set f0 = xor2c ;
set f1 = and2a ;
set f2 = and2c ;
set f3 = and2b ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2c ] & x <> [<*y,z*>,and2c ] & y <> [<*z,x*>,and2b ] & 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 A1: ( z <> [<*x,y*>,xor2c ] & x <> [<*y,z*>,and2c ] & y <> [<*z,x*>,and2b ] & 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 S1 = GFA2AdderStr x,y,z;
set S2 = GFA2CarryStr x,y,z;
set A = BitGFA2Circ x,y,z;
set A1 = GFA2AdderCirc x,y,z;
set A2 = GFA2CarryCirc 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 A2: ( a1 = s . x & a2 = s . y & 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))) )
A3: ( x in the carrier of (GFA2AdderStr x,y,z) & y in the carrier of (GFA2AdderStr x,y,z) & z in the carrier of (GFA2AdderStr x,y,z) ) by FACIRC_1:60;
A4: ( x in the carrier of (GFA2CarryStr x,y,z) & y in the carrier of (GFA2CarryStr x,y,z) & z in the carrier of (GFA2CarryStr x,y,z) ) by Th92;
reconsider s1 = s | the carrier of (GFA2AdderStr x,y,z) as State of (GFA2AdderCirc x,y,z) by FACIRC_1:26;
reconsider s2 = s | the carrier of (GFA2CarryStr x,y,z) as State of (GFA2CarryCirc x,y,z) by FACIRC_1:26;
reconsider t = s as State of ((GFA2AdderCirc x,y,z) +* (GFA2CarryCirc x,y,z)) ;
InputVertices (GFA2AdderStr x,y,z) = {x,y,z} by A1, FACIRC_1:57;
then A5: InputVertices (GFA2AdderStr x,y,z) = InputVertices (GFA2CarryStr x,y,z) by A1, Th90;
A6: ( InnerVertices (GFA2AdderStr x,y,z) misses InputVertices (GFA2AdderStr x,y,z) & InnerVertices (GFA2CarryStr x,y,z) misses InputVertices (GFA2CarryStr x,y,z) ) by XBOOLE_1:79;
dom s1 = the carrier of (GFA2AdderStr x,y,z) by CIRCUIT1:4;
then ( a1 = s1 . x & a2 = s1 . y & a3 = s1 . z ) by A2, A3, FUNCT_1:70;
then ( (Following t,2) . (GFA2AdderOutput x,y,z) = (Following s1,2) . (GFA2AdderOutput x,y,z) & (Following s1,2) . (GFA2AdderOutput x,y,z) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ) by A1, A5, A6, Th111, FACIRC_1:32;
hence (Following s,2) . (GFA2AdderOutput x,y,z) = (('not' a1) 'xor' a2) 'xor' ('not' a3) ; :: thesis: (Following s,2) . (GFA2CarryOutput x,y,z) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1)))
dom s2 = the carrier of (GFA2CarryStr x,y,z) by CIRCUIT1:4;
then ( a1 = s2 . x & a2 = s2 . y & a3 = s2 . z ) by A2, A4, FUNCT_1:70;
then ( (Following t,2) . (GFA2CarryOutput x,y,z) = (Following s2,2) . (GFA2CarryOutput x,y,z) & (Following s2,2) . (GFA2CarryOutput x,y,z) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ) by A1, A5, A6, Th98, FACIRC_1:33;
hence (Following s,2) . (GFA2CarryOutput x,y,z) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) ; :: thesis: verum