set f0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set g0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE );
set h0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )];
let n be Nat; :: thesis: for x, y being FinSeqLen of
for a, b being set holds
( (n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Str x,y) +* (BitGFA0Str a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Circ x,y) +* (BitGFA0Circ a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA0CarryOutput a,b,(n -BitGFA0CarryOutput x,y) )
let x, y be FinSeqLen of ; :: thesis: for a, b being set holds
( (n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Str x,y) +* (BitGFA0Str a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Circ x,y) +* (BitGFA0Circ a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA0CarryOutput a,b,(n -BitGFA0CarryOutput x,y) )
let a, b be set ; :: thesis: ( (n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Str x,y) +* (BitGFA0Str a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Circ x,y) +* (BitGFA0Circ a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA0CarryOutput a,b,(n -BitGFA0CarryOutput x,y) )
set p = x ^ <*a*>;
set q = y ^ <*b*>;
consider f, g, h being ManySortedSet of such that
A1: ( n -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = f . n & n -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = g . n ) and
A2: ( f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )] ) 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 +* (BitGFA0Str ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) & g . (n + 1) = A +* (BitGFA0Circ ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z) & h . (n + 1) = GFA0CarryOutput ((x ^ <*a*>) . (n + 1)),((y ^ <*b*>) . (n + 1)),z ) by Def2;
A4: ( n -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . n & (n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = f . (n + 1) & (n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = g . (n + 1) & (n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . (n + 1) ) by A2, A3, Th1;
( 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 -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0Str x,y & n -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0Circ x,y & n -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0CarryOutput x,y ) by Th5;
hence ( (n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Str x,y) +* (BitGFA0Str a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = (n -BitGFA0Circ x,y) +* (BitGFA0Circ a,b,(n -BitGFA0CarryOutput x,y)) & (n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = GFA0CarryOutput a,b,(n -BitGFA0CarryOutput x,y) ) by A1, A3, A4, A5; :: thesis: verum