Feature esum psum esum only

[affeldt-aist]

Motivation for this change

closes #1954

This PR intends to extract the reusable part of PR #1954

• new lemmas about esum
• removal of fpos and fneg (superseded by funrpos and funrneg)

TODO: completed (?)

• some scripts still need to be linted
• some lemmas are wrongly named
• ome lemmas are not in the right files

fyi: @lyonel2017 @strub

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

[affeldt-aist]

@t6s I am pinging you because this PR changes the definition of esum, I thought you may want to know

[affeldt-aist]

@lyonel2017 @strub rebased

[t6s]

It looks like there are several chunks of changes contained in this PR, making it difficult to review it at once. Can you provide a bit more detailed description of intention for each of them? For example, which lemmas do you consider are important?

[affeldt-aist]

It looks like there are several chunks of changes contained in this PR, making it difficult to review it at once.
Can you provide a bit more detailed description of intention for each of them?
For example, which lemmas do you consider are important?

This is maybe for @lyonel2017 and @strub to say because this PR is used to develop another one.

From my viewpoint, the main change is the definition of esum which has become:

Definition esum S f := pos_esum S f^\+ - pos_esum S f^\-.

Before that, it was only the positive part.

[lyonel2017]

This is maybe for @lyonel2017 and @strub to say because this PR is used to develop another one.

I started porting the distr.v file from experimental_real to use esum. The idea is to better integrate the result present in experimental_reals into mathcomp-analysis. The current status is available here counting_distr.v (the branch is not up to date with this PR). For me all the changes present in this PR are required for the porting.