feat(Logic/Relation): some EqvGen API

[mathlib-splicebot]

This PR was automatically created from PR #38086 by @dagurtomas via a review comment by @dagurtomas.

[github-actions]

PR summary 71b974ca1f

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ idempotent
+ map

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 71b974c).

• +2 new declarations
• −0 removed declarations

+Relation.EqvGen.idempotent
+Relation.EqvGen.map

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 71b974ca1f
Reference commit 26a595e711

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[dagurtomas]

This is not needed anymore in #38086, but might be useful regardless

[chenson2018]

Note that if placed at the very end of this module, we can write:

lemma idempotent (r : α → α → Prop) : EqvGen (EqvGen r) = EqvGen r :=
  Equivalence.eqvGen_eq (EqvGen.is_equivalence r)

I'm not 100% sure about the naming and statement. The analogous Relation.reflTransGen_idem uses a different naming, but this is also deprecated for the more general Relation.reflTransGen_eq_self. I think the situation is slightly different here because of Equivalence not being a typeclass that is inferred. If this were using IsEquiv, it'd be straightforward to just keep the generalization, but maybe here we need both.

[chenson2018]

The convention seems to suggest:

theorem lift {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ a b, r a b → p (f a) (f b))
    (hab : EqvGen r a b) : EqvGen p (f a) (f b) := sorry

theorem lift' {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ a b, r a b → EqvGen p (f a) (f b))
    (hab : EqvGen r a b) : EqvGen p (f a) (f b) := sorry

[dagurtomas]

@chenson2018 your suggestions seem very reasonable -- you or anyone else should feel free to take over this PR as I don't need it anymore and don't have time to work on it.

[chenson2018]

@chenson2018 your suggestions seem very reasonable -- you or anyone else should feel free to take over this PR as I don't need it anymore and don't have time to work on it.

Thanks, I've opened a new PR.

[dagurtomas]

See #40626