set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
set f0 = xor2c ;
let x, y, z be set ; ( z <> [<*x,y*>,xor2c] & x <> [<*y,z*>,and2a] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2c] implies for s being State of (BitGFA1Circ (x,y,z))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) ) )
assume that
A1:
z <> [<*x,y*>,xor2c]
and
A2:
( x <> [<*y,z*>,and2a] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2c] )
; for s being State of (BitGFA1Circ (x,y,z))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) )
set S2 = GFA1CarryStr (x,y,z);
set S1 = GFA1AdderStr (x,y,z);
InputVertices (GFA1AdderStr (x,y,z)) = {x,y,z}
by A1, FACIRC_1:57;
then A3:
InputVertices (GFA1AdderStr (x,y,z)) = InputVertices (GFA1CarryStr (x,y,z))
by A2, Th45;
set A2 = GFA1CarryCirc (x,y,z);
set A1 = GFA1AdderCirc (x,y,z);
set A = BitGFA1Circ (x,y,z);
let s be State of (BitGFA1Circ (x,y,z)); for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) )
let a1, a2, a3 be Element of BOOLEAN ; ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) ) )
assume that
A4:
a1 = s . x
and
A5:
a2 = s . y
and
A6:
a3 = s . z
; ( (Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) )
reconsider s1 = s | the carrier of (GFA1AdderStr (x,y,z)) as State of (GFA1AdderCirc (x,y,z)) by FACIRC_1:26;
A7:
dom s1 = the carrier of (GFA1AdderStr (x,y,z))
by CIRCUIT1:3;
z in the carrier of (GFA1AdderStr (x,y,z))
by FACIRC_1:60;
then A8:
a3 = s1 . z
by A6, A7, FUNCT_1:47;
y in the carrier of (GFA1AdderStr (x,y,z))
by FACIRC_1:60;
then A9:
a2 = s1 . y
by A5, A7, FUNCT_1:47;
reconsider t = s as State of ((GFA1AdderCirc (x,y,z)) +* (GFA1CarryCirc (x,y,z))) ;
InnerVertices (GFA1CarryStr (x,y,z)) misses InputVertices (GFA1CarryStr (x,y,z))
by XBOOLE_1:79;
then A10:
(Following (t,2)) . (GFA1AdderOutput (x,y,z)) = (Following (s1,2)) . (GFA1AdderOutput (x,y,z))
by A3, FACIRC_1:32;
reconsider s2 = s | the carrier of (GFA1CarryStr (x,y,z)) as State of (GFA1CarryCirc (x,y,z)) by FACIRC_1:26;
A11:
dom s2 = the carrier of (GFA1CarryStr (x,y,z))
by CIRCUIT1:3;
x in the carrier of (GFA1AdderStr (x,y,z))
by FACIRC_1:60;
then
a1 = s1 . x
by A4, A7, FUNCT_1:47;
hence
(Following (s,2)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3)
by A1, A9, A8, A10, Th62; (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1)
InnerVertices (GFA1AdderStr (x,y,z)) misses InputVertices (GFA1AdderStr (x,y,z))
by XBOOLE_1:79;
then A12:
(Following (t,2)) . (GFA1CarryOutput (x,y,z)) = (Following (s2,2)) . (GFA1CarryOutput (x,y,z))
by A3, FACIRC_1:33;
z in the carrier of (GFA1CarryStr (x,y,z))
by Th47;
then A13:
a3 = s2 . z
by A6, A11, FUNCT_1:47;
y in the carrier of (GFA1CarryStr (x,y,z))
by Th47;
then A14:
a2 = s2 . y
by A5, A11, FUNCT_1:47;
x in the carrier of (GFA1CarryStr (x,y,z))
by Th47;
then
a1 = s2 . x
by A4, A11, FUNCT_1:47;
hence
(Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1)
by A2, A14, A13, A12, Th53; verum