set f1 = nor2 ;
set f2 = nor2 ;
set f3 = nor2 ;
let x, y, z be set ; ( x <> [<*y,z*>,nor2] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,nor2] implies for s being State of (GFA3CarryCirc (x,y,z)) holds Following (s,2) is stable )
assume A1:
( x <> [<*y,z*>,nor2] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,nor2] )
; for s being State of (GFA3CarryCirc (x,y,z)) holds Following (s,2) is stable
set S = GFA3CarryStr (x,y,z);
reconsider xx = x, yy = y, zz = z as Vertex of (GFA3CarryStr (x,y,z)) by Th111;
let s be State of (GFA3CarryCirc (x,y,z)); Following (s,2) is stable
set a1 = s . xx;
set a2 = s . yy;
set a3 = s . zz;
set ffs = Following (s,2);
set fffs = Following (Following (s,2));
set xy = [<*x,y*>,nor2];
set yz = [<*y,z*>,nor2];
set zx = [<*z,x*>,nor2];
A2:
Following (s,2) = Following (Following s)
by FACIRC_1:15;
A3:
z in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then
(Following s) . z = s . zz
by CIRCUIT2:def 5;
then A4:
(Following (s,2)) . z = s . zz
by A2, A3, CIRCUIT2:def 5;
A5:
y in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then
(Following s) . y = s . yy
by CIRCUIT2:def 5;
then A6:
(Following (s,2)) . y = s . yy
by A2, A5, CIRCUIT2:def 5;
A7:
x in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then
(Following s) . x = s . xx
by CIRCUIT2:def 5;
then A8:
(Following (s,2)) . x = s . xx
by A2, A7, CIRCUIT2:def 5;
s . zz = s . z
;
then A9:
(Following (s,2)) . [<*x,y*>,nor2] = ('not' (s . xx)) '&' ('not' (s . yy))
by A1, Th117;
s . yy = s . y
;
then A10:
(Following (s,2)) . [<*z,x*>,nor2] = ('not' (s . xx)) '&' ('not' (s . zz))
by A1, Th117;
s . xx = s . x
;
then A11:
(Following (s,2)) . [<*y,z*>,nor2] = ('not' (s . yy)) '&' ('not' (s . zz))
by A1, Th117;
A12:
(Following (s,2)) . (GFA3CarryOutput (x,y,z)) = 'not' (((('not' (s . xx)) '&' ('not' (s . yy))) 'or' (('not' (s . yy)) '&' ('not' (s . zz)))) 'or' (('not' (s . zz)) '&' ('not' (s . xx))))
by A1, Th117;
A13:
now for a being object st a in the carrier of (GFA3CarryStr (x,y,z)) holds
(Following (s,2)) . a = (Following (Following (s,2))) . alet a be
object ;
( a in the carrier of (GFA3CarryStr (x,y,z)) implies (Following (s,2)) . a = (Following (Following (s,2))) . a )assume A14:
a in the
carrier of
(GFA3CarryStr (x,y,z))
;
(Following (s,2)) . a = (Following (Following (s,2))) . athen reconsider v =
a as
Vertex of
(GFA3CarryStr (x,y,z)) ;
A15:
v in (InputVertices (GFA3CarryStr (x,y,z))) \/ (InnerVertices (GFA3CarryStr (x,y,z)))
by A14, XBOOLE_1:45;
thus
(Following (s,2)) . a = (Following (Following (s,2))) . a
verumproof
per cases
( v in InputVertices (GFA3CarryStr (x,y,z)) or v in InnerVertices (GFA3CarryStr (x,y,z)) )
by A15, XBOOLE_0:def 3;
suppose
v in InnerVertices (GFA3CarryStr (x,y,z))
;
(Following (s,2)) . a = (Following (Following (s,2))) . athen
v in {[<*x,y*>,nor2],[<*y,z*>,nor2],[<*z,x*>,nor2]} \/ {(GFA3CarryOutput (x,y,z))}
by Th106;
then
(
v in {[<*x,y*>,nor2],[<*y,z*>,nor2],[<*z,x*>,nor2]} or
v in {(GFA3CarryOutput (x,y,z))} )
by XBOOLE_0:def 3;
then
(
v = [<*x,y*>,nor2] or
v = [<*y,z*>,nor2] or
v = [<*z,x*>,nor2] or
v = GFA3CarryOutput (
x,
y,
z) )
by ENUMSET1:def 1, TARSKI:def 1;
hence
(Following (s,2)) . a = (Following (Following (s,2))) . a
by A12, A9, A11, A10, A8, A6, A4, Th115, Th116;
verum end;
end; end;
( dom (Following (Following (s,2))) = the carrier of (GFA3CarryStr (x,y,z)) & dom (Following (s,2)) = the carrier of (GFA3CarryStr (x,y,z)) )
by CIRCUIT1:3;
hence
Following (s,2) = Following (Following (s,2))
by A13, FUNCT_1:2; CIRCUIT2:def 6 verum