BVFUNC26: Propositional Calculus for Boolean Valued Functions, {VIII}

:: Propositional Calculus for Boolean Valued Functions, {VIII}
:: by Shunichi Kobayashi
::
:: Received September 25, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

definition

let p, q be boolean-valued Function;
func p 'nand' q -> Function means :Def1: :: BVFUNC26:def 1
( dom it = (dom p) /\ (dom q) & ( for x being set st x in dom it holds
it . x = (p . x) 'nand' (q . x) ) );
existence

proof end;

proof end;

commutativity ;
func p 'nor' q -> Function means :Def2: :: BVFUNC26:def 2
( dom it = (dom p) /\ (dom q) & ( for x being set st x in dom it holds
it . x = (p . x) 'nor' (q . x) ) );
existence

proof end;

proof end;

commutativity ;

end;

:: deftheorem Def1 defines 'nand' BVFUNC26:def 1 :

:: deftheorem Def2 defines 'nor' BVFUNC26:def 2 :

registration

let p, q be boolean-valued Function;
cluster p 'nand' q -> boolean-valued ;
coherence

proof end;

cluster p 'nor' q -> boolean-valued ;
coherence

proof end;

end;

definition

let A be non empty set ;
let p, q be Element of Funcs (A,BOOLEAN);
:: original: 'nand'
redefine func p 'nand' q -> Element of Funcs (A,BOOLEAN) means :Def3: :: BVFUNC26:def 3
for x being Element of A holds it . x = (p . x) 'nand' (q . x);
coherence

proof end;

proof end;

:: original: 'nor'
redefine func p 'nor' q -> Element of Funcs (A,BOOLEAN) means :Def4: :: BVFUNC26:def 4
for x being Element of A holds it . x = (p . x) 'nor' (q . x);
coherence

proof end;

proof end;

end;

:: deftheorem Def3 defines 'nand' BVFUNC26:def 3 :

:: deftheorem Def4 defines 'nor' BVFUNC26:def 4 :

definition

let Y be non empty set ;
let a, b be Element of Funcs (Y,BOOLEAN);
:: original: 'nand'
redefine func a 'nand' b -> Element of Funcs (Y,BOOLEAN);
coherence

proof end;

:: original: 'nor'
redefine func a 'nor' b -> Element of Funcs (Y,BOOLEAN);
coherence

proof end;

end;

theorem Th1: :: BVFUNC26:1

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' b = 'not' (a '&' b)

proof end;

theorem Th2: :: BVFUNC26:2

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' b = 'not' (a 'or' b)

proof end;

theorem Th3: :: BVFUNC26:3

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds (I_el Y) 'nand' a = 'not' a

proof end;

theorem Th4: :: BVFUNC26:4

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds (O_el Y) 'nand' a = I_el Y

proof end;

theorem :: BVFUNC26:5

for Y being non empty set holds
( (O_el Y) 'nand' (O_el Y) = I_el Y & (O_el Y) 'nand' (I_el Y) = I_el Y & (I_el Y) 'nand' (I_el Y) = O_el Y )

proof end;

theorem :: BVFUNC26:6

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds
( a 'nand' a = 'not' a & 'not' (a 'nand' a) = a )

proof end;

theorem :: BVFUNC26:7

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds 'not' (a 'nand' b) = a '&' b

proof end;

theorem :: BVFUNC26:8

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds
( a 'nand' ('not' a) = I_el Y & 'not' (a 'nand' ('not' a)) = O_el Y )

proof end;

theorem Th9: :: BVFUNC26:9

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b '&' c) = 'not' ((a '&' b) '&' c)

proof end;

theorem Th10: :: BVFUNC26:10

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b '&' c) = (a '&' b) 'nand' c

proof end;

theorem Th11: :: BVFUNC26:11

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b 'or' c) = ('not' (a '&' b)) '&' ('not' (a '&' c))

proof end;

theorem :: BVFUNC26:12

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b 'xor' c) = (a '&' b) 'eqv' (a '&' c)

proof end;

theorem :: BVFUNC26:13

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds
( a 'nand' (b 'nand' c) = ('not' a) 'or' (b '&' c) & a 'nand' (b 'nand' c) = a 'imp' (b '&' c) )

proof end;

theorem :: BVFUNC26:14

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds
( a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c & a 'nand' (b 'nor' c) = a 'imp' (b 'or' c) )

proof end;

theorem :: BVFUNC26:15

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b 'eqv' c) = a 'imp' (b 'xor' c)

proof end;

theorem :: BVFUNC26:16

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a '&' b) = a 'nand' b

proof end;

theorem :: BVFUNC26:17

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'or' b) = ('not' a) '&' ('not' (a '&' b))

proof end;

theorem :: BVFUNC26:18

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'eqv' b) = a 'imp' (a 'xor' b)

proof end;

theorem :: BVFUNC26:19

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds
( a 'nand' (a 'nand' b) = ('not' a) 'or' b & a 'nand' (a 'nand' b) = a 'imp' b )

proof end;

theorem :: BVFUNC26:20

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'nor' b) = I_el Y

proof end;

theorem :: BVFUNC26:21

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'eqv' b) = ('not' a) 'or' ('not' b)

proof end;

theorem :: BVFUNC26:22

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a '&' b = (a 'nand' b) 'nand' (a 'nand' b)

proof end;

theorem :: BVFUNC26:23

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds (a 'nand' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)

proof end;

theorem Th24: :: BVFUNC26:24

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b 'imp' c) = (('not' a) 'or' b) '&' ('not' (a '&' c))

proof end;

theorem :: BVFUNC26:25

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'imp' b) = 'not' (a '&' b)

proof end;

theorem Th26: :: BVFUNC26:26

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds (I_el Y) 'nor' a = O_el Y

proof end;

theorem Th27: :: BVFUNC26:27

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds (O_el Y) 'nor' a = 'not' a

proof end;

theorem :: BVFUNC26:28

for Y being non empty set holds
( (O_el Y) 'nor' (O_el Y) = I_el Y & (O_el Y) 'nor' (I_el Y) = O_el Y & (I_el Y) 'nor' (I_el Y) = O_el Y )

proof end;

theorem :: BVFUNC26:29

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds
( a 'nor' a = 'not' a & 'not' (a 'nor' a) = a )

proof end;

theorem :: BVFUNC26:30

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds 'not' (a 'nor' b) = a 'or' b

proof end;

theorem :: BVFUNC26:31

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds
( a 'nor' ('not' a) = O_el Y & 'not' (a 'nor' ('not' a)) = I_el Y )

proof end;

theorem :: BVFUNC26:32

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds ('not' a) '&' (a 'xor' b) = ('not' a) '&' b

proof end;

theorem Th33: :: BVFUNC26:33

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b '&' c) = ('not' (a 'or' b)) 'or' ('not' (a 'or' c))

proof end;

theorem Th34: :: BVFUNC26:34

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'or' c) = 'not' ((a 'or' b) 'or' c)

proof end;

theorem Th35: :: BVFUNC26:35

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'eqv' c) = ('not' a) '&' (b 'xor' c)

proof end;

theorem Th36: :: BVFUNC26:36

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'imp' c) = (('not' a) '&' b) '&' ('not' c)

proof end;

theorem Th37: :: BVFUNC26:37

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'nand' c) = (('not' a) '&' b) '&' c

proof end;

theorem Th38: :: BVFUNC26:38

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'nor' c) = ('not' a) '&' (b 'or' c)

proof end;

theorem :: BVFUNC26:39

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a '&' b) = 'not' (a '&' (a 'or' b))

proof end;

theorem :: BVFUNC26:40

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'or' b) = 'not' (a 'or' b)

proof end;

theorem :: BVFUNC26:41

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'eqv' b) = ('not' a) '&' b

proof end;

theorem :: BVFUNC26:42

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'imp' b) = O_el Y

proof end;

theorem :: BVFUNC26:43

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'nand' b) = O_el Y

proof end;

theorem :: BVFUNC26:44

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'nor' b) = ('not' a) '&' b

proof end;

theorem :: BVFUNC26:45

for Y being non empty set holds (O_el Y) 'eqv' (O_el Y) = I_el Y

proof end;

theorem :: BVFUNC26:46

for Y being non empty set holds (O_el Y) 'eqv' (I_el Y) = O_el Y

proof end;

theorem :: BVFUNC26:47

for Y being non empty set holds (I_el Y) 'eqv' (I_el Y) = I_el Y

proof end;

theorem :: BVFUNC26:48

for Y being non empty set
for a being Element of Funcs (Y,BOOLEAN) holds
( a 'eqv' a = I_el Y & 'not' (a 'eqv' a) = O_el Y )

proof end;

theorem :: BVFUNC26:49

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'eqv' (a 'or' b) = a 'or' ('not' b)

proof end;

theorem Th50: :: BVFUNC26:50

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a '&' (b 'nand' c) = (a '&' ('not' b)) 'or' (a '&' ('not' c))

proof end;

theorem Th51: :: BVFUNC26:51

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'or' (b 'nand' c) = (a 'or' ('not' b)) 'or' ('not' c)

proof end;

theorem Th52: :: BVFUNC26:52

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'xor' (b 'nand' c) = (('not' a) '&' ('not' (b '&' c))) 'or' ((a '&' b) '&' c)

proof end;

theorem Th53: :: BVFUNC26:53

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'eqv' (b 'nand' c) = (a '&' ('not' (b '&' c))) 'or' ((('not' a) '&' b) '&' c)

proof end;

theorem Th54: :: BVFUNC26:54

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'imp' (b 'nand' c) = 'not' ((a '&' b) '&' c)

proof end;

theorem Th55: :: BVFUNC26:55

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nor' (b 'nand' c) = 'not' ((a 'or' ('not' b)) 'or' ('not' c))

proof end;

theorem :: BVFUNC26:56

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a '&' (a 'nand' b) = a '&' ('not' b)

proof end;

theorem :: BVFUNC26:57

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'or' (a 'nand' b) = I_el Y

proof end;

theorem :: BVFUNC26:58

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'xor' (a 'nand' b) = ('not' a) 'or' b

proof end;

theorem :: BVFUNC26:59

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'eqv' (a 'nand' b) = a '&' ('not' b)

proof end;

theorem :: BVFUNC26:60

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'imp' (a 'nand' b) = 'not' (a '&' b)

proof end;

theorem :: BVFUNC26:61

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nor' (a 'nand' b) = O_el Y

proof end;

theorem Th62: :: BVFUNC26:62

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a '&' (b 'nor' c) = (a '&' ('not' b)) '&' ('not' c)

proof end;

theorem Th63: :: BVFUNC26:63

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'or' (b 'nor' c) = (a 'or' ('not' b)) '&' (a 'or' ('not' c))

proof end;

theorem Th64: :: BVFUNC26:64

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'xor' (b 'nor' c) = (a 'or' ('not' (b 'or' c))) '&' ((('not' a) 'or' b) 'or' c)

proof end;

theorem Th65: :: BVFUNC26:65

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'eqv' (b 'nor' c) = ((a 'or' b) 'or' c) '&' (('not' a) 'or' ('not' (b 'or' c)))

proof end;

theorem Th66: :: BVFUNC26:66

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'imp' (b 'nor' c) = 'not' (a '&' (b 'or' c))

proof end;

theorem Th67: :: BVFUNC26:67

for Y being non empty set
for a, b, c being Element of Funcs (Y,BOOLEAN) holds a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c

proof end;

theorem :: BVFUNC26:68

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a '&' (a 'nor' b) = O_el Y

proof end;

theorem :: BVFUNC26:69

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'or' (a 'nor' b) = a 'or' ('not' b)

proof end;

theorem :: BVFUNC26:70

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'xor' (a 'nor' b) = a 'or' ('not' b)

proof end;

theorem :: BVFUNC26:71

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'eqv' (a 'nor' b) = ('not' a) '&' b

proof end;

theorem :: BVFUNC26:72

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'imp' (a 'nor' b) = 'not' (a 'or' (a '&' b))

proof end;

theorem :: BVFUNC26:73

for Y being non empty set
for a, b being Element of Funcs (Y,BOOLEAN) holds a 'nand' (a 'nor' b) = I_el Y

proof end;