feat(GroupTheory/GroupAction/DomAct): invariant functions are constant on orbits

[SNAPKITTYWEST]

Closes part of #5379 (dynamical properties item).

Summary

Adds Mathlib.GroupTheory.GroupAction.DomAct.Invariant with three results:

• DomMulAct.isFixedPt_smul_iff_constOn_orbit — f : α → β is fixed by c : Mᵈᵐᵃ iff f is constant on the orbit of mk.symm c
• DomMulAct.smul_eq_iff_constOn_orbits — f is invariant under all of Mᵈᵐᵃ iff it is constant on every M-orbit
• DomMulAct.eq_const_of_pretransitive — on a pretransitive M-action, every Mᵈᵐᵃ-invariant function is constant

Motivation

Issue #5379 lists "a function is invariant under the action if and only if it is constant on the orbits" as an open dynamical properties item. This PR closes that item.

Test plan

• lake build Mathlib.GroupTheory.GroupAction.DomAct.Invariant
• lake exe mk_all (already run locally — Mathlib.lean updated)

[github-actions]

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[felixpernegger]

LLM-generated

[github-actions]

PR summary f1174b4a7a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.GroupTheory.GroupAction.DomAct.Invariant (new file)	631

---

Declarations diff (regex)

+ eq_const_of_pretransitive
+ isFixedPt_smul_iff_constOn_orbit
+ smul_eq_iff_constOn_orbits

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 f1174b4a7a
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]

@SNAPKITTYWEST if you are using AI for your PR (which very much looks like the case), you must disclose it.