Goal:
  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  X : All (λ z : C, All2 (λ (x : A) (y : B), P x y z) [] []) []
  ============================
   (([] = []) + All2 (λ (x : A) (y : B), All (P x y) []) [] [])%type


DEBUG:Going to execute:
first
[ progress intros
| progress subst
| let is_ctor_list x := lazymatch x with
                        | [] => idtac
                        | _ :: _ => idtac
                        end in
  match goal with
  | H:All2 _ ?x ?y
    |- _ => (first [ is_ctor_list x | is_ctor_list y ]); inversion H; clear H
  | H:All2_fold _ ?x ?y
    |- _ => (first [ is_ctor_list x | is_ctor_list y ]); inversion H; clear H
  | H:All _ ?x |- _ => is_ctor_list x; inversion H; clear H
  | |- (_ :: _ = []) + _ => right
  | |- (?x = ?x) + _ => left
  end
| exactly_once constructor
| solve
[ eauto
| (first [ eapply All2_impl | eapply All2_fold_impl | eapply All_impl ]); tea;
   cbn; intros *; inversion 1 + constructor; subst; eauto ]
| congruence
| now
apply All_refl; constructor
| apply All2_from_nth_error => //; lia
| progress cbn in *
| match goal with
  | H:?x ≠ ?x ∨ ?A |- _ => assert A by now destruct H; clear H
  end ]