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; for x, y being FinSeqLen of n
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 n; 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 ; ( (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 NAT such that
A1:
n -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = f . n
and
A2:
n -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = g . n
and
A3:
f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> FALSE )
and
A4:
g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> FALSE )
and
A5:
h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> FALSE )]
and
A6:
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;
A7:
n -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . n
by A3, A4, A5, A6, Th1;
A8:
(n + 1) -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = f . (n + 1)
by A3, A4, A5, A6, Th1;
A9:
(n + 1) -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = g . (n + 1)
by A3, A4, A5, A6, Th1;
A10:
(n + 1) -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = h . (n + 1)
by A3, A4, A5, A6, Th1;
A11:
len x = n
by FINSEQ_1:def 18;
A12:
len y = n
by FINSEQ_1:def 18;
A13:
(x ^ <*a*>) . (n + 1) = a
by A11, FINSEQ_1:59;
A14:
(y ^ <*b*>) . (n + 1) = b
by A12, FINSEQ_1:59;
A15:
x ^ <*> = x
by FINSEQ_1:47;
A16:
y ^ <*> = y
by FINSEQ_1:47;
then A17:
n -BitGFA0Str (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0Str x,y
by A15, Th5;
A18:
n -BitGFA0Circ (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0Circ x,y
by A15, A16, Th5;
n -BitGFA0CarryOutput (x ^ <*a*>),(y ^ <*b*>) = n -BitGFA0CarryOutput x,y
by A15, A16, 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, A2, A6, A7, A8, A9, A10, A13, A14, A17, A18; verum