[ArtificialZeng]

A continuous family of injective continuous linear maps, with finite-dimensional source over a complete nontrivially normed field, has a uniform positive lower bound on a compact parameter set. A second lemma gives a uniform operator-norm perturbation radius preserving injectivity; the perturbing family need not be continuous.

These estimates support compact-family derivative perturbation arguments. The proof uses LinearMap.exists_antilipschitzWith and IsCompact.induction_on, adds no imports, and does not require the target to be finite-dimensional or complete.

---

Local validation on base 4c71e7e7e5f99e938e520c4242aaf2522a594953, Lean 4.36.0-rc1:

• lake build Mathlib.Analysis.Normed.Module.FiniteDimension passed after refreshing the base's dependency cache.
• Standard Mathlib source linters and examples covering a nonconstant compact family, empty sets, and a zero-dimensional source passed.
• A separate fresh compilation and axiom queries passed; both lemmas use only propext, Classical.choice, and Quot.sound.
• The same new proofs and a downstream derivative-estimate consumer also compiled against the pinned Lean 4.35.0 baseline.

AI disclosure: OpenAI Codex generated and adapted the proofs and performed automated source review and local checks. This draft is pending human review of the mathematics and API choices. Automated checks and AI review do not replace that review. Full upstream CI is still required.

[github-actions]

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[github-actions]

PR summary c9f852f03d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ IsCompact.exists_pos_forall_injective_of_norm_sub_lt
+ IsCompact.exists_pos_le_norm_apply

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 c9f852f).

• +2 new declarations
• −0 removed declarations

+IsCompact.exists_pos_forall_injective_of_norm_sub_lt
+IsCompact.exists_pos_le_norm_apply

---

No changes to strong technical debt.
No changes to weak technical debt.

Current commit c9f852f03d
Reference commit c21f07790b

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).

[ArtificialZeng]

i have read