Goal:
  
  A : Type
  B : Type
  P : A → B → Type
  x : A
  y : B
  l : list A
  l' : list B
  l0 : list A
  IHHF : Alli
           (λ (i : nat) (x : A),
              ∑ y : B,
                (nth_error l0 i = Some x ∧ nth_error l' i = Some y) × P x y)
           0 l
  ============================
   (Alli
      (λ (x : nat) (x0 : A),
         ∑ y : B,
           (nth_error l0 x = Some x0 ∧ nth_error l' x = Some y) × P x0 y) 0 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"