:: The canonical formulae
:: by Andrzej Trybulec
::
:: Received July 4, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users

registration

let m, n be non zero Nat;

cluster (0,n) --> (m,0) -> one-to-one ;

coherence
(0,n) --> (m,0) is one-to-one

proof end;

cluster (n,0) --> (0,m) -> one-to-one ;

coherence
(n,0) --> (0,m) is one-to-one

proof end;

end;

theorem :: HILBERT3:1

for i being Integer holds
( i is even iff i - 1 is odd ) ;

theorem :: HILBERT3:2

for i being Integer holds
( i is odd 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;

theorem :: HILBERT3:4

canceled;

::$CT

theorem Th4: :: 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 Th5: :: 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 :: HILBERT3:7

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

theorem Th7: :: 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 Th8: :: 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 Th9: :: HILBERT3:10

for a, b, x, y, x9, y9 being set st a <> b & (a,b) --> (x,y) = (a,b) --> (x9,y9) holds
( x = x9 & y = y9 )

proof end;

theorem Th10: :: 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 Th11: :: 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 Th12: :: 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 Th13: :: 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;

theorem Th14: :: HILBERT3:15

(0,1) --> (1,0) is Permutation of 2

proof end;

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: :: HILBERT3:16

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 (pr1 (A,B)) * [:f,g:] = f * (pr1 (C,D))

proof end;

theorem Th16: :: HILBERT3:17

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 :: HILBERT3:18

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

proof end;

theorem Th18: :: HILBERT3:19

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

proof end;

theorem :: HILBERT3:20

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: :: HILBERT3:21

for A, B, C being set st ( B = {} implies A = {} ) & ( C = {} implies B = {} ) 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: :: HILBERT3:22

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 Th22: :: 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 Th23: :: 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;

registration

let A, B be set ;
let P be Permutation of A;
let Q be Permutation of B;

cluster [:P,Q:] -> bijective for Function of [:A,B:],[:A,B:];

coherence
for b₁ being Function of [:A,B:],[:A,B:] st b₁ = [:P,Q:] holds
b₁ is bijective

proof end;

end;

theorem :: HILBERT3:25

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

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 Th25: :: 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 Th26: :: 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 Th27: :: 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;

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 for 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 Relation-like SetVal (V,(p => q)) -defined SetVal (V,(p => r)) -valued non empty Function-like total quasi_total Function-yielding V59() for 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 Relation-like Function-like Function-yielding V59() for 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;

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 p9 being Permutation of (SetVal (V,p)) ex q9 being Permutation of (SetVal (V,q)) st
( p9 = it . p & q9 = it . q & it . (p '&' q) = [:p9,q9:] & it . (p => q) = p9 => q9 ) ) );

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 p9 being Permutation of (SetVal (V,p)) ex q9 being Permutation of (SetVal (V,q)) st
( p9 = b₁ . p & q9 = b₁ . q & b₁ . (p '&' q) = [:p9,q9:] & b₁ . (p => q) = p9 => q9 ) ) )

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 p9 being Permutation of (SetVal (V,p)) ex q9 being Permutation of (SetVal (V,q)) st
( p9 = b₁ . p & q9 = b₁ . q & b₁ . (p '&' q) = [:p9,q9:] & b₁ . (p => q) = p9 => q9 ) ) & 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 p9 being Permutation of (SetVal (V,p)) ex q9 being Permutation of (SetVal (V,q)) st
( p9 = b₂ . p & q9 = b₂ . q & b₂ . (p '&' q) = [:p9,q9:] & b₂ . (p => q) = p9 => q9 ) ) 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 p9 being Permutation of (SetVal (V,p)) ex q9 being Permutation of (SetVal (V,q)) st
( p9 = b₃ . p & q9 = b₃ . q & b₃ . (p '&' q) = [:p9,q9:] & b₃ . (p => q) = p9 => q9 ) ) ) );

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 Th32: :: 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 Th34: :: 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 Th35: :: HILBERT3:36

for p, q being Element of HP-WFF
for V being SetValuation
for P being Permutation of V
for p9 being Permutation of (SetVal (V,p))
for q9 being Permutation of (SetVal (V,q)) st p9 = Perm (P,p) & q9 = Perm (P,q) holds
Perm (P,(p => q)) = p9 => q9

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 Th36: :: 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 Th37: :: 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 Th38: :: 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 Th39: :: 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;

definition

let p be Element of HP-WFF ;

attr p is canonical means :: 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 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;

registration

let p, q be Element of HP-WFF ;

cluster p => (q => p) -> canonical ;

coherence
p => (q => p) is canonical

proof end;

cluster (p '&' q) => p -> canonical ;

coherence
(p '&' q) => p is canonical

proof end;

cluster (p '&' q) => q -> canonical ;

coherence
(p '&' q) => q is canonical

proof end;

cluster p => (q => (p '&' q)) -> canonical ;

coherence
p => (q => (p '&' q)) is canonical

proof end;

end;

registration

let p, q, r be Element of HP-WFF ;

cluster (p => (q => r)) => ((p => q) => (p => r)) -> canonical ;

coherence
(p => (q => r)) => ((p => q) => (p => r)) is canonical

proof end;

end;

theorem Th40: :: HILBERT3:41

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

proof end;

theorem :: HILBERT3:42

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

proof end;

registration

cluster Relation-like Function-like V55() canonical for Element of HP-WFF ;

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

proof end;

end;

definition

let p be Element of HP-WFF ;

attr p is pseudo-canonical means :: 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 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 for 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 :: HILBERT3:43

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 Th43: :: HILBERT3:44

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 :: HILBERT3:45

for a, b being Element of NAT st a <> b holds
not (((prop a) => (prop b)) => (prop a)) => (prop a) is pseudo-canonical

proof end;