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,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . 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,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . 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,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . cin = a5 )
set A1 = GFA2AdderOutput (am,bp,cm);
set C1 = BitGFA2Circ (am,bp,cm);
set S1 = BitGFA2Str (am,bp,cm);
set S2 = BitGFA1Str ((GFA2AdderOutput (am,bp,cm)),cin,dp);
set C2 = BitGFA1Circ ((GFA2AdderOutput (am,bp,cm)),cin,dp);
set S = BitFTA2Str (am,bp,cm,dp,cin);
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,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . 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,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . cin = a5 ) )
assume that
A2: a1 = s . am and
A3: a2 = s . bp and
A4: a3 = s . cm and
A5: a4 = s . dp and
A6: a5 = s . cin ; :: thesis: ( (Following (s,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . cin = a5 )
reconsider s1 = s | the carrier of (BitGFA2Str (am,bp,cm)) as State of (BitGFA2Circ (am,bp,cm)) by FACIRC_1:26;
A7: dom s1 = the carrier of (BitGFA2Str (am,bp,cm)) by CIRCUIT1:3;
A8: dp in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
then A9: (Following s) . dp = a4 by A5, CIRCUIT2:def 5;
A10: cm in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
then A11: (Following s) . cm = a3 by A4, CIRCUIT2:def 5;
bp in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then A12: a2 = s1 . bp by A3, A7, FUNCT_1:47;
reconsider t = s as State of ((BitGFA2Circ (am,bp,cm)) +* (BitGFA1Circ ((GFA2AdderOutput (am,bp,cm)),cin,dp))) ;
( GFA2AdderOutput (am,bp,cm) in the carrier of (BitGFA2Str (am,bp,cm)) & InputVertices (BitGFA2Str (am,bp,cm)) misses InnerVertices (BitGFA1Str ((GFA2AdderOutput (am,bp,cm)),cin,dp)) ) by Lm22, GFACIRC1:100;
then A13: (Following (t,2)) . (GFA2AdderOutput (am,bp,cm)) = (Following (s1,2)) . (GFA2AdderOutput (am,bp,cm)) by FACIRC_1:32;
cm in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then A14: a3 = s1 . cm by A4, A7, FUNCT_1:47;
am in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then a1 = s1 . am by A2, A7, FUNCT_1:47;
hence (Following (s,2)) . (GFA2AdderOutput (am,bp,cm)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) by A12, A14, A13, GFACIRC1:103; :: thesis: ( (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . cin = a5 )
A15: bp in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
then A16: (Following s) . bp = a2 by A3, CIRCUIT2:def 5;
A17: cin in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
then A18: (Following s) . cin = a5 by A6, CIRCUIT2:def 5;
A19: am in InputVertices (BitFTA2Str (am,bp,cm,dp,cin)) by A1, Th26;
then ( Following (s,2) = Following (Following s) & (Following s) . am = a1 ) by A2, CIRCUIT2:def 5, FACIRC_1:15;
hence ( (Following (s,2)) . am = a1 & (Following (s,2)) . bp = a2 & (Following (s,2)) . cm = a3 & (Following (s,2)) . dp = a4 & (Following (s,2)) . cin = a5 ) by A19, A15, A10, A8, A17, A16, A11, A9, A18, CIRCUIT2:def 5; :: thesis: verum