Library prosa.util.seqset

From mathcomp Require Import ssreflect ssrbool ssrnat eqtype seq fintype.

In this section, we define a notion of a set (based on a sequence without duplicates).

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 to support _ == _ and _ ∈ _ operations.

  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 := PredType (fun (l : set) ⇒ mem_seq (_set_seq l)).
  Definition set_of of phant T := set.

End SeqSet.

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

Next we prove a basic lemma about sets.

Section Lemmas.

Consider a set s.

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

Then we show that element of s are unique.

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

End Lemmas.