let ap, bm, cp, dm be non pair set ; :: thesis: for cin being set st cin <> [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2] & not cin in InnerVertices (BitGFA1Str (ap,bm,cp)) holds
for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 )
let cin be set ; :: thesis: ( cin <> [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2] & not cin in InnerVertices (BitGFA1Str (ap,bm,cp)) implies for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 ) )
assume A1: ( cin <> [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2] & not cin in InnerVertices (BitGFA1Str (ap,bm,cp)) ) ; :: thesis: for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 )
set S = BitFTA1Str (ap,bm,cp,dm,cin);
A2: ( ap in InputVertices (BitFTA1Str (ap,bm,cp,dm,cin)) & bm in InputVertices (BitFTA1Str (ap,bm,cp,dm,cin)) ) by A1, Th16;
A3: ( cp in InputVertices (BitFTA1Str (ap,bm,cp,dm,cin)) & dm in InputVertices (BitFTA1Str (ap,bm,cp,dm,cin)) ) by A1, Th16;
set A1 = GFA1AdderOutput (ap,bm,cp);
let s be State of (BitFTA1Circ (ap,bm,cp,dm,cin)); :: thesis: for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin holds
( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 )
set dmA1 = [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2];
let a1, a2, a3, a4, a5 be Element of BOOLEAN ; :: thesis: ( a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin implies ( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 ) )
assume A4: ( a1 = s . ap & a2 = s . bm & a3 = s . cp & a4 = s . dm & a5 = s . cin ) ; :: thesis: ( (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) & (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 )
A5: (Following (s,3)) . [<*dm,(GFA1AdderOutput (ap,bm,cp))*>,nor2] = ('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3) by A1, A4, Lm15;
set cindm = [<*cin,dm*>,and2c];
set A1cin = [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,and2a];
set A2 = GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm);
A6: Following (s,(3 + 1)) = Following (Following (s,3)) by FACIRC_1:12;
( (Following (s,3)) . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,and2a] = ((a1 'xor' ('not' a2)) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dm*>,and2c] = a5 '&' ('not' a4) ) by A1, A4, Lm15;
hence (Following (s,4)) . (GFA2CarryOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = 'not' (((((a1 'xor' ('not' a2)) 'xor' a3) '&' a5) 'or' (a5 '&' ('not' a4))) 'or' (('not' a4) '&' ((a1 'xor' ('not' a2)) 'xor' a3))) by A6, A5, Lm17; :: thesis: ( (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 )
A7: ( (Following (s,3)) . cp = a3 & (Following (s,3)) . dm = a4 ) by A1, A4, Lm15;
A8: (Following (s,3)) . cin = a5 by A1, A4, Lm15;
A9: cin in InputVertices (BitFTA1Str (ap,bm,cp,dm,cin)) by A1, Th16;
( (Following (s,3)) . ap = a1 & (Following (s,3)) . bm = a2 ) by A1, A4, Lm15;
hence ( (Following (s,4)) . ap = a1 & (Following (s,4)) . bm = a2 & (Following (s,4)) . cp = a3 & (Following (s,4)) . dm = a4 & (Following (s,4)) . cin = a5 ) by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; :: thesis: verum