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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[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?)