:: On semilattice structure of {M}izar types
:: by Grzegorz Bancerek
::
:: Received August 8, 2003
:: Copyright (c) 2003 Association of Mizar Users
:: deftheorem Def1 defines Noetherian ABCMIZ_0:def 1 :
:: deftheorem Def2 defines Noetherian ABCMIZ_0:def 2 :
:: deftheorem defines Mizar-widening-like ABCMIZ_0:def 3 :
theorem Th1: :: ABCMIZ_0:1
:: deftheorem Def4 defines void ABCMIZ_0:def 4 :
theorem :: ABCMIZ_0:2
:: deftheorem defines non- ABCMIZ_0:def 5 :
theorem :: ABCMIZ_0:3
:: deftheorem Def6 defines involutive ABCMIZ_0:def 6 :
:: deftheorem defines without_fixpoints ABCMIZ_0:def 7 :
theorem Th4: :: ABCMIZ_0:4
theorem Th5: :: ABCMIZ_0:5
theorem Th6: :: ABCMIZ_0:6
:: deftheorem defines adjs ABCMIZ_0:def 8 :
theorem :: ABCMIZ_0:7
:: deftheorem Def9 defines consistent ABCMIZ_0:def 9 :
theorem Th8: :: ABCMIZ_0:8
:: deftheorem defines adj-structured ABCMIZ_0:def 10 :
theorem Th9: :: ABCMIZ_0:9
:: deftheorem Def11 defines adj-structured ABCMIZ_0:def 11 :
theorem Th10: :: ABCMIZ_0:10
:: deftheorem Def12 defines types ABCMIZ_0:def 12 :
:: deftheorem Def13 defines types ABCMIZ_0:def 13 :
theorem Th11: :: ABCMIZ_0:11
theorem :: ABCMIZ_0:12
theorem Th13: :: ABCMIZ_0:13
theorem Th14: :: ABCMIZ_0:14
theorem :: ABCMIZ_0:15
theorem Th16: :: ABCMIZ_0:16
:: deftheorem defines adjs-typed ABCMIZ_0:def 14 :
theorem Th17: :: ABCMIZ_0:17
theorem :: ABCMIZ_0:18
:: deftheorem Def15 defines is_applicable_to ABCMIZ_0:def 15 :
:: deftheorem Def16 defines is_applicable_to ABCMIZ_0:def 16 :
theorem :: ABCMIZ_0:19
canceled;
theorem Th20: :: ABCMIZ_0:20
:: deftheorem defines ast ABCMIZ_0:def 17 :
theorem Th21: :: ABCMIZ_0:21
theorem Th22: :: ABCMIZ_0:22
theorem Th23: :: ABCMIZ_0:23
theorem Th24: :: ABCMIZ_0:24
theorem Th25: :: ABCMIZ_0:25
theorem :: ABCMIZ_0:26
theorem Th27: :: ABCMIZ_0:27
:: deftheorem defines ast ABCMIZ_0:def 18 :
theorem Th28: :: ABCMIZ_0:28
:: deftheorem Def19 defines apply ABCMIZ_0:def 19 :
theorem :: ABCMIZ_0:29
theorem Th30: :: ABCMIZ_0:30
:: deftheorem defines ast ABCMIZ_0:def 20 :
theorem :: ABCMIZ_0:31
theorem Th32: :: ABCMIZ_0:32
theorem :: ABCMIZ_0:33
theorem Th34: :: ABCMIZ_0:34
theorem Th35: :: ABCMIZ_0:35
theorem Th36: :: ABCMIZ_0:36
theorem Th37: :: ABCMIZ_0:37
theorem Th38: :: ABCMIZ_0:38
:: deftheorem Def21 defines is_applicable_to ABCMIZ_0:def 21 :
theorem :: ABCMIZ_0:39
theorem Th40: :: ABCMIZ_0:40
theorem Th41: :: ABCMIZ_0:41
theorem Th42: :: ABCMIZ_0:42
theorem Th43: :: ABCMIZ_0:43
theorem Th44: :: ABCMIZ_0:44
theorem Th45: :: ABCMIZ_0:45
theorem Th46: :: ABCMIZ_0:46
theorem Th47: :: ABCMIZ_0:47
theorem Th48: :: ABCMIZ_0:48
theorem :: ABCMIZ_0:49
theorem Th50: :: ABCMIZ_0:50
theorem Th51: :: ABCMIZ_0:51
theorem :: ABCMIZ_0:52
theorem Th53: :: ABCMIZ_0:53
theorem :: ABCMIZ_0:54
theorem Th55: :: ABCMIZ_0:55
theorem Th56: :: ABCMIZ_0:56
theorem Th57: :: ABCMIZ_0:57
:: deftheorem Def22 defines sub ABCMIZ_0:def 22 :
:: deftheorem defines sub ABCMIZ_0:def 23 :
:: deftheorem defines non-absorbing ABCMIZ_0:def 24 :
:: deftheorem defines subjected ABCMIZ_0:def 25 :
:: deftheorem defines non-absorbing ABCMIZ_0:def 26 :
:: deftheorem Def27 defines is_properly_applicable_to ABCMIZ_0:def 27 :
:: deftheorem Def28 defines is_properly_applicable_to ABCMIZ_0:def 28 :
theorem Th58: :: ABCMIZ_0:58
theorem :: ABCMIZ_0:59
theorem :: ABCMIZ_0:60
theorem Th61: :: ABCMIZ_0:61
theorem Th62: :: ABCMIZ_0:62
:: deftheorem Def29 defines is_properly_applicable_to ABCMIZ_0:def 29 :
theorem Th63: :: ABCMIZ_0:63
theorem Th64: :: ABCMIZ_0:64
theorem Th65: :: ABCMIZ_0:65
:: deftheorem Def30 defines commutative ABCMIZ_0:def 30 :
theorem Th66: :: ABCMIZ_0:66
:: deftheorem Def31 defines @--> ABCMIZ_0:def 31 :
theorem Th67: :: ABCMIZ_0:67
scheme :: ABCMIZ_0:sch 2
RedInd{
F1()
-> non
empty set ,
P1[
set ,
set ],
F2()
-> Relation of
F1() } :
provided
A1:
for
x,
y being
Element of
F1() st
[x,y] in F2() holds
P1[
x,
y]
and A2:
for
x being
Element of
F1() holds
P1[
x,
x]
and A3:
for
x,
y,
z being
Element of
F1() st
P1[
x,
y] &
P1[
y,
z] holds
P1[
x,
z]
theorem Th68: :: ABCMIZ_0:68
theorem Th69: :: ABCMIZ_0:69
theorem Th70: :: ABCMIZ_0:70
theorem Th71: :: ABCMIZ_0:71
theorem Th72: :: ABCMIZ_0:72
theorem Th73: :: ABCMIZ_0:73
theorem Th74: :: ABCMIZ_0:74
theorem Th75: :: ABCMIZ_0:75
theorem Th76: :: ABCMIZ_0:76
theorem Th77: :: ABCMIZ_0:77
theorem Th78: :: ABCMIZ_0:78
:: deftheorem defines radix ABCMIZ_0:def 32 :
theorem Th79: :: ABCMIZ_0:79
theorem :: ABCMIZ_0:80
theorem Th81: :: ABCMIZ_0:81
theorem Th82: :: ABCMIZ_0:82
theorem :: ABCMIZ_0:83
theorem :: ABCMIZ_0:84