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affeldt-aist merged 4 commits into from
Jan 7, 2025
Merged

generalize indexing in lemmas for series #1397

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0ebe7c1
generalize some lemmas for series
t6s Nov 15, 2024
46c9399
changelog
t6s Nov 15, 2024
b21ee0c
undo eq_lt_total and le_lt_total
t6s Dec 18, 2024
b1dc185
fix changelog, more gen, rm deprecated
affeldt-aist Jan 7, 2025
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15 changes: 15 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -58,10 +58,25 @@

### Generalized

- in `sequences.v`,
+ generalized indexing from zero-based ones (`0 <= k < n` and `k <oo`)
to `m <= k < n` and `m <= k <oo`
* lemmas `nondecreasing_series`, `ereal_nondecreasing_series`,
`eseries_mkcondl`, `eseries_mkcondr`, `eq_eseriesl`,
`nneseries_lim_ge`, `adde_def_nneseries`,
`nneseriesD`, `nneseries_sum_nat`, `nneseries_sum`,
+ lemmas `nneseries_ge0`, `is_cvg_nneseries_cond`, `is_cvg_npeseries_cond`,
`is_cvg_nneseries`, `is_cvg_npeseries`, `nneseries_ge0`, `npeseries_le0`,
`lee_nneseries`

### Deprecated

### Removed

- in `sequences.v`:
+ notations `nneseries_pred0`, `eq_nneseries`, `nneseries0`,
`ereal_cvgPpinfty`, `ereal_cvgPninfty` (were deprecated since 0.6.0)

### Infrastructure

### Misc
4 changes: 2 additions & 2 deletions theories/charge.v
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Expand Up @@ -695,7 +695,7 @@ exists (D `\` Aoo).
have cvg_nuA : (\sum_(0 <= i < n) nu (A_ (v i))) @[n --> \oo]--> nu Aoo.
exact: charge_semi_sigma_additive.
have nuAoo : 0 <= nu Aoo.
move/cvg_lim : cvg_nuA => <-//=; apply: nneseries_ge0 => n _.
move/cvg_lim : cvg_nuA => <-//=; apply: nneseries_ge0 => n _ _.
exact: nuA_ge0.
have A_cvg_0 : nu (A_ (v n)) @[n --> \oo] --> 0.
rewrite [X in X @ _ --> _](_ : _ = (fun n => (fine (nu (A_ (v n))))%:E)); last first.
Expand Down Expand Up @@ -853,7 +853,7 @@ move=> /cvg_ex[[l| |]]; first last.
by rewrite leNgt => /negP; apply; rewrite ltNye_eq fin_num_measure.
- move/cvg_lim => limoo.
have := @npeseries_le0 _ (fun n => maxe (z_ (v n) * 2^-1%:E) (- 1%E)) xpredT 0.
by rewrite limoo// leNgt => /(_ (fun n _ => max_le0 n))/negP; apply.
by rewrite limoo// leNgt => /(_ (fun n _ _ => max_le0 n))/negP; exact.
move/fine_cvgP => [Hfin cvgl].
have : cvg (series (fun n => fine (maxe (z_ (v n) * 2^-1%:E) (- 1%E))) n @[n --> \oo]).
apply/cvg_ex; exists l; move: cvgl.
Expand Down
13 changes: 8 additions & 5 deletions theories/esum.v
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Expand Up @@ -300,7 +300,8 @@ Lemma lee_sum_fset_lim (R : realType) (f : (\bar R)^nat) (F : {fset nat})
\sum_(i <- F | P i) f i <= \sum_(i <oo | P i) f i.
Proof.
move=> f0; pose n := (\max_(k <- F) k).+1.
rewrite (le_trans (lee_sum_fset_nat F n _ _ _))//; last exact: nneseries_lim_ge.
rewrite (le_trans (lee_sum_fset_nat F n _ _ _))//; last first.
by apply: nneseries_lim_ge => i _; exact: f0.
move=> k /= kF; rewrite /n big_seq_fsetE/=.
by rewrite -[k]/(val [`kF]%fset) ltnS leq_bigmax.
Qed.
Expand All @@ -311,7 +312,7 @@ Lemma nneseries_esum (R : realType) (a : nat -> \bar R) (P : pred nat) :
\sum_(i <oo | P i) a i = \esum_(i in [set x | P x]) a i.
Proof.
move=> a0; apply/eqP; rewrite eq_le; apply/andP; split.
apply: (lime_le (is_cvg_nneseries_cond a0)); apply: nearW => n.
apply: (lime_le (is_cvg_nneseries_cond (fun n _ => a0 n))); apply: nearW => n.
apply: ereal_sup_ubound; exists [set` [fset val i | i in 'I_n & P i]%fset].
split; first exact: finite_fset.
by move=> /= k /imfsetP[/= i]; rewrite inE => + ->.
Expand Down Expand Up @@ -536,7 +537,7 @@ Lemma summable_cvg (P : pred nat) (f : (\bar R)^nat) :
cvg ((fun n => \sum_(0 <= k < n | P k) fine (f k))%R @ \oo).
Proof.
move=> f0 Pf; apply: nondecreasing_is_cvgn.
by apply: nondecreasing_series => n Pn; exact/fine_ge0/f0.
by apply: nondecreasing_series => n _ Pn; exact/fine_ge0/f0.
exists (fine (\sum_(i <oo | P i) `|f i|)) => x /= [n _ <-].
rewrite summable_fine_sum// -lee_fin fineK//; last first.
by apply/sum_fin_numP => i ni Pi; rewrite fin_num_abs (summable_pinfty Pf).
Expand Down Expand Up @@ -627,7 +628,8 @@ Lemma esumB D f g : summable D f -> summable D g ->
\esum_(i in D) f i - \esum_(i in D) g i.
Proof.
move=> Df Dg f0 g0.
have /eqP : esum D (f \- g)^\+ + esum_posneg D g = esum D (f \- g)^\- + esum_posneg D f.
have /eqP : esum D (f \- g)^\+ + esum_posneg D g =
esum D (f \- g)^\- + esum_posneg D f.
rewrite !ge0_esum_posneg// -!esumD//.
apply eq_esum => i Di; rewrite funeposE funenegE.
have [fg|fg] := leP 0 (f i - g i).
Expand All @@ -648,7 +650,8 @@ rewrite [X in _ == X -> _]addeC -sube_eq; last 2 first.
rewrite (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di.
by rewrite /= gee0_abs// f0.
rewrite -addeA addeCA eq_sym [X in _ == X -> _]addeC -sube_eq; last 2 first.
- rewrite ge0_esum_posneg// (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di.
- rewrite ge0_esum_posneg//.
rewrite (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di.
by rewrite /= gee0_abs// f0.
- rewrite fin_num_adde_defl// ge0_esum_posneg//.
rewrite (@eq_esum _ _ _ _ (abse \o g))// -?summableE// => i Di.
Expand Down
2 changes: 1 addition & 1 deletion theories/kernel.v
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Expand Up @@ -823,7 +823,7 @@ move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ =
apply: eq_integral => V _.
by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive.
apply/cvg_closeP; split.
by apply: is_cvg_nneseries => n _; exact: integral_ge0.
by apply: is_cvg_nneseries => n _ _; exact: integral_ge0.
rewrite closeE// integral_nneseries// => n.
exact: measurableT_comp (measurable_kernel k _ (mU n)) _.
Qed.
Expand Down
4 changes: 2 additions & 2 deletions theories/lebesgue_integral.v
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Expand Up @@ -1966,7 +1966,7 @@ rewrite [X in measurable_fun _ X](_ : _ =
rewrite [X in measurable_fun _ X](_ : _ =
(fun x => limn_esup (fun n => \sum_(0 <= i < n | P i) (h i) \_ D x))); last first.
apply/funext=> x; rewrite is_cvg_limn_esupE//.
apply: is_cvg_nneseries_cond => n Pn; rewrite patchE.
apply: is_cvg_nneseries_cond => n _ Pn; rewrite patchE.
by case: ifPn => // xD; rewrite h0//; exact/set_mem.
apply: measurable_fun_limn_esup => k.
under eq_fun do rewrite big_mkcond.
Expand Down Expand Up @@ -2926,7 +2926,7 @@ apply/eqP; rewrite eq_le; apply/andP; split; last first.
by apply: leeDl; exact: integral_ge0.
rewrite ge0_integralE//=; apply: ub_ereal_sup => /= _ [g /= gf] <-.
rewrite -integralT_nnsfun (integral_measure_series_nnsfun _ mD).
apply: lee_nneseries => n _.
apply: lee_nneseries => [n _ _|n _].
by apply: integral_ge0 => // x _; rewrite lee_fin.
rewrite [leRHS]integral_mkcond; apply: ge0_le_integral => //.
- by move=> x _; rewrite lee_fin.
Expand Down
2 changes: 1 addition & 1 deletion theories/lebesgue_measure.v
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Expand Up @@ -2920,7 +2920,7 @@ have [N F5e] : exists N, \sum_(N <= n <oo) \esum_(i in F n) mu (closure (B i)) <
apply: le_lt_trans foo.
by rewrite (nneseries_split _ N)// leeDr//; exact: sume_ge0.
rewrite fineK ?ge0_fin_numE//; last exact: nneseries_ge0.
apply: lee_nneseries => //; first by move=> i _; exact: esum_ge0.
apply: lee_nneseries => //; first by move=> i *; exact: esum_ge0.
move=> n Nn; rewrite measure_bigcup//=.
- by rewrite nneseries_esum// set_mem_set.
- by move=> i _; exact: measurable_closure.
Expand Down
15 changes: 8 additions & 7 deletions theories/measure.v
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Expand Up @@ -3021,7 +3021,7 @@ apply: (@le_trans _ _
move=> XD; have Xm := decomp_measurable Dm XD.
by apply: muS => // [i|]; [exact: mfD|exact: DXsub].
apply: lee_lim => /=; do ?apply: is_cvg_nneseries=> //.
by move=> n _; exact: sume_ge0.
by move=> n _ _; exact: sume_ge0.
near=> n; rewrite [n in _ <= n]big_mkcond; apply: lee_sum => i _.
rewrite ifT ?inE//.
under eq_big_seq.
Expand Down Expand Up @@ -4230,7 +4230,7 @@ have := outer_measure_sigma_subadditive mu
(fun n => if n \in ~` `I_N then F n else set0).
move/le_trans; apply.
rewrite [in leRHS]eseries_cond [in leRHS]eseries_mkcondr; apply: lee_nneseries.
- by move=> k _; exact: outer_measure_ge0.
- by move=> k _ _; exact: outer_measure_ge0.
- move=> k _; rewrite fun_if; case: ifPn => Nk; first by rewrite mem_not_I Nk.
by rewrite mem_not_I (negbTE Nk) outer_measure0.
Qed.
Expand Down Expand Up @@ -4510,7 +4510,8 @@ suff : forall X, mu X = \sum_(k <oo) mu (X `&` A k) + mu (X `&` ~` B).
rewrite (_ : (fun n => _) = fun n => \sum_(k < n) mu (A k)); last first.
rewrite funeqE => n; rewrite big_mkord; apply: eq_bigr => i _; congr (mu _).
by rewrite setIC; apply/setIidPl; exact: bigcup_sup.
move=> ->; have := fun n (_ : xpredT n) => outer_measure_ge0 mu (A n).
move=> ->.
have := fun n (_ : xpredT n) (_ : xpredT n) => outer_measure_ge0 mu (A n).
move/(@is_cvg_nneseries _ _ _ 0) => /cvg_ex[l] hl.
under [in X in _ --> X]eq_fun do rewrite -(big_mkord xpredT (mu \o A)).
by move/cvg_lim : (hl) => ->.
Expand Down Expand Up @@ -4687,9 +4688,9 @@ rewrite (_ : esum _ _ = \sum_(i <oo) \sum_(j <oo ) mu (G i j)); last first.
by move=> ? ? _ _; exact: (@can_inj _ _ _ snd).
by congr esum; rewrite predeqE => -[a b]; split; move=> [i _ <-]; exists i.
apply: lee_lim.
- apply: is_cvg_nneseries => n _.
by apply: nneseries_ge0 => m _; exact: (muG_ge0 (n, m)).
- by apply: is_cvg_nneseries => n _; apply: adde_ge0 => //; exact: mu_ext_ge0.
- apply: is_cvg_nneseries => n *.
by apply: nneseries_ge0 => m *; exact: (muG_ge0 (n, m)).
- by apply: is_cvg_nneseries => n *; apply: adde_ge0 => //; exact: mu_ext_ge0.
- by near=> n; apply: lee_sum => i _; exact: (PG i).2.
Unshelve. all: by end_near. Qed.

Expand Down Expand Up @@ -4923,7 +4924,7 @@ rewrite -(limeD cBA cBNA) // (_ : (fun _ => _) =
eseries (fun k => Rmu (B k `&` A) + Rmu (B k `&` ~` A))); last first.
by rewrite funeqE => n; rewrite -big_split /=; exact: eq_bigr.
apply/lee_lim => //.
by apply/is_cvg_nneseries => // n _; exact: adde_ge0.
by apply/is_cvg_nneseries => // n *; exact: adde_ge0.
near=> n; apply: lee_sum => i _; rewrite -measure_semi_additive2.
- apply: le_measure; rewrite /mkset ?inE//; [|by rewrite -setIUr setUCr setIT].
by apply: measurableU; [exact:measurableI|rewrite -setDE; exact:measurableD].
Expand Down
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