let am, bm, cm, dm be non pair set ; :: thesis: for cin being set st cin <> [<*dm,(GFA3AdderOutput am,bm,cm)*>,and2b ] & not cin in InnerVertices (BitGFA3Str am,bm,cm) holds
for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin holds
( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 )
let cin be set ; :: thesis: ( cin <> [<*dm,(GFA3AdderOutput am,bm,cm)*>,and2b ] & not cin in InnerVertices (BitGFA3Str am,bm,cm) implies for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin holds
( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 ) )
assume A1: ( cin <> [<*dm,(GFA3AdderOutput am,bm,cm)*>,and2b ] & not cin in InnerVertices (BitGFA3Str am,bm,cm) ) ; :: thesis: for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin holds
( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 )
let s be State of (BitFTA3Circ am,bm,cm,dm,cin); :: thesis: for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin holds
( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 )
set S = BitFTA3Str am,bm,cm,dm,cin;
set S1 = BitGFA3Str am,bm,cm;
set C1 = BitGFA3Circ am,bm,cm;
set A1 = GFA3AdderOutput am,bm,cm;
set S2 = BitGFA3Str (GFA3AdderOutput am,bm,cm),cin,dm;
set C2 = BitGFA3Circ (GFA3AdderOutput am,bm,cm),cin,dm;
let a1, a2, a3, a4, a5 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin implies ( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 ) )
assume A2: ( a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin ) ; :: thesis: ( (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 )
A3: ( am in the carrier of (BitGFA3Str am,bm,cm) & bm in the carrier of (BitGFA3Str am,bm,cm) & cm in the carrier of (BitGFA3Str am,bm,cm) ) by GFACIRC1:155;
reconsider s1 = s | the carrier of (BitGFA3Str am,bm,cm) as State of (BitGFA3Circ am,bm,cm) by FACIRC_1:26;
reconsider t = s as State of ((BitGFA3Circ am,bm,cm) +* (BitGFA3Circ (GFA3AdderOutput am,bm,cm),cin,dm)) ;
A4: GFA3AdderOutput am,bm,cm in the carrier of (BitGFA3Str am,bm,cm) by GFACIRC1:155;
A5: InputVertices (BitGFA3Str am,bm,cm) misses InnerVertices (BitGFA3Str (GFA3AdderOutput am,bm,cm),cin,dm) by LemmaX42;
dom s1 = the carrier of (BitGFA3Str am,bm,cm) by CIRCUIT1:4;
then ( a1 = s1 . am & a2 = s1 . bm & a3 = s1 . cm ) by A2, A3, FUNCT_1:70;
then ( (Following t,2) . (GFA3AdderOutput am,bm,cm) = (Following s1,2) . (GFA3AdderOutput am,bm,cm) & (Following s1,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ) by A4, A5, FACIRC_1:32, GFACIRC1:158;
hence (Following s,2) . (GFA3AdderOutput am,bm,cm) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) ; :: thesis: ( (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 )
A6: Following s,2 = Following (Following s) by FACIRC_1:15;
A7: ( am in InputVertices (BitFTA3Str am,bm,cm,dm,cin) & bm in InputVertices (BitFTA3Str am,bm,cm,dm,cin) & cm in InputVertices (BitFTA3Str am,bm,cm,dm,cin) & dm in InputVertices (BitFTA3Str am,bm,cm,dm,cin) & cin in InputVertices (BitFTA3Str am,bm,cm,dm,cin) ) by A1, ThFTA3S8;
then ( (Following s) . am = a1 & (Following s) . bm = a2 & (Following s) . cm = a3 & (Following s) . dm = a4 & (Following s) . cin = a5 ) by A2, CIRCUIT2:def 5;
hence ( (Following s,2) . am = a1 & (Following s,2) . bm = a2 & (Following s,2) . cm = a3 & (Following s,2) . dm = a4 & (Following s,2) . cin = a5 ) by A6, A7, CIRCUIT2:def 5; :: thesis: verum