:: FILTER_0 semantic presentation
theorem Th1: :: FILTER_0:1
theorem Th2: :: FILTER_0:2
theorem Th3: :: FILTER_0:3
theorem Th4: :: FILTER_0:4
theorem Th5: :: FILTER_0:5
theorem Th6: :: FILTER_0:6
theorem Th7: :: FILTER_0:7
:: deftheorem Def1 defines Filter FILTER_0:def 1 :
theorem Th8: :: FILTER_0:8
canceled;
theorem Th9: :: FILTER_0:9
theorem Th10: :: FILTER_0:10
theorem Th11: :: FILTER_0:11
theorem Th12: :: FILTER_0:12
theorem Th13: :: FILTER_0:13
theorem Th14: :: FILTER_0:14
theorem Th15: :: FILTER_0:15
:: deftheorem Def2 defines <. FILTER_0:def 2 :
:: deftheorem Def3 defines <. FILTER_0:def 3 :
theorem Th16: :: FILTER_0:16
canceled;
theorem Th17: :: FILTER_0:17
canceled;
theorem Th18: :: FILTER_0:18
theorem Th19: :: FILTER_0:19
theorem Th20: :: FILTER_0:20
:: deftheorem Def4 defines being_ultrafilter FILTER_0:def 4 :
theorem Th21: :: FILTER_0:21
canceled;
theorem Th22: :: FILTER_0:22
theorem Th23: :: FILTER_0:23
theorem Th24: :: FILTER_0:24
:: deftheorem Def5 defines <. FILTER_0:def 5 :
theorem Th25: :: FILTER_0:25
canceled;
theorem Th26: :: FILTER_0:26
theorem Th27: :: FILTER_0:27
theorem Th28: :: FILTER_0:28
canceled;
theorem Th29: :: FILTER_0:29
theorem Th30: :: FILTER_0:30
theorem Th31: :: FILTER_0:31
theorem Th32: :: FILTER_0:32
:: deftheorem Def6 defines prime FILTER_0:def 6 :
theorem Th33: :: FILTER_0:33
canceled;
theorem Th34: :: FILTER_0:34
:: deftheorem Def7 defines implicative FILTER_0:def 7 :
:: deftheorem Def8 defines => FILTER_0:def 8 :
theorem Th35: :: FILTER_0:35
canceled;
theorem Th36: :: FILTER_0:36
canceled;
theorem Th37: :: FILTER_0:37
theorem Th38: :: FILTER_0:38
theorem Th39: :: FILTER_0:39
theorem Th40: :: FILTER_0:40
theorem Th41: :: FILTER_0:41
theorem Th42: :: FILTER_0:42
:: deftheorem Def9 defines "/\" FILTER_0:def 9 :
theorem Th43: :: FILTER_0:43
canceled;
theorem Th44: :: FILTER_0:44
theorem Th45: :: FILTER_0:45
theorem Th46: :: FILTER_0:46
theorem Th47: :: FILTER_0:47
theorem Th48: :: FILTER_0:48
theorem Th49: :: FILTER_0:49
theorem Th50: :: FILTER_0:50
theorem Th51: :: FILTER_0:51
theorem Th52: :: FILTER_0:52
theorem Th53: :: FILTER_0:53
theorem Th54: :: FILTER_0:54
theorem Th55: :: FILTER_0:55
theorem Th56: :: FILTER_0:56
theorem Th57: :: FILTER_0:57
theorem Th58: :: FILTER_0:58
theorem Th59: :: FILTER_0:59
theorem Th60: :: FILTER_0:60
definition
let c
1 be
Lattice;
let c
2 be
Filter of c
1;
func latt c
2 -> Lattice means :
Def10:
:: FILTER_0:def 10
ex b
1, b
2 being
BinOp of a
2 st
( b
1 = the
L_join of a
1 || a
2 & b
2 = the
L_meet of a
1 || a
2 & a
3 = LattStr(# a
2,b
1,b
2 #) );
uniqueness
for b1, b2 being Lattice holds
( ex b3, b4 being BinOp of c2 st
( b3 = the L_join of c1 || c2 & b4 = the L_meet of c1 || c2 & b1 = LattStr(# c2,b3,b4 #) ) & ex b3, b4 being BinOp of c2 st
( b3 = the L_join of c1 || c2 & b4 = the L_meet of c1 || c2 & b2 = LattStr(# c2,b3,b4 #) ) implies b1 = b2 )
;
existence
ex b1 being Latticeex b2, b3 being BinOp of c2 st
( b2 = the L_join of c1 || c2 & b3 = the L_meet of c1 || c2 & b1 = LattStr(# c2,b2,b3 #) )
end;
:: deftheorem Def10 defines latt FILTER_0:def 10 :
theorem Th61: :: FILTER_0:61
canceled;
theorem Th62: :: FILTER_0:62
theorem Th63: :: FILTER_0:63
theorem Th64: :: FILTER_0:64
theorem Th65: :: FILTER_0:65
theorem Th66: :: FILTER_0:66
theorem Th67: :: FILTER_0:67
theorem Th68: :: FILTER_0:68
theorem Th69: :: FILTER_0:69
theorem Th70: :: FILTER_0:70
theorem Th71: :: FILTER_0:71
theorem Th72: :: FILTER_0:72
theorem Th73: :: FILTER_0:73
theorem Th74: :: FILTER_0:74
theorem Th75: :: FILTER_0:75
:: deftheorem Def11 defines <=> FILTER_0:def 11 :
theorem Th76: :: FILTER_0:76
canceled;
theorem Th77: :: FILTER_0:77
theorem Th78: :: FILTER_0:78
:: deftheorem Def12 defines equivalence_wrt FILTER_0:def 12 :
theorem Th79: :: FILTER_0:79
canceled;
theorem Th80: :: FILTER_0:80
theorem Th81: :: FILTER_0:81
theorem Th82: :: FILTER_0:82
theorem Th83: :: FILTER_0:83
theorem Th84: :: FILTER_0:84
theorem Th85: :: FILTER_0:85
:: deftheorem Def13 defines are_equivalence_wrt FILTER_0:def 13 :
theorem Th86: :: FILTER_0:86
canceled;
theorem Th87: :: FILTER_0:87
theorem Th88: :: FILTER_0:88
theorem Th89: :: FILTER_0:89
theorem Th90: :: FILTER_0:90
for b
1 being
B_Latticefor b
2 being
Filter of b
1for b
3 being
I_Latticefor b
4, b
5, b
6 being
Element of b
3for b
7 being
Filter of b
3for b
8, b
9, b
10 being
Element of b
1 holds
( ( b
4,b
5 are_equivalence_wrt b
7 & b
5,b
6 are_equivalence_wrt b
7 implies b
4,b
6 are_equivalence_wrt b
7 ) & ( b
8,b
9 are_equivalence_wrt b
2 & b
9,b
10 are_equivalence_wrt b
2 implies b
8,b
10 are_equivalence_wrt b
2 ) )