Goal:
  
  A : Type
  p : nat → A → bool
  q : nat → A → bool
  l : list A
  l' : list A
  B : Type
  f : B → A
  n : nat
  a : B
  bs : list B
  IHbs : ∀ n : nat,
           alli p n (map f bs) = alli (λ (i : nat) (x : B), p i (f x)) n bs
  ============================
   (p n (f a) && alli (λ (i : nat) (x : B), p i (f x)) (S n) bs =
    p n (f a) && alli (λ (i : nat) (x : B), p i (f x)) (S n) bs)


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"