Goal:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : Forall P xs -> forall x : elt, InA E.eq x xs -> P x
  H0 : (forall x : elt, InA E.eq x xs -> P x) -> Forall P xs
  ============================
   (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:
fun => <coq-core.plugins.ltac::split@0>

Goals:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : Forall P xs -> forall x : elt, InA E.eq x xs -> P x
  H0 : (forall x : elt, InA E.eq x xs -> P x) -> Forall P xs
  ============================
   (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)

  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : Forall P xs -> forall x : elt, InA E.eq x xs -> P x
  H0 : (forall x : elt, InA E.eq x xs -> P x) -> Forall P xs
  ============================
   ((forall x0 : elt, E.eq x0 x \/ InA E.eq x0 xs -> P x0) ->
    P x /\ (forall x : elt, InA E.eq x xs -> P x))


DEBUG:Going to execute:
<Coq.Init.Tauto.not_dep_intros>

Goals:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : Forall P xs -> forall x : elt, InA E.eq x xs -> P x
  H0 : (forall x : elt, InA E.eq x xs -> P x) -> Forall P xs
  ============================
   (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)

  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : Forall P xs -> forall x : elt, InA E.eq x xs -> P x
  H0 : (forall x : elt, InA E.eq x xs -> P x) -> Forall P xs
  ============================
   ((forall x0 : elt, E.eq x0 x \/ InA E.eq x0 xs -> P x0) ->
    P x /\ (forall x : elt, InA E.eq x xs -> P x))


DEBUG:Going to execute:
repeat
            match goal with
            | |- ?X1 -> ?X2 => intro
            | |- ~ _ => unfold not at 1; intro
            end