Goal:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt, InA E.eq x xs /\ P x -> List.Exists P xs
  x0 : elt
  H1 : InA E.eq x0 xs
  H2 : P x0
  ============================
   (E.eq x0 x)


DEBUG:Going to execute:
<coq-core.plugins.ltac::auto@0> $1 $2 $3

Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H1 : InA E.eq x0 xs
  H2 : P x0
  ============================
  InA E.eq x0 xs
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H1 : InA E.eq x0 xs
  H2 : P x0
  ============================
  P x0
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  ============================
  (exists x0 : elt, (E.eq x0 x \/ InA E.eq x0 xs) /\ P x0) ->
  P x \/ (exists x0 : elt, InA E.eq x0 xs /\ P x0)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  H1 : exists x0 : elt, (E.eq x0 x \/ InA E.eq x0 xs) /\ P x0
  ============================
  P x \/ (exists x0 : elt, InA E.eq x0 xs /\ P x0)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H1 : (E.eq x0 x \/ InA E.eq x0 xs) /\ P x0
  ============================
  P x \/ (exists x1 : elt, InA E.eq x1 xs /\ P x1)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H1 : E.eq x0 x \/ InA E.eq x0 xs
  H2 : P x0
  ============================
  P x \/ (exists x1 : elt, InA E.eq x1 xs /\ P x1)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  ============================
  P x \/ (exists x1 : elt, InA E.eq x1 xs /\ P x1)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  ============================
  P x \/ (exists x1 : elt, InA E.eq x1 xs /\ P x1)
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : (fun x : elt => InA E.eq x xs /\ P x) x0 -> List.Exists P xs
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : InA E.eq x0 xs -> P x0 -> List.Exists P xs
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : InA E.eq x0 xs -> P x0 -> List.Exists P xs
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : InA E.eq x0 xs -> P x0 -> List.Exists P xs
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : InA E.eq x0 xs -> P x0 -> List.Exists P xs
  ============================
  P x
Debug:
  
  P : E.t -> Prop
  P_Proper : Proper (E.eq ==> impl) P
  s : t
  x : elt
  xs : list elt
  H : List.Exists P xs -> exists x : elt, InA E.eq x xs /\ P x
  H0 : forall x : elt,
       (fun x0 : elt => InA E.eq x0 xs /\ P x0) x -> List.Exists P xs
  x0 : elt
  H2 : P x0
  H1 : E.eq x0 x
  H3 : forall y : E.t, E.eq x0 y -> impl (P x0) (P y)
  H4 : E.eq x0 x0 -> impl (P x0) (P x0)
  H5 : InA E.eq x0 xs -> P x0 -> List.Exists P xs
  ============================
  P x