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:
let rec use_hyps :=
            match goal with
            | H:_ ∧ _ |- _ => exact H || (destruct_hyp H; use_hyps)
            | H:_ * _ |- _ => exact H || (destruct_hyp H; use_hyps)
            | H:_ |- _ => solve [ inversion H ]
            | _ => idtac
            end
           in
           let do_atom := solve
           [ trivial
           with eq_true
           | reflexivity
           | symmetry; trivial
           | contradiction
           | congruence ]
           in
           let rec do_ccl :=
            try do_atom; repeat (do_intro; try do_atom); try arith_congr;
             (solve [ split; do_ccl ])
           in
           (solve [ do_atom | use_hyps; do_ccl ]) ||
             fail "Cannot solve this goal"