Library prosa.util.nondecreasing

From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype bigop.
Require Export prosa.util.epsilon.
Require Export prosa.util.nat.
Require Export prosa.util.list.

Non-Decreasing Sequence and Distances

In this file we introduce the notion of a non-decreasing sequence.
Section NondecreasingSequence.

First, let's introduce the notion of the nth element of a sequence.
  Notation "xs [| n |]" := (nth 0 xs n) (at level 30).

In this section we provide the notion of a non-decreasing sequence.
  Section Definitions.

We say that a sequence xs is non-decreasing iff for any two indices n1 and n2 such that n1 ≤ n2 < size [xs] condition [xs][n1] ≤ [xs][n2] holds.
    Definition nondecreasing_sequence (xs : seq nat) :=
      ∀ n1 n2,
        n1 ≤ n2 < size xs →
        xs[|n1|] ≤ xs[|n2|].

Similarly, we define an increasing sequence.
    Definition increasing_sequence (xs : seq nat) :=
      ∀ n1 n2,
        n1 < n2 < size xs →
        xs[|n1|] < xs[|n2|].

For a non-decreasing sequence we define the notion of distances between neighboring elements of the sequence. Example: Consider the following sequence of natural numbers: xs = :: 1; 10; 10; 17; 20; 41. Then drop 1 xs is equal to :: 10; 10; 17; 20; 41. Then zip xs (drop 1 xs) is equal to :: (1,10); (10,10); (10,17); (17,20); (20,41) And after the mapping map (fun '(x1, x2) ⇒ x2 - x1) we end up with :: 9; 0; 7; 3; 21.
    Definition distances (xs : seq nat) :=
      map (fun '(x1, x2) ⇒ x2 - x1) (zip xs (drop 1 xs)).

  End Definitions.

Properties of Increasing Sequence

In this section we prove a few lemmas about increasing sequences.
  Section IncreasingSequence.

Note that a filtered iota sequence is an increasing sequence.
    Lemma iota_is_increasing_sequence:
      ∀ a b P,
        increasing_sequence [seq x <- index_iota a b | P x].
    Proof.
      clear; intros ? ? ?.
      have EX : ∃ k, b - a ≤ k.
      { by ∃ (b-a). } destruct EX as [k BO].
      revert a b P BO; induction k.
      { move ⇒ a b P BO n1 n2.
        move: BO; rewrite leqn0; move ⇒ /eqP BO.
        rewrite /index_iota BO; simpl.
        by move ⇒ /andP [_ F].
      }
      { move ⇒ a b P BO n1 n2 /andP [GE LT].
        case: (leqP b a) ⇒ [N|N].
        - move: N; rewrite -subn_eq0; move ⇒ /eqP EQ.
          by rewrite /index_iota EQ //= in LT.
        - rewrite index_iota_lt_step; last by done.
          simpl; destruct (P a) eqn:PA.
          + destruct n1, n2; try done; simpl.
            × apply iota_filter_gt; first by done.
              by rewrite index_iota_lt_step // //= PA //= in LT.
            × apply IHk; try lia.
              apply/andP; split; first by done.
              by rewrite index_iota_lt_step // //= PA //= in LT.
          + apply IHk; try lia.
            apply/andP; split; first by done.
            by rewrite index_iota_lt_step // //= PA //= in LT.
      }
    Qed.

We prove that any increasing sequence is also a non-decreasing sequence.
    Lemma increasing_implies_nondecreasing:
      ∀ xs,
        increasing_sequence xs →
        nondecreasing_sequence xs.
    Proof.
      intros ? INC n1 n2.
      move ⇒ /andP; rewrite leq_eqVlt; move ⇒ [/orP [/eqP EQ| LT1] LT2].
      - by subst.
      - by apply ltnW; apply INC; apply/andP; split.
    Qed.

  End IncreasingSequence.

Properties of Non-Decreasing Sequence

In this section we prove a few lemmas about non-decreasing sequences.
  Section NonDecreasingSequence.

First we prove that if 0 ∈ xs, then 0 is the first element of xs.
    Lemma nondec_seq_zero_first:
      ∀ xs,
        (0 \in xs) →
        nondecreasing_sequence xs →
        first0 xs = 0.
    Proof.
      destruct xs as [ | x xs]; first by done.
      destruct x as [ | x]; first by done.
      rewrite in_cons; move ⇒ /orP [/eqP EQ | IN] ND; first by done.
      exfalso.
      have NTH := nth_index 0 IN.
      specialize (ND 0 (index 0 xs).+1).
      feed ND.
      { apply/andP; split; first by done.
          by simpl; rewrite ltnS index_mem. }
      by simpl in ND; rewrite NTH in ND.
    Qed.

If x1::x2::xs is a non-decreasing sequence, then either x1 = x2 or x1 < x2.
    Lemma nondecreasing_sequence_2cons_leVeq:
      ∀ x1 x2 xs,
        nondecreasing_sequence (x1 :: x2 :: xs) →
        x1 = x2 ∨ x1 < x2.
    Proof.
      intros ? ? ? ND.
      destruct (ltngtP x1 x2) as [LT | GT | EQ]; auto.
      move_neq_down GT.
      by specialize (ND 0 1); apply ND.
    Qed.

We prove that if x::xs is a non-decreasing sequence, then xs also is a non-decreasing sequence.
    Lemma nondecreasing_sequence_cons:
      ∀ x xs,
        nondecreasing_sequence (x :: xs) →
        nondecreasing_sequence xs.
    Proof.
      intros ? ? ND.
      move ⇒ n1 n2 /andP [LT1 LT2].
      specialize (ND n1.+1 n2.+1).
      apply ND.
      apply/andP; split.
      - by rewrite ltnS.
      - by simpl; rewrite ltnS.
    Qed.

Let xs be a non-decreasing sequence, then for x s.t. ∀ y ∈ xs, x ≤ y x::xs is a non-decreasing sequence.
    Lemma nondecreasing_sequence_add_min:
      ∀ x xs,
        (∀ y, y \in xs → x ≤ y) →
        nondecreasing_sequence xs →
        nondecreasing_sequence (x :: xs).
    Proof.
      intros x xs MIN ND [ |n1] [ | n2]; try done.
      - move ⇒ /andP [_ S] //=.
        apply leq_trans with (xs [| 0 |]).
        + apply MIN; apply mem_nth.
          simpl in *; rewrite ltnS in S.
          by apply leq_ltn_trans with n2.
        + apply ND; apply/andP; split; auto.
      - by rewrite //= !ltnS; apply ND.
    Qed.

We prove that if x::xs is a non-decreasing sequence, then x::x::xs also is a non-decreasing sequence.
    Lemma nondecreasing_sequence_cons_double:
      ∀ x xs,
        nondecreasing_sequence (x :: xs) →
        nondecreasing_sequence (x :: x :: xs).
    Proof.
      intros ? ? ND.
      move ⇒ [ | n1] [ | n2] /andP [LT1 LT2]; try done.
      - specialize (ND 0 n2).
        by apply ND; rewrite //= ltnS in LT2.
      - apply ND.
        apply/andP; split.
        + by rewrite ltnS in LT1.
        + by rewrite //= ltnS in LT2.
    Qed.

We prove that if x::xs is a non-decreasing sequence, then x is a minimal element of xs.
    Lemma nondecreasing_sequence_cons_min:
      ∀ x xs,
        nondecreasing_sequence (x :: xs) →
        (∀ y, y \in xs → x ≤ y).
    Proof.
      intros ? ? ND ? IN.
      have IDX := nth_index 0 IN.
      specialize (ND 0 (index y xs).+1).
      move: (IN) ⇒ IDL; rewrite -index_mem in IDL.
      feed ND.
      { by apply/andP; split; last rewrite //= ltnS. }
      eapply leq_trans; first by apply ND.
      by rewrite //= IDX.
    Qed.

We also prove a similar lemma for strict minimum.
    Corollary nondecreasing_sequence_cons_smin:
      ∀ x1 x2 xs,
        x1 < x2 →
        nondecreasing_sequence (x1 :: x2 :: xs) →
        (∀ y, y \in x2 :: xs → x1 < y).
    Proof.
      intros ? ? ? LT ND ? IN.
      eapply leq_trans; first by apply LT.
      apply nondecreasing_sequence_cons in ND.
      eapply nondecreasing_sequence_cons_min; last by apply IN.
      by apply nondecreasing_sequence_cons_double.
    Qed.

Next, we prove that no element can lie strictly between two neighboring elements and still belong to the list.
    Lemma antidensity_of_nondecreasing_seq:
      ∀ (xs : seq nat) (x : nat) (n : nat),
        nondecreasing_sequence xs →
        xs[|n|] < x < xs[|n.+1|] →
        ~~ (x \in xs).
    Proof.
      intros ? ? ? STR ?; apply/negP; intros ?.
      move: H0 ⇒ /nthP. intros GG.
      specialize (GG 0).
      move: GG ⇒ [ind LE HHH].
      subst x; rename ind into x.
      destruct (n.+1 < size xs) eqn:Bt; last first.
      { move: Bt ⇒ /negP /negP; rewrite -leqNgt; move ⇒ Bt.
        apply nth_default with (x0 := 0) in Bt.
        by rewrite Bt in H; move: H ⇒ /andP [_ T]. }
      have B1: n.+1 < size xs; first by done. clear Bt.
      have B2: n < size xs; first by apply leq_ltn_trans with n.+1.
      have GT: n < x.
      { move: H ⇒ /andP [T _].
        rewrite ltnNge; apply/negP; intros CONTR.
        specialize (STR x n).
        feed STR.
        { by apply/andP; split. }
        by move: STR; rewrite leqNgt; move ⇒ /negP STR; apply: STR.
      }
      have LT: x < n.+1.
      { clear GT.
        move: H ⇒ /andP [_ T].
        rewrite ltnNge; apply/negP; intros CONTR.
        move: CONTR; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ | CONTR].
        - by subst; rewrite ltnn in T.
        - specialize (STR n.+1 x).
          feed STR.
          { by apply/andP; split; first apply ltnW. }
          by move: STR; rewrite leqNgt; move ⇒ /negP STR; apply: STR.
      }
      by move: LT; rewrite ltnNge; move ⇒ /negP LT; apply: LT.
    Qed.

Alternatively, consider an arbitrary natural number x that is bounded by the first and the last element of a sequence xs. Then there is an index n such that xs[n] ≤ x < x[n+1].
    Lemma belonging_to_segment_of_seq_is_total:
      ∀ (xs : seq nat) (x : nat),
        2 ≤ size xs →
        first0 xs ≤ x < last0 xs →
        ∃ n,
          n.+1 < size xs ∧
          xs[|n|] ≤ x < xs[|n.+1|].
    Proof.
      intros ? ? SIZE LAST.
      have EX: ∃ n, size xs ≤ n by ∃ (size xs). move: EX ⇒ [n LE].
      destruct n; first by destruct xs.
      destruct n; first by destruct xs; last destruct xs.
      generalize dependent xs.
      induction n; intros.
      { by destruct xs; [ | destruct xs; [ | destruct xs; [∃ 0 | ] ] ]. }
      { destruct xs; [ | destruct xs; [ | ]]; try by done.
        destruct xs; first by (∃ 0).
        destruct (leqP n1 x) as [NEQ|NEQ]; last first.
        { ∃ 0; split; auto. move: LAST ⇒ /andP [LAST _].
          by apply/andP; split. }
        { specialize (IHn [:: n1, n2 & xs]).
          feed_n 3 IHn.
          { by done. }
          { move: LAST ⇒ /andP [LAST1 LAST2].
            apply/andP; split; first by done.
            apply leq_trans with (last0 [:: n0, n1, n2 & xs]); first by done.
            by rewrite /last0 !last_cons.
          }
          { by rewrite -(ltn_add2r 1) !addn1. }
          move: IHn ⇒ [idx [SI /andP [G L]]].
          ∃ idx.+1; split.
          - by simpl in *; rewrite -addn1 -[in X in _ ≤ X]addn1 leq_add2r.
          - by apply/andP; split.
        }
      }
    Qed.

Note that the last element of a non-decreasing sequence is the max element.
    Lemma last_is_max_in_nondecreasing_seq:
      ∀ (xs : seq nat) (x : nat),
        nondecreasing_sequence xs →
        (x \in xs) →
        x ≤ last0 xs.
    Proof.
      intros ? ? STR IN.
      have NEQ: ∀ x y, x = y ∨ x != y.
      { clear; intros.
        destruct (x == y) eqn:EQ.
        - by left; apply/eqP.
        - by right.
      }
      move: (NEQ _ x (last0 xs)); clear NEQ; move ⇒ [EQ|NEQ].
      { by subst x. }
      { move: IN ⇒ /nthP EX.
        specialize (EX 0).
        move: EX ⇒ [id SIZE EQ].
        rewrite /last0 -nth_last -EQ; subst x.
        rewrite -addn1 in SIZE.
        apply STR; apply/andP.
        have POS: 0 < size xs.
        { by apply leq_trans with (id + 1); first rewrite addn1. }
        split.
        - by rewrite -(leq_add2r 1) !addn1 prednK // -addn1.
        - by rewrite prednK.
      }
    Qed.

  End NonDecreasingSequence.

Properties of Undup of Non-Decreasing Sequence

In this section we prove a few lemmas about undup of non-decreasing sequences.
  Section Undup.

First we prove that undup x::x::xs is equal to undup x::xs.
    Remark nodup_sort_2cons_eq:
      ∀ {X : eqType} (x : X) (xs : seq X),
        undup (x::x::xs) = undup (x::xs).
    Proof.
      intros; rewrite {2 3}[cons] lock //= -lock.
      rewrite in_cons eq_refl; simpl.
      by destruct (x \in xs).
    Qed.

For non-decreasing sequences we show that the fact that x1 < x2 implies that undup x1::x2::xs = x1::undup x2::xs.
    Lemma nodup_sort_2cons_lt:
      ∀ x1 x2 xs,
        x1 < x2 →
        nondecreasing_sequence (x1::x2::xs) →
        undup (x1::x2::xs) = x1 :: undup (x2::xs).
    Proof.
      intros; rewrite {2 3 4}[cons] lock //= -lock.
      rewrite in_cons.
      have → : (x1 == x2) = false.
      { by apply/eqP/eqP; rewrite neq_ltn; rewrite H. }
      rewrite [cons]lock //= -lock.
      have → : (x1 \in xs) = false; last by done.
      apply/eqP; rewrite eqbF_neg.
      apply nondecreasing_sequence_cons in H0.
      apply/negP; intros ?; eauto 2.
      by eapply nondecreasing_sequence_cons_min in H0; eauto 2; lia.
    Qed.

Next we show that function undup doesn't change the last element of a sequence.
    Lemma last0_undup:
      ∀ xs,
        nondecreasing_sequence xs →
        last0 (undup xs) = last0 xs.
    Proof.
      induction xs as [ | x1 xs]; first by done.
      intros ND; destruct xs as [ |x2 xs]; first by done.
      destruct (nondecreasing_sequence_2cons_leVeq _ _ _ ND) as [EQ|LT].
      + subst; rename x2 into x.
        rewrite nodup_sort_2cons_eq IHxs.
        rewrite [in X in _ = X]last0_cons //.
        by eapply nondecreasing_sequence_cons; eauto 2.
      + rewrite nodup_sort_2cons_lt // last0_cons.
        rewrite IHxs //; eauto using nondecreasing_sequence_cons.
        by intros ?; apply undup_nil in H.
    Qed.

Non-decreasing sequence remains non-decreasing after application of undup.
    Lemma nondecreasing_sequence_undup:
      ∀ xs,
        nondecreasing_sequence xs →
        nondecreasing_sequence (undup xs).
    Proof.
      intros ?.
      have EX: ∃ len, size xs ≤ len.
      { by ∃ (size xs). } destruct EX as [n BO].
      revert xs BO; induction n.
      - by intros xs; rewrite leqn0 size_eq0; move ⇒ /eqP EQ; subst xs.
      - intros [ |x1 [ | x2 xs]] Size NonDec; try done.
        destruct (nondecreasing_sequence_2cons_leVeq _ _ _ NonDec) as [EQ|LT].
        + subst; rename x2 into x.
          by rewrite nodup_sort_2cons_eq; apply IHn; eauto using nondecreasing_sequence_cons.
        + rewrite nodup_sort_2cons_lt //.
          apply nondecreasing_sequence_add_min.
          intros ? ?.
          eapply nondecreasing_sequence_cons_min with (y := y) in NonDec; auto.
          rewrite -mem_undup; eauto using nondecreasing_sequence_cons.
          by apply IHn; eauto using nondecreasing_sequence_cons.
    Qed.

We also show that the penultimate element of a sequence undup xs is bounded by the penultimate element of sequence xs.
    Lemma undup_nth_le:
      ∀ xs,
        nondecreasing_sequence xs →
        undup xs [| (size (undup xs)).-2 |] ≤ xs [| (size xs).-2 |].
    Proof.
      Opaque undup.
      intros ?.
      have EX: ∃ len, size xs ≤ len by (∃ (size xs)).
      destruct EX as [n BO].
      revert xs BO; induction n.
      - by intros xs; rewrite leqn0 size_eq0; move ⇒ /eqP EQ; subst xs.
      - intros [ |x1 [ | x2 xs]] Size NonDec; try done.
        destruct (nondecreasing_sequence_2cons_leVeq _ _ _ NonDec) as [EQ|LT].
        × subst; rename x2 into x.
          rewrite nodup_sort_2cons_eq //.
          eapply leq_trans. apply IHn. by done. eapply nondecreasing_sequence_cons; eauto.
          destruct xs as [ | x1 xs]; first by done.
          by rewrite [in X in _ ≤ X]nth0_cons //.
        × rewrite nodup_sort_2cons_lt //; simpl.
          case (posnP (size (undup (x2 :: xs))).-1) as [Z|POS].
          -- rewrite Z; simpl.
             apply leq_trans with ([:: x1, x2 & xs] [|0|]); first by done.
             by apply NonDec; apply/andP; split; simpl.
          -- rewrite nth0_cons //.
             eapply leq_trans; first apply IHn; eauto using nondecreasing_sequence_cons.
             destruct xs.
             ++ by exfalso.
             ++ destruct xs; simpl in *; auto.
    Qed.

  End Undup.

Properties of Distances

In this section we prove a few lemmas about function distances.
  Section Distances.

We begin with a simple lemma that helps us unfold distances of lists with two consecutive cons x0::x1::xs.
    Lemma distances_unfold_2cons:
      ∀ x0 x1 xs,
        distances (x0::x1::xs) = (x1 - x0) :: distances (x1::xs).
    Proof. by intros; rewrite /distances //= drop0. Qed.

Similarly, we prove a lemma stating that two consecutive appends to a sequence in distances function (distances(xs ++ [:: a; b])) can be rewritten as distances(xs ++ [:: a]) ++ [:: b - a].
    Lemma distances_unfold_2app_last:
      ∀ (a b : nat) (xs : seq nat),
        distances (xs ++ [:: a; b])
        = distances (xs ++ [:: a]) ++ [:: b - a].
    Proof.
      intros.
      have EX: ∃ n, size xs ≤ n by (∃ (size xs)).
      destruct EX as [n LE].
      revert xs LE; induction n; intros.
      - by move: LE; rewrite leqn0 size_eq0; move ⇒ /eqP LE; subst.
      - destruct xs as [ | x0]; first by unfold distances.
        destruct xs as [ | x1]; first by unfold distances.
        have → : distances ([:: x0, x1 & xs] ++ [:: a; b]) = x1 - x0 :: distances ((x1 :: xs) ++ [:: a; b]).
        { by simpl; rewrite distances_unfold_2cons. }
        rewrite IHn; last by simpl in *; rewrite -(leq_add2r 1) !addn1.
        have → : distances ([:: x0, x1 & xs] ++ [:: a]) = x1 - x0 :: distances ((x1 :: xs) ++ [:: a]); last by done.
        by rewrite //= distances_unfold_2cons.
    Qed.

We also prove a lemma stating that one append to a sequence in the distances function (i.e., distances(xs ++ [:: x])) can be rewritten as distances xs ++ [:: x - last0 xs].
    Lemma distances_unfold_1app_last:
      ∀ x xs,
        size xs ≥ 1 →
        distances (xs ++ [:: x]) = distances xs ++ [:: x - last0 xs].
    Proof.
      intros x xs POS.
      have EX: ∃ n, size xs ≤ n by (∃ (size xs)).
      destruct EX as [n LE].
      revert x xs LE POS; induction n; intros.
      - by move: LE; rewrite leqn0 size_eq0; move ⇒ /eqP LE; subst.
      - move: LE; rewrite leq_eqVlt; move ⇒ /orP [/eqP LEN' | LE]; last first.
        + by rewrite ltnS in LE; apply IHn.
        + destruct (seq_elim_last _ _ LEN') as [x__new [xs__l [EQ2 LEN]]].
          subst xs; clear LEN' POS; rename xs__l into xs.
          rewrite -catA.
          rewrite distances_unfold_2app_last.
          destruct xs; first by done.
          rewrite IHn.
          × by rewrite /last0 last_cat.
          × by rewrite LEN.
          × by done.
    Qed.

We prove that the difference between any two neighboring elements is bounded by the max element of the distances-sequence.
    Lemma distance_between_neighboring_elements_le_max_distance_in_seq:
      ∀ (xs : seq nat) (n : nat),
        xs[|n.+1|] - xs[|n|] ≤ max0 (distances xs).
    Proof.
      intros xs id.
      apply leq_trans with (distances xs [| id |]).
      rewrite leq_eqVlt; apply/orP; left; apply/eqP.
      have EX: ∃ n, size xs ≤ n by (∃ (size xs)).
      move: EX ⇒ [n LE]; move: xs id LE.
      induction n; intros.
      { move: LE; rewrite leqn0 size_eq0; move ⇒ /eqP EQ; subst.
        by rewrite !nth_default.
      }
      { move: LE; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ|LT]; last first.
        { by apply IHn; rewrite ltnS in LT. }
        destruct xs; first by done.
        destruct xs; first by destruct id; [simpl |rewrite !nth_default].
        have ->: distances [:: n0, n1 & xs] = (n1 - n0) :: distances [:: n1 & xs].
        { by rewrite /distances //= drop0. }
        destruct id; first by done.
        rewrite //= -IHn //=.
        by move: EQ ⇒ /eqP; rewrite //= eqSS ⇒ /eqP EQ; rewrite -EQ.
      }
      { have Lem: ∀ xs x, x \in xs → x ≤ max0 xs.
        { clear; induction xs; intros ? IN; first by done.
          rewrite max0_cons leq_max; apply/orP.
          move: IN; rewrite in_cons; move ⇒ /orP [/eqP EQ| IN].
          - by left; subst.
          - by right; apply IHxs.
        }
        destruct (size (distances xs) ≤ id) eqn:SIZE.
        - by rewrite nth_default.
        - apply Lem; rewrite mem_nth //.
          move: SIZE ⇒ /negP /negP.
          by rewrite ltnNge.
      }
    Qed.

Note that the distances-function has the expected behavior indeed. I.e. an element on the position n of the distance-sequence is equal to the difference between elements on positions n+1 and n.
    Lemma function_of_distances_is_correct:
      ∀ (xs : seq nat) (n : nat),
        (distances xs)[|n|] = xs[|n.+1|] - xs[|n|].
    Proof.
      intros xs.
      have EX: ∃ len, size xs ≤ len by (∃ (size xs)).
      move: EX ⇒ [len LE]; move: xs LE.
      induction len; intros.
      { by move: LE; rewrite leqn0 size_eq0; move ⇒ /eqP EQ; subst; destruct n. }
      move: LE; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ| LE]; last by apply IHlen.
      destruct xs as [ | x1 xs]; first by done.
      destruct xs as [ | x2 xs]; first by destruct n as [ | [ | ]].
      destruct n; first by done.
      have ->: distances [:: x1, x2 & xs] [|n.+1|] = distances [::x2 & xs] [| n |].
      { have ->: distances [:: x1, x2 & xs] = (x2 - x1) :: distances [::x2 & xs]; last by done.
        { by rewrite /distances //= drop0. }
      }
      have ->: [:: x1, x2 & xs] [|n.+2|] - [:: x1, x2 & xs] [|n.+1|] = [:: x2 & xs] [|n.+1|] - [:: x2 & xs] [|n|] by done.
      apply IHlen.
      by move: EQ ⇒ /eqP; rewrite //= eqSS ⇒ /eqP <-.
    Qed.

We show that the size of a distances-sequence is one less than the size of the original sequence.
    Lemma size_of_seq_of_distances:
      ∀ (xs : seq nat),
        2 ≤ size xs →
        size xs = size (distances xs) + 1.
    Proof.
      intros xs.
      have EX: ∃ len, size xs ≤ len by (∃ (size xs)).
      move: EX ⇒ [len LE]; move: xs LE.
      induction len; intros × LE SIZE.
      { by move: LE; rewrite leqn0 size_eq0; move ⇒ /eqP EQ; subst. }
      { move: LE; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ| LE]; last by apply IHlen.
        destruct xs as [ | x1 xs]; first by inversion EQ.
        destruct xs as [ | x2 xs]; first by inversion SIZE.
        destruct xs as [ | x3 xs]; first by done.
        clear SIZE.
        have ->: size [:: x1, x2, x3 & xs] = size [:: x2, x3 & xs] + 1 by rewrite addn1.
        have ->: size (distances [:: x1, x2, x3 & xs]) = size (distances [:: x2, x3 & xs]) + 1 by rewrite addn1.
        apply/eqP; rewrite eqn_add2r; apply/eqP.
        apply IHlen; last by done.
        by move: EQ ⇒ /eqP; rewrite //= eqSS ⇒ /eqP <-.
      }
    Qed.

We prove that index_iota 0 n produces a sequence of numbers which are always one unit apart from each other.
    Lemma distances_of_iota_ε:
      ∀ n x, x \in distances (index_iota 0 n) → x = ε.
    Proof.
      intros n x IN; induction n.
      - by unfold index_iota, distances in IN.
      - destruct n; first by unfold distances, index_iota in IN.
        move: IN; rewrite -addn1 /index_iota subn0 iotaD add0n.
        rewrite distances_unfold_1app_last; last by rewrite size_iota.
        rewrite mem_cat ⇒ /orP [IN|IN].
        + by apply IHn; rewrite /index_iota subn0; simpl.
        + by move: IN;
            rewrite -addn1 iotaD /last0 last_cat add0n addn1 // subSnn in_cons;
            move ⇒ /orP [/eqP EQ|F]; subst.
    Qed.

  End Distances.

Properties of Distances of Non-Decreasing Sequence

In this section, we prove a few basic lemmas about distances of non-decreasing sequences.
  Section DistancesOfNonDecreasingSequence.

First we show that the max distance between elements of any nontrivial sequence (i.e. a sequence that contains at leas two distinct elements) is positive.
    Lemma max_distance_in_nontrivial_seq_is_positive:
      ∀ (xs : seq nat),
        nondecreasing_sequence xs →
        (∃ x y, x \in xs ∧ y \in xs ∧ x ≠ y) →
        0 < max0 (distances xs).
    Proof.
      move ⇒ xs SIZE [x [y [INx [INy NEQ]]]].
      move: INx INy ⇒ /nthP INx /nthP INy; specialize (INx 0); specialize (INy 0).
      move: INx INy ⇒ [indx SIZEx EQx] [indy SIZEy EQy].
      have L: ∀ (x y indx indy : nat),
          indx < size xs → indy < size xs →
          nondecreasing_sequence xs →
          x < y → xs [|indx|] = x → xs [|indy|] = y →
          0 < max0 (distances xs).
      { clear; intros x y indx indy SIZEx SIZEy SIZE LT EQx EQy.
        have LTind: indx < indy.
        { rewrite ltnNge; apply/negP; intros CONTR.
          subst x y; move: LT; rewrite ltnNge; move ⇒ /negP T; apply: T.
          by apply SIZE; apply/andP. }
        have EQ: ∃ Δ, indy = indx + Δ; [by ∃ (indy - indx); lia | move: EQ ⇒ [Δ EQ]; subst indy].
        have F: ∃ ind, indx ≤ ind < indx + Δ ∧ xs[|ind|] < xs[|ind.+1|].
        { subst x y; clear SIZEx SIZEy; revert xs indx LTind SIZE LT.
          induction Δ; intros; first by lia.
          destruct (posnP Δ) as [ZERO|POS].
          { by subst Δ; ∃ indx; split; [rewrite addn1; apply/andP | rewrite addn1 in LT]; auto. }
          have ALT: xs[|indx + Δ|] == xs[|indx + Δ.+1|] ∨ xs[|indx + Δ|] < xs[|indx + Δ.+1|].
          { apply/orP; rewrite -leq_eqVlt addnS.
            apply SIZE; apply/andP; split; first by done.
            rewrite ltnNge; apply/negP; intros CONTR.
            move: LT; rewrite ltnNge; move ⇒ /negP LT; apply: LT.
            by rewrite nth_default ?addnS. }
          move: ALT ⇒ [/eqP EQ|LT'].
          - edestruct (IHΔ) as [ind [B UP]]; eauto 5 using addn1, leq_add2l.
            ∃ ind; split; last by done.
            move: B ⇒ /andP [B1 B2]; apply/andP; split; first by done.
            by rewrite addnS ltnS ltnW.
          - ∃ (indx + Δ); split; last by rewrite -addnS.
            by apply/andP; split; [rewrite leq_addr | rewrite addnS]. }
        move: F ⇒ [ind [/andP [B1 B2] UP]].
        apply leq_trans with (xs [|ind.+1|] - xs [|ind|]).
        - by rewrite subn_gt0.
        - by apply distance_between_neighboring_elements_le_max_distance_in_seq.
      }
      move: NEQ ⇒ /eqP; rewrite neq_ltn; move ⇒ /orP [LT|LT].
      - by eapply L with (indx := indx) (x := x) (y := y); eauto.
      - by eapply L with (indx := indy) (indy := indx) (x := y) (y := x); eauto.
    Qed.

Given a non-decreasing sequence xs with length n, we show that the difference between the last element of xs and the last element of the distances-sequence of xs is equal to xs[n-2].
    Lemma last_seq_minus_last_distance_seq:
      ∀ (xs : seq nat),
        nondecreasing_sequence xs →
        last0 xs - last0 (distances xs) = xs [| (size xs).-2 |].
    Proof.
      unfold nondecreasing_sequence in ×.
      intros xs SIS.
      destruct xs as [ | x1 xs]; first by done.
      destruct xs as [ | x2 xs]; first by rewrite subn0.
      rewrite {2}/last0 -[in X in _ - X]nth_last function_of_distances_is_correct prednK; last by done.
      set [:: x1, x2 & xs] as X.
      rewrite /last0 -nth_last size_of_seq_of_distances; last by done.
      rewrite !addn1 -pred_Sn subKn; first by done.
      rewrite /X; apply SIS; apply/andP; split; first by simpl.
      by rewrite [in X in _ < X]size_of_seq_of_distances; first rewrite addn1.
    Qed.

The max element of the distances-sequence of a sequence xs is bounded by the last element of xs. Since all elements of xs are non-negative, they all lie within the interval 0, last xs.
    Lemma max_distance_in_seq_le_last_element_of_seq:
      ∀ (xs : seq nat),
        nondecreasing_sequence xs →
        max0 (distances xs) ≤ last0 xs.
    Proof.
      intros.
      have SIZE: size xs < 2 ∨ 2 ≤ size xs.
      { by destruct (size xs) as [ | n]; last destruct n; auto. }
      move: SIZE ⇒ [LT | SIZE2].
      { by destruct xs; last destruct xs. }
      apply leq_trans with (last0 xs - first0 xs); last by apply leq_subr.
      have F: ∀ xs c, (∀ x, x \in xs → x ≤ c) → max0 xs ≤ c.
      { clear; intros.
        induction xs; first by done.
        rewrite max0_cons geq_max; apply/andP; split.
        + by apply H; rewrite in_cons; apply/orP; left.
        + by apply IHxs; intros; apply H; rewrite in_cons; apply/orP; right.
      }
      apply: F ⇒ d IN.
      move: IN ⇒ /nthP T; specialize (T 0); move: T ⇒ [idx IN DIST].
      rewrite function_of_distances_is_correct in DIST.
      rewrite -DIST leq_sub //.
      { destruct (xs [|idx.+1|] == last0 xs) eqn:EQ.
        - by rewrite leq_eqVlt; apply/orP; left.
        - rewrite /last0 -nth_last. apply H.
          rewrite -(ltn_add2r 1) addn1 -size_of_seq_of_distances in IN; last by done.
          move: IN; rewrite leq_eqVlt; move ⇒ /orP [/eqP KK|KK].
          move: EQ; rewrite /last0 -nth_last -{1}KK -[_.+2.-1]pred_Sn; move ⇒ /eqP; by done.
          apply/andP; split; first rewrite -(ltn_add2r 1) !addn1 prednK //.
          + by apply ltn_trans with idx.+2.
          + by apply ltnW.
          + by rewrite prednK //; apply ltn_trans with idx.+2.
      }
      { destruct (first0 xs == xs [|idx|]) eqn:EQ.
        - by rewrite leq_eqVlt; apply/orP; left.
        - rewrite /first0 -nth0. apply H.
          rewrite -(ltn_add2r 1) addn1 -size_of_seq_of_distances in IN; last by done.
          destruct idx; first by move: EQ; rewrite /first0 -nth0; move ⇒ /eqP.
          apply/andP; split; first by done.
          by apply ltn_trans with idx.+2.
      }
    Qed.

Let xs be a non-decreasing sequence. We prove that distances of sequence [seq ρ <- index_iota 0 k.+1 | ρ \in xs] coincide with sequence [seq x <- distances xs | 0 < x]].
    Lemma distances_iota_filtered:
      ∀ xs k,
        (∀ x, x \in xs → x ≤ k) →
        nondecreasing_sequence xs →
        distances [seq ρ <- index_iota 0 k.+1 | ρ \in xs] = [seq x <- distances xs | 0 < x].
    Proof.
      intros xs.
      have EX: ∃ len, size xs ≤ len by (∃ (size xs)).
      destruct EX as [n BO].
      revert xs BO; induction n; intros ? Len k Bound NonDec.
      { move: Len; rewrite leqn0 size_eq0; move ⇒ /eqP T; subst.
        by rewrite filter_pred0.
      }
      { destruct xs as [ |x1 [ |x2 xs]].
        - by rewrite filter_pred0.
        - by rewrite index_iota_filter_singl; last (rewrite ltnS; apply Bound; rewrite in_cons; apply/orP; left).
        - have LEx1: x1 ≤ k. by apply Bound; rewrite in_cons eq_refl.
          have LEx2: x2 ≤ k. by apply Bound; rewrite !in_cons eq_refl orbT.
          have M1: ∀ y : nat, y \in x2 :: xs → x1 ≤ y. by apply nondecreasing_sequence_cons_min.
          have M2: ∀ x : nat, x \in x2 :: xs → x ≤ k. by intros; apply Bound; rewrite in_cons; apply/orP; right.
          have M3: ∀ y : nat, y \in xs → x2 ≤ y by apply nondecreasing_sequence_cons in NonDec; apply nondecreasing_sequence_cons_min.
          destruct (nondecreasing_sequence_2cons_leVeq _ _ _ NonDec).
          + subst; rewrite distances_unfold_2cons.
            replace ((@filter nat (fun x : nat ⇒ leq (S O) x) (@cons nat (subn x2 x2) (distances (@cons nat x2 xs)))))
              with ([seq x <- distances (x2 :: xs) | 0 < x]); last by rewrite subnn.
            rewrite range_filter_2cons.
              by apply IHn; eauto 2 using nondecreasing_sequence_cons.
          + have M4: ∀ y : nat, y \in x2 :: xs → x1 < y. by apply nondecreasing_sequence_cons_smin.
            rewrite distances_unfold_2cons.
            rewrite range_iota_filter_step // rem_lt_id //.
            replace ((@filter nat (fun x : nat ⇒ leq (S O) x) (@cons nat (subn x2 x1) (distances (@cons nat x2 xs)))))
              with (x2 - x1 :: [seq x <- distances (x2 :: xs) | 0 < x]); last first.
            { simpl; replace (0 < x2 - x1) with true; first by done.
                by apply/eqP; rewrite eq_sym; rewrite eqb_id subn_gt0. }
            rewrite -(IHn _ _ k) //; last eapply nondecreasing_sequence_cons; eauto 2.
            by rewrite {1}range_iota_filter_step // distances_unfold_2cons {1}range_iota_filter_step //.
      }
    Qed.

Let xs again be a non-decreasing sequence. We prove that distances of sequence undup xs coincide with sequence of positive distances of xs.
    Lemma distances_positive_undup:
      ∀ xs,
        nondecreasing_sequence xs →
        [seq d <- distances xs | 0 < d] = distances (undup xs).
    Proof.
      intros ? NonDec.
      rewrite -(distances_iota_filtered _ (max0 xs)); [ | by apply in_max0_le | by done].
      enough ([seq ρ <- index_iota 0 (max0 xs).+1 | ρ \in xs] = (undup xs)) as IN; first by rewrite IN.
      have EX: ∃ len, size xs ≤ len.
      { ∃ (size xs); now simpl. } destruct EX as [n BO].
      revert xs NonDec BO; induction n.
      - by intros xs _; rewrite leqn0 size_eq0 ⇒ /eqP ->; rewrite filter_pred0.
      - intros [ |x1 [ | x2 xs]].
        + by rewrite filter_pred0.
        + by rewrite index_iota_filter_singl ?L02 //; apply/andP; split; [ done | destruct x1].
        + intros NonDec Size.
          destruct (nondecreasing_sequence_2cons_leVeq _ _ _ NonDec) as [EQ|LT].
          × subst; rename x2 into x.
            rewrite nodup_sort_2cons_eq range_filter_2cons max0_2cons_eq.
            by apply IHn; [ eapply nondecreasing_sequence_cons; eauto | by done].
          × rewrite nodup_sort_2cons_lt // max0_2cons_le; last by rewrite ltnW.
            rewrite index_iota_filter_step; [ | | by apply nondecreasing_sequence_cons_min]; last first.
            { apply/andP; split; first by done.
              eapply leq_trans; first by eassumption.
              rewrite ltnW // ltnS; apply in_max0_le.
              by rewrite in_cons eq_refl.
            }
            rewrite rem_lt_id //; last by apply nondecreasing_sequence_cons_smin.
            by rewrite IHn //; eauto using nondecreasing_sequence_cons.
    Qed.


Consider two non-decreasing sequences xs and ys and assume that (1) first element of xs is at most the first element of ys and (2) distances-sequences of xs is dominated by distances-sequence of ys. Then xs is dominated by ys.
    Lemma domination_of_distances_implies_domination_of_seq:
      ∀ (xs ys : seq nat),
        first0 xs ≤ first0 ys →
        2 ≤ size xs →
        2 ≤ size ys →
        size xs = size ys →
        nondecreasing_sequence xs →
        nondecreasing_sequence ys →
        (∀ n, (distances xs)[|n|] ≤ (distances ys)[|n|]) →
        (∀ n, xs[|n|] ≤ ys[|n|]).
    Proof.
      intros xs ys.
      have EX: ∃ len, size xs ≤ len ∧ size ys ≤ len.
      { ∃ (maxn (size xs) (size ys)).
        by split; rewrite leq_max; apply/orP; [left | right].
      } move: EX ⇒ [len [LE1 LE2]].
      generalize dependent xs; generalize dependent ys.
      induction len.
      { by intros; move: LE1 LE2; rewrite !leqn0 !size_eq0 ⇒ /eqP → /eqP →. }
      { intros ? LycSIZE ? LxSIZE FLE Sxs Sys SIZEEQ STRxs STRys LE n.
        destruct xs as [ | x1 xs], ys as [ | y1 ys]; try by done.
        destruct xs as [ | x2 xs], ys as [ | y2 ys]; try by done.
        have F: x2 ≤ y2.
        { specialize (STRxs 0 1); feed STRxs; first by done.
          specialize (STRys 0 1); feed STRys; first by done.
          specialize (LE 0); simpl in LE, FLE.
          rewrite leqNgt; apply/negP; intros NEQ.
          move: LE; rewrite leqNgt; move ⇒ /negP LE; apply: LE.
          rewrite -(ltn_add2r x1) subnK // addnBAC // -(ltn_add2r y1) subnK.
          - by eapply leq_ltn_trans; [erewrite leq_add2l | erewrite ltn_add2r].
          - by apply leq_trans with y2; auto using leq_addr.
        }
        destruct xs as [ | x3 xs], ys as [ | y3 ys]; try by done.
        { by destruct n; [ | destruct n]. }
        destruct n; first by done.
        simpl; apply IHlen; try done.
        - by apply/eqP; rewrite -(eqn_add2r 1) !addn1; apply/eqP.
        - move ⇒ m1 m2 /andP [B1 B2].
          apply (STRxs m1.+1 m2.+1); apply/andP; split.
          + by rewrite ltnS.
          + by rewrite -(ltn_add2r 1) !addn1 in B2.
        - move ⇒ m1 m2 /andP [B1 B2].
          apply (STRys m1.+1 m2.+1); apply/andP; split.
          + by rewrite ltnS.
          + by rewrite -(ltn_add2r 1) !addn1 in B2.
        - by intros; specialize (LE n0.+1); simpl in LE.
      }
    Qed.

  End DistancesOfNonDecreasingSequence.

End NondecreasingSequence.