Goal:
  
  A : Type
  P : nat → A → Type
  x : A
  l : list A
  IHl : ∀ k : nat,
          Alli P k l
          → Alli (λ k' : nat, P (Nat.pred #|l| - k' + k)) 0 (List.rev l)
  k : nat
  X : Alli P k l × Alli P (#|l| + k) [x]
  ============================
   (Alli (λ k' : nat, P (Nat.pred #|l ++ [x]| - k' + k)) 0
      (List.rev (l ++ [x])))


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 := idtac; intuition_solver
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 := idtac; intuition_solver
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 := idtac; intuition_solver
of type
tacvalue
t_reduce := <coq-core.plugins.tauto::reduction_not_iff@0>
of type
tacvalue
flags := 
of type
tauto_flags of type tacvalue