Library prosa.util.bigcat

Require Export prosa.util.tactics prosa.util.notation.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.

In this section, we introduce useful lemmas about the concatenation operation performed over an arbitrary range of sequences.
Section BigCatLemmas.

Consider any type supporting equality comparisons...
  Variable T: eqType.

...and a function that, given an index, yields a sequence.
  Variable f: nat list T.

In this section, we prove that the concatenation over sequences works as expected: no element is lost during the concatenation, and no new element is introduced.
  Section BigCatElements.

First, we show that the concatenation comprises all the elements of each sequence; i.e. any element contained in one of the sequences will also be an element of the result of the concatenation.
    Lemma mem_bigcat_nat:
       x m n j,
        m j < n
        x \in f j
        x \in \cat_(m i < n) (f i).
    Proof.
      intros x m n j LE IN; move: LE ⇒ /andP [LE LE0].
      rewritebig_cat_nat with (n := j); simpl; [| by ins | by apply ltnW].
      rewrite mem_cat; apply/orP; right.
      destruct n; first by rewrite ltn0 in LE0.
      rewrite big_nat_recl; last by ins.
        by rewrite mem_cat; apply/orP; left.
    Qed.

Conversely, we prove that any element belonging to a concatenation of sequences must come from one of the sequences.
    Lemma mem_bigcat_nat_exists :
       x m n,
        x \in \cat_(m i < n) (f i)
         i,
          x \in f i m i < n.
    Proof.
      intros x m n IN.
      induction n; first by rewrite big_geq // in IN.
      destruct (leqP m n); last by rewrite big_geq ?in_nil // ltnW in IN.
      rewrite big_nat_recr // /= mem_cat in IN.
      move: IN ⇒ /orP [HEAD | TAIL].
      {
        apply IHn in HEAD; destruct HEAD; x0. move: H ⇒ [H /andP [H0 H1]].
        split; first by done.
          by apply/andP; split; [by done | by apply ltnW]. }
      {
         n; split; first by done.
        apply/andP; split; [by done | by apply ltnSn]. }
    Qed.

  End BigCatElements.

In this section, we show how we can preserve uniqueness of the elements (i.e. the absence of a duplicate) over a concatenation of sequences.
  Section BigCatDistinctElements.

Assume that there are no duplicates in each of the possible sequences to concatenate...
    Hypothesis H_uniq_seq: i, uniq (f i).

...and that there are no elements in common between the sequences.
    Hypothesis H_no_elements_in_common:
       x i1 i2, x \in f i1 x \in f i2 i1 = i2.

We prove that the concatenation will yield a sequence with unique elements.
    Lemma bigcat_nat_uniq :
       n1 n2,
        uniq (\cat_(n1 i < n2) (f i)).
    Proof.
      intros n1 n2.
      case (leqP n1 n2) ⇒ [LE | GT]; last by rewrite big_geq // ltnW.
      rewrite -[n2](addKn n1).
      rewrite -addnBA //; set delta := n2 - n1.
      induction delta; first by rewrite addn0 big_geq.
      rewrite addnS big_nat_recr /=; last by apply leq_addr.
      rewrite cat_uniq; apply/andP; split; first by apply IHdelta.
      apply /andP; split; last by apply H_uniq_seq.
      rewrite -all_predC; apply/allP; intros x INx.
      simpl; apply/negP; unfold not; intro BUG.
      apply mem_bigcat_nat_exists in BUG.
      move: BUG ⇒ [i [IN /andP [_ LTi]]].
      apply H_no_elements_in_common with (i1 := i) in INx; last by done.
      by rewrite INx ltnn in LTi.
    Qed.

  End BigCatDistinctElements.

End BigCatLemmas.