:: BVFUNC_1 semantic presentation
:: deftheorem Def1 defines 'imp' BVFUNC_1:def 1 :
:: deftheorem Def2 defines 'eqv' BVFUNC_1:def 2 :
:: deftheorem Def3 defines '<' BVFUNC_1:def 3 :
:: deftheorem Def4 defines BVF BVFUNC_1:def 4 :
:: deftheorem Def5 defines 'or' BVFUNC_1:def 5 :
:: deftheorem Def6 defines 'xor' BVFUNC_1:def 6 :
:: deftheorem Def7 defines 'or' BVFUNC_1:def 7 :
:: deftheorem Def8 defines 'xor' BVFUNC_1:def 8 :
:: deftheorem Def9 defines 'imp' BVFUNC_1:def 9 :
:: deftheorem Def10 defines 'eqv' BVFUNC_1:def 10 :
:: deftheorem Def11 defines 'imp' BVFUNC_1:def 11 :
:: deftheorem Def12 defines 'eqv' BVFUNC_1:def 12 :
:: deftheorem Def13 defines O_el BVFUNC_1:def 13 :
:: deftheorem Def14 defines I_el BVFUNC_1:def 14 :
theorem Th1: :: BVFUNC_1:1
canceled;
theorem Th2: :: BVFUNC_1:2
canceled;
theorem Th3: :: BVFUNC_1:3
canceled;
theorem Th4: :: BVFUNC_1:4
theorem Th5: :: BVFUNC_1:5
theorem Th6: :: BVFUNC_1:6
theorem Th7: :: BVFUNC_1:7
theorem Th8: :: BVFUNC_1:8
theorem Th9: :: BVFUNC_1:9
theorem Th10: :: BVFUNC_1:10
theorem Th11: :: BVFUNC_1:11
theorem Th12: :: BVFUNC_1:12
theorem Th13: :: BVFUNC_1:13
theorem Th14: :: BVFUNC_1:14
theorem Th15: :: BVFUNC_1:15
theorem Th16: :: BVFUNC_1:16
theorem Th17: :: BVFUNC_1:17
:: deftheorem Def15 defines '<' BVFUNC_1:def 15 :
theorem Th18: :: BVFUNC_1:18
theorem Th19: :: BVFUNC_1:19
theorem Th20: :: BVFUNC_1:20
theorem Th21: :: BVFUNC_1:21
definition
let c
1 be non
empty set ;
let c
2 be
Element of
Funcs c
1,
BOOLEAN ;
func B_INF c
2 -> Element of
Funcs a
1,
BOOLEAN means :
Def16:
:: BVFUNC_1:def 16
a
3 = I_el a
1 if for b
1 being
Element of a
1 holds a
2 . b
1 = TRUE otherwise a
3 = O_el a
1;
correctness
consistency
for b1 being Element of Funcs c1,BOOLEAN holds
verum;
existence
( ( for b1 being Element of Funcs c1,BOOLEAN holds
verum ) & not ( ( for b1 being Element of c1 holds c2 . b1 = TRUE ) & ( for b1 being Element of Funcs c1,BOOLEAN holds
not b1 = I_el c1 ) ) & not ( not for b1 being Element of c1 holds c2 . b1 = TRUE & ( for b1 being Element of Funcs c1,BOOLEAN holds
not b1 = O_el c1 ) ) );
uniqueness
for b1, b2 being Element of Funcs c1,BOOLEAN holds
( ( ( for b3 being Element of c1 holds c2 . b3 = TRUE ) & b1 = I_el c1 & b2 = I_el c1 implies b1 = b2 ) & ( not for b3 being Element of c1 holds c2 . b3 = TRUE & b1 = O_el c1 & b2 = O_el c1 implies b1 = b2 ) );
;
func B_SUP c
2 -> Element of
Funcs a
1,
BOOLEAN means :
Def17:
:: BVFUNC_1:def 17
a
3 = O_el a
1 if for b
1 being
Element of a
1 holds a
2 . b
1 = FALSE otherwise a
3 = I_el a
1;
correctness
consistency
for b1 being Element of Funcs c1,BOOLEAN holds
verum;
existence
( ( for b1 being Element of Funcs c1,BOOLEAN holds
verum ) & not ( ( for b1 being Element of c1 holds c2 . b1 = FALSE ) & ( for b1 being Element of Funcs c1,BOOLEAN holds
not b1 = O_el c1 ) ) & not ( not for b1 being Element of c1 holds c2 . b1 = FALSE & ( for b1 being Element of Funcs c1,BOOLEAN holds
not b1 = I_el c1 ) ) );
uniqueness
for b1, b2 being Element of Funcs c1,BOOLEAN holds
( ( ( for b3 being Element of c1 holds c2 . b3 = FALSE ) & b1 = O_el c1 & b2 = O_el c1 implies b1 = b2 ) & ( not for b3 being Element of c1 holds c2 . b3 = FALSE & b1 = I_el c1 & b2 = I_el c1 implies b1 = b2 ) );
;
end;
:: deftheorem Def16 defines B_INF BVFUNC_1:def 16 :
:: deftheorem Def17 defines B_SUP BVFUNC_1:def 17 :
theorem Th22: :: BVFUNC_1:22
theorem Th23: :: BVFUNC_1:23
theorem Th24: :: BVFUNC_1:24
theorem Th25: :: BVFUNC_1:25
theorem Th26: :: BVFUNC_1:26
theorem Th27: :: BVFUNC_1:27
theorem Th28: :: BVFUNC_1:28
theorem Th29: :: BVFUNC_1:29
theorem Th30: :: BVFUNC_1:30
:: deftheorem Def18 defines is_dependent_of BVFUNC_1:def 18 :
theorem Th31: :: BVFUNC_1:31
theorem Th32: :: BVFUNC_1:32
definition
let c
1 be non
empty set ;
let c
2 be
Element of
Funcs c
1,
BOOLEAN ;
let c
3 be
a_partition of c
1;
func B_INF c
2,c
3 -> Element of
Funcs a
1,
BOOLEAN means :
Def19:
:: BVFUNC_1:def 19
for b
1 being
Element of a
1 holds
( ( ( for b
2 being
Element of a
1 holds
( b
2 in EqClass b
1,a
3 implies a
2 . b
2 = TRUE ) ) implies a
4 . b
1 = TRUE ) & ( ex b
2 being
Element of a
1 st
( b
2 in EqClass b
1,a
3 & not a
2 . b
2 = TRUE ) implies a
4 . b
1 = FALSE ) );
existence
ex b1 being Element of Funcs c1,BOOLEAN st
for b2 being Element of c1 holds
( ( ( for b3 being Element of c1 holds
( b3 in EqClass b2,c3 implies c2 . b3 = TRUE ) ) implies b1 . b2 = TRUE ) & ( ex b3 being Element of c1 st
( b3 in EqClass b2,c3 & not c2 . b3 = TRUE ) implies b1 . b2 = FALSE ) )
uniqueness
for b1, b2 being Element of Funcs c1,BOOLEAN holds
( ( for b3 being Element of c1 holds
( ( ( for b4 being Element of c1 holds
( b4 in EqClass b3,c3 implies c2 . b4 = TRUE ) ) implies b1 . b3 = TRUE ) & ( ex b4 being Element of c1 st
( b4 in EqClass b3,c3 & not c2 . b4 = TRUE ) implies b1 . b3 = FALSE ) ) ) & ( for b3 being Element of c1 holds
( ( ( for b4 being Element of c1 holds
( b4 in EqClass b3,c3 implies c2 . b4 = TRUE ) ) implies b2 . b3 = TRUE ) & ( ex b4 being Element of c1 st
( b4 in EqClass b3,c3 & not c2 . b4 = TRUE ) implies b2 . b3 = FALSE ) ) ) implies b1 = b2 )
end;
:: deftheorem Def19 defines B_INF BVFUNC_1:def 19 :
definition
let c
1 be non
empty set ;
let c
2 be
Element of
Funcs c
1,
BOOLEAN ;
let c
3 be
a_partition of c
1;
func B_SUP c
2,c
3 -> Element of
Funcs a
1,
BOOLEAN means :
Def20:
:: BVFUNC_1:def 20
for b
1 being
Element of a
1 holds
( ( ex b
2 being
Element of a
1 st
( b
2 in EqClass b
1,a
3 & a
2 . b
2 = TRUE ) implies a
4 . b
1 = TRUE ) & ( ( for b
2 being
Element of a
1 holds
not ( b
2 in EqClass b
1,a
3 & a
2 . b
2 = TRUE ) ) implies a
4 . b
1 = FALSE ) );
existence
ex b1 being Element of Funcs c1,BOOLEAN st
for b2 being Element of c1 holds
( ( ex b3 being Element of c1 st
( b3 in EqClass b2,c3 & c2 . b3 = TRUE ) implies b1 . b2 = TRUE ) & ( ( for b3 being Element of c1 holds
not ( b3 in EqClass b2,c3 & c2 . b3 = TRUE ) ) implies b1 . b2 = FALSE ) )
uniqueness
for b1, b2 being Element of Funcs c1,BOOLEAN holds
( ( for b3 being Element of c1 holds
( ( ex b4 being Element of c1 st
( b4 in EqClass b3,c3 & c2 . b4 = TRUE ) implies b1 . b3 = TRUE ) & ( ( for b4 being Element of c1 holds
not ( b4 in EqClass b3,c3 & c2 . b4 = TRUE ) ) implies b1 . b3 = FALSE ) ) ) & ( for b3 being Element of c1 holds
( ( ex b4 being Element of c1 st
( b4 in EqClass b3,c3 & c2 . b4 = TRUE ) implies b2 . b3 = TRUE ) & ( ( for b4 being Element of c1 holds
not ( b4 in EqClass b3,c3 & c2 . b4 = TRUE ) ) implies b2 . b3 = FALSE ) ) ) implies b1 = b2 )
end;
:: deftheorem Def20 defines B_SUP BVFUNC_1:def 20 :
theorem Th33: :: BVFUNC_1:33
theorem Th34: :: BVFUNC_1:34
theorem Th35: :: BVFUNC_1:35
theorem Th36: :: BVFUNC_1:36
theorem Th37: :: BVFUNC_1:37
theorem Th38: :: BVFUNC_1:38
theorem Th39: :: BVFUNC_1:39
theorem Th40: :: BVFUNC_1:40
theorem Th41: :: BVFUNC_1:41
theorem Th42: :: BVFUNC_1:42
theorem Th43: :: BVFUNC_1:43
:: deftheorem Def21 defines GPart BVFUNC_1:def 21 :
theorem Th44: :: BVFUNC_1:44
theorem Th45: :: BVFUNC_1:45