Section Remove.
Hypothesis eqA_dec : forall x y : A, {eqA x y}+{~(eqA x y)}.
End Remove.
(** Results concerning lists modulo [eqA] and [ltA] *)
Variable ltA : A -> A -> Prop.
Hypothesis ltA_strorder : StrictOrder ltA.
Hypothesis ltA_compat : Proper (eqA==>eqA==>iff) ltA.
Let sotrans := (@StrictOrder_Transitive _ _ ltA_strorder).
Hint Resolve sotrans : core.
Notation InfA:=(lelistA ltA).
Notation SortA:=(sort ltA).
Hint Constructors lelistA sort : core.
Global Instance InfA_compat : Proper (eqA==>eqlistA==>iff) InfA.
Proof using eqA_equiv ltA_compat. (* and not ltA_strorder *)
intros x x' Hxx' l l' Hll'.
inversion_clear Hll'.
intuition.
split; intro; inv; constructor.
rewrite <- Hxx', <- H; auto.
rewrite Hxx', H; auto.
Qed.
Proof using eqA_equiv ltA_compat.
intros H; now rewrite H.
Qed.
Hint Immediate InfA_ltA InfA_eqA : core.

Lemma SortA_InfA_InA :
forall l x a, SortA l -> InfA a l -> InA x l -> ltA a x.