Lean 4 Theorem Proving

Use this skill whenever you're editing Lean 4 proofs, debugging Lean builds, formalizing mathematics in Lean, or learning Lean 4 concepts. 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

Command	Purpose
/lean4:formalize	Turn informal math into Lean statements
/lean4:prove	Guided cycle-by-cycle theorem proving
/lean4:autoprove	Autonomous multi-cycle proving with stop rules
/lean4:checkpoint	Verified save point (build + axiom check + commit)
/lean4:review	Quality audit (--mode=batch or --mode=stuck)
/lean4:refactor	Strategy-level proof simplification
/lean4:golf	Optimize proofs for brevity
/lean4:learn	Interactive teaching and mathlib exploration
/lean4:doctor	Plugin troubleshooting and migration help

Which Command?

Situation	Command
Draft a Lean statement from an informal claim	/lean4:formalize
Filling sorries (interactive)	/lean4:prove
Filling sorries (unattended)	/lean4:autoprove
Verified save point	/lean4:checkpoint
Quality check (read-only)	/lean4:review
Simplify proof strategies (mathlib leverage, helpers)	/lean4:refactor
Optimizing compiled proofs	/lean4:golf
New to this project / exploring	/lean4:learn --mode=repo
Navigating mathlib for a topic	/lean4:learn --mode=mathlib
Something not working	/lean4:doctor
Formalize + prove end-to-end	/lean4:autoprove --formalize=auto --source=... --claim-select=first --formalize-out=...

Typical Workflow

/lean4:formalize           Turn informal math into Lean statements (optional entry)
        ↓
/lean4:prove               Guided cycle-by-cycle proving (asks before each cycle)
/lean4:autoprove           Autonomous multi-cycle proving (runs with stop rules)
        ↓
/lean4:refactor            Simplify proof strategies (optional, or --dry-run to preview)
        ↓
/lean4:golf                Optimize proofs for tactic-level brevity (optional)
        ↓
/lean4:checkpoint          Create verified save point

Use /lean4:learn at any point to explore repo structure or navigate mathlib. Use /lean4:formalize standalone or via --formalize=auto on autoprove for end-to-end source-to-proof.

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 → continue. In prove, replan requires user approval; in autoprove, replan auto-continues
• Review supports --mode=batch (default) or --mode=stuck (triage); review is always read-only
• --formalize=auto on autoprove wraps formalize+prove in a single command (source → claims → skeletons → proofs)
• 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 (10/30s)
lean_leansearch("natural language")             # Semantic search (3/30s)
lean_loogle("?a → ?b → _")                      # Type-pattern (unlimited if local mode)
lean_hammer_premise(file, line, col)            # Premise suggestions for simp/aesop/grind (3/30s)
lean_state_search(file, line, col)              # Goal-conditioned lemma search (3/30s)
lean_multi_attempt(file, line, snippets=[...])  # Test multiple tactics

Core Primitives

Script	Purpose	Output
sorry_analyzer.py	Find sorries with context	text (default), json, markdown, summary
check_axioms_inline.sh	Check for non-standard axioms	text
smart_search.sh	Multi-source mathlib search	text
find_golfable.py	Detect optimization patterns	JSON
find_usages.sh	Find declaration usages	text

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.
• Keep stderr visible for Lean scripts (no /dev/null redirection), so real errors are not hidden.

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:

• Read the goal or error via lean_goal/lean_diagnostic_messages
• Search mathlib with up to 2 LSP tools (e.g. lean_local_search + lean_leanfinder/lean_leansearch/lean_loogle)
• Try the Automation Tactics cascade
• Validate with lean_diagnostic_messages (no project-gate lake build in this mode)
• No looping, no deep escalation, no multi-cycle behavior, no commits
• End with suggestions:

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

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

Verification ladder: lean_diagnostic_messages(file) per-edit → lake env lean <path/to/File.lean> file gate (run from project root) → lake build project gate only. See cycle-engine: Build Target Policy.

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. See grind-tactic for interactive workflows, annotation strategy, and simproc escalation.

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"
# One-pass discovery for troubleshooting (human-readable default text):
${LEAN4_PYTHON_BIN:-python3} "$LEAN4_SCRIPTS/sorry_analyzer.py" . --report-only
# Structured output (optional): --format=json
# Counts only (optional): --format=summary

Cold start / fresh worktree:

• Fresh worktree or after lake clean? Prime the cache in that worktree before the first real build.
• Use the project's cache command: lake cache get on newer Lake, or lake exe cache get where the project still uses the mathlib cache executable.
• If Lean LSP is cold or timing out on first use, run one lake build to bootstrap the workspace.
• After bootstrap, return to the normal verification ladder: lean_diagnostic_messages(file) → lake env lean <path/to/File.lean> (from project root) → lake build only at checkpoint/final gate.
• Do not symlink another worktree's .lake/build; use Lake cache/artifact mechanisms instead.

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), grind-tactic (SMT-style automation — when simp can't close), simp-reference (simp hygiene + custom simprocs), tactic-patterns, calc-patterns

Proof Development: proof-templates, proof-refactoring (28K — grep by topic), proof-simplification (strategy-level: mathlib search, congr lemmas, helper extraction), sorry-filling

Optimization: proof-golfing (includes safety rules, bounded LSP lemma replacement, bulk rewrites, anti-patterns; escalates to axiom-eliminator), proof-golfing-patterns, performance-optimization (grep by symptom), profiling-workflows (diagnose slow builds/proofs)

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

Style: mathlib-style, verso-docs (Verso doc comment roles and fixups)

Custom Syntax: lean4-custom-syntax (read when building notations, macros, elaborators, or DSLs), metaprogramming-patterns (MetaM/TacticM API — composable blocks, elaborators), scaffold-dsl (copy-paste DSL template), json-patterns (json% syntax + ToJson)

Quality: linter-authoring (project-specific linter rules), ffi-patterns (C/ObjC bindings via Lake)

Workflows: agent-workflows, subagent-workflows, command-examples, learn-pathways (intent taxonomy, game tracks, source handling)

Internals: review-hook-schema