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
```
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 : unit) (
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.
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.
Lemma set_itvoc0 x : `]x, x]%classic = set0.
Proof.
Lemma set_itvco0 x : `[x, x[%classic = set0.
Proof.
Lemma set_itv_infty_infty : `]-oo, +oo[%classic = @setT T.
Proof.
Lemma set_itv_o_infty x : `]x, +oo[%classic = [set z | (
x < z)
%O].
Proof.
Lemma set_itv_c_infty x : `[x, +oo[%classic = [set z | (
x <= z)
%O].
Proof.
Lemma set_itv_infty_o x : `]-oo, x[%classic = [set z | (
z < x)
%O].
Proof.
Lemma set_itv_infty_c x : `]-oo, x]%classic = [set z | (
z <= x)
%O].
Proof.
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 setUitv1 (
a : itv_bound T) (
x : T)
: (
a <= BLeft x)
%O ->
[set` Interval a (
BLeft x)
] `|` [set x] = [set` Interval a (
BRight x)
].
Proof.
Lemma setU1itv (
a : itv_bound T) (
x : T)
: (
BRight x <= a)
%O ->
x |` [set` Interval (
BRight x)
a] = [set` Interval (
BLeft x)
a].
Proof.
End set_itv_porderType.
Arguments neitv {d T} _.
Lemma set_itv_ge [disp : unit] [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 : unit) (
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.
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` (
0%O : interval R)
] = set0.
Proof.
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.
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.
Lemma memNE E x : E x = (
-%R @` E) (
- x).
Proof.
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.
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_maxr 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.
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.
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.
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 : unit) (
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.
Lemma set_itv_splitD i :
[set` i] = [set` Interval i.
1 +oo%O] `\` [set` Interval i.
2 +oo%O].
Proof.
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.
Definition line_path a b t : R := (
1 - t)
* a + t * b.
Lemma line_path_id : line_path 0 1 = id.
Proof.
Lemma line_pathEl a b t : line_path a b t = t * (
b - a)
+ a.
Proof.
Lemma line_pathEr a b t : line_path a b t = (
1 - t)
* (
a - b)
+ b.
Proof.
Lemma line_path10 t : line_path 1 0 t = 1 - t.
Proof.
Lemma line_path0 a b : line_path a b 0 = a.
Proof.
Lemma line_path1 a b : line_path a b 1 = b.
Proof.
Lemma line_path_sym a b t : line_path a b t = line_path b a (
1 - t).
Proof.
Lemma line_path_flat a : line_path a a = cst a.
Proof.
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.
Lemma factor_flat a : factor a a = cst 0.
Proof.
Lemma factorl a b : factor a b a = 0.
Proof.
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.
Lemma factorK a b : a != b -> cancel (
factor a b) (
line_path a b).
Proof.
Lemma line_pathK a b : a != b -> cancel (
line_path a b) (
factor a b).
Proof.
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.
Lemma factor_bij a b : a != b -> bijective (
factor a b).
Proof.
Lemma le_line_path a b : a < b -> {mono line_path a b : x y / x <= y}.
Proof.
Lemma le_factor a b : a < b -> {mono factor a b : x y / x <= y}.
Proof.
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.
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.
Lemma mem_line_path_itvcc a b : a <= b -> set_fun `[0, 1] `[a, b] (
line_path a b).
Proof.
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.
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.
Lemma onem_factor a b x : a != b -> `1-(
factor a b x)
= factor b a x.
Proof.
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.
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.
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.