Tail expectation formula for L1 random variable

[Yosuke-Ito-345]

Motivation for this change

Add the tail expectation formula expectation_cdf_ccdf for any L1 random variable.
This is a combined version of lemmas ge0_expectation_ccdf and le0_expectation_cdf.

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

[Yosuke-Ito-345]

@affeldt-aist @t6s
I’m not sure what caused this error. Is there anything I should do on my side?

[affeldt-aist]

Here is the error message:

File "./reals/signed.v", line 6, characters 29-37:
Error: Cannot find a physical path bound to logical path
unstable with prefix mathcomp.

and here is the problematic code:

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool.
From mathcomp Require Import ssrnat eqtype choice order ssralg ssrnum ssrint.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import mathcomp_extra.

I don't understand why it fails suddenly (maybe @proux01 knows)
but do not worry: signed.v is a deprecated file, I will test on my side ASAP,
worst case scenario: we don't find a fix, then we might simply give up on Coq 8.20.

[Yosuke-Ito-345]

@affeldt-aist
Thank you for investigating this error. I really appreciate your help.

[affeldt-aist]

Mmmhhh... I can't reproduce the error locally. :-(

[affeldt-aist]

(Sorry, I made a push force but at least it is updated w.r.t. master.)
I tried to simplify your proof using a theory for the positive/negative part of real functions taken from PR #1391 (not merged yet).
I think it is better to do the proof this way because it contributes to a theory that is useful elsewhere.
What do you think?
(I still need one pass to complete the review.)

[Yosuke-Ito-345]

@affeldt-aist
Thank you for your suggestion. I agree with you. The positive/negative part of real-valued functions is an important notion and appears everywhere in the Lebesgue integral theory.

[affeldt-aist]

It looks ok to me. This is great that you can use the library. I am pinging @t6s to have a second opinion (because I added a number of lines of code, orthogonal to your contribution though).

[affeldt-aist]

@t6s I am pinging to double-check the additions made to this PR. The duplication funepos vs. funrpos is a bit unfortunate but unavoidable as far as I can tell, and it is mitigated by the use of the same naming convention. Do you see a way to factorize?

[t6s]

what about parameterizing the definition (of funxpos) over realDomainTypes, so that it could work both for reals and ereals?

[t6s]

(I won't have time to check that myself today)

[affeldt-aist]

what about parameterizing the definition (of funxpos) over realDomainTypes, so that it could work both for reals and ereals?

realDomainType is maybe a bit too ambitious since with extended real numbers we have "oo - oo = -oo", we also need the parameter to be a measurable type, but, yes, you're right, we need to experiment with a common supertype.

I propose that we merge, so that @Yosuke-Ito-345 can continue working, and so that I can continue working on PR #1391, but that we work in parallel on experimenting the generalization. If it works, the changes will be manageable (mostly removing code and adjust naming), and I can take responsibility for it.

I'll create an issue for 1.17.0.

[Yosuke-Ito-345]

Thank you very much for the review.