let ap, bp, cp, dp be non pair set ; :: thesis: for cin being set st cin <> [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] & not cin in InnerVertices (BitGFA0Str (ap,bp,cp)) holds
for s being State of (BitFTA0Circ (ap,bp,cp,dp,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin holds
( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 )
let cin be set ; :: thesis: ( cin <> [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] & not cin in InnerVertices (BitGFA0Str (ap,bp,cp)) implies for s being State of (BitFTA0Circ (ap,bp,cp,dp,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin holds
( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 ) )
assume A1: ( cin <> [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] & not cin in InnerVertices (BitGFA0Str (ap,bp,cp)) ) ; :: thesis: for s being State of (BitFTA0Circ (ap,bp,cp,dp,cin))
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin holds
( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 )
set S = BitFTA0Str (ap,bp,cp,dp,cin);
A2: ( ap in InputVertices (BitFTA0Str (ap,bp,cp,dp,cin)) & bp in InputVertices (BitFTA0Str (ap,bp,cp,dp,cin)) ) by A1, Th6;
A3: ( cp in InputVertices (BitFTA0Str (ap,bp,cp,dp,cin)) & dp in InputVertices (BitFTA0Str (ap,bp,cp,dp,cin)) ) by A1, Th6;
let s be State of (BitFTA0Circ (ap,bp,cp,dp,cin)); :: thesis: for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin holds
( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 )
let a1, a2, a3, a4, a5 be Element of BOOLEAN ; :: thesis: ( a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin implies ( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 ) )
assume A4: ( a1 = s . ap & a2 = s . bp & a3 = s . cp & a4 = s . dp & a5 = s . cin ) ; :: thesis: ( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) & (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 )
A5: (Following (s,2)) . cin = a5 by A1, A4, Th8;
set cindp = [<*cin,dp*>,and2];
set A1 = GFA0AdderOutput (ap,bp,cp);
set A1cin = [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2];
set dpA1 = [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2];
A6: Following (s,(2 + 1)) = Following (Following (s,2)) by FACIRC_1:12;
( (Following (s,2)) . (GFA0AdderOutput (ap,bp,cp)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . dp = a4 ) by A1, A4, Th8;
hence ( (Following (s,3)) . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,and2] = ((a1 'xor' a2) 'xor' a3) '&' a5 & (Following (s,3)) . [<*cin,dp*>,and2] = a5 '&' a4 & (Following (s,3)) . [<*dp,(GFA0AdderOutput (ap,bp,cp))*>,and2] = a4 '&' ((a1 'xor' a2) 'xor' a3) ) by A6, A5, Lm3; :: thesis: ( (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 )
A7: ( (Following (s,2)) . cp = a3 & (Following (s,2)) . dp = a4 ) by A1, A4, Th8;
A8: (Following (s,2)) . cin = a5 by A1, A4, Th8;
A9: cin in InputVertices (BitFTA0Str (ap,bp,cp,dp,cin)) by A1, Th6;
( (Following (s,2)) . ap = a1 & (Following (s,2)) . bp = a2 ) by A1, A4, Th8;
hence ( (Following (s,3)) . ap = a1 & (Following (s,3)) . bp = a2 & (Following (s,3)) . cp = a3 & (Following (s,3)) . dp = a4 & (Following (s,3)) . cin = a5 ) by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; :: thesis: verum