fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
zprimitive_irrp a q :
p != 0 -> zprimitive p = a *: q -> a = sgz (lead_coef q).
Proof.
move=> nz_p Dp; have: p = (a * zcontents p) *: q.
by rewrite mulrC -scalerA -Dp -zpolyEprim.
case/zprimitive_min=> // b <- /eqP.
rewrite Dp -{1}[q]scale1r scalerA -subr_eq0 -scalerBl scale_poly_eq0 subr_eq0.
have{Dp} /negPf->: q ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] | algebra/intdiv.v | zprimitive_irr | |
zcontentsMp q : zcontents (p * q) = zcontents p * zcontents q.
Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !(mul0r, zcontents0).
have [-> | nz_q] := eqVneq q 0; first by rewrite !(mulr0, zcontents0).
rewrite -[zcontents q]mulr1 {1}[p]zpolyEprim {1}[q]zpolyEprim.
rewrite -scalerAl -scalerAr !zcontentsZ; cong... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] | algebra/intdiv.v | zcontentsM | |
zprimitiveMp q : zprimitive (p * q) = zprimitive p * zprimitive q.
Proof.
have [pq_0|] := eqVneq (p * q) 0.
rewrite pq_0; move/eqP: pq_0; rewrite mulf_eq0.
by case/pred2P=> ->; rewrite !zprimitive0 (mul0r, mulr0).
rewrite -zcontents_eq0 -polyC_eq0 => /mulfI; apply; rewrite !mul_polyC.
by rewrite -zpolyEprim zconten... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] | algebra/intdiv.v | zprimitiveM | |
dvdpP_intp q : p %| q -> {r | q = zprimitive p * r}.
Proof.
case/Pdiv.Idomain.dvdpP/sig2_eqW=> [[c r] /= nz_c Dpr].
exists (zcontents q *: zprimitive r); rewrite -scalerAr.
by rewrite -zprimitiveM mulrC -Dpr zprimitiveZ // -zpolyEprim.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] | algebra/intdiv.v | dvdpP_int | |
int_Smith_normal_formm n (M : 'M[int]_(m, n)) :
{L : 'M[int]_m & L \in unitmx &
{R : 'M[int]_n & R \in unitmx &
{d : seq int | sorted dvdz d &
M = L *m (\matrix_(i, j) (d`_i *+ (i == j :> nat))) *m R}}}.
Proof.
move: {2}_.+1 (ltnSn (m + n)) => mn.
elim: mn => // mn IHmn in m n M *; rewrite ltnS => le_mn.
have ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] | algebra/intdiv.v | int_Smith_normal_form | |
itv_bound(T : Type) : Type := BSide of bool & T | BInfty of bool. | Variant | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_bound | |
BLeft:= (BSide true). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | BLeft | |
BRight:= (BSide false). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | BRight | |
interval(T : Type) := Interval of itv_bound T & itv_bound T. | Variant | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | interval | |
pair_of_intervalT (I : interval T) : itv_bound T * itv_bound T :=
let: Interval b1 b2 := I in (b1, b2). | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | pair_of_interval | |
itv_bound_can:
cancel (fun b : itv_bound T =>
match b with BSide b x => (b, Some x) | BInfty b => (b, None) end)
(fun b =>
match b with (b, Some x) => BSide b x | (b, None) => BInfty _ b end).
Proof. by case. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_bound_can | |
interval_can:
@cancel _ (interval T)
(fun '(Interval b1 b2) => (b1, b2)) (fun '(b1, b2) => Interval b1 b2).
Proof. by case. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | interval_can | |
Definition_ (T : eqType) := Equality.copy (itv_bound T)
(can_type (@itv_bound_can T)).
#[export, hnf]
HB.instance Definition _ (T : eqType) := Equality.copy (interval T)
(can_type (@interval_can T)).
#[export, hnf]
HB.instance Definition _ (T : choiceType) := Choice.copy (itv_bound T)
(can_type (@itv_bound_can... | HB.instance | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | Definition | |
le_boundb1 b2 :=
match b1, b2 with
| -oo, _ | _, +oo => true
| BSide b1 x1, BSide b2 x2 => x1 < x2 ?<= if b2 ==> b1
| _, _ => false
end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | le_bound | |
lt_boundb1 b2 :=
match b1, b2 with
| -oo, +oo | -oo, BSide _ _ | BSide _ _, +oo => true
| BSide b1 x1, BSide b2 x2 => x1 < x2 ?<= if b1 && ~~ b2
| _, _ => false
end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lt_bound | |
lt_bound_defb1 b2 : lt_bound b1 b2 = (b2 != b1) && le_bound b1 b2.
Proof. by case: b1 b2 => [[]?|[]][[]?|[]] //=; rewrite lt_def. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lt_bound_def | |
le_bound_refl: reflexive le_bound.
Proof. by move=> [[]?|[]] /=. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | le_bound_refl | |
le_bound_anti: antisymmetric le_bound.
Proof. by case=> [[]?|[]] [[]?|[]] //=; case: comparableP => // ->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | le_bound_anti | |
le_bound_trans: transitive le_bound.
Proof.
by case=> [[]?|[]] [[]?|[]] [[]?|[]] lexy leyz //;
apply: (lteif_imply _ (lteif_trans lexy leyz)).
Qed.
HB.instance Definition _ :=
Order.isPOrder.Build (itv_bound_display disp) (itv_bound T)
lt_bound_def le_bound_refl le_bound_anti le_bound_trans. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | le_bound_trans | |
bound_lexxc1 c2 x : (BSide c1 x <= BSide c2 x) = (c2 ==> c1).
Proof. by rewrite /<=%O /= lteifxx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_lexx | |
bound_ltxxc1 c2 x : (BSide c1 x < BSide c2 x) = (c1 && ~~ c2).
Proof. by rewrite /<%O /= lteifxx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_ltxx | |
ge_pinftyb : (+oo <= b) = (b == +oo). Proof. by case: b => [|] []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | ge_pinfty | |
le_ninftyb : (b <= -oo) = (b == -oo). Proof. by case: b => // - []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | le_ninfty | |
gt_pinftyb : (+oo < b) = false. Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | gt_pinfty | |
lt_ninftyb : (b < -oo) = false. Proof. by case: b => // -[]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lt_ninfty | |
ltBSidex y (b b' : bool) :
BSide b x < BSide b' y = (x < y ?<= if b && ~~ b').
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | ltBSide | |
leBSidex y (b b' : bool) :
BSide b x <= BSide b' y = (x < y ?<= if b' ==> b).
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | leBSide | |
lteBSide:= (ltBSide, leBSide). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lteBSide | |
ltBRight_leBLeftb x : b < BRight x = (b <= BLeft x).
Proof. by move: b => [[] b|[]]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | ltBRight_leBLeft | |
leBRight_ltBLeftb x : BRight x <= b = (BLeft x < b).
Proof. by move: b => [[] b|[]]. Qed.
Let BLeft_ltE x y (b : bool) : BSide b x < BLeft y = (x < y).
Proof. by case: b. Qed.
Let BRight_leE x y (b : bool) : BSide b x <= BRight y = (x <= y).
Proof. by case: b. Qed.
Let BRight_BLeft_leE x y : BRight x <= BLeft y = (x < ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | leBRight_ltBLeft | |
bnd_simp:= (BLeft_ltE, BRight_leE,
BRight_BLeft_leE, BLeft_BRight_ltE,
BRight_BSide_ltE, BLeft_BSide_leE, BSide_ltE, BSide_leE,
BInfty_leE, BInfty_geE, BInfty_BInfty_ltE,
BInfty_le_eqE, BInfty_ge_eqE, BInfty_ltE, BInfty_gtE, BInfty_ltF, BInfty_gtF,
@lexx _ T, @ltxx _ T, @eqxx T). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bnd_simp | |
comparable_BSide_mins (x y : T) : (x >=< y)%O ->
BSide s (Order.min x y) = Order.min (BSide s x) (BSide s y).
Proof. by rewrite !minEle bnd_simp => /comparable_leP[]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | comparable_BSide_min | |
comparable_BSide_maxs (x y : T) : (x >=< y)%O ->
BSide s (Order.max x y) = Order.max (BSide s x) (BSide s y).
Proof. by rewrite !maxEle bnd_simp => /comparable_leP[]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | comparable_BSide_max | |
subitvi1 i2 :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in (b2l <= b1l) && (b1r <= b2r). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitv | |
subitv_refl: reflexive subitv.
Proof. by case=> /= ? ?; rewrite !lexx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitv_refl | |
subitv_anti: antisymmetric subitv.
Proof.
by case=> [? ?][? ?]; rewrite andbACA => /andP[] /le_anti -> /le_anti ->.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitv_anti | |
subitv_trans: transitive subitv.
Proof.
case=> [yl yr][xl xr][zl zr] /andP [Hl Hr] /andP [Hl' Hr'] /=.
by rewrite (le_trans Hl' Hl) (le_trans Hr Hr').
Qed.
HB.instance Definition _ :=
Order.isPOrder.Build (interval_display disp) (interval T)
(fun _ _ => erefl) subitv_refl subitv_anti subitv_trans. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitv_trans | |
pred_of_itvi : pred T := [pred x | `[x, x] <= i]. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | pred_of_itv | |
StructureitvPredType := PredType pred_of_itv. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | Structure | |
subitvEb1l b1r b2l b2r :
(Interval b1l b1r <= Interval b2l b2r) = (b2l <= b1l) && (b1r <= b2r).
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitvE | |
in_itvx i :
x \in i =
let: Interval l u := i in
match l with
| BSide b lb => lb < x ?<= if b
| BInfty b => b
end &&
match u with
| BSide b ub => x < ub ?<= if ~~ b
| BInfty b => ~~ b
end.
Proof. by case: i => [[? ?|[]][|[]]]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | in_itv | |
itv_boundlrbl br x :
(x \in Interval bl br) = (bl <= BLeft x) && (BRight x <= br).
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_boundlr | |
itv_splitIbl br x :
x \in Interval bl br = (x \in Interval bl +oo) && (x \in Interval -oo br).
Proof. by rewrite !itv_boundlr andbT. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_splitI | |
subitvPi1 i2 : i1 <= i2 -> {subset i1 <= i2}.
Proof. by move=> ? ? /le_trans; exact. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitvP | |
subset_itv(x y z u : itv_bound T) : x <= y -> z <= u ->
{subset Interval y z <= Interval x u}.
Proof. by move=> xy zu; apply: subitvP; rewrite subitvE xy zu. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use subset_itv instead.")] | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv | |
subset_itv_bound(r s u v : bool) x y : r <= u -> v <= s ->
{subset Interval (BSide r x) (BSide s y) <= Interval (BSide u x) (BSide v y)}.
Proof.
by move: r s u v=> [] [] [] []// *; apply: subset_itv; rewrite bnd_simp.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_bound | |
subset_itv_oo_ccx y : {subset `]x, y[ <= `[x, y]}.
Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_oo_cc | |
subset_itv_oo_ocx y : {subset `]x, y[ <= `]x, y]}.
Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_oo_oc | |
subset_itv_oo_cox y : {subset `]x, y[ <= `[x, y[}.
Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_oo_co | |
subset_itv_oc_ccx y : {subset `]x, y] <= `[x, y]}.
Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_oc_cc | |
subset_itv_co_ccx y : {subset `[x, y[ <= `[x, y]}.
Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subset_itv_co_cc | |
itvxxx : `[x, x] =i pred1 x.
Proof. by move=> y; rewrite in_itv/= -eq_le eq_sym. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itvxx | |
itvxxPy x : reflect (y = x) (y \in `[x, x]).
Proof. by rewrite itvxx; apply/eqP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itvxxP | |
subitvPlb1l b2l br :
b2l <= b1l -> {subset Interval b1l br <= Interval b2l br}.
Proof. by move=> ?; apply: subitvP; rewrite subitvE lexx andbT. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitvPl | |
subitvPrbl b1r b2r :
b1r <= b2r -> {subset Interval bl b1r <= Interval bl b2r}.
Proof. by move=> ?; apply: subitvP; rewrite subitvE lexx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | subitvPr | |
itv_xxx cl cr y :
y \in Interval (BSide cl x) (BSide cr x) = cl && ~~ cr && (y == x).
Proof. by case: cl cr => [] []; rewrite [LHS]lteif_anti // eq_sym. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_xx | |
boundl_in_itvc x b : x \in Interval (BSide c x) b = c && (BRight x <= b).
Proof. by rewrite itv_boundlr bound_lexx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | boundl_in_itv | |
boundr_in_itvc x b :
x \in Interval b (BSide c x) = ~~ c && (b <= BLeft x).
Proof. by rewrite itv_boundlr bound_lexx implybF andbC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | boundr_in_itv | |
bound_in_itv:= (boundl_in_itv, boundr_in_itv). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_in_itv | |
lt_in_itvbl br x : x \in Interval bl br -> bl < br.
Proof. by case/andP; apply/le_lt_trans. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lt_in_itv | |
lteif_in_itvcl cr yl yr x :
x \in Interval (BSide cl yl) (BSide cr yr) -> yl < yr ?<= if cl && ~~ cr.
Proof. exact: lt_in_itv. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | lteif_in_itv | |
itv_geb1 b2 : ~~ (b1 < b2) -> Interval b1 b2 =i pred0.
Proof. by move=> ltb12 y; apply/contraNF: ltb12; apply/lt_in_itv. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_ge | |
itv_decomposei x : Prop :=
let: Interval l u := i in
(match l return Prop with
| BSide b lb => lb < x ?<= if b
| BInfty b => b
end *
match u return Prop with
| BSide b ub => x < ub ?<= if ~~ b
| BInfty b => ~~ b
end)%type. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_decompose | |
itv_dec: forall x i, reflect (itv_decompose i x) (x \in i).
Proof. by move=> ? [[? ?|[]][? ?|[]]]; apply: (iffP andP); case. Qed.
Arguments itv_dec {x i}. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_dec | |
itv_rewritei x : Type :=
let: Interval l u := i in
(match l with
| BLeft a => (a <= x) * (x < a = false)
| BRight a => (a <= x) * (a < x) * (x <= a = false) * (x < a = false)
| -oo => forall x : T, x == x
| +oo => forall b : bool, unkeyed b = false
end *
match u with
|... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_rewrite | |
itvPx i : x \in i -> itv_rewrite i x.
Proof.
case: i => [[[]a|[]][[]b|[]]] /andP [] ha hb; rewrite /= ?bound_in_itv;
do ![split | apply/negbTE; rewrite (le_gtF, lt_geF)];
by [|apply: ltW | move: (lteif_trans ha hb) => //=; exact: ltW].
Qed.
Arguments itvP [x i]. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itvP | |
itv_splitU1b x : b <= BLeft x ->
Interval b (BRight x) =i [predU1 x & Interval b (BLeft x)].
Proof.
move=> bx z; rewrite !inE/= !subitvE ?bnd_simp//= lt_neqAle.
by case: (eqVneq z x) => [->|]//=; rewrite lexx bx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_splitU1 | |
itv_split1Ub x : BRight x <= b ->
Interval (BLeft x) b =i [predU1 x & Interval (BRight x) b].
Proof.
move=> bx z; rewrite !inE/= !subitvE ?bnd_simp//= lt_neqAle.
by case: (eqVneq z x) => [->|]//=; rewrite lexx bx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_split1U | |
bound_meetbl br : itv_bound T :=
match bl, br with
| -oo, _ | _, -oo => -oo
| +oo, b | b, +oo => b
| BSide xb x, BSide yb y =>
BSide (((x <= y) && xb) || ((y <= x) && yb)) (x `&` y)
end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_meet | |
bound_joinbl br : itv_bound T :=
match bl, br with
| -oo, b | b, -oo => b
| +oo, _ | _, +oo => +oo
| BSide xb x, BSide yb y =>
BSide ((~~ (x <= y) || yb) && (~~ (y <= x) || xb)) (x `|` y)
end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_join | |
bound_meetC: commutative bound_meet.
Proof.
case=> [? ?|[]][? ?|[]] //=; rewrite meetC; congr BSide.
by case: lcomparableP; rewrite ?orbF // orbC.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_meetC | |
bound_joinC: commutative bound_join.
Proof.
case=> [? ?|[]][? ?|[]] //=; rewrite joinC; congr BSide.
by case: lcomparableP; rewrite ?andbT // andbC.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_joinC | |
bound_meetA: associative bound_meet.
Proof.
case=> [? x|[]][? y|[]][? z|[]] //=; rewrite !lexI meetA; congr BSide.
by case: (lcomparableP x y) => [|||->]; case: (lcomparableP y z) => [|||->];
case: (lcomparableP x z) => [|||//<-]; case: (lcomparableP x y);
rewrite //= ?andbF ?orbF ?lexx ?orbA //; case: (lcomparable... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_meetA | |
bound_joinA: associative bound_join.
Proof.
case=> [? x|[]][? y|[]][? z|[]] //=; rewrite !leUx joinA; congr BSide.
by case: (lcomparableP x y) => [|||->]; case: (lcomparableP y z) => [|||->];
case: (lcomparableP x z) => [|||//<-]; case: (lcomparableP x y);
rewrite //= ?orbT ?andbT ?lexx ?andbA //; case: (lcomparabl... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_joinA | |
bound_meetKUb2 b1 : bound_join b1 (bound_meet b1 b2) = b1.
Proof.
case: b1 b2 => [? ?|[]][? ?|[]] //=;
rewrite ?meetKU ?joinxx ?leIl ?lexI ?lexx ?andbb //=; congr BSide.
by case: lcomparableP; rewrite ?orbF /= ?andbb ?orbK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_meetKU | |
bound_joinKIb2 b1 : bound_meet b1 (bound_join b1 b2) = b1.
Proof.
case: b1 b2 => [? ?|[]][? ?|[]] //=;
rewrite ?joinKI ?meetxx ?leUl ?leUx ?lexx ?orbb //=; congr BSide.
by case: lcomparableP; rewrite ?orbF ?orbb ?andKb.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_joinKI | |
bound_leEmeetb1 b2 : (b1 <= b2) = (bound_meet b1 b2 == b1).
Proof.
case: b1 b2 => [[]t[][]|[][][]] //=; rewrite ?eqxx// => t';
rewrite [LHS]/<=%O /eq_op ?andbT ?andbF ?orbF/= /eq_op/= /eq_op/=;
case: lcomparableP => //=; rewrite ?eqxx//=; [| | |].
- by move/lt_eqF.
- move=> ic; apply: esym; apply: contraNF ic.
by... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_leEmeet | |
bound_le0xb : -oo <= b. Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_le0x | |
bound_lex1b : b <= +oo. Proof. by case: b => [|[]]. Qed.
HB.instance Definition _ :=
Order.hasBottom.Build (itv_bound_display disp) (itv_bound T) bound_le0x.
HB.instance Definition _ :=
Order.hasTop.Build (itv_bound_display disp) (itv_bound T) bound_lex1. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | bound_lex1 | |
itv_meeti1 i2 : interval T :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in Interval (b1l `|` b2l) (b1r `&` b2r). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_meet | |
itv_joini1 i2 : interval T :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in Interval (b1l `&` b2l) (b1r `|` b2r). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_join | |
itv_meetC: commutative itv_meet.
Proof. by case=> [? ?][? ?] /=; rewrite meetC joinC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_meetC | |
itv_joinC: commutative itv_join.
Proof. by case=> [? ?][? ?] /=; rewrite meetC joinC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_joinC | |
itv_meetA: associative itv_meet.
Proof. by case=> [? ?][? ?][? ?] /=; rewrite meetA joinA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_meetA | |
itv_joinA: associative itv_join.
Proof. by case=> [? ?][? ?][? ?] /=; rewrite meetA joinA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_joinA | |
itv_meetKUi2 i1 : itv_join i1 (itv_meet i1 i2) = i1.
Proof. by case: i1 i2 => [? ?][? ?] /=; rewrite meetKU joinKI. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_meetKU | |
itv_joinKIi2 i1 : itv_meet i1 (itv_join i1 i2) = i1.
Proof. by case: i1 i2 => [? ?][? ?] /=; rewrite meetKU joinKI. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_joinKI | |
itv_leEmeeti1 i2 : (i1 <= i2) = (itv_meet i1 i2 == i1).
Proof.
by case: i1 i2 => [? ?] [? ?]; rewrite /eq_op/=/eq_op/= eq_meetl eq_joinl.
Qed.
HB.instance Definition _ :=
Order.POrder_isLattice.Build (interval_display disp) (interval T)
itv_meetC itv_joinC itv_meetA itv_joinA
itv_joinKI itv_meetKU itv_leEmeet... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_leEmeet | |
itv_le0xi : Interval +oo -oo <= i. Proof. by case: i => [[|[]]]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_le0x | |
itv_lex1i : i <= `]-oo, +oo[. Proof. by case: i => [?[|[]]]. Qed.
HB.instance Definition _ :=
Order.hasBottom.Build (interval_display disp) (interval T) itv_le0x.
HB.instance Definition _ :=
Order.hasTop.Build (interval_display disp) (interval T) itv_lex1. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_lex1 | |
in_itvIx i1 i2 : x \in i1 `&` i2 = (x \in i1) && (x \in i2).
Proof. exact: lexI. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | in_itvI | |
BSide_mins (x y : T) :
BSide s (Order.min x y) = Order.min (BSide s x) (BSide s y).
Proof. exact: comparable_BSide_min. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | BSide_min | |
BSide_maxs (x y : T) :
BSide s (Order.max x y) = Order.max (BSide s x) (BSide s y).
Proof. exact: comparable_BSide_max. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | BSide_max | |
itv_bound_total: total (<=%O : rel (itv_bound T)).
Proof. by move=> [[]?|[]][[]?|[]]; rewrite /<=%O //=; case: ltgtP. Qed.
HB.instance Definition _ :=
Order.Lattice_isTotal.Build
(itv_bound_display disp) (itv_bound T) itv_bound_total. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_bound_total | |
itv_meetUl: @left_distributive (interval T) _ Order.meet Order.join.
Proof.
by move=> [? ?][? ?][? ?]; rewrite /Order.meet /Order.join /= -meetUl -joinIl.
Qed.
HB.instance Definition _ :=
Order.Lattice_Meet_isDistrLattice.Build
(interval_display disp) (interval T) itv_meetUl. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_meetUl | |
itv_splitUc a b : a <= c <= b ->
forall y, y \in Interval a b = (y \in Interval a c) || (y \in Interval c b).
Proof.
case/andP => leac lecb y.
rewrite !itv_boundlr !(ltNge (BLeft y) _ : (BRight y <= _) = _).
case: (leP a) (leP b) (leP c) => leay [] leby [] lecy //=.
- by case: leP lecy (le_trans lecb leby).
- by case... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_splitU | |
itv_splitUeqx a b : x \in Interval a b ->
forall y, y \in Interval a b =
[|| y \in Interval a (BLeft x), y == x | y \in Interval (BRight x) b].
Proof.
case/andP => ax xb y; rewrite (@itv_splitU (BLeft x)) ?ax ?ltW //.
by congr orb; rewrite (@itv_splitU (BRight x)) ?bound_lexx // itv_xx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_splitUeq | |
itv_total_meet3Ei1 i2 i3 :
i1 `&` i2 `&` i3 \in [:: i1 `&` i2; i1 `&` i3; i2 `&` i3].
Proof.
case: i1 i2 i3 => [b1l b1r] [b2l b2r] [b3l b3r]; rewrite !inE /eq_op /=.
case: (leP b1l b2l); case: (leP b1l b3l); case: (leP b2l b3l);
case: (leP b1r b2r); case: (leP b1r b3r); case: (leP b2r b3r);
rewrite ?eqxx ?orbT //... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_total_meet3E | |
itv_total_join3Ei1 i2 i3 :
i1 `|` i2 `|` i3 \in [:: i1 `|` i2; i1 `|` i3; i2 `|` i3].
Proof.
case: i1 i2 i3 => [b1l b1r] [b2l b2r] [b3l b3r]; rewrite !inE /eq_op /=.
case: (leP b1l b2l); case: (leP b1l b3l); case: (leP b2l b3l);
case: (leP b1r b2r); case: (leP b1r b3r); case: (leP b2r b3r);
rewrite ?eqxx ?orbT //... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | itv_total_join3E | |
predC_itvla : [predC Interval -oo a] =i Interval a +oo.
Proof.
case: a => [b x|[]//] y.
by rewrite !inE !subitvE/= bnd_simp andbT !lteBSide/= lteifNE negbK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] | algebra/interval.v | predC_itvl |
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