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# Library prosa.util.seqset

From HB Require Import structures.
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
}.
Now we add the ssreflect boilerplate code to support _ == _ and _ ∈ _ operations.
Set Warnings "-redundant-canonical-projection".
HB.instance Definition _ setSubType := [isSub for _set_seq].
HB.instance Definition _ := [Equality of set by <:].
Canonical Structure mem_set_predType := PredType (fun (l : set) ⇒ mem_seq (_set_seq l)).
Definition set_of of phant T := set.
End SeqSet.
Set Warnings "redundant-canonical-projection".
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.