:: FILTER_1 semantic presentation
deffunc H1( LattStr ) -> M5([:the carrier of a1,the carrier of a1:],the carrier of a1) = the L_join of a1;
deffunc H2( LattStr ) -> M5([:the carrier of a1,the carrier of a1:],the carrier of a1) = the L_meet of a1;
theorem Th1: :: FILTER_1:1
theorem Th2: :: FILTER_1:2
definition
let D be non
empty set ;
let R be
Relation;
mode UnOp of
c1,
c2 -> UnOp of
a1 means :
Def1:
:: FILTER_1:def 1
for
x,
y being
Element of
D st
[x,y] in R holds
[(it . x),(it . y)] in R;
existence
ex b1 being UnOp of D st
for x, y being Element of D st [x,y] in R holds
[(b1 . x),(b1 . y)] in R
mode BinOp of
c1,
c2 -> BinOp of
a1 means :
Def2:
:: FILTER_1:def 2
for
x1,
y1,
x2,
y2 being
Element of
D st
[x1,y1] in R &
[x2,y2] in R holds
[(it . x1,x2),(it . y1,y2)] in R;
existence
ex b1 being BinOp of D st
for x1, y1, x2, y2 being Element of D st [x1,y1] in R & [x2,y2] in R holds
[(b1 . x1,x2),(b1 . y1,y2)] in R
end;
:: deftheorem Def1 defines UnOp FILTER_1:def 1 :
:: deftheorem Def2 defines BinOp FILTER_1:def 2 :
definition
let D be non
empty set ;
let R be
Equivalence_Relation of
D;
let u be
UnOp of
D;
assume E68:
u is
UnOp of
D,
R
;
func c3 /\/ c2 -> UnOp of
Class a2 means :: FILTER_1:def 3
for
x,
y being
set st
x in Class R &
y in x holds
it . x = Class R,
(u . y);
existence
ex b1 being UnOp of Class R st
for x, y being set st x in Class R & y in x holds
b1 . x = Class R,(u . y)
uniqueness
for b1, b2 being UnOp of Class R st ( for x, y being set st x in Class R & y in x holds
b1 . x = Class R,(u . y) ) & ( for x, y being set st x in Class R & y in x holds
b2 . x = Class R,(u . y) ) holds
b1 = b2
end;
:: deftheorem Def3 defines /\/ FILTER_1:def 3 :
definition
let D be non
empty set ;
let R be
Equivalence_Relation of
D;
let b be
BinOp of
D;
assume E68:
b is
BinOp of
D,
R
;
func c3 /\/ c2 -> BinOp of
Class a2 means :
Def4:
:: FILTER_1:def 4
for
x,
y,
x1,
y1 being
set st
x in Class R &
y in Class R &
x1 in x &
y1 in y holds
it . x,
y = Class R,
(b . x1,y1);
existence
ex b1 being BinOp of Class R st
for x, y, x1, y1 being set st x in Class R & y in Class R & x1 in x & y1 in y holds
b1 . x,y = Class R,(b . x1,y1)
uniqueness
for b1, b2 being BinOp of Class R st ( for x, y, x1, y1 being set st x in Class R & y in Class R & x1 in x & y1 in y holds
b1 . x,y = Class R,(b . x1,y1) ) & ( for x, y, x1, y1 being set st x in Class R & y in Class R & x1 in x & y1 in y holds
b2 . x,y = Class R,(b . x1,y1) ) holds
b1 = b2
end;
:: deftheorem Def4 defines /\/ FILTER_1:def 4 :
theorem Th3: :: FILTER_1:3
theorem Th4: :: FILTER_1:4
theorem Th5: :: FILTER_1:5
theorem Th6: :: FILTER_1:6
theorem Th7: :: FILTER_1:7
theorem Th8: :: FILTER_1:8
theorem Th9: :: FILTER_1:9
theorem Th10: :: FILTER_1:10
theorem Th11: :: FILTER_1:11
theorem Th12: :: FILTER_1:12
theorem Th13: :: FILTER_1:13
theorem Th14: :: FILTER_1:14
:: deftheorem Def5 defines /\/ FILTER_1:def 5 :
:: deftheorem Def6 defines /\/ FILTER_1:def 6 :
theorem Th15: :: FILTER_1:15
theorem Th16: :: FILTER_1:16
theorem Th17: :: FILTER_1:17
theorem Th18: :: FILTER_1:18
theorem Th19: :: FILTER_1:19
theorem Th20: :: FILTER_1:20
theorem Th21: :: FILTER_1:21
theorem Th22: :: FILTER_1:22
for
D1,
D2 being non
empty set for
a1,
b1 being
Element of
D1 for
a2,
b2 being
Element of
D2 for
f1 being
BinOp of
D1 for
f2 being
BinOp of
D2 holds
|:f1,f2:| . [a1,a2],
[b1,b2] = [(f1 . a1,b1),(f2 . a2,b2)]
scheme :: FILTER_1:sch 137
s137{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
P1[
set ,
set ] } :
for
d,
d' being
Element of
[:F1(),F2():] holds
P1[
d,
d']
provided
E68:
for
d1,
d1' being
Element of
F1()
for
d2,
d2' being
Element of
F2() holds
P1[
[d1,d2],
[d1',d2']]
scheme :: FILTER_1:sch 141
s141{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
P1[
set ,
set ,
set ] } :
for
a,
b,
c being
Element of
[:F1(),F2():] holds
P1[
a,
b,
c]
provided
E68:
for
a1,
b1,
c1 being
Element of
F1()
for
a2,
b2,
c2 being
Element of
F2() holds
P1[
[a1,a2],
[b1,b2],
[c1,c2]]
theorem Th23: :: FILTER_1:23
theorem Th24: :: FILTER_1:24
theorem Th25: :: FILTER_1:25
theorem Th26: :: FILTER_1:26
theorem Th27: :: FILTER_1:27
theorem Th28: :: FILTER_1:28
theorem Th29: :: FILTER_1:29
theorem Th30: :: FILTER_1:30
theorem Th31: :: FILTER_1:31
definition
let L1 be non
empty LattStr ,
L2 be non
empty LattStr ;
func [:c1,c2:] -> strict LattStr equals :: FILTER_1:def 7
LattStr(#
[:the carrier of L1,the carrier of L2:],
|:the L_join of L1,the L_join of L2:|,
|:the L_meet of L1,the L_meet of L2:| #);
correctness
coherence
LattStr(# [:the carrier of L1,the carrier of L2:],|:the L_join of L1,the L_join of L2:|,|:the L_meet of L1,the L_meet of L2:| #) is strict LattStr ;
;
end;
:: deftheorem Def7 defines [: FILTER_1:def 7 :
:: deftheorem Def8 defines LattRel FILTER_1:def 8 :
theorem Th32: :: FILTER_1:32
theorem Th33: :: FILTER_1:33
:: deftheorem Def9 defines are_isomorphic FILTER_1:def 9 :
theorem Th34: :: FILTER_1:34
theorem Th35: :: FILTER_1:35
theorem Th36: :: FILTER_1:36
theorem Th37: :: FILTER_1:37
theorem Th38: :: FILTER_1:38
theorem Th39: :: FILTER_1:39
theorem Th40: :: FILTER_1:40
theorem Th41: :: FILTER_1:41
theorem Th42: :: FILTER_1:42
theorem Th43: :: FILTER_1:43
theorem Th44: :: FILTER_1:44
theorem Th45: :: FILTER_1:45
theorem Th46: :: FILTER_1:46
theorem Th47: :: FILTER_1:47
theorem Th48: :: FILTER_1:48
theorem Th49: :: FILTER_1:49
theorem Th50: :: FILTER_1:50
theorem Th51: :: FILTER_1:51
theorem Th52: :: FILTER_1:52
theorem Th53: :: FILTER_1:53
theorem Th54: :: FILTER_1:54
theorem Th55: :: FILTER_1:55
theorem Th56: :: FILTER_1:56
Lemma188:
for I being I_Lattice
for FI being Filter of I
for i, j, k being Element of I st i => j in FI holds
( i => (j "\/" k) in FI & i => (k "\/" j) in FI & (i "/\" k) => j in FI & (k "/\" i) => j in FI )
theorem Th57: :: FILTER_1:57
Lemma190:
for I being I_Lattice
for FI being Filter of I
for i, k, j being Element of I st i => k in FI & j => k in FI holds
(i "\/" j) => k in FI
theorem Th58: :: FILTER_1:58
Lemma192:
for I being I_Lattice
for FI being Filter of I
for i, j, k being Element of I st i => j in FI & i => k in FI holds
i => (j "/\" k) in FI
theorem Th59: :: FILTER_1:59
Lemma194:
for I being I_Lattice
for FI being Filter of I
for i, j being Element of I holds
( i in Class (equivalence_wrt FI),j iff i <=> j in FI )
theorem Th60: :: FILTER_1:60
theorem Th61: :: FILTER_1:61
theorem Th62: :: FILTER_1:62