chore: clean up backward.privateInPublic around d₂₃

[thorimur]

Moves a single-use private declaration into a let declaration.

This reduces instructions in that file by ~29% (but this gets swallowed in the overall noise).

Note: currently LieModule.Cohomology.d₂₃_aux._proof_17 is the biggest proof in mathlib (counted with sharing) :) This PR changes it to CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite.

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[github-actions]

PR summary 50583db8d7

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

- d₂₃_aux

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 50583db).

• +0 new declarations
• −0 removed declarations

No declaration differences.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 50583db8d7
Reference commit 07c7e6d228

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).

[thorimur]

!bench

[leanprover-radar]

Benchmark results for fd4f5a5 against 07c7e6d are in. There are significant results. @thorimur

• 🟥 build//instructions: +20.2G (+0.01%)

Large changes (1✅)

• ✅ build/module/Mathlib.Algebra.Lie.Cochain//instructions: -18.2G (-28.62%)

[grunweg]

Nice find!

[grunweg]

Does this change the proof term? My rule of thumb was to always use letI --- but perhaps that's a bad idea here? (If so, can you add a comment about this, please, so this is not accidentally reverted?)

[b-mehta]

It does change the final term - let will create a let binding in the final term, whereas letI won't create the binding, it's inlined instead. In effect, let gives us a local definition, whereas letI is as if there's no auxiliary definition at all.

[thorimur]

letI seems good here, especially since this is data. Here's the size comparison; the definition only gets smaller with letI (with sharing), and no proofs change size:
let:

total: 40407
[(LieModule.Cohomology.d₂₃, 1219),
 (LieModule.Cohomology.d₂₃._abel_1, 1705),
 (LieModule.Cohomology.d₂₃._proof_12, 6880),
 (LieModule.Cohomology.d₂₃._abel_7, 1709),
 (LieModule.Cohomology.d₂₃._proof_8, 3789),
 (LieModule.Cohomology.d₂₃._proof_9, 6523),
 (LieModule.Cohomology.d₂₃._abel_10, 1825),
 (LieModule.Cohomology.d₂₃._abel_4, 1711),
 (LieModule.Cohomology.d₂₃._proof_5, 2163),
 (LieModule.Cohomology.d₂₃._proof_11, 5379),
 (LieModule.Cohomology.d₂₃._proof_3, 2460),
 (LieModule.Cohomology.d₂₃._proof_2, 1181),
 (LieModule.Cohomology.d₂₃._proof_6, 3667),
 (LieModule.Cohomology.d₂₃._proof_1, 196)]

letI

total: 40331
[(LieModule.Cohomology.d₂₃, 1143),
(LieModule.Cohomology.d₂₃._abel_1, 1705),
(LieModule.Cohomology.d₂₃._proof_12, 6880),
(LieModule.Cohomology.d₂₃._abel_7, 1709),
(LieModule.Cohomology.d₂₃._proof_8, 3789),
(LieModule.Cohomology.d₂₃._proof_9, 6523),
(LieModule.Cohomology.d₂₃._abel_10, 1825),
(LieModule.Cohomology.d₂₃._abel_4, 1711),
(LieModule.Cohomology.d₂₃._proof_5, 2163),
(LieModule.Cohomology.d₂₃._proof_11, 5379),
(LieModule.Cohomology.d₂₃._proof_3, 2460),
(LieModule.Cohomology.d₂₃._proof_2, 1181),
(LieModule.Cohomology.d₂₃._proof_6, 3667),
(LieModule.Cohomology.d₂₃._proof_1, 196)]

#time seems about the same. I'll !bench again just to be sure. :)

EDIT: oops, double-counted the main definition size.

[b-mehta]

Note: currently LieModule.Cohomology.d₂₃_aux._proof_17 is the biggest proof in mathlib (counted with sharing) :)

Interesting, is there an existing way of tracking this? Also, what's the biggest without sharing, out of curiosity?

[thorimur]

Note: currently LieModule.Cohomology.d₂₃_aux._proof_17 is the biggest proof in mathlib (counted with sharing) :)

Interesting, is there an existing way of tracking this? Also, what's the biggest without sharing, out of curiosity?

Without sharing, it's: InternalProjectivityProof.aux (124627237888851) in Mathlib.Condensed.Light.Sequence :) (I'll note that as you might know the "with sharing" one is what gets written to oleans, afaik)

For some numbers, we used to have LieModule.Cohomology.d₂₃_aux._proof_17 as the largest proof at size 107,260 with sharing; the total (shared) size for d_23_aux + d_23 was ~295k, and now it's ~40k.

No automated way to check this yet; this is just a script I wrote. It's fairly straightforward, though. Should we put it somewhere?

[thorimur]

!bench

[leanprover-radar]

Benchmark results for 50583db against 07c7e6d are in. There are significant results. @thorimur

• ✅ build//instructions: -28.7G (-0.02%)

Large changes (1✅)

• ✅ build/module/Mathlib.Algebra.Lie.Cochain//instructions: -18.7G (-29.46%)