Library rt.util.seqset

From mathcomp Require Import ssreflect ssrbool ssrnat eqtype seq fintype.

Section SeqSet.

  (* Let T be any type with decidable equality. *)
  Context {T: eqType}.

  (* We define a set as a sequence that has no duplicates. *)
  Record set :=
  {
    _set_seq :> seq T ;
    _ : uniq _set_seq (* no duplicates *)
  }.

  (* Now we add the ssreflect boilerplate code. *)
  Canonical Structure setSubType := [subType for _set_seq].
  Definition set_eqMixin := [eqMixin of set by <:].
  Canonical Structure set_eqType := EqType set set_eqMixin.
  Canonical Structure mem_set_predType := mkPredType (fun (l : set) ⇒ mem_seq (_set_seq l)).
  Definition set_of of phant T := set.

End SeqSet.

Notation " {set R } " := (set_of (Phant R)).

Section Lemmas.

  Context {T: eqType}.
  Variable s: {set T}.

  Lemma set_uniq : uniq s.
  Proof.
    by destruct s.
  Qed.

  Lemma set_mem : ∀ x, (x \in s) = (x \in _set_seq s).
  Proof.
    by intros x; destruct s.
  Qed.

End Lemmas.

Section LemmasFinType.

  Context {T: finType}.
  Variable s: {set T}.

  Lemma set_card : #|s| = size s.
  Proof.
    have UNIQ: uniq s by destruct s.
    by move: UNIQ ⇒ /card_uniqP →.
  Qed.

End LemmasFinType.