feat(GroupTheory): prove uniqueness of ℤᵐ⁰ automorphisms

[vaca22]

Prove that every order-preserving multiplicative automorphism of ℤᵐ⁰ is the identity. This also gives a Unique instance for its automorphism type and a Subsingleton instance for order-preserving multiplicative normalizations from G to ℤᵐ⁰.

The proof transports an automorphism through WithZero.exp and WithZero.log to an additive automorphism of ℤ, then excludes negation by monotonicity.

[github-actions]

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[github-actions]

PR summary 26d1c9d765

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ OrderMonoidIso.subsingleton_right_withZero_int
+ instance : Unique (ℤᵐ⁰ ≃*o ℤᵐ⁰)
+ logMapAddEquiv
+ orderMonoidIso_int_eq_refl

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 26d1c9d).

• +3 new declarations
• −0 removed declarations

+OrderMonoidIso.subsingleton_right_withZero_int
+WithZero.orderMonoidIso_int_eq_refl
+instUniqueOrderMonoidIsoWithZeroMultiplicativeInt

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 26d1c9d765
Reference commit 7fff4171d6

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).