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,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . 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,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . 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,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . 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;
set A1 = GFA2AdderOutput (am,bp,cm);
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,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . cin = a5 )
set dpA1 = [<*dp,(GFA2AdderOutput (am,bp,cm))*>,and2];
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,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . cin = a5 ) )
assume A4: ( a1 = s . am & a2 = s . bp & a3 = s . cm & a4 = s . dp & a5 = s . cin ) ; :: thesis: ( (Following (s,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) & (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . cin = a5 )
A5: (Following (s,3)) . [<*dp,(GFA2AdderOutput (am,bp,cm))*>,and2] = a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3)) by A1, A4, Lm25;
set cindp = [<*cin,dp*>,and2a];
set A1cin = [<*(GFA2AdderOutput (am,bp,cm)),cin*>,and2c];
set A2 = GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp);
A6: Following (s,(3 + 1)) = Following (Following (s,3)) by FACIRC_1:12;
( (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 ) by A1, A4, Lm25;
hence (Following (s,4)) . (GFA1CarryOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = ((((('not' a1) 'xor' a2) 'xor' ('not' a3)) '&' ('not' a5)) 'or' (('not' a5) '&' a4)) 'or' (a4 '&' ((('not' a1) 'xor' a2) 'xor' ('not' a3))) by A6, A5, Lm27; :: thesis: ( (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . cin = a5 )
A7: ( (Following (s,3)) . cm = a3 & (Following (s,3)) . dp = a4 ) by A1, A4, Lm25;
A8: (Following (s,3)) . cin = a5 by A1, A4, Lm25;
A9: cin in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
( (Following (s,3)) . am = a1 & (Following (s,3)) . bp = a2 ) by A1, A4, Lm25;
hence ( (Following (s,4)) . am = a1 & (Following (s,4)) . bp = a2 & (Following (s,4)) . cm = a3 & (Following (s,4)) . dp = a4 & (Following (s,4)) . cin = a5 ) by A6, A2, A3, A9, A7, A8, CIRCUIT2:def 5; :: thesis: verum