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change inequality in ereal_{d,}nbhs
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- from strict to large
- also in {p,n}infty_{d,}nbhs

closed #122
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affeldt-aist committed Dec 30, 2023
1 parent 489caf7 commit 1b0a58f
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10 changes: 10 additions & 0 deletions CHANGELOG_UNRELEASED.md
Original file line number Diff line number Diff line change
Expand Up @@ -76,6 +76,16 @@
`ae_eq_mul1l`,
`ae_eq_abse`

- moved from `derive.v` to `normedtype.v`:
+ lemmas `cvg_at_rightE`, `cvg_at_leftE`

- in `ereal.v`:
+ definitions `ereal_dnbhs` and `ereal_nbhs` changed to use large inequality instead
of strict inequality
- in `normedtype.v`:
+ definitions `pinfty_dnbhs` and `ninfty_nbhs` changed to use large inequality instead
of strict inequality

### Renamed

- in `exp.v`:
Expand Down
4 changes: 2 additions & 2 deletions theories/derive.v
Original file line number Diff line number Diff line change
Expand Up @@ -1319,8 +1319,8 @@ have imf_sup : has_sup imf.
split; first by exists (f a); apply/imageP; rewrite /= in_itv /= lexx.
have [M [Mreal imfltM]] : bounded_set (f @` `[a, b]).
by apply/compact_bounded/continuous_compact => //; exact: segment_compact.
exists (M + 1) => y /imfltM yleM.
by rewrite (le_trans _ (yleM _ _)) ?ler_norm ?ltr_addl.
exists (M + 1); apply/ubP => y /imfltM/= yleM.
by rewrite (le_trans _ (yleM _ _)) ?ler_addl ?ler_norm.
have [|imf_ltsup] := pselect (exists2 c, c \in `[a, b]%R & f c = sup imf).
move=> [c cab fceqsup]; exists c => // t tab; rewrite fceqsup.
by apply/sup_upper_bound => //; exact/imageP.
Expand Down
221 changes: 110 additions & 111 deletions theories/ereal.v
Original file line number Diff line number Diff line change
Expand Up @@ -633,17 +633,20 @@ Local Open Scope classical_set_scope.
Definition ereal_dnbhs (x : \bar R) (P : \bar R -> Prop) : Prop :=
match x with
| r%:E => r^' (fun r => P r%:E)
| +oo => exists M, M \is Num.real /\ forall y, M%:E < y -> P y
| -oo => exists M, M \is Num.real /\ forall y, y < M%:E -> P y
| +oo => exists M, M \is Num.real /\ forall y, M%:E <= y -> P y
| -oo => exists M, M \is Num.real /\ forall y, y <= M%:E -> P y
end.

Definition ereal_nbhs (x : \bar R) (P : \bar R -> Prop) : Prop :=
match x with
| x%:E => nbhs x (fun r => P r%:E)
| +oo => exists M, M \is Num.real /\ forall y, M%:E < y -> P y
| -oo => exists M, M \is Num.real /\ forall y, y < M%:E -> P y
| +oo => exists M, M \is Num.real /\ forall y, M%:E <= y -> P y
| -oo => exists M, M \is Num.real /\ forall y, y <= M%:E -> P y
end.

Canonical ereal_ereal_filter :=
FilteredType (extended R) (extended R) (ereal_nbhs).

End ereal_nbhs.

Section ereal_nbhs_instances.
Expand All @@ -652,70 +655,53 @@ Context {R : numFieldType}.
Global Instance ereal_dnbhs_filter :
forall x : \bar R, ProperFilter (ereal_dnbhs x).
Proof.
case=> [x||].
- case: (Proper_dnbhs_numFieldType x) => x0 [//= xT xI xS].
apply Build_ProperFilter' => //=; apply Build_Filter => //=.
move=> P Q lP lQ; exact: xI.
by move=> P Q PQ /xS; apply => y /PQ.
- apply Build_ProperFilter.
move=> P [x [xr xP]] //; exists (x + 1)%:E; apply xP => /=.
by rewrite lte_fin ltr_addl.
split=> /= [|P Q [MP [MPr gtMP]] [MQ [MQr gtMQ]] |P Q sPQ [M [Mr gtM]]].
move=> [r| |].
- by apply: Build_ProperFilter' => //=; exact: filter_not_empty.
- apply: Build_ProperFilter => [P [x [_ xP]]|].
by exists x%:E; exact: xP.
apply: Build_Filter => /= [|P Q [r +] [s +] |P Q PQ [r [rr rP]]].
+ by exists 0%R.
+ have [MP0|MP0] := eqVneq MP 0%R.
have [MQ0|MQ0] := eqVneq MQ 0%R.
by exists 0%R; split => // x x0; split;
[apply/gtMP; rewrite MP0 | apply/gtMQ; rewrite MQ0].
exists `|MQ|%R; rewrite realE normr_ge0; split => // x MQx; split.
by apply: gtMP; rewrite (le_lt_trans _ MQx) // MP0 lee_fin.
by apply gtMQ; rewrite (le_lt_trans _ MQx)// lee_fin real_ler_normr ?lexx.
have [MQ0|MQ0] := eqVneq MQ 0%R.
exists `|MP|%R; rewrite realE normr_ge0; split => // x MPx; split.
by apply gtMP; rewrite (le_lt_trans _ MPx)// lee_fin real_ler_normr ?lexx.
by apply gtMQ; rewrite (le_lt_trans _ MPx) // lee_fin MQ0.
have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0.
have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0.
exists (Num.max (PosNum MP0) (PosNum MQ0))%:num.
rewrite realE /= ge0 /=; split => //.
case=> [r| |//].
* rewrite lte_fin/= num_max num_lt_maxl /= => /andP[MPx MQx]; split.
by apply/gtMP; rewrite lte_fin (le_lt_trans _ MPx)// real_ler_normr ?lexx.
by apply/gtMQ; rewrite lte_fin (le_lt_trans _ MQx)// real_ler_normr ?lexx.
* by move=> _; split; [apply/gtMP | apply/gtMQ].
+ by exists M; split => // ? /gtM /sPQ.
- apply Build_ProperFilter.
+ move=> P [M [Mr ltMP]]; exists (M - 1)%:E.
by apply: ltMP; rewrite lte_fin gtr_addl oppr_lt0.
+ split=> /= [|P Q [MP [MPr ltMP]] [MQ [MQr ltMQ]] |P Q sPQ [M [Mr ltM]]].
+ have [-> [_ rP] [sr sQ]|r0 [rr rP]] := eqVneq r 0%R.
exists `|s|%R; rewrite normr_real; split => // x sx; split.
exact/rP/(le_trans _ sx).
by apply/sQ/(le_trans _ sx); rewrite lee_fin real_ler_normr ?lexx.
have [-> [_ sQ]|s0 [sr sQ]] := eqVneq s 0%R.
exists `|r|%R; rewrite normr_real; split => // x rx; split.
by apply/rP/(le_trans _ rx); rewrite lee_fin real_ler_normr ?lexx.
exact/sQ/(le_trans _ rx).
have /andP[{}r0 {}s0]: ((0 < `|r|) && (0 < `|s|))%R by rewrite !normr_gt0 r0.
exists (Num.max (PosNum r0) (PosNum s0))%:num.
rewrite realE /= ge0 /=; split => // -[t|_|//].
* rewrite lee_fin /= num_max num_le_maxl /= => /andP[rx sx]; split.
by apply/rP; rewrite lee_fin (le_trans _ rx)// real_ler_normr ?lexx.
by apply/sQ; rewrite lee_fin (le_trans _ sx)// real_ler_normr ?lexx.
* by split; [exact/rP|exact/sQ].
+ by exists r; split => // ? /rP /PQ.
- apply: Build_ProperFilter => [P [x [_ xP]]|].
+ by exists (x - 1)%:E; apply: xP; rewrite lee_fin ger_addl.
+ split=> /= [|P Q [r +] [s +] |P Q PQ [r [rr rP]]].
* by exists 0%R.
* have [MP0|MP0] := eqVneq MP 0%R.
have [MQ0|MQ0] := eqVneq MQ 0%R.
by exists 0%R; split => // x x0; split;
[apply/ltMP; rewrite MP0 | apply/ltMQ; rewrite MQ0].
exists (- `|MQ|)%R; rewrite realN realE normr_ge0; split => // x xMQ.
split.
by apply ltMP; rewrite (lt_le_trans xMQ)// lee_fin MP0 ler_oppl oppr0.
apply ltMQ; rewrite (lt_le_trans xMQ) // lee_fin ler_oppl -normrN.
by rewrite real_ler_normr ?realN // lexx.
* have [MQ0|MQ0] := eqVneq MQ 0%R.
exists (- `|MP|)%R; rewrite realN realE normr_ge0; split => // x MPx.
split.
apply ltMP; rewrite (lt_le_trans MPx) // lee_fin ler_oppl -normrN.
by rewrite real_ler_normr ?realN // lexx.
by apply ltMQ; rewrite (lt_le_trans MPx) // lee_fin MQ0 ler_oppl oppr0.
have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0.
have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0.
exists (- (Num.max (PosNum MP0) (PosNum MQ0))%:num)%R.
rewrite realN realE /= ge0 /=; split => //.
case=> [r|//|].
- rewrite lte_fin ltr_oppr num_max num_lt_maxl => /andP[].
rewrite ltr_oppr => MPx; rewrite ltr_oppr => MQx; split.
apply/ltMP; rewrite lte_fin (lt_le_trans MPx) //= ler_oppl -normrN.
* have [-> [_ rP] [sr sQ]|r0 [rr rP]] := eqVneq r 0%R.
exists (- `|s|)%R; rewrite realN normr_real; split => // x xs; split.
by apply/rP/(le_trans xs); rewrite lee_fin ler_oppl oppr0.
apply/sQ/(le_trans xs); rewrite lee_fin ler_oppl -normrN.
by rewrite real_ler_normr ?realN ?lexx.
have [-> [_ sQ]|s0 [sr sQ]] := eqVneq s 0%R.
exists (- `|r|)%R; rewrite realN normr_real; split => // x rx; split.
apply/rP/(le_trans rx); rewrite lee_fin ler_oppl -normrN.
by rewrite real_ler_normr ?realN // lexx.
apply/ltMQ; rewrite lte_fin (lt_le_trans MQx) //= ler_oppl -normrN.
by rewrite real_ler_normr ?realN // lexx.
- by move=> _; split; [apply/ltMP | apply/ltMQ].
* by exists M; split => // x /ltM /sPQ.
by apply/sQ/(le_trans rx); rewrite lee_fin ler_oppl oppr0.
have /andP[{}r0 {}s0]: ((0 < `|r|) && (0 < `|s|))%R by rewrite !normr_gt0 r0.
exists (- (Num.max (PosNum r0) (PosNum s0))%:num)%R.
rewrite realN realE /= ge0 /=; split => // -[t|//|_].
- rewrite lee_fin ler_oppr num_max num_le_maxl => /andP[].
rewrite ler_oppr => rx; rewrite ler_oppr => sx; split.
apply/rP; rewrite lee_fin (le_trans rx) //= ler_oppl -normrN.
by rewrite real_ler_normr ?realN ?lexx.
apply/sQ; rewrite lee_fin (le_trans sx) //= ler_oppl -normrN.
by rewrite real_ler_normr ?realN ?lexx.
- by split; [exact/rP|exact/sQ].
* by exists r; split => // ? /rP /PQ.
Qed.
Typeclasses Opaque ereal_dnbhs.

Expand All @@ -737,13 +723,19 @@ Context (R : numFieldType).
Implicit Type (r : R).

Lemma ereal_nbhs_pinfty_gt r : r \is Num.real -> \forall x \near +oo, r%:E < x.
Proof. by exists r. Qed.
Proof.
exists (r + 1)%R; split=> [|y]; first by rewrite realD.
by apply: lt_le_trans; rewrite EFinD lte_fin ltr_addl.
Qed.

Lemma ereal_nbhs_pinfty_ge r : r \is Num.real -> \forall x \near +oo, r%:E <= x.
Proof. by exists r; split => //; apply: ltW. Qed.

Lemma ereal_nbhs_ninfty_lt r : r \is Num.real -> \forall x \near -oo, r%:E > x.
Proof. by exists r. Qed.
Proof.
exists (r - 1)%R; split=> [|y]; first by rewrite realB.
by move=> /le_lt_trans; apply; rewrite lte_fin ltr_subl_addl ltr_addr.
Qed.

Lemma ereal_nbhs_ninfty_le r : r \is Num.real -> \forall x \near -oo, r%:E >= x.
Proof. by exists r; split => // ?; apply: ltW. Qed.
Expand Down Expand Up @@ -780,32 +772,34 @@ move: p => -[p| [M [Mreal MA]] | [M [Mreal MA]]] //=.
apply/nbhs_ballP; exists (e%:num / 2) => //= r per.
apply/nbhs_ballP; exists (e%:num / 2) => //= x rex.
apply/ballA/(@ball_splitl _ _ r) => //; exact/ball_sym.
- exists (M + 1)%R; split; first by rewrite realD.
move=> -[x| _ |_] //=; last by exists M.
rewrite lte_fin => M'x /=.
apply/nbhs_ballP; exists 1%R => //= y x1y.
apply MA; rewrite lte_fin.
rewrite addrC -ltr_subr_addl in M'x.
rewrite (lt_le_trans M'x) // ler_subl_addl addrC -ler_subl_addl.
rewrite (le_trans _ (ltW x1y)) // real_ler_norm // realB //.
rewrite ltr_subr_addr in M'x.
rewrite -comparabler0 (@comparabler_trans _ (M + 1)%R) //.
by rewrite /Order.comparable (ltW M'x) orbT.
by rewrite comparabler0 realD.
by rewrite num_real. (* where we really use realFieldType *)
- exists (M - 1)%R; split; first by rewrite realB.
move=> -[x| _ |_] //=; last by exists M.
rewrite lte_fin => M'x /=.
apply/nbhs_ballP; exists 1%R => //= y x1y.
apply MA; rewrite lte_fin.
rewrite ltr_subr_addl in M'x.
rewrite (le_lt_trans _ M'x) // addrC -ler_subl_addl.
rewrite (le_trans _ (ltW x1y)) // distrC real_ler_norm // realB //.
- exists (M + 1)%R; split; first by rewrite realD // real1.
move=> -[x| _ |] //=.
rewrite lee_fin => M'x /=.
apply/nbhs_ballP; exists 1%R => //= y x1y.
apply MA; rewrite lee_fin.
rewrite addrC -ler_subr_addl in M'x.
rewrite (le_trans M'x) // ler_subl_addl addrC -ler_subl_addl.
rewrite (le_trans _ (ltW x1y)) // real_ler_norm // realB //.
rewrite ler_subr_addr in M'x.
rewrite -comparabler0 (@comparabler_trans _ (M + 1)%R) //.
by rewrite /Order.comparable M'x orbT.
by rewrite comparabler0 realD // real1.
by rewrite num_real. (* where we really use realFieldType *)
rewrite addrC -ltr_subr_addr in M'x.
rewrite -comparabler0 (@comparabler_trans _ (M - 1)%R) //.
by rewrite /Order.comparable (ltW M'x).
by rewrite comparabler0 realB.
by exists M.
- exists (M - 1)%R; split; first by rewrite realB // real1.
move=> -[x| _ |] //=.
rewrite lee_fin => M'x /=.
apply/nbhs_ballP; exists 1%R => //= y x1y.
apply MA; rewrite lee_fin.
rewrite ler_subr_addl in M'x.
rewrite (le_trans _ M'x) // addrC -ler_subl_addl.
rewrite (le_trans _ (ltW x1y)) // distrC real_ler_norm // realB //.
by rewrite num_real. (* where we really use realFieldType *)
rewrite addrC -ler_subr_addr in M'x.
rewrite -comparabler0 (@comparabler_trans _ (M - 1)%R) //.
by rewrite /Order.comparable M'x.
by rewrite comparabler0 realB // real1.
by exists M.
Qed.

Definition ereal_topologicalMixin : Topological.mixin_of (@ereal_nbhs R) :=
Expand Down Expand Up @@ -833,19 +827,19 @@ case: x => [r /=| |].
- rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]].
exists (-%E @` S).
exists (- M)%R; rewrite realN Mreal; split => // x Mx.
by exists (- x); [apply MS; rewrite lte_oppl | rewrite oppeK].
by exists (- x); [apply MS; rewrite lee_oppl | rewrite oppeK].
rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK.
by exists (- x); [exists x | rewrite oppeK].
exists (- M)%R; rewrite realN; split => // y yM.
exists (- y); by [apply Mx; rewrite lte_oppr|rewrite oppeK].
exists (- y); by [apply Mx; rewrite lee_oppr|rewrite oppeK].
- rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]].
exists (-%E @` S).
exists (- M)%R; rewrite realN Mreal; split => // x Mx.
by exists (- x); [apply MS; rewrite lte_oppr | rewrite oppeK].
by exists (- x); [apply MS; rewrite lee_oppr | rewrite oppeK].
rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK.
by exists (- x); [exists x | rewrite oppeK].
exists (- M)%R; rewrite realN; split => // y yM.
exists (- y); by [apply Mx; rewrite lte_oppl|rewrite oppeK].
exists (- y); by [apply Mx; rewrite lee_oppl|rewrite oppeK].
Qed.

Lemma nbhsNKe (R : realFieldType) (z : \bar R) (A : set (\bar R)) :
Expand Down Expand Up @@ -1044,12 +1038,15 @@ Lemma nbhs_oo_up_e1 (A : set (\bar R)) (e : {posnum R}) : (e%:num <= 1)%R ->
ereal_ball +oo e%:num `<=` A -> nbhs +oo A.
Proof.
move=> e1 ooeA.
exists (fine (expand (1 - e%:num)%R)); rewrite num_real; split => //.
case => [r | | //].
- rewrite fine_expand; last first.
by rewrite ger0_norm ?ltr_subl_addl ?ltr_addr // subr_ge0.
by move=> ?; exact/ooeA/expand_ereal_ball_pinfty.
- by move=> _; exact/ooeA/ereal_ball_center.
exists (fine (expand (1 - e%:num / 2)%R)); rewrite num_real; split => //.
have e21 : (`|1 - e%:num / 2| < 1)%R.
rewrite ger0_norm; first by rewrite ltr_subl_addl ltr_addr.
by rewrite subr_ge0 ler_pdivr_mulr// mul1r (le_trans e1)// ler1n.
case => [r| _ |//]; last exact/ooeA/ereal_ball_center.
rewrite fine_expand // => er; apply/ooeA/expand_ereal_ball_pinfty => //.
rewrite (lt_le_trans _ er)// lt_expand ?inE; last exact/ltW.
by rewrite ler_lt_sub// ltr_pdivr_mulr// ltr_pmulr// ltr1n.
by rewrite ger0_norm ?subr_ge0// ler_subl_addl addrC -ler_subl_addl subrr.
Qed.

Lemma nbhs_oo_down_e1 (A : set (\bar R)) (e : {posnum R}) : (e%:num <= 1)%R ->
Expand Down Expand Up @@ -1312,12 +1309,12 @@ rewrite predeq2E => x A; split.
by rewrite subr_gt0 (le_lt_trans _ (contract_lt1 M)) // ler_norm.
case=> [r| |]/=.
* rewrite /ereal_ball [_ +oo]/= => rM1.
apply: MA; rewrite lte_fin.
apply: MA; rewrite lee_fin.
rewrite ger0_norm in rM1; last first.
by rewrite subr_ge0 // (le_trans _ (contract_le1 r%:E)) // ler_norm.
rewrite ltr_subl_addr addrC addrCA addrC -ltr_subl_addr subrr in rM1.
rewrite subr_gt0 in rM1.
by rewrite -lte_fin -lt_contract.
by rewrite -lee_fin -le_contract ltW.
* by rewrite /ereal_ball /= subrr normr0 => h; exact: MA.
* rewrite /ereal_ball /= opprK => h {MA}.
exfalso.
Expand All @@ -1332,12 +1329,12 @@ rewrite predeq2E => x A; split.
case=> [r| |].
* rewrite /ereal_ball => /= rM1.
apply MA.
rewrite lte_fin.
rewrite lee_fin.
rewrite ler0_norm in rM1; last first.
rewrite ler_subl_addl addr0 ltW //.
by move: (contract_lt1 r); rewrite ltr_norml => /andP[].
rewrite opprB opprK -ltr_subl_addl addrK in rM1.
by rewrite -lte_fin -lt_contract.
by rewrite -lee_fin -le_contract ltW.
* rewrite /ereal_ball /= -opprD normrN => h {MA}.
exfalso.
move: h; apply/negP.
Expand Down Expand Up @@ -1410,13 +1407,15 @@ case: x => /= [x [_/posnumP[d] dP] |[d [dreal dP]] |[d [dreal dP]]]; last 2 firs
have /ZnatP [N Nfloor] : floor (Num.max d 0%R) \is a Znat.
by rewrite Znat_def floor_ge0 le_maxr lexx orbC.
exists N.+1 => // n ltNn; apply: dP.
have /le_lt_trans : (d <= Num.max d 0)%R by rewrite le_maxr lexx.
by apply; rewrite (lt_le_trans (lt_succ_floor _))// Nfloor natr1 ler_nat.
have /le_trans : (d <= Num.max d 0)%R by rewrite le_maxr lexx.
apply; apply: le_trans (ltW (lt_succ_floor _)) _.
by rewrite Nfloor natr1 ler_nat.
have /ZnatP [N Nfloor] : floor (Num.max (- d)%R 0%R) \is a Znat.
by rewrite Znat_def floor_ge0 le_maxr lexx orbC.
exists N.+1 => // n ltNn; apply: dP; rewrite lte_fin ltr_oppl.
have /le_lt_trans : (- d <= Num.max (- d) 0)%R by rewrite le_maxr lexx.
by apply; rewrite (lt_le_trans (lt_succ_floor _))// Nfloor natr1 ler_nat.
exists N.+1 => // n ltNn; apply: dP; rewrite lee_fin ler_oppl.
have /le_trans : (- d <= Num.max (- d) 0)%R by rewrite le_maxr lexx.
apply; apply: le_trans (ltW (lt_succ_floor _)) _.
by rewrite Nfloor natr1 ler_nat.
have /ZnatP [N Nfloor] : floor (d%:num^-1) \is a Znat.
by rewrite Znat_def floor_ge0.
exists N => // n leNn; apply: dP; last first.
Expand Down
2 changes: 1 addition & 1 deletion theories/exp.v
Original file line number Diff line number Diff line change
Expand Up @@ -65,7 +65,7 @@ apply: series_le_cvg Kzxn _ _ => [//=| /= n|].
rewrite !normrM normr_id mulrAC mulfK // normr_eq0 expf_eq0 andbC.
by case: ltrgt0P zLx; rewrite //= normr_lt0.
do! (apply: ler_pmul || apply: mulr_ge0 || rewrite invr_ge0) => //.
by apply Kf => //; rewrite (lt_le_trans _ (ler_norm _))// ltr_addl.
by apply Kf => //; rewrite (le_trans _ (ler_norm _))// ler_addl.
have F : `|z / x| < 1.
by rewrite normrM normfV ltr_pdivr_mulr ?mul1r // (le_lt_trans _ zLx).
rewrite (_ : (fun _ => _) = geometric `|K + 1| `|z / x|); last first.
Expand Down
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