:: YELLOW21 semantic presentation
:: deftheorem Def1 defines as_1-sorted YELLOW21:def 1 :
:: deftheorem Def2 defines POSETS YELLOW21:def 2 :
:: deftheorem Def3 defines carrier-underlaid YELLOW21:def 3 :
:: deftheorem Def4 defines lattice-wise YELLOW21:def 4 :
:: deftheorem Def5 defines with_complete_lattices YELLOW21:def 5 :
scheme :: YELLOW21:sch 1
s1{ F
1()
-> non
empty set , P
1[
set ,
set ,
set ] } :
provided
E6:
for b
1 being
Element of F
1() holds
b
1 is
LATTICE
and
E7:
for b
1, b
2, b
3 being
LATTICE holds
( b
1 in F
1() & b
2 in F
1() & b
3 in F
1() implies for b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( P
1[b
1,b
2,b
4] & P
1[b
2,b
3,b
5] implies P
1[b
1,b
3,b
5 * b
4] ) )
and
E8:
for b
1 being
LATTICE holds
( b
1 in F
1() implies P
1[b
1,b
1,
id b
1] )
theorem Th1: :: YELLOW21:1
theorem Th2: :: YELLOW21:2
:: deftheorem Def6 defines latt YELLOW21:def 6 :
:: deftheorem Def7 defines @ YELLOW21:def 7 :
theorem Th3: :: YELLOW21:3
scheme :: YELLOW21:sch 2
s2{ F
1()
-> non
empty set , P
1[
set ,
set ,
set ] } :
provided
E9:
for b
1 being
Element of F
1() holds
b
1 is
LATTICE
and
E10:
for b
1, b
2, b
3 being
LATTICE holds
( b
1 in F
1() & b
2 in F
1() & b
3 in F
1() implies for b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( P
1[b
1,b
2,b
4] & P
1[b
2,b
3,b
5] implies P
1[b
1,b
3,b
5 * b
4] ) )
and
E11:
for b
1 being
LATTICE holds
( b
1 in F
1() implies P
1[b
1,b
1,
id b
1] )
scheme :: YELLOW21:sch 3
s3{ F
1()
-> non
empty set , P
1[
set ], P
2[
set ,
set ,
set ] } :
provided
E9:
ex b
1 being
strict LATTICE st
( P
1[b
1] & the
carrier of b
1 in F
1() )
and
E10:
for b
1, b
2, b
3 being
LATTICE holds
( P
1[b
1] & P
1[b
2] & P
1[b
3] implies for b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( P
2[b
1,b
2,b
4] & P
2[b
2,b
3,b
5] implies P
2[b
1,b
3,b
5 * b
4] ) )
and
E11:
for b
1 being
LATTICE holds
( P
1[b
1] implies P
2[b
1,b
1,
id b
1] )
scheme :: YELLOW21:sch 4
s4{ F
1()
-> non
empty set , P
1[
set ,
set ,
set ] } :
for b
1, b
2 being
lattice-wise category holds
( the
carrier of b
1 = F
1() & ( for b
3, b
4 being
object of b
1for b
5 being
monotone Function of
(latt b3),
(latt b4) holds
( b
5 in <^b3,b4^> iff P
1[b
3,b
4,b
5] ) ) & the
carrier of b
2 = F
1() & ( for b
3, b
4 being
object of b
2for b
5 being
monotone Function of
(latt b3),
(latt b4) holds
( b
5 in <^b3,b4^> iff P
1[b
3,b
4,b
5] ) ) implies
AltCatStr(# the
carrier of b
1,the
Arrows of b
1,the
Comp of b
1 #)
= AltCatStr(# the
carrier of b
2,the
Arrows of b
2,the
Comp of b
2 #) )
scheme :: YELLOW21:sch 5
s5{ F
1()
-> non
empty set , P
1[
set ], P
2[
set ,
set ,
set ] } :
for b
1, b
2 being
lattice-wise category holds
( ( for b
3 being
LATTICE holds
( b
3 is
object of b
1 iff ( b
3 is
strict & P
1[b
3] & the
carrier of b
3 in F
1() ) ) ) & ( for b
3, b
4 being
object of b
1for b
5 being
monotone Function of
(latt b3),
(latt b4) holds
( b
5 in <^b3,b4^> iff P
2[b
3,b
4,b
5] ) ) & ( for b
3 being
LATTICE holds
( b
3 is
object of b
2 iff ( b
3 is
strict & P
1[b
3] & the
carrier of b
3 in F
1() ) ) ) & ( for b
3, b
4 being
object of b
2for b
5 being
monotone Function of
(latt b3),
(latt b4) holds
( b
5 in <^b3,b4^> iff P
2[b
3,b
4,b
5] ) ) implies
AltCatStr(# the
carrier of b
1,the
Arrows of b
1,the
Comp of b
1 #)
= AltCatStr(# the
carrier of b
2,the
Arrows of b
2,the
Comp of b
2 #) )
scheme :: YELLOW21:sch 6
s6{ P
1[
set ,
set ,
set ], P
2[
set ,
set ,
set ], F
1()
-> lattice-wise category, F
2()
-> lattice-wise category, F
3(
set )
-> LATTICE, F
4(
set ,
set ,
set )
-> Function } :
provided
E9:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
1()
. b
1,b
2 iff ( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & P
1[b
1,b
2,b
3] ) )
and
E10:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
2()
. b
1,b
2 iff ( b
1 in the
carrier of F
2() & b
2 in the
carrier of F
2() & P
2[b
1,b
2,b
3] ) )
and
E11:
for b
1 being
LATTICE holds
( b
1 in the
carrier of F
1() implies F
3(b
1)
in the
carrier of F
2() )
and
E12:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( P
1[b
1,b
2,b
3] implies ( F
4(b
1,b
2,b
3) is
Function of F
3(b
1),F
3(b
2) & P
2[F
3(b
1),F
3(b
2),F
4(b
1,b
2,b
3)] ) )
and
E13:
for b
1 being
LATTICE holds
( b
1 in the
carrier of F
1() implies F
4(b
1,b
1,
(id b1))
= id F
3(b
1) )
and
E14:
for b
1, b
2, b
3 being
LATTICEfor b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( P
1[b
1,b
2,b
4] & P
1[b
2,b
3,b
5] implies F
4(b
1,b
3,
(b5 * b4))
= F
4(b
2,b
3,b
5)
* F
4(b
1,b
2,b
4) )
scheme :: YELLOW21:sch 7
s7{ P
1[
set ,
set ,
set ], P
2[
set ,
set ,
set ], F
1()
-> lattice-wise category, F
2()
-> lattice-wise category, F
3(
set )
-> LATTICE, F
4(
set ,
set ,
set )
-> Function } :
provided
E9:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
1()
. b
1,b
2 iff ( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & P
1[b
1,b
2,b
3] ) )
and
E10:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
2()
. b
1,b
2 iff ( b
1 in the
carrier of F
2() & b
2 in the
carrier of F
2() & P
2[b
1,b
2,b
3] ) )
and
E11:
for b
1 being
LATTICE holds
( b
1 in the
carrier of F
1() implies F
3(b
1)
in the
carrier of F
2() )
and
E12:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( P
1[b
1,b
2,b
3] implies ( F
4(b
1,b
2,b
3) is
Function of F
3(b
2),F
3(b
1) & P
2[F
3(b
2),F
3(b
1),F
4(b
1,b
2,b
3)] ) )
and
E13:
for b
1 being
LATTICE holds
( b
1 in the
carrier of F
1() implies F
4(b
1,b
1,
(id b1))
= id F
3(b
1) )
and
E14:
for b
1, b
2, b
3 being
LATTICEfor b
4 being
Function of b
1,b
2for b
5 being
Function of b
2,b
3 holds
( P
1[b
1,b
2,b
4] & P
1[b
2,b
3,b
5] implies F
4(b
1,b
3,
(b5 * b4))
= F
4(b
1,b
2,b
4)
* F
4(b
2,b
3,b
5) )
scheme :: YELLOW21:sch 8
s8{ P
1[
set ,
set ,
set ], P
2[
set ,
set ,
set ], F
1()
-> lattice-wise category, F
2()
-> lattice-wise category, F
3(
set )
-> LATTICE, F
4(
set ,
set ,
set )
-> Function } :
provided
E9:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
1()
. b
1,b
2 iff ( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & P
1[b
1,b
2,b
3] ) )
and
E10:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
2()
. b
1,b
2 iff ( b
1 in the
carrier of F
2() & b
2 in the
carrier of F
2() & P
2[b
1,b
2,b
3] ) )
and
E11:
ex b
1 being
covariant Functor of F
1(),F
2() st
( ( for b
2 being
object of F
1() holds b
1 . b
2 = F
3(b
2) ) & ( for b
2, b
3 being
object of F
1() holds
(
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
2,b
3 holds b
1 . b
4 = F
4(b
2,b
3,b
4) ) ) )
and
E12:
for b
1, b
2 being
LATTICE holds
( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & F
3(b
1)
= F
3(b
2) implies b
1 = b
2 )
and
E13:
for b
1, b
2 being
LATTICEfor b
3, b
4 being
Function of b
1,b
2 holds
( P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] & F
4(b
1,b
2,b
3)
= F
4(b
1,b
2,b
4) implies b
3 = b
4 )
and
E14:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
not ( P
2[b
1,b
2,b
3] & ( for b
4, b
5 being
LATTICEfor b
6 being
Function of b
4,b
5 holds
not ( b
4 in the
carrier of F
1() & b
5 in the
carrier of F
1() & P
1[b
4,b
5,b
6] & b
1 = F
3(b
4) & b
2 = F
3(b
5) & b
3 = F
4(b
4,b
5,b
6) ) ) )
scheme :: YELLOW21:sch 9
s9{ P
1[
set ,
set ,
set ], P
2[
set ,
set ,
set ], F
1()
-> lattice-wise category, F
2()
-> lattice-wise category, F
3(
set )
-> LATTICE, F
4(
set ,
set ,
set )
-> Function } :
provided
E9:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
1()
. b
1,b
2 iff ( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & P
1[b
1,b
2,b
3] ) )
and
E10:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
( b
3 in the
Arrows of F
2()
. b
1,b
2 iff ( b
1 in the
carrier of F
2() & b
2 in the
carrier of F
2() & P
2[b
1,b
2,b
3] ) )
and
E11:
ex b
1 being
contravariant Functor of F
1(),F
2() st
( ( for b
2 being
object of F
1() holds b
1 . b
2 = F
3(b
2) ) & ( for b
2, b
3 being
object of F
1() holds
(
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
2,b
3 holds b
1 . b
4 = F
4(b
2,b
3,b
4) ) ) )
and
E12:
for b
1, b
2 being
LATTICE holds
( b
1 in the
carrier of F
1() & b
2 in the
carrier of F
1() & F
3(b
1)
= F
3(b
2) implies b
1 = b
2 )
and
E13:
for b
1, b
2 being
LATTICEfor b
3, b
4 being
Function of b
1,b
2 holds
( F
4(b
1,b
2,b
3)
= F
4(b
1,b
2,b
4) implies b
3 = b
4 )
and
E14:
for b
1, b
2 being
LATTICEfor b
3 being
Function of b
1,b
2 holds
not ( P
2[b
1,b
2,b
3] & ( for b
4, b
5 being
LATTICEfor b
6 being
Function of b
4,b
5 holds
not ( b
4 in the
carrier of F
1() & b
5 in the
carrier of F
1() & P
1[b
4,b
5,b
6] & b
2 = F
3(b
4) & b
1 = F
3(b
5) & b
3 = F
4(b
4,b
5,b
6) ) ) )
:: deftheorem Def8 defines with_all_isomorphisms YELLOW21:def 8 :
theorem Th4: :: YELLOW21:4
theorem Th5: :: YELLOW21:5
scheme :: YELLOW21:sch 10
s10{ P
1[
set ,
set ,
set ], P
2[
set ,
set ,
set ], F
1()
-> lattice-wise category, F
2()
-> lattice-wise category, F
3(
set )
-> LATTICE, F
4(
set )
-> LATTICE, F
5(
set ,
set ,
set )
-> Function, F
6(
set ,
set ,
set )
-> Function, F
7(
set )
-> Function, F
8(
set )
-> Function } :
provided
E10:
for b
1, b
2 being
object of F
1()
for b
3 being
monotone Function of
(latt b1),
(latt b2) holds
( b
3 in <^b1,b2^> iff P
1[
latt b
1,
latt b
2,b
3] )
and
E11:
for b
1, b
2 being
object of F
2()
for b
3 being
monotone Function of
(latt b1),
(latt b2) holds
( b
3 in <^b1,b2^> iff P
2[
latt b
1,
latt b
2,b
3] )
and
E12:
ex b
1 being
covariant Functor of F
1(),F
2() st
( ( for b
2 being
object of F
1() holds b
1 . b
2 = F
3(b
2) ) & ( for b
2, b
3 being
object of F
1() holds
(
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
2,b
3 holds b
1 . b
4 = F
5(b
2,b
3,b
4) ) ) )
and
E13:
ex b
1 being
covariant Functor of F
2(),F
1() st
( ( for b
2 being
object of F
2() holds b
1 . b
2 = F
4(b
2) ) & ( for b
2, b
3 being
object of F
2() holds
(
<^b2,b3^> <> {} implies for b
4 being
Morphism of b
2,b
3 holds b
1 . b
4 = F
6(b
2,b
3,b
4) ) ) )
and
E14:
for b
1 being
LATTICE holds
not ( b
1 in the
carrier of F
1() & ( for b
2 being
monotone Function of F
4(F
3(b
1)),b
1 holds
not ( b
2 = F
7(b
1) & b
2 is
isomorphic & P
1[F
4(F
3(b
1)),b
1,b
2] & P
1[b
1,F
4(F
3(b
1)),b
2 " ] ) ) )
and
E15:
for b
1 being
LATTICE holds
not ( b
1 in the
carrier of F
2() & ( for b
2 being
monotone Function of b
1,F
3(F
4(b
1)) holds
not ( b
2 = F
8(b
1) & b
2 is
isomorphic & P
2[b
1,F
3(F
4(b
1)),b
2] & P
2[F
3(F
4(b
1)),b
1,b
2 " ] ) ) )
and
E16:
for b
1, b
2 being
object of F
1() holds
(
<^b1,b2^> <> {} implies for b
3 being
Morphism of b
1,b
2 holds F
7(b
2)
* F
6(F
3(b
1),F
3(b
2),F
5(b
1,b
2,b
3))
= (@ b3) * F
7(b
1) )
and
E17:
for b
1, b
2 being
object of F
2() holds
(
<^b1,b2^> <> {} implies for b
3 being
Morphism of b
1,b
2 holds F
5(F
4(b
1),F
4(b
2),F
6(b
1,b
2,b
3))
* F
8(b
1)
= F
8(b
2)
* (@ b3) )
:: deftheorem Def9 defines upper-bounded YELLOW21:def 9 :
Lemma11:
for b1, b2 being set holds
( b1 in b2 iff b2 = (b2 \ {b1}) \/ {b1} )
theorem Th6: :: YELLOW21:6
theorem Th7: :: YELLOW21:7
theorem Th8: :: YELLOW21:8
theorem Th9: :: YELLOW21:9
theorem Th10: :: YELLOW21:10
theorem Th11: :: YELLOW21:11
definition
let c
1 be non
empty set ;
given c
2 being
Element of c
1 such that E18:
not c
2 is
empty
;
defpred S
1[
LATTICE] means a
1 is
complete;
defpred S
2[
LATTICE,
LATTICE,
Function of a
1,a
2] means a
3 is
directed-sups-preserving;
func c
1 -UPS_category -> strict lattice-wise category means :
Def10:
:: YELLOW21:def 10
( ( for b
1 being
LATTICE holds
( b
1 is
object of a
2 iff ( b
1 is
strict & b
1 is
complete & the
carrier of b
1 in a
1 ) ) ) & ( for b
1, b
2 being
object of a
2for b
3 being
monotone Function of
(latt b1),
(latt b2) holds
( b
3 in <^b1,b2^> iff b
3 is
directed-sups-preserving ) ) );
existence
ex b1 being strict lattice-wise category st
( ( for b2 being LATTICE holds
( b2 is object of b1 iff ( b2 is strict & b2 is complete & the carrier of b2 in c1 ) ) ) & ( for b2, b3 being object of b1
for b4 being monotone Function of (latt b2),(latt b3) holds
( b4 in <^b2,b3^> iff b4 is directed-sups-preserving ) ) )
correctness
uniqueness
for b1, b2 being strict lattice-wise category holds
( ( for b3 being LATTICE holds
( b3 is object of b1 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b1
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is directed-sups-preserving ) ) & ( for b3 being LATTICE holds
( b3 is object of b2 iff ( b3 is strict & b3 is complete & the carrier of b3 in c1 ) ) ) & ( for b3, b4 being object of b2
for b5 being monotone Function of (latt b3),(latt b4) holds
( b5 in <^b3,b4^> iff b5 is directed-sups-preserving ) ) implies b1 = b2 );
end;
:: deftheorem Def10 defines -UPS_category YELLOW21:def 10 :
theorem Th12: :: YELLOW21:12
theorem Th13: :: YELLOW21:13
theorem Th14: :: YELLOW21:14
theorem Th15: :: YELLOW21:15
theorem Th16: :: YELLOW21:16
:: deftheorem Def11 defines -CONT_category YELLOW21:def 11 :
:: deftheorem Def12 defines -ALG_category YELLOW21:def 12 :
theorem Th17: :: YELLOW21:17
theorem Th18: :: YELLOW21:18
theorem Th19: :: YELLOW21:19
theorem Th20: :: YELLOW21:20