let am, bm, cm, dm be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = ('not' a123) '&' ('not' a5) & (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) )
let cin be set ; :: thesis: for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = ('not' a123) '&' ('not' a5) & (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) )
set S = BitFTA3Str (am,bm,cm,dm,cin);
set C = BitFTA3Circ (am,bm,cm,dm,cin);
set A1 = GFA3AdderOutput (am,bm,cm);
set A1cin = [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b];
set cindm = [<*cin,dm*>,and2b];
set dmA1 = [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b];
let s be State of (BitFTA3Circ (am,bm,cm,dm,cin)); :: thesis: for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = ('not' a123) '&' ('not' a5) & (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) )
let a123, a4, a5 be Element of BOOLEAN ; :: thesis: ( a123 = s . (GFA3AdderOutput (am,bm,cm)) & a4 = s . dm & a5 = s . cin implies ( (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = ('not' a123) '&' ('not' a5) & (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) ) )
assume that
A1: a123 = s . (GFA3AdderOutput (am,bm,cm)) and
A2: a4 = s . dm and
A3: a5 = s . cin ; :: thesis: ( (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = ('not' a123) '&' ('not' a5) & (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) )
A4: dom s = the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by CIRCUIT1:3;
A5: cin in the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by Th34;
A6: GFA3AdderOutput (am,bm,cm) in the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by Th34;
A7: InnerVertices (BitFTA3Str (am,bm,cm,dm,cin)) = the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by FACIRC_1:37;
then [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] in the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by Th35;
hence (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,and2b] = and2b . (s * <*(GFA3AdderOutput (am,bm,cm)),cin*>) by FACIRC_1:35
.= and2b . <*a123,a5*> by A1, A3, A6, A5, A4, FINSEQ_2:125
.= ('not' a123) '&' ('not' a5) by TWOSCOMP:def 3 ;
:: thesis: ( (Following s) . [<*cin,dm*>,and2b] = ('not' a5) '&' ('not' a4) & (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123) )
A8: dm in the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by Th34;
[<*cin,dm*>,and2b] in the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by A7, Th35;
hence (Following s) . [<*cin,dm*>,and2b] = and2b . (s * <*cin,dm*>) by FACIRC_1:35
.= and2b . <*a5,a4*> by A2, A3, A8, A5, A4, FINSEQ_2:125
.= ('not' a5) '&' ('not' a4) by TWOSCOMP:def 3 ;
:: thesis: (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = ('not' a4) '&' ('not' a123)
[<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] in the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by A7, Th35;
hence (Following s) . [<*dm,(GFA3AdderOutput (am,bm,cm))*>,and2b] = and2b . (s * <*dm,(GFA3AdderOutput (am,bm,cm))*>) by FACIRC_1:35
.= and2b . <*a4,a123*> by A1, A2, A6, A8, A4, FINSEQ_2:125
.= ('not' a4) '&' ('not' a123) by TWOSCOMP:def 3 ;
:: thesis: verum