refactor(RepresentationTheory/Induced): avoid unfolding resIndHomEquiv and resIndAdjunction

[JX-Mo]

This is split from #41808. We descend core computation lemmas around resIndHomEquiv to unbundled Representation level, so that downstream proofs will not unfold resIndHomEquiv and resIndAdjunction anymore. Moreover, Radar shows an actual dramatic speed up without deprecation from #41808
✅ build/module/Mathlib.RepresentationTheory.Induced//instructions: -33.2G (-47.79%)

• depends on: chore(RepresentationTheory/Rep/Res): remove a layer on Rep.res #43983
• depends on: refactor(RepresentationTheory/Induced): add induced representations API and remove tech debt #43949

[github-actions]

PR summary d9e5f5e5ed

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ IndV.inductionOn
+ IndV.lift
+ IndV.lift_apply_mk
+ IndV.mk_map_eq
+ IndV.mk_map_inv_eq
+ IndV.mk_map_inv_mul
+ IndV.mk_map_mul
+ coe_coindMap_apply
+ coe_coindMap_apply_apply
+ coe_coind_apply
+ coinvariantsTensorIndEquiv
+ coinvariantsTensorIndHom
+ coinvariantsTensorIndHom_apply_mk
+ coinvariantsTensorIndHom_inv
+ coinvariantsTensorIndInv
+ coinvariantsTensorIndInv_apply_mk
+ coinvariantsTensorIndInv_hom
+ coinvariantsTensorIndIso_hom
+ coinvariantsTensorIndIso_inv
+ coinvariantsTensorIndNatIso_hom_app
+ coinvariantsTensorIndNatIso_inv_app
+ ind.counit
+ ind.counit_apply_mk
+ ind.evalOne
+ ind.evalOne_apply
+ ind.hom_ext
+ ind.lift
+ ind.lift_apply_mk
+ ind.unit
+ ind.unit_apply
+ indMap_apply_mk
+ indResAdjunction_counit_app_hom
+ indResAdjunction_homEquiv
+ indResAdjunction_unit_app_hom
+ indResHomEquiv
+ indResHomEquiv_apply_hom
+ indResHomEquiv_symm_apply_hom
+ indToCoind_mk
+ ind_apply_mk
+ instance : AddCommGroup (IndV φ ρ) := by unfold IndV; infer_instance
+ instance : CoeFun (coindV φ σ) (fun _ => H → A) := ⟨Subtype.val⟩
+ instance : Module k (IndV φ ρ) := by unfold IndV; infer_instance
++- indMap

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 d9e5f5e).

• +49 new declarations
• −2 removed declarations

+Rep.coinvariantsTensorIndIso_hom_hom
+Rep.coinvariantsTensorIndIso_inv_hom
+Rep.coinvariantsTensorIndNatIso_hom_app_hom
+Rep.coinvariantsTensorIndNatIso_inv_app_hom
+Rep.indResAdjunction_counit_app_hom
+Rep.indResAdjunction_homEquiv
+Rep.indResAdjunction_unit_app_hom
-Rep.indResHomEquiv_apply
+Rep.indResHomEquiv_apply_hom
-Rep.indResHomEquiv_symm_apply
+Rep.indResHomEquiv_symm_apply_hom
+Rep.indToCoind_mk
+Rep.resFunctor_map
+Rep.resFunctor_obj
+Representation.IndV.inductionOn
+Representation.IndV.lift
+Representation.IndV.lift.congr_simp
+Representation.IndV.lift_apply_mk
+Representation.IndV.mk_map_eq
+Representation.IndV.mk_map_inv_eq
+Representation.IndV.mk_map_inv_mul
+Representation.IndV.mk_map_mul
+Representation.coe_coindMap_apply
+Representation.coe_coindMap_apply_apply
+Representation.coe_coind_apply
+Representation.coinvariantsTensorIndEquiv
+Representation.coinvariantsTensorIndHom
+Representation.coinvariantsTensorIndHom_apply_mk
+Representation.coinvariantsTensorIndHom_inv
+Representation.coinvariantsTensorIndInv
+Representation.coinvariantsTensorIndInv_apply_mk
+Representation.coinvariantsTensorIndInv_hom
+Representation.ind.counit
+Representation.ind.counit_apply_mk
+Representation.ind.evalOne
+Representation.ind.evalOne_apply
+Representation.ind.hom_ext
+Representation.ind.hom_ext_iff
+Representation.ind.lift
+Representation.ind.lift_apply_mk
+Representation.ind.unit
+Representation.ind.unit_apply
+Representation.indMap
+Representation.indMap_apply_mk
+Representation.indResHomEquiv
+Representation.indResHomEquiv_apply
+Representation.indResHomEquiv_symm_apply
+Representation.ind_apply_mk
+Representation.instAddCommGroupIndV
+Representation.instCoeFunSubtypeForallMemSubmoduleCoindV
+Representation.instModuleIndV

---

Decrease in strong tech debt: (relative, absolute) = (8.57, 0.01)

Current number	Change	Type (strong)
4133	-6	backward.defeqAttrib.useBackward
2307	-10	backward.isDefEq.respectTransparency.types

No changes to weak technical debt.

Current commit d9e5f5e5ed
Reference commit b63f6e8a68

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.py 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).

[mathlib-dependent-issues]

This PR/issue depends on:

• chore(RepresentationTheory/Rep/Res): remove a layer on Rep.res #43983
• refactor(RepresentationTheory/Induced): add induced representations API and remove tech debt #43949
  By Dependent Issues (🤖). Happy coding!

[JX-Mo]

!radar

[leanprover-radar]

Benchmark results for d9e5f5e against b63f6e8 are in. There are significant results. @JX-Mo

• ✅ build//instructions: -44.6G (-0.03%)

Large changes (2✅)

• ✅ build/module/Mathlib.RepresentationTheory.FiniteIndex//instructions: -9.8G (-22.83%)
• ✅ build/module/Mathlib.RepresentationTheory.Induced//instructions: -29.7G (-42.80%)

Small changes (3✅)

• ✅ build/module/Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality//instructions: -1.5G (-2.34%)
• ✅ build/module/Mathlib.RepresentationTheory.Homological.GroupCohomology.Shapiro//instructions: -1.1G (-8.94%)
• ✅ build/module/Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality//instructions: -1.9G (-1.46%)