let am, bp, cm, dp be non pair set ; :: thesis: for cin being set st cin <> [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] & not cin in InnerVertices (BitGFA2Str am,bp,cm) holds
for s being State of (BitFTA2Circ am,bp,cm,dp,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin holds
( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 )
let cin be set ; :: thesis: ( cin <> [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] & not cin in InnerVertices (BitGFA2Str am,bp,cm) implies for s being State of (BitFTA2Circ am,bp,cm,dp,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin holds
( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 ) )
assume A1: ( cin <> [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] & not cin in InnerVertices (BitGFA2Str am,bp,cm) ) ; :: thesis: for s being State of (BitFTA2Circ am,bp,cm,dp,cin)
for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin holds
( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 )
set S = BitFTA2Str am,bp,cm,dp,cin;
A2: ( am in InputVertices (BitFTA2Str am,bp,cm,dp,cin) & bp in InputVertices (BitFTA2Str am,bp,cm,dp,cin) ) by A1, Th26;
A3: ( cm in InputVertices (BitFTA2Str am,bp,cm,dp,cin) & dp in InputVertices (BitFTA2Str am,bp,cm,dp,cin) ) by A1, Th26;
let s be State of (BitFTA2Circ am,bp,cm,dp,cin); :: thesis: for a1, a2, a3, a4, a5 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin holds
( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 )
let a1, a2, a3, a4, a5 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin implies ( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 ) )
assume A4: ( a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin ) ; :: thesis: ( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) & (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 )
A5: (Following s,2) . cin = a5 by A1, A4, Th28;
set cindp = [<*cin,dp*>,and2a ];
set A1 = GFA2AdderOutput am,bp,cm;
set A1cin = [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ];
set dpA1 = [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ];
A6: Following s,(2 + 1) = Following (Following s,2) by FACIRC_1:12;
( (Following s,2) . (GFA2AdderOutput am,bp,cm) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following s,2) . dp = a4 ) by A1, A4, Th28;
hence ( (Following s,3) . [<*(GFA2AdderOutput am,bp,cm),cin*>,and2c ] = ((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5) & (Following s,3) . [<*cin,dp*>,and2a ] = ('not' a5) '&' a4 & (Following s,3) . [<*dp,(GFA2AdderOutput am,bp,cm)*>,and2 ] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) ) by A6, A5, Lm23; :: thesis: ( (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 )
A7: ( (Following s,2) . cm = a3 & (Following s,2) . dp = a4 ) by A1, A4, Th28;
A8: (Following s,2) . cin = a5 by A1, A4, Th28;
A9: cin in InputVertices (BitFTA2Str am,bp,cm,dp,cin) by A1, Th26;
( (Following s,2) . am = a1 & (Following s,2) . bp = a2 ) by A1, A4, Th28;
hence ( (Following s,3) . am = a1 & (Following s,3) . bp = a2 & (Following s,3) . cm = a3 & (Following s,3) . dp = a4 & (Following s,3) . cin = a5 ) by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; :: thesis: verum
set C2 = BitGFA1Circ (GFA2AdderOutput am,bp,cm),cin,dp;
set S2 = BitGFA1Str (GFA2AdderOutput am,bp,cm),cin,dp;
set C1 = BitGFA2Circ am,bp,cm;
set S1 = BitGFA2Str am,bp,cm;
set C = BitFTA2Circ am,bp,cm,dp,cin;