Goal:
  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  a : C
  ls1 : list C
  IH : ∀ (ls2 : list A) (ls3 : list B),
         All (λ z : C, All2 (λ (x : A) (y : B), P x y z) ls2 ls3) ls1
         → (ls1 = []) + All2 (λ (x : A) (y : B), All (P x y) ls1) ls2 ls3
  a0 : A
  ls2 : list A
  X0 : All2 (λ (x : A) (y : B), P x y a) (a0 :: ls2) []
  X1 : All (λ z : C, All2 (λ (x : A) (y : B), P x y z) (a0 :: ls2) []) ls1
  ============================
   ((a :: ls1 = []) +
    All2 (λ (x : A) (y : B), All (P x y) (a :: ls1)) (a0 :: ls2) [])%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 ]