:: COHSP_1 semantic presentation
Lemma1:
for b1, b2 being non empty set
for b3 being Function of b1,b2
for b4 being Element of b1
for b5 being set holds
( b5 in b3 . b4 implies b5 in union b2 )
by TARSKI:def 4;
:: deftheorem Def1 defines binary_complete COHSP_1:def 1 :
:: deftheorem Def2 defines FlatCoh COHSP_1:def 2 :
:: deftheorem Def3 defines Sub_of_Fin COHSP_1:def 3 :
theorem Th1: :: COHSP_1:1
for b
1, b
2 being
set holds
( b
2 in FlatCoh b
1 iff not ( not b
2 = {} & ( for b
3 being
set holds
not ( b
2 = {b3} & b
3 in b
1 ) ) ) )
theorem Th2: :: COHSP_1:2
theorem Th3: :: COHSP_1:3
theorem Th4: :: COHSP_1:4
:: deftheorem Def4 defines c=directed COHSP_1:def 4 :
:: deftheorem Def5 defines c=filtered COHSP_1:def 5 :
theorem Th5: :: COHSP_1:5
for b
1 being
set holds
( b
1 is
c=directed implies for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & ( for b
4 being
set holds
not ( b
2 \/ b
3 c= b
4 & b
4 in b
1 ) ) ) )
theorem Th6: :: COHSP_1:6
for b
1 being non
empty set holds
( ( for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & ( for b
4 being
set holds
not ( b
2 \/ b
3 c= b
4 & b
4 in b
1 ) ) ) ) implies b
1 is
c=directed )
theorem Th7: :: COHSP_1:7
for b
1 being
set holds
( b
1 is
c=filtered implies for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & ( for b
4 being
set holds
not ( b
4 c= b
2 /\ b
3 & b
4 in b
1 ) ) ) )
theorem Th8: :: COHSP_1:8
for b
1 being non
empty set holds
( ( for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & ( for b
4 being
set holds
not ( b
4 c= b
2 /\ b
3 & b
4 in b
1 ) ) ) ) implies b
1 is
c=filtered )
theorem Th9: :: COHSP_1:9
theorem Th10: :: COHSP_1:10
theorem Th11: :: COHSP_1:11
theorem Th12: :: COHSP_1:12
theorem Th13: :: COHSP_1:13
:: deftheorem Def6 COHSP_1:def 6 :
canceled;
:: deftheorem Def7 defines d.union-closed COHSP_1:def 7 :
theorem Th14: :: COHSP_1:14
canceled;
theorem Th15: :: COHSP_1:15
:: deftheorem Def8 defines includes_lattice_of COHSP_1:def 8 :
theorem Th16: :: COHSP_1:16
:: deftheorem Def9 defines includes_lattice_of COHSP_1:def 9 :
theorem Th17: :: COHSP_1:17
:: deftheorem Def10 defines union-distributive COHSP_1:def 10 :
:: deftheorem Def11 defines d.union-distributive COHSP_1:def 11 :
:: deftheorem Def12 defines c=-monotone COHSP_1:def 12 :
:: deftheorem Def13 defines cap-distributive COHSP_1:def 13 :
theorem Th18: :: COHSP_1:18
theorem Th19: :: COHSP_1:19
:: deftheorem Def14 defines U-continuous COHSP_1:def 14 :
:: deftheorem Def15 defines U-stable COHSP_1:def 15 :
:: deftheorem Def16 defines U-linear COHSP_1:def 16 :
theorem Th20: :: COHSP_1:20
theorem Th21: :: COHSP_1:21
theorem Th22: :: COHSP_1:22
theorem Th23: :: COHSP_1:23
theorem Th24: :: COHSP_1:24
:: deftheorem Def17 defines graph COHSP_1:def 17 :
theorem Th25: :: COHSP_1:25
theorem Th26: :: COHSP_1:26
theorem Th27: :: COHSP_1:27
theorem Th28: :: COHSP_1:28
theorem Th29: :: COHSP_1:29
Lemma35:
for b1, b2 being Coherence_Space
for b3 being Subset of [:b1,(union b2):] holds
not ( ( for b4 being set holds
( b4 in b3 implies b4 `1 is finite ) ) & ( for b4, b5 being finite Element of b1 holds
( b4 c= b5 implies for b6 being set holds
( [b4,b6] in b3 implies [b5,b6] in b3 ) ) ) & ( for b4 being finite Element of b1
for b5, b6 being set holds
( [b4,b5] in b3 & [b4,b6] in b3 implies {b5,b6} in b2 ) ) & ( for b4 being U-continuous Function of b1,b2 holds
not ( b3 = graph b4 & ( for b5 being Element of b1 holds b4 . b5 = b3 .: (Fin b5) ) ) ) )
theorem Th30: :: COHSP_1:30
for b
1, b
2 being
Coherence_Spacefor b
3 being
Subset of
[:b1,(union b2):] holds
not ( ( for b
4 being
set holds
( b
4 in b
3 implies b
4 `1 is
finite ) ) & ( for b
4, b
5 being
finite Element of b
1 holds
( b
4 c= b
5 implies for b
6 being
set holds
(
[b4,b6] in b
3 implies
[b5,b6] in b
3 ) ) ) & ( for b
4 being
finite Element of b
1for b
5, b
6 being
set holds
(
[b4,b5] in b
3 &
[b4,b6] in b
3 implies
{b5,b6} in b
2 ) ) & ( for b
4 being
U-continuous Function of b
1,b
2 holds
not b
3 = graph b
4 ) )
theorem Th31: :: COHSP_1:31
:: deftheorem Def18 defines Trace COHSP_1:def 18 :
for b
1 being
Functionfor b
2 being
set holds
( b
2 = Trace b
1 iff for b
3 being
set holds
( b
3 in b
2 iff ex b
4, b
5 being
set st
( b
3 = [b4,b5] & b
4 in dom b
1 & b
5 in b
1 . b
4 & ( for b
6 being
set holds
( b
6 in dom b
1 & b
6 c= b
4 & b
5 in b
1 . b
6 implies b
4 = b
6 ) ) ) ) );
theorem Th32: :: COHSP_1:32
theorem Th33: :: COHSP_1:33
theorem Th34: :: COHSP_1:34
theorem Th35: :: COHSP_1:35
theorem Th36: :: COHSP_1:36
theorem Th37: :: COHSP_1:37
theorem Th38: :: COHSP_1:38
Lemma43:
for b1, b2 being Coherence_Space
for b3 being Subset of [:b1,(union b2):] holds
not ( ( for b4 being set holds
( b4 in b3 implies b4 `1 is finite ) ) & ( for b4, b5 being Element of b1 holds
( b4 \/ b5 in b1 implies for b6, b7 being set holds
( [b4,b6] in b3 & [b5,b7] in b3 implies {b6,b7} in b2 ) ) ) & ( for b4, b5 being Element of b1 holds
( b4 \/ b5 in b1 implies for b6 being set holds
( [b4,b6] in b3 & [b5,b6] in b3 implies b4 = b5 ) ) ) & ( for b4 being U-stable Function of b1,b2 holds
not ( b3 = Trace b4 & ( for b5 being Element of b1 holds b4 . b5 = b3 .: (Fin b5) ) ) ) )
theorem Th39: :: COHSP_1:39
for b
1, b
2 being
Coherence_Spacefor b
3 being
Subset of
[:b1,(union b2):] holds
not ( ( for b
4 being
set holds
( b
4 in b
3 implies b
4 `1 is
finite ) ) & ( for b
4, b
5 being
Element of b
1 holds
( b
4 \/ b
5 in b
1 implies for b
6, b
7 being
set holds
(
[b4,b6] in b
3 &
[b5,b7] in b
3 implies
{b6,b7} in b
2 ) ) ) & ( for b
4, b
5 being
Element of b
1 holds
( b
4 \/ b
5 in b
1 implies for b
6 being
set holds
(
[b4,b6] in b
3 &
[b5,b6] in b
3 implies b
4 = b
5 ) ) ) & ( for b
4 being
U-stable Function of b
1,b
2 holds
not b
3 = Trace b
4 ) )
theorem Th40: :: COHSP_1:40
theorem Th41: :: COHSP_1:41
theorem Th42: :: COHSP_1:42
theorem Th43: :: COHSP_1:43
theorem Th44: :: COHSP_1:44
theorem Th45: :: COHSP_1:45
theorem Th46: :: COHSP_1:46
:: deftheorem Def19 defines StabCoh COHSP_1:def 19 :
theorem Th47: :: COHSP_1:47
theorem Th48: :: COHSP_1:48
theorem Th49: :: COHSP_1:49
theorem Th50: :: COHSP_1:50
definition
let c
1 be
Function;
func LinTrace c
1 -> set means :
Def20:
:: COHSP_1:def 20
for b
1 being
set holds
( b
1 in a
2 iff ex b
2, b
3 being
set st
( b
1 = [b2,b3] &
[{b2},b3] in Trace a
1 ) );
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being set st
( b3 = [b4,b5] & [{b4},b5] in Trace c1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being set st
( b3 = [b4,b5] & [{b4},b5] in Trace c1 ) ) ) implies b1 = b2 )
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being set st
( b2 = [b3,b4] & [{b3},b4] in Trace c1 ) )
end;
:: deftheorem Def20 defines LinTrace COHSP_1:def 20 :
theorem Th51: :: COHSP_1:51
theorem Th52: :: COHSP_1:52
theorem Th53: :: COHSP_1:53
:: deftheorem Def21 defines LinCoh COHSP_1:def 21 :
theorem Th54: :: COHSP_1:54
theorem Th55: :: COHSP_1:55
theorem Th56: :: COHSP_1:56
Lemma61:
for b1, b2 being Coherence_Space
for b3 being Subset of [:(union b1),(union b2):] holds
not ( ( for b4, b5 being set holds
( {b4,b5} in b1 implies for b6, b7 being set holds
( [b4,b6] in b3 & [b5,b7] in b3 implies {b6,b7} in b2 ) ) ) & ( for b4, b5 being set holds
( {b4,b5} in b1 implies for b6 being set holds
( [b4,b6] in b3 & [b5,b6] in b3 implies b4 = b5 ) ) ) & ( for b4 being U-linear Function of b1,b2 holds
not ( b3 = LinTrace b4 & ( for b5 being Element of b1 holds b4 . b5 = b3 .: b5 ) ) ) )
theorem Th57: :: COHSP_1:57
for b
1, b
2 being
Coherence_Spacefor b
3 being
Subset of
[:(union b1),(union b2):] holds
not ( ( for b
4, b
5 being
set holds
(
{b4,b5} in b
1 implies for b
6, b
7 being
set holds
(
[b4,b6] in b
3 &
[b5,b7] in b
3 implies
{b6,b7} in b
2 ) ) ) & ( for b
4, b
5 being
set holds
(
{b4,b5} in b
1 implies for b
6 being
set holds
(
[b4,b6] in b
3 &
[b5,b6] in b
3 implies b
4 = b
5 ) ) ) & ( for b
4 being
U-linear Function of b
1,b
2 holds
not b
3 = LinTrace b
4 ) )
theorem Th58: :: COHSP_1:58
theorem Th59: :: COHSP_1:59
theorem Th60: :: COHSP_1:60
theorem Th61: :: COHSP_1:61
theorem Th62: :: COHSP_1:62
theorem Th63: :: COHSP_1:63
theorem Th64: :: COHSP_1:64
theorem Th65: :: COHSP_1:65
for b
1, b
2 being
Coherence_Spacefor b
3, b
4, b
5, b
6 being
set holds
(
[[b3,b5],[b4,b6]] in Web (LinCoh b1,b2) iff ( b
3 in union b
1 & b
4 in union b
1 & ( ( not
[b3,b4] in Web b
1 & b
5 in union b
2 & b
6 in union b
2 ) or (
[b5,b6] in Web b
2 & ( b
5 = b
6 implies b
3 = b
4 ) ) ) ) )
:: deftheorem Def22 defines 'not' COHSP_1:def 22 :
theorem Th66: :: COHSP_1:66
theorem Th67: :: COHSP_1:67
theorem Th68: :: COHSP_1:68
theorem Th69: :: COHSP_1:69
theorem Th70: :: COHSP_1:70
Lemma70:
for b1 being Coherence_Space holds 'not' ('not' b1) c= b1
theorem Th71: :: COHSP_1:71
theorem Th72: :: COHSP_1:72
theorem Th73: :: COHSP_1:73
:: deftheorem Def23 defines U+ COHSP_1:def 23 :
theorem Th74: :: COHSP_1:74
theorem Th75: :: COHSP_1:75
theorem Th76: :: COHSP_1:76
for b
1, b
2, b
3 being
set holds
( b
3 in b
1 U+ b
2 implies ( b
3 = [(b3 `1 ),(b3 `2 )] & ( ( b
3 `2 = 1 & b
3 `1 in b
1 ) or ( b
3 `2 = 2 & b
3 `1 in b
2 ) ) ) )
theorem Th77: :: COHSP_1:77
for b
1, b
2, b
3 being
set holds
(
[b3,1] in b
1 U+ b
2 iff b
3 in b
1 )
theorem Th78: :: COHSP_1:78
for b
1, b
2, b
3 being
set holds
(
[b3,2] in b
1 U+ b
2 iff b
3 in b
2 )
theorem Th79: :: COHSP_1:79
for b
1, b
2, b
3, b
4 being
set holds
( b
1 U+ b
2 c= b
3 U+ b
4 iff ( b
1 c= b
3 & b
2 c= b
4 ) )
theorem Th80: :: COHSP_1:80
for b
1, b
2, b
3 being
set holds
not ( b
3 c= b
1 U+ b
2 & ( for b
4, b
5 being
set holds
not ( b
3 = b
4 U+ b
5 & b
4 c= b
1 & b
5 c= b
2 ) ) )
theorem Th81: :: COHSP_1:81
for b
1, b
2, b
3, b
4 being
set holds
( b
1 U+ b
2 = b
3 U+ b
4 iff ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th82: :: COHSP_1:82
for b
1, b
2, b
3, b
4 being
set holds
(b1 U+ b2) \/ (b3 U+ b4) = (b1 \/ b3) U+ (b2 \/ b4)
theorem Th83: :: COHSP_1:83
for b
1, b
2, b
3, b
4 being
set holds
(b1 U+ b2) /\ (b3 U+ b4) = (b1 /\ b3) U+ (b2 /\ b4)
:: deftheorem Def24 defines "/\" COHSP_1:def 24 :
:: deftheorem Def25 defines "\/" COHSP_1:def 25 :
theorem Th84: :: COHSP_1:84
theorem Th85: :: COHSP_1:85
theorem Th86: :: COHSP_1:86
theorem Th87: :: COHSP_1:87
theorem Th88: :: COHSP_1:88
theorem Th89: :: COHSP_1:89
theorem Th90: :: COHSP_1:90
theorem Th91: :: COHSP_1:91
theorem Th92: :: COHSP_1:92
theorem Th93: :: COHSP_1:93
theorem Th94: :: COHSP_1:94
theorem Th95: :: COHSP_1:95
theorem Th96: :: COHSP_1:96
:: deftheorem Def26 defines [*] COHSP_1:def 26 :
theorem Th97: :: COHSP_1:97
theorem Th98: :: COHSP_1:98
theorem Th99: :: COHSP_1:99
theorem Th100: :: COHSP_1:100