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    Module mathcomp.classical.set_interval

    From mathcomp Require Import all_ssreflect ssralg ssrnum interval.
    From mathcomp Require Import mathcomp_extra boolp classical_sets.
    From HB Require Import structures.
    From mathcomp Require Import functions.

    # Sets and Intervals This files contains lemmas about sets and intervals. ``` neitv i == the interval i is non-empty when the support type is a numFieldType, this is equivalent to (i.1 < i.2)%O (lemma neitvE) set_itv_infty_set0 == multirule to simplify empty intervals line_path a b t := (1 - t) * a + t * b, convexity operator over a numDomainType ndline_path == line_path a b with the constraint that a < b factor a b x := (x - a) / (b - a) set_itvE == multirule to turn intervals into inequalities disjoint_itv i j == intervals i and j are disjoint itv_is_ray i == i is either `]x,+oo[ or `]-oo,x[ itv_is_bd_open i == i is `]x,y[ itv_open_ends i == i has open endpoints, E.G. is one of the two above is_open_itv A == the set A can be written as an open interval open_itv_cover A == the set A can be covered by open intervals ```

    Set Implicit Arguments.
    Unset Strict Implicit.
    Unset Printing Implicit Defensive.
    Import Order.TTheory GRing.Theory Num.Def Num.Theory.

    Local Open Scope classical_set_scope.
    Local Open Scope ring_scope.

    definitions and lemmas to make a bridge between MathComp intervals and classical sets
    Section set_itv_porderType.
    Variables (d : Order.disp_t) (T : porderType d).
    Implicit Types (i j : interval T) (x y : T) (a : itv_bound T).

    Definition neitv i := [set` i] != set0.

    Lemma neitv_lt_bnd i : neitv i -> (i.1 < i.2)%O.
    Proof.
    case: i => a b; apply: contraNT => /= /itv_ge ab0.
    by apply/eqP; rewrite predeqE => t; split => //=; rewrite ab0.
    Qed.

    Lemma set_itvP i j : [set` i] = [set` j] :> set _ <-> i =i j.
    Proof.
    split => [ij x|ij]; first by have /(congr1 (@^~ x))/=/is_true_inj := ij.
    by rewrite predeqE => r /=; rewrite ij.
    Qed.

    Lemma subset_itvP i j : {subset i <= j} <-> [set` i] `<=` [set` j].
    Proof.
    by []. Qed.

    Lemma in1_subset_itv (P : T -> Prop) i j :
      [set` j] `<=` [set` i] -> {in i, forall x, P x} -> {in j, forall x, P x}.
    Proof.
    by move=> /subset_itvP ji iP z zB; apply: iP; exact: ji. Qed.

    Lemma subset_itvW x y z u b0 b1 :
        (x <= y)%O -> (z <= u)%O ->
      `]y, z[ `<=` [set` Interval (BSide b0 x) (BSide b1 u)].
    Proof.
    move=> xy zu; apply: (@subset_trans _ `]x, u[%classic).
      move=> x0/=; rewrite 2!in_itv/= => /andP[].
      by move=> /(le_lt_trans xy) ->/= /lt_le_trans; exact.
    by move: b0 b1 => [] [] /=; [exact: subset_itv_oo_co|exact: subset_itv_oo_cc|
      exact: subset_refl|exact: subset_itv_oo_oc].
    Qed.

    Lemma subset_itvl (a b c : itv_bound T) : (b <= c)%O ->
      [set` Interval a b] `<=` [set` Interval a c].
    Proof.
    case: c => [[|] c bc x/=|[//|_] x/=].
    - rewrite !in_itv/= => /andP[->/=].
      case: b bc => [[|]/=|[|]//] b bc.
        by move=> /lt_le_trans; exact.
      by move=> /le_lt_trans; exact.
    - rewrite !in_itv/= => /andP[->/=].
      case: b bc => [[|]/=|[|]//] b bc.
        by move=> /ltW /le_trans; apply.
      by move=> /le_trans; apply.
    - by move: x; rewrite le_ninfty => /eqP ->.
    - by rewrite !in_itv/=; case: a => [[|]/=|[|]//] a /andP[->].
    Qed.

    Lemma subset_itvr (a b c : itv_bound T) : (c <= a)%O ->
      [set` Interval a b] `<=` [set` Interval c b].
    Proof.
    move=> ac x/=; rewrite !in_itv/= => /andP[ax ->]; rewrite andbT.
    move: c a ax ac => [[|] c [[|]/= a ax|[|]//=]|[//|]]; rewrite ?bnd_simp.
    - by move=> /le_trans; exact.
    - by move=> /le_trans; apply; exact/ltW.
    - by move=> /lt_le_trans; exact.
    - by move=> /le_lt_trans; exact.
    - by move=> [[|]|[|]//].
    Qed.

    Lemma subset_itvScc (a b : itv_bound T) (c e : T) :
        (BLeft c <= a)%O -> (b <= BRight e)%O ->
      [set` Interval a b] `<=` [set` `[c, e]].
    Proof.
    move=> ca be z/=; rewrite !in_itv/==> /andP[az zb].
    case: a ca az => [[|]/=|[|]//] a; rewrite bnd_simp => ca az.
    - rewrite (le_trans ca az)/=.
      move: b be zb => [[|]/= b|[|]//]; rewrite bnd_simp => be.
        by move=> /ltW/le_trans; exact.
      by move=> /le_trans; exact.
    - rewrite (le_trans ca (ltW az))/=.
      move: b be zb => [[|]/= b|[|]//]; rewrite bnd_simp => be.
        by move=> /ltW/le_trans; exact.
      by move=> /le_trans; exact.
    Qed.

    Lemma subset_itvSoo (a b : itv_bound T) (c e : T) :
        (BLeft c < a)%O -> (b < BRight e)%O ->
      [set` Interval a b] `<=` [set` `]c, e[].
    Proof.
    move=> ca be z/=; rewrite !in_itv/= => /andP[az zb].
    case: a ca az => [[|]/=|[|]//] a; rewrite bnd_simp => ca az.
      rewrite (lt_le_trans ca az)/=.
      move: b be zb => [[|]/= b|[|]//]; rewrite bnd_simp => be.
        by move=> /lt_le_trans; exact.
      by move=> /le_lt_trans; exact.
    rewrite (le_lt_trans ca az)/=.
    move: b be zb => [[|]/= b|[|]//]; rewrite bnd_simp => be.
      by move=> /lt_le_trans; exact.
    by move=> /le_lt_trans; exact.
    Qed.

    Lemma interval_set1 x : `[x, x]%classic = [set x] :> set T.
    Proof.
    apply/seteqP; split => [y/=|y <-]; last by rewrite /= in_itv/= lexx.
    by rewrite in_itv/= => /andP[yx xy]; apply/eqP; rewrite eq_le yx xy.
    Qed.

    Lemma set_itvoo x y : `]x, y[%classic = [set z | (x < z < y)%O].
    Proof.
    by []. Qed.

    Lemma set_itvco x y : `[x, y[%classic = [set z | (x <= z < y)%O].
    Proof.
    by []. Qed.

    Lemma set_itvcc x y : `[x, y]%classic = [set z | (x <= z <= y)%O].
    Proof.
    by []. Qed.

    Lemma set_itvoc x y : `]x, y]%classic = [set z | (x < z <= y)%O].
    Proof.
    by []. Qed.

    Lemma set_itv1 x : `[x, x]%classic = [set x].
    Proof.
    by apply/seteqP; split=> y /=; rewrite itvxx ?inE (rwP eqP). Qed.

    Lemma set_itvoo0 x : `]x, x[%classic = set0.
    Proof.
    by rewrite -subset0 => y /=; rewrite itv_ge//= bnd_simp ltxx. Qed.

    Lemma set_itvoc0 x : `]x, x]%classic = set0.
    Proof.
    by rewrite -subset0 => y /=; rewrite itv_ge//= bnd_simp ltxx. Qed.

    Lemma set_itvco0 x : `[x, x[%classic = set0.
    Proof.
    by rewrite -subset0 => y /=; rewrite itv_ge//= bnd_simp ltxx. Qed.

    Lemma set_itv_infty_infty : `]-oo, +oo[%classic = @setT T.
    Proof.
    by rewrite predeqE. Qed.

    Lemma set_itv_o_infty x : `]x, +oo[%classic = [set z | (x < z)%O].
    Proof.
    by rewrite predeqE => r /=; rewrite in_itv andbT. Qed.

    Lemma set_itv_c_infty x : `[x, +oo[%classic = [set z | (x <= z)%O].
    Proof.
    by rewrite predeqE /mkset => r; rewrite in_itv andbT. Qed.

    Lemma set_itv_infty_o x : `]-oo, x[%classic = [set z | (z < x)%O].
    Proof.
    by rewrite predeqE /mkset => r; rewrite in_itv. Qed.

    Lemma set_itv_infty_c x : `]-oo, x]%classic = [set z | (z <= x)%O].
    Proof.
    by rewrite predeqE /mkset => r; rewrite in_itv. Qed.

    Lemma set_itv_pinfty_bnd a : [set` Interval +oo%O a] = set0.
    Proof.
    by apply/eqP/negPn/negP => /neitv_lt_bnd. Qed.

    Lemma set_itv_bnd_ninfty a : [set` Interval a -oo%O] = set0.
    Proof.
    by apply/eqP/negPn/negP => /neitv_lt_bnd /=; case: a => [[]a|[]]. Qed.

    Definition set_itv_infty_set0 := (set_itv_bnd_ninfty, set_itv_pinfty_bnd).

    Definition set_itvE := (set_itv1, set_itvoo0, set_itvoc0, set_itvco0, set_itvoo,
      set_itvcc, set_itvoc, set_itvco, set_itv_infty_infty, set_itv_o_infty,
      set_itv_c_infty, set_itv_infty_o, set_itv_infty_c, set_itv_infty_set0).

    Lemma set_itvxx a : [set` Interval a a] = set0.
    Proof.
    by move: a => [[|] a |[|]]; rewrite !set_itvE. Qed.

    Lemma setUitv1 a x : (a <= BLeft x)%O ->
      [set` Interval a (BLeft x)] `|` [set x] = [set` Interval a (BRight x)].
    Proof.
    move=> ax; apply/predeqP => z /=; rewrite itv_splitU1// [in X in _ <-> X]inE.
    by rewrite (rwP eqP) (rwP orP) orbC.
    Qed.

    Lemma setU1itv a x : (BRight x <= a)%O ->
      x |` [set` Interval (BRight x) a] = [set` Interval (BLeft x) a].
    Proof.
    move=> ax; apply/predeqP => z /=; rewrite itv_split1U// [in X in _ <-> X]inE.
    by rewrite (rwP eqP) (rwP orP) orbC.
    Qed.

    Lemma setDitv1r a x :
      [set` Interval a (BRight x)] `\ x = [set` Interval a (BLeft x)].
    Proof.
    apply/seteqP; split => [z|z] /=; rewrite !in_itv/=.
      by move=> [/andP[-> /= zx] /eqP xz]; rewrite lt_neqAle xz.
    by rewrite lt_neqAle => /andP[-> /andP[/eqP ? ->]].
    Qed.

    Lemma setDitv1l a x :
      [set` Interval (BLeft x) a] `\ x = [set` Interval (BRight x) a].
    Proof.
    apply/seteqP; split => [z|z] /=; rewrite !in_itv/=.
      move=> [/andP[xz ->]]; rewrite andbT => /eqP.
      by rewrite lt_neqAle eq_sym => ->.
    move=> /andP[]; rewrite lt_neqAle => /andP[xz zx ->].
    by rewrite andbT; split => //; exact/nesym/eqP.
    Qed.

    End set_itv_porderType.
    Arguments neitv {d T} _.
    #[deprecated(since="mathcomp-analysis 1.4.0", note="renamed to subset_itvScc")]
    Notation subset_itvS := subset_itvScc (only parsing).

    Section set_itv_orderType.
    Variables (d : Order.disp_t) (T : orderType d).
    Implicit Types a x y : itv_bound T.

    Lemma itv_bndbnd_setU a x y : (a <= x)%O -> (x <= y)%O ->
      ([set` Interval a y] = [set` Interval a x] `|` [set` Interval x y])%classic.
    Proof.
    rewrite le_eqVlt => /predU1P[<-{x} ay|]; first by rewrite set_itvxx set0U.
    move=> /[swap].
    rewrite le_eqVlt => /predU1P[-> ay|]; first by rewrite set_itvxx setU0.
    move: y => [yb y/=|[|]]; last 2 first.
      by case: x => [|[|]].
      move=> _ ax; apply/seteqP; split => [z|z] /=.
        rewrite !in_itv/= !andbT => -> /=; apply/orP.
        by move: x => [[|] x/=|[|]//] in ax *; rewrite leNgt ?(orbN,orNb).
      rewrite !in_itv/= !andbT => -[/andP[]|]//.
      move: x => [[|] x/=|[|]//] in ax *; move: a => [[|] a/=|[|]//] in ax * => //.
      - by apply/le_trans; exact/ltW.
      - exact/lt_le_trans.
      - by move=> /(le_lt_trans ax) /ltW.
      - exact/lt_trans.
    move=> xy ax; apply/seteqP; split => [z|z] /=.
      rewrite !in_itv /= => /andP[].
      move: a ax => [b t /=|[]//= oox _].
        move=> tx -> zxy /=; rewrite zxy andbT/=; apply/orP.
        by case: x xy tx => [[|] x/=|[|]//] xy tx; rewrite leNgt ?(orbN,orNb).
      move=> ->; rewrite andbT; apply/orP.
      by move: x => [[|] x/=|[|]//] in oox xy *; rewrite leNgt ?(orbN,orNb).
    rewrite !in_itv/=.
    move: a ax => [b t /= tx| [/= oox|/= oox]].
    - move=> [/andP[-> zx]|].
        move: x => [[|] x|[|]//]/= in xy tx zx *.
          case: yb => /= in xy *.
            by rewrite (lt_trans zx _).
          by rewrite (ltW (lt_le_trans zx _)).
        rewrite bnd_simp in xy.
        case: yb => /=.
          by rewrite (le_lt_trans zx _).
        by rewrite (ltW (le_lt_trans zx _)).
      move: x => [[|] x|[|]//]/= in xy tx *; rewrite bnd_simp in xy tx.
      + move=> /andP[xz ->]; rewrite andbT.
        case: b => /=.
          by rewrite (le_trans _ xz)// ltW.
        by rewrite (lt_le_trans tx).
      move=> /andP[xz ->]; rewrite andbT.
      case: b tx => /= tx; rewrite bnd_simp in tx.
        by rewrite ltW// (le_lt_trans _ xz).
      by rewrite (lt_trans tx).
    - move: x => [[|] x|[|]//]/= in xy oox *; move=> [|].
      + case: yb => /= in xy *.
          by move=> /lt_trans; exact.
        rewrite bnd_simp in xy.
        by move=> /lt_le_trans => /(_ _ xy)/ltW.
      + by move=> /andP[].
      + case: yb => /= in xy *.
          by move=> /le_lt_trans; apply.
        by move=> /le_trans; apply; exact/ltW.
      + by move=> /andP[].
    - by move: x => [[|] x|[|]//]/= in xy oox *.
    Qed.

    End set_itv_orderType.

    Lemma set_itv_ge disp [T : porderType disp] [b1 b2 : itv_bound T] :
      ~~ (b1 < b2)%O -> [set` Interval b1 b2] = set0.
    Proof.
    by move=> Nb12; rewrite -subset0 => x /=; rewrite itv_ge. Qed.

    Section set_itv_latticeType.
    Variables (d : Order.disp_t) (T : latticeType d).
    Implicit Types (i j : interval T) (x y : T) (a : itv_bound T).

    Lemma set_itvI i j : [set` (i `&` j)%O] = [set` i] `&` [set` j].
    Proof.
    by apply/seteqP; split=> x /=; rewrite in_itvI (rwP andP). Qed.

    End set_itv_latticeType.

    Section set_itv_numFieldType.
    Variable R : numFieldType.
    Implicit Types i : interval R.

    Lemma neitvE i : neitv i = (i.1 < i.2)%O.
    Proof.
    apply/idP/idP; first exact: neitv_lt_bnd.
    by move=> /mem_miditv ii; apply/set0P; exists (miditv i).
    Qed.

    Lemma neitvP i : reflect (i.1 < i.2)%O (neitv i).
    Proof.
    by apply: (iffP idP); rewrite -neitvE. Qed.

    End set_itv_numFieldType.

    Lemma setitv0 (R : realDomainType) : [set` (\bot%O : interval R)] = set0.
    Proof.
    by rewrite predeqE. Qed.

    Section interval_has_bound.
    Variable R : numDomainType.

    Lemma has_lbound_itv (x : R) b (a : itv_bound R) :
      has_lbound [set` Interval (BSide b x) a].
    Proof.
    by case: b; exists x => r /andP[]; rewrite bnd_simp // => /ltW. Qed.

    Lemma has_ubound_itv (x : R) b (a : itv_bound R) :
      has_ubound [set` Interval a (BSide b x)].
    Proof.
    by case: b; exists x => r /andP[]; rewrite bnd_simp // => _ /ltW. Qed.

    End interval_has_bound.

    Section subr_image.
    Variable R : numDomainType.
    Implicit Types E : set R.
    Implicit Types x : R.

    Lemma setNK : involutive (fun E => -%R @` E).
    Proof.
    move=> A; rewrite image_comp (_ : _ \o _ = id) ?image_id//.
    by rewrite funeqE => r /=; rewrite opprK.
    Qed.

    Lemma lb_ubN E x : lbound E x <-> ubound (-%R @` E) (- x).
    Proof.
    split=> [/lbP xlbE|/ubP xlbE].
    by move=> _ [z Ez <-]; rewrite lerNr opprK; apply xlbE.
    by move=> y Ey; rewrite -(opprK x) lerNl; apply xlbE; exists y.
    Qed.

    Lemma ub_lbN E x : ubound E x <-> lbound (-%R @` E) (- x).
    Proof.
    split=> [? | /lb_ubN]; first by apply/lb_ubN; rewrite opprK setNK.
    by rewrite opprK setNK.
    Qed.

    Lemma memNE E x : E x = (-%R @` E) (- x).
    Proof.
    by rewrite image_inj //; exact: oppr_inj. Qed.

    Lemma nonemptyN E : nonempty (-%R @` E) <-> nonempty E.
    Proof.
    split=> [[x ENx]|[x Ex]]; exists (- x); last by rewrite -memNE.
    by rewrite memNE opprK.
    Qed.

    Lemma opp_set_eq0 E : (-%R @` E) = set0 <-> E = set0.
    Proof.
    by split; apply: contraPP => /eqP/set0P/nonemptyN/set0P/eqP. Qed.

    Lemma has_lb_ubN E : has_lbound E <-> has_ubound (-%R @` E).
    Proof.
    by split=> [[x /lb_ubN] | [x /ub_lbN]]; [|rewrite setNK]; exists (- x).
    Qed.

    End subr_image.

    Section interval_hasNbound.
    Variable R : realDomainType.
    Implicit Types E : set R.
    Implicit Types x : R.

    Lemma has_ubPn {E} : ~ has_ubound E <-> (forall x, exists2 y, E y & x < y).
    Proof.
    split; last first.
      move=> h [x] /ubP hle; case/(_ x): h => y /hle.
      by rewrite leNgt => /negbTE ->.
    move/forallNP => h x; have {h} := h x.
    move=> /ubP /existsNP => -[y /not_implyP[Ey /negP]].
    by rewrite -ltNge => ltx; exists y.
    Qed.

    Lemma has_lbPn E : ~ has_lbound E <-> (forall x, exists2 y, E y & y < x).
    Proof.
    split=> [/has_lb_ubN /has_ubPn NEnub x|Enlb /has_lb_ubN].
      have [y ENy ltxy] := NEnub (- x); exists (- y); rewrite 1?ltrNl //.
      by case: ENy => z Ez <-; rewrite opprK.
    apply/has_ubPn => x; have [y Ey ltyx] := Enlb (- x).
    exists (- y); last by rewrite ltrNr.
    by exists y => //; rewrite opprK.
    Qed.

    Lemma hasNlbound_itv (a : itv_bound R) : a != -oo%O ->
      ~ has_lbound [set` Interval -oo%O a].
    Proof.
    move: a => [b r|[|]] _ //.
      suff: ~ has_lbound `]-oo, r[%classic.
        by case: b => //; apply/contra_not/subset_has_lbound => x /ltW.
      apply/has_lbPn => x; exists (minr (r - 1) (x - 1)).
        by rewrite !set_itvE/= gt_min ltrBlDr ltrDl ltr01.
      by rewrite gt_min orbC ltrBlDr ltrDl ltr01.
    case=> r /(_ (r - 1)) /=; rewrite in_itv /= => /(_ erefl).
    by apply/negP; rewrite -ltNge ltrBlDr ltrDl.
    Qed.

    Lemma hasNubound_itv (a : itv_bound R) : a != +oo%O ->
      ~ has_ubound [set` Interval a +oo%O].
    Proof.
    move: a => [b r|[|]] _ //.
      suff: ~ has_ubound `]r, +oo[%classic.
        case: b => //; apply/contra_not/subset_has_ubound => x.
        by rewrite !set_itvE => /ltW.
      apply/has_ubPn => x; rewrite !set_itvE; exists (maxr (r + 1) (x + 1));
      by rewrite ?in_itv /= ?andbT lt_max ltrDl ltr01 // orbT.
    case=> r /(_ (r + 1)) /=; rewrite in_itv /= => /(_ erefl).
    by apply/negP; rewrite -ltNge ltrDl.
    Qed.

    End interval_hasNbound.

    #[global] Hint Extern 0 (has_lbound _) => solve[apply: has_lbound_itv] : core.
    #[global] Hint Extern 0 (has_ubound _) => solve[apply: has_ubound_itv] : core.
    #[global]
    Hint Extern 0 (~ has_lbound _) => solve[by apply: hasNlbound_itv] : core.
    #[global]
    Hint Extern 0 (~ has_ubound _) => solve[by apply: hasNubound_itv] : core.

    Lemma opp_itv_bnd_infty (R : numDomainType) (x : R) b :
      -%R @` [set` Interval (BSide b x) +oo%O] =
      [set` Interval -oo%O (BSide (negb b) (- x))].
    Proof.
    rewrite predeqE => /= r; split=> [[y xy <-]|xr].
      by case: b xy; rewrite !in_itv/= andbT (lerN2, ltrN2).
    exists (- r); rewrite ?opprK //.
    by case: b xr; rewrite !in_itv/= andbT (lerNr, ltrNr).
    Qed.

    Lemma opp_itv_infty_bnd (R : numDomainType) (x : R) b :
      -%R @` [set` Interval -oo%O (BSide b x)] =
      [set` Interval (BSide (negb b) (- x)) +oo%O].
    Proof.
    rewrite predeqE => /= r; split=> [[y xy <-]|xr].
      by case: b xy; rewrite !in_itv/= andbT (lerN2, ltrN2).
    exists (- r); rewrite ?opprK //.
    by case: b xr; rewrite !in_itv/= andbT (lerNl, ltrNl).
    Qed.

    Lemma opp_itv_bnd_bnd (R : numDomainType) a b (x y : R) :
      -%R @` [set` Interval (BSide a x) (BSide b y)] =
      [set` Interval (BSide (~~ b) (- y)) (BSide (~~ a) (- x))].
    Proof.
    rewrite predeqE => /= r; split => [[{}r + <-]|].
      by rewrite !in_itv/= 2!lteifN2 negbK andbC.
    rewrite in_itv/= negbK => yrab.
    by exists (- r); rewrite ?opprK// !in_itv lteifNr andbC lteifNl.
    Qed.

    Lemma opp_itvoo (R : numDomainType) (x y : R) :
      -%R @` `]x, y[%classic = `](- y), (- x)[%classic.
    Proof.
    rewrite predeqE => /= r; split => [[{}r + <-]|].
      by rewrite !in_itv/= !ltrN2 andbC.
    by exists (- r); rewrite ?opprK// !in_itv/= ltrNl ltrNr andbC.
    Qed.

    lemmas between itv and set-theoretic operations
    Section set_itv_porderType.
    Variables (d : Order.disp_t) (T : orderType d).
    Implicit Types (a : itv_bound T) (x y : T) (i j : interval T) (b : bool).

    Lemma setCitvl a : ~` [set` Interval -oo%O a] = [set` Interval a +oo%O].
    Proof.
    by apply/predeqP => y /=; rewrite -predC_itvl (rwP negP). Qed.

    Lemma setCitvr a : ~` [set` Interval a +oo%O] = [set` Interval -oo%O a].
    Proof.
    by rewrite -setCitvl setCK. Qed.

    Lemma set_itv_splitI i : [set` i] = [set` Interval i.1 +oo%O] `&` [set` Interval -oo%O i.2].
    Proof.
    case: i => [a a']; apply/predeqP=> x/=.
    by rewrite [in X in X <-> _]itv_splitI (rwP andP).
    Qed.

    Lemma setCitv i :
      ~` [set` i] = [set` Interval -oo%O i.1] `|` [set` Interval i.2 +oo%O].
    Proof.
    by apply/predeqP => x /=; rewrite (rwP orP) (rwP negP) [x \notin i]predC_itv.
    Qed.

    Lemma set_itv_splitD i :
      [set` i] = [set` Interval i.1 +oo%O] `\` [set` Interval i.2 +oo%O].
    Proof.
    by rewrite set_itv_splitI/= setDE setCitvr. Qed.

    End set_itv_porderType.

    Section line_path_factor_numDomainType.
    Variable R : numDomainType.
    Implicit Types (a b t r : R) (A : set R).

    Lemma mem_1B_itvcc t : (1 - t \in `[0, 1]) = (t \in `[0, 1]).
    Proof.
    by rewrite !in_itv/= subr_ge0 gerDl oppr_le0 andbC. Qed.

    Definition line_path a b t : R := (1 - t) * a + t * b.

    Lemma line_path_id : line_path 0 1 = id.
    Proof.
    by apply/funext => t; rewrite /line_path mulr0 add0r mulr1. Qed.

    Lemma line_pathEl a b t : line_path a b t = t * (b - a) + a.
    Proof.
    by rewrite /line_path mulrBl mul1r mulrBr addrAC [RHS]addrC addrA. Qed.

    Lemma line_pathEr a b t : line_path a b t = (1 - t) * (a - b) + b.
    Proof.
    rewrite /line_path mulrBr -addrA; congr (_ + _).
    by rewrite mulrBl opprB mul1r addrNK.
    Qed.

    Lemma line_path10 t : line_path 1 0 t = 1 - t.
    Proof.
    by rewrite /line_path mulr0 addr0 mulr1. Qed.

    Lemma line_path0 a b : line_path a b 0 = a.
    Proof.
    by rewrite /line_path subr0 mul1r mul0r addr0. Qed.

    Lemma line_path1 a b : line_path a b 1 = b.
    Proof.
    by rewrite /line_path subrr mul0r add0r mul1r. Qed.

    Lemma line_path_sym a b t : line_path a b t = line_path b a (1 - t).
    Proof.
    by rewrite /line_path opprB addrCA subrr addr0 addrC. Qed.

    Lemma line_path_flat a : line_path a a = cst a.
    Proof.
    by apply/funext => t; rewrite line_pathEl subrr mulr0 add0r. Qed.

    Lemma leW_line_path a b : a <= b -> {homo line_path a b : x y / x <= y}.
    Proof.
    by move=> ? ? ? ?; rewrite !line_pathEl lerD ?ler_wpM2r// subr_ge0.
    Qed.

    Definition factor a b x := (x - a) / (b - a).

    Lemma leW_factor a b : a <= b -> {homo factor a b : x y / x <= y}.
    Proof.
    by move=> ? ? ? ?; rewrite /factor ler_wpM2r ?lerD// invr_ge0 subr_ge0.
    Qed.

    Lemma factor_flat a : factor a a = cst 0.
    Proof.
    by apply/funext => x; rewrite /factor subrr invr0 mulr0. Qed.

    Lemma factorl a b : factor a b a = 0.
    Proof.
    by rewrite /factor subrr mul0r. Qed.

    Definition ndline_path a b of a < b := line_path a b.

    Lemma ndline_pathE a b (ab : a < b) : ndline_path ab = line_path a b.
    Proof.
    by []. Qed.

    End line_path_factor_numDomainType.

    Section line_path_factor_numFieldType.
    Variable R : numFieldType.
    Implicit Types (a b t r : R) (A : set R).

    Lemma factorr a b : a != b -> factor a b b = 1.
    Proof.
    by move=> Nab; rewrite /factor divff// subr_eq0 eq_sym. Qed.

    Lemma factorK a b : a != b -> cancel (factor a b) (line_path a b).
    Proof.
    by move=> ? x; rewrite line_pathEl mulfVK ?addrNK// subr_eq0 eq_sym. Qed.

    Lemma line_pathK a b : a != b -> cancel (line_path a b) (factor a b).
    Proof.
    by move=> ? x; rewrite /factor line_pathEl addrK mulfK// subr_eq0 eq_sym.
    Qed.

    Lemma line_path_inj a b : a != b -> injective (line_path a b).
    Proof.
    by move/line_pathK/can_inj. Qed.

    Lemma factor_inj a b : a != b -> injective (factor a b).
    Proof.
    by move/factorK/can_inj. Qed.

    Lemma line_path_bij a b : a != b -> bijective (line_path a b).
    Proof.
    by move=> ab; apply: Bijective (line_pathK ab) (factorK ab). Qed.

    Lemma factor_bij a b : a != b -> bijective (factor a b).
    Proof.
    by move=> ab; apply: Bijective (factorK ab) (line_pathK ab). Qed.

    Lemma le_line_path a b : a < b -> {mono line_path a b : x y / x <= y}.
    Proof.
    move=> ltab; have leab := ltW ltab.
    apply: homo_mono (line_pathK _) (leW_factor _) (leW_line_path _) => //.
    by rewrite lt_eqF.
    Qed.

    Lemma le_factor a b : a < b -> {mono factor a b : x y / x <= y}.
    Proof.
    move=> ltab; have leab := ltW ltab.
    apply: homo_mono (factorK _) (leW_line_path _) (leW_factor _) => //.
    by rewrite lt_eqF.
    Qed.

    Lemma lt_line_path a b : a < b -> {mono line_path a b : x y / x < y}.
    Proof.
    by move/le_line_path/leW_mono. Qed.

    Lemma lt_factor a b : a < b -> {mono factor a b : x y / x < y}.
    Proof.
    by move/le_factor/leW_mono. Qed.

    Let ltNeq a b : a < b -> a != b
    Proof.
    by move=> /lt_eqF->. Qed.

    HB.instance Definition _ a b (ab : a < b) :=
      @Can2.Build _ _ setT setT (ndline_path ab) (factor a b)
        (fun _ _ => I) (fun _ _ => I)
        (in1W (line_pathK (ltNeq ab))) (in1W (factorK (ltNeq ab))).

    Lemma line_path_itv_bij ba bb a b : a < b ->
      set_bij [set` Interval (BSide ba 0) (BSide bb 1)]
              [set` Interval (BSide ba a) (BSide bb b)] (line_path a b).
    Proof.
    move=> ltab; rewrite -ndline_pathE.
    apply: bij_subr => //=; rewrite setTI ?ndline_pathE.
    apply/predeqP => t /=; rewrite !in_itv/= {1}line_pathEl line_pathEr.
    rewrite -lteifBlDr subrr -lteif_pdivrMr ?subr_gt0// mul0r.
    rewrite -lteifBrDr subrr -lteif_ndivrMr ?subr_lt0// mul0r.
    by rewrite lteifBrDl addr0.
    Qed.

    Lemma factor_itv_bij ba bb a b : a < b ->
      set_bij [set` Interval (BSide ba a) (BSide bb b)]
              [set` Interval (BSide ba 0) (BSide bb 1)] (factor a b).
    Proof.
    move=> ltab; have -> : factor a b = (ndline_path ltab)^-1%FUN by [].
    by apply/splitbij_sub_sym => //; apply: line_path_itv_bij.
    Qed.

    Lemma mem_line_path_itv ba bb a b : a < b ->
      set_fun [set` Interval (BSide ba 0) (BSide bb 1)]
              [set` Interval (BSide ba a) (BSide bb b)] (line_path a b).
    Proof.
    by case/(line_path_itv_bij ba bb). Qed.

    Lemma mem_line_path_itvcc a b : a <= b -> set_fun `[0, 1] `[a, b] (line_path a b).
    Proof.
    rewrite le_eqVlt => /predU1P[<-|]; first by rewrite set_itv1 line_path_flat.
    by move=> lt_ab; case: (line_path_itv_bij true false lt_ab).
    Qed.

    Lemma range_line_path ba bb a b : a < b ->
       line_path a b @` [set` Interval (BSide ba 0) (BSide bb 1)] =
                   [set` Interval (BSide ba a) (BSide bb b)].
    Proof.
    by move=> /(line_path_itv_bij ba bb)/Pbij[f ->]; rewrite image_eq. Qed.

    Lemma range_factor ba bb a b : a < b ->
       factor a b @` [set` Interval (BSide ba a) (BSide bb b)] =
                     [set` Interval (BSide ba 0) (BSide bb 1)].
    Proof.
    by move=> /(factor_itv_bij ba bb)/Pbij[f ->]; rewrite image_eq. Qed.

    Lemma onem_factor a b x : a != b -> `1-(factor a b x) = factor b a x.
    Proof.
    rewrite eq_sym -subr_eq0 => ab; rewrite /onem /factor -(divff ab) -mulrBl.
    by rewrite opprB addrA subrK -mulrNN opprB -invrN opprB.
    Qed.

    End line_path_factor_numFieldType.

    Lemma mem_factor_itv (R : realFieldType) ba bb (a b : R) :
      set_fun [set` Interval (BSide ba a) (BSide bb b)]
              [set` Interval (BSide ba 0) (BSide bb 1)] (factor a b).
    Proof.
    have [|leba] := ltP a b; first by case/(factor_itv_bij ba bb).
    move=> x /=; have [|/itv_ge->//] := (boolP (BSide ba a < BSide bb b)%O).
    rewrite lteBSide; case: ba bb => [] []//=; rewrite ?le_gtF//.
    by case: ltgtP leba => // -> _ _ _; rewrite factor_flat in_itv/= lexx ler01.
    Qed.

    Lemma neitv_bnd1 (R : numFieldType) (s : seq (interval R)) :
      all neitv s -> forall i, i \in s -> i.1 != +oo%O.
    Proof.
    move=> /allP sne [a b] si /=; apply/negP => /eqP boo; move: si.
    by rewrite boo => /sne /negP; apply; rewrite set_itv_infty_set0.
    Qed.

    Lemma neitv_bnd2 (R : numFieldType) (s : seq (interval R)) :
      all neitv s -> forall i, i \in s -> i.2 != -oo%O.
    Proof.
    move=> /allP sne [a b] si /=; apply/negP => /eqP boo; move: si.
    by rewrite boo => /sne /negP; apply; rewrite set_itv_infty_set0.
    Qed.

    Lemma trivIset_set_itv_nth (R : numDomainType) def (s : seq (interval R))
      (D : set nat) : [set` def] = set0 ->
      trivIset D (fun i => [set` nth def s i]) <->
        trivIset D (fun i => nth set0 [seq [set` j] | j <- s] i).
    Proof.
    move=> def0; split=> /trivIsetP ss; apply/trivIsetP => i j Di Dj ij.
    - have [si|si] := ltP i (size s); last first.
        by rewrite (nth_default set0) ?size_map// set0I.
      have [sj|sj] := ltP j (size s); last first.
        by rewrite setIC (nth_default set0) ?size_map// set0I.
      by rewrite (nth_map def) // (nth_map def) // ss.
    - have [?|h] := ltP i (size s); last by rewrite (nth_default def h) def0 set0I.
      have [?|h] := ltP j (size s); last by rewrite (nth_default def h) def0 setI0.
      by have := ss _ _ Di Dj ij; rewrite (nth_map def) // (nth_map def).
    Qed.
    Arguments trivIset_set_itv_nth {R} _ {s}.

    Section disjoint_itv.
    Context {R : numDomainType}.

    Definition disjoint_itv : rel (interval R) :=
      fun a b => [disjoint [set` a] & [set` b]].

    Lemma disjoint_itvxx (i : interval R) : neitv i -> ~~ disjoint_itv i i.
    Proof.
    by move=> i0; rewrite /disjoint_itv/= /disj_set /= setIid. Qed.

    Lemma lt_disjoint (i j : interval R) :
      (forall x y, x \in i -> y \in j -> x < y) -> disjoint_itv i j.
    Proof.
    move=> ij; apply/eqP; rewrite predeqE => x; split => // -[xi xj].
    by have := ij _ _ xi xj; rewrite ltxx.
    Qed.

    End disjoint_itv.

    Lemma disjoint_neitv {R : realFieldType} (i j : interval R) :
      disjoint_itv i j <-> ~~ neitv (itv_meet i j).
    Proof.
    case: i j => [a b] [c d]; rewrite /disjoint_itv/disj_set /= -set_itvI.
    by split => [/negPn//|?]; apply/negPn.
    Qed.

    Section open_endpoints.
    Context {d} {T : porderType d}.

    Definition is_open_itv (A : set T) := exists ab, A = `]ab.1, ab.2[%classic.

    Definition open_itv_cover (A : set T) := [set F : nat -> set T |
      (forall i, is_open_itv (F i)) /\ A `<=` \bigcup_k (F k)].

    Definition itv_is_ray (i : interval T) : Prop :=
      match i with
      | Interval -oo%O (BLeft _) => True
      | Interval (BRight _) +oo%O => True
      | Interval -oo%O +oo%O => True
      | _ => False
      end.

    Definition itv_is_bd_open (i : interval T) : Prop :=
      match i with
      | Interval (BRight _) (BLeft _) => True
      | _ => False
      end.

    Definition itv_open_ends (i : interval T) : Prop :=
      itv_is_ray i \/ itv_is_bd_open i.

    Lemma itv_open_ends_rside l b (t : T) :
      itv_open_ends (Interval l (BSide b t)) -> b = true.
    Proof.
    by case: b; move: l => [[]?|[]] // [] //. Qed.

    Lemma itv_open_ends_rinfty l b :
      itv_open_ends (Interval l (BInfty T b)) -> b = false.
    Proof.
    by case: b => //; move: l => [[]?|[]] // []. Qed.

    Lemma itv_open_ends_lside l b (t : T) :
      itv_open_ends (Interval (BSide b t) l) -> b = false.
    Proof.
    by case: b; move: l => [[]?|[]] // []. Qed.

    Lemma itv_open_ends_linfty l b :
      itv_open_ends (Interval (BInfty T b) l) -> b = true.
    Proof.
    by case: b => //; move: l => [[]?|[]] // []. Qed.

    Lemma is_open_itv_itv_is_bd_openP (i : interval T) :
      itv_is_bd_open i -> is_open_itv [set` i].
    Proof.
    by case: i=> [] [[]l|[]] // [[]r|[]] // ?; exists (l,r).
    Qed.

    End open_endpoints.

    Lemma itv_open_endsI {d} {T : orderType d} (i j : interval T) :
      itv_open_ends i -> itv_open_ends j -> itv_open_ends (i `&` j)%O.
    Proof.
    move: i => [][[]a|[]] [[]b|[]] []//= _; move: j => [][[]x|[]] [[]y|[]] []//= _;
    by rewrite /itv_open_ends/= ?orbF ?andbT -?negb_or ?le_total//=;
      try ((by left)||(by right)).
    Qed.

    Lemma itv_setU {d} {T : orderType d} (i j : interval T) :
      [set` i] `&` [set` j] !=set0 -> [set` (i `|` j)%O] = [set` i] `|` [set` j].
    Proof.
    case=> p [ip jp]; have pij : p \in (i `|` j)%O by exact/(le_trans ip)/leUl.
    move: i j ip jp pij => [x y] [a b] /andP[xp py] /andP[ap pb] pab.
    rewrite eqEsubset; split => /= r /=; first last.
      by move=> -[ra|rb]; [exact/(le_trans ra)/leUl|exact/(le_trans rb)/leUr].
    rewrite (@itv_splitUeq _ T p (x `&` a)%O)// => /orP[].
    - move=> /andP[xar rp]; have /orP[ax|xa] := le_total a x.
      + right; apply/andP; split; first by rewrite (le_trans _ xar)// leIidr.
        by rewrite (le_trans rp)// (le_trans _ pb)// bnd_simp.
      + left; apply/andP; split; first by rewrite (le_trans _ xar)// leIidl.
        by rewrite (le_trans rp)// (le_trans _ py)//= bnd_simp.
    - move=> /predU1P[->|/andP[pr ryb]]; first by left; apply/andP.
      have /orP[bly|ylb] := le_total b y.
      + left; apply/andP; split; last by rewrite (le_trans ryb)// leUidr.
        by rewrite (le_trans _ pr)// (le_trans xp)//= bnd_simp.
      + right; apply/andP; split; last by rewrite (le_trans ryb)// leUidl.
        by rewrite (le_trans ap)// (le_trans _ pr)//= bnd_simp.
    Qed.

    Lemma itv_setI {d} {T : orderType d} (i j : interval T) :
      [set` (i `&` j)%O] = [set` i] `&` [set` j].
    Proof.
    by rewrite eqEsubset; split => z; rewrite /in_mem/= /pred_of_itv/= lexI=> /andP.
    Qed.
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