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:
let rec t_tauto_intuit :=
            <Coq.Init.Tauto.simplif> flags;
             <Coq.Init.Tauto.axioms> flags ||
               match reverse goal with
               | id:(?X1 → ?X2) → ?X3
                 |- _ =>
                     Ltac.cut X3;
                      [ intro; clear id; t_tauto_intuit
                      | Ltac.cut (X1 → X2);
                         [ exact
                         id
                         | generalize (λ y : X2, id (λ x : X1, y)); intro;
                            clear id; (solve [ t_tauto_intuit ]) ] ]
               | id:¬ ?X1 → ?X3
                 |- _ =>
                     Ltac.cut X3;
                      [ intro; clear id; t_tauto_intuit
                      | Ltac.cut (¬ X1);
                         [ exact
                         id
                         | clear id; intro; (solve [ t_tauto_intuit ]) ] ]
               | |- ?X1 =>
                     <Coq.Init.Tauto.is_disj> flags X1; (solve
                      [ Ltac.left; t_tauto_intuit
                      | Ltac.right; t_tauto_intuit ])
               end ||
                 match goal with
                 | |- _ → _ => intro; t_tauto_intuit
                 | |- _ => t_reduce; t_solver
                 end || t_solver
           in
           t_tauto_intuit