Goal:
  
  A : Type
  B : Type
  P : A → nat → B → Type
  a : B
  ls1 : list B
  a0 : A
  ls2 : list A
  n : nat
  X2 : Alli (P a0) (S n) ls1
  X3 : All (λ x : A, Alli (P x) (S n) ls1) ls2
  X1 : P a0 n a
  X4 : All (λ x : A, P x n a) ls2
  X0 : All (λ x : A, P x n a) (a0 :: ls2)
  X5 : Alli (λ (n : nat) (y : B), All (λ x : A, P x n y) (a0 :: ls2)) 
         (S n) ls1
  ============================
   (All (λ x : A, Alli (P x) n (a :: ls1)) ls2)


DEBUG:Going to execute:
<Coq.Init.Tauto.not_dep_intros>
Level 1: evaluation returns
<Coq.Init.Tauto.not_dep_intros>;
 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>)
where
flags := 
of type
tauto_flags of type tacvalue