:: ARMSTRNG semantic presentation
theorem Th1: :: ARMSTRNG:1
theorem Th2: :: ARMSTRNG:2
:: deftheorem Def1 defines Maximal_in ARMSTRNG:def 1 :
:: deftheorem Def2 defines is_/\-irreducible_in ARMSTRNG:def 2 :
:: deftheorem Def3 defines /\-IRR ARMSTRNG:def 3 :
theorem Th3: :: ARMSTRNG:3
:: deftheorem Def4 defines (B1) ARMSTRNG:def 4 :
theorem Th4: :: ARMSTRNG:4
:: deftheorem Def5 defines '&' ARMSTRNG:def 5 :
theorem Th5: :: ARMSTRNG:5
theorem Th6: :: ARMSTRNG:6
theorem Th7: :: ARMSTRNG:7
canceled;
theorem Th8: :: ARMSTRNG:8
:: deftheorem Def6 ARMSTRNG:def 6 :
canceled;
:: deftheorem Def7 defines Dependencies ARMSTRNG:def 7 :
theorem Th9: :: ARMSTRNG:9
theorem Th10: :: ARMSTRNG:10
:: deftheorem Def8 defines >|> ARMSTRNG:def 8 :
:: deftheorem Def9 defines Dependency-str ARMSTRNG:def 9 :
theorem Th11: :: ARMSTRNG:11
canceled;
theorem Th12: :: ARMSTRNG:12
:: deftheorem Def10 defines >= ARMSTRNG:def 10 :
theorem Th13: :: ARMSTRNG:13
for b
1 being
set for b
2, b
3 being
Dependency of b
1 holds
( b
2 <= b
3 & b
3 <= b
2 implies b
2 = b
3 )
theorem Th14: :: ARMSTRNG:14
for b
1 being
set for b
2, b
3, b
4 being
Dependency of b
1 holds
( b
2 <= b
3 & b
3 <= b
4 implies b
2 <= b
4 )
theorem Th15: :: ARMSTRNG:15
for b
1 being
set for b
2, b
3, b
4, b
5 being
Subset of b
1 holds
(
[b2,b3] >= [b4,b5] iff ( b
2 c= b
4 & b
5 c= b
3 ) )
:: deftheorem Def11 defines Dependencies-Order ARMSTRNG:def 11 :
theorem Th16: :: ARMSTRNG:16
theorem Th17: :: ARMSTRNG:17
theorem Th18: :: ARMSTRNG:18
theorem Th19: :: ARMSTRNG:19
:: deftheorem Def12 defines (F1) ARMSTRNG:def 12 :
theorem Th20: :: ARMSTRNG:20
definition
let c
1 be
set ;
let c
2 be
Dependency-set of c
1;
attr a
2 is
(F3) means :
Def13:
:: ARMSTRNG:def 13
for b
1, b
2, b
3, b
4 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[b1,b2] >= [b3,b4] implies
[b3,b4] in a
2 );
attr a
2 is
(F4) means :
Def14:
:: ARMSTRNG:def 14
for b
1, b
2, b
3, b
4 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[b3,b4] in a
2 implies
[(b1 \/ b3),(b2 \/ b4)] in a
2 );
end;
:: deftheorem Def13 defines (F3) ARMSTRNG:def 13 :
:: deftheorem Def14 defines (F4) ARMSTRNG:def 14 :
theorem Th21: :: ARMSTRNG:21
:: deftheorem Def15 defines full_family ARMSTRNG:def 15 :
theorem Th22: :: ARMSTRNG:22
:: deftheorem Def16 defines (DC3) ARMSTRNG:def 16 :
theorem Th23: :: ARMSTRNG:23
theorem Th24: :: ARMSTRNG:24
theorem Th25: :: ARMSTRNG:25
theorem Th26: :: ARMSTRNG:26
:: deftheorem Def17 defines Maximal_wrt ARMSTRNG:def 17 :
theorem Th27: :: ARMSTRNG:27
:: deftheorem Def18 defines ^|^ ARMSTRNG:def 18 :
theorem Th28: :: ARMSTRNG:28
theorem Th29: :: ARMSTRNG:29
definition
let c
1 be
set ;
let c
2 be
Dependency-set of c
1;
attr a
2 is
(M1) means :
Def19:
:: ARMSTRNG:def 19
for b
1 being
Subset of a
1 holds
ex b
2, b
3 being
Subset of a
1 st
(
[b2,b3] >= [b1,b1] &
[b2,b3] in a
2 );
attr a
2 is
(M2) means :
Def20:
:: ARMSTRNG:def 20
for b
1, b
2, b
3, b
4 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[b3,b4] in a
2 &
[b1,b2] >= [b3,b4] implies ( b
1 = b
3 & b
2 = b
4 ) );
attr a
2 is
(M3) means :
Def21:
:: ARMSTRNG:def 21
for b
1, b
2, b
3, b
4 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[b3,b4] in a
2 & b
3 c= b
2 implies b
4 c= b
2 );
end;
:: deftheorem Def19 defines (M1) ARMSTRNG:def 19 :
:: deftheorem Def20 defines (M2) ARMSTRNG:def 20 :
:: deftheorem Def21 defines (M3) ARMSTRNG:def 21 :
theorem Th30: :: ARMSTRNG:30
theorem Th31: :: ARMSTRNG:31
theorem Th32: :: ARMSTRNG:32
:: deftheorem Def22 defines saturated-subsets ARMSTRNG:def 22 :
theorem Th33: :: ARMSTRNG:33
theorem Th34: :: ARMSTRNG:34
:: deftheorem Def23 defines deps_encl_by ARMSTRNG:def 23 :
theorem Th35: :: ARMSTRNG:35
theorem Th36: :: ARMSTRNG:36
theorem Th37: :: ARMSTRNG:37
:: deftheorem Def24 defines enclosure_of ARMSTRNG:def 24 :
theorem Th38: :: ARMSTRNG:38
theorem Th39: :: ARMSTRNG:39
:: deftheorem Def25 defines Dependency-closure ARMSTRNG:def 25 :
theorem Th40: :: ARMSTRNG:40
theorem Th41: :: ARMSTRNG:41
theorem Th42: :: ARMSTRNG:42
theorem Th43: :: ARMSTRNG:43
:: deftheorem Def26 defines is_generator-set_of ARMSTRNG:def 26 :
theorem Th44: :: ARMSTRNG:44
theorem Th45: :: ARMSTRNG:45
theorem Th46: :: ARMSTRNG:46
theorem Th47: :: ARMSTRNG:47
theorem Th48: :: ARMSTRNG:48
Lemma56:
for b1, b2, b3 being set holds
( b2 = { [b4,b5] where B is Subset of b1, B is Subset of b1 : for b1 being set holds
( b6 in b3 & b4 c= b6 implies b5 c= b6 ) } implies for b4, b5 being Subset of b1 holds
( [b4,b5] in b2 iff for b6 being set holds
( b6 in b3 & b4 c= b6 implies b5 c= b6 ) ) )
:: deftheorem Def27 defines candidate-keys ARMSTRNG:def 27 :
theorem Th49: :: ARMSTRNG:49
:: deftheorem Def28 defines without_proper_subsets ARMSTRNG:def 28 :
theorem Th50: :: ARMSTRNG:50
theorem Th51: :: ARMSTRNG:51
theorem Th52: :: ARMSTRNG:52
theorem Th53: :: ARMSTRNG:53
definition
let c
1 be
set ;
let c
2 be
Dependency-set of c
1;
attr a
2 is
(DC4) means :
Def29:
:: ARMSTRNG:def 29
for b
1, b
2, b
3 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[b1,b3] in a
2 implies
[b1,(b2 \/ b3)] in a
2 );
attr a
2 is
(DC5) means :
Def30:
:: ARMSTRNG:def 30
for b
1, b
2, b
3, b
4 being
Subset of a
1 holds
(
[b1,b2] in a
2 &
[(b2 \/ b3),b4] in a
2 implies
[(b1 \/ b3),b4] in a
2 );
attr a
2 is
(DC6) means :
Def31:
:: ARMSTRNG:def 31
for b
1, b
2, b
3 being
Subset of a
1 holds
(
[b1,b2] in a
2 implies
[(b1 \/ b3),b2] in a
2 );
end;
:: deftheorem Def29 defines (DC4) ARMSTRNG:def 29 :
:: deftheorem Def30 defines (DC5) ARMSTRNG:def 30 :
:: deftheorem Def31 defines (DC6) ARMSTRNG:def 31 :
theorem Th54: :: ARMSTRNG:54
theorem Th55: :: ARMSTRNG:55
theorem Th56: :: ARMSTRNG:56
:: deftheorem Def32 defines charact_set ARMSTRNG:def 32 :
theorem Th57: :: ARMSTRNG:57
theorem Th58: :: ARMSTRNG:58
theorem Th59: :: ARMSTRNG:59
theorem Th60: :: ARMSTRNG:60
theorem Th61: :: ARMSTRNG:61
:: deftheorem Def33 defines is_p_i_w_ncv_of ARMSTRNG:def 33 :
theorem Th62: :: ARMSTRNG:62