set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
set f0 = xor2 ;
let x, y, z be set ; :: thesis: ( z <> [<*x,y*>,xor2] & x <> [<*y,z*>,and2] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] implies for s being State of (BitGFA0Circ (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)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) ) )
assume that
A1: z <> [<*x,y*>,xor2] and
A2: ( x <> [<*y,z*>,and2] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] ) ; :: thesis: for s being State of (BitGFA0Circ (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)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) )
set S2 = GFA0CarryStr (x,y,z);
set S1 = GFA0AdderStr (x,y,z);
InputVertices (GFA0AdderStr (x,y,z)) = {x,y,z} by A1, FACIRC_1:57;
then A3: InputVertices (GFA0AdderStr (x,y,z)) = InputVertices (GFA0CarryStr (x,y,z)) by A2, Th14;
set A2 = GFA0CarryCirc (x,y,z);
set A1 = GFA0AdderCirc (x,y,z);
set A = BitGFA0Circ (x,y,z);
let s be State of (BitGFA0Circ (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)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) )
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following (s,2)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) ) )
assume that
A4: a1 = s . x and
A5: a2 = s . y and
A6: a3 = s . z ; :: thesis: ( (Following (s,2)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) )
reconsider s1 = s | the carrier of (GFA0AdderStr (x,y,z)) as State of (GFA0AdderCirc (x,y,z)) by FACIRC_1:26;
A7: dom s1 = the carrier of (GFA0AdderStr (x,y,z)) by CIRCUIT1:3;
z in the carrier of (GFA0AdderStr (x,y,z)) by FACIRC_1:60;
then A8: a3 = s1 . z by A6, A7, FUNCT_1:47;
y in the carrier of (GFA0AdderStr (x,y,z)) by FACIRC_1:60;
then A9: a2 = s1 . y by A5, A7, FUNCT_1:47;
reconsider t = s as State of ((GFA0AdderCirc (x,y,z)) +* (GFA0CarryCirc (x,y,z))) ;
InnerVertices (GFA0CarryStr (x,y,z)) misses InputVertices (GFA0CarryStr (x,y,z)) by XBOOLE_1:79;
then A10: (Following (t,2)) . (GFA0AdderOutput (x,y,z)) = (Following (s1,2)) . (GFA0AdderOutput (x,y,z)) by A3, FACIRC_1:32;
reconsider s2 = s | the carrier of (GFA0CarryStr (x,y,z)) as State of (GFA0CarryCirc (x,y,z)) by FACIRC_1:26;
A11: dom s2 = the carrier of (GFA0CarryStr (x,y,z)) by CIRCUIT1:3;
x in the carrier of (GFA0AdderStr (x,y,z)) by FACIRC_1:60;
then a1 = s1 . x by A4, A7, FUNCT_1:47;
hence (Following (s,2)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 by A1, A9, A8, A10, Th30; :: thesis: (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)
InnerVertices (GFA0AdderStr (x,y,z)) misses InputVertices (GFA0AdderStr (x,y,z)) by XBOOLE_1:79;
then A12: (Following (t,2)) . (GFA0CarryOutput (x,y,z)) = (Following (s2,2)) . (GFA0CarryOutput (x,y,z)) by A3, FACIRC_1:33;
z in the carrier of (GFA0CarryStr (x,y,z)) by Th16;
then A13: a3 = s2 . z by A6, A11, FUNCT_1:47;
y in the carrier of (GFA0CarryStr (x,y,z)) by Th16;
then A14: a2 = s2 . y by A5, A11, FUNCT_1:47;
x in the carrier of (GFA0CarryStr (x,y,z)) by Th16;
then a1 = s2 . x by A4, A11, FUNCT_1:47;
hence (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A2, A14, A13, A12, Th22; :: thesis: verum