:: The canonical formulae
:: by Andrzej Trybulec
::
:: Received July 4, 2000
:: Copyright (c) 2000 Association of Mizar Users

begin

theorem :: HILBERT3:1

for i being Integer holds
( i is even iff not i - 1 is even ) ;

theorem :: HILBERT3:2

for i being Integer holds
( not i is even iff i - 1 is even ) ;

theorem Th3: :: HILBERT3:3

for X being trivial set
for x being set st x in X holds
for f being Function of X,X holds x is_a_fixpoint_of f

proof end;

registration

let A, B, C be set ;
cluster -> Function-yielding Element of bool [:A,(Funcs B,C):];
coherence
for b₁ being Function of A,(Funcs B,C) holds b₁ is Function-yielding

proof end;

end;

theorem Th4: :: HILBERT3:4

for f being Function-yielding Function holds SubFuncs (rng f) = rng f

proof end;

theorem Th5: :: HILBERT3:5

for A, B, x being set
for f being Function st x in A & f in Funcs A,B holds
f . x in B

proof end;

theorem Th6: :: HILBERT3:6

for A, B, C being set st ( not C = {} or B = {} or A = {} ) holds
for f being Function of A,(Funcs B,C) holds doms f = A --> B

proof end;

theorem Th7: :: HILBERT3:7

for x being set holds {} . x = {}

proof end;

registration

let A be set ;
let B be functional set ;
cluster -> Function-yielding Element of bool [:A,B:];
coherence
for b₁ being Function of A,B holds b₁ is Function-yielding

proof end;

end;

theorem Th8: :: HILBERT3:8

for X being set
for A being Subset of X holds (0 ,1 --> 1,0 ) * (chi A,X) = chi (A ` ),X

proof end;

theorem Th9: :: HILBERT3:9

for X being set
for A being Subset of X holds (0 ,1 --> 1,0 ) * (chi (A ` ),X) = chi A,X

proof end;

theorem Th10: :: HILBERT3:10

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 Th11: :: HILBERT3:11

for a, b, x, y, X, Y being set st a <> b & x in X & y in Y holds
a,b --> x,y in product (a,b --> X,Y)

proof end;

theorem Th12: :: HILBERT3:12

for D being non empty set
for f being Function of 2,D ex d1, d2 being Element of D st f = 0 ,1 --> d1,d2

proof end;

theorem Th13: :: HILBERT3:13

for a, b, c, d being set st a <> b holds
(a,b --> c,d) * (a,b --> b,a) = a,b --> d,c

proof end;

theorem Th14: :: HILBERT3:14

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

proof end;

begin

registration

let f, g be one-to-one Function;
cluster [:f,g:] -> one-to-one ;
coherence
[:f,g:] is one-to-one

proof end;

end;

theorem Th15: :: HILBERT3:15

for A, B being non empty set
for C, D being set
for f being Function of C,A
for g being Function of D,B holds (pr1 A,B) * [:f,g:] = f * (pr1 C,D)

proof end;

theorem Th16: :: HILBERT3:16

for A, B being non empty set
for C, D being set
for f being Function of C,A
for g being Function of D,B holds (pr2 A,B) * [:f,g:] = g * (pr2 C,D)

proof end;

theorem :: HILBERT3:17

for g being Function holds {} .. g = {}

proof end;

theorem Th18: :: HILBERT3:18

for f being Function-yielding Function
for g, h being Function holds (f .. g) * h = (f * h) .. (g * h)

proof end;

theorem :: HILBERT3:19

for C being set
for A being non empty set
for f being Function of A,(Funcs {} ,C)
for g being Function of A,{} holds rng (f .. g) = {{} }

proof end;

theorem Th20: :: HILBERT3:20

for A, B, C being set st ( B = {} implies A = {} ) holds
for f being Function of A,(Funcs B,C)
for g being Function of A,B holds rng (f .. g) c= C

proof end;

theorem Th21: :: HILBERT3:21

for A, B, C being set st ( not C = {} or B = {} or A = {} ) holds
for f being Function of A,(Funcs B,C) holds dom (Frege f) = Funcs A,B

proof end;

theorem :: HILBERT3:22

canceled;

theorem Th23: :: HILBERT3:23

for A, B, C being set st ( not C = {} or B = {} or A = {} ) holds
for f being Function of A,(Funcs B,C) holds rng (Frege f) c= Funcs A,C

proof end;

theorem Th24: :: HILBERT3:24

for A, B, C being set st ( not C = {} or B = {} or A = {} ) holds
for f being Function of A,(Funcs B,C) holds Frege f is Function of (Funcs A,B),(Funcs A,C)

proof end;

begin

theorem Th25: :: HILBERT3:25

for A, B being set
for P being Permutation of A
for Q being Permutation of B holds [:P,Q:] is bijective

proof end;

definition

let A, B be non empty set ;
let P be Permutation of A;
let Q be Function of B,B;
func P => Q -> Function of (Funcs A,B),(Funcs A,B) means :Def1: :: HILBERT3:def 1
for f being Function of A,B holds it . f = (Q * f) * (P " );
existence
ex b₁ being Function of (Funcs A,B),(Funcs A,B) st
for f being Function of A,B holds b₁ . f = (Q * f) * (P " )

proof end;

uniqueness
for b₁, b₂ being Function of (Funcs A,B),(Funcs A,B) st ( for f being Function of A,B holds b₁ . f = (Q * f) * (P " ) ) & ( for f being Function of A,B holds b₂ . f = (Q * f) * (P " ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines => HILBERT3:def 1 :

for A, B being non empty set
for P being Permutation of A
for Q being Function of B,B
for b₅ being Function of (Funcs A,B),(Funcs A,B) holds
( b₅ = P => Q iff for f being Function of A,B holds b₅ . f = (Q * f) * (P " ) );

registration

let A, B be non empty set ;
let P be Permutation of A;
let Q be Permutation of B;
cluster P => Q -> bijective ;
coherence
P => Q is bijective

proof end;

end;

theorem Th26: :: HILBERT3:26

for A, B being non empty set
for P being Permutation of A
for Q being Permutation of B
for f being Function of A,B holds ((P => Q) " ) . f = ((Q " ) * f) * P

proof end;

theorem Th27: :: HILBERT3:27

for A, B being non empty set
for P being Permutation of A
for Q being Permutation of B holds (P => Q) " = (P " ) => (Q " )

proof end;

theorem Th28: :: HILBERT3:28

for A, B, C being non empty set
for f being Function of A,(Funcs B,C)
for g being Function of A,B
for P being Permutation of B
for Q being Permutation of C holds ((P => Q) * f) .. (P * g) = Q * (f .. g)

proof end;

begin

definition

mode SetValuation is V5() ManySortedSet of NAT ;

end;

definition

let V be SetValuation;
func SetVal V -> ManySortedSet of HP-WFF means :Def2: :: HILBERT3:def 2
( it . VERUM = 1 & ( for n being Element of NAT holds it . (prop n) = V . n ) & ( for p, q being Element of HP-WFF holds
( it . (p '&' q) = [:(it . p),(it . q):] & it . (p => q) = Funcs (it . p),(it . q) ) ) );
existence
ex b₁ being ManySortedSet of HP-WFF st
( b₁ . VERUM = 1 & ( for n being Element of NAT holds b₁ . (prop n) = V . n ) & ( for p, q being Element of HP-WFF holds
( b₁ . (p '&' q) = [:(b₁ . p),(b₁ . q):] & b₁ . (p => q) = Funcs (b₁ . p),(b₁ . q) ) ) )

proof end;

uniqueness
for b₁, b₂ being ManySortedSet of HP-WFF st b₁ . VERUM = 1 & ( for n being Element of NAT holds b₁ . (prop n) = V . n ) & ( for p, q being Element of HP-WFF holds
( b₁ . (p '&' q) = [:(b₁ . p),(b₁ . q):] & b₁ . (p => q) = Funcs (b₁ . p),(b₁ . q) ) ) & b₂ . VERUM = 1 & ( for n being Element of NAT holds b₂ . (prop n) = V . n ) & ( for p, q being Element of HP-WFF holds
( b₂ . (p '&' q) = [:(b₂ . p),(b₂ . q):] & b₂ . (p => q) = Funcs (b₂ . p),(b₂ . q) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines SetVal HILBERT3:def 2 :

for V being SetValuation
for b₂ being ManySortedSet of HP-WFF holds
( b₂ = SetVal V iff ( b₂ . VERUM = 1 & ( for n being Element of NAT holds b₂ . (prop n) = V . n ) & ( for p, q being Element of HP-WFF holds
( b₂ . (p '&' q) = [:(b₂ . p),(b₂ . q):] & b₂ . (p => q) = Funcs (b₂ . p),(b₂ . q) ) ) ) );

definition

let V be SetValuation;
let p be Element of HP-WFF ;
func SetVal V,p -> set equals :: HILBERT3:def 3
(SetVal V) . p;
correctness
coherence
(SetVal V) . p is set ;
;

end;

:: deftheorem defines SetVal HILBERT3:def 3 :

for V being SetValuation
for p being Element of HP-WFF holds SetVal V,p = (SetVal V) . p;

registration

let V be SetValuation;
let p be Element of HP-WFF ;
cluster SetVal V,p -> non empty ;
coherence
not SetVal V,p is empty

proof end;

end;

theorem :: HILBERT3:29

for V being SetValuation holds SetVal V,VERUM = 1 by Def2;

theorem :: HILBERT3:30

for n being Element of NAT
for V being SetValuation holds SetVal V,(prop n) = V . n by Def2;

theorem :: HILBERT3:31

for p, q being Element of HP-WFF
for V being SetValuation holds SetVal V,(p '&' q) = [:(SetVal V,p),(SetVal V,q):] by Def2;

theorem :: HILBERT3:32

for p, q being Element of HP-WFF
for V being SetValuation holds SetVal V,(p => q) = Funcs (SetVal V,p),(SetVal V,q) by Def2;

registration

let V be SetValuation;
let p, q be Element of HP-WFF ;
cluster SetVal V,(p => q) -> functional ;
coherence
SetVal V,(p => q) is functional

proof end;

end;

registration

let V be SetValuation;
let p, q, r be Element of HP-WFF ;
cluster -> Function-yielding Element of SetVal V,(p => (q => r));
coherence
for b₁ being Element of SetVal V,(p => (q => r)) holds b₁ is Function-yielding

proof end;

end;

registration

let V be SetValuation;
let p, q, r be Element of HP-WFF ;
cluster Function-yielding Element of bool [:(SetVal V,(p => q)),(SetVal V,(p => r)):];
existence
ex b₁ being Function of (SetVal V,(p => q)),(SetVal V,(p => r)) st b₁ is Function-yielding

proof end;

cluster Function-yielding Element of SetVal V,(p => (q => r));
existence
ex b₁ being Element of SetVal V,(p => (q => r)) st b₁ is Function-yielding

proof end;

end;

begin

definition

let V be SetValuation;
mode Permutation of V -> Function means :Def4: :: HILBERT3:def 4
( dom it = NAT & ( for n being Element of NAT holds it . n is Permutation of (V . n) ) );
existence
ex b₁ being Function st
( dom b₁ = NAT & ( for n being Element of NAT holds b₁ . n is Permutation of (V . n) ) )

proof end;

end;

:: deftheorem Def4 defines Permutation HILBERT3:def 4 :

for V being SetValuation
for b₂ being Function holds
( b₂ is Permutation of V iff ( dom b₂ = NAT & ( for n being Element of NAT holds b₂ . n is Permutation of (V . n) ) ) );

definition

let V be SetValuation;
let P be Permutation of V;
func Perm P -> ManySortedFunction of SetVal V, SetVal V means :Def5: :: HILBERT3:def 5
( it . VERUM = id 1 & ( for n being Element of NAT holds it . (prop n) = P . n ) & ( for p, q being Element of HP-WFF ex p' being Permutation of (SetVal V,p) ex q' being Permutation of (SetVal V,q) st
( p' = it . p & q' = it . q & it . (p '&' q) = [:p',q':] & it . (p => q) = p' => q' ) ) );
existence
ex b₁ being ManySortedFunction of SetVal V, SetVal V st
( b₁ . VERUM = id 1 & ( for n being Element of NAT holds b₁ . (prop n) = P . n ) & ( for p, q being Element of HP-WFF ex p' being Permutation of (SetVal V,p) ex q' being Permutation of (SetVal V,q) st
( p' = b₁ . p & q' = b₁ . q & b₁ . (p '&' q) = [:p',q':] & b₁ . (p => q) = p' => q' ) ) )

proof end;

uniqueness
for b₁, b₂ being ManySortedFunction of SetVal V, SetVal V st b₁ . VERUM = id 1 & ( for n being Element of NAT holds b₁ . (prop n) = P . n ) & ( for p, q being Element of HP-WFF ex p' being Permutation of (SetVal V,p) ex q' being Permutation of (SetVal V,q) st
( p' = b₁ . p & q' = b₁ . q & b₁ . (p '&' q) = [:p',q':] & b₁ . (p => q) = p' => q' ) ) & b₂ . VERUM = id 1 & ( for n being Element of NAT holds b₂ . (prop n) = P . n ) & ( for p, q being Element of HP-WFF ex p' being Permutation of (SetVal V,p) ex q' being Permutation of (SetVal V,q) st
( p' = b₂ . p & q' = b₂ . q & b₂ . (p '&' q) = [:p',q':] & b₂ . (p => q) = p' => q' ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines Perm HILBERT3:def 5 :

for V being SetValuation
for P being Permutation of V
for b₃ being ManySortedFunction of SetVal V, SetVal V holds
( b₃ = Perm P iff ( b₃ . VERUM = id 1 & ( for n being Element of NAT holds b₃ . (prop n) = P . n ) & ( for p, q being Element of HP-WFF ex p' being Permutation of (SetVal V,p) ex q' being Permutation of (SetVal V,q) st
( p' = b₃ . p & q' = b₃ . q & b₃ . (p '&' q) = [:p',q':] & b₃ . (p => q) = p' => q' ) ) ) );

definition

let V be SetValuation;
let P be Permutation of V;
let p be Element of HP-WFF ;
func Perm P,p -> Function of (SetVal V,p),(SetVal V,p) equals :: HILBERT3:def 6
(Perm P) . p;
correctness
coherence
(Perm P) . p is Function of (SetVal V,p),(SetVal V,p);
;

end;

:: deftheorem defines Perm HILBERT3:def 6 :

for V being SetValuation
for P being Permutation of V
for p being Element of HP-WFF holds Perm P,p = (Perm P) . p;

theorem Th33: :: HILBERT3:33

for V being SetValuation
for P being Permutation of V holds Perm P,VERUM = id (SetVal V,VERUM )

proof end;

theorem :: HILBERT3:34

for n being Element of NAT
for V being SetValuation
for P being Permutation of V holds Perm P,(prop n) = P . n by Def5;

theorem Th35: :: HILBERT3:35

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V holds Perm P,(p '&' q) = [:(Perm P,p),(Perm P,q):]

proof end;

theorem Th36: :: HILBERT3:36

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for p' being Permutation of (SetVal V,p)
for q' being Permutation of (SetVal V,q) st p' = Perm P,p & q' = Perm P,q holds
Perm P,(p => q) = p' => q'

proof end;

registration

let V be SetValuation;
let P be Permutation of V;
let p be Element of HP-WFF ;
cluster Perm P,p -> bijective ;
coherence
Perm P,p is bijective

proof end;

end;

theorem Th37: :: HILBERT3:37

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for g being Function of (SetVal V,p),(SetVal V,q) holds (Perm P,(p => q)) . g = ((Perm P,q) * g) * ((Perm P,p) " )

proof end;

theorem Th38: :: HILBERT3:38

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for g being Function of (SetVal V,p),(SetVal V,q) holds ((Perm P,(p => q)) " ) . g = (((Perm P,q) " ) * g) * (Perm P,p)

proof end;

theorem Th39: :: HILBERT3:39

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for f, g being Function of (SetVal V,p),(SetVal V,q) st f = (Perm P,(p => q)) . g holds
(Perm P,q) * g = f * (Perm P,p)

proof end;

theorem Th40: :: HILBERT3:40

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for x being set st x is_a_fixpoint_of Perm P,p holds
for f being Function st f is_a_fixpoint_of Perm P,(p => q) holds
f . x is_a_fixpoint_of Perm P,q

proof end;

begin

definition

let p be Element of HP-WFF ;
attr p is canonical means :Def7: :: HILBERT3:def 7
for V being SetValuation ex x being set st
for P being Permutation of V holds x is_a_fixpoint_of Perm P,p;

end;

:: deftheorem Def7 defines canonical HILBERT3:def 7 :

for p being Element of HP-WFF holds
( p is canonical iff for V being SetValuation ex x being set st
for P being Permutation of V holds x is_a_fixpoint_of Perm P,p );

registration

cluster VERUM -> canonical ;
coherence
VERUM is canonical

proof end;

end;

theorem Th41: :: HILBERT3:41

for p, q being Element of HP-WFF holds p => (q => p) is canonical

proof end;

theorem Th42: :: HILBERT3:42

for p, q, r being Element of HP-WFF holds (p => (q => r)) => ((p => q) => (p => r)) is canonical

proof end;

theorem Th43: :: HILBERT3:43

for p, q being Element of HP-WFF holds (p '&' q) => p is canonical

proof end;

theorem Th44: :: HILBERT3:44

for p, q being Element of HP-WFF holds (p '&' q) => q is canonical

proof end;

theorem Th45: :: HILBERT3:45

for p, q being Element of HP-WFF holds p => (q => (p '&' q)) is canonical

proof end;

theorem Th46: :: HILBERT3:46

for p, q being Element of HP-WFF st p is canonical & p => q is canonical holds
q is canonical

proof end;

theorem :: HILBERT3:47

for p being Element of HP-WFF st p in HP_TAUT holds
p is canonical

proof end;

registration

cluster canonical Element of HP-WFF ;
existence
ex b₁ being Element of HP-WFF st b₁ is canonical

proof end;

end;

begin

definition

let p be Element of HP-WFF ;
attr p is pseudo-canonical means :Def8: :: HILBERT3:def 8
for V being SetValuation
for P being Permutation of V ex x being set st x is_a_fixpoint_of Perm P,p;

end;

:: deftheorem Def8 defines pseudo-canonical HILBERT3:def 8 :

for p being Element of HP-WFF holds
( p is pseudo-canonical iff for V being SetValuation
for P being Permutation of V ex x being set st x is_a_fixpoint_of Perm P,p );

registration

cluster canonical -> pseudo-canonical Element of HP-WFF ;
coherence
for b₁ being Element of HP-WFF st b₁ is canonical holds
b₁ is pseudo-canonical

proof end;

end;

theorem :: HILBERT3:48

for p, q being Element of HP-WFF holds p => (q => p) is pseudo-canonical

proof end;

theorem :: HILBERT3:49

for p, q, r being Element of HP-WFF holds (p => (q => r)) => ((p => q) => (p => r)) is pseudo-canonical

proof end;

theorem :: HILBERT3:50

for p, q being Element of HP-WFF holds (p '&' q) => p is pseudo-canonical

proof end;

theorem :: HILBERT3:51

for p, q being Element of HP-WFF holds (p '&' q) => q is pseudo-canonical

proof end;

theorem :: HILBERT3:52

for p, q being Element of HP-WFF holds p => (q => (p '&' q)) is pseudo-canonical

proof end;

theorem :: HILBERT3:53

for p, q being Element of HP-WFF st p is pseudo-canonical & p => q is pseudo-canonical holds
q is pseudo-canonical

proof end;

theorem Th54: :: HILBERT3:54

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V st ex f being set st f is_a_fixpoint_of Perm P,p & ( for f being set holds not f is_a_fixpoint_of Perm P,q ) holds
not p => q is pseudo-canonical

proof end;

theorem :: HILBERT3:55

not (((prop 0 ) => (prop 1)) => (prop 0 )) => (prop 0 ) is pseudo-canonical

proof end;