in Z holds
ex x, y being set st z = [x,y] ) holds
ex X, Y being set st Z c= [:X,Y:]

proof end;

theorem :: FUNCT_4:2

for g, f being Function holds g * f = (g | (rng f)) * f

proof end;

theorem :: FUNCT_4:3

canceled;

theorem :: FUNCT_4:4

for X, Y being set holds
( id X c= id Y iff X c= Y )

proof end;

theorem :: FUNCT_4:5

for X, Y, a being set st X c= Y holds
X --> a c= Y --> a

proof end;

theorem Th6: :: FUNCT_4:6

for X, a, Y, b being set st X --> a c= Y --> b holds
X c= Y

proof end;

theorem :: FUNCT_4:7

for X, a, Y, b being set st X <> {} & X --> a c= Y --> b holds
a = b

proof end;

theorem :: FUNCT_4:8

for x being set
for f being Function st x in dom f holds
x .--> (f . x) c= f

proof end;

theorem :: FUNCT_4:9

for Y, X being set
for f being Function holds (Y | f) | X c= f

proof end;

theorem :: FUNCT_4:10

for Y, X being set
for f, g being Function st f c= g holds
(Y | f) | X c= (Y | g) | X

proof end;

definition

let f, g be Function;
func f +* g -> Function means :Def1: :: FUNCT_4:def 1
( dom it = (dom f) \/ (dom g) & ( for x being set st x in (dom f) \/ (dom g) holds
( ( x in dom g implies it . x = g . x ) & ( not x in dom g implies it . x = f . x ) ) ) );
existence

proof end;

uniqueness

proof end;

idempotence ;

end;

:: deftheorem Def1 defines +* FUNCT_4:def 1 :

theorem Th11: :: FUNCT_4:11

for f, g being Function holds
( dom f c= dom (f +* g) & dom g c= dom (f +* g) )

proof end;

theorem Th12: :: FUNCT_4:12

for x being set
for g, f being Function st not x in dom g holds
(f +* g) . x = f . x

proof end;

theorem Th13: :: FUNCT_4:13

for x being set
for f, g being Function holds
( x in dom (f +* g) iff ( x in dom f or x in dom g ) )

proof end;

theorem Th14: :: FUNCT_4:14

for x being set
for g, f being Function st x in dom g holds
(f +* g) . x = g . x

proof end;

theorem Th15: :: FUNCT_4:15

for f, g, h being Function holds (f +* g) +* h = f +* (g +* h)

proof end;

theorem Th16: :: FUNCT_4:16

for x being set
for f, g being Function st f tolerates g & x in dom f holds
(f +* g) . x = f . x

proof end;

theorem :: FUNCT_4:17

for x being set
for f, g being Function st dom f misses dom g & x in dom f holds
(f +* g) . x = f . x

proof end;

theorem Th18: :: FUNCT_4:18

for f, g being Function holds rng (f +* g) c= (rng f) \/ (rng g)

proof end;

theorem :: FUNCT_4:19

for g, f being Function holds rng g c= rng (f +* g)

proof end;

theorem Th20: :: FUNCT_4:20

for f, g being Function st dom f c= dom g holds
f +* g = g

proof end;

theorem Th21: :: FUNCT_4:21

for f being Function holds {} +* f = f

proof end;

theorem Th22: :: FUNCT_4:22

for f being Function holds f +* {} = f

proof end;

theorem :: FUNCT_4:23

for X, Y being set holds (id X) +* (id Y) = id (X \/ Y)

proof end;

theorem :: FUNCT_4:24

for f, g being Function holds (f +* g) | (dom g) = g

proof end;

theorem Th25: :: FUNCT_4:25

for f, g being Function holds (f +* g) | ((dom f) \ (dom g)) c= f

proof end;

theorem Th26: :: FUNCT_4:26

for g, f being Function holds g c= f +* g

proof end;

theorem :: FUNCT_4:27

for f, g, h being Function st f tolerates g +* h holds
f | ((dom f) \ (dom h)) tolerates g

proof end;

theorem Th28: :: FUNCT_4:28

for f, g, h being Function st f tolerates g +* h holds
f tolerates h

proof end;

theorem Th29: :: FUNCT_4:29

for f, g being Function holds
( f tolerates g iff f c= f +* g )

proof end;

theorem Th30: :: FUNCT_4:30

for f, g being Function holds f +* g c= f \/ g

proof end;

theorem Th31: :: FUNCT_4:31

for f, g being Function holds
( f tolerates g iff f \/ g = f +* g )

proof end;

theorem Th32: :: FUNCT_4:32

for f, g being Function st dom f misses dom g holds
f \/ g = f +* g

proof end;

theorem Th33: :: FUNCT_4:33

for f, g being Function st dom f misses dom g holds
f c= f +* g

proof end;

theorem :: FUNCT_4:34

for f, g being Function st dom f misses dom g holds
(f +* g) | (dom f) = f

proof end;

theorem Th35: :: FUNCT_4:35

for f, g being Function holds
( f tolerates g iff f +* g = g +* f )

proof end;

theorem :: FUNCT_4:36

for f, g being Function st dom f misses dom g holds
f +* g = g +* f

proof end;

theorem :: FUNCT_4:37

for X, Y being set
for f, g being PartFunc of X,Y st g is total holds
f +* g = g

proof end;

theorem Th38: :: FUNCT_4:38

for X, Y being set
for f, g being Function of X,Y st ( Y = {} implies X = {} ) holds
f +* g = g

proof end;

theorem :: FUNCT_4:39

for X being set
for f, g being Function of X,X holds f +* g = g by Th38;

theorem :: FUNCT_4:40

for X being set
for D being non empty set
for f, g being Function of X,D holds f +* g = g by Th38;

theorem :: FUNCT_4:41

for X, Y being set
for f, g being PartFunc of X,Y holds f +* g is PartFunc of X,Y

proof end;

definition

let f be Function;
func ~ f -> Function means :Def2: :: FUNCT_4:def 2
( ( for x being set holds
( x in dom it iff ex y, z being set st
( x = [z,y] & [y,z] in dom f ) ) ) & ( for y, z being set st [y,z] in dom f holds
it . z,y = f . y,z ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines ~ FUNCT_4:def 2 :

theorem Th42: :: FUNCT_4:42

for f being Function holds rng (~ f) c= rng f

proof end;

theorem Th43: :: FUNCT_4:43

for x, y being set
for f being Function holds
( [x,y] in dom f iff [y,x] in dom (~ f) )

proof end;

theorem :: FUNCT_4:44

for y, x being set
for f being Function st [y,x] in dom (~ f) holds
(~ f) . y,x = f . x,y

proof end;

theorem :: FUNCT_4:45

for f being Function ex X, Y being set st dom (~ f) c= [:X,Y:]

proof end;

theorem Th46: :: FUNCT_4:46

for X, Y being set
for f being Function st dom f c= [:X,Y:] holds
dom (~ f) c= [:Y,X:]

proof end;

theorem Th47: :: FUNCT_4:47

for X, Y being set
for f being Function st dom f = [:X,Y:] holds
dom (~ f) = [:Y,X:]

proof end;

theorem Th48: :: FUNCT_4:48

for X, Y being set
for f being Function st dom f c= [:X,Y:] holds
rng (~ f) = rng f

proof end;

theorem :: FUNCT_4:49

for X, Y, Z being set
for f being PartFunc of [:X,Y:],Z holds ~ f is PartFunc of [:Y,X:],Z

proof end;

theorem Th50: :: FUNCT_4:50

for X, Y, Z being set
for f being Function of [:X,Y:],Z st Z <> {} holds
~ f is Function of [:Y,X:],Z

proof end;

theorem :: FUNCT_4:51

for X, Y being set
for D being non empty set
for f being Function of [:X,Y:],D holds ~ f is Function of [:Y,X:],D by Th50;

theorem Th52: :: FUNCT_4:52

for f being Function holds ~ (~ f) c= f

proof end;

theorem Th53: :: FUNCT_4:53

for X, Y being set
for f being Function st dom f c= [:X,Y:] holds
~ (~ f) = f

proof end;

theorem :: FUNCT_4:54

for X, Y, Z being set
for f being PartFunc of [:X,Y:],Z holds ~ (~ f) = f

proof end;

theorem :: FUNCT_4:55

canceled;

theorem :: FUNCT_4:56

canceled;

definition

let f, g be Function;
func |:f,g:| -> Function means :Def3: :: FUNCT_4:def 3
( ( for z being set holds
( z in dom it iff ex x, y, x', y' being set st
( z = [[x,x'],[y,y']] & [x,y] in dom f & [x',y'] in dom g ) ) ) & ( for x, y, x', y' being set st [x,y] in dom f & [x',y'] in dom g holds
it . [x,x'],[y,y'] = [(f . x,y),(g . x',y')] ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines |: FUNCT_4:def 3 :

theorem Th57: :: FUNCT_4:57

for x, x', y, y' being set
for f, g being Function holds
( [[x,x'],[y,y']] in dom |:f,g:| iff ( [x,y] in dom f & [x',y'] in dom g ) )

proof end;

theorem :: FUNCT_4:58

for x, x', y, y' being set
for f, g being Function st [[x,x'],[y,y']] in dom |:f,g:| holds
|:f,g:| . [x,x'],[y,y'] = [(f . x,y),(g . x',y')]

proof end;

theorem Th59: :: FUNCT_4:59

for f, g being Function holds rng |:f,g:| c= [:(rng f),(rng g):]

proof end;

theorem Th60: :: FUNCT_4:60

for X, Y, X', Y' being set
for f, g being Function st dom f c= [:X,Y:] & dom g c= [:X',Y':] holds
dom |:f,g:| c= [:[:X,X':],[:Y,Y':]:]

proof end;

theorem Th61: :: FUNCT_4:61

for X, Y, X', Y' being set
for f, g being Function st dom f = [:X,Y:] & dom g = [:X',Y':] holds
dom |:f,g:| = [:[:X,X':],[:Y,Y':]:]

proof end;

theorem :: FUNCT_4:62

for X, Y, Z, X', Y', Z' being set
for f being PartFunc of [:X,Y:],Z
for g being PartFunc of [:X',Y':],Z' holds |:f,g:| is PartFunc of [:[:X,X':],[:Y,Y':]:],[:Z,Z':]

proof end;

theorem Th63: :: FUNCT_4:63

for X, Y, Z, X', Y', Z' being set
for f being Function of [:X,Y:],Z
for g being Function of [:X',Y':],Z' st Z <> {} & Z' <> {} holds
|:f,g:| is Function of [:[:X,X':],[:Y,Y':]:],[:Z,Z':]

proof end;

theorem :: FUNCT_4:64

for X, Y, X', Y' being set
for D, D' being non empty set
for f being Function of [:X,Y:],D
for g being Function of [:X',Y':],D' holds |:f,g:| is Function of [:[:X,X':],[:Y,Y':]:],[:D,D':] by Th63;

definition

let x, y, a, b be set ;
func x,y --> a,b -> set equals :: FUNCT_4:def 4
(x .--> a) +* (y .--> b);
correctness ;

end;

:: deftheorem defines --> FUNCT_4:def 4 :

registration

let x, y, a, b be set ;
cluster x,y --> a,b -> Relation-like Function-like ;
coherence ;

end;

theorem Th65: :: FUNCT_4:65

for x1, x2, y1, y2 being set holds
( dom (x1,x2 --> y1,y2) = {x1,x2} & rng (x1,x2 --> y1,y2) c= {y1,y2} )

proof end;

theorem Th66: :: FUNCT_4:66

for x1, x2, y1, y2 being set holds
( ( x1 <> x2 implies (x1,x2 --> y1,y2) . x1 = y1 ) & (x1,x2 --> y1,y2) . x2 = y2 )

proof end;

theorem :: FUNCT_4:67

for x1, x2, y1, y2 being set st x1 <> x2 holds
rng (x1,x2 --> y1,y2) = {y1,y2}

proof end;

theorem :: FUNCT_4:68

for x1, x2, y being set holds x1,x2 --> y,y = {x1,x2} --> y

proof end;

definition

let A be non empty set ;
let x1, x2 be set ;
let y1, y2 be Element of A;
:: original: -->
redefine func x1,x2 --> y1,y2 -> Function of {x1,x2},A;
coherence

proof end;

end;

theorem :: FUNCT_4:69

for a, b, c, d being set
for g being Function st dom g = {a,b} & g . a = c & g . b = d holds
g = a,b --> c,d

proof end;

theorem :: FUNCT_4:70

canceled;

theorem Th71: :: FUNCT_4:71

for a, b, c, d being set st a <> c holds
a,c --> b,d = {[a,b],[c,d]}

proof end;

theorem :: FUNCT_4:72

for a, b, x, y, x', y' being set st a <> b & a,b --> x,y = a,b --> x',y' holds
( x = x' & y = y' )

proof end;

theorem :: FUNCT_4:73

for f1, f2, g1, g2 being Function st rng g1 c= dom f1 & rng g2 c= dom f2 & f1 tolerates f2 holds
(f1 +* f2) * (g1 +* g2) = (f1 * g1) +* (f2 * g2)

proof end;

theorem Th74: :: FUNCT_4:74

for f being Function
for A, B being set st dom f c= A \/ B holds
(f | A) +* (f | B) = f

proof end;

theorem Th75: :: FUNCT_4:75

for p, q being Function
for A being set holds (p +* q) | A = (p | A) +* (q | A)

proof end;

theorem Th76: :: FUNCT_4:76

for f, g being Function
for A being set st A misses dom g holds
(f +* g) | A = f | A

proof end;

theorem :: FUNCT_4:77

for f, g being Function
for A being set st dom f misses A holds
(f +* g) | A = g | A

proof end;

theorem :: FUNCT_4:78

for f, g, h being Function st dom g = dom h holds
(f +* g) +* h = f +* h

proof end;

theorem Th79: :: FUNCT_4:79

for f, g being Function st f c= g holds
( f +* g = g & g +* f = g )

proof end;

theorem :: FUNCT_4:80

for f being Function
for A being set holds f +* (f | A) = f

proof end;

theorem :: FUNCT_4:81

for f, g being Function
for B, C being set st dom f c= B & dom g c= C & B misses C holds
( (f +* g) | B = f & (f +* g) | C = g )

proof end;

theorem :: FUNCT_4:82

for p, q being Function
for A being set st dom p c= A & dom q misses A holds
(p +* q) | A = p

proof end;

theorem :: FUNCT_4:83

for f being Function
for A, B being set holds f | (A \/ B) = (f | A) +* (f | B)

proof end;

theorem :: FUNCT_4:84

for i, j, k being set holds i,j :-> k = [i,j] .--> k ;

theorem :: FUNCT_4:85

for i, j, k being set holds (i,j :-> k) . i,j = k by FUNCOP_1:86;

theorem :: FUNCT_4:86

for a, b, c being set holds a,a --> b,c = a .--> c

proof end;

theorem :: FUNCT_4:87

for x, y being set holds x .--> y = {[x,y]} by ZFMISC_1:35;

theorem :: FUNCT_4:88

for f being Function
for a, b, c being set st a <> c holds
(f +* (a .--> b)) . c = f . c

proof end;

theorem :: FUNCT_4:89

for f being Function
for a, b, c, d being set st a <> b holds
( (f +* (a,b --> c,d)) . a = c & (f +* (a,b --> c,d)) . b = d )

proof end;

theorem Th90: :: FUNCT_4:90

for a, b being set
for f being Function st a in dom f & f . a = b holds
a .--> b c= f

proof end;

theorem :: FUNCT_4:91

for a, b, c, d being set
for f being Function st a in dom f & c in dom f & f . a = b & f . c = d holds
a,c --> b,d c= f

proof end;

theorem :: FUNCT_4:92

for f, g, h being Function st f c= h & g c= h holds
f +* g c= h

proof end;

theorem :: FUNCT_4:93

for f, g being Function
for A being set st A /\ (dom f) c= A /\ (dom g) holds
(f +* (g | A)) | A = g | A

proof end;

theorem :: FUNCT_4:94

for f, g being Function
for D being set st dom g misses D holds
(f +* g) | D = f | D

proof end;

theorem :: FUNCT_4:95

for f being Function
for a, b, n, m being set holds ((f +* (a .--> b)) +* (m .--> n)) . m = n

proof end;

theorem :: FUNCT_4:96

for f being Function
for n, m being set holds ((f +* (n .--> m)) +* (m .--> n)) . n = m

proof end;

theorem :: FUNCT_4:97

for f being Function
for a, b, n, m, x being set st x <> m & x <> a holds
((f +* (a .--> b)) +* (m .--> n)) . x = f . x

proof end;

theorem :: FUNCT_4:98

for f, g being Function st f is one-to-one & g is one-to-one & rng f misses rng g holds
f +* g is one-to-one

proof end;

theorem :: FUNCT_4:99

for f, g being Function holds (f +* g) +* g = f +* g

proof end;

theorem :: FUNCT_4:100

for f, g, h being Function
for D being set st (f +* g) | D = h | D holds
(h +* g) | D = (f +* g) | D

proof end;

theorem :: FUNCT_4:101

for f, g, h being Function
for D being set st f | D = h | D holds
(h +* g) | D = (f +* g) | D

proof end;

theorem Th102: :: FUNCT_4:102

for x being set holds x .--> x = id {x}

proof end;

theorem :: FUNCT_4:103

for f, g being Function st f c= g holds
f +* g = g

proof end;

theorem Th104: :: FUNCT_4:104

for f, g being Function st f c= g holds
g +* f = g

proof end;

definition

let f be Function;
let x, y be set ;
func f +~ x,y -> set equals :: FUNCT_4:def 5
f +* ((x .--> y) * f);
coherence ;

end;

:: deftheorem defines +~ FUNCT_4:def 5 :

registration

let f be Function;
let x, y be set ;
cluster f +~ x,y -> Relation-like Function-like ;
coherence ;

end;

theorem Tw1: :: FUNCT_4:105

for f being Function
for x, y being set holds dom (f +~ x,y) = dom f

proof end;

theorem Tw2: :: FUNCT_4:106

for f being Function
for x, y being set st x <> y holds
not x in rng (f +~ x,y)

proof end;

theorem :: FUNCT_4:107

for f being Function
for x, y being set st x in rng f holds
y in rng (f +~ x,y)

proof end;

theorem TH1: :: FUNCT_4:108

for f being Function
for x being set holds f +~ x,x = f

proof end;

theorem TH2: :: FUNCT_4:109

for f being Function
for x, y being set st not x in rng f holds
f +~ x,y = f

proof end;

theorem :: FUNCT_4:110

for f being Function
for x, y being set holds rng (f +~ x,y) c= ((rng f) \ {x}) \/ {y}

proof end;

theorem :: FUNCT_4:111

for z being set
for f being Function
for x, y being set st f . z <> x holds
(f +~ x,y) . z = f . z

proof end;

theorem :: FUNCT_4:112

for z being set
for f being Function
for x, y being set st z in dom f & f . z = x holds
(f +~ x,y) . z = y

proof end;

theorem :: FUNCT_4:113

canceled;

theorem :: FUNCT_4:114

canceled;

theorem :: FUNCT_4:115

for f being Function
for x, y being set st not x in dom f holds
f c= f +* (x .--> y)

proof end;

theorem :: FUNCT_4:116

for X, Y being set
for f being PartFunc of X,Y
for x, y being set st x in X & y in Y holds
f +* (x .--> y) is PartFunc of X,Y

proof end;

registration

let f be Function;
let g be non empty Function;
cluster f +* g -> non empty ;
coherence

proof end;

cluster g +* f -> non empty ;
coherence

proof end;

end;