:: Algebra of Normal Forms
:: by Andrzej Trybulec
::
:: Copyright (c) 1990-2021 Association of Mizar Users

definition
let A, B be non empty preBoolean set ;
let x, y be Element of [:A,B:];
pred x c= y means :: NORMFORM:def 1
( x 1 c= y 1 & x 2 c= y 2 );
reflexivity
for x being Element of [:A,B:] holds
( x 1 c= x 1 & x 2 c= x 2 )
;
func x \/ y -> Element of [:A,B:] equals :: NORMFORM:def 2
[((x 1) \/ (y 1)),((x 2) \/ (y 2))];
correctness
coherence
[((x 1) \/ (y 1)),((x 2) \/ (y 2))] is Element of [:A,B:]
;
;
commutativity
for b1, x, y being Element of [:A,B:] st b1 = [((x 1) \/ (y 1)),((x 2) \/ (y 2))] holds
b1 = [((y 1) \/ (x 1)),((y 2) \/ (x 2))]
;
idempotence
for x being Element of [:A,B:] holds x = [((x 1) \/ (x 1)),((x 2) \/ (x 2))]
by MCART_1:21;
func x /\ y -> Element of [:A,B:] equals :: NORMFORM:def 3
[((x 1) /\ (y 1)),((x 2) /\ (y 2))];
correctness
coherence
[((x 1) /\ (y 1)),((x 2) /\ (y 2))] is Element of [:A,B:]
;
;
commutativity
for b1, x, y being Element of [:A,B:] st b1 = [((x 1) /\ (y 1)),((x 2) /\ (y 2))] holds
b1 = [((y 1) /\ (x 1)),((y 2) /\ (x 2))]
;
idempotence
for x being Element of [:A,B:] holds x = [((x 1) /\ (x 1)),((x 2) /\ (x 2))]
by MCART_1:21;
func x \ y -> Element of [:A,B:] equals :: NORMFORM:def 4
[((x 1) \ (y 1)),((x 2) \ (y 2))];
correctness
coherence
[((x 1) \ (y 1)),((x 2) \ (y 2))] is Element of [:A,B:]
;
;
func x \+\ y -> Element of [:A,B:] equals :: NORMFORM:def 5
[((x 1) \+\ (y 1)),((x 2) \+\ (y 2))];
correctness
coherence
[((x 1) \+\ (y 1)),((x 2) \+\ (y 2))] is Element of [:A,B:]
;
;
commutativity
for b1, x, y being Element of [:A,B:] st b1 = [((x 1) \+\ (y 1)),((x 2) \+\ (y 2))] holds
b1 = [((y 1) \+\ (x 1)),((y 2) \+\ (x 2))]
;
end;

:: deftheorem defines c= NORMFORM:def 1 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds
( x c= y iff ( x 1 c= y 1 & x 2 c= y 2 ) );

:: deftheorem defines \/ NORMFORM:def 2 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \/ y = [((x 1) \/ (y 1)),((x 2) \/ (y 2))];

:: deftheorem defines /\ NORMFORM:def 3 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x /\ y = [((x 1) /\ (y 1)),((x 2) /\ (y 2))];

:: deftheorem defines \ NORMFORM:def 4 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \ y = [((x 1) \ (y 1)),((x 2) \ (y 2))];

:: deftheorem defines \+\ NORMFORM:def 5 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \+\ y = [((x 1) \+\ (y 1)),((x 2) \+\ (y 2))];

theorem Th1: :: NORMFORM:1
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] st a c= b & b c= a holds
a = b
proof end;

theorem Th2: :: NORMFORM:2
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b & b c= c holds
a c= c
proof end;

theorem Th3: :: NORMFORM:3
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds (a \/ b) \/ c = a \/ (b \/ c)
proof end;

theorem :: NORMFORM:4
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds (a /\ b) /\ c = a /\ (b /\ c)
proof end;

theorem Th5: :: NORMFORM:5
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds a /\ (b \/ c) = (a /\ b) \/ (a /\ c)
proof end;

theorem Th6: :: NORMFORM:6
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds a \/ (b /\ a) = a
proof end;

theorem Th7: :: NORMFORM:7
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds a /\ (b \/ a) = a
proof end;

theorem :: NORMFORM:8
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds a \/ (b /\ c) = (a \/ b) /\ (a \/ c)
proof end;

theorem Th9: :: NORMFORM:9
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= c & b c= c holds
a \/ b c= c by XBOOLE_1:8;

theorem Th10: :: NORMFORM:10
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds
( a c= a \/ b & b c= a \/ b ) by XBOOLE_1:7;

theorem :: NORMFORM:11
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] st a = a \/ b holds
b c= a by Th10;

theorem Th12: :: NORMFORM:12
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b holds
( c \/ a c= c \/ b & a \/ c c= b \/ c ) by XBOOLE_1:9;

theorem :: NORMFORM:13
for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds (a \ b) \/ b = a \/ b
proof end;

theorem :: NORMFORM:14
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a \ b c= c holds
a c= b \/ c by XBOOLE_1:44;

theorem :: NORMFORM:15
for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b \/ c holds
a \ c c= b by XBOOLE_1:43;

definition
let A be set ;
func FinPairUnion A -> BinOp of [:(Fin A),(Fin A):] means :Def6: :: NORMFORM:def 6
for x, y being Element of [:(Fin A),(Fin A):] holds it . (x,y) = x \/ y;
existence
ex b1 being BinOp of [:(Fin A),(Fin A):] st
for x, y being Element of [:(Fin A),(Fin A):] holds b1 . (x,y) = x \/ y
proof end;
uniqueness
for b1, b2 being BinOp of [:(Fin A),(Fin A):] st ( for x, y being Element of [:(Fin A),(Fin A):] holds b1 . (x,y) = x \/ y ) & ( for x, y being Element of [:(Fin A),(Fin A):] holds b2 . (x,y) = x \/ y ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines FinPairUnion NORMFORM:def 6 :
for A being set
for b2 being BinOp of [:(Fin A),(Fin A):] holds
( b2 = FinPairUnion A iff for x, y being Element of [:(Fin A),(Fin A):] holds b2 . (x,y) = x \/ y );

definition
let X be non empty set ;
let A be set ;
let B be Element of Fin X;
let f be Function of X,[:(Fin A),(Fin A):];
func FinPairUnion (B,f) -> Element of [:(Fin A),(Fin A):] equals :: NORMFORM:def 7
() $$(B,f); correctness coherence ()$$ (B,f) is Element of [:(Fin A),(Fin A):]
;
;
end;

:: deftheorem defines FinPairUnion NORMFORM:def 7 :
for X being non empty set
for A being set
for B being Element of Fin X
for f being Function of X,[:(Fin A),(Fin A):] holds FinPairUnion (B,f) = ()  (B,f);

Lm1: for A being set holds FinPairUnion A is idempotent
proof end;

Lm2: for A being set holds FinPairUnion A is commutative
proof end;

Lm3: for A being set holds FinPairUnion A is associative
proof end;

registration
let A be set ;
coherence by ;
end;

theorem :: NORMFORM:16
for A being set
for X being non empty set
for f being Function of X,[:(Fin A),(Fin A):]
for B being Element of Fin X
for x being Element of X st x in B holds
f . x c= FinPairUnion (B,f)
proof end;

theorem Th17: :: NORMFORM:17
for A being set holds [({}. A),({}. A)] is_a_unity_wrt FinPairUnion A
proof end;

theorem Th18: :: NORMFORM:18
for A being set holds FinPairUnion A is having_a_unity
proof end;

theorem Th19: :: NORMFORM:19
for A being set holds the_unity_wrt () = [({}. A),({}. A)]
proof end;

theorem Th20: :: NORMFORM:20
for A being set
for x being Element of [:(Fin A),(Fin A):] holds the_unity_wrt () c= x
proof end;

theorem :: NORMFORM:21
for A being set
for X being non empty set
for f being Function of X,[:(Fin A),(Fin A):]
for B being Element of Fin X
for c being Element of [:(Fin A),(Fin A):] st ( for x being Element of X st x in B holds
f . x c= c ) holds
FinPairUnion (B,f) c= c
proof end;

theorem :: NORMFORM:22
for A being set
for X being non empty set
for B being Element of Fin X
for f, g being Function of X,[:(Fin A),(Fin A):] st f | B = g | B holds
FinPairUnion (B,f) = FinPairUnion (B,g)
proof end;

:: Part 2. Disjoint pairs of finite sets
definition
let X be set ;
func DISJOINT_PAIRS X -> Subset of [:(Fin X),(Fin X):] equals :: NORMFORM:def 8
{ a where a is Element of [:(Fin X),(Fin X):] : a 1 misses a 2 } ;
coherence
{ a where a is Element of [:(Fin X),(Fin X):] : a 1 misses a 2 } is Subset of [:(Fin X),(Fin X):]
proof end;
end;

:: deftheorem defines DISJOINT_PAIRS NORMFORM:def 8 :
for X being set holds DISJOINT_PAIRS X = { a where a is Element of [:(Fin X),(Fin X):] : a 1 misses a 2 } ;

registration
let X be set ;
cluster DISJOINT_PAIRS X -> non empty ;
coherence
not DISJOINT_PAIRS X is empty
proof end;
end;

theorem Th23: :: NORMFORM:23
for X being set
for y being Element of [:(Fin X),(Fin X):] holds
( y in DISJOINT_PAIRS X iff y 1 misses y 2 )
proof end;

theorem :: NORMFORM:24
for X being set
for x, y being Element of [:(Fin X),(Fin X):] st y in DISJOINT_PAIRS X & x in DISJOINT_PAIRS X holds
( y \/ x in DISJOINT_PAIRS X iff ((y 1) /\ (x 2)) \/ ((x 1) /\ (y 2)) = {} )
proof end;

theorem :: NORMFORM:25
for X being set
for a being Element of DISJOINT_PAIRS X holds a 1 misses a 2 by Th23;

theorem Th26: :: NORMFORM:26
for X being set
for x being Element of [:(Fin X),(Fin X):]
for b being Element of DISJOINT_PAIRS X st x c= b holds
x is Element of DISJOINT_PAIRS X
proof end;

theorem Th27: :: NORMFORM:27
for X being set
for a being Element of DISJOINT_PAIRS X
for x being set holds
( not x in a 1 or not x in a 2 ) by ;

theorem :: NORMFORM:28
for X being set
for a, b being Element of DISJOINT_PAIRS X st not a \/ b in DISJOINT_PAIRS X holds
ex p being Element of X st
( ( p in a 1 & p in b 2 ) or ( p in b 1 & p in a 2 ) )
proof end;

theorem :: NORMFORM:29
for X being set
for x being Element of [:(Fin X),(Fin X):] st x 1 misses x 2 holds
x is Element of DISJOINT_PAIRS X by Th23;

theorem :: NORMFORM:30
for X being set
for a being Element of DISJOINT_PAIRS X
for V, W being set st V c= a 1 & W c= a 2 holds
[V,W] is Element of DISJOINT_PAIRS X
proof end;

Lm4: for A being set holds {} in { B where B is Element of Fin () : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b
}

proof end;

definition
let A be set ;
func Normal_forms_on A -> Subset of () equals :: NORMFORM:def 9
{ B where B is Element of Fin () : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b
}
;
coherence
{ B where B is Element of Fin () : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b
}
is Subset of ()
proof end;
end;

:: deftheorem defines Normal_forms_on NORMFORM:def 9 :
for A being set holds Normal_forms_on A = { B where B is Element of Fin () : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b
}
;

registration
let A be set ;
cluster Normal_forms_on A -> non empty ;
coherence
not Normal_forms_on A is empty
by Lm4;
end;

theorem :: NORMFORM:31
for A being set holds {} in Normal_forms_on A by Lm4;

theorem Th32: :: NORMFORM:32
for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin () st B in Normal_forms_on A & a in B & b in B & a c= b holds
a = b
proof end;

theorem Th33: :: NORMFORM:33
for A being set
for B being Element of Fin () st ( for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b ) holds
B in Normal_forms_on A ;

definition
let A be set ;
let B be Element of Fin ();
func mi B -> Element of Normal_forms_on A equals :: NORMFORM:def 10
{ t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t )
}
;
coherence
{ t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t )
}
is Element of Normal_forms_on A
proof end;
correctness
;
let C be Element of Fin ();
func B ^ C -> Element of Fin () equals :: NORMFORM:def 11
() /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } ;
coherence
() /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } is Element of Fin ()
proof end;
correctness
;
end;

:: deftheorem defines mi NORMFORM:def 10 :
for A being set
for B being Element of Fin () holds mi B = { t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t )
}
;

:: deftheorem defines ^ NORMFORM:def 11 :
for A being set
for B, C being Element of Fin () holds B ^ C = () /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } ;

theorem Th34: :: NORMFORM:34
for A being set
for x being Element of [:(Fin A),(Fin A):]
for B, C being Element of Fin () st x in B ^ C holds
ex b, c being Element of DISJOINT_PAIRS A st
( b in B & c in C & x = b \/ c )
proof end;

theorem Th35: :: NORMFORM:35
for A being set
for b, c being Element of DISJOINT_PAIRS A
for B, C being Element of Fin () st b in B & c in C & b \/ c in DISJOINT_PAIRS A holds
b \/ c in B ^ C
proof end;

theorem Th36: :: NORMFORM:36
for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin () st a in mi B holds
( a in B & ( b in B & b c= a implies b = a ) )
proof end;

theorem :: NORMFORM:37
for A being set
for a being Element of DISJOINT_PAIRS A
for B being Element of Fin () st a in mi B holds
a in B by Th36;

theorem :: NORMFORM:38
for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin () st a in mi B & b in B & b c= a holds
b = a by Th36;

theorem Th39: :: NORMFORM:39
for A being set
for a being Element of DISJOINT_PAIRS A
for B being Element of Fin () st a in B & ( for b being Element of DISJOINT_PAIRS A st b in B & b c= a holds
b = a ) holds
a in mi B
proof end;

Lm5: for A being set
for B, C being Element of Fin () st ( for a being Element of DISJOINT_PAIRS A st a in B holds
a in C ) holds
B c= C

proof end;

theorem Th40: :: NORMFORM:40
for A being set
for B being Element of Fin () holds mi B c= B
proof end;

theorem Th41: :: NORMFORM:41
for A being set
for b being Element of DISJOINT_PAIRS A
for B being Element of Fin () st b in B holds
ex c being Element of DISJOINT_PAIRS A st
( c c= b & c in mi B )
proof end;

theorem Th42: :: NORMFORM:42
for A being set
for K being Element of Normal_forms_on A holds mi K = K
proof end;

theorem Th43: :: NORMFORM:43
for A being set
for B, C being Element of Fin () holds mi (B \/ C) c= (mi B) \/ C
proof end;

theorem Th44: :: NORMFORM:44
for A being set
for B, C being Element of Fin () holds mi ((mi B) \/ C) = mi (B \/ C)
proof end;

theorem :: NORMFORM:45
for A being set
for B, C being Element of Fin () holds mi (B \/ (mi C)) = mi (B \/ C) by Th44;

theorem Th46: :: NORMFORM:46
for A being set
for B, C, D being Element of Fin () st B c= C holds
B ^ D c= C ^ D
proof end;

Lm6: for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin () st a in B ^ C holds
ex b being Element of DISJOINT_PAIRS A st
( b c= a & b in (mi B) ^ C )

proof end;

theorem Th47: :: NORMFORM:47
for A being set
for B, C being Element of Fin () holds mi (B ^ C) c= (mi B) ^ C
proof end;

theorem Th48: :: NORMFORM:48
for A being set
for B, C being Element of Fin () holds B ^ C = C ^ B
proof end;

theorem Th49: :: NORMFORM:49
for A being set
for B, C, D being Element of Fin () st B c= C holds
D ^ B c= D ^ C
proof end;

theorem Th50: :: NORMFORM:50
for A being set
for B, C being Element of Fin () holds mi ((mi B) ^ C) = mi (B ^ C)
proof end;

theorem Th51: :: NORMFORM:51
for A being set
for B, C being Element of Fin () holds mi (B ^ (mi C)) = mi (B ^ C)
proof end;

theorem Th52: :: NORMFORM:52
for A being set
for K, L, M being Element of Normal_forms_on A holds K ^ (L ^ M) = (K ^ L) ^ M
proof end;

theorem Th53: :: NORMFORM:53
for A being set
for K, L, M being Element of Normal_forms_on A holds K ^ (L \/ M) = (K ^ L) \/ (K ^ M)
proof end;

Lm7: for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin () st a in B ^ C holds
ex c being Element of DISJOINT_PAIRS A st
( c in C & c c= a )

proof end;

Lm8: for A being set
for K, L being Element of Normal_forms_on A holds mi ((K ^ L) \/ L) = mi L

proof end;

theorem Th54: :: NORMFORM:54
for A being set
for B being Element of Fin () holds B c= B ^ B
proof end;

theorem Th55: :: NORMFORM:55
for A being set
for K being Element of Normal_forms_on A holds mi (K ^ K) = mi K
proof end;

definition
let A be set ;
func NormForm A -> strict LattStr means :Def12: :: NORMFORM:def 12
( the carrier of it = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of it . (B,C) = mi (B \/ C) & the L_meet of it . (B,C) = mi (B ^ C) ) ) );
existence
ex b1 being strict LattStr st
( the carrier of b1 = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b1 . (B,C) = mi (B \/ C) & the L_meet of b1 . (B,C) = mi (B ^ C) ) ) )
proof end;
uniqueness
for b1, b2 being strict LattStr st the carrier of b1 = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b1 . (B,C) = mi (B \/ C) & the L_meet of b1 . (B,C) = mi (B ^ C) ) ) & the carrier of b2 = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b2 . (B,C) = mi (B \/ C) & the L_meet of b2 . (B,C) = mi (B ^ C) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines NormForm NORMFORM:def 12 :
for A being set
for b2 being strict LattStr holds
( b2 = NormForm A iff ( the carrier of b2 = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b2 . (B,C) = mi (B \/ C) & the L_meet of b2 . (B,C) = mi (B ^ C) ) ) ) );

registration
let A be set ;
cluster NormForm A -> non empty strict ;
coherence
not NormForm A is empty
proof end;
end;

Lm9: for A being set
for a, b being Element of () holds a "\/" b = b "\/" a

proof end;

Lm10: for A being set
for a, b, c being Element of () holds a "\/" (b "\/" c) = (a "\/" b) "\/" c

proof end;

Lm11: for A being set
for K, L being Element of Normal_forms_on A holds the L_join of () . (( the L_meet of () . (K,L)),L) = L

proof end;

Lm12: for A being set
for a, b being Element of () holds (a "/\" b) "\/" b = b

proof end;

Lm13: for A being set
for a, b being Element of () holds a "/\" b = b "/\" a

proof end;

Lm14: for A being set
for a, b, c being Element of () holds a "/\" (b "/\" c) = (a "/\" b) "/\" c

proof end;

Lm15: for A being set
for K, L, M being Element of Normal_forms_on A holds the L_meet of () . (K,( the L_join of () . (L,M))) = the L_join of () . (( the L_meet of () . (K,L)),( the L_meet of () . (K,M)))

proof end;

Lm16: for A being set
for a, b being Element of () holds a "/\" (a "\/" b) = a

proof end;

registration
let A be set ;
coherence
proof end;
end;

registration
let A be set ;
coherence
proof end;
end;

theorem :: NORMFORM:56
for A being set holds {} is Element of ()
proof end;

theorem :: NORMFORM:57
for A being set holds Bottom () = {}
proof end;