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

Merge policy

As a rule of thumb:

• 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.

Reminder to reviewers

• Read this Checklist
• Put a milestone if possible
• Check labels

[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.