Goal:
  
  A : Type
  B : Type
  P : A → nat → B → Type
  a : B
  ls1 : list B
  IH : ∀ (ls2 : list A) (n : nat),
         All (λ x : A, Alli (P x) n ls1) ls2 <~>
         Alli (λ (n0 : nat) (y : B), All (λ x : A, P x n0 y) ls2) n ls1
  a0 : A
  ls2 : list A
  n : nat
  X1 : All (λ x : A, Alli (P x) n (a :: ls1)) ls2
  X3 : All (λ x : A, Alli (P x) n (a :: ls1)) ls2
  X : P a0 n a
  X4 : Alli (P a0) (S n) ls1
  X2 : P a0 n a
  X5 : Alli (P a0) (S n) ls1
  ============================
   (All (λ x : A, P x n a) ls2)


DEBUG:Going to execute:
repeat
            (match reverse goal with
             | id:?X1
               |- _ =>
                   <Coq.Init.Tauto.is_conj> flags X1; elim id; do 2 intro;
                    clear id
             | id:_ ↔ _ |- _ => elim id; do 2 intro; clear id
             | id:¬ _ |- _ => red in id
             | id:?X1
               |- _ =>
                   <Coq.Init.Tauto.is_disj> flags X1; elim id; intro; clear
                    id
             | _ =>
                 <Coq.Init.Tauto.find_cut>
                  ltac:(fun id0 id1 X2 =>
                          assert X2; [ exact (id0 id1) | clear id0 ])
             | id:?X1 → ?X2
               |- _ =>
                   <Coq.Init.Tauto.is_unit_or_eq> flags X1; Ltac.cut X2;
                    [ intro; clear id
                    | Ltac.cut X1; [ exact id | constructor 1; fail ] ]
             | id:?X1 → ?X2
               |- _ =>
                   <Coq.Init.Tauto.flatten_contravariant_conj> flags X1 X2 id
             | id:?X1 ↔ ?X2 → ?X3
               |- _ =>
                   assert ((X1 → X2) → (X2 → X1) → X3) by
                    (do 2 intro; apply id; Ltac.split; Ltac.assumption);
                    clear id
             | id:?X1 → ?X2
               |- _ =>
                   <Coq.Init.Tauto.flatten_contravariant_disj> flags X1 X2 id
             | |- ?X1 => <Coq.Init.Tauto.is_conj> flags X1; Ltac.split
             | |- _ ↔ _ => Ltac.split
             | |- ¬ _ => red
             end; <Coq.Init.Tauto.not_dep_intros>)
Level 0: Backtrace:

No matching clauses for match.