Lean4 Theorem Proving

Lean 4 Theorem Proving

Core Principle

Build incrementally, structure before solving, trust the type checker. Lean's type checker is your test suite.

Success = lake build passes + zero sorries + zero custom axioms. Theorems with sorries/axioms are scaffolding, not results.

Quick Reference

When to Use

Use for ANY Lean 4 development: pure/applied math, program verification, mathlib contributions.

Critical for: Type class synthesis errors, sorry/axiom management, mathlib search, measure theory/probability work.

Tools & Workflows

7 slash commands for search, analysis, and optimization - type /lean in Claude Code. See COMMANDS.md for full guide with examples and workflows.

16 automation scripts for search, verification, and refactoring. See scripts/README.md for complete documentation.

Lean LSP Server (optional) provides 30x faster feedback with instant proof state and parallel tactic testing. See lean-lsp-server.md for setup and workflows.

Subagent delegation (optional, Claude Code users) enables batch automation. See subagent-workflows.md for patterns.

Build-First Principle

ALWAYS compile before committing. Run lake build to verify. "Compiles" ≠ "Complete" - files can compile with sorries/axioms but aren't done until those are eliminated.

The 4-Phase Workflow

1. Structure Before Solving - Outline proof strategy with have statements and documented sorries before writing tactics
2. Helper Lemmas First - Build infrastructure bottom-up, extract reusable components as separate lemmas
3. Incremental Filling - Fill ONE sorry at a time, compile after each, commit working code
4. Type Class Management - Add explicit instances with haveI/letI when synthesis fails, respect binder order for sub-structures

Finding and Using Mathlib Lemmas

Philosophy: Search before prove. Mathlib has 100,000+ theorems.

Use /search-mathlib slash command, LSP server search tools, or automation scripts. See mathlib-guide.md for detailed search techniques, naming conventions, and import organization.

Essential Tactics

Key tactics: simp only, rw, apply, exact, refine, by_cases, rcases, ext/funext. See tactics-reference.md for comprehensive guide with examples and decision trees.

Domain-Specific Patterns

Analysis & Topology: Integrability, continuity, compactness patterns. Tactics: continuity, fun_prop.

Algebra: Instance building, quotient constructions. Tactics: ring, field_simp, group.

Measure Theory & Probability (emphasis in this skill): Conditional expectation, sub-σ-algebras, a.e. properties. Tactics: measurability, positivity. See measure-theory.md for detailed patterns.

Complete domain guide: domain-patterns.md

Managing Incomplete Proofs

Standard mathlib axioms (acceptable): Classical.choice, propext, quot.sound. Check with #print axioms theorem_name or /check-axioms.

CRITICAL: Sorries/axioms are NOT complete work. A theorem that compiles with sorries is scaffolding, not a result. Document every sorry with concrete strategy and dependencies. Search mathlib exhaustively before adding custom axioms.

When sorries are acceptable: (1) Active work in progress with documented plan, (2) User explicitly approves temporary axioms with elimination strategy.

Not acceptable: "Should be in mathlib", "infrastructure lemma", "will prove later" without concrete plan.

Compiler-Guided Proof Repair

When proofs fail to compile, use iterative compiler-guided repair instead of blind resampling.

Quick repair: /lean4-theorem-proving:repair-file FILE.lean

How it works:

1. Compile → extract structured error (type, location, goal, context)
2. Try automated solver cascade first (many simple cases handled mechanically, zero LLM cost)
   • Order: rfl → simp → ring → linarith → nlinarith → omega → exact? → apply? → aesop
3. If solvers fail → call lean4-proof-repair agent:
   • Stage 1: Haiku (fast, most common cases) - 6 attempts
   • Stage 2: Sonnet (precise, complex cases) - 18 attempts
4. Apply minimal patch (1-5 lines), recompile, repeat (max 24 attempts)

Key benefits:

• Low sampling budget (K=1 per attempt, not K=100)
• Error-driven action selection (specific fix per error type, not random guessing)
• Fast model first (Haiku), escalate only when needed (Sonnet)
• Solver cascade handles simple cases mechanically (zero LLM cost)
• Early stopping prevents runaway costs (bail after 3 identical errors)

Expected outcomes: Success improves over time as structured logging enables learning from attempts. Cost optimized through solver cascade (free) and multi-stage escalation.

Commands:

• /repair-file FILE.lean - Full file repair
• /repair-goal FILE.lean LINE - Specific goal repair
• /repair-interactive FILE.lean - Interactive with confirmations

Detailed guide: compiler-guided-repair.md

Inspired by: APOLLO (https://arxiv.org/abs/2505.05758) - compiler-guided repair with multi-stage models and low sampling budgets.

Common Compilation Errors

See compilation-errors.md for detailed debugging workflows.

Documentation Conventions

• Write timeless documentation (describe what code is, not development history)
• Don't highlight "axiom-free" status after proofs are complete
• Mark internal helpers as private or in dedicated sections
• Use example for educational code, not lemma/theorem

Quality Checklist

Before commit:

• lake build succeeds on full project
• All sorries documented with concrete strategy
• No new axioms without elimination plan
• Imports minimal

Doing it right: Sorries/axioms decrease over time, each commit completes one lemma, proofs build on mathlib.

Red flags: Sorries multiply, claiming "complete" with sorries/axioms, fighting type checker for hours, monolithic proofs (>100 lines), long have blocks (>30 lines should be extracted as lemmas - see proof-refactoring.md).

Reference Files

Core references: lean-phrasebook.md, mathlib-guide.md, tactics-reference.md, compilation-errors.md

Domain-specific: domain-patterns.md, measure-theory.md, instance-pollution.md, calc-patterns.md

Incomplete proofs: sorry-filling.md, axiom-elimination.md

Optimization & refactoring: performance-optimization.md, proof-golfing.md, proof-refactoring.md, mathlib-style.md

Automation: compiler-guided-repair.md, lean-lsp-server.md, lean-lsp-tools-api.md, subagent-workflows.md