[[[ Show that H(Omega_n) > n - 8000. Omega_n is the first n bits of Omega, where we choose Omega = xxx0111111... instead of Omega = xxx1000000... if necessary. ]]] [Here is the prefix.] define pi let (count-halt prefix bits-left-to-extend) if = bits-left-to-extend 0 if = success car try t 'eval read-exp prefix 1 0 + (count-halt append prefix '(0) - bits-left-to-extend 1) (count-halt append prefix '(1) - bits-left-to-extend 1) let (omega t) cons (count-halt nil t) cons / cons ^ 2 t nil let w eval read-exp let n length w let w cons base2-to-10 w cons / cons ^ 2 n nil let (loop t) if (<=rat w (omega t)) (big nil n) (loop + t 1) let (<=rat x y) <= * car x caddr y * caddr x car y let (big prefix bits-left-to-add) if = 0 bits-left-to-add cons cadr try t 'eval read-exp prefix nil append (big append prefix '(0) - bits-left-to-add 1) (big append prefix '(1) - bits-left-to-add 1) (loop 0) [Run pi.] cadr try no-time-limit 'eval read-exp append bits pi bits ' [Program to compute first n = 8 bits of Omega] '(0 0 0 0 0 0 0 1) [Size pi.] length bits pi