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r-irber-irbe/proof-skills/skills/lean-math-foundations/SKILL.md

lean-math-foundations

Skill3 starsChanged 24 days agoApache-2.0View on GitHub

USE FOR: foundational mathematics in Lean 4: Lean's type system (Prop / Type / universes). DO NOT USE FOR: applied lattice instances such as severity / quality-gate lattices. TRIGGERS: type universe, Prop vs Type, classical reasoning.

The file

We split this skill into the rules it gives an agent

3 rules. About 0 are common, 0 are distinctive, and 3 are very specific to this project.

  • ≈0

    Common · estimate

    Many files like this have a rule of this kind.

  • ≈0

    Distinctive · estimate

    This author's own choices. Worth a look.

  • 3

    Very specific

    Tied to their project's names, files or tools. Probably skip.

  • Common
  • Distinctive
  • Very specific
  1. Very specific:

    If the answer is a specific proof, handoff to @lean-proof.

    line 46
  2. Very specific:

    STOP if a foundational choice silently introduces a new axiom — #print axioms must show only propext, Classical.choice, Quot.sound (and …

    line 52
  3. Very specific:

    STOP and handoff to @lean-research if the question is about a result not present in Mathlib at the current pin (e.g., a new …

    line 57

≈ “Common” is an estimate: at least 1 in 5 of the 350,495 skill files we read have a rule of the same kind, not the same words. “Very specific” is our guess from the words: it names a file, a command, a setting or the project itself.

Keep what fits and drop the rest. Your version, ready for your agents.
Files
SKILL.md

Lean 4 Mathematical Foundations

Guide to formalizing foundational mathematics in Lean 4, covering the structures and patterns that underpin all domain-specific formalization work.

Routing

  • USE FOR: foundational mathematics in Lean 4 — Lean's type system (Prop / Type / universes), classical vs constructive choices, Mathlib's algebraic typeclass hierarchy (Group / Ring / Order / Lattice / Module / Algebra), category theory basics, and any proof that needs foundational reasoning about types, propositions, or universes.
  • DO NOT USE FOR: applied lattice instances such as severity / quality-gate lattices (delegate to @lean-math-discrete); analysis-flavoured algebra such as normed-space structure (delegate to @lean-math-analysis); writing a specific proof (delegate to @lean-proof); reviewing a finished proof

Workflow

  1. Identify whether the question is about Lean's logical foundation (Part 1), Mathlib's typeclass hierarchy (Parts 2–5), or a category-theoretic construction (later parts).
  2. Locate the relevant Part below and read the project-relevance callouts before applying any pattern.
  3. If the answer is a specific proof, handoff to @lean-proof; if it is a soundness-affecting choice (e.g., adding a new Classical import), handoff to @lean-proof-review and @lean-enforcement.

Recovery & STOP

  • STOP if a foundational choice silently introduces a new axiom — #print axioms must show only propext, Classical.choice, Quot.sound (and funext is a theorem, not an axiom). Escalate to @lean-review-council if uncertain.
  • STOP and handoff to @lean-research if the question is about a result not present in Mathlib at the current pin (e.g., a new universe-polymorphism API).

Handoffs

  • Predecessors: agent:gateway, skill:lean-proof (mid-proof typeclass synthesis failure), skill:lean-research (when a survey turns up a foundational construct).
  • Successors: skill:lean-proof (apply the foundational pattern in a concrete proof), skill:lean-proof-review (verify axiom impact), skill:lean-enforcement (run native-decide / #print axioms checks after the change).

Detailed reference

Full encyclopedic reference, decision trees, and lookup tables live in references/lean-math-foundations-handbook.md .

See also

  • ../../references/lean-math-foundations-handbook.md -- Complete lean-math-foundations Handbook
  • ../lean-gateway/SKILL.md -- Top-level skill router

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