let am, bp, cm, dp be non pair set ; 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 ; ( 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)) )
; 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)); 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 ; ( 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 )
; ( (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; ( (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; verum