:: KNASTER semantic presentation
theorem Th1: :: KNASTER:1
theorem Th2: :: KNASTER:2
canceled;
theorem Th3: :: KNASTER:3
theorem Th4: :: KNASTER:4
theorem Th5: :: KNASTER:5
:: deftheorem Def1 KNASTER:def 1 :
canceled;
:: deftheorem Def2 KNASTER:def 2 :
canceled;
:: deftheorem Def3 defines c=-monotone KNASTER:def 3 :
:: deftheorem Def4 defines lfp KNASTER:def 4 :
:: deftheorem Def5 defines gfp KNASTER:def 5 :
theorem Th6: :: KNASTER:6
theorem Th7: :: KNASTER:7
theorem Th8: :: KNASTER:8
theorem Th9: :: KNASTER:9
theorem Th10: :: KNASTER:10
theorem Th11: :: KNASTER:11
theorem Th12: :: KNASTER:12
theorem Th13: :: KNASTER:13
theorem Th14: :: KNASTER:14
definition
let c
1 be
Lattice;
let c
2 be
Function of the
carrier of c
1,the
carrier of c
1;
let c
3 be
Element of c
1;
let c
4 be
Ordinal;
func c
2,c
4 +. c
3 -> set means :
Def6:
:: KNASTER:def 6
ex b
1 being
T-Sequence st
( a
5 = last b
1 &
dom b
1 = succ a
4 & b
1 . {} = a
3 & ( for b
2 being
Ordinal holds
(
succ b
2 in succ a
4 implies b
1 . (succ b2) = a
2 . (b1 . b2) ) ) & ( for b
2 being
Ordinal holds
( b
2 in succ a
4 & b
2 <> {} & b
2 is_limit_ordinal implies b
1 . b
2 = "\/" (rng (b1 | b2)),a
1 ) ) );
correctness
existence
ex b1 being set ex b2 being T-Sequence st
( b1 = last b2 & dom b2 = succ c4 & b2 . {} = c3 & ( for b3 being Ordinal holds
( succ b3 in succ c4 implies b2 . (succ b3) = c2 . (b2 . b3) ) ) & ( for b3 being Ordinal holds
( b3 in succ c4 & b3 <> {} & b3 is_limit_ordinal implies b2 . b3 = "\/" (rng (b2 | b3)),c1 ) ) );
uniqueness
for b1, b2 being set holds
( ex b3 being T-Sequence st
( b1 = last b3 & dom b3 = succ c4 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c4 implies b3 . (succ b4) = c2 . (b3 . b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c4 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = "\/" (rng (b3 | b4)),c1 ) ) ) & ex b3 being T-Sequence st
( b2 = last b3 & dom b3 = succ c4 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c4 implies b3 . (succ b4) = c2 . (b3 . b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c4 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = "\/" (rng (b3 | b4)),c1 ) ) ) implies b1 = b2 );
func c
2,c
4 -. c
3 -> set means :
Def7:
:: KNASTER:def 7
ex b
1 being
T-Sequence st
( a
5 = last b
1 &
dom b
1 = succ a
4 & b
1 . {} = a
3 & ( for b
2 being
Ordinal holds
(
succ b
2 in succ a
4 implies b
1 . (succ b2) = a
2 . (b1 . b2) ) ) & ( for b
2 being
Ordinal holds
( b
2 in succ a
4 & b
2 <> {} & b
2 is_limit_ordinal implies b
1 . b
2 = "/\" (rng (b1 | b2)),a
1 ) ) );
correctness
existence
ex b1 being set ex b2 being T-Sequence st
( b1 = last b2 & dom b2 = succ c4 & b2 . {} = c3 & ( for b3 being Ordinal holds
( succ b3 in succ c4 implies b2 . (succ b3) = c2 . (b2 . b3) ) ) & ( for b3 being Ordinal holds
( b3 in succ c4 & b3 <> {} & b3 is_limit_ordinal implies b2 . b3 = "/\" (rng (b2 | b3)),c1 ) ) );
uniqueness
for b1, b2 being set holds
( ex b3 being T-Sequence st
( b1 = last b3 & dom b3 = succ c4 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c4 implies b3 . (succ b4) = c2 . (b3 . b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c4 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = "/\" (rng (b3 | b4)),c1 ) ) ) & ex b3 being T-Sequence st
( b2 = last b3 & dom b3 = succ c4 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c4 implies b3 . (succ b4) = c2 . (b3 . b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c4 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = "/\" (rng (b3 | b4)),c1 ) ) ) implies b1 = b2 );
end;
:: deftheorem Def6 defines +. KNASTER:def 6 :
:: deftheorem Def7 defines -. KNASTER:def 7 :
theorem Th15: :: KNASTER:15
canceled;
theorem Th16: :: KNASTER:16
theorem Th17: :: KNASTER:17
theorem Th18: :: KNASTER:18
theorem Th19: :: KNASTER:19
theorem Th20: :: KNASTER:20
theorem Th21: :: KNASTER:21
theorem Th22: :: KNASTER:22
theorem Th23: :: KNASTER:23
:: deftheorem Def8 defines with_suprema KNASTER:def 8 :
:: deftheorem Def9 defines with_infima KNASTER:def 9 :
:: deftheorem Def10 defines latt KNASTER:def 10 :
theorem Th24: :: KNASTER:24
theorem Th25: :: KNASTER:25
theorem Th26: :: KNASTER:26
theorem Th27: :: KNASTER:27
theorem Th28: :: KNASTER:28
theorem Th29: :: KNASTER:29
theorem Th30: :: KNASTER:30
theorem Th31: :: KNASTER:31
theorem Th32: :: KNASTER:32
Lemma32:
for b1, b2 being Ordinal holds
not ( not b1 c< b2 & not b1 = b2 & not b2 c< b1 )
theorem Th33: :: KNASTER:33
theorem Th34: :: KNASTER:34
theorem Th35: :: KNASTER:35
theorem Th36: :: KNASTER:36
theorem Th37: :: KNASTER:37
theorem Th38: :: KNASTER:38
:: deftheorem Def11 defines FixPoints KNASTER:def 11 :
theorem Th39: :: KNASTER:39
theorem Th40: :: KNASTER:40
theorem Th41: :: KNASTER:41
theorem Th42: :: KNASTER:42
theorem Th43: :: KNASTER:43
:: deftheorem Def12 defines lfp KNASTER:def 12 :
:: deftheorem Def13 defines gfp KNASTER:def 13 :
theorem Th44: :: KNASTER:44
theorem Th45: :: KNASTER:45
theorem Th46: :: KNASTER:46
theorem Th47: :: KNASTER:47
theorem Th48: :: KNASTER:48
theorem Th49: :: KNASTER:49
theorem Th50: :: KNASTER:50
theorem Th51: :: KNASTER:51