Lean 4 Theorem Proving

Use this skill whenever you’re editing Lean 4 proofs or debugging Lean builds. It prioritizes LSP-based inspection and mathlib search, with scripted primitives for sorry analysis, axiom checking, and error parsing.

Core Principles

Search before prove. Many mathematical facts already exist in mathlib. Search exhaustively before writing tactics.

Build incrementally. Lean’s type checker is your test suite—if it compiles with no sorries and standard axioms only, the proof is sound.

Respect scope. Follow the user’s preference: fill one sorry, its transitive dependencies, all sorries in a file, or everything. Ask if unclear.

Never change statements or add axioms without explicit permission. Theorem/lemma statements, type signatures, and docstrings are off-limits unless the user requests changes. Inline comments may be adjusted; docstrings may not (they’re part of the API). Custom axioms require explicit approval—if a proof seems to need one, stop and discuss.

Commands

Typical Workflow

/lean4:prove               Guided cycle-by-cycle proving (asks before each cycle)
/lean4:autoprove           Autonomous multi-cycle proving (runs with stop rules)
        ↓
/lean4:golf                Optimize proofs (optional, prompted at end)
        ↓
/lean4:checkpoint          Create verified save point

Notes:

• /lean4:prove asks before each cycle; /lean4:autoprove loops autonomously with hard stop conditions
• Both trigger /lean4:review at configured intervals (--review-every)
• When reviews run (via --review-every), they act as gates: review → replan → approval → continue
• Review supports --mode=batch (default) or --mode=stuck (triage)
• If you hit environment issues, run /lean4:doctor to diagnose

LSP Tools (Preferred)

Sub-second feedback and search tools (LeanSearch, Loogle, LeanFinder) via Lean LSP MCP:

lean_goal(file, line)                           # See exact goal
lean_hover_info(file, line, col)                # Understand types
lean_local_search("keyword")                    # Fast local + mathlib (unlimited)
lean_leanfinder("goal or query")                # Semantic, goal-aware (rate-limited)
lean_leansearch("natural language")             # Semantic search (rate-limited)
lean_loogle("?a → ?b → _")                      # Type-pattern (rate-limited)
lean_multi_attempt(file, line, snippets=[...])  # Test multiple tactics

Core Primitives

Usage: Invoked by commands automatically. See references/ for details.

Invocation contract: Never run bare script names. Always use:

• Python: ${LEAN4_PYTHON_BIN:-python3} "$LEAN4_SCRIPTS/script.py" ...
• Shell: bash "$LEAN4_SCRIPTS/script.sh" ...
• Report-only calls: add --report-only to sorry_analyzer.py, check_axioms_inline.sh, unused_declarations.sh — suppresses exit 1 on findings; real errors still exit 1. Do not use in gate commands like /lean4:checkpoint.

If $LEAN4_SCRIPTS is unset or missing, run /lean4:doctor and stay LSP-only until resolved.

Automation

/lean4:prove and /lean4:autoprove handle most tasks:

• prove — guided, asks before each cycle. Ideal for interactive sessions.
• autoprove — autonomous, loops with hard stop rules. Ideal for unattended runs.

Both share the same cycle engine (plan → work → checkpoint → review → replan → continue/stop) and follow the LSP-first protocol: LSP tools are normative for discovery and search; script fallback only when LSP is unavailable or exhausted. Compiler-guided repair is escalation-only — not the first response to build errors. For complex proofs, they may delegate to internal workflows for deep sorry-filling (with snapshot, rollback, and scope budgets), proof repair, or axiom elimination. You don’t invoke these directly.

Skill-Only Behavior

When editing .lean files without invoking a command, the skill runs one bounded pass:

• Attempt to fix the immediate issue (build error, single sorry)
• No looping, no deep escalation, no multi-cycle behavior
• End with suggestions:

  Use /lean4:prove for guided cycle-by-cycle help. Use /lean4:autoprove for autonomous cycles with stop safeguards.

Common Fixes

See compilation-errors for error-by-error guidance (type mismatch, unknown identifier, failed to synthesize, timeout, etc.).

Type Class Patterns

-- Local instance for this proof block
haveI : MeasurableSpace Ω := inferInstance
letI : Fintype α := ⟨...⟩

-- Scoped instances (affects current section)
open scoped Topology MeasureTheory

Order matters: provide outer structures before inner ones.

Automation Tactics

Try in order (stop on first success): rfl → simp → ring → linarith → nlinarith → omega → exact? → apply? → grind → aesop

Note: exact?/apply? query mathlib (slow). grind and aesop are powerful but may timeout.

Troubleshooting

If LSP tools aren’t responding, scripts provide fallback for all operations. If environment variables (LEAN4_SCRIPTS, LEAN4_REFS) are missing, run /lean4:doctor to diagnose.

Script environment check:

echo "$LEAN4_SCRIPTS"
ls -l "$LEAN4_SCRIPTS/sorry_analyzer.py"
${LEAN4_PYTHON_BIN:-python3} "$LEAN4_SCRIPTS/sorry_analyzer.py" . --format=summary --report-only

Quality Gate

A proof is complete when:

• lake build passes
• Zero sorries in agreed scope
• Only standard axioms (propext, Classical.choice, Quot.sound)
• No statement changes without permission

References

Cycle Engine: cycle-engine — shared prove/autoprove logic (stuck, deep mode, falsification, safety)

LSP Tools: lean-lsp-server (quick start), lean-lsp-tools-api (full API — grep ^## for tool names)

Search: mathlib-guide (read when searching for existing lemmas), lean-phrasebook (math→Lean translations)

Errors: compilation-errors (read first for any build error), instance-pollution (typeclass conflicts — grep ## Sub- for patterns), compiler-guided-repair (escalation-only repair — not first-pass)

Tactics: tactics-reference (tactic lookup — grep ^### TacticName), tactic-patterns, calc-patterns, simp-hygiene

Proof Development: proof-templates, proof-refactoring (28K — grep by topic), sorry-filling

Optimization: proof-golfing (includes bounded LSP lemma replacement; bulk rewrites are context-filtered and regression-reverting; escalates to axiom-eliminator), proof-golfing-patterns, proof-golfing-safety, performance-optimization (grep by symptom)

Domain: domain-patterns (25K — grep ## Area), measure-theory (28K), axiom-elimination

Style: mathlib-style

Custom Syntax: lean4-custom-syntax (read when building notations, macros, elaborators, or DSLs), scaffold-dsl (copy-paste DSL template)

Workflows: agent-workflows, subagent-workflows, command-examples

Internals: review-hook-schema