Goal:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  IH : Forall P xs <-> (forall x : elt, InA E.eq x xs -> P x)
  ============================
   (P x /\ (forall x : elt, InA E.eq x xs -> P x) <->
    (forall x0 : elt, E.eq x0 x \/ InA E.eq x0 xs -> P x0))


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
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (X1 -> X2);
             [ exact
             id
             | generalize (fun y : X2 => id (fun x : X1 => y)); intro; clear
                id; (solve [ t_tauto_intuit ]) ] ]
   | id:~ ?X1 -> ?X3
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (~ X1);
             [ exact id | clear id; intro; (solve [ t_tauto_intuit ]) ] ]
   | |- ?X1 =>
         <Coq.Init.Tauto.is_disj> flags X1; (solve
          [ left; t_tauto_intuit | 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 := auto
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
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (X1 -> X2);
             [ exact
             id
             | generalize (fun y : X2 => id (fun x : X1 => y)); intro; clear
                id; (solve [ t_tauto_intuit ]) ] ]
   | id:~ ?X1 -> ?X3
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (~ X1);
             [ exact id | clear id; intro; (solve [ t_tauto_intuit ]) ] ]
   | |- ?X1 =>
         <Coq.Init.Tauto.is_disj> flags X1; (solve
          [ left; t_tauto_intuit | 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 := auto
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
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (X1 -> X2);
             [ exact
             id
             | generalize (fun y : X2 => id (fun x : X1 => y)); intro; clear
                id; (solve [ t_tauto_intuit ]) ] ]
   | id:~ ?X1 -> ?X3
     |- _ =>
         cut X3;
          [ intro; clear id; t_tauto_intuit
          | cut (~ X1);
             [ exact id | clear id; intro; (solve [ t_tauto_intuit ]) ] ]
   | |- ?X1 =>
         <Coq.Init.Tauto.is_disj> flags X1; (solve
          [ left; t_tauto_intuit | 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 := auto
of type
tacvalue
t_reduce := <coq-core.plugins.tauto::reduction_not_iff@0>
of type
tacvalue
flags := 
of type
tauto_flags of type tacvalue