:: LATTICE8 semantic presentation
:: deftheorem Def1 defines c= LATTICE8:def 1 :
:: deftheorem Def2 defines finitely_typed LATTICE8:def 2 :
:: deftheorem Def3 defines has_a_representation_of_type<= LATTICE8:def 3 :
Lemma3:
not 1 is even
Lemma4:
2 is even
theorem Th1: :: LATTICE8:1
theorem Th2: :: LATTICE8:2
theorem Th3: :: LATTICE8:3
theorem Th4: :: LATTICE8:4
theorem Th5: :: LATTICE8:5
theorem Th6: :: LATTICE8:6
theorem Th7: :: LATTICE8:7
theorem Th8: :: LATTICE8:8
theorem Th9: :: LATTICE8:9
:: deftheorem Def4 defines new_set2 LATTICE8:def 4 :
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_fun2 c
3,c
4 -> BiFunction of
(new_set2 a1),a
2 means :
Def5:
:: LATTICE8:def 5
( ( 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}} = ((a3 . (a4 `1 ),(a4 `2 )) "\/" (a4 `3 )) "/\" (a4 `4 ) & a
5 . {{a1}},
{a1} = ((a3 . (a4 `1 ),(a4 `2 )) "\/" (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 `2 )) "\/" (a4 `3 ) & a
5 . {{a1}},b
1 = (a3 . b1,(a4 `2 )) "\/" (a4 `3 ) ) ) );
existence
ex b1 being BiFunction of (new_set2 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}} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (c4 `3 )) "/\" (c4 `4 ) & b1 . {{c1}},{c1} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (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 `2 )) "\/" (c4 `3 ) & b1 . {{c1}},b2 = (c3 . b2,(c4 `2 )) "\/" (c4 `3 ) ) ) )
uniqueness
for b1, b2 being BiFunction of (new_set2 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}} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (c4 `3 )) "/\" (c4 `4 ) & b1 . {{c1}},{c1} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (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 `2 )) "\/" (c4 `3 ) & b1 . {{c1}},b3 = (c3 . b3,(c4 `2 )) "\/" (c4 `3 ) ) ) & ( 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}} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (c4 `3 )) "/\" (c4 `4 ) & b2 . {{c1}},{c1} = ((c3 . (c4 `1 ),(c4 `2 )) "\/" (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 `2 )) "\/" (c4 `3 ) & b2 . {{c1}},b3 = (c3 . b3,(c4 `2 )) "\/" (c4 `3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines new_bi_fun2 LATTICE8:def 5 :
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_set2 b1),b
2 holds
( b
5 = new_bi_fun2 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}} = ((b3 . (b4 `1 ),(b4 `2 )) "\/" (b4 `3 )) "/\" (b4 `4 ) & b
5 . {{b1}},
{b1} = ((b3 . (b4 `1 ),(b4 `2 )) "\/" (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 `2 )) "\/" (b4 `3 ) & b
5 . {{b1}},b
6 = (b3 . b6,(b4 `2 )) "\/" (b4 `3 ) ) ) ) );
theorem Th10: :: LATTICE8:10
theorem Th11: :: LATTICE8:11
theorem Th12: :: LATTICE8:12
theorem Th13: :: LATTICE8:13
theorem Th14: :: LATTICE8:14
:: deftheorem Def6 defines ConsecutiveSet2 LATTICE8:def 6 :
theorem Th15: :: LATTICE8:15
theorem Th16: :: LATTICE8:16
theorem Th17: :: LATTICE8:17
theorem Th18: :: LATTICE8:18
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 E23:
c
5 in dom c
4
;
func Quadr2 c
4,c
5 -> Element of
[:(ConsecutiveSet2 a1,a5),(ConsecutiveSet2 a1,a5),the carrier of a2,the carrier of a2:] equals :
Def7:
:: LATTICE8:def 7
a
4 . a
5;
correctness
coherence
c4 . c5 is Element of [:(ConsecutiveSet2 c1,c5),(ConsecutiveSet2 c1,c5),the carrier of c2,the carrier of c2:];
end;
:: deftheorem Def7 defines Quadr2 LATTICE8:def 7 :
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 ConsecutiveDelta2 c
4,c
5 -> set means :
Def8:
:: LATTICE8:def 8
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_fun2 (BiFun (b1 . b2),(ConsecutiveSet2 a1,b2),a2),
(Quadr2 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_fun2 (BiFun (b2 . b3),(ConsecutiveSet2 c1,b3),c2),(Quadr2 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_fun2 (BiFun (b3 . b4),(ConsecutiveSet2 c1,b4),c2),(Quadr2 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_fun2 (BiFun (b3 . b4),(ConsecutiveSet2 c1,b4),c2),(Quadr2 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 Def8 defines ConsecutiveDelta2 LATTICE8:def 8 :
theorem Th19: :: LATTICE8:19
theorem Th20: :: LATTICE8:20
theorem Th21: :: LATTICE8:21
theorem Th22: :: LATTICE8:22
theorem Th23: :: LATTICE8:23
theorem Th24: :: LATTICE8:24
theorem Th25: :: LATTICE8:25
theorem Th26: :: LATTICE8:26
theorem Th27: :: LATTICE8:27
theorem Th28: :: LATTICE8:28
theorem Th29: :: LATTICE8:29
:: deftheorem Def9 defines NextSet2 LATTICE8:def 9 :
:: deftheorem Def10 defines NextDelta2 LATTICE8:def 10 :
:: deftheorem Def11 defines is_extension2_of LATTICE8:def 11 :
theorem Th30: :: LATTICE8:30
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_extension2_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 being
Element of b
4 holds
not ( b
5 . b
6,b
10 = b
8 & b
5 . b
10,b
11 = ((b3 . b6,b7) "\/" b8) "/\" b
9 & b
5 . b
11,b
7 = b
8 ) ) ) )
definition
let c
1 be non
empty set ;
let c
2 be
lower-bounded modular LATTICE;
let c
3 be
distance_function of c
1,c
2;
mode ExtensionSeq2 of c
1,c
3 -> Function means :
Def12:
:: LATTICE8:def 12
(
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_extension2_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_extension2_of b3,b4 & b1 . b2 = [b3,b4] & b1 . (b2 + 1) = [b5,b6] ) ) )
end;
:: deftheorem Def12 defines ExtensionSeq2 LATTICE8:def 12 :
theorem Th31: :: LATTICE8:31
theorem Th32: :: LATTICE8:32
theorem Th33: :: LATTICE8:33
theorem Th34: :: LATTICE8:34
for b
1 being
lower-bounded modular LATTICEfor b
2 being
ExtensionSeq2 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 being
Element of b
3 holds
not ( b
4 . b
5,b
9 = b
7 & b
4 . b
9,b
10 = ((b4 . b5,b6) "\/" b7) "/\" b
8 & b
4 . b
10,b
6 = b
7 ) ) )
Lemma42:
for b1 being Nat holds
not ( b1 in Seg 4 & not b1 = 1 & not b1 = 2 & not b1 = 3 & not b1 = 4 )
Lemma43:
for b1 being Nat holds
not ( 1 <= b1 & b1 < 4 & not b1 = 1 & not b1 = 2 & not b1 = 3 )
theorem Th35: :: LATTICE8:35
theorem Th36: :: LATTICE8:36
theorem Th37: :: LATTICE8:37