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.
    by destruct s.

  Lemma set_mem : x, (x s) = (x _set_seq s).
    by intros x; destruct s.

End Lemmas.

Section LemmasFinType.

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

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

End LemmasFinType.