Goal:
  
  A : Type
  p : A → bool
  a : A
  l : list A
  n : nat
  x : A
  Hnth : nth_error (a :: l) (S n) = Some x
  IHl : ∀ n : nat, nth_error l n = Some x → forallb p l → p x
  ============================
   (p a
    ∧ (fix forallb (l : list A) : bool :=
         match l with
         | [] => true
         | a :: l0 => p a && forallb l0
         end) l → forallb p l)


DEBUG:Going to execute:
let rec use_hyp H :=
            match type of H with
            | _ ∧ _ => exact H || destruct_hyp H
            | _ * _ => exact H || destruct_hyp H
            | _ => try (solve [ inversion H ])
            end
           with do_intro := let H := fresh in
                            intro H; use_hyp H
           with destruct_hyp H := case H; clear H; do_intro; do_intro
           in
           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"