Local Notation "[ ]" := nil (at level 0).Local Notation "[ a ; .. ; b ]" := (a :: .. (b :: []) ..) (at level 0).(** Locally sorted: consecutive elements of the list are ordered *)

  Inductive LocallySorted : list A -> Prop :=
    | LSorted_nil : LocallySorted []
    | LSorted_cons1 a : LocallySorted [a]
    | LSorted_consn a b l :
        LocallySorted (b :: l) -> R a b -> LocallySorted (a :: b :: l).(** Locally sorted: consecutive elements of the list are ordered *)

  Inductive LocallySorted : list A -> Prop :=
    | LSorted_nil : LocallySorted []
    | LSorted_cons1 a : LocallySorted [a]
    | LSorted_consn a b l :
        LocallySorted (b :: l) -> R a b -> LocallySorted (a :: b :: l).(** Locally sorted: consecutive elements of the list are ordered *)

  Inductive LocallySorted : list A -> Prop :=
    | LSorted_nil : LocallySorted []
    | LSorted_cons1 a : LocallySorted [a]
    | LSorted_consn a b l :
        LocallySorted (b :: l) -> R a b -> LocallySorted (a :: b :: l).(** Locally sorted: consecutive elements of the list are ordered *)

  Inductive LocallySorted : list A -> Prop :=
    | LSorted_nil : LocallySorted []
    | LSorted_cons1 a : LocallySorted [a]
    | LSorted_consn a b l :
        LocallySorted (b :: l) -> R a b -> LocallySorted (a :: b :: l).(** Alternative two-step definition of being locally sorted *)

  Inductive HdRel a : list A -> Prop :=
    | HdRel_nil : HdRel a []
    | HdRel_cons b l : R a b -> HdRel a (b :: l).(** Alternative two-step definition of being locally sorted *)

  Inductive HdRel a : list A -> Prop :=
    | HdRel_nil : HdRel a []
    | HdRel_cons b l : R a b -> HdRel a (b :: l).(** Alternative two-step definition of being locally sorted *)

  Inductive HdRel a : list A -> Prop :=
    | HdRel_nil : HdRel a []
    | HdRel_cons b l : R a b -> HdRel a (b :: l).Inductive Sorted : list A -> Prop :=
    | Sorted_nil : Sorted []
    | Sorted_cons a l : Sorted l -> HdRel a l -> Sorted (a :: l).Inductive Sorted : list A -> Prop :=
    | Sorted_nil : Sorted []
    | Sorted_cons a l : Sorted l -> HdRel a l -> Sorted (a :: l).Inductive Sorted : list A -> Prop :=
    | Sorted_nil : Sorted []
    | Sorted_cons a l : Sorted l -> HdRel a l -> Sorted (a :: l).

Lemma Sorted_LocallySorted_iff : forall l, Sorted l <-> LocallySorted l.