let A be non empty set ; for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is symmetric
let L be lower-bounded LATTICE; for O being Ordinal
for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is symmetric
let O be Ordinal; for d being BiFunction of A,L st d is symmetric holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is symmetric
let d be BiFunction of A,L; ( d is symmetric implies for q being QuadrSeq of d holds ConsecutiveDelta q,O is symmetric )
assume A1:
d is symmetric
; for q being QuadrSeq of d holds ConsecutiveDelta q,O is symmetric
let q be QuadrSeq of d; ConsecutiveDelta q,O is symmetric
defpred S1[ Ordinal] means ConsecutiveDelta q,$1 is symmetric ;
A2:
for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be
Ordinal;
( S1[O1] implies S1[ succ O1] )
assume
ConsecutiveDelta q,
O1 is
symmetric
;
S1[ succ O1]
then A3:
new_bi_fun (ConsecutiveDelta q,O1),
(Quadr q,O1) is
symmetric
by Th19;
let x,
y be
Element of
ConsecutiveSet A,
(succ O1);
LATTICE5:def 6 (ConsecutiveDelta q,(succ O1)) . x,y = (ConsecutiveDelta q,(succ O1)) . y,x
reconsider x9 =
x,
y9 =
y as
Element of
new_set (ConsecutiveSet A,O1) by Th25;
A4:
ConsecutiveDelta q,
(succ O1) =
new_bi_fun (BiFun (ConsecutiveDelta q,O1),(ConsecutiveSet A,O1),L),
(Quadr q,O1)
by Th30
.=
new_bi_fun (ConsecutiveDelta q,O1),
(Quadr q,O1)
by Def16
;
hence (ConsecutiveDelta q,(succ O1)) . x,
y =
(new_bi_fun (ConsecutiveDelta q,O1),(Quadr q,O1)) . y9,
x9
by A3, Def6
.=
(ConsecutiveDelta q,(succ O1)) . y,
x
by A4
;
verum
end;
A5:
for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) holds
S1[O2]
proof
deffunc H1(
Ordinal)
-> BiFunction of
(ConsecutiveSet A,$1),
L =
ConsecutiveDelta q,$1;
let O2 be
Ordinal;
( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that A6:
(
O2 <> {} &
O2 is
limit_ordinal )
and A7:
for
O1 being
Ordinal st
O1 in O2 holds
ConsecutiveDelta q,
O1 is
symmetric
;
S1[O2]
set CS =
ConsecutiveSet A,
O2;
consider Ls being
T-Sequence such that A8:
(
dom Ls = O2 & ( for
O1 being
Ordinal st
O1 in O2 holds
Ls . O1 = H1(
O1) ) )
from ORDINAL2:sch 2();
ConsecutiveDelta q,
O2 = union (rng Ls)
by A6, A8, Th31;
then reconsider f =
union (rng Ls) as
BiFunction of
(ConsecutiveSet A,O2),
L ;
deffunc H2(
Ordinal)
-> set =
ConsecutiveSet A,$1;
consider Ts being
T-Sequence such that A9:
(
dom Ts = O2 & ( for
O1 being
Ordinal st
O1 in O2 holds
Ts . O1 = H2(
O1) ) )
from ORDINAL2:sch 2();
A10:
ConsecutiveSet A,
O2 = union (rng Ts)
by A6, A9, Th26;
f is
symmetric
proof
let x,
y be
Element of
ConsecutiveSet A,
O2;
LATTICE5:def 6 f . x,y = f . y,x
consider x1 being
set such that A11:
x in x1
and A12:
x1 in rng Ts
by A10, TARSKI:def 4;
consider o1 being
set such that A13:
o1 in dom Ts
and A14:
x1 = Ts . o1
by A12, FUNCT_1:def 5;
consider y1 being
set such that A15:
y in y1
and A16:
y1 in rng Ts
by A10, TARSKI:def 4;
consider o2 being
set such that A17:
o2 in dom Ts
and A18:
y1 = Ts . o2
by A16, FUNCT_1:def 5;
reconsider o1 =
o1,
o2 =
o2 as
Ordinal by A13, A17;
A19:
x in ConsecutiveSet A,
o1
by A9, A11, A13, A14;
A20:
Ls . o1 = ConsecutiveDelta q,
o1
by A8, A9, A13;
then reconsider h1 =
Ls . o1 as
BiFunction of
(ConsecutiveSet A,o1),
L ;
A21:
h1 is
symmetric
proof
let x,
y be
Element of
ConsecutiveSet A,
o1;
LATTICE5:def 6 h1 . x,y = h1 . y,x
A22:
ConsecutiveDelta q,
o1 is
symmetric
by A7, A9, A13;
thus h1 . x,
y =
(ConsecutiveDelta q,o1) . x,
y
by A8, A9, A13
.=
(ConsecutiveDelta q,o1) . y,
x
by A22, Def6
.=
h1 . y,
x
by A8, A9, A13
;
verum
end;
A23:
dom h1 = [:(ConsecutiveSet A,o1),(ConsecutiveSet A,o1):]
by FUNCT_2:def 1;
A24:
y in ConsecutiveSet A,
o2
by A9, A15, A17, A18;
A25:
Ls . o2 = ConsecutiveDelta q,
o2
by A8, A9, A17;
then reconsider h2 =
Ls . o2 as
BiFunction of
(ConsecutiveSet A,o2),
L ;
A26:
h2 is
symmetric
proof
let x,
y be
Element of
ConsecutiveSet A,
o2;
LATTICE5:def 6 h2 . x,y = h2 . y,x
A27:
ConsecutiveDelta q,
o2 is
symmetric
by A7, A9, A17;
thus h2 . x,
y =
(ConsecutiveDelta q,o2) . x,
y
by A8, A9, A17
.=
(ConsecutiveDelta q,o2) . y,
x
by A27, Def6
.=
h2 . y,
x
by A8, A9, A17
;
verum
end;
A28:
dom h2 = [:(ConsecutiveSet A,o2),(ConsecutiveSet A,o2):]
by FUNCT_2:def 1;
per cases
( o1 c= o2 or o2 c= o1 )
;
suppose
o1 c= o2
;
f . x,y = f . y,xthen A29:
ConsecutiveSet A,
o1 c= ConsecutiveSet A,
o2
by Th32;
then A30:
[y,x] in dom h2
by A19, A24, A28, ZFMISC_1:106;
ConsecutiveDelta q,
o2 in rng Ls
by A8, A9, A17, A25, FUNCT_1:def 5;
then A31:
h2 c= f
by A25, ZFMISC_1:92;
reconsider x9 =
x,
y9 =
y as
Element of
ConsecutiveSet A,
o2 by A9, A15, A17, A18, A19, A29;
[x,y] in dom h2
by A19, A24, A28, A29, ZFMISC_1:106;
hence f . x,
y =
h2 . x9,
y9
by A31, GRFUNC_1:8
.=
h2 . y9,
x9
by A26, Def6
.=
f . y,
x
by A31, A30, GRFUNC_1:8
;
verum end; suppose
o2 c= o1
;
f . x,y = f . y,xthen A32:
ConsecutiveSet A,
o2 c= ConsecutiveSet A,
o1
by Th32;
then A33:
[y,x] in dom h1
by A19, A24, A23, ZFMISC_1:106;
ConsecutiveDelta q,
o1 in rng Ls
by A8, A9, A13, A20, FUNCT_1:def 5;
then A34:
h1 c= f
by A20, ZFMISC_1:92;
reconsider x9 =
x,
y9 =
y as
Element of
ConsecutiveSet A,
o1 by A9, A11, A13, A14, A24, A32;
[x,y] in dom h1
by A19, A24, A23, A32, ZFMISC_1:106;
hence f . x,
y =
h1 . x9,
y9
by A34, GRFUNC_1:8
.=
h1 . y9,
x9
by A21, Def6
.=
f . y,
x
by A34, A33, GRFUNC_1:8
;
verum end;
end;
hence
S1[
O2]
by A6, A8, Th31;
verum
end;
A35:
S1[ {} ]
proof
let x,
y be
Element of
ConsecutiveSet A,
{} ;
LATTICE5:def 6 (ConsecutiveDelta q,{} ) . x,y = (ConsecutiveDelta q,{} ) . y,x
reconsider x9 =
x,
y9 =
y as
Element of
A by Th24;
thus (ConsecutiveDelta q,{} ) . x,
y =
d . x9,
y9
by Th29
.=
d . y9,
x9
by A1, Def6
.=
(ConsecutiveDelta q,{} ) . y,
x
by Th29
;
verum
end;
for O being Ordinal holds S1[O]
from ORDINAL2:sch 1(A35, A2, A5);
hence
ConsecutiveDelta q,O is symmetric
; verum