let am, bm, cm, dm be non pair set ; for cin being set st cin <> [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] & 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,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
let cin be set ; ( cin <> [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] & 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,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 ) )
assume A1:
( cin <> [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] & not cin in InnerVertices (BitGFA3Str (am,bm,cm)) )
; 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,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
set S = BitFTA3Str (am,bm,cm,dm,cin);
A2:
( am in InputVertices (BitFTA3Str (am,bm,cm,dm,cin)) & bm in InputVertices (BitFTA3Str (am,bm,cm,dm,cin)) )
by A1, Th36;
A3:
( cm in InputVertices (BitFTA3Str (am,bm,cm,dm,cin)) & dm in InputVertices (BitFTA3Str (am,bm,cm,dm,cin)) )
by A1, Th36;
let s be 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,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
let a1, a2, a3, a4, a5 be Element of BOOLEAN ; ( a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin implies ( (Following (s,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 ) )
assume A4:
( a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin )
; ( (Following (s,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
A5:
(Following (s,2)) . cin = a5
by A1, A4, Th38;
set cindm = [<*cin,dm*>,nor2];
set A1 = GFA3AdderOutput (am,bm,cm);
set A1cin = [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2];
set dmA1 = [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2];
A6:
Following (s,(2 + 1)) = Following (Following (s,2))
by FACIRC_1:12;
( (Following (s,2)) . (GFA3AdderOutput (am,bm,cm)) = 'not' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) & (Following (s,2)) . dm = a4 )
by A1, A4, Th38;
hence
( (Following (s,3)) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,nor2] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following (s,3)) . [<*cin,dm*>,nor2] = ('not' a5) '&' ('not' a4) & (Following (s,3)) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,nor2] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) )
by A6, A5, Lm33; ( (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
A7:
( (Following (s,2)) . cm = a3 & (Following (s,2)) . dm = a4 )
by A1, A4, Th38;
A8:
(Following (s,2)) . cin = a5
by A1, A4, Th38;
A9:
cin in InputVertices (BitFTA3Str (am,bm,cm,dm,cin))
by A1, Th36;
( (Following (s,2)) . am = a1 & (Following (s,2)) . bm = a2 )
by A1, A4, Th38;
hence
( (Following (s,3)) . am = a1 & (Following (s,3)) . bm = a2 & (Following (s,3)) . cm = a3 & (Following (s,3)) . dm = a4 & (Following (s,3)) . cin = a5 )
by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; verum