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,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . 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,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . 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,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . 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;
set A1 = GFA3AdderOutput am,bm,cm;
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,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . cin = a5 )
set dmA1 = [<*dm,(GFA3AdderOutput am,bm,cm)*>,and2b ];
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,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . cin = a5 ) )
assume A4: ( a1 = s . am & a2 = s . bm & a3 = s . cm & a4 = s . dm & a5 = s . cin ) ; :: thesis: ( (Following s,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) & (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . cin = a5 )
A5: (Following s,3) . [<*dm,(GFA3AdderOutput am,bm,cm)*>,and2b ] = ('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) by A1, A4, Lm35;
set cindm = [<*cin,dm*>,and2b ];
set A1cin = [<*(GFA3AdderOutput am,bm,cm),cin*>,and2b ];
set A2 = GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm;
A6: Following s,(3 + 1) = Following (Following s,3) by FACIRC_1:12;
( (Following s,3) . [<*(GFA3AdderOutput am,bm,cm),cin*>,and2b ] = ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dm*>,and2b ] = ('not' a5) '&' ('not' a4) ) by A1, A4, Lm35;
hence (Following s,4) . (GFA3CarryOutput (GFA3AdderOutput am,bm,cm),cin,dm) = 'not' (((((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' ('not' a4))) 'or' (('not' a4) '&' ((('not' a1) 'xor' ('not' a2)) 'xor' ('not' a3)))) by A6, A5, Lm37; :: thesis: ( (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . cin = a5 )
A7: ( (Following s,3) . cm = a3 & (Following s,3) . dm = a4 ) by A1, A4, Lm35;
A8: (Following s,3) . cin = a5 by A1, A4, Lm35;
A9: cin in InputVertices (BitFTA3Str am,bm,cm,dm,cin) by A1, Th36;
( (Following s,3) . am = a1 & (Following s,3) . bm = a2 ) by A1, A4, Lm35;
hence ( (Following s,4) . am = a1 & (Following s,4) . bm = a2 & (Following s,4) . cm = a3 & (Following s,4) . dm = a4 & (Following s,4) . cin = a5 ) by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; :: thesis: verum
set C2 = BitGFA3Circ (GFA3AdderOutput am,bm,cm),cin,dm;
set S2 = BitGFA3Str (GFA3AdderOutput am,bm,cm),cin,dm;
set C1 = BitGFA3Circ am,bm,cm;
set S1 = BitGFA3Str am,bm,cm;
set C = BitFTA3Circ am,bm,cm,dm,cin;