:: NORMFORM semantic presentation
definition
let A be non
empty preBoolean set ;
let B be non
empty preBoolean set ;
let x be
Element of
[:A,B:];
let y be
Element of
[:A,B:];
pred c3 c= c4 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 c3 \/ c4 -> Element of
[:a1,a2:] 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_1:23;
func c3 /\ c4 -> Element of
[:a1,a2:] 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_1:23;
func c3 \ c4 -> Element of
[:a1,a2:] 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 c3 \+\ c4 -> Element of
[:a1,a2:] 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 Def1 defines c= NORMFORM:def 1 :
:: deftheorem Def2 defines \/ NORMFORM:def 2 :
:: deftheorem Def3 defines /\ NORMFORM:def 3 :
:: deftheorem Def4 defines \ NORMFORM:def 4 :
:: deftheorem Def5 defines \+\ NORMFORM:def 5 :
theorem Th1: :: NORMFORM:1
canceled;
theorem Th2: :: NORMFORM:2
canceled;
theorem Th3: :: NORMFORM:3
canceled;
theorem Th4: :: NORMFORM:4
theorem Th5: :: NORMFORM:5
theorem Th6: :: NORMFORM:6
canceled;
theorem Th7: :: NORMFORM:7
canceled;
theorem Th8: :: NORMFORM:8
canceled;
theorem Th9: :: NORMFORM:9
canceled;
theorem Th10: :: NORMFORM:10
theorem Th11: :: NORMFORM:11
theorem Th12: :: NORMFORM:12
theorem Th13: :: NORMFORM:13
theorem Th14: :: NORMFORM:14
canceled;
theorem Th15: :: NORMFORM:15
canceled;
theorem Th16: :: NORMFORM:16
theorem Th17: :: NORMFORM:17
canceled;
theorem Th18: :: NORMFORM:18
canceled;
theorem Th19: :: NORMFORM:19
theorem Th20: :: NORMFORM:20
theorem Th21: :: NORMFORM:21
theorem Th22: :: NORMFORM:22
theorem Th23: :: NORMFORM:23
canceled;
theorem Th24: :: NORMFORM:24
theorem Th25: :: NORMFORM:25
theorem Th26: :: NORMFORM:26
theorem Th27: :: NORMFORM:27
theorem Th28: :: NORMFORM:28
theorem Th29: :: NORMFORM:29
theorem Th30: :: NORMFORM:30
theorem Th31: :: NORMFORM:31
definition
let A be
set ;
func FinPairUnion c1 -> BinOp of
[:(Fin a1),(Fin a1):] 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
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
end;
:: deftheorem Def6 defines FinPairUnion NORMFORM:def 6 :
:: deftheorem Def7 defines FinPairUnion NORMFORM:def 7 :
Lemma36:
for A being set holds FinPairUnion A is idempotent
Lemma37:
for A being set holds FinPairUnion A is commutative
Lemma38:
for A being set holds FinPairUnion A is associative
theorem Th32: :: NORMFORM:32
canceled;
theorem Th33: :: NORMFORM:33
canceled;
theorem Th34: :: NORMFORM:34
canceled;
theorem Th35: :: NORMFORM:35
theorem Th36: :: NORMFORM:36
theorem Th37: :: NORMFORM:37
theorem Th38: :: NORMFORM:38
theorem Th39: :: NORMFORM:39
theorem Th40: :: NORMFORM:40
theorem Th41: :: NORMFORM:41
:: deftheorem Def8 defines DISJOINT_PAIRS NORMFORM:def 8 :
theorem Th42: :: NORMFORM:42
theorem Th43: :: NORMFORM:43
theorem Th44: :: NORMFORM:44
theorem Th45: :: NORMFORM:45
theorem Th46: :: NORMFORM:46
theorem Th47: :: NORMFORM:47
theorem Th48: :: NORMFORM:48
canceled;
theorem Th49: :: NORMFORM:49
theorem Th50: :: NORMFORM:50
Lemma63:
for A being set holds {} in { B where B is Element of Fin (DISJOINT_PAIRS A) : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b }
:: deftheorem Def9 defines Normal_forms_on NORMFORM:def 9 :
theorem Th51: :: NORMFORM:51
theorem Th52: :: NORMFORM:52
theorem Th53: :: NORMFORM:53
:: deftheorem Def10 defines mi NORMFORM:def 10 :
:: deftheorem Def11 defines ^ NORMFORM:def 11 :
theorem Th54: :: NORMFORM:54
canceled;
theorem Th55: :: NORMFORM:55
theorem Th56: :: NORMFORM:56
theorem Th57: :: NORMFORM:57
canceled;
theorem Th58: :: NORMFORM:58
theorem Th59: :: NORMFORM:59
theorem Th60: :: NORMFORM:60
theorem Th61: :: NORMFORM:61
Lemma76:
for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) st ( for a being Element of DISJOINT_PAIRS A st a in B holds
a in C ) holds
B c= C
theorem Th62: :: NORMFORM:62
canceled;
theorem Th63: :: NORMFORM:63
canceled;
theorem Th64: :: NORMFORM:64
theorem Th65: :: NORMFORM:65
theorem Th66: :: NORMFORM:66
theorem Th67: :: NORMFORM:67
theorem Th68: :: NORMFORM:68
theorem Th69: :: NORMFORM:69
theorem Th70: :: NORMFORM:70
Lemma88:
for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin (DISJOINT_PAIRS A) st a in B ^ C holds
ex b being Element of DISJOINT_PAIRS A st
( b c= a & b in (mi B) ^ C )
theorem Th71: :: NORMFORM:71
theorem Th72: :: NORMFORM:72
theorem Th73: :: NORMFORM:73
theorem Th74: :: NORMFORM:74
theorem Th75: :: NORMFORM:75
theorem Th76: :: NORMFORM:76
theorem Th77: :: NORMFORM:77
Lemma100:
for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin (DISJOINT_PAIRS A) st a in B ^ C holds
ex c being Element of DISJOINT_PAIRS A st
( c in C & c c= a )
Lemma101:
for A being set
for K, L being Element of Normal_forms_on A holds mi ((K ^ L) \/ L) = mi L
theorem Th78: :: NORMFORM:78
theorem Th79: :: NORMFORM:79
definition
let A be
set ;
canceled;canceled;func NormForm c1 -> strict LattStr means :
Def14:
:: NORMFORM:def 14
( 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) ) ) )
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
end;
:: deftheorem Def12 NORMFORM:def 12 :
canceled;
:: deftheorem Def13 NORMFORM:def 13 :
canceled;
:: deftheorem Def14 defines NormForm NORMFORM:def 14 :
Lemma114:
for A being set
for a, b being Element of (NormForm A) holds a "\/" b = b "\/" a
Lemma118:
for A being set
for a, b, c being Element of (NormForm A) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
Lemma120:
for A being set
for K, L being Element of Normal_forms_on A holds the L_join of (NormForm A) . (the L_meet of (NormForm A) . K,L),L = L
Lemma121:
for A being set
for a, b being Element of (NormForm A) holds (a "/\" b) "\/" b = b
Lemma122:
for A being set
for a, b being Element of (NormForm A) holds a "/\" b = b "/\" a
Lemma123:
for A being set
for a, b, c being Element of (NormForm A) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
Lemma124:
for A being set
for K, L, M being Element of Normal_forms_on A holds the L_meet of (NormForm A) . K,(the L_join of (NormForm A) . L,M) = the L_join of (NormForm A) . (the L_meet of (NormForm A) . K,L),(the L_meet of (NormForm A) . K,M)
Lemma128:
for A being set
for a, b being Element of (NormForm A) holds a "/\" (a "\/" b) = a
theorem Th80: :: NORMFORM:80
canceled;
theorem Th81: :: NORMFORM:81
canceled;
theorem Th82: :: NORMFORM:82
canceled;
theorem Th83: :: NORMFORM:83
canceled;
theorem Th84: :: NORMFORM:84
canceled;
theorem Th85: :: NORMFORM:85
theorem Th86: :: NORMFORM:86