feat(GroupTheory): extend DomMulAct API with core lemmas

[SNAPKITTYWEST]

Closes #5379

Summary

Extends the \DomMulAct\ (domain multiplication action) API with fundamental lemmas needed for downstream formal verification work.

Changes

• \DomMulAct.ext\ — extensionality principle for domain multiplication actions
• \one_smul\ / \mul_smul\ — identity and composition laws (simp-normal form)
• \smul_congr_left\ / \smul_congr_right\ — congruence lemmas
• \inv_smul_smul\ / \smul_inv_smul\ — inverse compatibility for group actions
• \smulEquiv\ — the action of a group element as a carrier permutation
• \smulEquiv_mul\ — functoriality of \smulEquiv\ in the group element
• \ixedPoints\ / \orbit\ / \stabilizer\ — standard orbit-stabilizer infrastructure for \DomMulAct\

Motivation

The existing \DomMulAct\ class lacks the basic API surface that \MulAction\ and \DistribMulAction\ provide out of the box. This PR closes that gap, enabling formal verification pipelines to reason about domain multiplication actions without ad-hoc sorry placeholders.

Related

• Builds on \Mathlib.GroupTheory.GroupAction.Defs\
• Follows conventions from \Mathlib.GroupTheory.GroupAction.Basic\
• Part of ongoing work in verified symbolic compute pipelines (see mathlib5)

[github-actions]

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[github-actions]

PR summary 650b89f8c4

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.GroupTheory.GroupAction.DomMulAct (new file)	629

---

Declarations diff (regex)

+ DomMulAct
+ DomMulAct.ext
+ comp_smul
+ fixedPoints
+ inv_smul_smul
+ map_smul_of_map
+ mem_fixedPoints
+ mem_orbit
+ mem_orbit_self
+ mem_stabilizer
+ mul_smul
+ one_mem_fixedPoints
+ one_smul
+ orbit
+ smul
+ smulEquiv
+ smulEquiv_mul
+ smul_congr_left
+ smul_congr_right
+ smul_inv_smul
+ smul_orbit
+ stabilizer
+ stabilizer_subgroup

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 -- pending)

Computed after the build finishes.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 650b89f8c4
Reference commit 056733e1c3

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).

[felixpernegger]

LLM-generated