set B = {0,1} \/ ].(1 / 2),1.[;
set L = RealSubLatt ((In (0,REAL)),(In (1,REAL)));
set R = Real_Lattice ;
A1:
{0,1} \/ ].(1 / 2),1.[ c= {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
{0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) } c= {0,1} \/ ].(1 / 2),1.[
then A9:
{0,1} \/ ].(1 / 2),1.[ = {0} \/ { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
by A1, XBOOLE_0:def 10;
A10:
0 in { r where r is Real : ( 0 <= r & r <= 1 ) }
;
then reconsider A = [.0,jj.] as non empty set by RCOMP_1:def 1;
A11:
the L_meet of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = minreal || [.0,jj.]
by Def4;
reconsider B = {0,1} \/ ].(1 / 2),1.[ as non empty set ;
A12:
the L_join of (RealSubLatt ((In (0,REAL)),(In (1,REAL)))) = maxreal || [.0,jj.]
by Def4;
A13:
A = the carrier of (RealSubLatt ((In (0,REAL)),(In (1,REAL))))
by Def4;
then reconsider Ma = maxreal || A, Mi = minreal || A as BinOp of A by Def4;
A14:
now for x1 being object st x1 in B holds
x1 in Alet x1 be
object ;
( x1 in B implies x1 in A )assume A15:
x1 in B
;
x1 in Ahence
x1 in A
;
verum end;
then A16:
B c= A
;
then A17:
[:B,B:] c= [:A,A:]
by ZFMISC_1:96;
then reconsider ma = Ma || B, mi = Mi || B as Function of [:B,B:],A by FUNCT_2:32;
A18:
for x1 being Element of A holds x1 is Element of Real_Lattice
by REAL_LAT:def 3, XREAL_0:def 1;
A19:
dom mi = [:B,B:]
by FUNCT_2:def 1;
A20:
now for x9 being object st x9 in dom mi holds
mi . x9 in Blet x9 be
object ;
( x9 in dom mi implies mi . x9 in B )assume A21:
x9 in dom mi
;
mi . x9 in Bthen consider x1,
x2 being
object such that A22:
x9 = [x1,x2]
by RELAT_1:def 1;
A23:
x2 in B
by A21, A22, ZFMISC_1:87;
A24:
x1 in B
by A21, A22, ZFMISC_1:87;
now mi . x9 in Bper cases
( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A24, XBOOLE_0:def 3;
suppose
x1 in {0}
;
mi . x9 in Bthen A25:
x1 = 0
by TARSKI:def 1;
now mi . x9 in Bper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A23, XBOOLE_0:def 3;
suppose
x2 in {0}
;
mi . x9 in Bthen A26:
x2 = 0
by TARSKI:def 1;
mi . x9 =
Mi . [x1,x2]
by A21, A22, FUNCT_1:49
.=
minreal . (
x1,
x2)
by A17, A21, A22, FUNCT_1:49
.=
min (
0,
0)
by A25, A26, REAL_LAT:def 1
.=
0
;
then
mi . x9 in {0}
by TARSKI:def 1;
hence
mi . x9 in B
by A9, XBOOLE_0:def 3;
verum end; suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
mi . x9 in Bthen consider y2 being
Real such that A27:
x2 = y2
and A28:
( 1
/ 2
< y2 &
y2 <= 1 )
;
reconsider y2 =
y2 as
Real ;
mi . x9 =
Mi . [x1,x2]
by A21, A22, FUNCT_1:49
.=
minreal . (
x1,
x2)
by A17, A21, A22, FUNCT_1:49
.=
min (
0,
y2)
by A25, A27, REAL_LAT:def 1
;
then
(
mi . x9 = 0 or
mi . x9 = y2 )
by XXREAL_0:15;
then
(
mi . x9 in {0} or
mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A28, TARSKI:def 1;
hence
mi . x9 in B
by A9, XBOOLE_0:def 3;
verum end; end; end; hence
mi . x9 in B
;
verum end; suppose
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
mi . x9 in Bthen consider y1 being
Real such that A29:
x1 = y1
and A30:
( 1
/ 2
< y1 &
y1 <= 1 )
;
reconsider y1 =
y1 as
Real ;
now mi . x9 in Bper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A23, XBOOLE_0:def 3;
suppose
x2 in {0}
;
mi . x9 in Bthen A31:
x2 = 0
by TARSKI:def 1;
mi . x9 =
Mi . [x1,x2]
by A21, A22, FUNCT_1:49
.=
minreal . (
x1,
x2)
by A17, A21, A22, FUNCT_1:49
.=
min (
y1,
0)
by A29, A31, REAL_LAT:def 1
;
then
(
mi . x9 = y1 or
mi . x9 = 0 )
by XXREAL_0:15;
then
(
mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or
mi . x9 in {0} )
by A30, TARSKI:def 1;
hence
mi . x9 in B
by A9, XBOOLE_0:def 3;
verum end; suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
mi . x9 in Bthen consider y2 being
Real such that A32:
x2 = y2
and A33:
( 1
/ 2
< y2 &
y2 <= 1 )
;
reconsider y2 =
y2 as
Real ;
mi . x9 =
Mi . [x1,x2]
by A21, A22, FUNCT_1:49
.=
minreal . (
x1,
x2)
by A17, A21, A22, FUNCT_1:49
.=
min (
y1,
y2)
by A29, A32, REAL_LAT:def 1
;
then
(
mi . x9 = y1 or
mi . x9 = y2 )
by XXREAL_0:15;
then
mi . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
by A30, A33;
hence
mi . x9 in B
by A9, XBOOLE_0:def 3;
verum end; end; end; hence
mi . x9 in B
;
verum end; end; end; hence
mi . x9 in B
;
verum end;
A34:
dom ma = [:B,B:]
by FUNCT_2:def 1;
A35:
now for x9 being object st x9 in dom ma holds
ma . x9 in Blet x9 be
object ;
( x9 in dom ma implies ma . x9 in B )assume A36:
x9 in dom ma
;
ma . x9 in Bthen consider x1,
x2 being
object such that A37:
x9 = [x1,x2]
by RELAT_1:def 1;
A38:
x2 in B
by A36, A37, ZFMISC_1:87;
A39:
x1 in B
by A36, A37, ZFMISC_1:87;
now ma . x9 in Bper cases
( x1 in {0} or x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A39, XBOOLE_0:def 3;
suppose
x1 in {0}
;
ma . x9 in Bthen A40:
x1 = 0
by TARSKI:def 1;
now ma . x9 in Bper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A38, XBOOLE_0:def 3;
suppose
x2 in {0}
;
ma . x9 in Bthen A41:
x2 = 0
by TARSKI:def 1;
ma . x9 =
Ma . [x1,x2]
by A36, A37, FUNCT_1:49
.=
maxreal . (
x1,
x2)
by A17, A36, A37, FUNCT_1:49
.=
max (
0,
0)
by A40, A41, REAL_LAT:def 2
.=
0
;
then
ma . x9 in {0}
by TARSKI:def 1;
hence
ma . x9 in B
by A9, XBOOLE_0:def 3;
verum end; suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
ma . x9 in Bthen consider y2 being
Real such that A42:
x2 = y2
and A43:
( 1
/ 2
< y2 &
y2 <= 1 )
;
reconsider y2 =
y2 as
Real ;
ma . x9 =
Ma . [x1,x2]
by A36, A37, FUNCT_1:49
.=
maxreal . (
x1,
x2)
by A17, A36, A37, FUNCT_1:49
.=
max (
0,
y2)
by A40, A42, REAL_LAT:def 2
;
then
(
ma . x9 = 0 or
ma . x9 = y2 )
by XXREAL_0:16;
then
(
ma . x9 in {0} or
ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A43, TARSKI:def 1;
hence
ma . x9 in B
by A9, XBOOLE_0:def 3;
verum end; end; end; hence
ma . x9 in B
;
verum end; suppose
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
ma . x9 in Bthen consider y1 being
Real such that A44:
x1 = y1
and A45:
( 1
/ 2
< y1 &
y1 <= 1 )
;
reconsider y1 =
y1 as
Real ;
now ma . x9 in Bper cases
( x2 in {0} or x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by A1, A38, XBOOLE_0:def 3;
suppose
x2 in {0}
;
ma . x9 in Bthen A46:
x2 = 0
by TARSKI:def 1;
ma . x9 =
Ma . [x1,x2]
by A36, A37, FUNCT_1:49
.=
maxreal . (
x1,
x2)
by A17, A36, A37, FUNCT_1:49
.=
max (
y1,
0)
by A44, A46, REAL_LAT:def 2
;
then
(
ma . x9 = y1 or
ma . x9 = 0 )
by XXREAL_0:16;
then
(
ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } or
ma . x9 in {0} )
by A45, TARSKI:def 1;
hence
ma . x9 in B
by A9, XBOOLE_0:def 3;
verum end; suppose
x2 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
ma . x9 in Bthen consider y2 being
Real such that A47:
x2 = y2
and A48:
( 1
/ 2
< y2 &
y2 <= 1 )
;
reconsider y2 =
y2 as
Real ;
ma . x9 =
Ma . [x1,x2]
by A36, A37, FUNCT_1:49
.=
maxreal . (
x1,
x2)
by A17, A36, A37, FUNCT_1:49
.=
max (
y1,
y2)
by A44, A47, REAL_LAT:def 2
;
then
(
ma . x9 = y1 or
ma . x9 = y2 )
by XXREAL_0:16;
then
ma . x9 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
by A45, A48;
hence
ma . x9 in B
by A9, XBOOLE_0:def 3;
verum end; end; end; hence
ma . x9 in B
;
verum end; end; end; hence
ma . x9 in B
;
verum end;
reconsider mi = mi as BinOp of B by A19, A20, FUNCT_2:3;
reconsider ma = ma as BinOp of B by A34, A35, FUNCT_2:3;
reconsider L9 = LattStr(# B,ma,mi #) as non empty LattStr ;
A49:
now for a, b being Element of L9 holds
( a "\/" b = maxreal . (a,b) & a "/\" b = minreal . (a,b) )let a,
b be
Element of
L9;
( a "\/" b = maxreal . (a,b) & a "/\" b = minreal . (a,b) )thus a "\/" b =
the
L_join of
L9 . (
a,
b)
by LATTICES:def 1
.=
(maxreal || A) . [a,b]
by FUNCT_1:49
.=
maxreal . (
a,
b)
by A17, FUNCT_1:49
;
a "/\" b = minreal . (a,b)thus a "/\" b =
the
L_meet of
L9 . (
a,
b)
by LATTICES:def 2
.=
(minreal || A) . [a,b]
by FUNCT_1:49
.=
minreal . (
a,
b)
by A17, FUNCT_1:49
;
verum end;
reconsider L = RealSubLatt ((In (0,REAL)),(In (1,REAL))) as complete Lattice by Th20;
A55:
now for p, q, r being Element of L9 holds p "/\" (q "/\" r) = (p "/\" q) "/\" rlet p,
q,
r be
Element of
L9;
p "/\" (q "/\" r) = (p "/\" q) "/\" rreconsider p9 =
p,
q9 =
q,
r9 =
r as
Element of
L by A50;
reconsider p9 =
p9,
q9 =
q9,
r9 =
r9 as
Element of
Real_Lattice by A13, A18;
thus p "/\" (q "/\" r) =
minreal . (
p,
(q "/\" r))
by A49
.=
minreal . (
p,
(minreal . (q,r)))
by A49
.=
minreal . (
(minreal . (p9,q9)),
r9)
by REAL_LAT:4
.=
minreal . (
(p "/\" q),
r)
by A49
.=
(p "/\" q) "/\" r
by A49
;
verum end;
now for p, q, r being Element of L9 holds p "\/" (q "\/" r) = (p "\/" q) "\/" rlet p,
q,
r be
Element of
L9;
p "\/" (q "\/" r) = (p "\/" q) "\/" rreconsider p9 =
p,
q9 =
q,
r9 =
r as
Element of
L by A50;
reconsider p9 =
p9,
q9 =
q9,
r9 =
r9 as
Element of
Real_Lattice by A13, A18;
thus p "\/" (q "\/" r) =
maxreal . (
p,
(q "\/" r))
by A49
.=
maxreal . (
p,
(maxreal . (q,r)))
by A49
.=
maxreal . (
(maxreal . (p9,q9)),
r9)
by REAL_LAT:3
.=
maxreal . (
(p "\/" q),
r)
by A49
.=
(p "\/" q) "\/" r
by A49
;
verum end;
then
( L9 is join-commutative & L9 is join-associative & L9 is meet-absorbing & L9 is meet-commutative & L9 is meet-associative & L9 is join-absorbing )
by A51, A54, A53, A55, A52, LATTICES:def 4, LATTICES:def 5, LATTICES:def 6, LATTICES:def 7, LATTICES:def 8, LATTICES:def 9;
then reconsider L9 = L9 as Lattice ;
reconsider L9 = L9 as SubLattice of RealSubLatt ((In (0,REAL)),(In (1,REAL))) by A13, A12, A11, A16, NAT_LAT:def 12;
take
L9
; ( L9 is \/-inheriting & not L9 is /\-inheriting )
now for X being Subset of L9 holds "\/" (X,L) in the carrier of L9let X be
Subset of
L9;
"\/" (X,L) in the carrier of L9thus
"\/" (
X,
L)
in the
carrier of
L9
verumproof
0 in { r where r is Real : ( 0 <= r & r <= 1 ) }
;
then reconsider w =
0 as
Element of
L by A13, RCOMP_1:def 1;
A56:
X is_less_than "\/" (
X,
L)
by LATTICE3:def 21;
"\/" (
X,
L)
in [.0,1.]
by A13;
then
"\/" (
X,
L)
in { r where r is Real : ( 0 <= r & r <= 1 ) }
by RCOMP_1:def 1;
then consider y being
Real such that A57:
y = "\/" (
X,
L)
and
0 <= y
and A58:
y <= 1
;
reconsider y =
y as
Real ;
assume A59:
not
"\/" (
X,
L)
in the
carrier of
L9
;
contradiction
then
not
"\/" (
X,
L)
in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
by A9, XBOOLE_0:def 3;
then A60:
y <= 1
/ 2
by A57, A58;
now for z9 being object st z9 in X holds
z9 in {0}let z9 be
object ;
( z9 in X implies z9 in {0} )assume A61:
z9 in X
;
z9 in {0}then reconsider z =
z9 as
Element of
L9 ;
reconsider z =
z as
Element of
L by A13, A14;
A62:
z [= "\/" (
X,
L)
by A56, A61;
reconsider z1 =
z as
Real ;
reconsider z1 =
z1 as
Real ;
min (
z1,
y) =
minreal . (
z1,
("\/" (X,L)))
by A57, REAL_LAT:def 1
.=
(minreal || A) . [z,("\/" (X,L))]
by A13, FUNCT_1:49
.=
(minreal || A) . (
z,
("\/" (X,L)))
.=
z "/\" ("\/" (X,L))
by A11, LATTICES:def 2
.=
z1
by A62, LATTICES:4
;
then
z1 <= y
by XXREAL_0:def 9;
then
y + z1 <= (1 / 2) + y
by A60, XREAL_1:7;
then
for
v being
Real holds
( not
z1 = v or not 1
/ 2
< v or not
v <= 1 )
by XREAL_1:6;
then
not
z1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) }
;
hence
z9 in {0}
by A9, XBOOLE_0:def 3;
verum end;
then A63:
X c= {0}
;
hence
contradiction
;
verum
end; end;
hence
L9 is \/-inheriting
; not L9 is /\-inheriting
now not for X being Subset of L9 holds "/\" (X,L) in the carrier of L9
1
/ 2
in { x where x is Real : ( 0 <= x & x <= 1 ) }
;
then reconsider z = 1
/ 2 as
Element of
L by A13, RCOMP_1:def 1;
set X =
{ x where x is Real : ( 1 / 2 < x & x <= 1 ) } ;
for
y being
Real holds
( not
y = 1
/ 2 or not 1
/ 2
< y or not
y <= 1 )
;
then A68:
( not 1
/ 2
in {0} & not 1
/ 2
in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } )
by TARSKI:def 1;
for
x1 being
object st
x1 in { x where x is Real : ( 1 / 2 < x & x <= 1 ) } holds
x1 in B
by A9, XBOOLE_0:def 3;
then reconsider X =
{ x where x is Real : ( 1 / 2 < x & x <= 1 ) } as
Subset of
L9 by TARSKI:def 3;
take X =
X;
not "/\" (X,L) in the carrier of L9A69:
now for b being Element of L st b is_less_than X holds
b [= zlet b be
Element of
L;
( b is_less_than X implies b [= z )
b in A
by A13;
then
b in { r where r is Real : ( 0 <= r & r <= 1 ) }
by RCOMP_1:def 1;
then consider b9 being
Real such that A70:
b = b9
and A71:
0 <= b9
and A72:
b9 <= 1
;
reconsider b9 =
b9 as
Real ;
assume A73:
b is_less_than X
;
b [= zA74:
b9 <= 1
/ 2
proof
assume A75:
1
/ 2
< b9
;
contradiction
then
b9 in X
by A72;
then A76:
b = "/\" (
X,
L)
by A70, A73, LATTICE3:41;
(1 / 2) + b9 < b9 + b9
by A75, XREAL_1:6;
then A77:
((1 / 2) + b9) / 2
< (b9 + b9) / 2
by XREAL_1:74;
then
(((1 / 2) + b9) / 2) + b9 <= b9 + 1
by A72, XREAL_1:7;
then A78:
((1 / 2) + b9) / 2
<= 1
by XREAL_1:6;
then
((1 / 2) + b9) / 2
in { r where r is Real : ( 0 <= r & r <= 1 ) }
by A71;
then reconsider c =
((1 / 2) + b9) / 2 as
Element of
L by A13, RCOMP_1:def 1;
reconsider cc =
c as
Real ;
(1 / 2) + (1 / 2) < b9 + (1 / 2)
by A75, XREAL_1:6;
then
1
/ 2
< ((1 / 2) + b9) / 2
by XREAL_1:74;
then
((1 / 2) + b9) / 2
in X
by A78;
then
b [= c
by A76, LATTICE3:38;
then b9 =
b "/\" c
by A70, LATTICES:4
.=
(minreal || A) . (
b,
c)
by A11, LATTICES:def 2
.=
minreal . [b,c]
by A13, FUNCT_1:49
.=
minreal . (
b,
cc)
.=
min (
b9,
(((1 / 2) + b9) / 2))
by A70, REAL_LAT:def 1
;
hence
contradiction
by A77, XXREAL_0:def 9;
verum
end; b "/\" z =
(minreal || A) . (
b,
z)
by A11, LATTICES:def 2
.=
minreal . [b,z]
by A13, FUNCT_1:49
.=
minreal . (
b,
z)
.=
min (
b9,
jd)
by A70, REAL_LAT:def 1
.=
b
by A70, A74, XXREAL_0:def 9
;
hence
b [= z
by LATTICES:4;
verum end; then
z is_less_than X
;
then
"/\" (
X,
L)
= jd
by A69, LATTICE3:34;
hence
not
"/\" (
X,
L)
in the
carrier of
L9
by A1, A68, XBOOLE_0:def 3;
verum end;
hence
not L9 is /\-inheriting
; verum