:: PARTIT1 semantic presentation
theorem Th1: :: PARTIT1:1
theorem Th2: :: PARTIT1:2
canceled;
theorem Th3: :: PARTIT1:3
theorem Th4: :: PARTIT1:4
theorem Th5: :: PARTIT1:5
theorem Th6: :: PARTIT1:6
canceled;
theorem Th7: :: PARTIT1:7
:: deftheorem Def1 defines is_a_dependent_set_of PARTIT1:def 1 :
:: deftheorem Def2 defines is_min_depend PARTIT1:def 2 :
theorem Th8: :: PARTIT1:8
theorem Th9: :: PARTIT1:9
theorem Th10: :: PARTIT1:10
theorem Th11: :: PARTIT1:11
theorem Th12: :: PARTIT1:12
theorem Th13: :: PARTIT1:13
theorem Th14: :: PARTIT1:14
:: deftheorem Def3 defines PARTITIONS PARTIT1:def 3 :
:: deftheorem Def4 defines '/\' PARTIT1:def 4 :
theorem Th15: :: PARTIT1:15
theorem Th16: :: PARTIT1:16
theorem Th17: :: PARTIT1:17
definition
let c
1 be non
empty set ;
let c
2, c
3 be
a_partition of c
1;
func c
2 '\/' c
3 -> a_partition of a
1 means :
Def5:
:: PARTIT1:def 5
for b
1 being
set holds
( b
1 in a
4 iff b
1 is_min_depend a
2,a
3 );
existence
ex b1 being a_partition of c1 st
for b2 being set holds
( b2 in b1 iff b2 is_min_depend c2,c3 )
uniqueness
for b1, b2 being a_partition of c1 holds
( ( for b3 being set holds
( b3 in b1 iff b3 is_min_depend c2,c3 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is_min_depend c2,c3 ) ) implies b1 = b2 )
commutativity
for b1, b2, b3 being a_partition of c1 holds
( ( for b4 being set holds
( b4 in b1 iff b4 is_min_depend b2,b3 ) ) implies for b4 being set holds
( b4 in b1 iff b4 is_min_depend b3,b2 ) )
end;
:: deftheorem Def5 defines '\/' PARTIT1:def 5 :
theorem Th18: :: PARTIT1:18
canceled;
theorem Th19: :: PARTIT1:19
theorem Th20: :: PARTIT1:20
theorem Th21: :: PARTIT1:21
theorem Th22: :: PARTIT1:22
theorem Th23: :: PARTIT1:23
:: deftheorem Def6 defines ERl PARTIT1:def 6 :
:: deftheorem Def7 defines Rel PARTIT1:def 7 :
theorem Th24: :: PARTIT1:24
theorem Th25: :: PARTIT1:25
theorem Th26: :: PARTIT1:26
theorem Th27: :: PARTIT1:27
theorem Th28: :: PARTIT1:28
theorem Th29: :: PARTIT1:29
theorem Th30: :: PARTIT1:30
theorem Th31: :: PARTIT1:31
theorem Th32: :: PARTIT1:32
theorem Th33: :: PARTIT1:33
theorem Th34: :: PARTIT1:34
:: deftheorem Def8 PARTIT1:def 8 :
canceled;
:: deftheorem Def9 defines %O PARTIT1:def 9 :
theorem Th35: :: PARTIT1:35
theorem Th36: :: PARTIT1:36
theorem Th37: :: PARTIT1:37
theorem Th38: :: PARTIT1:38
theorem Th39: :: PARTIT1:39
theorem Th40: :: PARTIT1:40
theorem Th41: :: PARTIT1:41