:: WAYBEL_4 semantic presentation
:: deftheorem Def1 WAYBEL_4:def 1 :
canceled;
:: deftheorem Def2 defines -waybelow WAYBEL_4:def 2 :
:: deftheorem Def3 defines IntRel WAYBEL_4:def 3 :
Lemma2:
for b1 being RelStr
for b2, b3 being Element of b1 holds
( [b2,b3] in IntRel b1 iff b2 <= b3 )
by ORDERS_2:def 9;
:: deftheorem Def4 defines auxiliary(i) WAYBEL_4:def 4 :
:: deftheorem Def5 defines auxiliary(ii) WAYBEL_4:def 5 :
:: deftheorem Def6 defines auxiliary(iii) WAYBEL_4:def 6 :
:: deftheorem Def7 defines auxiliary(iv) WAYBEL_4:def 7 :
:: deftheorem Def8 defines auxiliary WAYBEL_4:def 8 :
theorem Th1: :: WAYBEL_4:1
Lemma9:
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) auxiliary(ii) Relation of b1 holds b2 is transitive
:: deftheorem Def9 defines Aux WAYBEL_4:def 9 :
Lemma11:
for b1 being lower-bounded sup-Semilattice
for b2 being auxiliary(i) Relation of b1
for b3, b4 being set holds
( [b3,b4] in b2 implies [b3,b4] in IntRel b1 )
theorem Th2: :: WAYBEL_4:2
theorem Th3: :: WAYBEL_4:3
:: deftheorem Def10 defines AuxBottom WAYBEL_4:def 10 :
theorem Th4: :: WAYBEL_4:4
theorem Th5: :: WAYBEL_4:5
theorem Th6: :: WAYBEL_4:6
theorem Th7: :: WAYBEL_4:7
theorem Th8: :: WAYBEL_4:8
theorem Th9: :: WAYBEL_4:9
theorem Th10: :: WAYBEL_4:10
theorem Th11: :: WAYBEL_4:11
:: deftheorem Def11 defines -below WAYBEL_4:def 11 :
:: deftheorem Def12 defines -above WAYBEL_4:def 12 :
theorem Th12: :: WAYBEL_4:12
:: deftheorem Def13 defines -below WAYBEL_4:def 13 :
theorem Th13: :: WAYBEL_4:13
theorem Th14: :: WAYBEL_4:14
Lemma24:
for b1 being lower-bounded with_suprema Poset
for b2 being Relation of b1
for b3 being set
for b4 being Element of b1 holds
( b3 in downarrow b4 iff [b3,b4] in the InternalRel of b1 )
theorem Th15: :: WAYBEL_4:15
definition
let c
1 be non
empty Poset;
func MonSet c
1 -> strict RelStr means :
Def14:
:: WAYBEL_4:def 14
for b
1 being
set holds
( not ( b
1 in the
carrier of a
2 & ( for b
2 being
Function of a
1,
(InclPoset (Ids a1)) holds
not ( b
1 = b
2 & b
2 is
monotone & ( for b
3 being
Element of a
1 holds b
2 . b
3 c= downarrow b
3 ) ) ) ) & ( ex b
2 being
Function of a
1,
(InclPoset (Ids a1)) st
( b
1 = b
2 & b
2 is
monotone & ( for b
3 being
Element of a
1 holds b
2 . b
3 c= downarrow b
3 ) ) implies b
1 in the
carrier of a
2 ) & ( for b
2, b
3 being
set holds
(
[b2,b3] in the
InternalRel of a
2 iff ex b
4, b
5 being
Function of a
1,
(InclPoset (Ids a1)) st
( b
2 = b
4 & b
3 = b
5 & b
2 in the
carrier of a
2 & b
3 in the
carrier of a
2 & b
4 <= b
5 ) ) ) );
existence
ex b1 being strict RelStr st
for b2 being set holds
( not ( b2 in the carrier of b1 & ( for b3 being Function of c1,(InclPoset (Ids c1)) holds
not ( b2 = b3 & b3 is monotone & ( for b4 being Element of c1 holds b3 . b4 c= downarrow b4 ) ) ) ) & ( ex b3 being Function of c1,(InclPoset (Ids c1)) st
( b2 = b3 & b3 is monotone & ( for b4 being Element of c1 holds b3 . b4 c= downarrow b4 ) ) implies b2 in the carrier of b1 ) & ( for b3, b4 being set holds
( [b3,b4] in the InternalRel of b1 iff ex b5, b6 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b5 & b4 = b6 & b3 in the carrier of b1 & b4 in the carrier of b1 & b5 <= b6 ) ) ) )
uniqueness
for b1, b2 being strict RelStr holds
( ( for b3 being set holds
( not ( b3 in the carrier of b1 & ( for b4 being Function of c1,(InclPoset (Ids c1)) holds
not ( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) ) ) & ( ex b4 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) implies b3 in the carrier of b1 ) & ( for b4, b5 being set holds
( [b4,b5] in the InternalRel of b1 iff ex b6, b7 being Function of c1,(InclPoset (Ids c1)) st
( b4 = b6 & b5 = b7 & b4 in the carrier of b1 & b5 in the carrier of b1 & b6 <= b7 ) ) ) ) ) & ( for b3 being set holds
( not ( b3 in the carrier of b2 & ( for b4 being Function of c1,(InclPoset (Ids c1)) holds
not ( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) ) ) & ( ex b4 being Function of c1,(InclPoset (Ids c1)) st
( b3 = b4 & b4 is monotone & ( for b5 being Element of c1 holds b4 . b5 c= downarrow b5 ) ) implies b3 in the carrier of b2 ) & ( for b4, b5 being set holds
( [b4,b5] in the InternalRel of b2 iff ex b6, b7 being Function of c1,(InclPoset (Ids c1)) st
( b4 = b6 & b5 = b7 & b4 in the carrier of b2 & b5 in the carrier of b2 & b6 <= b7 ) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def14 defines MonSet WAYBEL_4:def 14 :
theorem Th16: :: WAYBEL_4:16
theorem Th17: :: WAYBEL_4:17
theorem Th18: :: WAYBEL_4:18
theorem Th19: :: WAYBEL_4:19
theorem Th20: :: WAYBEL_4:20
theorem Th21: :: WAYBEL_4:21
theorem Th22: :: WAYBEL_4:22
theorem Th23: :: WAYBEL_4:23
theorem Th24: :: WAYBEL_4:24
Lemma31:
for b1 being lower-bounded sup-Semilattice
for b2 being Ideal of b1 holds {(Bottom b1)} c= b2
theorem Th25: :: WAYBEL_4:25
Lemma33:
for b1 being lower-bounded sup-Semilattice
for b2 being Function of b1,(InclPoset (Ids b1)) holds
( b2 = the carrier of b1 --> {(Bottom b1)} implies b2 is monotone )
theorem Th26: :: WAYBEL_4:26
theorem Th27: :: WAYBEL_4:27
Lemma35:
for b1 being lower-bounded sup-Semilattice holds
ex b2 being Element of (MonSet b1) st b2 is_>=_than the carrier of (MonSet b1)
:: deftheorem Def15 defines Rel2Map WAYBEL_4:def 15 :
theorem Th28: :: WAYBEL_4:28
theorem Th29: :: WAYBEL_4:29
Lemma38:
for b1 being lower-bounded sup-Semilattice holds Rel2Map b1 is monotone
definition
let c
1 be
lower-bounded sup-Semilattice;
func Map2Rel c
1 -> Function of
(MonSet a1),
(InclPoset (Aux a1)) means :
Def16:
:: WAYBEL_4:def 16
for b
1 being
set holds
not ( b
1 in the
carrier of
(MonSet a1) & ( for b
2 being
auxiliary Relation of a
1 holds
not ( b
2 = a
2 . b
1 & ( for b
3, b
4 being
set holds
(
[b3,b4] in b
2 iff ex b
5, b
6 being
Element of a
1ex b
7 being
Function of a
1,
(InclPoset (Ids a1)) st
( b
5 = b
3 & b
6 = b
4 & b
7 = b
1 & b
5 in b
7 . b
6 ) ) ) ) ) );
existence
ex b1 being Function of (MonSet c1),(InclPoset (Aux c1)) st
for b2 being set holds
not ( b2 in the carrier of (MonSet c1) & ( for b3 being auxiliary Relation of c1 holds
not ( b3 = b1 . b2 & ( for b4, b5 being set holds
( [b4,b5] in b3 iff ex b6, b7 being Element of c1ex b8 being Function of c1,(InclPoset (Ids c1)) st
( b6 = b4 & b7 = b5 & b8 = b2 & b6 in b8 . b7 ) ) ) ) ) )
uniqueness
for b1, b2 being Function of (MonSet c1),(InclPoset (Aux c1)) holds
( ( for b3 being set holds
not ( b3 in the carrier of (MonSet c1) & ( for b4 being auxiliary Relation of c1 holds
not ( b4 = b1 . b3 & ( for b5, b6 being set holds
( [b5,b6] in b4 iff ex b7, b8 being Element of c1ex b9 being Function of c1,(InclPoset (Ids c1)) st
( b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8 ) ) ) ) ) ) ) & ( for b3 being set holds
not ( b3 in the carrier of (MonSet c1) & ( for b4 being auxiliary Relation of c1 holds
not ( b4 = b2 . b3 & ( for b5, b6 being set holds
( [b5,b6] in b4 iff ex b7, b8 being Element of c1ex b9 being Function of c1,(InclPoset (Ids c1)) st
( b7 = b5 & b8 = b6 & b9 = b3 & b7 in b9 . b8 ) ) ) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def16 defines Map2Rel WAYBEL_4:def 16 :
Lemma40:
for b1 being lower-bounded sup-Semilattice holds Map2Rel b1 is monotone
theorem Th30: :: WAYBEL_4:30
theorem Th31: :: WAYBEL_4:31
theorem Th32: :: WAYBEL_4:32
theorem Th33: :: WAYBEL_4:33
theorem Th34: :: WAYBEL_4:34
:: deftheorem Def17 defines DownMap WAYBEL_4:def 17 :
Lemma46:
for b1 being Semilattice
for b2 being Ideal of b1 holds DownMap b2 is monotone
Lemma47:
for b1 being Semilattice
for b2 being Element of b1
for b3 being Ideal of b1 holds (DownMap b3) . b2 c= downarrow b2
theorem Th35: :: WAYBEL_4:35
theorem Th36: :: WAYBEL_4:36
:: deftheorem Def18 defines approximating WAYBEL_4:def 18 :
:: deftheorem Def19 defines approximating WAYBEL_4:def 19 :
theorem Th37: :: WAYBEL_4:37
Lemma53:
for b1 being complete LATTICE
for b2 being Element of b1
for b3 being directed Subset of b1 holds sup ({b2} "/\" b3) <= b2
theorem Th38: :: WAYBEL_4:38
theorem Th39: :: WAYBEL_4:39
theorem Th40: :: WAYBEL_4:40
theorem Th41: :: WAYBEL_4:41
Lemma57:
for b1 being lower-bounded continuous LATTICE holds b1 -waybelow is approximating
:: deftheorem Def20 defines App WAYBEL_4:def 20 :
theorem Th42: :: WAYBEL_4:42
theorem Th43: :: WAYBEL_4:43
theorem Th44: :: WAYBEL_4:44
theorem Th45: :: WAYBEL_4:45
theorem Th46: :: WAYBEL_4:46
:: deftheorem Def21 defines satisfying_SI WAYBEL_4:def 21 :
:: deftheorem Def22 defines satisfying_INT WAYBEL_4:def 22 :
theorem Th47: :: WAYBEL_4:47
canceled;
theorem Th48: :: WAYBEL_4:48
theorem Th49: :: WAYBEL_4:49
theorem Th50: :: WAYBEL_4:50
theorem Th51: :: WAYBEL_4:51
theorem Th52: :: WAYBEL_4:52
theorem Th53: :: WAYBEL_4:53
theorem Th54: :: WAYBEL_4:54
:: deftheorem Def23 defines is_directed_wrt WAYBEL_4:def 23 :
theorem Th55: :: WAYBEL_4:55
:: deftheorem Def24 defines is_maximal_wrt WAYBEL_4:def 24 :
:: deftheorem Def25 defines is_maximal_in WAYBEL_4:def 25 :
theorem Th56: :: WAYBEL_4:56
:: deftheorem Def26 defines is_minimal_wrt WAYBEL_4:def 26 :
:: deftheorem Def27 defines is_minimal_in WAYBEL_4:def 27 :
theorem Th57: :: WAYBEL_4:57
theorem Th58: :: WAYBEL_4:58
theorem Th59: :: WAYBEL_4:59
theorem Th60: :: WAYBEL_4:60
theorem Th61: :: WAYBEL_4:61
theorem Th62: :: WAYBEL_4:62