let am, bp, cm be non pair set ; :: thesis: for dp, cin being set
for s being State of (BitFTA2Circ (am,bp,cm,dp,cin))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
let dp, cin be set ; :: thesis: for s being State of (BitFTA2Circ (am,bp,cm,dp,cin))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
let s be State of (BitFTA2Circ (am,bp,cm,dp,cin)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . am & a2 = s . bp & a3 = s . cm holds
( (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
set S1 = BitGFA2Str (am,bp,cm);
set C1 = BitGFA2Circ (am,bp,cm);
set A1 = GFA2AdderOutput (am,bp,cm);
set A2 = GFA2CarryOutput (am,bp,cm);
set S2 = BitGFA1Str ((GFA2AdderOutput (am,bp,cm)),cin,dp);
set C2 = BitGFA1Circ ((GFA2AdderOutput (am,bp,cm)),cin,dp);
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . am & a2 = s . bp & a3 = s . cm implies ( (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ) )
assume that
A1: a1 = s . am and
A2: a2 = s . bp and
A3: a3 = s . cm ; :: thesis: ( (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) )
reconsider s1 = s | the carrier of (BitGFA2Str (am,bp,cm)) as State of (BitGFA2Circ (am,bp,cm)) by FACIRC_1:26;
A4: dom s1 = the carrier of (BitGFA2Str (am,bp,cm)) by CIRCUIT1:3;
am in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then A5: a1 = s1 . am by A1, A4, FUNCT_1:47;
reconsider t = s as State of ((BitGFA2Circ (am,bp,cm)) +* (BitGFA1Circ ((GFA2AdderOutput (am,bp,cm)),cin,dp))) ;
A6: InputVertices (BitGFA2Str (am,bp,cm)) misses InnerVertices (BitGFA1Str ((GFA2AdderOutput (am,bp,cm)),cin,dp)) by Lm22;
cm in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then A7: a3 = s1 . cm by A3, A4, FUNCT_1:47;
bp in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then A8: a2 = s1 . bp by A2, A4, FUNCT_1:47;
GFA2CarryOutput (am,bp,cm) in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then (Following (t,2)) . (GFA2CarryOutput (am,bp,cm)) = (Following (s1,2)) . (GFA2CarryOutput (am,bp,cm)) by A6, FACIRC_1:32;
hence (Following (s,2)) . (BitFTA2CarryOutput (am,bp,cm,dp,cin)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) by A5, A8, A7, GFACIRC1:103; :: thesis: (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3)
GFA2AdderOutput (am,bp,cm) in the carrier of (BitGFA2Str (am,bp,cm)) by GFACIRC1:100;
then (Following (t,2)) . (GFA2AdderOutput (am,bp,cm)) = (Following (s1,2)) . (GFA2AdderOutput (am,bp,cm)) by A6, FACIRC_1:32;
hence (Following (s,2)) . (BitFTA2AdderOutputI (am,bp,cm,dp,cin)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) by A5, A8, A7, GFACIRC1:103; :: thesis: verum