set f1 = and2a ;
set f2 = and2c ;
set f3 = nor2 ;
set f0 = xor2c ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2c] & x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] implies for s being State of (BitGFA2Circ (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)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ) )
assume that
A1: z <> [<*x,y*>,xor2c] and
A2: ( x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] ) ; :: thesis: for s being State of (BitGFA2Circ (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)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) )
set S2 = GFA2CarryStr (x,y,z);
set S1 = GFA2AdderStr (x,y,z);
InputVertices (GFA2AdderStr (x,y,z)) = {x,y,z} by A1, FACIRC_1:57;
then A3: InputVertices (GFA2AdderStr (x,y,z)) = InputVertices (GFA2CarryStr (x,y,z)) by A2, Th77;
set A2 = GFA2CarryCirc (x,y,z);
set A1 = GFA2AdderCirc (x,y,z);
set A = BitGFA2Circ (x,y,z);
let s be State of (BitGFA2Circ (x,y,z)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following (s,2)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) )
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following (s,2)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ) )
assume that
A4: a1 = s . x and
A5: a2 = s . y and
A6: a3 = s . z ; :: thesis: ( (Following (s,2)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) )
reconsider s1 = s | the carrier of (GFA2AdderStr (x,y,z)) as State of (GFA2AdderCirc (x,y,z)) by FACIRC_1:26;
A7: dom s1 = the carrier of (GFA2AdderStr (x,y,z)) by CIRCUIT1:3;
z in the carrier of (GFA2AdderStr (x,y,z)) by FACIRC_1:60;
then A8: a3 = s1 . z by A6, A7, FUNCT_1:47;
y in the carrier of (GFA2AdderStr (x,y,z)) by FACIRC_1:60;
then A9: a2 = s1 . y by A5, A7, FUNCT_1:47;
reconsider t = s as State of ((GFA2AdderCirc (x,y,z)) +* (GFA2CarryCirc (x,y,z))) ;
InnerVertices (GFA2CarryStr (x,y,z)) misses InputVertices (GFA2CarryStr (x,y,z)) by XBOOLE_1:79;
then A10: (Following (t,2)) . (GFA2AdderOutput (x,y,z)) = (Following (s1,2)) . (GFA2AdderOutput (x,y,z)) by A3, FACIRC_1:32;
reconsider s2 = s | the carrier of (GFA2CarryStr (x,y,z)) as State of (GFA2CarryCirc (x,y,z)) by FACIRC_1:26;
A11: dom s2 = the carrier of (GFA2CarryStr (x,y,z)) by CIRCUIT1:3;
x in the carrier of (GFA2AdderStr (x,y,z)) by FACIRC_1:60;
then a1 = s1 . x by A4, A7, FUNCT_1:47;
hence (Following (s,2)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) by A1, A9, A8, A10, Th94; :: thesis: (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1)))
InnerVertices (GFA2AdderStr (x,y,z)) misses InputVertices (GFA2AdderStr (x,y,z)) by XBOOLE_1:79;
then A12: (Following (t,2)) . (GFA2CarryOutput (x,y,z)) = (Following (s2,2)) . (GFA2CarryOutput (x,y,z)) by A3, FACIRC_1:33;
z in the carrier of (GFA2CarryStr (x,y,z)) by Th79;
then A13: a3 = s2 . z by A6, A11, FUNCT_1:47;
y in the carrier of (GFA2CarryStr (x,y,z)) by Th79;
then A14: a2 = s2 . y by A5, A11, FUNCT_1:47;
x in the carrier of (GFA2CarryStr (x,y,z)) by Th79;
then a1 = s2 . x by A4, A11, FUNCT_1:47;
hence (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) by A2, A14, A13, A12, Th85; :: thesis: verum