:: LATTICE5 semantic presentation
theorem Th1: :: LATTICE5:1
theorem Th2: :: LATTICE5:2
theorem Th3: :: LATTICE5:3
:: deftheorem Def1 defines EqRelLATT LATTICE5:def 1 :
theorem Th4: :: LATTICE5:4
theorem Th5: :: LATTICE5:5
theorem Th6: :: LATTICE5:6
theorem Th7: :: LATTICE5:7
theorem Th8: :: LATTICE5:8
theorem Th9: :: LATTICE5:9
theorem Th10: :: LATTICE5:10
:: deftheorem Def2 defines complete LATTICE5:def 2 :
:: deftheorem Def3 defines are_joint_by LATTICE5:def 3 :
theorem Th11: :: LATTICE5:11
canceled;
theorem Th12: :: LATTICE5:12
E13:
now
let c
1, c
2, c
3 be
Nat;
assume
( 1
<= c
1 & c
1 < c
2 + c
3 )
;
then
not ( not ( 1
<= c
1 & c
1 < c
2 ) & not ( c
2 = c
1 & c
1 < c
2 + c
3 ) & not ( c
2 < c
1 & c
1 < c
2 + c
3 ) )
by REAL_1:def 5;
hence
not ( not ( 1
<= c
1 & c
1 < c
2 ) & not ( c
2 = c
1 & c
1 < c
2 + c
3 ) & not ( c
2 + 1
<= c
1 & c
1 < c
2 + c
3 ) )
by NAT_1:38;
end;
theorem Th13: :: LATTICE5:13
canceled;
theorem Th14: :: LATTICE5:14
definition
let c
1 be non
empty set ;
let c
2 be
Sublattice of
EqRelLATT c
1;
given c
3 being
Equivalence_Relation of c
1 such that E15:
c
3 in the
carrier of c
2
and E16:
c
3 <> id c
1
;
given c
4 being
Nat such that E17:
for b
1, b
2 being
Equivalence_Relation of c
1for b
3, b
4 being
set holds
not ( b
1 in the
carrier of c
2 & b
2 in the
carrier of c
2 &
[b3,b4] in b
1 "\/" b
2 & ( for b
5 being non
empty FinSequence of c
1 holds
not (
len b
5 = c
4 & b
3,b
4 are_joint_by b
5,b
1,b
2 ) ) )
;
func type_of c
2 -> Nat means :
Def4:
:: LATTICE5:def 4
( ( for b
1, b
2 being
Equivalence_Relation of a
1for b
3, b
4 being
set holds
not ( b
1 in the
carrier of a
2 & b
2 in the
carrier of a
2 &
[b3,b4] in b
1 "\/" b
2 & ( for b
5 being non
empty FinSequence of a
1 holds
not (
len b
5 = a
3 + 2 & b
3,b
4 are_joint_by b
5,b
1,b
2 ) ) ) ) & ex b
1, b
2 being
Equivalence_Relation of a
1ex b
3, b
4 being
set st
( b
1 in the
carrier of a
2 & b
2 in the
carrier of a
2 &
[b3,b4] in b
1 "\/" b
2 & ( for b
5 being non
empty FinSequence of a
1 holds
not (
len b
5 = a
3 + 1 & b
3,b
4 are_joint_by b
5,b
1,b
2 ) ) ) );
existence
ex b1 being Nat st
( ( for b2, b3 being Equivalence_Relation of c1
for b4, b5 being set holds
not ( b2 in the carrier of c2 & b3 in the carrier of c2 & [b4,b5] in b2 "\/" b3 & ( for b6 being non empty FinSequence of c1 holds
not ( len b6 = b1 + 2 & b4,b5 are_joint_by b6,b2,b3 ) ) ) ) & ex b2, b3 being Equivalence_Relation of c1ex b4, b5 being set st
( b2 in the carrier of c2 & b3 in the carrier of c2 & [b4,b5] in b2 "\/" b3 & ( for b6 being non empty FinSequence of c1 holds
not ( len b6 = b1 + 1 & b4,b5 are_joint_by b6,b2,b3 ) ) ) )
uniqueness
for b1, b2 being Nat holds
( ( for b3, b4 being Equivalence_Relation of c1
for b5, b6 being set holds
not ( b3 in the carrier of c2 & b4 in the carrier of c2 & [b5,b6] in b3 "\/" b4 & ( for b7 being non empty FinSequence of c1 holds
not ( len b7 = b1 + 2 & b5,b6 are_joint_by b7,b3,b4 ) ) ) ) & ex b3, b4 being Equivalence_Relation of c1ex b5, b6 being set st
( b3 in the carrier of c2 & b4 in the carrier of c2 & [b5,b6] in b3 "\/" b4 & ( for b7 being non empty FinSequence of c1 holds
not ( len b7 = b1 + 1 & b5,b6 are_joint_by b7,b3,b4 ) ) ) & ( for b3, b4 being Equivalence_Relation of c1
for b5, b6 being set holds
not ( b3 in the carrier of c2 & b4 in the carrier of c2 & [b5,b6] in b3 "\/" b4 & ( for b7 being non empty FinSequence of c1 holds
not ( len b7 = b2 + 2 & b5,b6 are_joint_by b7,b3,b4 ) ) ) ) & ex b3, b4 being Equivalence_Relation of c1ex b5, b6 being set st
( b3 in the carrier of c2 & b4 in the carrier of c2 & [b5,b6] in b3 "\/" b4 & ( for b7 being non empty FinSequence of c1 holds
not ( len b7 = b2 + 1 & b5,b6 are_joint_by b7,b3,b4 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines type_of LATTICE5:def 4 :
for b
1 being non
empty set for b
2 being
Sublattice of
EqRelLATT b
1 holds
( ex b
3 being
Equivalence_Relation of b
1 st
( b
3 in the
carrier of b
2 & b
3 <> id b
1 ) & ex b
3 being
Nat st
for b
4, b
5 being
Equivalence_Relation of b
1for b
6, b
7 being
set holds
not ( b
4 in the
carrier of b
2 & b
5 in the
carrier of b
2 &
[b6,b7] in b
4 "\/" b
5 & ( for b
8 being non
empty FinSequence of b
1 holds
not (
len b
8 = b
3 & b
6,b
7 are_joint_by b
8,b
4,b
5 ) ) ) implies for b
3 being
Nat holds
( b
3 = type_of b
2 iff ( ( for b
4, b
5 being
Equivalence_Relation of b
1for b
6, b
7 being
set holds
not ( b
4 in the
carrier of b
2 & b
5 in the
carrier of b
2 &
[b6,b7] in b
4 "\/" b
5 & ( for b
8 being non
empty FinSequence of b
1 holds
not (
len b
8 = b
3 + 2 & b
6,b
7 are_joint_by b
8,b
4,b
5 ) ) ) ) & ex b
4, b
5 being
Equivalence_Relation of b
1ex b
6, b
7 being
set st
( b
4 in the
carrier of b
2 & b
5 in the
carrier of b
2 &
[b6,b7] in b
4 "\/" b
5 & ( for b
8 being non
empty FinSequence of b
1 holds
not (
len b
8 = b
3 + 1 & b
6,b
7 are_joint_by b
8,b
4,b
5 ) ) ) ) ) );
theorem Th15: :: LATTICE5:15
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
canceled;attr a
3 is
symmetric means :
Def6:
:: LATTICE5:def 6
for b
1, b
2 being
Element of a
1 holds a
3 . b
1,b
2 = a
3 . b
2,b
1;
attr a
3 is
zeroed means :
Def7:
:: LATTICE5:def 7
for b
1 being
Element of a
1 holds a
3 . b
1,b
1 = Bottom a
2;
attr a
3 is
u.t.i. means :
Def8:
:: LATTICE5:def 8
for b
1, b
2, b
3 being
Element of a
1 holds
(a3 . b1,b2) "\/" (a3 . b2,b3) >= a
3 . b
1,b
3;
end;
:: deftheorem Def5 LATTICE5:def 5 :
canceled;
:: deftheorem Def6 defines symmetric LATTICE5:def 6 :
:: deftheorem Def7 defines zeroed LATTICE5:def 7 :
:: deftheorem Def8 defines u.t.i. LATTICE5:def 8 :
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
distance_function of c
1,c
2;
func alpha c
3 -> Function of a
2,
(EqRelLATT a1) means :
Def9:
:: LATTICE5:def 9
for b
1 being
Element of a
2 holds
ex b
2 being
Equivalence_Relation of a
1 st
( b
2 = a
4 . b
1 & ( for b
3, b
4 being
Element of a
1 holds
(
[b3,b4] in b
2 iff a
3 . b
3,b
4 <= b
1 ) ) );
existence
ex b1 being Function of c2,(EqRelLATT c1) st
for b2 being Element of c2 holds
ex b3 being Equivalence_Relation of c1 st
( b3 = b1 . b2 & ( for b4, b5 being Element of c1 holds
( [b4,b5] in b3 iff c3 . b4,b5 <= b2 ) ) )
uniqueness
for b1, b2 being Function of c2,(EqRelLATT c1) holds
( ( for b3 being Element of c2 holds
ex b4 being Equivalence_Relation of c1 st
( b4 = b1 . b3 & ( for b5, b6 being Element of c1 holds
( [b5,b6] in b4 iff c3 . b5,b6 <= b3 ) ) ) ) & ( for b3 being Element of c2 holds
ex b4 being Equivalence_Relation of c1 st
( b4 = b2 . b3 & ( for b5, b6 being Element of c1 holds
( [b5,b6] in b4 iff c3 . b5,b6 <= b3 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines alpha LATTICE5:def 9 :
theorem Th16: :: LATTICE5:16
theorem Th17: :: LATTICE5:17
:: deftheorem Def10 defines new_set LATTICE5:def 10 :
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
let c
4 be
Element of
[:c1,c1,the carrier of c2,the carrier of c2:];
func new_bi_fun c
3,c
4 -> BiFunction of
(new_set a1),a
2 means :
Def11:
:: LATTICE5:def 11
( ( for b
1, b
2 being
Element of a
1 holds a
5 . b
1,b
2 = a
3 . b
1,b
2 ) & a
5 . {a1},
{a1} = Bottom a
2 & a
5 . {{a1}},
{{a1}} = Bottom a
2 & a
5 . {{{a1}}},
{{{a1}}} = Bottom a
2 & a
5 . {{a1}},
{{{a1}}} = a
4 `3 & a
5 . {{{a1}}},
{{a1}} = a
4 `3 & a
5 . {a1},
{{a1}} = a
4 `4 & a
5 . {{a1}},
{a1} = a
4 `4 & a
5 . {a1},
{{{a1}}} = (a4 `3 ) "\/" (a4 `4 ) & a
5 . {{{a1}}},
{a1} = (a4 `3 ) "\/" (a4 `4 ) & ( for b
1 being
Element of a
1 holds
( a
5 . b
1,
{a1} = (a3 . b1,(a4 `1 )) "\/" (a4 `3 ) & a
5 . {a1},b
1 = (a3 . b1,(a4 `1 )) "\/" (a4 `3 ) & a
5 . b
1,
{{a1}} = ((a3 . b1,(a4 `1 )) "\/" (a4 `3 )) "\/" (a4 `4 ) & a
5 . {{a1}},b
1 = ((a3 . b1,(a4 `1 )) "\/" (a4 `3 )) "\/" (a4 `4 ) & a
5 . b
1,
{{{a1}}} = (a3 . b1,(a4 `2 )) "\/" (a4 `4 ) & a
5 . {{{a1}}},b
1 = (a3 . b1,(a4 `2 )) "\/" (a4 `4 ) ) ) );
existence
ex b1 being BiFunction of (new_set c1),c2 st
( ( for b2, b3 being Element of c1 holds b1 . b2,b3 = c3 . b2,b3 ) & b1 . {c1},{c1} = Bottom c2 & b1 . {{c1}},{{c1}} = Bottom c2 & b1 . {{{c1}}},{{{c1}}} = Bottom c2 & b1 . {{c1}},{{{c1}}} = c4 `3 & b1 . {{{c1}}},{{c1}} = c4 `3 & b1 . {c1},{{c1}} = c4 `4 & b1 . {{c1}},{c1} = c4 `4 & b1 . {c1},{{{c1}}} = (c4 `3 ) "\/" (c4 `4 ) & b1 . {{{c1}}},{c1} = (c4 `3 ) "\/" (c4 `4 ) & ( for b2 being Element of c1 holds
( b1 . b2,{c1} = (c3 . b2,(c4 `1 )) "\/" (c4 `3 ) & b1 . {c1},b2 = (c3 . b2,(c4 `1 )) "\/" (c4 `3 ) & b1 . b2,{{c1}} = ((c3 . b2,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b1 . {{c1}},b2 = ((c3 . b2,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b1 . b2,{{{c1}}} = (c3 . b2,(c4 `2 )) "\/" (c4 `4 ) & b1 . {{{c1}}},b2 = (c3 . b2,(c4 `2 )) "\/" (c4 `4 ) ) ) )
uniqueness
for b1, b2 being BiFunction of (new_set c1),c2 holds
( ( for b3, b4 being Element of c1 holds b1 . b3,b4 = c3 . b3,b4 ) & b1 . {c1},{c1} = Bottom c2 & b1 . {{c1}},{{c1}} = Bottom c2 & b1 . {{{c1}}},{{{c1}}} = Bottom c2 & b1 . {{c1}},{{{c1}}} = c4 `3 & b1 . {{{c1}}},{{c1}} = c4 `3 & b1 . {c1},{{c1}} = c4 `4 & b1 . {{c1}},{c1} = c4 `4 & b1 . {c1},{{{c1}}} = (c4 `3 ) "\/" (c4 `4 ) & b1 . {{{c1}}},{c1} = (c4 `3 ) "\/" (c4 `4 ) & ( for b3 being Element of c1 holds
( b1 . b3,{c1} = (c3 . b3,(c4 `1 )) "\/" (c4 `3 ) & b1 . {c1},b3 = (c3 . b3,(c4 `1 )) "\/" (c4 `3 ) & b1 . b3,{{c1}} = ((c3 . b3,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b1 . {{c1}},b3 = ((c3 . b3,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b1 . b3,{{{c1}}} = (c3 . b3,(c4 `2 )) "\/" (c4 `4 ) & b1 . {{{c1}}},b3 = (c3 . b3,(c4 `2 )) "\/" (c4 `4 ) ) ) & ( for b3, b4 being Element of c1 holds b2 . b3,b4 = c3 . b3,b4 ) & b2 . {c1},{c1} = Bottom c2 & b2 . {{c1}},{{c1}} = Bottom c2 & b2 . {{{c1}}},{{{c1}}} = Bottom c2 & b2 . {{c1}},{{{c1}}} = c4 `3 & b2 . {{{c1}}},{{c1}} = c4 `3 & b2 . {c1},{{c1}} = c4 `4 & b2 . {{c1}},{c1} = c4 `4 & b2 . {c1},{{{c1}}} = (c4 `3 ) "\/" (c4 `4 ) & b2 . {{{c1}}},{c1} = (c4 `3 ) "\/" (c4 `4 ) & ( for b3 being Element of c1 holds
( b2 . b3,{c1} = (c3 . b3,(c4 `1 )) "\/" (c4 `3 ) & b2 . {c1},b3 = (c3 . b3,(c4 `1 )) "\/" (c4 `3 ) & b2 . b3,{{c1}} = ((c3 . b3,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b2 . {{c1}},b3 = ((c3 . b3,(c4 `1 )) "\/" (c4 `3 )) "\/" (c4 `4 ) & b2 . b3,{{{c1}}} = (c3 . b3,(c4 `2 )) "\/" (c4 `4 ) & b2 . {{{c1}}},b3 = (c3 . b3,(c4 `2 )) "\/" (c4 `4 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines new_bi_fun LATTICE5:def 11 :
for b
1 being non
empty set for b
2 being
lower-bounded LATTICEfor b
3 being
BiFunction of b
1,b
2for b
4 being
Element of
[:b1,b1,the carrier of b2,the carrier of b2:]for b
5 being
BiFunction of
(new_set b1),b
2 holds
( b
5 = new_bi_fun b
3,b
4 iff ( ( for b
6, b
7 being
Element of b
1 holds b
5 . b
6,b
7 = b
3 . b
6,b
7 ) & b
5 . {b1},
{b1} = Bottom b
2 & b
5 . {{b1}},
{{b1}} = Bottom b
2 & b
5 . {{{b1}}},
{{{b1}}} = Bottom b
2 & b
5 . {{b1}},
{{{b1}}} = b
4 `3 & b
5 . {{{b1}}},
{{b1}} = b
4 `3 & b
5 . {b1},
{{b1}} = b
4 `4 & b
5 . {{b1}},
{b1} = b
4 `4 & b
5 . {b1},
{{{b1}}} = (b4 `3 ) "\/" (b4 `4 ) & b
5 . {{{b1}}},
{b1} = (b4 `3 ) "\/" (b4 `4 ) & ( for b
6 being
Element of b
1 holds
( b
5 . b
6,
{b1} = (b3 . b6,(b4 `1 )) "\/" (b4 `3 ) & b
5 . {b1},b
6 = (b3 . b6,(b4 `1 )) "\/" (b4 `3 ) & b
5 . b
6,
{{b1}} = ((b3 . b6,(b4 `1 )) "\/" (b4 `3 )) "\/" (b4 `4 ) & b
5 . {{b1}},b
6 = ((b3 . b6,(b4 `1 )) "\/" (b4 `3 )) "\/" (b4 `4 ) & b
5 . b
6,
{{{b1}}} = (b3 . b6,(b4 `2 )) "\/" (b4 `4 ) & b
5 . {{{b1}}},b
6 = (b3 . b6,(b4 `2 )) "\/" (b4 `4 ) ) ) ) );
theorem Th18: :: LATTICE5:18
theorem Th19: :: LATTICE5:19
theorem Th20: :: LATTICE5:20
theorem Th21: :: LATTICE5:21
theorem Th22: :: LATTICE5:22
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
func DistEsti c
3 -> Cardinal means :
Def12:
:: LATTICE5:def 12
a
4,
{ [b1,b2,b3,b4] where B is Element of a1, B is Element of a1, B is Element of a2, B is Element of a2 : a3 . b1,b2 <= b3 "\/" b4 } are_equipotent ;
existence
ex b1 being Cardinal st b1,{ [b2,b3,b4,b5] where B is Element of c1, B is Element of c1, B is Element of c2, B is Element of c2 : c3 . b2,b3 <= b4 "\/" b5 } are_equipotent
uniqueness
for b1, b2 being Cardinal holds
( b1,{ [b3,b4,b5,b6] where B is Element of c1, B is Element of c1, B is Element of c2, B is Element of c2 : c3 . b3,b4 <= b5 "\/" b6 } are_equipotent & b2,{ [b3,b4,b5,b6] where B is Element of c1, B is Element of c1, B is Element of c2, B is Element of c2 : c3 . b3,b4 <= b5 "\/" b6 } are_equipotent implies b1 = b2 )
end;
:: deftheorem Def12 defines DistEsti LATTICE5:def 12 :
theorem Th23: :: LATTICE5:23
:: deftheorem Def13 defines ConsecutiveSet LATTICE5:def 13 :
theorem Th24: :: LATTICE5:24
theorem Th25: :: LATTICE5:25
theorem Th26: :: LATTICE5:26
theorem Th27: :: LATTICE5:27
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
mode QuadrSeq of c
3 -> T-Sequence of
[:a1,a1,the carrier of a2,the carrier of a2:] means :
Def14:
:: LATTICE5:def 14
(
dom a
4 is
Cardinal & a
4 is
one-to-one &
rng a
4 = { [b1,b2,b3,b4] where B is Element of a1, B is Element of a1, B is Element of a2, B is Element of a2 : a3 . b1,b2 <= b3 "\/" b4 } );
existence
ex b1 being T-Sequence of [:c1,c1,the carrier of c2,the carrier of c2:] st
( dom b1 is Cardinal & b1 is one-to-one & rng b1 = { [b2,b3,b4,b5] where B is Element of c1, B is Element of c1, B is Element of c2, B is Element of c2 : c3 . b2,b3 <= b4 "\/" b5 } )
end;
:: deftheorem Def14 defines QuadrSeq LATTICE5:def 14 :
for b
1 being non
empty set for b
2 being
lower-bounded LATTICEfor b
3 being
BiFunction of b
1,b
2for b
4 being
T-Sequence of
[:b1,b1,the carrier of b2,the carrier of b2:] holds
( b
4 is
QuadrSeq of b
3 iff (
dom b
4 is
Cardinal & b
4 is
one-to-one &
rng b
4 = { [b5,b6,b7,b8] where B is Element of b1, B is Element of b1, B is Element of b2, B is Element of b2 : b3 . b5,b6 <= b7 "\/" b8 } ) );
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
let c
4 be
QuadrSeq of c
3;
let c
5 be
Ordinal;
assume E37:
c
5 in dom c
4
;
func Quadr c
4,c
5 -> Element of
[:(ConsecutiveSet a1,a5),(ConsecutiveSet a1,a5),the carrier of a2,the carrier of a2:] equals :
Def15:
:: LATTICE5:def 15
a
4 . a
5;
correctness
coherence
c4 . c5 is Element of [:(ConsecutiveSet c1,c5),(ConsecutiveSet c1,c5),the carrier of c2,the carrier of c2:];
end;
:: deftheorem Def15 defines Quadr LATTICE5:def 15 :
theorem Th28: :: LATTICE5:28
:: deftheorem Def16 defines BiFun LATTICE5:def 16 :
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
BiFunction of c
1,c
2;
let c
4 be
QuadrSeq of c
3;
let c
5 be
Ordinal;
func ConsecutiveDelta c
4,c
5 -> set means :
Def17:
:: LATTICE5:def 17
ex b
1 being
T-Sequence st
( a
6 = last b
1 &
dom b
1 = succ a
5 & b
1 . {} = a
3 & ( for b
2 being
Ordinal holds
(
succ b
2 in succ a
5 implies b
1 . (succ b2) = new_bi_fun (BiFun (b1 . b2),(ConsecutiveSet a1,b2),a2),
(Quadr a4,b2) ) ) & ( for b
2 being
Ordinal holds
( b
2 in succ a
5 & b
2 <> {} & b
2 is_limit_ordinal implies b
1 . b
2 = union (rng (b1 | b2)) ) ) );
correctness
existence
ex b1 being set ex b2 being T-Sequence st
( b1 = last b2 & dom b2 = succ c5 & b2 . {} = c3 & ( for b3 being Ordinal holds
( succ b3 in succ c5 implies b2 . (succ b3) = new_bi_fun (BiFun (b2 . b3),(ConsecutiveSet c1,b3),c2),(Quadr c4,b3) ) ) & ( for b3 being Ordinal holds
( b3 in succ c5 & b3 <> {} & b3 is_limit_ordinal implies b2 . b3 = union (rng (b2 | b3)) ) ) );
uniqueness
for b1, b2 being set holds
( ex b3 being T-Sequence st
( b1 = last b3 & dom b3 = succ c5 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c5 implies b3 . (succ b4) = new_bi_fun (BiFun (b3 . b4),(ConsecutiveSet c1,b4),c2),(Quadr c4,b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c5 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = union (rng (b3 | b4)) ) ) ) & ex b3 being T-Sequence st
( b2 = last b3 & dom b3 = succ c5 & b3 . {} = c3 & ( for b4 being Ordinal holds
( succ b4 in succ c5 implies b3 . (succ b4) = new_bi_fun (BiFun (b3 . b4),(ConsecutiveSet c1,b4),c2),(Quadr c4,b4) ) ) & ( for b4 being Ordinal holds
( b4 in succ c5 & b4 <> {} & b4 is_limit_ordinal implies b3 . b4 = union (rng (b3 | b4)) ) ) ) implies b1 = b2 );
end;
:: deftheorem Def17 defines ConsecutiveDelta LATTICE5:def 17 :
theorem Th29: :: LATTICE5:29
theorem Th30: :: LATTICE5:30
theorem Th31: :: LATTICE5:31
theorem Th32: :: LATTICE5:32
theorem Th33: :: LATTICE5:33
theorem Th34: :: LATTICE5:34
theorem Th35: :: LATTICE5:35
theorem Th36: :: LATTICE5:36
theorem Th37: :: LATTICE5:37
theorem Th38: :: LATTICE5:38
theorem Th39: :: LATTICE5:39
:: deftheorem Def18 defines NextSet LATTICE5:def 18 :
:: deftheorem Def19 defines NextDelta LATTICE5:def 19 :
:: deftheorem Def20 defines is_extension_of LATTICE5:def 20 :
theorem Th40: :: LATTICE5:40
for b
1 being non
empty set for b
2 being
lower-bounded LATTICEfor b
3 being
distance_function of b
1,b
2for b
4 being non
empty set for b
5 being
distance_function of b
4,b
2 holds
( b
4,b
5 is_extension_of b
1,b
3 implies for b
6, b
7 being
Element of b
1for b
8, b
9 being
Element of b
2 holds
not ( b
3 . b
6,b
7 <= b
8 "\/" b
9 & ( for b
10, b
11, b
12 being
Element of b
4 holds
not ( b
5 . b
6,b
10 = b
8 & b
5 . b
11,b
12 = b
8 & b
5 . b
10,b
11 = b
9 & b
5 . b
12,b
7 = b
9 ) ) ) )
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded LATTICE;
let c
3 be
distance_function of c
1,c
2;
mode ExtensionSeq of c
1,c
3 -> Function means :
Def21:
:: LATTICE5:def 21
(
dom a
4 = NAT & a
4 . 0
= [a1,a3] & ( for b
1 being
Nat holds
ex b
2 being non
empty set ex b
3 being
distance_function of b
2,a
2ex b
4 being non
empty set ex b
5 being
distance_function of b
4,a
2 st
( b
4,b
5 is_extension_of b
2,b
3 & a
4 . b
1 = [b2,b3] & a
4 . (b1 + 1) = [b4,b5] ) ) );
existence
ex b1 being Function st
( dom b1 = NAT & b1 . 0 = [c1,c3] & ( for b2 being Nat holds
ex b3 being non empty set ex b4 being distance_function of b3,c2ex b5 being non empty set ex b6 being distance_function of b5,c2 st
( b5,b6 is_extension_of b3,b4 & b1 . b2 = [b3,b4] & b1 . (b2 + 1) = [b5,b6] ) ) )
end;
:: deftheorem Def21 defines ExtensionSeq LATTICE5:def 21 :
theorem Th41: :: LATTICE5:41
theorem Th42: :: LATTICE5:42
definition
let c
1 be
lower-bounded LATTICE;
func BasicDF c
1 -> distance_function of the
carrier of a
1,a
1 means :
Def22:
:: LATTICE5:def 22
for b
1, b
2 being
Element of a
1 holds
( ( b
1 <> b
2 implies a
2 . b
1,b
2 = b
1 "\/" b
2 ) & ( b
1 = b
2 implies a
2 . b
1,b
2 = Bottom a
1 ) );
existence
ex b1 being distance_function of the carrier of c1,c1 st
for b2, b3 being Element of c1 holds
( ( b2 <> b3 implies b1 . b2,b3 = b2 "\/" b3 ) & ( b2 = b3 implies b1 . b2,b3 = Bottom c1 ) )
uniqueness
for b1, b2 being distance_function of the carrier of c1,c1 holds
( ( for b3, b4 being Element of c1 holds
( ( b3 <> b4 implies b1 . b3,b4 = b3 "\/" b4 ) & ( b3 = b4 implies b1 . b3,b4 = Bottom c1 ) ) ) & ( for b3, b4 being Element of c1 holds
( ( b3 <> b4 implies b2 . b3,b4 = b3 "\/" b4 ) & ( b3 = b4 implies b2 . b3,b4 = Bottom c1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def22 defines BasicDF LATTICE5:def 22 :
theorem Th43: :: LATTICE5:43
E58:
now
let c
1 be
Nat;
assume E59:
( 1
<= c
1 & c
1 < 5 )
;
then
c
1 < 4
+ 1
;
then
c
1 <= 4
by NAT_1:38;
then
not ( not c
1 = 0 & not c
1 = 1 & not c
1 = 2 & not c
1 = 3 & not c
1 = 4 )
by CQC_THE1:5;
hence
not ( not c
1 = 1 & not c
1 = 2 & not c
1 = 3 & not c
1 = 4 )
by E59;
end;
theorem Th44: :: LATTICE5:44
theorem Th45: :: LATTICE5:45
for b
1 being
lower-bounded LATTICEfor b
2 being
ExtensionSeq of the
carrier of b
1,
BasicDF b
1for b
3 being non
empty set for b
4 being
distance_function of b
3,b
1for b
5, b
6 being
Element of b
3for b
7, b
8 being
Element of b
1 holds
not ( b
3 = union { ((b2 . b9) `1 ) where B is Nat : verum } & b
4 = union { ((b2 . b9) `2 ) where B is Nat : verum } & b
4 . b
5,b
6 <= b
7 "\/" b
8 & ( for b
9, b
10, b
11 being
Element of b
3 holds
not ( b
4 . b
5,b
9 = b
7 & b
4 . b
10,b
11 = b
7 & b
4 . b
9,b
10 = b
8 & b
4 . b
11,b
6 = b
8 ) ) )
theorem Th46: :: LATTICE5:46
for b
1 being
lower-bounded LATTICEfor b
2 being
ExtensionSeq of the
carrier of b
1,
BasicDF b
1for b
3 being non
empty set for b
4 being
distance_function of b
3,b
1for b
5 being
Homomorphism of b
1,
(EqRelLATT b3)for b
6, b
7 being
Element of b
3for b
8, b
9 being
Equivalence_Relation of b
3for b
10, b
11 being
set holds
not ( b
5 = alpha b
4 & b
3 = union { ((b2 . b12) `1 ) where B is Nat : verum } & b
4 = union { ((b2 . b12) `2 ) where B is Nat : verum } & b
8 in the
carrier of
(Image b5) & b
9 in the
carrier of
(Image b5) &
[b10,b11] in b
8 "\/" b
9 & ( for b
12 being non
empty FinSequence of b
3 holds
not (
len b
12 = 3
+ 2 & b
10,b
11 are_joint_by b
12,b
8,b
9 ) ) )
theorem Th47: :: LATTICE5:47