Generalize integration_by_parts

[IshiguroYoshihiro]

Motivation for this change

Add a part of generalized version of integration by parts for unbound intervals `[a, +oo[.
This PR is over #1656 and #1662.
I don't sure that I should add 4 more lemmas and complete all variation of integration by parts for unbound intervals.
I think that 8 similar lemmas would be a mess. So, I'd like to somehow make them up into one or two lemmas, but I don't have an idea.

Checklist

• added corresponding entries in CHANGELOG_UNRELEASED.md

• added corresponding documentation in the headers

Reference: How to document

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• PRs with several commits that make sense individually and that
  all compile are preferentially merged into master.
• PRs with disorganized commits are very likely to be squash-rebased.

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[Tragicus]

Are not all 4 versions the same, the only difference being whether the minus is inside or outside the integral?

[IshiguroYoshihiro]

Are not all 4 versions the same, the only difference being whether the minus is inside or outside the integral?

If you said that these lemmas can be proved with one of them, I think it is no.
As you can see, the version of le0_ge0 and le0_le0 can be proved with ge0_ge0 or ge0_le0, but ge0_le0 can not proved from ge0_ge0.
Because that integration by substitusion using a transformation F -> F \o -%R change the unbounded side of the interval from `[a, +oo[ to `[-oo, - a[, I may have to set 4 more variations of this lemma for `[-oo, -a[.
It will makes messy for the use to set 8 variation of the lemma. So I will consider some good way.

[affeldt-aist]

Are not all 4 versions the same, the only difference being whether the minus is inside or outside the integral?

Indeed le0_le0 is the same as ge0_ge0 (and it is proved as such).
Similarly, for ge0_le0 vs. le0_ge0.
I don't think we can have more sharing.
Do you think that we should work towards a general version? A single lemma encompassing them all?

[affeldt-aist]

Are not all 4 versions the same, the only difference being whether the minus is inside or outside the integral?

Indeed le0_le0 is the same as ge0_ge0 (and it is proved as such). Similarly, for ge0_le0 vs. le0_ge0. I don't think we can have more sharing. Do you think that we should work towards a general version? A single lemma encompassing them all?

To give a bit of context, these lemmas have been written this way after having been tested to formalize the Gamma function. Maybe their apparent specialization is the result of practical considerations? @IshiguroYoshihiro can you confirm? (Or am I remembering wrong?)

[Tragicus]

Sorry, I am back.
One way to factorize would be to ask for the relevant functions to have constant sign. I was also wondering whether we want to generalize over the interval, instead of [a, +oo[. We would require that the relevant functions are derivable on the interior of the interval and have limits at the boundaries.
@affeldt-aist Some of the following suggestions are quite ugly but save a lot of time, what do you think?

[Tragicus]

Suggested change

	rewrite (@integration_by_partsy_le0_ge0 R (- F)%R G (- f)%R g a (- FGoo))//.
	- rewrite oppeD; last exact: fin_num_adde_defr.
	rewrite -EFinN opprD 2!opprK mulNr.
	by under eq_integral do rewrite mulNr EFinN oppeK.
	- by move=> ?; apply: cvgN; exact: cf.
	- exact: derivable_oy_continuous_bndN.
	- by move=> ? ?; rewrite fctE derive1N ?Ff//; move: Foy => [+ _]; apply.
	rewrite (@integration_by_partsy_le0_ge0 R (- F)%R G (- f)%R g a (- FGoo))//.
	- rewrite oppeD; last exact: fin_num_adde_defr.
	rewrite -EFinN opprD 2!opprK mulNr.
	by under eq_integral do rewrite mulNr EFinN oppeK.
	- by move=> ?; apply: cvgN; exact: cf.
	- exact: derivable_oy_continuous_bndN.
	- by move=> ? ?; rewrite fctE derive1N ?Ff => [//|//|]; apply: Foy.1.

.6s faster

[affeldt-aist]

To give a bit of context, these lemmas have been written this way after having been tested to formalize the Gamma function.

FYI, here is the PR that formalizes the Gamma function:
#1762

One way to factorize would be to ask for the relevant functions to have constant sign. I was also wondering whether we want to generalize over the interval, instead of [a, +oo[. We would require that the relevant functions are derivable on the interior of the interval and have limits at the boundaries.

This has not been tested yet (by lack of time). :-(

Some of the following suggestions are quite ugly but save a lot of time, what do you think?

We have merged all of them. Thanks for identifying the performance issues, that helps.

[t6s]

I suspect that this minus sign can be brought out of the integral, so that the statements become more uniform.

[t6s]

like this:

Lemma integration_by_partsy_le0_ge0 :
  {in `]a, +oo[, forall x, (f x * G x)%:E <= 0} ->
  {in `]a, +oo[, forall x, 0 <= (F x * g x)%:E} ->
  \int[mu]_(x in `[a, +oo[) (F x * g x)%:E = (FGoo - F a * G a)%:E -
  \int[mu]_(x in `[a, +oo[) (f x * G x)%:E.
Proof.
move=> fG0 Fg0.
have /= mMfG : measurable_fun `]a, +oo[ (fun x => (- (f x * G x))%R).
  exact: measurableT_comp.
rewrite -integral_itv_obnd_cbnd// -[in RHS]integral_itv_obnd_cbnd//.
rewrite -integralN/=; last first.
  rewrite fin_num_adde_defr//.
  under eq_integral => ? ?.
    by rewrite (le0_funeposE fG0); first over; rewrite inE.
  by rewrite integral_cst// mul0e.
rewrite itv_bndy_bigcup_BRight seqDU_bigcup_eq.
rewrite 2?ge0_integral_bigcup//= -?seqDU_bigcup_eq -?itv_bndy_bigcup_BRight.
- rewrite [LHS]sum_integral_limn; apply: cvg_lim => //.
  apply: cvgeD; first exact: fin_num_adde_defr; first exact: FGaoo.
  under eq_cvg do rewrite sumNint_sumintN_fG.
  apply: is_cvg_nneseries_cond => n _ _; apply: integral_ge0 => x.
  rewrite seqDUE => anx; rewrite EFinN oppe_ge0.
  apply: fG0; rewrite inE/=.
  by apply: subset_itv anx; rewrite bnd_simp// lerDl.
- by move=> k; apply: measurableD => //; apply: bigsetU_measurable.
- exact/measurable_EFinP/measurableT_comp.
- by move=> ? ?; rewrite EFinN oppe_ge0 fG0// inE.
- by move=> k; apply: measurableD => //; apply: bigsetU_measurable.
- exact/measurable_EFinP.
- by move=> ? ?; rewrite Fg0// inE.
Qed.

[t6s]

I suggest keeping 4 lemmas as is, and additionally, provide a unified interface lemma that use the constant_sign predicate

[t6s]

The character immediately after apply: (that looks like a whitespace) seems to be 0xa0, not an whitespace.
This causes a build failure in my env.

[affeldt-aist]

It looks like this Let is almost useless.

[affeldt-aist]

This could be reduced to a mere application of integralN.