Goal:
  
  A : Type
  B : Type
  f : nat → A → option B
  g : nat → A → option B
  P : nat → A → Type
  n : nat
  hd : A
  tl : list A
  p : P n hd
  Ha : Alli P (S n) tl
  Hfg : ∀ (i : nat) (x : A) (t : B), P i x → f i x = Some t → g i x = Some t
  IHHa : ∀ t : list B,
           map_option_out (mapi_rec f tl (S n)) = Some t
           → map_option_out (mapi_rec g tl (S n)) = Some t
  t : list B
  b : B
  E : f n hd = Some b
  b' : list B
  E' : map_option_out (mapi_rec f tl (S n)) = Some b'
  ============================
   (Some (b :: b') = Some (b :: b'))


DEBUG:Going to execute:
do_atom