let ap, bm, cp, dm be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA1Circ ap,bm,cp,dm,cin)
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = ('not' a123) '&' a5 & (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) )
let cin be set ; :: thesis: for s being State of (BitFTA1Circ ap,bm,cp,dm,cin)
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = ('not' a123) '&' a5 & (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) )
set S = BitFTA1Str ap,bm,cp,dm,cin;
set C = BitFTA1Circ ap,bm,cp,dm,cin;
set A1 = GFA1AdderOutput ap,bm,cp;
set S2 = BitGFA2Str (GFA1AdderOutput ap,bm,cp),cin,dm;
set C2 = BitGFA2Circ (GFA1AdderOutput ap,bm,cp),cin,dm;
set A1cin = [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ];
set cindm = [<*cin,dm*>,and2c ];
set dmA1 = [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ];
let s be State of (BitFTA1Circ ap,bm,cp,dm,cin); :: thesis: for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a4 = s . dm & a5 = s . cin holds
( (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = ('not' a123) '&' a5 & (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) )
let a123, a4, a5 be Element of BOOLEAN ; :: thesis: ( a123 = s . (GFA1AdderOutput ap,bm,cp) & a4 = s . dm & a5 = s . cin implies ( (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = ('not' a123) '&' a5 & (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) ) )
assume A1: ( a123 = s . (GFA1AdderOutput ap,bm,cp) & a4 = s . dm & a5 = s . cin ) ; :: thesis: ( (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = ('not' a123) '&' a5 & (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) )
A2: ( GFA1AdderOutput ap,bm,cp in the carrier of (BitFTA1Str ap,bm,cp,dm,cin) & dm in the carrier of (BitFTA1Str ap,bm,cp,dm,cin) & cin in the carrier of (BitFTA1Str ap,bm,cp,dm,cin) ) by ThFTA1S6;
A3: dom s = the carrier of (BitFTA1Str ap,bm,cp,dm,cin) by CIRCUIT1:4;
InnerVertices (BitFTA1Str ap,bm,cp,dm,cin) = the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) by FACIRC_1:37;
then A4: ( [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] in the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) & [<*cin,dm*>,and2c ] in the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) & [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] in the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) ) by ThFTA1S7;
hence (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,and2a ] = and2a . (s * <*(GFA1AdderOutput ap,bm,cp),cin*>) by FACIRC_1:35
.= and2a . <*a123,a5*> by A1, A2, A3, FINSEQ_2:145
.= ('not' a123) '&' a5 by TWOSCOMP:def 2 ;
:: thesis: ( (Following s) . [<*cin,dm*>,and2c ] = a5 '&' ('not' a4) & (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123) )
thus (Following s) . [<*cin,dm*>,and2c ] = and2c . (s * <*cin,dm*>) by A4, FACIRC_1:35
.= and2c . <*a5,a4*> by A1, A2, A3, FINSEQ_2:145
.= a5 '&' ('not' a4) by GFACIRC1:def 3 ; :: thesis: (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = ('not' a4) '&' ('not' a123)
thus (Following s) . [<*dm,(GFA1AdderOutput ap,bm,cp)*>,and2b ] = and2b . (s * <*dm,(GFA1AdderOutput ap,bm,cp)*>) by A4, FACIRC_1:35
.= and2b . <*a4,a123*> by A1, A2, A3, FINSEQ_2:145
.= ('not' a4) '&' ('not' a123) by TWOSCOMP:def 3 ; :: thesis: verum