set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
let x, y, z be set ; :: thesis:
assume A1: ( x <> [<*y,z*>,and2] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] ) ; :: thesis:
set S = GFA0CarryStr (x,y,z);
reconsider x9 = x, y9 = y, z9 = z as Vertex of (GFA0CarryStr (x,y,z)) by Th16;
let s be State of (GFA0CarryCirc (x,y,z)); :: thesis:
y in InputVertices (GFA0CarryStr (x,y,z)) by A1, Th18;
then A2: (Following s) . y9 = s . y by CIRCUIT2:def 5;
z in InputVertices (GFA0CarryStr (x,y,z)) by A1, Th18;
then A3: (Following s) . z9 = s . z by CIRCUIT2:def 5;
set xy = [<*x,y*>,and2];
set yz = [<*y,z*>,and2];
set zx = [<*z,x*>,and2];
A4: Following (s,2) = Following (Following s) by FACIRC_1:15;
let a1, a2, a3 be Element of BOOLEAN ; :: thesis:
assume A5: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis:
A6: (Following s) . [<*z,x*>,and2] = a3 '&' a1 by A5, Th20;
( (Following s) . [<*x,y*>,and2] = a1 '&' a2 & (Following s) . [<*y,z*>,and2] = a2 '&' a3 ) by A5, Th20;
hence (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A4, A6, Th21; :: thesis:
x in InputVertices (GFA0CarryStr (x,y,z)) by A1, Th18;
then (Following s) . x9 = s . x by CIRCUIT2:def 5;
hence ( (Following (s,2)) . [<*x,y*>,and2] = a1 '&' a2 & (Following (s,2)) . [<*y,z*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*z,x*>,and2] = a3 '&' a1 ) by A5, A2, A3, A4, Th20; :: thesis: