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:
<Coq.Init.Tauto.simplif> flags
Level 3: evaluation returns
<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
where
t_tauto_intuit := <tactic closure>
of type
tacvalue
t_solver := eauto  3 with core terms
of type
tacvalue
t_reduce := <coq-core.plugins.tauto::reduction_not_iff@0>
of type
tacvalue
flags := 
of type
tauto_flags of type tacvalue

Level 2: evaluation returns
<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
where
t_tauto_intuit := <tactic closure>
of type
tacvalue
t_solver := eauto  3 with core terms
of type
tacvalue
t_reduce := <coq-core.plugins.tauto::reduction_not_iff@0>
of type
tacvalue
flags := 
of type
tauto_flags of type tacvalue

Level 1: evaluation returns
<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
where
t_tauto_intuit := <tactic closure>
of type
tacvalue
t_solver := eauto  3 with core terms
of type
tacvalue
t_reduce := <coq-core.plugins.tauto::reduction_not_iff@0>
of type
tacvalue
flags := 
of type
tauto_flags of type tacvalue