set f0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set g0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set h0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )];
let n be Nat; :: thesis: for x, y being FinSeqLen of
for a, b being set holds
( (n + 1) -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Str x,y) +* (BitGFA1Str a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Circ x,y) +* (BitGFA1Circ a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA1CarryOutput a,b,(n -BitGFA1CarryOutput x,y) )
let x, y be FinSeqLen of ; :: thesis: for a, b being set holds
( (n + 1) -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Str x,y) +* (BitGFA1Str a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Circ x,y) +* (BitGFA1Circ a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA1CarryOutput a,b,(n -BitGFA1CarryOutput x,y) )
let a, b be set ; :: thesis: ( (n + 1) -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Str x,y) +* (BitGFA1Str a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Circ x,y) +* (BitGFA1Circ a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA1CarryOutput a,b,(n -BitGFA1CarryOutput x,y) )
set p = x ^ <*a*>;
set q = y ^ <*b*>;
consider f, g, h being ManySortedSet of such that
A1: ( n -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = f . n & n -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = g . n ) and
A2: ( f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] ) and
A3: for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra of S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitGFA1Str ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) & g . (n + 1) = A +* (BitGFA1Circ ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) & h . (n + 1) = GFA1CarryOutput ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z ) by Def6;
A4: ( n -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . n & (n + 1) -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = f . (n + 1) & (n + 1) -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = g . (n + 1) & (n + 1) -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . (n + 1) ) by A2, A3, Th15;
( len x = n & len y = n ) by FINSEQ_1:def 18;
then A5: ( (x ^ <*a*>) . (n + 1) = a & (y ^ <*b*>) . (n + 1) = b ) by FINSEQ_1:59;
( x ^ <*> = x & y ^ <*> = y ) by FINSEQ_1:47;
then ( n -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA1Str x,y & n -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA1Circ x,y & n -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA1CarryOutput x,y ) by Th19;
hence ( (n + 1) -BitGFA1Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Str x,y) +* (BitGFA1Str a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA1Circ x,y) +* (BitGFA1Circ a,b,(n -BitGFA1CarryOutput x,y)) & (n + 1) -BitGFA1CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA1CarryOutput a,b,(n -BitGFA1CarryOutput x,y) ) by A1, A3, A4, A5; :: thesis: verum